getting started with abaqus standard

Getting Started with ABAQUS/StandardGetting Started with ABAQUS/Standard

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ADAMS is a registered United States trademark of Mechanical Dynamics, Inc.

ADAMS/Flex and ADAMS/View are trademarks of Mechanical Dynamics, Inc.

CATIA is a registered trademark of Dassault Systémes.

C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD.

Compaq Alpha is registered in the U.S. Patent and Trademark Office.

FE-SAFE is a trademark of Safe Technology, Ltd.

Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited.

Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co.

Hitachi is a registered trademark of Hitachi, Ltd.

IBM RS/6000 is a trademark of IBM.

Intel is a registered trademark of the Intel Corporation.

NEC is a trademark of the NEC Corporation.

PostScript is a registered trademark of Adobe Systems, Inc.

Silicon Graphics is a registered trademark of Silicon Graphics, Inc.

SUN is a registered trademark of Sun Microsystems, Inc.

TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Groupin the U.S. and other countries.

Windows NT is a registered trademark of the Microsoft Corporation.

ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACISis a registered trademark of SPATIAL TECHNOLOGY INC.

This release of ABAQUS on Windows NT includes the diff program obtained from the Free SoftwareFoundation. You may freely distribute the diff program and/or modify it under the terms of the GNULibrary General Public License as published by the Free Software Foundation, Inc., 59 Temple Place,Suite 330, Boston, MA 02111-1307 USA.

This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integerprogramming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, theNetherlands.

Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. AllRights Reserved. Permission to use, copy, modify, and distribute the Python software and itsdocumentation for any purpose and without fee is hereby granted, provided that the above copyrightnotice appear in all copies and that both that copyright notice and this permission notice appear insupporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or

ADAMS is a registered United States trademark of Mechanical Dynamics, Inc.

ADAMS/Flex and ADAMS/View are trademarks of Mechanical Dynamics, Inc.

CATIA is a registered trademark of Dassault Systémes.

C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD.

Compaq Alpha is registered in the U.S. Patent and Trademark Office.

FE-SAFE is a trademark of Safe Technology, Ltd.

Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited.

Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co.

Hitachi is a registered trademark of Hitachi, Ltd.

IBM RS/6000 is a trademark of IBM.

Intel is a registered trademark of the Intel Corporation.

NEC is a trademark of the NEC Corporation.

PostScript is a registered trademark of Adobe Systems, Inc.

Silicon Graphics is a registered trademark of Silicon Graphics, Inc.

SUN is a registered trademark of Sun Microsystems, Inc.

TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Groupin the U.S. and other countries.

Windows NT is a registered trademark of the Microsoft Corporation.

ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACISis a registered trademark of SPATIAL TECHNOLOGY INC.

This release of ABAQUS on Windows NT includes the diff program obtained from the Free SoftwareFoundation. You may freely distribute the diff program and/or modify it under the terms of the GNULibrary General Public License as published by the Free Software Foundation, Inc., 59 Temple Place,Suite 330, Boston, MA 02111-1307 USA.

This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integerprogramming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, theNetherlands.

Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. AllRights Reserved. Permission to use, copy, modify, and distribute the Python software and itsdocumentation for any purpose and without fee is hereby granted, provided that the above copyrightnotice appear in all copies and that both that copyright notice and this permission notice appear insupporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or

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Corporation for National Research Initiatives or CNRI not be used in advertising or publicitypertaining to distribution of the software without specific, written prior permission.

All other brand or product names are trademarks or registered trademarks of their respectivecompanies or organizations.

Corporation for National Research Initiatives or CNRI not be used in advertising or publicitypertaining to distribution of the software without specific, written prior permission.

All other brand or product names are trademarks or registered trademarks of their respectivecompanies or organizations.

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General conversion factors (to five significant digits)Quantity U.S. unit SI equivalent

Length 1 in 0.025400 m1 ft 0.30480 m1 mile 1609.3 m

Area 1 in2 0.64516 � 10-3 m2

1 ft2 0.092903 m2

1 acre 4046.9 m2

Volume 1 in3 0.016387 � 10-3 m3

1 ft3 0.028317 m3

1 US gallon 3.7854 � 10-3 m3

Conversion factors for stress analysisQuantity U.S. unit SI equivalent

Density 1 slug/ft3 = 1 lbf s2/ft4 515.38 kg/m3

1 lbf s2/in4 10.687 � 106 kg/m3

Energy 1 ft lbf 1.3558 J (N m)Force 1 lbf 4.4482 N (kg m/s2)Mass 1 slug = 1 lbf s2/ft 14.594 kg (N s2/m)

1 lbf s2/in 175.13 kgPower 1 ft lbf/s 1.3558 W (N m/s)Pressure, Stress 1 psi (lbf/in2) 6894.8 Pa (N/m2)

Conversion factors for heat transfer analysisQuantity U.S. unit SI equivalent

Conductivity 1 Btu/ft hr °F 1.7307 W/m °C1 Btu/in hr °F 20.769 W/m °C

Density 1 lbm/in3 27680. kg/m3

Energy 1 Btu 1055.1 JHeat flux density 1 Btu/in2 hr 454.26 W/m2

Power 1 Btu/hr 0.29307 WSpecific heat 1 Btu/lbm °F 4186.8 J/kg °CTemperature 1 °F 5/9 °C

Temp °F 9/5 � Temp °C + 32°9/5 � Temp °K - 459.67°

Important constantsConstant U.S. unit SI unit

Absolute zero -459.67 °F -273.15 °CAcceleration of gravity 32.174 ft/s2 9.8066 m/s2

Atmospheric pressure 14.694 psi 0.10132 � 106 PaStefan-Boltzmannconstant

0.1714 � 10-8 Btu/hr ft2

°R4 5.669 � 10-8 W/m2 °K4

where °R = °F + 459.67 where °K = °C + 273.15

Approximate properties of mild steel at room temperatureQuantity U.S. unit SI unit

Conductivity 28.9 Btu/ft hr °F 50 W/m °C2.4 Btu/in hr °F

Density 15.13 slug/ft3 (lbf s2/ft4) 7800 kg/m3

0.730 � 10-3 lbf s2/in4

General conversion factors (to five significant digits)Quantity U.S. unit SI equivalent

Length 1 in 0.025400 m1 ft 0.30480 m1 mile 1609.3 m

Area 1 in2 0.64516 � 10-3 m2

1 ft2 0.092903 m2

1 acre 4046.9 m2

Volume 1 in3 0.016387 � 10-3 m3

1 ft3 0.028317 m3

1 US gallon 3.7854 � 10-3 m3

Conversion factors for stress analysisQuantity U.S. unit SI equivalent

Density 1 slug/ft3 = 1 lbf s2/ft4 515.38 kg/m3

1 lbf s2/in4 10.687 � 106 kg/m3

Energy 1 ft lbf 1.3558 J (N m)Force 1 lbf 4.4482 N (kg m/s2)Mass 1 slug = 1 lbf s2/ft 14.594 kg (N s2/m)

1 lbf s2/in 175.13 kgPower 1 ft lbf/s 1.3558 W (N m/s)Pressure, Stress 1 psi (lbf/in2) 6894.8 Pa (N/m2)

Conversion factors for heat transfer analysisQuantity U.S. unit SI equivalent

Conductivity 1 Btu/ft hr °F 1.7307 W/m °C1 Btu/in hr °F 20.769 W/m °C

Density 1 lbm/in3 27680. kg/m3

Energy 1 Btu 1055.1 JHeat flux density 1 Btu/in2 hr 454.26 W/m2

Power 1 Btu/hr 0.29307 WSpecific heat 1 Btu/lbm °F 4186.8 J/kg °CTemperature 1 °F 5/9 °C

Temp °F 9/5 � Temp °C + 32°9/5 � Temp °K - 459.67°

Important constantsConstant U.S. unit SI unit

Absolute zero -459.67 °F -273.15 °CAcceleration of gravity 32.174 ft/s2 9.8066 m/s2

Atmospheric pressure 14.694 psi 0.10132 � 106 PaStefan-Boltzmannconstant

0.1714 � 10-8 Btu/hr ft2

°R4 5.669 � 10-8 W/m2 °K4

where °R = °F + 459.67 where °K = °C + 273.15

Approximate properties of mild steel at room temperatureQuantity U.S. unit SI unit

Conductivity 28.9 Btu/ft hr °F 50 W/m °C2.4 Btu/in hr °F

Density 15.13 slug/ft3 (lbf s2/ft4) 7800 kg/m3

0.730 � 10-3 lbf s2/in4

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0.282 lbm/in3

Elastic modulus 30 � 106 psi 207 � 109 PaSpecific heat 0.11 Btu/lbm °F 460 J/kg °CYield stress 30 � 103 psi 207 � 106 Pa

0.282 lbm/in3

Elastic modulus 30 � 106 psi 207 � 109 PaSpecific heat 0.11 Btu/lbm °F 460 J/kg °CYield stress 30 � 103 psi 207 � 106 Pa

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E-mail: [email protected]

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ABAQUS Solutions Northeast, LLC

39221 Paseo Padre Parkway, Suite F Summit Office Park, West BuildingFremont, CA 94538-1611 300 Centerville Road, Suite 209WTel: 510 794 5891 Warwick, RI 02886-0201Fax: 510 794 1194 Tel: 401 739 3637E-mail: [email protected] Fax: 401 739 3302

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UNITED STATESHibbitt, Karlsson & Sorensen, Inc. Hibbitt, Karlsson & Sorensen (Michigan),

Inc.1080 Main Street 14500 Sheldon Road, Suite 160Pawtucket, RI 02860-4847 Plymouth, MI 48170-2408Tel: 401 727 4200 Tel: 734 451 0217Fax: 401 727 4208 Fax: 734 451 0458E-mail: [email protected],[email protected]


http://www.abaqus.comHibbitt, Karlsson & Sorensen (West),Inc.

ABAQUS Solutions Northeast, LLC

39221 Paseo Padre Parkway, Suite F Summit Office Park, West BuildingFremont, CA 94538-1611 300 Centerville Road, Suite 209WTel: 510 794 5891 Warwick, RI 02886-0201Fax: 510 794 1194 Tel: 401 739 3637E-mail: [email protected] Fax: 401 739 3302

E-mail: [email protected] Engineering, Inc.1440 Innovation PlaceWest Lafayette, IN 47906-1000Tel: 765 497 1373Fax: 765 497 4444E-mail: [email protected] AUSTRALIAKB Engineering S. R. L. Compumod Pty. Ltd.Florida 274, Of. 37 Level 13, 309 Pitt Street(1005) Buenos Aires, Argentina Sydney 2000Tel: +54 11 4393 8444 P.O. Box A807Fax: +54 11 4326 2424 Sydney South 1235E-mail: [email protected] Tel: 02 9283 2577

Fax: 02 9283 2585E-mail: [email protected]://www.compumod.com.au

AUSTRIA BENELUXVOEST-ALPINE STAHL LINZ GmbH ABAQUS Benelux BVDepartment WFE Huizermaatweg 576Postfach 3 1276 LN HuizenA-4031 Linz The NetherlandsTel: 0732 6585 9919 Tel: +31 35 52 58 424Fax: 0732 6980 4338 Fax: +31 35 52 44 257E-mail: [email protected] E-mail: [email protected] CZECH REPUBLIC AND SLOVAK

REPUBLICAdvanced Finite Element Services ASATTEDepartment of Engineering Mechanics Technická 4, 166 07 Praha 6Tsinghua University Czech RepublicBeijing 100084, P. R. China Tel: 420 2 24352654Tel: 010 62783986 Fax: 420 2 33322482

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Auckland Tel: 61 852 31 19Tel: +64 9 623 1223 Fax: 61 852 31 19Fax: +64 9 623 1134 E-mail: [email protected]: [email protected] SOUTH AFRICACompumod (Singapore) Pte Ltd Finite Element Analysis Services (Pty) Ltd.#17-05 Asia Chambers Suite 20-303C, The Waverley20 McCallum Street Wyecroft RoadSingapore 069046 Mowbray 7700Tel: 223 2996 Tel: 021 448 7608Fax: 226 0336 Fax: 021 448 7679E-mail:[email protected]


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1. IntroductionABAQUS is a suite of powerful engineering simulation programs, based on the finite element method,that can solve problems ranging from relatively simple linear analyses to the most challengingnonlinear simulations. ABAQUS contains an extensive library of elements that can model virtually anygeometry. It has an equally extensive list of material models that can simulate the behavior of mosttypical engineering materials including metals, rubber, polymers, composites, reinforced concrete,crushable and resilient foams, and geotechnical materials such as soils and rock. Designed as ageneral-purpose simulation tool, ABAQUS can be used to study more than just structural(stress/displacement) problems. It can simulate problems in such diverse areas as heat transfer, massdiffusion, thermal management of electrical components (coupled thermal-electrical analyses),acoustics, soil mechanics (coupled pore fluid-stress analyses), and piezoelectric analysis.

ABAQUS is simple to use even though it offers the user a wide range of capabilities. The mostcomplicated problems can be modeled easily. For example, problems with multiple components aremodeled by associating the geometry defining each component with the appropriate material models.In most simulations, even highly nonlinear ones, the user need only provide the engineering data suchas the geometry of the structure, its material behavior, its boundary conditions, and the loads applied toit. In a nonlinear analysis ABAQUS automatically chooses appropriate load increments andconvergence tolerances. Not only does it choose the values for these parameters, it also continuallyadjusts them during the analysis to ensure that an accurate solution is obtained efficiently. The userrarely has to define parameters for controlling the numerical solution of the problem.

1.1 The ABAQUS modulesABAQUS consists of two main analysis modules--ABAQUS/Standard and ABAQUS/Explicit. Thereare also three special-purpose add-on analysis products for ABAQUS/Standard--ABAQUS/Aqua,ABAQUS/Design, and ABAQUS/USA. In addition, ABAQUS/Safe provides fatigue postprocessing;while ABAQUS/ADAMS, ABAQUS/CAT, and ABAQUS/C-MOLD are interfaces to ADAMS/Flex,CATIA, and C-MOLD, respectively. ABAQUS/CAE is a complete ABAQUS environment thatincludes capabilities for creating ABAQUS models, submitting and monitoring ABAQUS jobs, andevaluating results. ABAQUS/Viewer is a subset of ABAQUS/CAE that includes just thepostprocessing functionality. The relationship between these modules is shown in Figure 1-1.

Figure 1-1 ABAQUS modules.

1. IntroductionABAQUS is a suite of powerful engineering simulation programs, based on the finite element method,that can solve problems ranging from relatively simple linear analyses to the most challengingnonlinear simulations. ABAQUS contains an extensive library of elements that can model virtually anygeometry. It has an equally extensive list of material models that can simulate the behavior of mosttypical engineering materials including metals, rubber, polymers, composites, reinforced concrete,crushable and resilient foams, and geotechnical materials such as soils and rock. Designed as ageneral-purpose simulation tool, ABAQUS can be used to study more than just structural(stress/displacement) problems. It can simulate problems in such diverse areas as heat transfer, massdiffusion, thermal management of electrical components (coupled thermal-electrical analyses),acoustics, soil mechanics (coupled pore fluid-stress analyses), and piezoelectric analysis.

ABAQUS is simple to use even though it offers the user a wide range of capabilities. The mostcomplicated problems can be modeled easily. For example, problems with multiple components aremodeled by associating the geometry defining each component with the appropriate material models.In most simulations, even highly nonlinear ones, the user need only provide the engineering data suchas the geometry of the structure, its material behavior, its boundary conditions, and the loads applied toit. In a nonlinear analysis ABAQUS automatically chooses appropriate load increments andconvergence tolerances. Not only does it choose the values for these parameters, it also continuallyadjusts them during the analysis to ensure that an accurate solution is obtained efficiently. The userrarely has to define parameters for controlling the numerical solution of the problem.

1.1 The ABAQUS modulesABAQUS consists of two main analysis modules--ABAQUS/Standard and ABAQUS/Explicit. Thereare also three special-purpose add-on analysis products for ABAQUS/Standard--ABAQUS/Aqua,ABAQUS/Design, and ABAQUS/USA. In addition, ABAQUS/Safe provides fatigue postprocessing;while ABAQUS/ADAMS, ABAQUS/CAT, and ABAQUS/C-MOLD are interfaces to ADAMS/Flex,CATIA, and C-MOLD, respectively. ABAQUS/CAE is a complete ABAQUS environment thatincludes capabilities for creating ABAQUS models, submitting and monitoring ABAQUS jobs, andevaluating results. ABAQUS/Viewer is a subset of ABAQUS/CAE that includes just thepostprocessing functionality. The relationship between these modules is shown in Figure 1-1.

Figure 1-1 ABAQUS modules.

Introduction

1-9

ABAQUS/Standard

ABAQUS/Standard is a general-purpose analysis module that can solve a wide range of linearand nonlinear problems involving the static, dynamic, thermal, and electrical response ofcomponents. This module is the main subject of this guide.

ABAQUS/Explicit

ABAQUS/Explicit is a special-purpose analysis module that uses an explicit dynamic finiteelement formulation. It is suitable for short, transient dynamic events, such as impact and blastproblems, and is also very efficient for highly nonlinear problems involving changing contactconditions, such as forming simulations. ABAQUS/Explicit is not specifically covered in thisguide but is described in the companion publication Getting Started with ABAQUS/Explicit.The basic format of the input file is the same for both analysis modules. In addition, the outputis similar for the two modules, and ABAQUS/Viewer can be used to postprocess the resultsfrom either module.

ABAQUS/CAE

ABAQUS/CAE is an interactive, graphical environment for ABAQUS. It allows models to becreated quickly and easily by producing or importing the geometry of the structure to beanalyzed and decomposing the geometry into meshable regions. Physical and materialproperties can be assigned to the geometry, together with loads and boundary conditions.ABAQUS/CAE contains very powerful options to mesh the geometry and to verify theresulting analysis model. Once the model is complete, ABAQUS/CAE can submit, monitor,and control the analysis jobs. The visualization module can then be used to interpret theresults.

Only the visualization module of ABAQUS/CAE (ABAQUS/Viewer) is covered in this guide.If you want to learn how to use other aspects of ABAQUS/CAE but are unable to attend aseminar, you can obtain a copy of the ABAQUS/CAE training material.

ABAQUS/Viewer

ABAQUS/Standard

ABAQUS/Standard is a general-purpose analysis module that can solve a wide range of linearand nonlinear problems involving the static, dynamic, thermal, and electrical response ofcomponents. This module is the main subject of this guide.

ABAQUS/Explicit

ABAQUS/Explicit is a special-purpose analysis module that uses an explicit dynamic finiteelement formulation. It is suitable for short, transient dynamic events, such as impact and blastproblems, and is also very efficient for highly nonlinear problems involving changing contactconditions, such as forming simulations. ABAQUS/Explicit is not specifically covered in thisguide but is described in the companion publication Getting Started with ABAQUS/Explicit.The basic format of the input file is the same for both analysis modules. In addition, the outputis similar for the two modules, and ABAQUS/Viewer can be used to postprocess the resultsfrom either module.

ABAQUS/CAE

ABAQUS/CAE is an interactive, graphical environment for ABAQUS. It allows models to becreated quickly and easily by producing or importing the geometry of the structure to beanalyzed and decomposing the geometry into meshable regions. Physical and materialproperties can be assigned to the geometry, together with loads and boundary conditions.ABAQUS/CAE contains very powerful options to mesh the geometry and to verify theresulting analysis model. Once the model is complete, ABAQUS/CAE can submit, monitor,and control the analysis jobs. The visualization module can then be used to interpret theresults.

Only the visualization module of ABAQUS/CAE (ABAQUS/Viewer) is covered in this guide.If you want to learn how to use other aspects of ABAQUS/CAE but are unable to attend aseminar, you can obtain a copy of the ABAQUS/CAE training material.

ABAQUS/Viewer

Introduction

1-10

ABAQUS/Viewer is the visualization module of ABAQUS/CAE; it is an interactive, graphicalpostprocessor that supports all of the capabilities in the ABAQUS analysis modules andprovides a wide range of options for interpreting the results. ABAQUS/Viewer is discussed inthis guide.

ABAQUS/Aqua

ABAQUS/Aqua is a set of optional capabilities that can be added to ABAQUS/Standard. It isintended for the simulation of offshore structures, such as oil platforms. Some of the optionalcapabilities include the effects of wave and wind loading and buoyancy. ABAQUS/Aqua isnot discussed in this guide.

ABAQUS/USA

ABAQUS/USA is an interface between ABAQUS/Standard and the USA code written byLockheed. It is intended for underwater shock analysis. ABAQUS/USA is not discussed in thisguide.

ABAQUS/ADAMS

ABAQUS/ADAMS allows ABAQUS finite element models to be included as flexiblecomponents within the MDI ADAMS family of products. The interface is based on thecomponent mode synthesis formulation of ADAMS/Flex. ABAQUS/ADAMS is not discussedin this guide.

ABAQUS/CAT

ABAQUS/CAT allows ABAQUS analyses to be set up and postprocessed entirely in CATIA.ABAQUS/CAT is not discussed in this guide.

ABAQUS/C-MOLD

ABAQUS/C-MOLD translates finite element mesh, material property, and initial stress datafrom a C-MOLD mold filling analysis to an ABAQUS input file. ABAQUS/C-MOLD is notdiscussed in this guide.

ABAQUS/Design

ABAQUS/Design is a set of optional capabilities that can be added to ABAQUS/Standard toperform design sensitivity calculations. ABAQUS/Design is not discussed in this guide.

ABAQUS/Safe

ABAQUS/Safe is the fatigue analysis module in ABAQUS. Using results from ABAQUSanalyses, it determines the fatigue life of a component. ABAQUS/Safe is not discussed in thisguide.

1.2 Getting started with ABAQUS/StandardThis is an introductory text designed to give new users guidance in analyzing continuum, shell, andframework problems with ABAQUS/Standard and viewing the results in ABAQUS/Viewer. You do

ABAQUS/Viewer is the visualization module of ABAQUS/CAE; it is an interactive, graphicalpostprocessor that supports all of the capabilities in the ABAQUS analysis modules andprovides a wide range of options for interpreting the results. ABAQUS/Viewer is discussed inthis guide.

ABAQUS/Aqua

ABAQUS/Aqua is a set of optional capabilities that can be added to ABAQUS/Standard. It isintended for the simulation of offshore structures, such as oil platforms. Some of the optionalcapabilities include the effects of wave and wind loading and buoyancy. ABAQUS/Aqua isnot discussed in this guide.

ABAQUS/USA

ABAQUS/USA is an interface between ABAQUS/Standard and the USA code written byLockheed. It is intended for underwater shock analysis. ABAQUS/USA is not discussed in thisguide.

ABAQUS/ADAMS

ABAQUS/ADAMS allows ABAQUS finite element models to be included as flexiblecomponents within the MDI ADAMS family of products. The interface is based on thecomponent mode synthesis formulation of ADAMS/Flex. ABAQUS/ADAMS is not discussedin this guide.

ABAQUS/CAT

ABAQUS/CAT allows ABAQUS analyses to be set up and postprocessed entirely in CATIA.ABAQUS/CAT is not discussed in this guide.

ABAQUS/C-MOLD

ABAQUS/C-MOLD translates finite element mesh, material property, and initial stress datafrom a C-MOLD mold filling analysis to an ABAQUS input file. ABAQUS/C-MOLD is notdiscussed in this guide.

ABAQUS/Design

ABAQUS/Design is a set of optional capabilities that can be added to ABAQUS/Standard toperform design sensitivity calculations. ABAQUS/Design is not discussed in this guide.

ABAQUS/Safe

ABAQUS/Safe is the fatigue analysis module in ABAQUS. Using results from ABAQUSanalyses, it determines the fatigue life of a component. ABAQUS/Safe is not discussed in thisguide.

1.2 Getting started with ABAQUS/StandardThis is an introductory text designed to give new users guidance in analyzing continuum, shell, andframework problems with ABAQUS/Standard and viewing the results in ABAQUS/Viewer. You do

Introduction

1-11

not need any previous knowledge of ABAQUS to benefit from this guide, although some previousexposure to the finite-element method is recommended.

This document covers only stress/displacement simulations, concentrating on both linear and nonlinearstatic analyses. Some very simple types of dynamic analyses are introduced, but dynamic simulationsare not covered comprehensively. Other types of simulations, such as heat transfer and mass diffusion,are not covered.

1.2.1 How to use this guideThere are 11 chapters in this guide, each of which introduces one or more topics. Most chapterscontain a short discussion of the topic or topics being considered and one or two tutorial examples.You should work through the examples carefully since they contain a great deal of practical advice onusing ABAQUS. The capabilities of ABAQUS/Viewer are introduced gradually in these examples.

This chapter is a short introduction to ABAQUS and this guide. Chapter 2, "ABAQUS Basics," whichis centered around a simple example, covers the basics of using ABAQUS. By the end of Chapter 2,"ABAQUS Basics," you will know the fundamentals of how to prepare input data for an ABAQUSsimulation, check the data, solve the analysis, and view the results in ABAQUS/Viewer.

Chapter 3, "ABAQUS Elements - A Brief Introduction," presents an overview of the main elementfamilies available in ABAQUS. The use of continuum (solid) elements, shell elements, and beamelements is discussed in Chapter 4, "Using Continuum Elements;" Chapter 5, "Using Shell Elements;"and Chapter 6, "Using Beam Elements;" respectively.

Chapter 7, "Nonlinearity," introduces the concept of nonlinearity in general, and geometricnonlinearity in particular, and contains the first nonlinear ABAQUS simulation. Material nonlinearityis introduced in Chapter 8, "Materials," and dynamic analyses are discussed in Chapter 9, "Dynamics."Chapter 10, "Multiple Step Analysis," introduces the concept of multi-step simulations, and Chapter11, "Contact," discusses the many issues that arise in contact analyses.

1.2.2 Conventions used in this guideDifferent text styles used in the tutorial examples follow:

� Input in COURIER FONT should be typed into the input file, ABAQUS/Viewer, or your computerexactly as shown. For example,

*SOLID SECTION, ELSET=LUG, MATERIAL=STEEL

would be entered into an input file.

abaqus job=skew_nl

would be typed on your computer to run ABAQUS.

� Items in <italics> indicate that user data such as a name or value should be substituted:

*SOLID SECTION, ELSET=<element set>, MATERIAL=<material name>

� Menu selections, tabs within dialog boxes, and labels of items on the screen in ABAQUS/Viewer

not need any previous knowledge of ABAQUS to benefit from this guide, although some previousexposure to the finite-element method is recommended.

This document covers only stress/displacement simulations, concentrating on both linear and nonlinearstatic analyses. Some very simple types of dynamic analyses are introduced, but dynamic simulationsare not covered comprehensively. Other types of simulations, such as heat transfer and mass diffusion,are not covered.

1.2.1 How to use this guideThere are 11 chapters in this guide, each of which introduces one or more topics. Most chapterscontain a short discussion of the topic or topics being considered and one or two tutorial examples.You should work through the examples carefully since they contain a great deal of practical advice onusing ABAQUS. The capabilities of ABAQUS/Viewer are introduced gradually in these examples.

This chapter is a short introduction to ABAQUS and this guide. Chapter 2, "ABAQUS Basics," whichis centered around a simple example, covers the basics of using ABAQUS. By the end of Chapter 2,"ABAQUS Basics," you will know the fundamentals of how to prepare input data for an ABAQUSsimulation, check the data, solve the analysis, and view the results in ABAQUS/Viewer.

Chapter 3, "ABAQUS Elements - A Brief Introduction," presents an overview of the main elementfamilies available in ABAQUS. The use of continuum (solid) elements, shell elements, and beamelements is discussed in Chapter 4, "Using Continuum Elements;" Chapter 5, "Using Shell Elements;"and Chapter 6, "Using Beam Elements;" respectively.

Chapter 7, "Nonlinearity," introduces the concept of nonlinearity in general, and geometricnonlinearity in particular, and contains the first nonlinear ABAQUS simulation. Material nonlinearityis introduced in Chapter 8, "Materials," and dynamic analyses are discussed in Chapter 9, "Dynamics."Chapter 10, "Multiple Step Analysis," introduces the concept of multi-step simulations, and Chapter11, "Contact," discusses the many issues that arise in contact analyses.

1.2.2 Conventions used in this guideDifferent text styles used in the tutorial examples follow:

� Input in COURIER FONT should be typed into the input file, ABAQUS/Viewer, or your computerexactly as shown. For example,


would be entered into an input file.

abaqus job=skew_nl

would be typed on your computer to run ABAQUS.

� Items in <italics> indicate that user data such as a name or value should be substituted:

*SOLID SECTION, ELSET=<element set>, MATERIAL=<material name>

� Menu selections, tabs within dialog boxes, and labels of items on the screen in ABAQUS/Viewer

Introduction

1-12

are indicated in bold:

View->View Options->HardwareDeformed Plot Options

1.3 ABAQUS documentationThe documentation for ABAQUS is extensive and complete. You will need to refer to the manualsregularly as you use ABAQUS. The following documentation and publications are available fromHKS, unless otherwise specified, in printed form and through our online documentation server. Formore information on accessing the online books, refer to the discussion of execution procedures in theABAQUS/Standard User's Manual.

ABAQUS/Standard User's Manual

This is the ABAQUS manual that you will use most often. Split into three volumes, it is areference manual for all of the capabilities in ABAQUS/Standard and contains a completedescription of the elements, material models, procedures, input specifications, etc. This guideregularly refers to the ABAQUS/Standard User's Manual, so you should have a copy availableas you work through the examples.

ABAQUS Keywords Manual

This manual contains a complete decscription of all the input options available in ABAQUS,including any parameters that may be used with a given option.

ABAQUS/Viewer User's Manual

This basic reference document for ABAQUS/Viewer includes an introductory tutorial as wellas a complete description of how to use ABAQUS/Viewer to display your model and results.

Using ABAQUS Online Documentation

This online manual contains instructions on using the ABAQUS online documentation serverto read the online manuals.

Other documentation available from HKS:

ABAQUS Example Problems Manual

This manual contains more than 75 detailed examples designed to illustrate the approachesand decisions needed to perform meaningful linear and nonlinear analysis. Typical cases arelarge motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of athin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation undera footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. Thedescription of each problem includes a discussion of the choice of element type and meshdensity.

When you want to use a feature that you have not used before, you should look up one or more

are indicated in bold:

View->View Options->HardwareDeformed Plot Options

1.3 ABAQUS documentationThe documentation for ABAQUS is extensive and complete. You will need to refer to the manualsregularly as you use ABAQUS. The following documentation and publications are available fromHKS, unless otherwise specified, in printed form and through our online documentation server. Formore information on accessing the online books, refer to the discussion of execution procedures in theABAQUS/Standard User's Manual.

ABAQUS/Standard User's Manual

This is the ABAQUS manual that you will use most often. Split into three volumes, it is areference manual for all of the capabilities in ABAQUS/Standard and contains a completedescription of the elements, material models, procedures, input specifications, etc. This guideregularly refers to the ABAQUS/Standard User's Manual, so you should have a copy availableas you work through the examples.

ABAQUS Keywords Manual

This manual contains a complete decscription of all the input options available in ABAQUS,including any parameters that may be used with a given option.

ABAQUS/Viewer User's Manual

This basic reference document for ABAQUS/Viewer includes an introductory tutorial as wellas a complete description of how to use ABAQUS/Viewer to display your model and results.

Using ABAQUS Online Documentation

This online manual contains instructions on using the ABAQUS online documentation serverto read the online manuals.

Other documentation available from HKS:


This manual contains more than 75 detailed examples designed to illustrate the approachesand decisions needed to perform meaningful linear and nonlinear analysis. Typical cases arelarge motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of athin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation undera footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. Thedescription of each problem includes a discussion of the choice of element type and meshdensity.

When you want to use a feature that you have not used before, you should look up one or more

Introduction

1-13

examples that use that feature. Then, use the example to familiarize yourself with the correctusage of the capability. To find an example that uses a certain feature, search the onlinedocumentation or use the abaqus findkeyword utility (see ``Execution procedure for queryingthe keyword/problem database,'' Section 3.2.8 of the ABAQUS/Standard User's Manual, formore information). All the input files associated with the examples are provided as part of theABAQUS installation. A utility that gives you access to these files, called ABAQUS/Fetch, isprovided with every ABAQUS release. The syntax for this utility is

abaqus fetch job=<file name>

You may fetch any of the example files from the installation directory, so that you can run thesimulations yourself and review the results. You can also access the input files through theABAQUS Input Files electronic book.

ABAQUS Benchmarks Manual

This volume (available online and, if requested, in print) contains benchmark problems andstandard analyses used to evaluate the performance of ABAQUS; results are compared againstexact calculations and other published results. These problems, like the example problems,offer a good starting point for learning about the behavior of elements and material models.The NAFEMS benchmark problems are included in this manual. The input files are alsosupplied as part of the installation and can be accessed just like example problem input files.

ABAQUS Verification Manual

This online-only volume contains basic test problems that provide quality assurance testing ofeach individual program feature (procedures, output options, MPCs, etc.). It may be useful torun these problems when learning to use a new capability. The input files are also supplied aspart of the installation and can be accessed just like example problem input files.

ABAQUS Input Files

This online manual contains all the input files that are included with the ABAQUS release andreferred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual,and the ABAQUS Verification Manual. They are listed under the title of the problem thatrefers to them. The input file references in the manuals hyperlink directly to this book.Although the input files in this book cannot be saved directly to a file, they can be saved bycopying and pasting them into an editor.

ABAQUS Theory Manual

This volume (available online and, if requested, in print) contains detailed, precise discussionsof the theoretical aspects of ABAQUS. It is written to be understood by users with anengineering background and is not required on a routine basis.

ABAQUS/CAE User's Manual

This reference document for ABAQUS/CAE includes three comprehensive tutorials as well asdetailed descriptions of how to use ABAQUS/CAE for model generation, analysis, and resultsevaluation.

examples that use that feature. Then, use the example to familiarize yourself with the correctusage of the capability. To find an example that uses a certain feature, search the onlinedocumentation or use the abaqus findkeyword utility (see ``Execution procedure for queryingthe keyword/problem database,'' Section 3.2.8 of the ABAQUS/Standard User's Manual, formore information). All the input files associated with the examples are provided as part of theABAQUS installation. A utility that gives you access to these files, called ABAQUS/Fetch, isprovided with every ABAQUS release. The syntax for this utility is


You may fetch any of the example files from the installation directory, so that you can run thesimulations yourself and review the results. You can also access the input files through theABAQUS Input Files electronic book.

ABAQUS Benchmarks Manual

This volume (available online and, if requested, in print) contains benchmark problems andstandard analyses used to evaluate the performance of ABAQUS; results are compared againstexact calculations and other published results. These problems, like the example problems,offer a good starting point for learning about the behavior of elements and material models.The NAFEMS benchmark problems are included in this manual. The input files are alsosupplied as part of the installation and can be accessed just like example problem input files.

ABAQUS Verification Manual

This online-only volume contains basic test problems that provide quality assurance testing ofeach individual program feature (procedures, output options, MPCs, etc.). It may be useful torun these problems when learning to use a new capability. The input files are also supplied aspart of the installation and can be accessed just like example problem input files.

ABAQUS Input Files

This online manual contains all the input files that are included with the ABAQUS release andreferred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual,and the ABAQUS Verification Manual. They are listed under the title of the problem thatrefers to them. The input file references in the manuals hyperlink directly to this book.Although the input files in this book cannot be saved directly to a file, they can be saved bycopying and pasting them into an editor.

ABAQUS Theory Manual

This volume (available online and, if requested, in print) contains detailed, precise discussionsof the theoretical aspects of ABAQUS. It is written to be understood by users with anengineering background and is not required on a routine basis.

ABAQUS/CAE User's Manual

This reference document for ABAQUS/CAE includes three comprehensive tutorials as well asdetailed descriptions of how to use ABAQUS/CAE for model generation, analysis, and resultsevaluation.

Introduction

1-14

ABAQUS/Explicit User's Manual

This is the reference manual for all of the capabilities in ABAQUS/Explicit.

Release Notes

This online-only document contains brief descriptions of the new features available in thelatest release of the ABAQUS product line.

Site Guide

This document, which is provided with each ABAQUS license, describes how to installABAQUS and how to configure the installation for particular circumstances. Some of thisinformation, of most relevance to users, is also included in the User's manuals.

Quality Assurance Plan

This document describes HKS's QA procedures. It is a controlled document, provided tocustomers who subscribe to either HKS's Nuclear QA Program or the Quality MonitoringService.

Lecture Notes

Notes and workshops are available for many features and applications for which ABAQUS isused, such as metal forming or heat transfer. The notes are used in the technical seminars thatHKS offers to help users improve their understanding and usage of ABAQUS. While they arenot intended as stand-alone tutorial material, they are sufficiently comprehensive that they canbe used in that mode. Frequently, the lecture notes on a topic serve as a good introduction tothe topic, making the User's Manual easier to understand. The list of available lecture notes isincluded in the Documentation Price List or can be found on the ABAQUS Home Page atwww.abaqus.com.

Newsletters

HKS publishes ABAQUS/News and ABAQUS/Answers on a regular schedule.ABAQUS/News includes topical information about program releases, training seminars, etc.ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage.These newsletters are distributed at no cost to users who wish to subscribe. Please contactyour local ABAQUS support office if you wish to be added to the mailing list for thesepublications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

ABAQUS Web server

HKS has a home page on the World Wide Web (www.abaqus.com), containing a variety ofuseful information about the ABAQUS suite of programs, including:

� Frequently asked questions

� ABAQUS systems information and machine requirements

� Benchmark timing documents

ABAQUS/Explicit User's Manual

This is the reference manual for all of the capabilities in ABAQUS/Explicit.

Release Notes

This online-only document contains brief descriptions of the new features available in thelatest release of the ABAQUS product line.

Site Guide

This document, which is provided with each ABAQUS license, describes how to installABAQUS and how to configure the installation for particular circumstances. Some of thisinformation, of most relevance to users, is also included in the User's manuals.

Quality Assurance Plan

This document describes HKS's QA procedures. It is a controlled document, provided tocustomers who subscribe to either HKS's Nuclear QA Program or the Quality MonitoringService.

Lecture Notes

Notes and workshops are available for many features and applications for which ABAQUS isused, such as metal forming or heat transfer. The notes are used in the technical seminars thatHKS offers to help users improve their understanding and usage of ABAQUS. While they arenot intended as stand-alone tutorial material, they are sufficiently comprehensive that they canbe used in that mode. Frequently, the lecture notes on a topic serve as a good introduction tothe topic, making the User's Manual easier to understand. The list of available lecture notes isincluded in the Documentation Price List or can be found on the ABAQUS Home Page atwww.abaqus.com.

Newsletters

HKS publishes ABAQUS/News and ABAQUS/Answers on a regular schedule.ABAQUS/News includes topical information about program releases, training seminars, etc.ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage.These newsletters are distributed at no cost to users who wish to subscribe. Please contactyour local ABAQUS support office if you wish to be added to the mailing list for thesepublications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

ABAQUS Web server

HKS has a home page on the World Wide Web (www.abaqus.com), containing a variety ofuseful information about the ABAQUS suite of programs, including:

� Frequently asked questions

� ABAQUS systems information and machine requirements

� Benchmark timing documents

Introduction

1-15

� Error status reports

� ABAQUS documentation price list

� Training seminar schedule

� Newsletters

User's manuals for ABAQUS/ADAMS, ABAQUS/CAT, ABAQUS/C-MOLD, and ABAQUS/Safe arealso available.

1.4 SupportHKS offers both technical (engineering) support and systems support for ABAQUS. Technical andsystems support are provided through the nearest local support office. We regard technical support asan important part of the service we offer and encourage you to contact us with any questions orconcerns that you have about your ABAQUS analyses. You can contact our offices by telephone, fax,electronic mail, or regular mail. Information on how to contact each office is listed in the front of eachABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on theWorld Wide Web. When contacting your local support office, please specify whether you would liketechnical support (you have encountered problems performing an ABAQUS analysis) or systemssupport (ABAQUS will not install correctly, licensing does not work correctly, or otherhardware-related issues have arisen).

We welcome any suggestions for improvements to the support program or documentation. We willensure that any enhancement requests you make are considered for future releases. If you wish to file acomplaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

1.4.1 Technical supportHKS technical support engineers can assist in clarifying ABAQUS features and checking errors bygiving both general information on using ABAQUS and information on its application to specificanalyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, sinceit is usually easier to solve problems at the beginning of a project rather than trying to correct ananalysis at the end.

Please have the following information ready before calling the technical support hotline, and include itin any written contacts:

� The version of ABAQUS that are you using.

� The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of thedata (.dat) file.

� The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selectingHelp->On version from the main menu bar.

� The version number for ABAQUS/CAT is given at the top of the input (.inp) file as well as

� Error status reports

� ABAQUS documentation price list

� Training seminar schedule

� Newsletters

User's manuals for ABAQUS/ADAMS, ABAQUS/CAT, ABAQUS/C-MOLD, and ABAQUS/Safe arealso available.

1.4 SupportHKS offers both technical (engineering) support and systems support for ABAQUS. Technical andsystems support are provided through the nearest local support office. We regard technical support asan important part of the service we offer and encourage you to contact us with any questions orconcerns that you have about your ABAQUS analyses. You can contact our offices by telephone, fax,electronic mail, or regular mail. Information on how to contact each office is listed in the front of eachABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on theWorld Wide Web. When contacting your local support office, please specify whether you would liketechnical support (you have encountered problems performing an ABAQUS analysis) or systemssupport (ABAQUS will not install correctly, licensing does not work correctly, or otherhardware-related issues have arisen).

We welcome any suggestions for improvements to the support program or documentation. We willensure that any enhancement requests you make are considered for future releases. If you wish to file acomplaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

1.4.1 Technical supportHKS technical support engineers can assist in clarifying ABAQUS features and checking errors bygiving both general information on using ABAQUS and information on its application to specificanalyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, sinceit is usually easier to solve problems at the beginning of a project rather than trying to correct ananalysis at the end.

Please have the following information ready before calling the technical support hotline, and include itin any written contacts:

� The version of ABAQUS that are you using.

� The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of thedata (.dat) file.

� The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selectingHelp->On version from the main menu bar.

� The version number for ABAQUS/CAT is given at the top of the input (.inp) file as well as

Introduction

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the data file.

� The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to thescreen.

� The version number for ABAQUS/Safe is given under the ABAQUS logo in the mainwindow.

� The type of computer on which you are running ABAQUS.

� The symptoms of any problems, including the exact error messages, if any.

� Workarounds or tests that you have already tried.

For support about a specific problem, any available ABAQUS output files may be helpful in answeringquestions that the support engineer may ask you.

The support engineer will try to diagnose your problem from the model description and a descriptionof the difficulties you are having. Frequently, the support engineer will need model sketches, whichcan be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that theanalysis terminated may also be needed to understand what may have caused the problem.

If the support engineer cannot diagnose your problem from this information, you may be asked to sendthe input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUSHome Page at www.abaqus.com for the media formats that are currently accepted.

All support calls are logged into a database, which enables us to monitor the progress of a particularproblem and to check that we are resolving support issues efficiently. If you would like to know the lognumber of your particular call for future reference, please ask the support engineer. If you are calling todiscuss an existing support problem and you know the log number, please mention it so that we canconsult the database to see what the latest action has been and, thus, avoid duplication of effort. Inaddition, please give the receptionist the support engineer's name (or include it at the top of any e-mailcorrespondence).

1.4.2 Systems supportHKS systems support engineers can help you resolve issues related to the installation and running ofABAQUS, including licensing difficulties, that are not covered by technical support.

You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. Ifyou encounter problems with the installation or licensing, first review the instructions in the ABAQUSSite Guide to ensure that they have been followed correctly. If this does not resolve the problems, lookon the ABAQUS Home Page under Customer Support for information about known installationproblems. If this does not address your situation, please contact your local support office. Sendwhatever information is available to define the problem: error messages from an aborted analysis or adetailed explanation of the problems encountered. Whenever possible, please send the output from theabaqus info=env and abaqus info=sys commands.

1.4.3 Support for academic institutions

the data file.

� The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to thescreen.

� The version number for ABAQUS/Safe is given under the ABAQUS logo in the mainwindow.

� The type of computer on which you are running ABAQUS.

� The symptoms of any problems, including the exact error messages, if any.

� Workarounds or tests that you have already tried.

For support about a specific problem, any available ABAQUS output files may be helpful in answeringquestions that the support engineer may ask you.

The support engineer will try to diagnose your problem from the model description and a descriptionof the difficulties you are having. Frequently, the support engineer will need model sketches, whichcan be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that theanalysis terminated may also be needed to understand what may have caused the problem.

If the support engineer cannot diagnose your problem from this information, you may be asked to sendthe input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUSHome Page at www.abaqus.com for the media formats that are currently accepted.

All support calls are logged into a database, which enables us to monitor the progress of a particularproblem and to check that we are resolving support issues efficiently. If you would like to know the lognumber of your particular call for future reference, please ask the support engineer. If you are calling todiscuss an existing support problem and you know the log number, please mention it so that we canconsult the database to see what the latest action has been and, thus, avoid duplication of effort. Inaddition, please give the receptionist the support engineer's name (or include it at the top of any e-mailcorrespondence).

1.4.2 Systems supportHKS systems support engineers can help you resolve issues related to the installation and running ofABAQUS, including licensing difficulties, that are not covered by technical support.

You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. Ifyou encounter problems with the installation or licensing, first review the instructions in the ABAQUSSite Guide to ensure that they have been followed correctly. If this does not resolve the problems, lookon the ABAQUS Home Page under Customer Support for information about known installationproblems. If this does not address your situation, please contact your local support office. Sendwhatever information is available to define the problem: error messages from an aborted analysis or adetailed explanation of the problems encountered. Whenever possible, please send the output from theabaqus info=env and abaqus info=sys commands.

1.4.3 Support for academic institutions

Introduction

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Under the terms of the Academic License Agreement, we do not provide support to users at academicinstitutions unless the institution has also purchased technical support. Please contact us for moreinformation.

1.5 A quick review of the finite element methodThis section reviews the basics of the finite element method. A simple example of a truss, constrainedat one end and loaded at the other end as shown in Figure 1-2, is used to introduce some terms andconventions used in the rest of this document. The objective of the analysis is to find the displacementof the free end of the truss, the stress in the truss, and the reaction force at the constrained end of thetruss.

Figure 1-2 Truss problem.

The first step of any finite element simulation is to discretize the actual geometry of the structure usinga collection of finite elements. Each finite element represents a discrete portion of the physicalstructure. The finite elements are joined by shared nodes.The collection of nodes and finite elements iscalled the mesh. The number of elements used in a particular mesh is referred to as the mesh density. Ina stress analysis the displacements of the nodes are the fundamental variables that ABAQUScalculates. Once the nodal displacements are known, the stresses and strains in each finite element canbe determined easily.

In this case the rod shown in Figure 1-2 will be modeled with two truss elements. In ABAQUS trusselements can carry axial loads only. The discretized model is shown in Figure 1-3 together with thenode and element labels.

Figure 1-3 Discretized model of the truss problem.

Free-body diagrams for each node in the model are shown in Figure 1-4. In general each node willcarry an external load applied to the model, P , and internal loads, I , caused by stresses in the elementsattached to that node. For a model to be in static equilibrium, the net force acting on each node must bezero; i.e., the internal and external loads at each node must balance each other. For node a this

Under the terms of the Academic License Agreement, we do not provide support to users at academicinstitutions unless the institution has also purchased technical support. Please contact us for moreinformation.

1.5 A quick review of the finite element methodThis section reviews the basics of the finite element method. A simple example of a truss, constrainedat one end and loaded at the other end as shown in Figure 1-2, is used to introduce some terms andconventions used in the rest of this document. The objective of the analysis is to find the displacementof the free end of the truss, the stress in the truss, and the reaction force at the constrained end of thetruss.

Figure 1-2 Truss problem.

The first step of any finite element simulation is to discretize the actual geometry of the structure usinga collection of finite elements. Each finite element represents a discrete portion of the physicalstructure. The finite elements are joined by shared nodes.The collection of nodes and finite elements iscalled the mesh. The number of elements used in a particular mesh is referred to as the mesh density. Ina stress analysis the displacements of the nodes are the fundamental variables that ABAQUScalculates. Once the nodal displacements are known, the stresses and strains in each finite element canbe determined easily.

In this case the rod shown in Figure 1-2 will be modeled with two truss elements. In ABAQUS trusselements can carry axial loads only. The discretized model is shown in Figure 1-3 together with thenode and element labels.

Figure 1-3 Discretized model of the truss problem.

Free-body diagrams for each node in the model are shown in Figure 1-4. In general each node willcarry an external load applied to the model, P , and internal loads, I , caused by stresses in the elementsattached to that node. For a model to be in static equilibrium, the net force acting on each node must bezero; i.e., the internal and external loads at each node must balance each other. For node a this

Introduction

1-18

equilibrium equation can be obtained as follows.

Figure 1-4 Free-body diagram for each node.

Assuming that the change in length of the rod is small, the strain in element 1 is given by

"11 =ub ¡ ua

L;

where ua and ub are the displacements at nodes a and b, respectively, and L is the original length ofthe element.

Assuming that the material is elastic, the stress in the rod is given by the strain multiplied by theYoung's modulus, E:

¾11 = E"11 :

The axial force acting on the end node is equivalent to the stress in the rod multiplied by itscross-sectional area, A. Thus, a relationship between internal force, material properties, anddisplacements is obtained:

I1a = ¾11A = E"11A =(EA)

L(ub ¡ ua):

Equilibrium at node a can, therefore, be written as

Pa +(EA)

L(ub ¡ ua) = 0:

Equilibrium at node b must take into account the internal forces acting from both elements joined atthat node. The internal force from element 1 is now acting in the opposite direction and so becomesnegative. The resulting equation is

Pb¡EA

L(ub ¡ ua) +

EA

L(uc ¡ ub) = 0:

For node c the equilibrium equation is

equilibrium equation can be obtained as follows.

Figure 1-4 Free-body diagram for each node.

Assuming that the change in length of the rod is small, the strain in element 1 is given by

"11 =ub ¡ ua

L;

where ua and ub are the displacements at nodes a and b, respectively, and L is the original length ofthe element.

Assuming that the material is elastic, the stress in the rod is given by the strain multiplied by theYoung's modulus, E:

¾11 = E"11 :

The axial force acting on the end node is equivalent to the stress in the rod multiplied by itscross-sectional area, A. Thus, a relationship between internal force, material properties, anddisplacements is obtained:

I1a = ¾11A = E"11A =(EA)

L(ub ¡ ua):

Equilibrium at node a can, therefore, be written as

Pa +(EA)

L(ub ¡ ua) = 0:

Equilibrium at node b must take into account the internal forces acting from both elements joined atthat node. The internal force from element 1 is now acting in the opposite direction and so becomesnegative. The resulting equation is

Pb¡EA

L(ub ¡ ua) +

EA

L(uc ¡ ub) = 0:

For node c the equilibrium equation is

Introduction

1-19

Pc ¡ EA

L(uc ¡ ub) = 0:

These equilibrium equations need to be solved simultaneously to obtain the displacements of all thenodes. This requirement is best achieved by matrix techniques; therefore, write the internal andexternal force contributions as matrices. If the properties and dimensions of the two elements are thesame, the equilibrium equations can be simplified as follows:8<:Pa

Pb

Pc

9=;+

µEA

L

¶24¡1 1 01 ¡2 10 1 ¡1

358<: ua

ub

uc

9=; = 0:

In general, it may be that the element stiffnesses, the EA=L terms, are different from element toelement; therefore, write the element stiffnesses as K1 and K2 for the two elements in the model. Weare interested in obtaining the solution to the equilibrium equation in which the externally appliedforces, P , are in equilibrium with the internally generated forces, I . When discussing this equationwith reference to convergence and nonlinearity, we tend to write it as

fPg ¡ fIg = 0:

For the complete two-element, three-node structure we, therefore, modify the signs and rewrite theequilibrium equation as8<:Pa

Pb

Pc

9=;¡24 K1 ¡K1 0¡K1 (K1 +K2) ¡K2

0 ¡K2 K2

358<: ua

ub

uc

9=; = 0:

This system of equations can then be solved to obtain values for the three unknown variables: ub, uc,and Pa (ua is specified in the problem as 0.0). Once the displacements are known, we can go back anduse them to calculate the stresses in the truss elements.

Pc ¡ EA

L(uc ¡ ub) = 0:

These equilibrium equations need to be solved simultaneously to obtain the displacements of all thenodes. This requirement is best achieved by matrix techniques; therefore, write the internal andexternal force contributions as matrices. If the properties and dimensions of the two elements are thesame, the equilibrium equations can be simplified as follows:8<:Pa

Pb

Pc

9=;+

µEA

L

¶24¡1 1 01 ¡2 10 1 ¡1

358<: ua

ub

uc

9=; = 0:

In general, it may be that the element stiffnesses, the EA=L terms, are different from element toelement; therefore, write the element stiffnesses as K1 and K2 for the two elements in the model. Weare interested in obtaining the solution to the equilibrium equation in which the externally appliedforces, P , are in equilibrium with the internally generated forces, I . When discussing this equationwith reference to convergence and nonlinearity, we tend to write it as

fPg ¡ fIg = 0:

For the complete two-element, three-node structure we, therefore, modify the signs and rewrite theequilibrium equation as8<:Pa

Pb

Pc

9=;¡24 K1 ¡K1 0¡K1 (K1 +K2) ¡K2

0 ¡K2 K2

358<: ua

ub

uc

9=; = 0:

This system of equations can then be solved to obtain values for the three unknown variables: ub, uc,and Pa (ua is specified in the problem as 0.0). Once the displacements are known, we can go back anduse them to calculate the stresses in the truss elements.

Introduction

1-20

2. ABAQUS BasicsA complete ABAQUS/Standard analysis usually consists of three distinct stages: preprocessing,simulation, and postprocessing. These three stages are linked together by files as shown below:

Preprocessing (ABAQUS/CAE)

In this stage you must define the model of the physical problem and create an ABAQUS inputfile. The model is usually created graphically using ABAQUS/CAE or another preprocessor,although the ABAQUS input file for a simple analysis can be created directly using a texteditor.

Simulation (ABAQUS/Standard)

The simulation, which normally is run as a background process, is the stage in whichABAQUS/Standard solves the numerical problem defined in the input file. Examples of outputfrom a stress analysis include displacements and stresses that are stored in binary files readyfor postpocessing. Depending on the complexity of the problem being analyzed and the powerof the computer being used, it may take anywhere from seconds to days to complete ananalysis run.

Postprocessing (ABAQUS/Viewer)

You can evaluate the results once the simulation has been completed and the displacements,stresses, or other fundamental variables have been calculated. The evaluation is generally doneinteractively using ABAQUS/Viewer or another postprocessor. ABAQUS/Viewer, whichreads the neutral binary output database file, has a variety of options for displaying the results,including color contour plots, animations, deformed shape plots, and X-Y plots.

2.1 Components of an ABAQUS model

2. ABAQUS BasicsA complete ABAQUS/Standard analysis usually consists of three distinct stages: preprocessing,simulation, and postprocessing. These three stages are linked together by files as shown below:

Preprocessing (ABAQUS/CAE)

In this stage you must define the model of the physical problem and create an ABAQUS inputfile. The model is usually created graphically using ABAQUS/CAE or another preprocessor,although the ABAQUS input file for a simple analysis can be created directly using a texteditor.

Simulation (ABAQUS/Standard)

The simulation, which normally is run as a background process, is the stage in whichABAQUS/Standard solves the numerical problem defined in the input file. Examples of outputfrom a stress analysis include displacements and stresses that are stored in binary files readyfor postpocessing. Depending on the complexity of the problem being analyzed and the powerof the computer being used, it may take anywhere from seconds to days to complete ananalysis run.

Postprocessing (ABAQUS/Viewer)

You can evaluate the results once the simulation has been completed and the displacements,stresses, or other fundamental variables have been calculated. The evaluation is generally doneinteractively using ABAQUS/Viewer or another postprocessor. ABAQUS/Viewer, whichreads the neutral binary output database file, has a variety of options for displaying the results,including color contour plots, animations, deformed shape plots, and X-Y plots.

2.1 Components of an ABAQUS model

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An ABAQUS model is composed of several different components that together describe the physicalproblem to be analyzed and the results to be obtained. At a minimum the model consists of thefollowing information: geometry, element section properties, material data, loads and boundaryconditions, analysis type, and output requests.

Geometry

Finite elements and nodes define the basic geometry of the physical structure being modeled inABAQUS. Each element in the model represents a discrete portion of the physical structure,which is, in turn, represented by many interconnected elements. Elements are connected to oneanother by shared nodes. The coordinates of the nodes and the connectivity of theelements--that is, which nodes belong to which elements--comprise the model geometry. Thecollection of all the elements and nodes in a model is called the mesh. Generally, the mesh willbe only an approximation of the actual geometry of the structure.

The element type, shape, and location, as well as the overall number of elements used in themesh, affect the results obtained from a simulation. The greater the mesh density (i.e., thegreater the number of elements in the mesh), the more accurate the results. As the meshdensity increases, the analysis results converge to a unique solution, and the computer timerequired for the analysis increases. The solution obtained from the numerical model isgenerally an approximation to the solution of the physical problem being simulated. Theextent of the approximations made in the model's geometry, material behavior, boundaryconditions, and loading determines how well the numerical simulation matches the physicalproblem.

Element section properties

ABAQUS has a wide range of elements, many of which have geometry not defined completelyby the coordinates of their nodes. For example, the layers of a composite shell or thedimensions of an I-beam section are not defined by the nodes of the element. Such additionalgeometric data are defined as physical properties of the element and are necessary to define themodel geometry completely (see Chapter 3, "ABAQUS Elements - A Brief Introduction").

Material data

Material properties for all elements must be specified. While high-quality material data areoften difficult to obtain, particularly for the more complex material models, the validity of theABAQUS results is limited by the accuracy and extent of the material data.

Loads and boundary conditions

Loads distort the physical structure and, thus, create stress in it. The most common forms ofloading include:

� Point loads.

� Pressure loads on surfaces.

� Body forces, such as the force of gravity.

An ABAQUS model is composed of several different components that together describe the physicalproblem to be analyzed and the results to be obtained. At a minimum the model consists of thefollowing information: geometry, element section properties, material data, loads and boundaryconditions, analysis type, and output requests.

Geometry

Finite elements and nodes define the basic geometry of the physical structure being modeled inABAQUS. Each element in the model represents a discrete portion of the physical structure,which is, in turn, represented by many interconnected elements. Elements are connected to oneanother by shared nodes. The coordinates of the nodes and the connectivity of theelements--that is, which nodes belong to which elements--comprise the model geometry. Thecollection of all the elements and nodes in a model is called the mesh. Generally, the mesh willbe only an approximation of the actual geometry of the structure.

The element type, shape, and location, as well as the overall number of elements used in themesh, affect the results obtained from a simulation. The greater the mesh density (i.e., thegreater the number of elements in the mesh), the more accurate the results. As the meshdensity increases, the analysis results converge to a unique solution, and the computer timerequired for the analysis increases. The solution obtained from the numerical model isgenerally an approximation to the solution of the physical problem being simulated. Theextent of the approximations made in the model's geometry, material behavior, boundaryconditions, and loading determines how well the numerical simulation matches the physicalproblem.


ABAQUS has a wide range of elements, many of which have geometry not defined completelyby the coordinates of their nodes. For example, the layers of a composite shell or thedimensions of an I-beam section are not defined by the nodes of the element. Such additionalgeometric data are defined as physical properties of the element and are necessary to define themodel geometry completely (see Chapter 3, "ABAQUS Elements - A Brief Introduction").

Material data

Material properties for all elements must be specified. While high-quality material data areoften difficult to obtain, particularly for the more complex material models, the validity of theABAQUS results is limited by the accuracy and extent of the material data.

Loads and boundary conditions

Loads distort the physical structure and, thus, create stress in it. The most common forms ofloading include:

� Point loads.

� Pressure loads on surfaces.

� Body forces, such as the force of gravity.

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� Thermal loads.

Boundary conditions are used to constrain portions of the model to remain fixed (zerodisplacements) or to move by a prescribed amount (nonzero displacements). In a staticanalysis enough boundary conditions must be used to prevent the model from moving as arigid body in any direction; otherwise, unrestrained rigid body motion causes the stiffnessmatrix to be singular. A solver problem will occur during the solution stage and may cause thesimulation to stop prematurely.

ABAQUS will issue a warning message if it detects a solver problem during a simulation. It isimportant that you learn to interpret such error messages issued by ABAQUS. If you see a"numerical singularity" or "zero pivot" warning message during a static stress analysis, youshould check whether all or part of your model lacks constraints against rigid body translationsor rotations. In a dynamic analysis inertia forces prevent the model from undergoing infinitemotion instantaneously as long as all separate parts in the model have some mass; therefore,solver problem warnings in a dynamic analysis usually indicate some other modeling problem,such as excessive plasticity.

Analysis type

The most common type of simulation is a static analysis, where the long-term response of thestructure to the applied loads is obtained. In other cases the dynamic response of a structure tothe loads may be of interest: for example, the effect of a sudden load on a component, such asoccurs during an impact, or the response of a building in an earthquake.

ABAQUS can carry out many different types of simulations, but this guide only covers the twomost common: static and dynamic stress analysis.

Output requests

An ABAQUS simulation can generate a vast amount of output. To avoid using excessive diskspace, you can use options that limit the output to that required for interpreting the results.

2.2 Format of the input fileThe input file is the means of communication between the preprocessor, usually ABAQUS/CAE, andthe solver, ABAQUS/Standard. It contains a complete description of the numerical model. The inputfile is a text file that has an intuitive, keyword-based format, so it is easy to modify using a text editorif necessary; if a preprocessor such as ABAQUS/CAE is used, modifications should be made using it.Indeed, small analyses can be specified easily by typing the input file directly.

The example of an overhead hoist, shown in Figure 2-1, is used to illustrate the basic format of theABAQUS input file. The hoist is a simple, pin-jointed framework that is constrained at the left-handend and mounted on rollers at the right-hand end. The members can rotate freely at the joints. Theframe is prevented from moving out of plane. A simulation is performed to determine the structure'sdeflection and the peak stress in its members when a 10 kN load is applied as shown in Figure 2-1.

� Thermal loads.

Boundary conditions are used to constrain portions of the model to remain fixed (zerodisplacements) or to move by a prescribed amount (nonzero displacements). In a staticanalysis enough boundary conditions must be used to prevent the model from moving as arigid body in any direction; otherwise, unrestrained rigid body motion causes the stiffnessmatrix to be singular. A solver problem will occur during the solution stage and may cause thesimulation to stop prematurely.

ABAQUS will issue a warning message if it detects a solver problem during a simulation. It isimportant that you learn to interpret such error messages issued by ABAQUS. If you see a"numerical singularity" or "zero pivot" warning message during a static stress analysis, youshould check whether all or part of your model lacks constraints against rigid body translationsor rotations. In a dynamic analysis inertia forces prevent the model from undergoing infinitemotion instantaneously as long as all separate parts in the model have some mass; therefore,solver problem warnings in a dynamic analysis usually indicate some other modeling problem,such as excessive plasticity.

Analysis type

The most common type of simulation is a static analysis, where the long-term response of thestructure to the applied loads is obtained. In other cases the dynamic response of a structure tothe loads may be of interest: for example, the effect of a sudden load on a component, such asoccurs during an impact, or the response of a building in an earthquake.

ABAQUS can carry out many different types of simulations, but this guide only covers the twomost common: static and dynamic stress analysis.

Output requests

An ABAQUS simulation can generate a vast amount of output. To avoid using excessive diskspace, you can use options that limit the output to that required for interpreting the results.

2.2 Format of the input fileThe input file is the means of communication between the preprocessor, usually ABAQUS/CAE, andthe solver, ABAQUS/Standard. It contains a complete description of the numerical model. The inputfile is a text file that has an intuitive, keyword-based format, so it is easy to modify using a text editorif necessary; if a preprocessor such as ABAQUS/CAE is used, modifications should be made using it.Indeed, small analyses can be specified easily by typing the input file directly.

The example of an overhead hoist, shown in Figure 2-1, is used to illustrate the basic format of theABAQUS input file. The hoist is a simple, pin-jointed framework that is constrained at the left-handend and mounted on rollers at the right-hand end. The members can rotate freely at the joints. Theframe is prevented from moving out of plane. A simulation is performed to determine the structure'sdeflection and the peak stress in its members when a 10 kN load is applied as shown in Figure 2-1.

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Figure 2-1 Schematic of an overhead hoist.

Since this problem is very simple, the ABAQUS input file is compact and easily understood. Thecomplete ABAQUS input file for this example, which is shown in Figure 2-2 and also in ``Overheadhoist frame,'' Section A.1, is split into two distinct parts. The first section contains model data andincludes all the information required to define the structure being analyzed. The second sectioncontains history data that define what happens to the model: the sequence of loading or events forwhich the response of the structure is required. This history is divided into a sequence of steps, eachdefining a separate part of the simulation. For example, the first step may define a static loading whilethe second step may define a dynamic loading, etc.

Figure 2-2 Input for overhead hoist model.

Figure 2-1 Schematic of an overhead hoist.

Since this problem is very simple, the ABAQUS input file is compact and easily understood. Thecomplete ABAQUS input file for this example, which is shown in Figure 2-2 and also in ``Overheadhoist frame,'' Section A.1, is split into two distinct parts. The first section contains model data andincludes all the information required to define the structure being analyzed. The second sectioncontains history data that define what happens to the model: the sequence of loading or events forwhich the response of the structure is required. This history is divided into a sequence of steps, eachdefining a separate part of the simulation. For example, the first step may define a static loading whilethe second step may define a dynamic loading, etc.

Figure 2-2 Input for overhead hoist model.

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The input file is composed of a number of option blocks that contain data describing a part of themodel. Each option block begins with a keyword line, which is usually followed by one or more datalines. These lines cannot exceed 256 characters.

2.2.1 Keyword linesKeywords (or options) always begin with a star or asterisk (*). For example, *NODE is the keywordfor specifying the nodal coordinates, and *ELEMENT is the keyword for specifying the elementconnectivity. Keywords are often followed by parameters, some of which may be required. Theparameter TYPE is required with the *ELEMENT option because the element type must always begiven when defining elements. For example,

*ELEMENT, TYPE=T2D2

indicates that we are defining the connectivity of T2D2 elements (two-dimensional truss elements withtwo nodes). Many parameters are optional and are defined only in certain circumstances. For example,

The input file is composed of a number of option blocks that contain data describing a part of themodel. Each option block begins with a keyword line, which is usually followed by one or more datalines. These lines cannot exceed 256 characters.

2.2.1 Keyword linesKeywords (or options) always begin with a star or asterisk (*). For example, *NODE is the keywordfor specifying the nodal coordinates, and *ELEMENT is the keyword for specifying the elementconnectivity. Keywords are often followed by parameters, some of which may be required. Theparameter TYPE is required with the *ELEMENT option because the element type must always begiven when defining elements. For example,

*ELEMENT, TYPE=T2D2

indicates that we are defining the connectivity of T2D2 elements (two-dimensional truss elements withtwo nodes). Many parameters are optional and are defined only in certain circumstances. For example,

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2-25

*NODE, NSET=PART1

indicates that all the nodes defined in this option block will be put into a set called PART1. It is notessential to put nodes into sets, although it is convenient in many instances.

Keywords and parameters are case independent and must use enough characters to make them unique.Parameters are separated by commas. If a parameter has a value, an equal sign (=) is used to associatethe value with the parameter.

Occasionally, so many parameters are required that they will not fit on a single 256-character line. Inthis case a comma is placed at the end of the line to indicate that the next line is a continuation line.For example,

*ELEMENT, TYPE = T2D2,

ELSET = FRAME

is a valid keyword line.

Details of the keywords are documented in the ABAQUS Keywords Manual.

2.2.2 Data linesKeyword lines are usually followed by data lines, which provide data that are more easily specified aslists than as parameters on the keyword line. Examples of such data include nodal coordinates; elementconnectivities; or tables of material properties, such as stress-strain curves. The data required forparticular option blocks are specified in the ABAQUS Keywords Manual. For example, the optionblock defining the nodes for the overhead hoist model is:

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.

The first piece of data in each data line is an integer that defines the node number. The second, third,and fourth entries are floating-point numbers that specify the x1 , x2 , x3 coordinates of the node.

The data can consist of a mixture of integer, floating point, or alphanumeric values. Floating pointvalues can be entered in a variety of ways; for example, ABAQUS interprets all of the following as thenumber four:

4.0 4. 44.0E+0 .4E+1 40.E-1

Data items are separated by commas, as in Figure 2-2, which allows fairly arbitrary spacing of theinput values on the data line. If there is only one item on a data line, it must be followed by acomma.

2.3 Creating an input fileThe simulation of the pin-jointed, overhead hoist in Figure 2-1 is used to illustrate the creation of an

*NODE, NSET=PART1

indicates that all the nodes defined in this option block will be put into a set called PART1. It is notessential to put nodes into sets, although it is convenient in many instances.

Keywords and parameters are case independent and must use enough characters to make them unique.Parameters are separated by commas. If a parameter has a value, an equal sign (=) is used to associatethe value with the parameter.

Occasionally, so many parameters are required that they will not fit on a single 256-character line. Inthis case a comma is placed at the end of the line to indicate that the next line is a continuation line.For example,

*ELEMENT, TYPE = T2D2,

ELSET = FRAME

is a valid keyword line.

Details of the keywords are documented in the ABAQUS Keywords Manual.

2.2.2 Data linesKeyword lines are usually followed by data lines, which provide data that are more easily specified aslists than as parameters on the keyword line. Examples of such data include nodal coordinates; elementconnectivities; or tables of material properties, such as stress-strain curves. The data required forparticular option blocks are specified in the ABAQUS Keywords Manual. For example, the optionblock defining the nodes for the overhead hoist model is:

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.

The first piece of data in each data line is an integer that defines the node number. The second, third,and fourth entries are floating-point numbers that specify the x1 , x2 , x3 coordinates of the node.

The data can consist of a mixture of integer, floating point, or alphanumeric values. Floating pointvalues can be entered in a variety of ways; for example, ABAQUS interprets all of the following as thenumber four:

4.0 4. 44.0E+0 .4E+1 40.E-1

Data items are separated by commas, as in Figure 2-2, which allows fairly arbitrary spacing of theinput values on the data line. If there is only one item on a data line, it must be followed by acomma.

2.3 Creating an input fileThe simulation of the pin-jointed, overhead hoist in Figure 2-1 is used to illustrate the creation of an

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ABAQUS input file using an editor. As you read through this section, you should type the data into afile using one of the editors available on your computer. The ABAQUS input file must have an .inpfile extension. For convenience, name the input file frame.inp. The file identifier, which can bechosen to identify the analysis, is called the jobname. In this case use the jobname "frame" to easilyassociate it with the input file called frame.inp.

All of the other examples in this guide assume that you will be using a preprocessor, such asABAQUS/CAE, to generate the mesh if you are going to create the model from scratch. Input files forall the examples are available. See ``Execution procedure for ABAQUS/Fetch,'' Section 3.2.9 of theABAQUS/Standard User's Manual, for instructions on how to retrieve these input files. However,since the purpose of this example is to help you understand the structure and format of the ABAQUSinput file, you should type this input file in directly, rather than use a preprocessor or copy the inputfile that is provided.

2.3.1 UnitsBefore starting to define this or any model, you need to decide which system of units you will use.ABAQUS has no built-in system of units. All input data must be specified in consistent units. Somecommon systems of consistent units are shown in Table 2-1.

Table 2-1. Consistent units.Quantity SI SI (mm) US Unit (ft) US Unit (inch)

Length m mm ft inForce N N lbf lbfMass kg tonne (103 kg) slug lbf s2 /inTime s s s sStress Pa (N/m2 ) MPa (N/mm2 ) lbf/ft2 psi (lbf/in2)Energy J mJ (10-3 J) ft lbf in lbfDensity kg/m3 tonne/mm3 slug/ft3 lbf s2 /in4

The SI system of units is used throughout this guide. Users working in the systems labeled " US Unit"should be careful with the units of density; often the densities given in handbooks of materialproperties are multiplied by the acceleration due to gravity.

2.3.2 Coordinate systemsYou also need to decide which coordinate system to use. The global coordinate system in ABAQUS isa right-handed, rectangular (Cartesian) system. For this example define the global 1-axis to be thehorizontal axis of the hoist and the global 2-axis to be the vertical axis ( Figure 2-3). The global 3-axisis normal to the plane of the framework. The origin (x1=0, x2=0, x3=0) is the bottom left-hand cornerof the frame.

Figure 2-3 Coordinate system and origin for model.

ABAQUS input file using an editor. As you read through this section, you should type the data into afile using one of the editors available on your computer. The ABAQUS input file must have an .inpfile extension. For convenience, name the input file frame.inp. The file identifier, which can bechosen to identify the analysis, is called the jobname. In this case use the jobname "frame" to easilyassociate it with the input file called frame.inp.

All of the other examples in this guide assume that you will be using a preprocessor, such asABAQUS/CAE, to generate the mesh if you are going to create the model from scratch. Input files forall the examples are available. See ``Execution procedure for ABAQUS/Fetch,'' Section 3.2.9 of theABAQUS/Standard User's Manual, for instructions on how to retrieve these input files. However,since the purpose of this example is to help you understand the structure and format of the ABAQUSinput file, you should type this input file in directly, rather than use a preprocessor or copy the inputfile that is provided.

2.3.1 UnitsBefore starting to define this or any model, you need to decide which system of units you will use.ABAQUS has no built-in system of units. All input data must be specified in consistent units. Somecommon systems of consistent units are shown in Table 2-1.

Table 2-1. Consistent units.Quantity SI SI (mm) US Unit (ft) US Unit (inch)

Length m mm ft inForce N N lbf lbfMass kg tonne (103 kg) slug lbf s2 /inTime s s s sStress Pa (N/m2 ) MPa (N/mm2 ) lbf/ft2 psi (lbf/in2)Energy J mJ (10-3 J) ft lbf in lbfDensity kg/m3 tonne/mm3 slug/ft3 lbf s2 /in4

The SI system of units is used throughout this guide. Users working in the systems labeled " US Unit"should be careful with the units of density; often the densities given in handbooks of materialproperties are multiplied by the acceleration due to gravity.

2.3.2 Coordinate systemsYou also need to decide which coordinate system to use. The global coordinate system in ABAQUS isa right-handed, rectangular (Cartesian) system. For this example define the global 1-axis to be thehorizontal axis of the hoist and the global 2-axis to be the vertical axis ( Figure 2-3). The global 3-axisis normal to the plane of the framework. The origin (x1=0, x2=0, x3=0) is the bottom left-hand cornerof the frame.

Figure 2-3 Coordinate system and origin for model.

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For two-dimensional problems, such as this one, ABAQUS requires that the model lie in a planeparallel to the global 1-2 plane.

2.3.3 MeshYou must select the element types and design the mesh. Creating a proper mesh for a given problemrequires experience. For this example you will use a single truss element to model each member of theframe, as shown in Figure 2-4.

Figure 2-4 Finite element mesh.

A truss element, which can carry only tensile and compressive axial loads, is ideal for modelingpin-jointed frameworks, such as the overhead hoist. Truss elements are described in ``Truss elements,''Section 3.5, and also in the ABAQUS/Standard User's Manual, which describes every elementavailable in ABAQUS. The index of element types (Chapter i, "Element Index," of theABAQUS/Standard User's Manual) makes locating a particular element easy. Whenever you are usingan element for the first time, you should read the description, which includes the element connectivityand any element section properties needed to define the element's geometry.

The connectivity for the truss elements used in the overhead hoist model is shown in Figure 2-5.

Figure 2-5 Connectivity for the 2-node truss element (T2D2).

For two-dimensional problems, such as this one, ABAQUS requires that the model lie in a planeparallel to the global 1-2 plane.

2.3.3 MeshYou must select the element types and design the mesh. Creating a proper mesh for a given problemrequires experience. For this example you will use a single truss element to model each member of theframe, as shown in Figure 2-4.

Figure 2-4 Finite element mesh.

A truss element, which can carry only tensile and compressive axial loads, is ideal for modelingpin-jointed frameworks, such as the overhead hoist. Truss elements are described in ``Truss elements,''Section 3.5, and also in the ABAQUS/Standard User's Manual, which describes every elementavailable in ABAQUS. The index of element types (Chapter i, "Element Index," of theABAQUS/Standard User's Manual) makes locating a particular element easy. Whenever you are usingan element for the first time, you should read the description, which includes the element connectivityand any element section properties needed to define the element's geometry.

The connectivity for the truss elements used in the overhead hoist model is shown in Figure 2-5.

Figure 2-5 Connectivity for the 2-node truss element (T2D2).

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Node and element numbers are merely identification labels. They are usually generated automaticallyby ABAQUS/CAE or another preprocessor. The only requirement for node and element numbers isthat they must be positive integers. Gaps in the numbering are allowed, and the order in which nodesand elements are defined does not matter. Any nodes that are defined but not associated with anelement are removed automatically and are not included in the simulation.

In this case we use the node and element numbers shown in Figure 2-6.

Figure 2-6 Node and element numbers for the hoist model.

2.3.4 Model dataThe first part of the input file must contain all of the model data. These data define the structure beinganalyzed. In the overhead hoist example the model data consist of the following:

� Geometry:

� Nodal coordinates.

� Element connectivity.

� Element section properties.

� Material properties.

Heading

The first option in any ABAQUS input file must be *HEADING. The data lines that follow the*HEADING option are lines of text describing the problem being simulated. You shouldprovide an accurate description to allow the input file to be identified at a later date.Moreover, it is often helpful to specify the system of units, directions of the global coordinatesystem, etc. For example, the *HEADING option block for the hoist problem contains thefollowing:

*HEADING

Two-dimensional overhead hoist frame

SI Units

1-axis horizontal, 2-axis vertical

Node and element numbers are merely identification labels. They are usually generated automaticallyby ABAQUS/CAE or another preprocessor. The only requirement for node and element numbers isthat they must be positive integers. Gaps in the numbering are allowed, and the order in which nodesand elements are defined does not matter. Any nodes that are defined but not associated with anelement are removed automatically and are not included in the simulation.

In this case we use the node and element numbers shown in Figure 2-6.

Figure 2-6 Node and element numbers for the hoist model.

2.3.4 Model dataThe first part of the input file must contain all of the model data. These data define the structure beinganalyzed. In the overhead hoist example the model data consist of the following:

� Geometry:

� Nodal coordinates.

� Element connectivity.

� Element section properties.

� Material properties.

Heading

The first option in any ABAQUS input file must be *HEADING. The data lines that follow the*HEADING option are lines of text describing the problem being simulated. You shouldprovide an accurate description to allow the input file to be identified at a later date.Moreover, it is often helpful to specify the system of units, directions of the global coordinatesystem, etc. For example, the *HEADING option block for the hoist problem contains thefollowing:

*HEADING


SI Units


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Data file printing options

By default, ABAQUS will not print an echo of the input file or the model and historydefinition data to the printed output (.dat) file. However, it is recommended that you checkyour model and history definition in a datacheck run before performing the analysis. Thedatacheck run is discussed later in this chapter.

To request a printout of the input file and of the model and history definition data, add

*PREPRINT, ECHO=YES, MODEL=YES, HISTORY=YES

to the input file.

Nodal coordinates

The coordinates of each node can be defined once you select the mesh design and nodenumbering scheme. For this problem use the numbering shown in Figure 2-6. The coordinatesof nodes are defined using the *NODE option. Each data line of this option block has theform

<node number>,<x1-coordinate>,<x2-coordinate>,<x3-coordinate>

The nodes for the hoist model are defined as follows:

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.

Element connectivity

The members of the overhead hoist are modeled with truss elements. The format of each dataline for a truss element is

<element number>, <node 1>, <node 2>

where node 1 and node 2 are at the ends of the element (see Figure 2-5). For example, element16 connects nodes 103 and 105 (see Figure 2-6), so the data line defining this element is

16, 103, 105

The TYPE parameter on the *ELEMENT option must be used to specify the kind of elementbeing defined. In this case you will use T2D2 truss elements.

One of the most useful features in ABAQUS is the availability of node and element sets thatare referred to by name. By using the ELSET parameter on the *ELEMENT option, all of theelements defined in the option block are added to an element set called FRAME. A set namecan have as many as 80 characters and must start with a letter. Since element sectionproperties are assigned through element set names, all elements in the model must belong to atleast one element set.

Data file printing options

By default, ABAQUS will not print an echo of the input file or the model and historydefinition data to the printed output (.dat) file. However, it is recommended that you checkyour model and history definition in a datacheck run before performing the analysis. Thedatacheck run is discussed later in this chapter.

To request a printout of the input file and of the model and history definition data, add


to the input file.

Nodal coordinates

The coordinates of each node can be defined once you select the mesh design and nodenumbering scheme. For this problem use the numbering shown in Figure 2-6. The coordinatesof nodes are defined using the *NODE option. Each data line of this option block has theform

<node number>,<x1-coordinate>,<x2-coordinate>,<x3-coordinate>

The nodes for the hoist model are defined as follows:

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.


The members of the overhead hoist are modeled with truss elements. The format of each dataline for a truss element is

<element number>, <node 1>, <node 2>

where node 1 and node 2 are at the ends of the element (see Figure 2-5). For example, element16 connects nodes 103 and 105 (see Figure 2-6), so the data line defining this element is

16, 103, 105

The TYPE parameter on the *ELEMENT option must be used to specify the kind of elementbeing defined. In this case you will use T2D2 truss elements.

One of the most useful features in ABAQUS is the availability of node and element sets thatare referred to by name. By using the ELSET parameter on the *ELEMENT option, all of theelements defined in the option block are added to an element set called FRAME. A set namecan have as many as 80 characters and must start with a letter. Since element sectionproperties are assigned through element set names, all elements in the model must belong to atleast one element set.

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The complete *ELEMENT option block for the overhead hoist model (see Figure 2-6) isshown below:

*ELEMENT, TYPE=T2D2, ELSET=FRAME

11, 101,102

12, 102,103

13, 101,104

14, 102,104

15, 102,105

16, 103,105

17, 104,105


Each element must refer to an element section property. The appropriate element sectionoption for each element and the additional geometric data (if any) needed for each element aredescribed in the ABAQUS/Standard User's Manual.

For the T2D2 element you must use the *SOLID SECTION option and give one data line withthe cross-sectional area of the element. If you leave the data line blank, the cross-sectional areais assumed to be 1.0.

In this case all the members are circular bars that are 5 mm in diameter. Their cross-sectionalarea is 1.963 � 10-5 m2.

The MATERIAL parameter, which is required for most element section options, refers to thename of a material property definition that is to be used with the elements. The name can haveup to 80 characters and must begin with a letter.

In this example all of the elements have the same section properties and are made of the samematerial. Typically, there will be several different element section properties in an analysis; forexample, different components in a model may be made of different materials. The elementsare associated with material properties through element sets. For the overhead hoist model theelements are added to an element set called FRAME. Element set FRAME is then used as thevalue of the ELSET parameter on the element section option. Add the following option blockto your input file:

Materials

One of the features that makes ABAQUS a truly general-purpose finite element program isthat almost any material model can be used with any element. Once the mesh has been created,material models can be associated, as appropriate, with the elements in the mesh.

ABAQUS has a large number of material models, many of which include nonlinear behavior.

The complete *ELEMENT option block for the overhead hoist model (see Figure 2-6) isshown below:


11, 101,102

12, 102,103

13, 101,104

14, 102,104

15, 102,105

16, 103,105

17, 104,105


Each element must refer to an element section property. The appropriate element sectionoption for each element and the additional geometric data (if any) needed for each element aredescribed in the ABAQUS/Standard User's Manual.

For the T2D2 element you must use the *SOLID SECTION option and give one data line withthe cross-sectional area of the element. If you leave the data line blank, the cross-sectional areais assumed to be 1.0.

In this case all the members are circular bars that are 5 mm in diameter. Their cross-sectionalarea is 1.963 � 10-5 m2.

The MATERIAL parameter, which is required for most element section options, refers to thename of a material property definition that is to be used with the elements. The name can haveup to 80 characters and must begin with a letter.

In this example all of the elements have the same section properties and are made of the samematerial. Typically, there will be several different element section properties in an analysis; forexample, different components in a model may be made of different materials. The elementsare associated with material properties through element sets. For the overhead hoist model theelements are added to an element set called FRAME. Element set FRAME is then used as thevalue of the ELSET parameter on the element section option. Add the following option blockto your input file:

Materials

One of the features that makes ABAQUS a truly general-purpose finite element program isthat almost any material model can be used with any element. Once the mesh has been created,material models can be associated, as appropriate, with the elements in the mesh.

ABAQUS has a large number of material models, many of which include nonlinear behavior.

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In this overhead hoist example we use the simplest form of material behavior: linear elasticity.In Chapter 8, "Materials," two of the most common forms of nonlinear material behavior areconsidered: metal plasticity and rubber elasticity. A discussion of all the material modelsavailable in ABAQUS/Standard can be found in the ABAQUS/Standard User's Manual.

Linear elasticity is appropriate for many materials at small strains, particularly for metals up totheir yield point. It is characterized by a linear relationship between stress and strain (Hooke'slaw), as shown in Figure 2-7.

Figure 2-7 Linear elastic material.

The material behavior is characterized by two constants: Young's modulus, E, and Poisson'sratio, º.

A material definition in the ABAQUS input file starts with a *MATERIAL option. Theparameter NAME is used to associate a material with an element section property. Forexample,

Material suboptions directly follow their associated *MATERIAL option. Several suboptionsmay be required to complete the material definition. All material suboptions are associatedwith the material that is listed on the most recent *MATERIAL option until another*MATERIAL option or a non-material option block is given.

Without considering thermal expansion effects (which would be defined with the*EXPANSION material suboption), one material suboption, *ELASTIC, is required to definea linear elastic material. The form of this option block is

*ELASTIC

<E>,<º>

Therefore, the complete, isotropic, linear elastic material definition for the hoist members,which are made of steel, should be entered into your input file as

*MATERIAL, NAME=STEEL

*ELASTIC

In this overhead hoist example we use the simplest form of material behavior: linear elasticity.In Chapter 8, "Materials," two of the most common forms of nonlinear material behavior areconsidered: metal plasticity and rubber elasticity. A discussion of all the material modelsavailable in ABAQUS/Standard can be found in the ABAQUS/Standard User's Manual.

Linear elasticity is appropriate for many materials at small strains, particularly for metals up totheir yield point. It is characterized by a linear relationship between stress and strain (Hooke'slaw), as shown in Figure 2-7.

Figure 2-7 Linear elastic material.

The material behavior is characterized by two constants: Young's modulus, E, and Poisson'sratio, º.

A material definition in the ABAQUS input file starts with a *MATERIAL option. Theparameter NAME is used to associate a material with an element section property. Forexample,

Material suboptions directly follow their associated *MATERIAL option. Several suboptionsmay be required to complete the material definition. All material suboptions are associatedwith the material that is listed on the most recent *MATERIAL option until another*MATERIAL option or a non-material option block is given.

Without considering thermal expansion effects (which would be defined with the*EXPANSION material suboption), one material suboption, *ELASTIC, is required to definea linear elastic material. The form of this option block is

*ELASTIC

<E>,<º>

Therefore, the complete, isotropic, linear elastic material definition for the hoist members,which are made of steel, should be entered into your input file as


*ELASTIC

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200.E9, 0.3

The model definition portion of this problem is now complete since all the componentsdescribing the structure have been specified.

2.3.5 History dataThe history data define the sequence of events for the simulation. This loading history is divided into aseries of steps, each defining a different portion of the structure's loading. Each step contains thefollowing information:

� the type of simulation (static, dynamic, etc.);

� the loads and constraints; and

� the output required.

In this example we are interested in the static response of the overhead hoist to a 10 kN load applied atthe center, with the left-hand end fully constrained and a roller constraint on the right-hand end (seeFigure 2-1). This is a single event, so only a single step is needed for the simulation.

The *STEP option is used to mark the start of a step. Like the *HEADING option, this option may befollowed by data lines containing a title for the step. In your hoist model use the following *STEPoption block:

*STEP,PERTURBATION

10kN central load

The PERTURBATION parameter indicates that this is a linear analysis. If this parameter is omitted,the analysis may be linear or nonlinear. The use of the PERTURBATION parameter is discussedfurther in Chapter 10, "Multiple Step Analysis."

Analysis procedure

The analysis procedure (the type of simulation) must be defined immediately following the*STEP option block. In this case we want the long-term static response of the structure. Theoption for a static simulation is *STATIC. For linear analysis this option has no parameters ordata lines, so add the following line to your input file:

*STATIC

The remaining input data in the step define the boundary conditions (constraints), loads, andoutput required and can be given in any order that is convenient.

Boundary conditions

Boundary conditions are applied to those parts of the model where the displacements areknown. Such parts may be constrained to remain fixed (have zero displacement) during thesimulation or may have specified, nonzero displacements. In either situation the constraints areapplied directly to the nodes of the model.

200.E9, 0.3

The model definition portion of this problem is now complete since all the componentsdescribing the structure have been specified.

2.3.5 History dataThe history data define the sequence of events for the simulation. This loading history is divided into aseries of steps, each defining a different portion of the structure's loading. Each step contains thefollowing information:

� the type of simulation (static, dynamic, etc.);

� the loads and constraints; and

� the output required.

In this example we are interested in the static response of the overhead hoist to a 10 kN load applied atthe center, with the left-hand end fully constrained and a roller constraint on the right-hand end (seeFigure 2-1). This is a single event, so only a single step is needed for the simulation.

The *STEP option is used to mark the start of a step. Like the *HEADING option, this option may befollowed by data lines containing a title for the step. In your hoist model use the following *STEPoption block:

*STEP,PERTURBATION

10kN central load

The PERTURBATION parameter indicates that this is a linear analysis. If this parameter is omitted,the analysis may be linear or nonlinear. The use of the PERTURBATION parameter is discussedfurther in Chapter 10, "Multiple Step Analysis."

Analysis procedure

The analysis procedure (the type of simulation) must be defined immediately following the*STEP option block. In this case we want the long-term static response of the structure. Theoption for a static simulation is *STATIC. For linear analysis this option has no parameters ordata lines, so add the following line to your input file:

*STATIC

The remaining input data in the step define the boundary conditions (constraints), loads, andoutput required and can be given in any order that is convenient.

Boundary conditions

Boundary conditions are applied to those parts of the model where the displacements areknown. Such parts may be constrained to remain fixed (have zero displacement) during thesimulation or may have specified, nonzero displacements. In either situation the constraints areapplied directly to the nodes of the model.

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In some cases a node may be constrained completely and, thus, cannot move in any direction(for example, node 101 in our case). In other cases a node is constrained in some directionsbut is free to move in others. For example, node 103 is fixed in the vertical direction but isfree to move in the horizontal direction. The directions in which a node is able to move arecalled degrees of freedom (dof). In the case of our two-dimensional hoist, each node can movein the global 1- and 2-directions; therefore, there are two degrees of freedom at each node. Ifthe hoist could move out of plane, the problem would be three-dimensional, and each nodewould have three degrees of freedom. Nodes attached to beam and shell elements haveadditional degrees of freedom representing the components of rotation and, thus, may have upto six degrees of freedom.

The labeling convention used for the degrees of freedom in ABAQUS is as follows:

The degrees of freedom active at a node depend on the type of elements attached to that node.Chapter 3, "ABAQUS Elements - A Brief Introduction," describes the active degrees offreedom for some of the elements available in ABAQUS. The two-dimensional truss element,T2D2, has two degrees of freedom active at each node--translation in the 1- and 2-directions(dof 1 and dof 2).

Constraints on nodes are defined by using the *BOUNDARY option and specifying theconstrained degrees of freedom. Each data line is of the form:

<node number>, <first dof>, <last dof>, <magnitude of displacement>

The first degree of freedom and last degree of freedom are used to give a range of degrees offreedom that will be constrained. For example,

101, 1, 3, 0.0

constrains degrees of freedom 1, 2, and 3 at node 101 to have zero displacement (the nodecannot move in either the global 1-, 2-, or 3-direction).

If the magnitude of the displacement is not specified on the data line, it is assumed to be zero.If the node is constrained in one direction only, the third field should be blank or equal to thesecond field. For example, to constrain node 103 in the 2-direction (degree of freedom 2) only,any of the following data line formats can be used:

In some cases a node may be constrained completely and, thus, cannot move in any direction(for example, node 101 in our case). In other cases a node is constrained in some directionsbut is free to move in others. For example, node 103 is fixed in the vertical direction but isfree to move in the horizontal direction. The directions in which a node is able to move arecalled degrees of freedom (dof). In the case of our two-dimensional hoist, each node can movein the global 1- and 2-directions; therefore, there are two degrees of freedom at each node. Ifthe hoist could move out of plane, the problem would be three-dimensional, and each nodewould have three degrees of freedom. Nodes attached to beam and shell elements haveadditional degrees of freedom representing the components of rotation and, thus, may have upto six degrees of freedom.

The labeling convention used for the degrees of freedom in ABAQUS is as follows:

The degrees of freedom active at a node depend on the type of elements attached to that node.Chapter 3, "ABAQUS Elements - A Brief Introduction," describes the active degrees offreedom for some of the elements available in ABAQUS. The two-dimensional truss element,T2D2, has two degrees of freedom active at each node--translation in the 1- and 2-directions(dof 1 and dof 2).

Constraints on nodes are defined by using the *BOUNDARY option and specifying theconstrained degrees of freedom. Each data line is of the form:

<node number>, <first dof>, <last dof>, <magnitude of displacement>

The first degree of freedom and last degree of freedom are used to give a range of degrees offreedom that will be constrained. For example,

101, 1, 3, 0.0

constrains degrees of freedom 1, 2, and 3 at node 101 to have zero displacement (the nodecannot move in either the global 1-, 2-, or 3-direction).

If the magnitude of the displacement is not specified on the data line, it is assumed to be zero.If the node is constrained in one direction only, the third field should be blank or equal to thesecond field. For example, to constrain node 103 in the 2-direction (degree of freedom 2) only,any of the following data line formats can be used:

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103, 2,2, 0.0

or

103, 2,2

or

103, 2

Boundary conditions on a node are cumulative. Thus, the following input constrains node 101in both directions 1 and 2:

101, 1

101, 2

Rather than specifying each constrained degree of freedom, some of the more commonconstraints can be given directly using the following named constraints:ENCASTRE Constraint on all displacements and rotations at a

node.PINNED Constraint on all translational degrees of freedom.XSYMM Symmetry constraint about a plane of constant x1 .YSYMM Symmetry constraint about a plane of constant x2 .ZSYMM Symmetry constraint about a plane of constant x3.XASYMM Antisymmetry constraint about a plane of constant x1.YASYMM Antisymmetry constraint about a plane of constant x2.ZASYMM Antisymmetry constraint about a plane of constant x3.

Thus, another way to constrain all the active degrees of freedom at node 101 in the hoistmodel is

101, ENCASTRE

The complete *BOUNDARY option block for our hoist problem is:

*BOUNDARY

101,ENCASTRE

103,2

In this example all of the constraints are in the global 1- or 2-directions. In many casesconstraints are required in directions that are not aligned with the global directions. The*TRANSFORM option can be used in such cases to define a local coordinate system forboundary condition application. The skew plate example in Chapter 5, "Using ShellElements," demonstrates how to use this option in such cases.

Loading

Loading is anything that causes the displacement or deformation of the structure, including:

� Concentrated loads.

� Pressure loads.

103, 2,2, 0.0

or

103, 2,2

or

103, 2

Boundary conditions on a node are cumulative. Thus, the following input constrains node 101in both directions 1 and 2:

101, 1

101, 2

Rather than specifying each constrained degree of freedom, some of the more commonconstraints can be given directly using the following named constraints:ENCASTRE Constraint on all displacements and rotations at a

node.PINNED Constraint on all translational degrees of freedom.XSYMM Symmetry constraint about a plane of constant x1 .YSYMM Symmetry constraint about a plane of constant x2 .ZSYMM Symmetry constraint about a plane of constant x3.XASYMM Antisymmetry constraint about a plane of constant x1.YASYMM Antisymmetry constraint about a plane of constant x2.ZASYMM Antisymmetry constraint about a plane of constant x3.

Thus, another way to constrain all the active degrees of freedom at node 101 in the hoistmodel is

101, ENCASTRE

The complete *BOUNDARY option block for our hoist problem is:

*BOUNDARY

101,ENCASTRE

103,2

In this example all of the constraints are in the global 1- or 2-directions. In many casesconstraints are required in directions that are not aligned with the global directions. The*TRANSFORM option can be used in such cases to define a local coordinate system forboundary condition application. The skew plate example in Chapter 5, "Using ShellElements," demonstrates how to use this option in such cases.

Loading

Loading is anything that causes the displacement or deformation of the structure, including:

� Concentrated loads.

� Pressure loads.

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� Nonzero boundary conditions.

� Body loads.

� Temperature (with thermal expansion of the material defined).

In reality there is no such thing as a concentrated, or point, load; the load will always beapplied over some finite area. However, if the area being loaded is similar to or smaller thanthe elements in that area, it is an appropriate idealization to treat the load as a concentratedload applied to a node.

Concentrated loads are specified using the *CLOAD option. The data lines for this optionhave the form:

<node number>, <dof>, <load magnitude>

In this simulation a load of -10 kN is applied in the 2-direction to node 102. The option blockis:

*CLOAD

102, 2, -10.E3

Output requests

Finite element analyses can create very large amounts of output. ABAQUS allows you tocontrol and manage this output so that only data required to interpret the results of yoursimulation are produced. Four types of output are available:

� Results stored in a neutral binary file used by ABAQUS/Viewer for postprocessing: outputdatabase (.odb) file.

� Printed tables of results.

� Restart data, used to continue the analysis.

� Results stored in binary files for subsequent postprocessing with third-party software.

You will use the first two of these in the overhead hoist simulation.

By default, an output database file, which includes a preselected set of the most commonlyused output variables for a given type of analysis, is created. A list of preselected variables fordefault output database output is given in the ABAQUS/Standard User's Manual. You do notneed to add any output requests to accept these defaults. For this example the default outputdatabase output includes the deformed configuration and the applied nodal loads.

Selected results also can be written in tabular form to the ABAQUS data file. By default, noprintout is written to the ABAQUS data file. The *NODE PRINT option controls the printingof nodal results (for example, displacements and reaction forces), while the *EL PRINT optioncontrols the printing of element results. A comprehensive list of the output variables availableis given in the ABAQUS/Standard User's Manual.

� Nonzero boundary conditions.

� Body loads.

� Temperature (with thermal expansion of the material defined).

In reality there is no such thing as a concentrated, or point, load; the load will always beapplied over some finite area. However, if the area being loaded is similar to or smaller thanthe elements in that area, it is an appropriate idealization to treat the load as a concentratedload applied to a node.

Concentrated loads are specified using the *CLOAD option. The data lines for this optionhave the form:

<node number>, <dof>, <load magnitude>

In this simulation a load of -10 kN is applied in the 2-direction to node 102. The option blockis:

*CLOAD

102, 2, -10.E3

Output requests

Finite element analyses can create very large amounts of output. ABAQUS allows you tocontrol and manage this output so that only data required to interpret the results of yoursimulation are produced. Four types of output are available:

� Results stored in a neutral binary file used by ABAQUS/Viewer for postprocessing: outputdatabase (.odb) file.

� Printed tables of results.

� Restart data, used to continue the analysis.

� Results stored in binary files for subsequent postprocessing with third-party software.

You will use the first two of these in the overhead hoist simulation.

By default, an output database file, which includes a preselected set of the most commonlyused output variables for a given type of analysis, is created. A list of preselected variables fordefault output database output is given in the ABAQUS/Standard User's Manual. You do notneed to add any output requests to accept these defaults. For this example the default outputdatabase output includes the deformed configuration and the applied nodal loads.

Selected results also can be written in tabular form to the ABAQUS data file. By default, noprintout is written to the ABAQUS data file. The *NODE PRINT option controls the printingof nodal results (for example, displacements and reaction forces), while the *EL PRINT optioncontrols the printing of element results. A comprehensive list of the output variables availableis given in the ABAQUS/Standard User's Manual.

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The data lines for either of these options list the output to appear in the columns of the table.Each data line creates a separate table of data that can have a maximum of nine columns.

For this analysis we are interested in the displacements of the nodes (output variable U), thereaction forces at the constrained nodes (output variable RF), and the stress in the members(output variable S). Use the following in your input file:

*NODE PRINT

U,

RF,

*EL PRINT

S,

to request that ABAQUS generate three tables of output data in the data file.

Since you have now finished the definition of all the data required for the step, use the *ENDSTEP option to mark the end of the step:

*END STEP

The input file is now complete. Compare the input file you have generated to the completeinput file given in Figure 2-2. Save the data as frame.inp, and exit the editor.

2.4 Checking the input dataHaving generated the input file for this simulation, you are ready to run the analysis. Unfortunately, itis possible to have errors in the input file because of typing errors or incorrect or missing data. Youshould perform a datacheck analysis first before running the simulation. To run a datacheck analysis,make sure that you are in the directory where the input file frame.inp is located, and type thefollowing command:

abaqus job=frame datacheck interactive

If this command results in an error message, the ABAQUS installation on your computer has beencustomized. You should contact your systems administrator to find out the appropriate command torun ABAQUS. The job=frame parameter specifies that the jobname for this analysis is frame. Allthe files associated with this analysis will have this jobname as their identifier, which allows them tobe recognized easily.

The analysis will run interactively, and messages similar to those shown below will appear on yourscreen:

ABAQUS JOB frame

BEGIN USER INPUT PROCESSING

Fri May 14 14:34:51 EDT 1999

Run /usr/abaqus/bin/pre.x

ABAQUS/STANDARD is running on a Category A machine.

ABAQUS License Server checked out 4 Network Tokens

Fri May 14 14:34:52 EDT 1999

The data lines for either of these options list the output to appear in the columns of the table.Each data line creates a separate table of data that can have a maximum of nine columns.

For this analysis we are interested in the displacements of the nodes (output variable U), thereaction forces at the constrained nodes (output variable RF), and the stress in the members(output variable S). Use the following in your input file:

*NODE PRINT

U,

RF,

*EL PRINT

S,

to request that ABAQUS generate three tables of output data in the data file.

Since you have now finished the definition of all the data required for the step, use the *ENDSTEP option to mark the end of the step:

*END STEP

The input file is now complete. Compare the input file you have generated to the completeinput file given in Figure 2-2. Save the data as frame.inp, and exit the editor.

2.4 Checking the input dataHaving generated the input file for this simulation, you are ready to run the analysis. Unfortunately, itis possible to have errors in the input file because of typing errors or incorrect or missing data. Youshould perform a datacheck analysis first before running the simulation. To run a datacheck analysis,make sure that you are in the directory where the input file frame.inp is located, and type thefollowing command:

abaqus job=frame datacheck interactive

If this command results in an error message, the ABAQUS installation on your computer has beencustomized. You should contact your systems administrator to find out the appropriate command torun ABAQUS. The job=frame parameter specifies that the jobname for this analysis is frame. Allthe files associated with this analysis will have this jobname as their identifier, which allows them tobe recognized easily.

The analysis will run interactively, and messages similar to those shown below will appear on yourscreen:

ABAQUS JOB frame

BEGIN USER INPUT PROCESSING

Fri May 14 14:34:51 EDT 1999

Run /usr/abaqus/bin/pre.x



Fri May 14 14:34:52 EDT 1999

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END OF USER INPUT PROCESSING

BEGIN IMPLICIT DATACHECK

Fri May 14 14:34:52 EDT 1999

Run /usr/abaqus/bin/standard.x



Fri May 14 14:35:02 EDT 1999

END OF DATACHECK

ABAQUS JOB frame COMPLETED

When the datacheck analysis is complete, you will find that a number of additional files have beencreated by ABAQUS. If any errors are encountered during the datacheck analysis, messages will bewritten to the data file, frame.dat. This data file is a text file that can be viewed in an editor orprinted. Part of this data file is shown in Figure 2-8 through Figure 2-12.

Try viewing the data file in your editor. The file contains lines up to 256 characters long, so make surethat the window is wide enough to accommodate that many characters. At the top of the file is a headerpage that contains information about the version of ABAQUS used to run the analysis. The headerpage also contains the phone number, address, and contact information of your local ABAQUS officeor representative who can offer technical support and advice.

Following this header page is an echo of the input file (Figure 2-8). The input data echo is generated byadding the option *PREPRINT, ECHO=YES to the input file. By default, the parameter ECHO is set toNO.

Figure 2-8 Input file echo in the data file.

END OF USER INPUT PROCESSING

BEGIN IMPLICIT DATACHECK

Fri May 14 14:34:52 EDT 1999




Fri May 14 14:35:02 EDT 1999

END OF DATACHECK


When the datacheck analysis is complete, you will find that a number of additional files have beencreated by ABAQUS. If any errors are encountered during the datacheck analysis, messages will bewritten to the data file, frame.dat. This data file is a text file that can be viewed in an editor orprinted. Part of this data file is shown in Figure 2-8 through Figure 2-12.

Try viewing the data file in your editor. The file contains lines up to 256 characters long, so make surethat the window is wide enough to accommodate that many characters. At the top of the file is a headerpage that contains information about the version of ABAQUS used to run the analysis. The headerpage also contains the phone number, address, and contact information of your local ABAQUS officeor representative who can offer technical support and advice.

Following this header page is an echo of the input file (Figure 2-8). The input data echo is generated byadding the option *PREPRINT, ECHO=YES to the input file. By default, the parameter ECHO is set toNO.

Figure 2-8 Input file echo in the data file.

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Following the input data echo is a list of the options processed by ABAQUS (Figure 2-9). This is thefirst point at which error and warning messages appear. All error messages are prefixed with ***ERROR, while warnings begin with ***WARNING. Since these messages always begin the same way,searching the data file for warning and error messages is straightforward. When the error is a syntaxproblem (i.e., when ABAQUS cannot understand the input), the error message is followed by the linefrom the input file that is causing the error.

Figure 2-9 List of options processed by ABAQUS.

Following the input data echo is a list of the options processed by ABAQUS (Figure 2-9). This is thefirst point at which error and warning messages appear. All error messages are prefixed with ***ERROR, while warnings begin with ***WARNING. Since these messages always begin the same way,searching the data file for warning and error messages is straightforward. When the error is a syntaxproblem (i.e., when ABAQUS cannot understand the input), the error message is followed by the linefrom the input file that is causing the error.

Figure 2-9 List of options processed by ABAQUS.

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The rest of the data file is a series of tables containing all of the model data ( Figure 2-10) and thehistory data (Figure 2-11) that should be checked for any obvious errors or omissions. These tables aregenerated by including the option *PREPRINT, MODEL=YES, HISTORY=YES in the input file.However, these tables may take up a large amount of disk space for large models. By default, theparameters MODEL and HISTORY are set to NO.

Figure 2-10 Model data in data file.

The rest of the data file is a series of tables containing all of the model data ( Figure 2-10) and thehistory data (Figure 2-11) that should be checked for any obvious errors or omissions. These tables aregenerated by including the option *PREPRINT, MODEL=YES, HISTORY=YES in the input file.However, these tables may take up a large amount of disk space for large models. By default, theparameters MODEL and HISTORY are set to NO.

Figure 2-10 Model data in data file.

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Figure 2-11 History data in data file.Figure 2-11 History data in data file.

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If there are any error messages, the number of such messages produced during the datacheck analysisis listed at the end of the data file. If there are only warning messages, the number of these messages islisted at the bottom of the data file after any of the requested output.

If error messages are generated during the datacheck analysis, it will not be possible to perform theanalysis until the causes of the error messages are corrected. The causes of warning messages shouldalways be investigated. Sometimes, warning messages are indications of mistakes in the input data;other times they are harmless and can be ignored safely.

Finally, there is a summary of the size of the numerical model and an estimate of the file sizes requiredfor the simulation (Figure 2-12). When analyzing large models, use this output to ensure that you haveenough disk space available to perform the analysis.

If there are any error messages, the number of such messages produced during the datacheck analysisis listed at the end of the data file. If there are only warning messages, the number of these messages islisted at the bottom of the data file after any of the requested output.

If error messages are generated during the datacheck analysis, it will not be possible to perform theanalysis until the causes of the error messages are corrected. The causes of warning messages shouldalways be investigated. Sometimes, warning messages are indications of mistakes in the input data;other times they are harmless and can be ignored safely.

Finally, there is a summary of the size of the numerical model and an estimate of the file sizes requiredfor the simulation (Figure 2-12). When analyzing large models, use this output to ensure that you haveenough disk space available to perform the analysis.

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Figure 2-12 Summary of problem and file sizes.

2.5 Running the analysisMake any necessary corrections to your input file. When the datacheck analysis completes with noerror messages, run the analysis itself by using the command

abaqus job=frame continue interactive

Messages like those below will appear on the screen:


ABAQUS JOB frame

BEGIN IMPLICIT ANALYSIS

Fri May 14 15:01:20 EDT 1999




Fri May 14 15:01:21 EDT 1999

END OF ANALYSIS


You should always perform a datacheck analysis before running a simulation to ensure that the inputdata are correct and to check that there is enough disk space and memory available to complete the

Figure 2-12 Summary of problem and file sizes.

2.5 Running the analysisMake any necessary corrections to your input file. When the datacheck analysis completes with noerror messages, run the analysis itself by using the command

abaqus job=frame continue interactive

Messages like those below will appear on the screen:


ABAQUS JOB frame

BEGIN IMPLICIT ANALYSIS

Fri May 14 15:01:20 EDT 1999




Fri May 14 15:01:21 EDT 1999

END OF ANALYSIS


You should always perform a datacheck analysis before running a simulation to ensure that the inputdata are correct and to check that there is enough disk space and memory available to complete the

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analysis. However, it is possible to combine the datacheck and analysis phases of the simulation byusing the command

abaqus job=frame interactive

If a simulation is expected to take a substantial amount of time, it is convenient to run it in thebackground by omitting the interactive parameter:

abaqus job=frame

(The above commands apply for the standard ABAQUS installation on a workstation. However,ABAQUS jobs may be run in batch queues on some computers. If you have any questions, ask yoursystems administrator how to run ABAQUS on your system.)

2.6 ResultsAfter the analysis is completed, the data file, frame.dat, will contain the tables of results requestedwith the *NODE PRINT and *EL PRINT options. The tables of results follow the output from thedatacheck analysis. The results from the overhead hoist simulation are shown in Figure 2-13 andFigure 2-14.

Figure 2-13 Element output.

analysis. However, it is possible to combine the datacheck and analysis phases of the simulation byusing the command

abaqus job=frame interactive

If a simulation is expected to take a substantial amount of time, it is convenient to run it in thebackground by omitting the interactive parameter:

abaqus job=frame

(The above commands apply for the standard ABAQUS installation on a workstation. However,ABAQUS jobs may be run in batch queues on some computers. If you have any questions, ask yoursystems administrator how to run ABAQUS on your system.)

2.6 ResultsAfter the analysis is completed, the data file, frame.dat, will contain the tables of results requestedwith the *NODE PRINT and *EL PRINT options. The tables of results follow the output from thedatacheck analysis. The results from the overhead hoist simulation are shown in Figure 2-13 andFigure 2-14.

Figure 2-13 Element output.

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Figure 2-14 Node output.

Are the nodal displacements and peak stresses in the individual members reasonable for this hoist andthese applied loads?

It is always a good idea to check that the results of the simulation satisfy basic physical principles. Inthis case check that the external forces applied to the hoist sum to zero in both the vertical andhorizontal directions.

What nodes have vertical forces applied to them? What nodes have horizontal forces? Do the resultsfrom your simulation match those shown here?

ABAQUS also creates several other files during a simulation. One such file--the output database file,frame.odb--can be used to visualize the results graphically using ABAQUS/Viewer.

2.7 Postprocessing with ABAQUS/ViewerGraphical postprocessing is important because of the great volume of data created during a simulation.For any realistic model it is impractical for you to try to interpret results in the tabular form of the datafile. ABAQUS/Viewer allows you to view the results graphically using a variety of methods, includingdeformed shape plots, contour plots, vector plots, animations, and X-Y plots. All of these methods arediscussed in this guide. For more information on any of the postprocessing features discussed in thisguide, consult the ABAQUS/Viewer User's Manual. For this example you will use ABAQUS/Viewer

Figure 2-14 Node output.

Are the nodal displacements and peak stresses in the individual members reasonable for this hoist andthese applied loads?

It is always a good idea to check that the results of the simulation satisfy basic physical principles. Inthis case check that the external forces applied to the hoist sum to zero in both the vertical andhorizontal directions.

What nodes have vertical forces applied to them? What nodes have horizontal forces? Do the resultsfrom your simulation match those shown here?

ABAQUS also creates several other files during a simulation. One such file--the output database file,frame.odb--can be used to visualize the results graphically using ABAQUS/Viewer.

2.7 Postprocessing with ABAQUS/ViewerGraphical postprocessing is important because of the great volume of data created during a simulation.For any realistic model it is impractical for you to try to interpret results in the tabular form of the datafile. ABAQUS/Viewer allows you to view the results graphically using a variety of methods, includingdeformed shape plots, contour plots, vector plots, animations, and X-Y plots. All of these methods arediscussed in this guide. For more information on any of the postprocessing features discussed in thisguide, consult the ABAQUS/Viewer User's Manual. For this example you will use ABAQUS/Viewer

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to do some basic model checks and to display the deformed shape of the frame.

Start ABAQUS/Viewer by typing

abaqus viewer

at the operating system prompt. The ABAQUS/Viewer window appears.

Reading the output database file

To begin this exercise, open the output database file that ABAQUS/Standard generated duringthe analysis of the problem.

To open the output database file:

1. From the main menu bar, select File->Open; or use the tool in the toolbar.

The Open Database dialog box appears.

2. From the list of available output database files, select frame.odb.

3. Click OK.Tip: You can also open the output database frame.odb by typing

abaqus viewer odb=frame

at the operating system prompt.

ABAQUS/Viewer displays a fast plot of the model. A fast plot is a basic representation ofthe undeformed model shape and is an indication that you have opened the desired file.

Important: The fast plot does not display results and cannot be customized, forexample, to display element and node numbers. You must display the undeformedmodel shape to customize the appearance of the model.

The title block at the bottom of the viewport indicates the following:

� The description of the model (from the first line of the *HEADING option in the inputfile).

� The name of the output database (from the name of the analysis job).

� The product name (ABAQUS/Standard or ABAQUS/Explicit) and version used togenerate the output database.

� The date the output database was last modified.

The state block at the bottom of the viewport indicates the following:

� Which step is being displayed.

� The increment within the step.

� The step time.

to do some basic model checks and to display the deformed shape of the frame.


abaqus viewer

at the operating system prompt. The ABAQUS/Viewer window appears.

Reading the output database file

To begin this exercise, open the output database file that ABAQUS/Standard generated duringthe analysis of the problem.

To open the output database file:

1. From the main menu bar, select File->Open; or use the tool in the toolbar.

The Open Database dialog box appears.

2. From the list of available output database files, select frame.odb.

3. Click OK.Tip: You can also open the output database frame.odb by typing

abaqus viewer odb=frame


ABAQUS/Viewer displays a fast plot of the model. A fast plot is a basic representation ofthe undeformed model shape and is an indication that you have opened the desired file.

Important: The fast plot does not display results and cannot be customized, forexample, to display element and node numbers. You must display the undeformedmodel shape to customize the appearance of the model.

The title block at the bottom of the viewport indicates the following:

� The description of the model (from the first line of the *HEADING option in the inputfile).

� The name of the output database (from the name of the analysis job).

� The product name (ABAQUS/Standard or ABAQUS/Explicit) and version used togenerate the output database.

� The date the output database was last modified.

The state block at the bottom of the viewport indicates the following:

� Which step is being displayed.

� The increment within the step.

� The step time.

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The view orientation triad indicates the orientation of the model in the global coordinatesystem.

Displaying and customizing an undeformed shape plot

You will now display the undeformed model shape and use the plot options to enable thedisplay of node and element numbering in the plot.

From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox. ABAQUS/Viewer displays the undeformed model shape, as shown in Figure 2-15.

Figure 2-15 Undeformed model shape.

To display node numbers:

1. From the main menu bar, select Options->Undeformed.

The Undeformed Plot Options dialog box appears.

2. Click the Labels tab.

3. Toggle on Show node labels.

4. Click Apply.

ABAQUS/Viewer applies the change and keeps the dialog box open.

The customized undeformed plot is shown in Figure 2-16.

Figure 2-16 Node number plot.

The view orientation triad indicates the orientation of the model in the global coordinatesystem.

Displaying and customizing an undeformed shape plot

You will now display the undeformed model shape and use the plot options to enable thedisplay of node and element numbering in the plot.

From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox. ABAQUS/Viewer displays the undeformed model shape, as shown in Figure 2-15.

Figure 2-15 Undeformed model shape.

To display node numbers:



2. Click the Labels tab.

3. Toggle on Show node labels.

4. Click Apply.

ABAQUS/Viewer applies the change and keeps the dialog box open.

The customized undeformed plot is shown in Figure 2-16.

Figure 2-16 Node number plot.

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To display element numbers:

1. In the Undeformed Plot Options dialog box, toggle on Show element labels.

2. Click OK.

ABAQUS/Viewer applies the change and closes the dialog box.

The resulting plot is shown in Figure 2-17.

Figure 2-17 Node and element number plot.

To disable the display of node and element numbers in the undeformed shape plot, repeat theabove procedure and, under Labels, toggle off Show node labels and Show element labels.

Displaying and customizing a deformed shape plot

You will now display the deformed model shape and use the plot options to change thedeformation scale factor and overlay the undeformed model shape on the deformed modelshape.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.ABAQUS/Viewer displays the deformed model shape, as shown in Figure 2-18.

Figure 2-18 Deformed model shape.

To display element numbers:

1. In the Undeformed Plot Options dialog box, toggle on Show element labels.

2. Click OK.

ABAQUS/Viewer applies the change and closes the dialog box.

The resulting plot is shown in Figure 2-17.

Figure 2-17 Node and element number plot.

To disable the display of node and element numbers in the undeformed shape plot, repeat theabove procedure and, under Labels, toggle off Show node labels and Show element labels.

Displaying and customizing a deformed shape plot

You will now display the deformed model shape and use the plot options to change thedeformation scale factor and overlay the undeformed model shape on the deformed modelshape.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.ABAQUS/Viewer displays the deformed model shape, as shown in Figure 2-18.

Figure 2-18 Deformed model shape.

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For small-displacement analyses the displacements are scaled automatically to ensure that theyare clearly visible. The scale factor is displayed in the state block. In this case thedisplacements have been scaled by a factor of 42.83.

To change the deformation scale factor:

1. From the main menu bar, select Options->Deformed.

2. From the Deformed Plot Options dialog box, click the Basic tab if it is not alreadyselected.

3. From the Deformation Scale Factor area, toggle on Uniform and enter 10.0 in theValue field.

4. Click Apply.

The state block displays the new scale factor.

To return to automatic scaling of the displacements, repeat the above procedure and, in theDeformation Scale Factor field, toggle on Auto-compute.

To overlay the undeformed model shape on the deformed model shape:

1. In the Basic options in the Deformed Plot Options dialog box, toggle on Superimposeundeformed plot.

2. Click OK.

By default, ABAQUS/Viewer plots the undeformed model shape in green and the deformedmodel shape in white. The plot is shown in Figure 2-19.

Figure 2-19 Undeformed and deformed model shapes.

For small-displacement analyses the displacements are scaled automatically to ensure that theyare clearly visible. The scale factor is displayed in the state block. In this case thedisplacements have been scaled by a factor of 42.83.

To change the deformation scale factor:


2. From the Deformed Plot Options dialog box, click the Basic tab if it is not alreadyselected.

3. From the Deformation Scale Factor area, toggle on Uniform and enter 10.0 in theValue field.

4. Click Apply.

The state block displays the new scale factor.

To return to automatic scaling of the displacements, repeat the above procedure and, in theDeformation Scale Factor field, toggle on Auto-compute.

To overlay the undeformed model shape on the deformed model shape:

1. In the Basic options in the Deformed Plot Options dialog box, toggle on Superimposeundeformed plot.

2. Click OK.

By default, ABAQUS/Viewer plots the undeformed model shape in green and the deformedmodel shape in white. The plot is shown in Figure 2-19.

Figure 2-19 Undeformed and deformed model shapes.

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Checking history data with ABAQUS/Viewer

By default, both the model data and history data are written to the output database file duringthe datacheck phase. Thus, you can use ABAQUS/Viewer to check that the input data arecorrect before running the simulation. You have already learned how to draw plots of themodel and to display the node and element numbers. These are useful tools for checking thatABAQUS is using the correct mesh.

The boundary conditions applied to the overhead hoist model can also be displayed andchecked using ABAQUS/Viewer.

To display boundary conditions on the undeformed model:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox.

2. From the main menu bar, select View->ODB Display Options.

3. In the ODB Display Options dialog box, click the BC tab.

4. Toggle on Show boundary conditions.

5. Click OK.

ABAQUS/Viewer displays symbols to indicate the applied boundary conditions, as shownin Figure 2-20.

Figure 2-20 Applied boundary conditions on the overhead hoist.

Checking history data with ABAQUS/Viewer

By default, both the model data and history data are written to the output database file duringthe datacheck phase. Thus, you can use ABAQUS/Viewer to check that the input data arecorrect before running the simulation. You have already learned how to draw plots of themodel and to display the node and element numbers. These are useful tools for checking thatABAQUS is using the correct mesh.

The boundary conditions applied to the overhead hoist model can also be displayed andchecked using ABAQUS/Viewer.

To display boundary conditions on the undeformed model:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox.


3. In the ODB Display Options dialog box, click the BC tab.

4. Toggle on Show boundary conditions.

5. Click OK.

ABAQUS/Viewer displays symbols to indicate the applied boundary conditions, as shownin Figure 2-20.

Figure 2-20 Applied boundary conditions on the overhead hoist.

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Exiting ABAQUS/Viewer

From the main menu bar, select File->Exit to exit ABAQUS/Viewer.

2.8 Summary� The ABAQUS input file contains a complete description of the analysis model. It is the means of

communication between the preprocessor (ABAQUS/CAE) and the solver (ABAQUS/Standard).

� The input file contains two sections: the model data defining the structure being analyzed and thehistory data defining what happens to the structure.

� Each section of the input file comprises a number of option blocks, each consisting of a keywordline, which may be followed by data lines.

� You can perform a datacheck analysis once you have created the input file. Error and warningmessages are printed to the data file. After a successful datacheck analysis, estimates of thecomputer resources required for the simulation are printed to the data file.

� Use ABAQUS/Viewer to verify the model geometry and boundary conditions graphically, usingthe output database file generated during the datacheck phase.

� It is often easiest to check for mistakes in material properties in the data (.dat) file; geometry,loads, and boundary conditions are more easily checked with a graphical postprocessor such asABAQUS/Viewer.

� To run the analysis after a successful datacheck analysis without repeating the datacheck phase,use the following command: abaqus job=<jobname> continue.

� Tables of results are printed in the data file. The output database file contains results for graphicalpostprocessing with ABAQUS/Viewer.

� Always check that the results satisfy basic engineering principles, such as equilibrium.

� ABAQUS/Viewer allows you to visualize analysis results graphically in a variety of ways.

Exiting ABAQUS/Viewer

From the main menu bar, select File->Exit to exit ABAQUS/Viewer.

2.8 Summary� The ABAQUS input file contains a complete description of the analysis model. It is the means of

communication between the preprocessor (ABAQUS/CAE) and the solver (ABAQUS/Standard).

� The input file contains two sections: the model data defining the structure being analyzed and thehistory data defining what happens to the structure.

� Each section of the input file comprises a number of option blocks, each consisting of a keywordline, which may be followed by data lines.

� You can perform a datacheck analysis once you have created the input file. Error and warningmessages are printed to the data file. After a successful datacheck analysis, estimates of thecomputer resources required for the simulation are printed to the data file.

� Use ABAQUS/Viewer to verify the model geometry and boundary conditions graphically, usingthe output database file generated during the datacheck phase.

� It is often easiest to check for mistakes in material properties in the data (.dat) file; geometry,loads, and boundary conditions are more easily checked with a graphical postprocessor such asABAQUS/Viewer.

� To run the analysis after a successful datacheck analysis without repeating the datacheck phase,use the following command: abaqus job=<jobname> continue.

� Tables of results are printed in the data file. The output database file contains results for graphicalpostprocessing with ABAQUS/Viewer.

� Always check that the results satisfy basic engineering principles, such as equilibrium.

� ABAQUS/Viewer allows you to visualize analysis results graphically in a variety of ways.

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3. ABAQUS Elements - A Brief IntroductionThere is a wide range of elements that are available in ABAQUS. This extensive element libraryprovides you with a powerful set of tools for solving many different problems. This chapter introducesyou to the five aspects of an element that influence how it behaves.

3.1 Characterizing elementsEach element is characterized by the following:

� Family

� Degrees of freedom (directly related to the element family)

� Number of nodes

� Formulation

� Integration

Each element in ABAQUS has a unique name, such as T2D2, S4R, or C3D8I. The element name, asyou saw in the overhead hoist example in Chapter 2, "ABAQUS Basics," is used as the value of theTYPE parameter on the *ELEMENT option in the input file. The element name identifies each of thefive aspects of an element. The naming convention is explained in this chapter.

3.1.1 FamilyFigure 3-1 shows the element families most commonly used in a stress analysis. One of the majordistinctions between different element families is the geometry type that each family assumes.

Figure 3-1 Commonly used element families.

The element families that you will use in this guide--continuum, shell, beam, truss, and rigidelements--are discussed in detail in other chapters. The other element families are not covered in thisguide; if you are interested in using them in your models, read about them in the ABAQUS/StandardUser's Manual.

3. ABAQUS Elements - A Brief IntroductionThere is a wide range of elements that are available in ABAQUS. This extensive element libraryprovides you with a powerful set of tools for solving many different problems. This chapter introducesyou to the five aspects of an element that influence how it behaves.

3.1 Characterizing elementsEach element is characterized by the following:

� Family

� Degrees of freedom (directly related to the element family)

� Number of nodes

� Formulation

� Integration

Each element in ABAQUS has a unique name, such as T2D2, S4R, or C3D8I. The element name, asyou saw in the overhead hoist example in Chapter 2, "ABAQUS Basics," is used as the value of theTYPE parameter on the *ELEMENT option in the input file. The element name identifies each of thefive aspects of an element. The naming convention is explained in this chapter.

3.1.1 FamilyFigure 3-1 shows the element families most commonly used in a stress analysis. One of the majordistinctions between different element families is the geometry type that each family assumes.

Figure 3-1 Commonly used element families.

The element families that you will use in this guide--continuum, shell, beam, truss, and rigidelements--are discussed in detail in other chapters. The other element families are not covered in thisguide; if you are interested in using them in your models, read about them in the ABAQUS/StandardUser's Manual.

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The first letter or letters of an element's name indicate to which family the element belongs. Forexample, S4R is a shell element, C3D8I is a continuum element, and CINPE4 is an infinite element.

3.1.2 Degrees of freedomThe degrees of freedom (dof) are the fundamental variables calculated during the analysis. For astress/displacement simulation the degrees of freedom are the translations as well as, for shell andbeam elements, the rotations at each node. For a heat transfer simulation the degrees of freedom are thetemperatures at each node; a heat transfer analysis, therefore, requires the use of different elementsthan a stress analysis, since the degrees of freedom are not the same.

The following numbering convention is used for the degrees of freedom in ABAQUS:1 Translation in direction 12 Translation in direction 23 Translation in direction 34 Rotation about the 1-axis5 Rotation about the 2-axis6 Rotation about the 3-axis7 Warping in open-section beam elements8 Acoustic pressure or pore pressure9 Electric potential11 Temperature (or normalized concentration in mass diffusion

analysis) for continuum elements or temperature at the firstpoint through the thickness of beams and shells

12+ Temperature at other points through the thickness of beamsand shells

Directions 1, 2, and 3 correspond to the global 1-, 2-, and 3-directions, respectively, unless a localcoordinate system has been defined at the nodes.

Axisymmetric elements are the exception, with the displacement and rotation degrees of freedomreferred to as follows:1 Translation in the r-direction2 Translation in the z-direction6 Rotation in the r-z plane

Directions r and z correspond to the global 1- and 2-directions, respectively, unless a local coordinatesystem has been defined at the nodes.

See Chapter 5, "Using Shell Elements," for a discussion of defining a local coordinate system at thenodes.

In this guide our attention is restricted to structural applications. As such, only elements withtranslational and rotational degrees of freedom are discussed. For information on other types ofelements (for example, heat transfer elements) consult the ABAQUS/Standard User's Manual.

3.1.3 Number of nodes--order of interpolationDisplacements, rotations, temperatures, and the other degrees of freedom mentioned in the previoussection are calculated only at the nodes of the element. At any other point in the element, thedisplacements are obtained by interpolating from the nodal displacements. Usually the interpolation

The first letter or letters of an element's name indicate to which family the element belongs. Forexample, S4R is a shell element, C3D8I is a continuum element, and CINPE4 is an infinite element.

3.1.2 Degrees of freedomThe degrees of freedom (dof) are the fundamental variables calculated during the analysis. For astress/displacement simulation the degrees of freedom are the translations as well as, for shell andbeam elements, the rotations at each node. For a heat transfer simulation the degrees of freedom are thetemperatures at each node; a heat transfer analysis, therefore, requires the use of different elementsthan a stress analysis, since the degrees of freedom are not the same.

The following numbering convention is used for the degrees of freedom in ABAQUS:1 Translation in direction 12 Translation in direction 23 Translation in direction 34 Rotation about the 1-axis5 Rotation about the 2-axis6 Rotation about the 3-axis7 Warping in open-section beam elements8 Acoustic pressure or pore pressure9 Electric potential11 Temperature (or normalized concentration in mass diffusion

analysis) for continuum elements or temperature at the firstpoint through the thickness of beams and shells

12+ Temperature at other points through the thickness of beamsand shells

Directions 1, 2, and 3 correspond to the global 1-, 2-, and 3-directions, respectively, unless a localcoordinate system has been defined at the nodes.

Axisymmetric elements are the exception, with the displacement and rotation degrees of freedomreferred to as follows:1 Translation in the r-direction2 Translation in the z-direction6 Rotation in the r-z plane

Directions r and z correspond to the global 1- and 2-directions, respectively, unless a local coordinatesystem has been defined at the nodes.

See Chapter 5, "Using Shell Elements," for a discussion of defining a local coordinate system at thenodes.

In this guide our attention is restricted to structural applications. As such, only elements withtranslational and rotational degrees of freedom are discussed. For information on other types ofelements (for example, heat transfer elements) consult the ABAQUS/Standard User's Manual.

3.1.3 Number of nodes--order of interpolationDisplacements, rotations, temperatures, and the other degrees of freedom mentioned in the previoussection are calculated only at the nodes of the element. At any other point in the element, thedisplacements are obtained by interpolating from the nodal displacements. Usually the interpolation


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order is determined by the number of nodes used in the element. Elements that have nodes only at theircorners, such as the 8-node brick shown in Figure 3-2(a), use linear interpolation in each direction andare often called linear elements or first-order elements. Elements with midside nodes, such as the20-node brick shown in Figure 3-2(b), use quadratic interpolation and are often called quadraticelements or second-order elements.

Figure 3-2 Linear and quadratic brick elements.

Typically, the number of nodes in an element is clearly identified in its name. The 8-node brickelement, as you have seen, is called C3D8; and the 8-node general shell element is called S8R. Thebeam element family uses a slightly different convention: the order of interpolation is identified in thename. Thus, a first-order, three-dimensional beam element is called B31, whereas a second-order,three-dimensional beam element is called B32.

3.1.4 FormulationAn element's formulation refers to the mathematical theory used to define the element's behavior. Allof the stress/displacement elements in ABAQUS are based on the Lagrangian or material descriptionof behavior: the material associated with an element remains associated with the element throughoutthe analysis, and material cannot move among elements. In the alternative, Eulerian or spatialdescription, elements are fixed in space as the material flows through them. Eulerian methods are usedcommonly in fluid mechanics simulations. ABAQUS uses Eulerian elements to model convective heattransfer, a topic not discussed in this guide.

To accommodate different types of behavior, some element families in ABAQUS include elementswith several different formulations. For example, the shell element family has three classes: one with ageneral-purpose formulation, another with a thin-only formulation, and yet another with a thick-onlyformulation. (These shell element formulations are explained in Chapter 5, "Using Shell Elements.")

Some element families have a standard formulation as well as some alternative formulations. Elementswith alternative formulations are identified by an additional character at the end of the element name.For example, the continuum, beam, and truss element families include members with a hybridformulation in which the pressure (continuum elements) or axial force (beam and truss elements) istreated as an additional unknown; these elements are identified by the letter "H" at the end of the name(C3D8H or B31H).

Several of the most commonly used element formulations are discussed later in this guide.

order is determined by the number of nodes used in the element. Elements that have nodes only at theircorners, such as the 8-node brick shown in Figure 3-2(a), use linear interpolation in each direction andare often called linear elements or first-order elements. Elements with midside nodes, such as the20-node brick shown in Figure 3-2(b), use quadratic interpolation and are often called quadraticelements or second-order elements.

Figure 3-2 Linear and quadratic brick elements.

Typically, the number of nodes in an element is clearly identified in its name. The 8-node brickelement, as you have seen, is called C3D8; and the 8-node general shell element is called S8R. Thebeam element family uses a slightly different convention: the order of interpolation is identified in thename. Thus, a first-order, three-dimensional beam element is called B31, whereas a second-order,three-dimensional beam element is called B32.

3.1.4 FormulationAn element's formulation refers to the mathematical theory used to define the element's behavior. Allof the stress/displacement elements in ABAQUS are based on the Lagrangian or material descriptionof behavior: the material associated with an element remains associated with the element throughoutthe analysis, and material cannot move among elements. In the alternative, Eulerian or spatialdescription, elements are fixed in space as the material flows through them. Eulerian methods are usedcommonly in fluid mechanics simulations. ABAQUS uses Eulerian elements to model convective heattransfer, a topic not discussed in this guide.

To accommodate different types of behavior, some element families in ABAQUS include elementswith several different formulations. For example, the shell element family has three classes: one with ageneral-purpose formulation, another with a thin-only formulation, and yet another with a thick-onlyformulation. (These shell element formulations are explained in Chapter 5, "Using Shell Elements.")

Some element families have a standard formulation as well as some alternative formulations. Elementswith alternative formulations are identified by an additional character at the end of the element name.For example, the continuum, beam, and truss element families include members with a hybridformulation in which the pressure (continuum elements) or axial force (beam and truss elements) istreated as an additional unknown; these elements are identified by the letter "H" at the end of the name(C3D8H or B31H).

Several of the most commonly used element formulations are discussed later in this guide.


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3.1.5 IntegrationABAQUS uses numerical techniques to integrate various quantities over the volume of each element.Using Gaussian quadrature for most elements, ABAQUS evaluates the material response at eachintegration point in each element. When using continuum elements, you must choose between full orreduced integration, a choice that can have a significant effect on the accuracy of the element for agiven problem, as discussed in detail in ``Element formulation and integration,'' Section 4.1.

ABAQUS uses the letter "R" at the end of the element name to distinguish reduced-integrationelements. For example, CAX4 is the fully integrated, linear, axisymmetric solid element; and CAX4Ris the reduced-integration version of the same element.

3.2 Continuum elementsAmong the different element families, continuum or solid elements can be used to model the widestvariety of components. Conceptually, continuum elements simply model small blocks of material in acomponent. Since they may be connected to other elements on any of their faces, continuum elements,like bricks in a building or tiles in a mosaic, can be used to build models of nearly any shape, subjectedto nearly any loading.

Continuum stress/displacement elements in ABAQUS have names that begin with the letter "C." Thenext two letters indicate the dimensionality and usually, but not always, the active degrees of freedomin the element. The letters "3D" indicate a three-dimensional element; " AX," an axisymmetric element;"PE," a plane strain element; and "PS," a plane stress element.

The use of continuum elements is discussed further in Chapter 4, "Using Continuum Elements."

3.2.1 Three-dimensional continuum element libraryThree-dimensional continuum elements can be hexahedra (bricks), wedges, or tetrahedra. The fullinventory of three-dimensional continuum elements and the nodal connectivity for each type can befound in the ABAQUS/Standard User's Manual.

3.2.2 Two-dimensional continuum element libraryABAQUS has several classes of two-dimensional continuum elements that differ from each other intheir out-of-plane behavior. Two-dimensional elements can be quadrilateral or triangular. Figure 3-3shows the three classes that are used most commonly.

Figure 3-3 Plane strain, plane stress, and axisymmetric elements without twist.

3.1.5 IntegrationABAQUS uses numerical techniques to integrate various quantities over the volume of each element.Using Gaussian quadrature for most elements, ABAQUS evaluates the material response at eachintegration point in each element. When using continuum elements, you must choose between full orreduced integration, a choice that can have a significant effect on the accuracy of the element for agiven problem, as discussed in detail in ``Element formulation and integration,'' Section 4.1.

ABAQUS uses the letter "R" at the end of the element name to distinguish reduced-integrationelements. For example, CAX4 is the fully integrated, linear, axisymmetric solid element; and CAX4Ris the reduced-integration version of the same element.

3.2 Continuum elementsAmong the different element families, continuum or solid elements can be used to model the widestvariety of components. Conceptually, continuum elements simply model small blocks of material in acomponent. Since they may be connected to other elements on any of their faces, continuum elements,like bricks in a building or tiles in a mosaic, can be used to build models of nearly any shape, subjectedto nearly any loading.

Continuum stress/displacement elements in ABAQUS have names that begin with the letter "C." Thenext two letters indicate the dimensionality and usually, but not always, the active degrees of freedomin the element. The letters "3D" indicate a three-dimensional element; " AX," an axisymmetric element;"PE," a plane strain element; and "PS," a plane stress element.

The use of continuum elements is discussed further in Chapter 4, "Using Continuum Elements."

3.2.1 Three-dimensional continuum element libraryThree-dimensional continuum elements can be hexahedra (bricks), wedges, or tetrahedra. The fullinventory of three-dimensional continuum elements and the nodal connectivity for each type can befound in the ABAQUS/Standard User's Manual.

3.2.2 Two-dimensional continuum element libraryABAQUS has several classes of two-dimensional continuum elements that differ from each other intheir out-of-plane behavior. Two-dimensional elements can be quadrilateral or triangular. Figure 3-3shows the three classes that are used most commonly.

Figure 3-3 Plane strain, plane stress, and axisymmetric elements without twist.


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Plane strain elements assume that the out-of-plane strain, "33 , is zero; they can be used to model thickstructures.

Plane stress elements assume that the out-of-plane stress, ¾33 , is zero; they are suitable for modelingthin structures.

Axisymmetric elements without twist, the "CAX" class of elements, model a 360° ring; they aresuitable for analyzing structures with axisymmetric geometry subjected to axisymmetric loading.

ABAQUS also provides generalized plane strain elements, axisymmetric elements with twist, andaxisymmetric elements with asymmetric deformation.

Generalized plane strain elements include the additional generalization that the out-of-plane strain mayvary linearly with position in the plane of the model. This formulation is particularly suited to thethermal-stress analysis of thick sections.

Axisymmetric elements with twist model an initially axisymmetric geometry that can twist about theaxis of symmetry. These elements are useful for modeling the torsion of cylindrical structures, such asaxisymmetric rubber bushings.

Axisymmetric elements with asymmetric deformation model an initially axisymmetric geometry thatcan deform asymmetrically. They are useful for simulating problems such as an axisymmetric rubbermount that is subjected to shear loads.

The latter three classes of two-dimensional continuum elements are not discussed in this guide.

Two-dimensional solid elements must be defined in the 1-2 plane so that the node order iscounterclockwise around the element perimeter, as shown in Figure 3-4.

Plane strain elements assume that the out-of-plane strain, "33 , is zero; they can be used to model thickstructures.

Plane stress elements assume that the out-of-plane stress, ¾33 , is zero; they are suitable for modelingthin structures.

Axisymmetric elements without twist, the "CAX" class of elements, model a 360° ring; they aresuitable for analyzing structures with axisymmetric geometry subjected to axisymmetric loading.

ABAQUS also provides generalized plane strain elements, axisymmetric elements with twist, andaxisymmetric elements with asymmetric deformation.

Generalized plane strain elements include the additional generalization that the out-of-plane strain mayvary linearly with position in the plane of the model. This formulation is particularly suited to thethermal-stress analysis of thick sections.

Axisymmetric elements with twist model an initially axisymmetric geometry that can twist about theaxis of symmetry. These elements are useful for modeling the torsion of cylindrical structures, such asaxisymmetric rubber bushings.

Axisymmetric elements with asymmetric deformation model an initially axisymmetric geometry thatcan deform asymmetrically. They are useful for simulating problems such as an axisymmetric rubbermount that is subjected to shear loads.

The latter three classes of two-dimensional continuum elements are not discussed in this guide.

Two-dimensional solid elements must be defined in the 1-2 plane so that the node order iscounterclockwise around the element perimeter, as shown in Figure 3-4.


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Figure 3-4 Correct nodal connectivity for two-dimensional elements.

When using a preprocessor to generate the mesh, ensure that the element normals all point in the samedirection as the positive, global 3-axis. Failure to provide the correct element connectivity will causeABAQUS to issue an error message stating that elements have negative area.

3.2.3 Degrees of freedomAll of the stress/displacement continuum elements have translational degrees of freedom at each node.Correspondingly, degrees of freedom 1, 2, and 3 are active in three-dimensional elements, while onlydegrees of freedom 1 and 2 are active in plane strain elements, plane stress elements, and axisymmetricelements without twist. Consult the ABAQUS/Standard User's Manual to find the active degrees offreedom in the other classes of two-dimensional solid elements.

3.2.4 Element propertiesThe *SOLID SECTION option defines the material and any additional geometric data associated witha set of continuum elements. For three-dimensional and axisymmetric elements no additionalgeometric information is required: the nodal coordinates completely define the element geometry. Forplane stress and plane strain elements the thickness of the elements must be specified on the data line.For example, if the elements are 0.2 m thick, the element property definition would be the following:

*SOLID SECTION, ELSET=<element set name>, MATERIAL=<material name>

0.2,

3.2.5 Formulation and integrationAlternative formulations available for the continuum family of elements include an incompatible modeformulation and a hybrid element formulation, both of which are discussed in detail later in this guide.The solid elements can use full or reduced integration. Both the formulation and type of integration canhave a significant effect on the accuracy of solid elements, as discussed in `Èlement formulation andintegration,'' Section 4.1.

3.2.6 Element output variablesBy default, element output variables such as stress and strain refer to the global Cartesian coordinatesystem. Thus, the ¾11-component of stress at the integration point shown in Figure 3-5(a) acts in theglobal 1-direction. Even if the element rotates during a large-displacement simulation, as shown inFigure 3-5(b), the default is still to use the global Cartesian system as the basis for defining the elementvariables. However, ABAQUS allows you to define a local coordinate system for element variables,using the *ORIENTATION option (see `Èxample: skew plate,'' Section 5.5). This local coordinatesystem rotates with the motion of the element in large-displacement simulations. A local coordinate

Figure 3-4 Correct nodal connectivity for two-dimensional elements.

When using a preprocessor to generate the mesh, ensure that the element normals all point in the samedirection as the positive, global 3-axis. Failure to provide the correct element connectivity will causeABAQUS to issue an error message stating that elements have negative area.

3.2.3 Degrees of freedomAll of the stress/displacement continuum elements have translational degrees of freedom at each node.Correspondingly, degrees of freedom 1, 2, and 3 are active in three-dimensional elements, while onlydegrees of freedom 1 and 2 are active in plane strain elements, plane stress elements, and axisymmetricelements without twist. Consult the ABAQUS/Standard User's Manual to find the active degrees offreedom in the other classes of two-dimensional solid elements.

3.2.4 Element propertiesThe *SOLID SECTION option defines the material and any additional geometric data associated witha set of continuum elements. For three-dimensional and axisymmetric elements no additionalgeometric information is required: the nodal coordinates completely define the element geometry. Forplane stress and plane strain elements the thickness of the elements must be specified on the data line.For example, if the elements are 0.2 m thick, the element property definition would be the following:

*SOLID SECTION, ELSET=<element set name>, MATERIAL=<material name>

0.2,

3.2.5 Formulation and integrationAlternative formulations available for the continuum family of elements include an incompatible modeformulation and a hybrid element formulation, both of which are discussed in detail later in this guide.The solid elements can use full or reduced integration. Both the formulation and type of integration canhave a significant effect on the accuracy of solid elements, as discussed in `Èlement formulation andintegration,'' Section 4.1.

3.2.6 Element output variablesBy default, element output variables such as stress and strain refer to the global Cartesian coordinatesystem. Thus, the ¾11-component of stress at the integration point shown in Figure 3-5(a) acts in theglobal 1-direction. Even if the element rotates during a large-displacement simulation, as shown inFigure 3-5(b), the default is still to use the global Cartesian system as the basis for defining the elementvariables. However, ABAQUS allows you to define a local coordinate system for element variables,using the *ORIENTATION option (see `Èxample: skew plate,'' Section 5.5). This local coordinatesystem rotates with the motion of the element in large-displacement simulations. A local coordinate


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system can be very useful if the object being modeled has some natural material orientation, such asthe fiber directions in a composite material.

Figure 3-5 Default material directions for continuum elements.

3.3 Shell elementsShell elements are used to model structures in which one dimension (the thickness) is significantlysmaller than the other dimensions, and the stresses normal to the thickness direction are negligible.

Shell element names in ABAQUS begin with the letter "S." Axisymmetric shells all begin with theletters "SAX," while axisymmetric shells with asymmetric deformations begin with the letters"SAXA." The first number in a shell element name indicates the number of nodes in the element,except for the case of axisymmetric shells, for which the first number indicates the order ofinterpolation. If the last character in the name is the number "5," the element uses only two of the threerotational degrees of freedom whenever possible.

The use of shell elements is discussed in detail in Chapter 5, "Using Shell Elements."

3.3.1 Shell element libraryGeneral three-dimensional shell elements are available with three different formulations:general-purpose, thin-only, and thick-only. Triangular and quadrilateral elements with linear andquadratic interpolation are available. Both linear and quadratic axisymmetric shell elements areavailable. Table 3-1 summarizes the available shell elements.

Table 3-1. Three classes of shell elements in ABAQUS.General-Purpose Shells Thin-Only Shells Thick-Only ShellsS4, S4R, S3/S3R, SAX1,

SAX2, SAX2TSTRI3, STRI65,

S4R5, S8R5, S9R5,SAXA

S8R, S8RT

3.3.2 Degrees of freedomThree-dimensional elements whose names end in the number "5" (e.g., S4R5, STRI65) have 5 degreesof freedom at each node: three translations and two in-plane rotations (i.e., no rotations about the shellnormal). However, all six degrees of freedom are activated at a node if required; for example, if

system can be very useful if the object being modeled has some natural material orientation, such asthe fiber directions in a composite material.

Figure 3-5 Default material directions for continuum elements.

3.3 Shell elementsShell elements are used to model structures in which one dimension (the thickness) is significantlysmaller than the other dimensions, and the stresses normal to the thickness direction are negligible.

Shell element names in ABAQUS begin with the letter "S." Axisymmetric shells all begin with theletters "SAX," while axisymmetric shells with asymmetric deformations begin with the letters"SAXA." The first number in a shell element name indicates the number of nodes in the element,except for the case of axisymmetric shells, for which the first number indicates the order ofinterpolation. If the last character in the name is the number "5," the element uses only two of the threerotational degrees of freedom whenever possible.

The use of shell elements is discussed in detail in Chapter 5, "Using Shell Elements."

3.3.1 Shell element libraryGeneral three-dimensional shell elements are available with three different formulations:general-purpose, thin-only, and thick-only. Triangular and quadrilateral elements with linear andquadratic interpolation are available. Both linear and quadratic axisymmetric shell elements areavailable. Table 3-1 summarizes the available shell elements.




S8R, S8RT

3.3.2 Degrees of freedomThree-dimensional elements whose names end in the number "5" (e.g., S4R5, STRI65) have 5 degreesof freedom at each node: three translations and two in-plane rotations (i.e., no rotations about the shellnormal). However, all six degrees of freedom are activated at a node if required; for example, if


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rotational boundary conditions are applied or the node is on a fold line of the shell.

The remaining three-dimensional shell elements (e.g., S4R and S8R) have six degrees of freedom ateach node (three translations and three rotations).

The axisymmetric shells have three degrees of freedom associated with each node:1 Translation in the r-direction.2 Translation in the z-direction.6 Rotation in the r-z plane.

3.3.3 Element propertiesUse either the *SHELL GENERAL SECTION or the *SHELL SECTION option to define thethickness and material properties for a set of shell elements. These two options have similar formats:

*SHELL SECTION, ELSET=<element set name>, MATERIAL=<material name>

<thickness>,<number of section points>

or

*SHELL GENERAL SECTION, ELSET=<element set name>, MATERIAL=<material name

<thickness>

If you specify the *SHELL SECTION option, ABAQUS uses numerical integration to calculate thebehavior at selected points through the thickness of the shell. These points are called section points.The MATERIAL parameter refers to a material property definition, which may be linear or nonlinear.You can specify any odd number of section points through the shell thickness.

The *SHELL GENERAL SECTION option allows you to define the cross-section behavior in anumber of general ways to model linear or nonlinear behavior. Since ABAQUS models the shell'scross-section behavior directly in terms of section engineering quantities (area, moments of inertia,etc.) with this option, there is no need for ABAQUS to integrate any quantities over the elementcross-section. Therefore, *SHELL GENERAL SECTION is less expensive computationally than*SHELL SECTION. The response is calculated in terms of force and moment resultants; the stressesand strains are calculated only when they are requested for output.

3.3.4 Formulation and integrationABAQUS has three classes of shell elements distinguishable by their element formulations:general-purpose, thin-only, and thick-only. The shell formulations are discussed in detail in Chapter 5,"Using Shell Elements." The S4, S4R, S3/S3R, and the SAX axisymmetric shell elements account forfinite membrane strains and allow the shell thickness to change with the element deformation. All ofthe other shell elements assume small strains and no change in shell thickness, even though theelement's nodes may undergo finite rotations.

All of the quadrilateral shell elements (except for S4) and the triangular shell element, S3/S3R, usereduced integration. The S4 element and the other triangular shell elements use full integration.

3.3.5 Element output variables

rotational boundary conditions are applied or the node is on a fold line of the shell.

The remaining three-dimensional shell elements (e.g., S4R and S8R) have six degrees of freedom ateach node (three translations and three rotations).

The axisymmetric shells have three degrees of freedom associated with each node:1 Translation in the r-direction.2 Translation in the z-direction.6 Rotation in the r-z plane.

3.3.3 Element propertiesUse either the *SHELL GENERAL SECTION or the *SHELL SECTION option to define thethickness and material properties for a set of shell elements. These two options have similar formats:

*SHELL SECTION, ELSET=<element set name>, MATERIAL=<material name>

<thickness>,<number of section points>

or

*SHELL GENERAL SECTION, ELSET=<element set name>, MATERIAL=<material name

<thickness>

If you specify the *SHELL SECTION option, ABAQUS uses numerical integration to calculate thebehavior at selected points through the thickness of the shell. These points are called section points.The MATERIAL parameter refers to a material property definition, which may be linear or nonlinear.You can specify any odd number of section points through the shell thickness.

The *SHELL GENERAL SECTION option allows you to define the cross-section behavior in anumber of general ways to model linear or nonlinear behavior. Since ABAQUS models the shell'scross-section behavior directly in terms of section engineering quantities (area, moments of inertia,etc.) with this option, there is no need for ABAQUS to integrate any quantities over the elementcross-section. Therefore, *SHELL GENERAL SECTION is less expensive computationally than*SHELL SECTION. The response is calculated in terms of force and moment resultants; the stressesand strains are calculated only when they are requested for output.

3.3.4 Formulation and integrationABAQUS has three classes of shell elements distinguishable by their element formulations:general-purpose, thin-only, and thick-only. The shell formulations are discussed in detail in Chapter 5,"Using Shell Elements." The S4, S4R, S3/S3R, and the SAX axisymmetric shell elements account forfinite membrane strains and allow the shell thickness to change with the element deformation. All ofthe other shell elements assume small strains and no change in shell thickness, even though theelement's nodes may undergo finite rotations.

All of the quadrilateral shell elements (except for S4) and the triangular shell element, S3/S3R, usereduced integration. The S4 element and the other triangular shell elements use full integration.

3.3.5 Element output variables


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The element output variables for shells are defined in terms of local material directions that lie on thesurface of each shell element. In large-displacement simulations these axes rotate with the element'sdeformation.

3.4 Beam elementsBeam elements are used to model components in which one dimension (the length) is significantlygreater than the other two dimensions and only the stress in the direction along the axis of the beam issignificant.

Beam element names in ABAQUS begin with the letter "B." The next character indicates thedimensionality of the element: "2" for two-dimensional beams and "3" for three-dimensional beams.The third character indicates the interpolation used: "1" for linear interpolation, "2" for quadraticinterpolation, and "3" for cubic interpolation.

The use of beam elements is discussed in Chapter 6, "Using Beam Elements."

3.4.1 Beam element libraryLinear, quadratic, and cubic beams are available in two and three dimensions.

3.4.2 Degrees of freedomThree-dimensional beams have six degrees of freedom at each node: three translational degrees offreedom (1-3) and three rotational degrees of freedom (4-6). "Open-section"-type beams (such asB31OS) have an additional degree of freedom (7) that represents the warping of the beamcross-section.

Two-dimensional beams have three degrees of freedom at each node: two translational degrees offreedom (1 and 2) and one rotational degree of freedom (6) about the normal to the plane of the model.

3.4.3 Element propertiesUse either the *BEAM SECTION or the *BEAM GENERAL SECTION option to define the geometryof the beam cross-section; the nodal coordinates define only the length.

If you specify the *BEAM SECTION option, the beam cross-section is defined geometrically, and theMATERIAL parameter refers to a material property definition. ABAQUS calculates the cross-sectionbehavior of the beam by numerical integration over the cross-section, allowing both linear andnonlinear material behavior.

The *BEAM GENERAL SECTION option allows you to define the cross-section behavior in a numberof general ways to model linear or nonlinear behavior. Since ABAQUS models the beam'scross-section behavior directly in terms of section engineering quantities (area, moments of inertia,etc.) with this option, there is no need for ABAQUS to integrate any quantities over the elementcross-section. Therefore, *BEAM GENERAL SECTION is less expensive computationally than*BEAM SECTION. The response is calculated in terms of the force and moment resultants; thestresses and strains are calculated only when they are requested for output.

The element output variables for shells are defined in terms of local material directions that lie on thesurface of each shell element. In large-displacement simulations these axes rotate with the element'sdeformation.

3.4 Beam elementsBeam elements are used to model components in which one dimension (the length) is significantlygreater than the other two dimensions and only the stress in the direction along the axis of the beam issignificant.

Beam element names in ABAQUS begin with the letter "B." The next character indicates thedimensionality of the element: "2" for two-dimensional beams and "3" for three-dimensional beams.The third character indicates the interpolation used: "1" for linear interpolation, "2" for quadraticinterpolation, and "3" for cubic interpolation.

The use of beam elements is discussed in Chapter 6, "Using Beam Elements."

3.4.1 Beam element libraryLinear, quadratic, and cubic beams are available in two and three dimensions.

3.4.2 Degrees of freedomThree-dimensional beams have six degrees of freedom at each node: three translational degrees offreedom (1-3) and three rotational degrees of freedom (4-6). "Open-section"-type beams (such asB31OS) have an additional degree of freedom (7) that represents the warping of the beamcross-section.

Two-dimensional beams have three degrees of freedom at each node: two translational degrees offreedom (1 and 2) and one rotational degree of freedom (6) about the normal to the plane of the model.

3.4.3 Element propertiesUse either the *BEAM SECTION or the *BEAM GENERAL SECTION option to define the geometryof the beam cross-section; the nodal coordinates define only the length.

If you specify the *BEAM SECTION option, the beam cross-section is defined geometrically, and theMATERIAL parameter refers to a material property definition. ABAQUS calculates the cross-sectionbehavior of the beam by numerical integration over the cross-section, allowing both linear andnonlinear material behavior.

The *BEAM GENERAL SECTION option allows you to define the cross-section behavior in a numberof general ways to model linear or nonlinear behavior. Since ABAQUS models the beam'scross-section behavior directly in terms of section engineering quantities (area, moments of inertia,etc.) with this option, there is no need for ABAQUS to integrate any quantities over the elementcross-section. Therefore, *BEAM GENERAL SECTION is less expensive computationally than*BEAM SECTION. The response is calculated in terms of the force and moment resultants; thestresses and strains are calculated only when they are requested for output.


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3.4.4 Formulation and integrationThe linear beams (B21 and B31) and the quadratic beams (B22 and B32) are shear deformable andaccount for finite axial strains; therefore, they are suitable for modeling both slender and nonslenderbeams. The cubic beam elements (B23 and B33) do not account for shear flexibility and assume smallaxial strain, although large displacements and rotations of the beams are valid. They are, therefore,suitable for modeling slender beams. ABAQUS provides variants of linear and quadratic beamelements that are suitable for modeling thin-walled, open-section beams ( B31OS and B32OS). Theseelements correctly model the effects of torsion and warping in open cross-sections, such as I-beams orU-section channels. Open-section beams are not covered in this guide.

ABAQUS also has hybrid beam elements that are used for modeling very slender members, such asflexible risers on offshore oil installations, or for modeling very rigid links. Hybrid beams are notcovered in this guide.

3.4.5 Element output variablesThe stress components in three-dimensional, shear-deformable beam elements are the axial stress ( ¾11)and the shear stress due to torsion (¾12). The shear stress acts about the section wall in a thin-walledsection. Corresponding strain measures are also available. The shear deformable beams also provideestimates of transverse shear forces on the section. The slender (cubic) beams have only the axialvariables as output. Open-section beams in space also have only the axial variables as output, since thetorsional shear stresses are negligible in this case.

All two-dimensional beams use only axial stress and strain.

The axial force, bending moments, and curvatures about the local beam axes can also be requested foroutput. For details of what components are available with which elements, refer to theABAQUS/Standard User's Manual. Details of how the local beam axes are defined are given inChapter 6, "Using Beam Elements."

3.5 Truss elementsTruss elements are rods that can carry only tensile or compressive loads. They have no resistance tobending; therefore, they are useful for modeling pin-jointed frames. Moreover, truss elements can beused as an approximation for cables or strings (for example, in a tennis racket). Trusses are alsosometimes used to represent reinforcement within other elements. The overhead hoist model inChapter 2, "ABAQUS Basics," uses truss elements.

All truss element names begin with the letter "T." The next two characters indicate the dimensionalityof the element--"2D" for two-dimensional trusses and "3D" for three-dimensional trusses. The finalcharacter represents the number of nodes in the element.

3.5.1 Truss element libraryLinear and quadratic trusses are available in two and three dimensions.

3.5.2 Degrees of freedom

3.4.4 Formulation and integrationThe linear beams (B21 and B31) and the quadratic beams (B22 and B32) are shear deformable andaccount for finite axial strains; therefore, they are suitable for modeling both slender and nonslenderbeams. The cubic beam elements (B23 and B33) do not account for shear flexibility and assume smallaxial strain, although large displacements and rotations of the beams are valid. They are, therefore,suitable for modeling slender beams. ABAQUS provides variants of linear and quadratic beamelements that are suitable for modeling thin-walled, open-section beams ( B31OS and B32OS). Theseelements correctly model the effects of torsion and warping in open cross-sections, such as I-beams orU-section channels. Open-section beams are not covered in this guide.

ABAQUS also has hybrid beam elements that are used for modeling very slender members, such asflexible risers on offshore oil installations, or for modeling very rigid links. Hybrid beams are notcovered in this guide.

3.4.5 Element output variablesThe stress components in three-dimensional, shear-deformable beam elements are the axial stress ( ¾11)and the shear stress due to torsion (¾12). The shear stress acts about the section wall in a thin-walledsection. Corresponding strain measures are also available. The shear deformable beams also provideestimates of transverse shear forces on the section. The slender (cubic) beams have only the axialvariables as output. Open-section beams in space also have only the axial variables as output, since thetorsional shear stresses are negligible in this case.

All two-dimensional beams use only axial stress and strain.

The axial force, bending moments, and curvatures about the local beam axes can also be requested foroutput. For details of what components are available with which elements, refer to theABAQUS/Standard User's Manual. Details of how the local beam axes are defined are given inChapter 6, "Using Beam Elements."

3.5 Truss elementsTruss elements are rods that can carry only tensile or compressive loads. They have no resistance tobending; therefore, they are useful for modeling pin-jointed frames. Moreover, truss elements can beused as an approximation for cables or strings (for example, in a tennis racket). Trusses are alsosometimes used to represent reinforcement within other elements. The overhead hoist model inChapter 2, "ABAQUS Basics," uses truss elements.

All truss element names begin with the letter "T." The next two characters indicate the dimensionalityof the element--"2D" for two-dimensional trusses and "3D" for three-dimensional trusses. The finalcharacter represents the number of nodes in the element.

3.5.1 Truss element libraryLinear and quadratic trusses are available in two and three dimensions.

3.5.2 Degrees of freedom


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Truss elements have only translational degrees of freedom at each node. Three-dimensional trusselements have degrees of freedom 1, 2, and 3, while two-dimensional truss elements have degrees offreedom 1 and 2.

3.5.3 Element propertiesThe *SOLID SECTION option is used to specify the name of the material property definitionassociated with the given set of truss elements. The cross-sectional area is given on the data line:

*SOLID SECTION, ELSET=<element set name>, MATERIAL=<material>

<cross-sectional area>

3.5.4 Formulation and integrationIn addition to the standard formulation a hybrid truss element formulation is available. It is useful formodeling very rigid links whose stiffness is much greater than that of the overall structure.

3.5.5 Element output variablesAxial stress and strain are available as output.

3.6 Rigid elementsRigid elements define rigid bodies. The nodes on a rigid element are different from other nodes in themodel because they have no independent degrees of freedom. The motion of the entire rigid body iscontrolled by the motion of a single node, the rigid body reference node (see Figure 3-6).

Figure 3-6 Rigid elements forming a rigid body.

The motion of the entire rigid body is calculated from the motion of this node. The nodes definingrigid elements can have loads applied to them or can be connected to other elements, but they cannothave any boundary conditions (*BOUNDARY), multi-point constraints (*MPC), or constraintequations (*EQUATION) applied to them.

The rigid body reference node that is associated with the rigid elements is defined on the *RIGIDBODY option. All rigid elements defining a surface must share a common rigid body reference nodesince they form a single rigid body. The same node can be used as the rigid body reference node formore than one rigid body. Boundary conditions, multi-point constraints, constraint equations, andloads can be applied to the rigid body reference node.

The names of all rigid elements begin with the letter "R." The next characters indicate the

Truss elements have only translational degrees of freedom at each node. Three-dimensional trusselements have degrees of freedom 1, 2, and 3, while two-dimensional truss elements have degrees offreedom 1 and 2.

3.5.3 Element propertiesThe *SOLID SECTION option is used to specify the name of the material property definitionassociated with the given set of truss elements. The cross-sectional area is given on the data line:

*SOLID SECTION, ELSET=<element set name>, MATERIAL=<material>

<cross-sectional area>

3.5.4 Formulation and integrationIn addition to the standard formulation a hybrid truss element formulation is available. It is useful formodeling very rigid links whose stiffness is much greater than that of the overall structure.

3.5.5 Element output variablesAxial stress and strain are available as output.

3.6 Rigid elementsRigid elements define rigid bodies. The nodes on a rigid element are different from other nodes in themodel because they have no independent degrees of freedom. The motion of the entire rigid body iscontrolled by the motion of a single node, the rigid body reference node (see Figure 3-6).

Figure 3-6 Rigid elements forming a rigid body.

The motion of the entire rigid body is calculated from the motion of this node. The nodes definingrigid elements can have loads applied to them or can be connected to other elements, but they cannothave any boundary conditions (*BOUNDARY), multi-point constraints (*MPC), or constraintequations (*EQUATION) applied to them.

The rigid body reference node that is associated with the rigid elements is defined on the *RIGIDBODY option. All rigid elements defining a surface must share a common rigid body reference nodesince they form a single rigid body. The same node can be used as the rigid body reference node formore than one rigid body. Boundary conditions, multi-point constraints, constraint equations, andloads can be applied to the rigid body reference node.

The names of all rigid elements begin with the letter "R." The next characters indicate the


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dimensionality of the element. For example, "2D" indicates that the element is planar; and " AX," thatthe element is axisymmetric. The final character represents the number of nodes in the element.

The use of rigid elements and rigid bodies is illustrated in Chapter 11, "Contact."

3.6.1 Rigid element libraryThe three-dimensional quadrilateral (R3D4) and triangular (R3D3) rigid elements are used to modelthe two-dimensional surfaces of a three-dimensional rigid body. Another element--a two-node,rigid-beam element (RB3D2)--is provided mainly to model components of offshore structures to whichfluid drag and buoyancy loads must be applied.

Two-node, rigid elements are available for plane strain, plane stress, and axisymmetric models. Thereis also a planar, two-node rigid-beam element, which is used mainly to model offshore structures intwo dimensions.

The connectivity of a rigid element determines its "top" and "bottom" side. It is important to knowwhich side of the element forms the outer surface of the rigid body when defining contact conditions inABAQUS. Creating a surface on the "inside" of the rigid body rather than the desired "outside" surfacecauses problems.

3.6.2 Degrees of freedomOnly the rigid body reference node has independent degrees of freedom. For three-dimensionalelements the reference node has three translational and three rotational degrees of freedom; for planarand axisymmetric elements the reference node has degrees of freedom 1, 2, and 6 (rotation about the3-axis).

The nodes attached to rigid elements have only slave degrees of freedom. The motion of the slavedegrees of freedom is determined entirely from the motion of the rigid-body reference node. For planarand three-dimensional rigid elements the only slave degrees of freedom are translations. Therigid-beam elements have the same slave degrees of freedom as the corresponding deformable-beamelements: 1-6 for the three-dimensional rigid beam and 1, 2, and 6 for the planar rigid beam.

3.6.3 Physical propertiesAll rigid elements must refer to a *RIGID BODY option. For the planar and rigid beam elements thecross-sectional area can be defined on the data line. For the axisymmetric and three-dimensionalelements the thickness can be defined on the data line. These data are required only if you apply bodyforces to the rigid elements.

3.6.4 Formulation and integrationSince the rigid elements are not deformable, they do not use numerical integration points, and there areno optional formulations.

3.6.5 Element output variablesThere are no element output variables. The only output from rigid elements is the motion of the nodes.

dimensionality of the element. For example, "2D" indicates that the element is planar; and " AX," thatthe element is axisymmetric. The final character represents the number of nodes in the element.

The use of rigid elements and rigid bodies is illustrated in Chapter 11, "Contact."

3.6.1 Rigid element libraryThe three-dimensional quadrilateral (R3D4) and triangular (R3D3) rigid elements are used to modelthe two-dimensional surfaces of a three-dimensional rigid body. Another element--a two-node,rigid-beam element (RB3D2)--is provided mainly to model components of offshore structures to whichfluid drag and buoyancy loads must be applied.

Two-node, rigid elements are available for plane strain, plane stress, and axisymmetric models. Thereis also a planar, two-node rigid-beam element, which is used mainly to model offshore structures intwo dimensions.

The connectivity of a rigid element determines its "top" and "bottom" side. It is important to knowwhich side of the element forms the outer surface of the rigid body when defining contact conditions inABAQUS. Creating a surface on the "inside" of the rigid body rather than the desired "outside" surfacecauses problems.

3.6.2 Degrees of freedomOnly the rigid body reference node has independent degrees of freedom. For three-dimensionalelements the reference node has three translational and three rotational degrees of freedom; for planarand axisymmetric elements the reference node has degrees of freedom 1, 2, and 6 (rotation about the3-axis).

The nodes attached to rigid elements have only slave degrees of freedom. The motion of the slavedegrees of freedom is determined entirely from the motion of the rigid-body reference node. For planarand three-dimensional rigid elements the only slave degrees of freedom are translations. Therigid-beam elements have the same slave degrees of freedom as the corresponding deformable-beamelements: 1-6 for the three-dimensional rigid beam and 1, 2, and 6 for the planar rigid beam.

3.6.3 Physical propertiesAll rigid elements must refer to a *RIGID BODY option. For the planar and rigid beam elements thecross-sectional area can be defined on the data line. For the axisymmetric and three-dimensionalelements the thickness can be defined on the data line. These data are required only if you apply bodyforces to the rigid elements.

3.6.4 Formulation and integrationSince the rigid elements are not deformable, they do not use numerical integration points, and there areno optional formulations.

3.6.5 Element output variablesThere are no element output variables. The only output from rigid elements is the motion of the nodes.


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In addition, reaction forces and reaction moments are available at the rigid body reference node.

3.7 Summary� ABAQUS has an extensive library of elements that can be used for a wide range of structural

applications. Your choice of element type has important consequences regarding the accuracy andefficiency of your simulation.

� The degrees of freedom active at a node depend on the element types attached to the node.

� The element name completely identifies the element's family, formulation, number of nodes, andtype of integration.

� All elements must refer to an element property option. The element property option provides anyadditional data required to define the geometry of the element and also identifies the associatedmaterial property definition.

� For continuum elements ABAQUS defines the element output variables, such as stress and strain,with respect to the global Cartesian coordinate system. You can change to a local coordinatesystem by using the *ORIENTATION option.

� For three-dimensional shell elements ABAQUS defines the element output variables with respectto a coordinate system based on the surface of the shell. You can change the coordinate system byusing the *ORIENTATION option.

In addition, reaction forces and reaction moments are available at the rigid body reference node.

3.7 Summary� ABAQUS has an extensive library of elements that can be used for a wide range of structural

applications. Your choice of element type has important consequences regarding the accuracy andefficiency of your simulation.

� The degrees of freedom active at a node depend on the element types attached to the node.

� The element name completely identifies the element's family, formulation, number of nodes, andtype of integration.

� All elements must refer to an element property option. The element property option provides anyadditional data required to define the geometry of the element and also identifies the associatedmaterial property definition.

� For continuum elements ABAQUS defines the element output variables, such as stress and strain,with respect to the global Cartesian coordinate system. You can change to a local coordinatesystem by using the *ORIENTATION option.

� For three-dimensional shell elements ABAQUS defines the element output variables with respectto a coordinate system based on the surface of the shell. You can change the coordinate system byusing the *ORIENTATION option.


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4. Using Continuum ElementsThe continuum family of stress/displacement elements is the most comprehensive of the elementlibraries in ABAQUS. For three-dimensional elements you have a choice of hexahedrons, tetrahedrons,and wedges; for two-dimensional elements the choice is between triangles and quadrilaterals. There arelinear and quadratic versions for each of these basic element shapes. You have the choice between full-and reduced-integration elements for hexahedrons and quadrilaterals. Finally, you have the option ofusing either the standard or hybrid element formulation; for linear hexahedral or quadrilateral elementsyou have the additional possibility of using an incompatible mode formulation, while for quadratictriangular or tetrahedral elements you have the additional possibility of using a "modified"formulation.

When the permutations of all these various element options are made, the total number of solidelements available to you is large--over 20 just for three-dimensional models. The accuracy of yoursimulation will depend strongly on the type of element you use in your model. The thought of choosingwhich of these elements is best for your model may seem daunting, especially at first. However, youwill come to view this selection as a 20+ piece tool set that provides you with the ability to choose justthe right tool, or element, for a particular job.

This chapter discusses the effect that different element formulations and levels of integration have onthe accuracy of a particular analysis. Some general guidelines for selecting continuum elements arealso given. These provide the foundation upon which you can build your knowledge as you gain moreexperience using ABAQUS. The example at the end of this section will allow you to put thisknowledge to use as you build and analyze a connecting lug.

4.1 Element formulation and integrationThe influence that the order of the element (linear or quadratic), the element formulation, and the levelof integration have on the accuracy of a structural simulation will be demonstrated by considering thecantilever beam shown in Figure 4-1. This is a classic test used to assess the behavior of a given finiteelement. Since the beam is relatively slender, we would normally model it with beam elements.However, it is used here to help assess the effectiveness of various solid elements.

Figure 4-1 Cantilever beam under a point load P at its free end.

The beam is 150 mm long, 2.5 mm wide, and 5 mm deep; built-in at one end; and carrying a tip load of5 N at the free end. The material has a Young's modulus, E, of 70 GPa and a Poisson's ratio of 0.0.

4. Using Continuum ElementsThe continuum family of stress/displacement elements is the most comprehensive of the elementlibraries in ABAQUS. For three-dimensional elements you have a choice of hexahedrons, tetrahedrons,and wedges; for two-dimensional elements the choice is between triangles and quadrilaterals. There arelinear and quadratic versions for each of these basic element shapes. You have the choice between full-and reduced-integration elements for hexahedrons and quadrilaterals. Finally, you have the option ofusing either the standard or hybrid element formulation; for linear hexahedral or quadrilateral elementsyou have the additional possibility of using an incompatible mode formulation, while for quadratictriangular or tetrahedral elements you have the additional possibility of using a "modified"formulation.

When the permutations of all these various element options are made, the total number of solidelements available to you is large--over 20 just for three-dimensional models. The accuracy of yoursimulation will depend strongly on the type of element you use in your model. The thought of choosingwhich of these elements is best for your model may seem daunting, especially at first. However, youwill come to view this selection as a 20+ piece tool set that provides you with the ability to choose justthe right tool, or element, for a particular job.

This chapter discusses the effect that different element formulations and levels of integration have onthe accuracy of a particular analysis. Some general guidelines for selecting continuum elements arealso given. These provide the foundation upon which you can build your knowledge as you gain moreexperience using ABAQUS. The example at the end of this section will allow you to put thisknowledge to use as you build and analyze a connecting lug.

4.1 Element formulation and integrationThe influence that the order of the element (linear or quadratic), the element formulation, and the levelof integration have on the accuracy of a structural simulation will be demonstrated by considering thecantilever beam shown in Figure 4-1. This is a classic test used to assess the behavior of a given finiteelement. Since the beam is relatively slender, we would normally model it with beam elements.However, it is used here to help assess the effectiveness of various solid elements.

Figure 4-1 Cantilever beam under a point load P at its free end.

The beam is 150 mm long, 2.5 mm wide, and 5 mm deep; built-in at one end; and carrying a tip load of5 N at the free end. The material has a Young's modulus, E, of 70 GPa and a Poisson's ratio of 0.0.

Using Continuum Elements

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Using beam theory, the deflection of the tip of the beam for a load P is given as

±tip =(Pl3)

3EI;

where I = bd3=12, l is the length, b is the width, and d is the depth of the beam.

For P = 5 N the tip deflection is 3.09 mm.

4.1.1 Full integrationThe expression "full integration" refers to the number of Gauss points required to integrate thepolynomial terms in an element's stiffness matrix exactly when the element has a regular shape. Forhexahedral and quadrilateral elements a "regular shape" means that the edges meet at right angles andthat any edge nodes are at the midpoint of the edge. Fully integrated, linear elements use twointegration points in each direction. Thus, the three-dimensional element C3D8 uses a 2 � 2 � 2 arrayof integration points in the element. Fully integrated, quadratic elements use three integration points ineach direction. The locations of the integration points in fully integrated, two-dimensional,quadrilateral elements are shown in Figure 4-2.

Figure 4-2 Integration points in fully integrated, two-dimensional, quadrilateral elements.

Several different finite element meshes were used in ABAQUS simulations of the cantilever beamproblem, as shown in Figure 4-3. The simulations use either linear or quadratic, fully integratedelements and illustrate the effects of both the order of the element (first versus second) and the meshdensity on the accuracy of the results.

Figure 4-3 Meshes used for the cantilever beam simulations.

Using beam theory, the deflection of the tip of the beam for a load P is given as

±tip =(Pl3)

3EI;

where I = bd3=12, l is the length, b is the width, and d is the depth of the beam.

For P = 5 N the tip deflection is 3.09 mm.

4.1.1 Full integrationThe expression "full integration" refers to the number of Gauss points required to integrate thepolynomial terms in an element's stiffness matrix exactly when the element has a regular shape. Forhexahedral and quadrilateral elements a "regular shape" means that the edges meet at right angles andthat any edge nodes are at the midpoint of the edge. Fully integrated, linear elements use twointegration points in each direction. Thus, the three-dimensional element C3D8 uses a 2 � 2 � 2 arrayof integration points in the element. Fully integrated, quadratic elements use three integration points ineach direction. The locations of the integration points in fully integrated, two-dimensional,quadrilateral elements are shown in Figure 4-2.

Figure 4-2 Integration points in fully integrated, two-dimensional, quadrilateral elements.

Several different finite element meshes were used in ABAQUS simulations of the cantilever beamproblem, as shown in Figure 4-3. The simulations use either linear or quadratic, fully integratedelements and illustrate the effects of both the order of the element (first versus second) and the meshdensity on the accuracy of the results.

Figure 4-3 Meshes used for the cantilever beam simulations.


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The ratios of the tip displacements for the various simulations to the beam-theory value of 3.09 mm areshown in Table 4-1.

Table 4-1. Normalized beam deflections with integrated elements.Element Mesh Size (Depth � Length)

1 � 6 2 � 12 4 � 12 8 � 24CPS4 0.074 0.242 0.242 0.561CPS8 0.994 1.000 1.000 1.000C3D8 0.077 0.248 0.243 0.563

C3D20 0.994 1.000 1.000 1.000

The linear elements CPS4 and C3D8 underpredict the deflection so badly that the results are unusable.The results are least accurate with coarse meshes, but even a fine mesh (8 � 24) still predicts a tipdisplacement that is only 56% of the theoretical value. Notice that for the linear, fully integratedelements it makes no difference how many elements there are through the thickness of the beam. Thisis caused by shear locking, which is a problem with all fully integrated, first-order, solid elements.

As we have seen, shear locking causes the elements to be too stiff in bending. It is explained asfollows. Consider a small piece of material in a structure subject to pure bending. The material willdistort as shown in Figure 4-4. Lines initially parallel to the horizontal axis take on constant curvature,and lines through the thickness remain straight. The angle between the horizontal and vertical linesremains at 90°.

Figure 4-4 Deformation of material subjected to bending moment M.

The edges of a linear element are unable to curve; therefore, if the small piece of material is modeledusing a single element, its deformed shape is like that shown in Figure 4-5.

Figure 4-5 Deformation of a fully integrated, linear element subjected to bending moment M.

The ratios of the tip displacements for the various simulations to the beam-theory value of 3.09 mm areshown in Table 4-1.

Table 4-1. Normalized beam deflections with integrated elements.Element Mesh Size (Depth � Length)

1 � 6 2 � 12 4 � 12 8 � 24CPS4 0.074 0.242 0.242 0.561CPS8 0.994 1.000 1.000 1.000C3D8 0.077 0.248 0.243 0.563

C3D20 0.994 1.000 1.000 1.000

The linear elements CPS4 and C3D8 underpredict the deflection so badly that the results are unusable.The results are least accurate with coarse meshes, but even a fine mesh (8 � 24) still predicts a tipdisplacement that is only 56% of the theoretical value. Notice that for the linear, fully integratedelements it makes no difference how many elements there are through the thickness of the beam. Thisis caused by shear locking, which is a problem with all fully integrated, first-order, solid elements.

As we have seen, shear locking causes the elements to be too stiff in bending. It is explained asfollows. Consider a small piece of material in a structure subject to pure bending. The material willdistort as shown in Figure 4-4. Lines initially parallel to the horizontal axis take on constant curvature,and lines through the thickness remain straight. The angle between the horizontal and vertical linesremains at 90°.

Figure 4-4 Deformation of material subjected to bending moment M.

The edges of a linear element are unable to curve; therefore, if the small piece of material is modeledusing a single element, its deformed shape is like that shown in Figure 4-5.

Figure 4-5 Deformation of a fully integrated, linear element subjected to bending moment M.


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For visualization, dotted lines that pass through the integration points are plotted. It is apparent that theupper line has increased in length, indicating that the direct stress in the 1-direction, ¾11 , is tensile.Similarly, the length of the lower dotted line has decreased, indicating that ¾11 is compressive. Thelength of the vertical dotted lines has not changed (assuming that displacements are small); therefore,¾22 at all integration points is zero. All this is consistent with the expected state of stress of a smallpiece of material subjected to pure bending. However, at each integration point the angle between thevertical and horizontal lines, which was initially 90°, has changed. This indicates that the shear stress,¾12 , at these points is nonzero. This is incorrect: the shear stress in a piece of material under purebending is zero.

This spurious shear stress arises because the edges of the element are unable to curve. Its presencemeans that strain energy is creating shearing deformation rather than the intended bendingdeformation, so the overall deflections are less: the element is too stiff.

Shear locking only affects the performance of fully integrated, linear elements subjected to bendingloads. These elements function perfectly well under direct or shear loads. Shear locking is not aproblem for quadratic elements since their edges are able to curve (see Figure 4-6). The predicted tipdisplacements for the quadratic elements shown in Table 4-1 are close to the theoretical value.However, quadratic elements will also exhibit some locking if they are distorted or if the bendingstress has a gradient, both of which can occur in practical problems.

Figure 4-6 Deformation of a fully integrated, quadratic element subjected to bending moment M.

Fully integrated, linear elements should be used only when you are fairly certain that the loads willproduce minimal bending in your model. Use a different element type if you have doubts about thetype of deformation the loading will create. Fully integrated, quadratic elements can also lock undercomplex states of stress; thus, you should check the results carefully if they are used exclusively inyour model. However, they are very useful for modeling areas where there are local stressconcentrations.

4.1.2 Reduced integrationOnly quadrilateral and hexahedral elements can use a reduced-integration scheme; all wedge,tetrahedral, and triangular solid elements use full integration, although they can be used in the samemesh with reduced-integration hexahedral or quadrilateral elements.

Reduced-integration elements use one fewer integration point in each direction than the fully

For visualization, dotted lines that pass through the integration points are plotted. It is apparent that theupper line has increased in length, indicating that the direct stress in the 1-direction, ¾11 , is tensile.Similarly, the length of the lower dotted line has decreased, indicating that ¾11 is compressive. Thelength of the vertical dotted lines has not changed (assuming that displacements are small); therefore,¾22 at all integration points is zero. All this is consistent with the expected state of stress of a smallpiece of material subjected to pure bending. However, at each integration point the angle between thevertical and horizontal lines, which was initially 90°, has changed. This indicates that the shear stress,¾12 , at these points is nonzero. This is incorrect: the shear stress in a piece of material under purebending is zero.

This spurious shear stress arises because the edges of the element are unable to curve. Its presencemeans that strain energy is creating shearing deformation rather than the intended bendingdeformation, so the overall deflections are less: the element is too stiff.

Shear locking only affects the performance of fully integrated, linear elements subjected to bendingloads. These elements function perfectly well under direct or shear loads. Shear locking is not aproblem for quadratic elements since their edges are able to curve (see Figure 4-6). The predicted tipdisplacements for the quadratic elements shown in Table 4-1 are close to the theoretical value.However, quadratic elements will also exhibit some locking if they are distorted or if the bendingstress has a gradient, both of which can occur in practical problems.

Figure 4-6 Deformation of a fully integrated, quadratic element subjected to bending moment M.

Fully integrated, linear elements should be used only when you are fairly certain that the loads willproduce minimal bending in your model. Use a different element type if you have doubts about thetype of deformation the loading will create. Fully integrated, quadratic elements can also lock undercomplex states of stress; thus, you should check the results carefully if they are used exclusively inyour model. However, they are very useful for modeling areas where there are local stressconcentrations.

4.1.2 Reduced integrationOnly quadrilateral and hexahedral elements can use a reduced-integration scheme; all wedge,tetrahedral, and triangular solid elements use full integration, although they can be used in the samemesh with reduced-integration hexahedral or quadrilateral elements.

Reduced-integration elements use one fewer integration point in each direction than the fully


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integrated elements. Reduced-integration, linear elements have just a single integration point located atthe element's centroid. (Actually, these first-order elements in ABAQUS use the more accurate"uniform strain" formulation, where average values of the strain components are computed for theelement. This distinction is not important for this discussion.) The locations of the integration pointsfor reduced-integration, quadrilateral elements are shown in Figure 4-7.

Figure 4-7 Integration points in two-dimensional elements with reduced integration.

ABAQUS simulations of the cantilever beam problem were performed using the reduced-integrationversions of the same four elements utilized previously and using the four finite element meshes shownin Figure 4-3. The results from these simulations are presented in Table 4-2.

Table 4-2. Normalized beam deflections with reduced-integration elements.Element Mesh Size (Depth � Length)

1 � 6 2 � 12 4 � 12 8 � 24CPS4R 20.3* 1.308 1.051 1.012CPS8R 1.000 1.000 1.000 1.000C3D8R 70.1* 1.323 1.063 1.015

C3D20R 1.000 1.000 1.000 1.000* no stiffness to resist the applied load

Linear reduced-integration elements tend to be too flexible because they suffer from their ownnumerical problem called hourglassing. Again, consider a single reduced-integration element modelinga small piece of material subjected to pure bending (see Figure 4-8).

Figure 4-8 Deformation of a linear element with reduced integration subjected to bending moment M.

Neither of the dotted visualization lines has changed in length, and the angle between them is alsounchanged, which means that all components of stress at the element's single integration point are zero.This bending mode of deformation is thus a zero-energy mode because no strain energy is generated bythis element distortion. The element is unable to resist this type of deformation since it has no stiffnessin this mode. In coarse meshes this zero-energy mode can propagate through the mesh, producing

integrated elements. Reduced-integration, linear elements have just a single integration point located atthe element's centroid. (Actually, these first-order elements in ABAQUS use the more accurate"uniform strain" formulation, where average values of the strain components are computed for theelement. This distinction is not important for this discussion.) The locations of the integration pointsfor reduced-integration, quadrilateral elements are shown in Figure 4-7.

Figure 4-7 Integration points in two-dimensional elements with reduced integration.

ABAQUS simulations of the cantilever beam problem were performed using the reduced-integrationversions of the same four elements utilized previously and using the four finite element meshes shownin Figure 4-3. The results from these simulations are presented in Table 4-2.

Table 4-2. Normalized beam deflections with reduced-integration elements.Element Mesh Size (Depth � Length)

1 � 6 2 � 12 4 � 12 8 � 24CPS4R 20.3* 1.308 1.051 1.012CPS8R 1.000 1.000 1.000 1.000C3D8R 70.1* 1.323 1.063 1.015

C3D20R 1.000 1.000 1.000 1.000* no stiffness to resist the applied load

Linear reduced-integration elements tend to be too flexible because they suffer from their ownnumerical problem called hourglassing. Again, consider a single reduced-integration element modelinga small piece of material subjected to pure bending (see Figure 4-8).

Figure 4-8 Deformation of a linear element with reduced integration subjected to bending moment M.

Neither of the dotted visualization lines has changed in length, and the angle between them is alsounchanged, which means that all components of stress at the element's single integration point are zero.This bending mode of deformation is thus a zero-energy mode because no strain energy is generated bythis element distortion. The element is unable to resist this type of deformation since it has no stiffnessin this mode. In coarse meshes this zero-energy mode can propagate through the mesh, producing


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meaningless results.

In ABAQUS a small amount of artificial "hourglass stiffness" is introduced in reduced-integrationelements to limit the propagation of hourglass modes. This stiffness is more effective at limiting thehourglass modes when more elements are used in the model, which means that linearreduced-integration elements can give acceptable results as long as a reasonably fine mesh is used. Theerrors seen with the finer meshes of linear, reduced-integration elements (see Table 4-2) are within anacceptable range for many applications. The results suggest that at least four elements should be usedthrough the thickness when modeling any structures carrying bending loads with this type of element.When a single linear, reduced-integration element is used through the thickness of the beam, all theintegration points lie on the neutral axis and the model is unable to resist bending loads. (These casesare marked with a * in Table 4-2.)

Linear, reduced-integration elements are very tolerant of distortion; therefore, use a fine mesh of theseelements in any simulation where the distortion levels may be very high.

Quadratic, reduced-integration elements also have hourglass modes. However, the modes are almostimpossible to propagate in a normal mesh and are rarely a problem if the mesh is sufficiently fine. The1 � 6 mesh of C3D20R elements does show some hourglassing, but the more refined meshes do not.Quadratic, reduced-integration elements are not susceptible to locking, even when subjected tocomplicated states of stress. Therefore, these elements are generally the best choice for most generalstress/displacement simulations, except in large-displacement simulations involving very large strainsand in some types of contact analyses.

4.1.3 Incompatible mode elementsThe incompatible mode elements are an attempt to overcome the problems of shear locking in fullyintegrated, first-order elements. Since shear locking is caused by the inability of the element'sdisplacement field to model the kinematics associated with bending, additional degrees of freedom,which enhance the element's deformation gradients, are introduced into the first-order element. Theseenhancements to the deformation gradients allow a first-order element to have a linear variation of thedeformation gradient across the element's domain as shown in Figure 4-9(a). The standard elementformulation results in a constant deformation gradient across the element as shown in Figure 4-9(b),resulting in the nonzero shear stress associated with shear locking.

Figure 4-9 Variation of deformation gradient in (a) an incompatible mode (enhanced deformation

gradient) element and (b) a first-order element using a standard formulation.

These enhancements to the deformation gradients are entirely internal to an element and are not

meaningless results.

In ABAQUS a small amount of artificial "hourglass stiffness" is introduced in reduced-integrationelements to limit the propagation of hourglass modes. This stiffness is more effective at limiting thehourglass modes when more elements are used in the model, which means that linearreduced-integration elements can give acceptable results as long as a reasonably fine mesh is used. Theerrors seen with the finer meshes of linear, reduced-integration elements (see Table 4-2) are within anacceptable range for many applications. The results suggest that at least four elements should be usedthrough the thickness when modeling any structures carrying bending loads with this type of element.When a single linear, reduced-integration element is used through the thickness of the beam, all theintegration points lie on the neutral axis and the model is unable to resist bending loads. (These casesare marked with a * in Table 4-2.)

Linear, reduced-integration elements are very tolerant of distortion; therefore, use a fine mesh of theseelements in any simulation where the distortion levels may be very high.

Quadratic, reduced-integration elements also have hourglass modes. However, the modes are almostimpossible to propagate in a normal mesh and are rarely a problem if the mesh is sufficiently fine. The1 � 6 mesh of C3D20R elements does show some hourglassing, but the more refined meshes do not.Quadratic, reduced-integration elements are not susceptible to locking, even when subjected tocomplicated states of stress. Therefore, these elements are generally the best choice for most generalstress/displacement simulations, except in large-displacement simulations involving very large strainsand in some types of contact analyses.

4.1.3 Incompatible mode elementsThe incompatible mode elements are an attempt to overcome the problems of shear locking in fullyintegrated, first-order elements. Since shear locking is caused by the inability of the element'sdisplacement field to model the kinematics associated with bending, additional degrees of freedom,which enhance the element's deformation gradients, are introduced into the first-order element. Theseenhancements to the deformation gradients allow a first-order element to have a linear variation of thedeformation gradient across the element's domain as shown in Figure 4-9(a). The standard elementformulation results in a constant deformation gradient across the element as shown in Figure 4-9(b),resulting in the nonzero shear stress associated with shear locking.

Figure 4-9 Variation of deformation gradient in (a) an incompatible mode (enhanced deformation

gradient) element and (b) a first-order element using a standard formulation.

These enhancements to the deformation gradients are entirely internal to an element and are not


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associated with nodes positioned along the element edges. Unlike incompatible mode formulations thatenhance the displacement field directly, the formulation used in ABAQUS does not result inoverlapping material or a hole along the boundary between two elements, as shown in Figure 4-10.Furthermore, the formulation used in ABAQUS is extended easily to nonlinear, finite-strainsimulations, something which is not as easy with the enhanced displacement field elements.

Figure 4-10 Potential kinematic incompatibility between incompatible mode elements that use

enhanced displacement fields rather than enhanced deformation gradients. ABAQUS uses the latter

formulation for its incompatible mode elements.

Incompatible mode elements can produce results in bending problems that are comparable to quadraticelements but at significantly lower computational cost. However, they are sensitive to elementdistortions. Figure 4-11 shows the cantilever beam modeled with deliberately distorted incompatiblemode elements: in one case with "parallel" distortion and in the other with "trapezoidal" distortion.

Figure 4-11 Distorted meshes of incompatible mode elements.

Figure 4-12 shows the normalized tip displacement for the cantilever beam models plotted against thelevel of element distortion.

Figure 4-12 Effect of parallel and trapezoidal distortion of incompatible mode elements.

associated with nodes positioned along the element edges. Unlike incompatible mode formulations thatenhance the displacement field directly, the formulation used in ABAQUS does not result inoverlapping material or a hole along the boundary between two elements, as shown in Figure 4-10.Furthermore, the formulation used in ABAQUS is extended easily to nonlinear, finite-strainsimulations, something which is not as easy with the enhanced displacement field elements.

Figure 4-10 Potential kinematic incompatibility between incompatible mode elements that use

enhanced displacement fields rather than enhanced deformation gradients. ABAQUS uses the latter

formulation for its incompatible mode elements.

Incompatible mode elements can produce results in bending problems that are comparable to quadraticelements but at significantly lower computational cost. However, they are sensitive to elementdistortions. Figure 4-11 shows the cantilever beam modeled with deliberately distorted incompatiblemode elements: in one case with "parallel" distortion and in the other with "trapezoidal" distortion.

Figure 4-11 Distorted meshes of incompatible mode elements.

Figure 4-12 shows the normalized tip displacement for the cantilever beam models plotted against thelevel of element distortion.

Figure 4-12 Effect of parallel and trapezoidal distortion of incompatible mode elements.


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Three types of plane stress elements are compared: the fully integrated, linear element; thereduced-integration, quadratic element; and the linear, incompatible mode element. The fullyintegrated, linear elements produce poor results in all cases, as expected. On the other hand, thereduced-integration, quadratic elements give very good results that do not deteriorate until theelements are badly distorted.

When the incompatible mode elements are rectangular, even a mesh with just one element through thethickness of the cantilever gives results that are very close to the theoretical value. However, evenquite small levels of trapezoidal distortion make the elements much too stiff. Parallel distortion alsoreduces the accuracy of the element but to a lesser extent.

Incompatible mode elements are useful because they can provide high accuracy at a low cost if they areused appropriately. However, care must be taken to ensure that the element distortions are small,which may be difficult when meshing complex geometries; therefore, you should again consider usingthe reduced-integration, quadratic elements in models with such geometries because they show muchless sensitivity to mesh distortion. In a severely distorted mesh, however, simply changing the elementtype will generally not produce accurate results. The mesh distortion should be minimized as much aspossible to improve the accuracy of the results.

4.1.4 Hybrid elementsA hybrid element formulation is available for just about every type of continuum element in ABAQUS,including all reduced-integration and incompatible mode elements. Elements using this formulationhave the letter "H" in their names.

Hybrid elements are used when the material behavior is incompressible (Poisson's ratio = 0.5) or veryclose to incompressible (Poisson's ratio > 0.475). Rubber is an example of a material withincompressible material behavior. An incompressible material response cannot be modeled withregular elements (except in the case of plane stress) because the pressure stress in the element isindeterminate. Consider an element under uniform hydrostatic pressure ( Figure 4-13).

Figure 4-13 Element under hydrostatic pressure.

Three types of plane stress elements are compared: the fully integrated, linear element; thereduced-integration, quadratic element; and the linear, incompatible mode element. The fullyintegrated, linear elements produce poor results in all cases, as expected. On the other hand, thereduced-integration, quadratic elements give very good results that do not deteriorate until theelements are badly distorted.

When the incompatible mode elements are rectangular, even a mesh with just one element through thethickness of the cantilever gives results that are very close to the theoretical value. However, evenquite small levels of trapezoidal distortion make the elements much too stiff. Parallel distortion alsoreduces the accuracy of the element but to a lesser extent.

Incompatible mode elements are useful because they can provide high accuracy at a low cost if they areused appropriately. However, care must be taken to ensure that the element distortions are small,which may be difficult when meshing complex geometries; therefore, you should again consider usingthe reduced-integration, quadratic elements in models with such geometries because they show muchless sensitivity to mesh distortion. In a severely distorted mesh, however, simply changing the elementtype will generally not produce accurate results. The mesh distortion should be minimized as much aspossible to improve the accuracy of the results.

4.1.4 Hybrid elementsA hybrid element formulation is available for just about every type of continuum element in ABAQUS,including all reduced-integration and incompatible mode elements. Elements using this formulationhave the letter "H" in their names.

Hybrid elements are used when the material behavior is incompressible (Poisson's ratio = 0.5) or veryclose to incompressible (Poisson's ratio > 0.475). Rubber is an example of a material withincompressible material behavior. An incompressible material response cannot be modeled withregular elements (except in the case of plane stress) because the pressure stress in the element isindeterminate. Consider an element under uniform hydrostatic pressure ( Figure 4-13).

Figure 4-13 Element under hydrostatic pressure.


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If the material is incompressible, its volume cannot change under this loading. Therefore, the pressurestress cannot be computed from the displacements of the nodes; and, thus, a pure displacementformulation is inadequate for any element with incompressible material behavior.

Hybrid elements include an additional degree of freedom that determines the pressure stress in theelement directly. The nodal displacements are used only to calculate the deviatoric (shear) strains andstresses.

A more detailed description of the analysis of rubber materials is given in Chapter 8, "Materials."

4.2 Selecting continuum elementsThe correct choice of element for a particular simulation is vital if accurate results are to be obtained ata reasonable cost. You will undoubtedly develop your own guidelines for selecting elements for yourown particular applications as you become more experienced in using ABAQUS. However, as youbegin to use ABAQUS, you should consider the following guidelines:

� Use quadratic, reduced-integration elements (CAX8R, CPE8R, CPS8R, C3D20R, etc.) for generalanalysis work, unless you need to model very large strains or have a simulation with complex,changing contact conditions.

� Use quadratic, fully integrated elements (CAX8, CPE8, CPS8, C3D20, etc.) locally where stressconcentrations may exist. They provide the best resolution of the stress gradients at the lowestcost.

� Use a fine mesh of linear, reduced-integration elements (CAX4R, CPE4R, CPS4R, C3D8R, etc.)for simulations involving very large mesh distortions (large-strain analysis).

� For contact problems use a fine mesh of linear, reduced-integration elements or incompatiblemode elements (CAX4I, CPE4I, CPS4I, C3D8I, etc.). See Chapter 11, "Contact."

� Minimize the mesh distortion as much as possible. Coarse meshes with distorted linear elementscan give very poor results.

� In three dimensions use hexahedral (brick-shaped) elements wherever possible. They give the bestresults for the minimum cost. Complex geometries can be difficult to mesh completely withhexahedrons; therefore, wedge and tetrahedral elements may be necessary. The linear versions ofthese elements, C3D6 and C3D4, are poor elements (fine meshes are needed to obtain accurateresults); as a result, these elements should generally be used only when necessary to complete amesh, and, even then, they should be far from any areas where accurate results are needed.

If the material is incompressible, its volume cannot change under this loading. Therefore, the pressurestress cannot be computed from the displacements of the nodes; and, thus, a pure displacementformulation is inadequate for any element with incompressible material behavior.

Hybrid elements include an additional degree of freedom that determines the pressure stress in theelement directly. The nodal displacements are used only to calculate the deviatoric (shear) strains andstresses.

A more detailed description of the analysis of rubber materials is given in Chapter 8, "Materials."

4.2 Selecting continuum elementsThe correct choice of element for a particular simulation is vital if accurate results are to be obtained ata reasonable cost. You will undoubtedly develop your own guidelines for selecting elements for yourown particular applications as you become more experienced in using ABAQUS. However, as youbegin to use ABAQUS, you should consider the following guidelines:

� Use quadratic, reduced-integration elements (CAX8R, CPE8R, CPS8R, C3D20R, etc.) for generalanalysis work, unless you need to model very large strains or have a simulation with complex,changing contact conditions.

� Use quadratic, fully integrated elements (CAX8, CPE8, CPS8, C3D20, etc.) locally where stressconcentrations may exist. They provide the best resolution of the stress gradients at the lowestcost.

� Use a fine mesh of linear, reduced-integration elements (CAX4R, CPE4R, CPS4R, C3D8R, etc.)for simulations involving very large mesh distortions (large-strain analysis).

� For contact problems use a fine mesh of linear, reduced-integration elements or incompatiblemode elements (CAX4I, CPE4I, CPS4I, C3D8I, etc.). See Chapter 11, "Contact."

� Minimize the mesh distortion as much as possible. Coarse meshes with distorted linear elementscan give very poor results.

� In three dimensions use hexahedral (brick-shaped) elements wherever possible. They give the bestresults for the minimum cost. Complex geometries can be difficult to mesh completely withhexahedrons; therefore, wedge and tetrahedral elements may be necessary. The linear versions ofthese elements, C3D6 and C3D4, are poor elements (fine meshes are needed to obtain accurateresults); as a result, these elements should generally be used only when necessary to complete amesh, and, even then, they should be far from any areas where accurate results are needed.


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� Some preprocessors contain free-meshing algorithms that mesh arbitrary geometries withtetrahedral elements. As long as you use quadratic tetrahedral elements ( C3D10), you should getreasonable results for small-displacement problems without contact. An alternative to this elementis the modified quadratic tetrahedral element (C3D10M), which is robust for large-deformationand contact problems and exhibits minimal shear and volumetric locking. With either element,however, the analysis will take longer to run than an equivalent mesh of hexahedral elements. Youshould not use a mesh containing only linear tetrahedral elements ( C3D4): the results will beinaccurate unless you use an extremely large number of elements.

4.3 Example: connecting lugIn this example you will use three-dimensional, continuum elements to model the connecting lugshown in Figure 4-14.

Figure 4-14 Sketch of the connecting lug.

The lug is welded firmly to a massive structure at one end. The other end contains a hole. When it is inservice, a bolt will be placed through the hole of the lug. You have been asked to determine thedeflection of the lug when a 30 kN load is applied to the bolt in the negative 2-direction. You decide tosimplify this problem by making the following assumptions:

� Rather than include the complex bolt-lug interaction in the model, you will use a distributedpressure over the bottom half of the hole to load the connecting lug (see Figure 4-14).

� You will neglect the variation of pressure magnitude around the circumference of the hole and usea uniform pressure.

� Some preprocessors contain free-meshing algorithms that mesh arbitrary geometries withtetrahedral elements. As long as you use quadratic tetrahedral elements ( C3D10), you should getreasonable results for small-displacement problems without contact. An alternative to this elementis the modified quadratic tetrahedral element (C3D10M), which is robust for large-deformationand contact problems and exhibits minimal shear and volumetric locking. With either element,however, the analysis will take longer to run than an equivalent mesh of hexahedral elements. Youshould not use a mesh containing only linear tetrahedral elements ( C3D4): the results will beinaccurate unless you use an extremely large number of elements.

4.3 Example: connecting lugIn this example you will use three-dimensional, continuum elements to model the connecting lugshown in Figure 4-14.

Figure 4-14 Sketch of the connecting lug.

The lug is welded firmly to a massive structure at one end. The other end contains a hole. When it is inservice, a bolt will be placed through the hole of the lug. You have been asked to determine thedeflection of the lug when a 30 kN load is applied to the bolt in the negative 2-direction. You decide tosimplify this problem by making the following assumptions:

� Rather than include the complex bolt-lug interaction in the model, you will use a distributedpressure over the bottom half of the hole to load the connecting lug (see Figure 4-14).

� You will neglect the variation of pressure magnitude around the circumference of the hole and usea uniform pressure.


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� The magnitude of the applied uniform pressure will be 50 MPa (30 kN/ 2 � 0.015 m � 0.02 m).

4.3.1 Coordinate systemIn your model define the global 1-axis to lie along the length of the lug, the global 2-axis to be vertical,and the global 3-axis to lie in the thickness direction. Place the origin of the global coordinate system( x = 0; y = 0; z = 0) at the center of the hole on the z = 0 face (see Figure 4-14).

4.3.2 Mesh designYou need to consider the type of element that will be used before you start building the mesh for aparticular problem. A suitable mesh design that uses quadratic elements may very well be unsuitable ifyou change to linear, reduced-integration elements. For this example use 20-node hexahedral elementswith reduced integration (C3D20R). With the element type selected, you can design the mesh for theconnecting lug. The most important decision regarding the mesh design for this application is howmany elements to use around the circumference of the lug's hole. A possible mesh for the connectinglug is shown in Figure 4-15; you should build your model to be similar to it.

Figure 4-15 Suggested mesh of C3D20R elements for the connecting lug model.

Another thing to consider when designing a mesh is what type of results you want from the simulation.The mesh in Figure 4-15 is rather coarse and, therefore, unlikely to yield accurate stresses. Fourquadratic elements per 90° is the minimum number that should be considered for a problem like thisone; using twice that many is recommended to obtain reasonably accurate stress results. However, thismesh should be adequate to predict the overall level of deformation in the lug under the applied loads,which is what you were asked to determine. The influence of increasing the mesh density used in thissimulation is discussed in ``Mesh convergence,'' Section 4.4.

You need to decide what system of units to use in your model. The SI system of meters, seconds, andkilograms is recommended, but use another system if you prefer.

4.3.3 Preprocessing--creating the modelThe model for the overhead hoist in Chapter 2, "ABAQUS Basics," was simple enough that theABAQUS input file could be created by typing the input directly into a text editor. This clearly is

� The magnitude of the applied uniform pressure will be 50 MPa (30 kN/ 2 � 0.015 m � 0.02 m).

4.3.1 Coordinate systemIn your model define the global 1-axis to lie along the length of the lug, the global 2-axis to be vertical,and the global 3-axis to lie in the thickness direction. Place the origin of the global coordinate system( x = 0; y = 0; z = 0) at the center of the hole on the z = 0 face (see Figure 4-14).

4.3.2 Mesh designYou need to consider the type of element that will be used before you start building the mesh for aparticular problem. A suitable mesh design that uses quadratic elements may very well be unsuitable ifyou change to linear, reduced-integration elements. For this example use 20-node hexahedral elementswith reduced integration (C3D20R). With the element type selected, you can design the mesh for theconnecting lug. The most important decision regarding the mesh design for this application is howmany elements to use around the circumference of the lug's hole. A possible mesh for the connectinglug is shown in Figure 4-15; you should build your model to be similar to it.

Figure 4-15 Suggested mesh of C3D20R elements for the connecting lug model.

Another thing to consider when designing a mesh is what type of results you want from the simulation.The mesh in Figure 4-15 is rather coarse and, therefore, unlikely to yield accurate stresses. Fourquadratic elements per 90° is the minimum number that should be considered for a problem like thisone; using twice that many is recommended to obtain reasonably accurate stress results. However, thismesh should be adequate to predict the overall level of deformation in the lug under the applied loads,which is what you were asked to determine. The influence of increasing the mesh density used in thissimulation is discussed in ``Mesh convergence,'' Section 4.4.

You need to decide what system of units to use in your model. The SI system of meters, seconds, andkilograms is recommended, but use another system if you prefer.

4.3.3 Preprocessing--creating the modelThe model for the overhead hoist in Chapter 2, "ABAQUS Basics," was simple enough that theABAQUS input file could be created by typing the input directly into a text editor. This clearly is


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impractical for most real problems; the input for the examples in this guide can be created easily usingABAQUS/CAE or another preprocessor.

Use a preprocessor to generate the mesh, the node and element sets shown in Figure 4-16, and thepressure load and boundary conditions shown in Figure 4-14. The following discussion assumes thatyou have done this. Although most preprocessors can create the entire input file for this simulation, itis suggested that you generate only the mesh, the element and node sets shown, and the correctpressure loads and boundary conditions using the preprocessor. You should then add the additionaldata needed for the model with an editor to obtain a better understanding of the format of an ABAQUSinput file. If you do not have a preprocessor, use the ABAQUS mesh generation options in``Connecting lug,'' Section A.2.

Figure 4-16 Useful node and element sets for the connecting lug simulation.

In the description of this simulation that follows, the node and element numbers used are from themodel found in ``Connecting lug,'' Section A.2. These node and element numbers are shown in Figure4-17 and Figure 4-18. If you use a preprocessor, the node and element numbering in your model willalmost certainly differ from that shown here. As you make modifications to your input file, rememberto use the node and element numbers in your model and not those given here.

Figure 4-17 Node numbers in the plane z = 0.

impractical for most real problems; the input for the examples in this guide can be created easily usingABAQUS/CAE or another preprocessor.

Use a preprocessor to generate the mesh, the node and element sets shown in Figure 4-16, and thepressure load and boundary conditions shown in Figure 4-14. The following discussion assumes thatyou have done this. Although most preprocessors can create the entire input file for this simulation, itis suggested that you generate only the mesh, the element and node sets shown, and the correctpressure loads and boundary conditions using the preprocessor. You should then add the additionaldata needed for the model with an editor to obtain a better understanding of the format of an ABAQUSinput file. If you do not have a preprocessor, use the ABAQUS mesh generation options in``Connecting lug,'' Section A.2.

Figure 4-16 Useful node and element sets for the connecting lug simulation.

In the description of this simulation that follows, the node and element numbers used are from themodel found in ``Connecting lug,'' Section A.2. These node and element numbers are shown in Figure4-17 and Figure 4-18. If you use a preprocessor, the node and element numbering in your model willalmost certainly differ from that shown here. As you make modifications to your input file, rememberto use the node and element numbers in your model and not those given here.

Figure 4-17 Node numbers in the plane z = 0.


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Figure 4-18 Element numbers in the plane z = 0.

4.3.4 Reviewing the input file--the model dataThe model data--including the node and element definitions, set definitions, and section and materialproperties--are discussed in the following sections.

Model description

An ABAQUS input file always starts with the *HEADING option. Often the description givenin this option by the preprocessor is not very informative, although it might give the date andtime when the file was generated. You should provide a suitable title on the data lines of thisoption so that someone looking at this file can tell what is being modeled and what units youused. You should modify the *HEADING option block in your file to look like thefollowing:

*HEADING

Linear Elastic Steel Connecting Lug

S.I. Units (N, kg, m, s)

Figure 4-18 Element numbers in the plane z = 0.

4.3.4 Reviewing the input file--the model dataThe model data--including the node and element definitions, set definitions, and section and materialproperties--are discussed in the following sections.

Model description

An ABAQUS input file always starts with the *HEADING option. Often the description givenin this option by the preprocessor is not very informative, although it might give the date andtime when the file was generated. You should provide a suitable title on the data lines of thisoption so that someone looking at this file can tell what is being modeled and what units youused. You should modify the *HEADING option block in your file to look like thefollowing:

*HEADING




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Nodal coordinates and element connectivity

In input files created by a preprocessor, the model's nodal coordinates usually are in one large*NODE option block, with the coordinates specified for each node individually.

The element definitions generated by the preprocessor usually are contained in several*ELEMENT option blocks. Typically, each block contains elements that have the sameelement section and material properties. In the connecting lug model only one element type isused, and all the elements have the same properties. Therefore, there will probably be a single*ELEMENT option block in your input file. It will look similar to

*ELEMENT, TYPE=C3D20R, ELSET=LUG

1, 1, 401, 405, 5, 10001, 10401, 10405, 10005,

201, 403, 205, 3, 10201, 10403, 10205, 10003,

5001, 5401, 5405, 5005

2, 5, 405, 409, 9, 10005, 10405, 10409, 10009,

205, 407, 209, 7, 10205, 10407, 10209, 10007,

5005, 5405, 5409, 5009

.......

It takes three data lines to define the connectivity of one C3D20R element completely. If adata line in an *ELEMENT option block ends with a comma, it indicates that the next dataline contains more nodes defining the current element. The parameter ELSET=LUG indicatesthat all the elements defined in the following data lines will be stored in an element set calledLUG. If your model does not have a descriptive element set name in the *ELEMENT option,change it to LUG.

Node and element sets

The node and element sets are important components of an ABAQUS input file because theyallow you to assign loads, boundary conditions, and material properties efficiently. They alsooffer great flexibility in defining the output that your simulation will produce and make itmuch easier to understand the input file.

Some preprocessors, such as ABAQUS/CAE, will allow you to select and name groups ofentities, such as nodes and elements, as you build the model; when the ABAQUS input file iscreated, node and element sets are generated from these groups.

You define sets using the *NSET and *ELSET options in the input file. The name of a set isspecified with either the NSET or ELSET parameter. The data lines list the nodes or elementsthat are included in the set. Each data line can contain up to 16 numbers, and there can be asmany data lines as required. For example, the node set LHEND (see Figure 4-16) can bedefined as

*NSET, NSET=LHEND

3241, 3243, 3245, 3247, 3249, 3251, 3253, 3255, 3257,

8241, 8245, 8249, 8253, 8257, 13241, 13243, 13245, 13247,

13249, 13251, 13253, 13255, 13257, 18241, 18245, 18249, 18253,


In input files created by a preprocessor, the model's nodal coordinates usually are in one large*NODE option block, with the coordinates specified for each node individually.

The element definitions generated by the preprocessor usually are contained in several*ELEMENT option blocks. Typically, each block contains elements that have the sameelement section and material properties. In the connecting lug model only one element type isused, and all the elements have the same properties. Therefore, there will probably be a single*ELEMENT option block in your input file. It will look similar to

*ELEMENT, TYPE=C3D20R, ELSET=LUG

1, 1, 401, 405, 5, 10001, 10401, 10405, 10005,

201, 403, 205, 3, 10201, 10403, 10205, 10003,

5001, 5401, 5405, 5005

2, 5, 405, 409, 9, 10005, 10405, 10409, 10009,

205, 407, 209, 7, 10205, 10407, 10209, 10007,

5005, 5405, 5409, 5009

.......

It takes three data lines to define the connectivity of one C3D20R element completely. If adata line in an *ELEMENT option block ends with a comma, it indicates that the next dataline contains more nodes defining the current element. The parameter ELSET=LUG indicatesthat all the elements defined in the following data lines will be stored in an element set calledLUG. If your model does not have a descriptive element set name in the *ELEMENT option,change it to LUG.

Node and element sets

The node and element sets are important components of an ABAQUS input file because theyallow you to assign loads, boundary conditions, and material properties efficiently. They alsooffer great flexibility in defining the output that your simulation will produce and make itmuch easier to understand the input file.

Some preprocessors, such as ABAQUS/CAE, will allow you to select and name groups ofentities, such as nodes and elements, as you build the model; when the ABAQUS input file iscreated, node and element sets are generated from these groups.

You define sets using the *NSET and *ELSET options in the input file. The name of a set isspecified with either the NSET or ELSET parameter. The data lines list the nodes or elementsthat are included in the set. Each data line can contain up to 16 numbers, and there can be asmany data lines as required. For example, the node set LHEND (see Figure 4-16) can bedefined as

*NSET, NSET=LHEND

3241, 3243, 3245, 3247, 3249, 3251, 3253, 3255, 3257,

8241, 8245, 8249, 8253, 8257, 13241, 13243, 13245, 13247,

13249, 13251, 13253, 13255, 13257, 18241, 18245, 18249, 18253,


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18257, 23241, 23243, 23245, 23247, 23249, 23251, 23253, 23255,

23257

If you are adding a node or element set to the input file with an editor and the identificationnumbers follow a regular pattern, the GENERATE parameter allows a range of nodes to beincluded by specifying the beginning node number, ending node number, and the increment innode numbers. For example, the node set LHEND could be defined as follows:

*NSET, NSET=LHEND, GENERATE

3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

18241, 18257, 4

23241, 23257, 2

Sets can also be created by referring to other sets. If the preprocessor that you used did notcreate the element set BUILTIN or the node set HOLEBOT that are shown in Figure 4-16, addthem to your input file using an editor; they will be essential in limiting the output during thesimulation. You should also create the element set PRESS shown in Figure 4-16. Rememberto use the node and element numbers in your model and not those shown in the figure.

Section properties

Look up the C3D20R element in Chapter 14, "Continuum Elements," of theABAQUS/Standard User's Manual to determine the correct element section option and therequired data that must be specified for this element. You will discover that the C3D20Relement uses the *SOLID SECTION option to define the element's section properties. Becausethis is a three-dimensional element, ABAQUS needs no additional geometric data for theelement section.

Element set LUG contains all the elements, so add this element section option to your model


If you did not define an element set with the name LUG, use the name of whatever element setcontains all the elements in your model as the value of the ELSET parameter. The materialproperty definition in the model will have the name STEEL.

Materials

The connecting lug is made of a mild steel and, thus, has isotropic, linear elastic materialbehavior under the loads being applied. Assume that E = 200 GPa and that º = 0.3. These aregiven on the data line of the *ELASTIC option (remember the overhead hoist example inChapter 2, "ABAQUS Basics"). Put the following material property definition in your inputfile:


*ELASTIC

200.E9, 0.3

18257, 23241, 23243, 23245, 23247, 23249, 23251, 23253, 23255,

23257

If you are adding a node or element set to the input file with an editor and the identificationnumbers follow a regular pattern, the GENERATE parameter allows a range of nodes to beincluded by specifying the beginning node number, ending node number, and the increment innode numbers. For example, the node set LHEND could be defined as follows:


3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

18241, 18257, 4

23241, 23257, 2

Sets can also be created by referring to other sets. If the preprocessor that you used did notcreate the element set BUILTIN or the node set HOLEBOT that are shown in Figure 4-16, addthem to your input file using an editor; they will be essential in limiting the output during thesimulation. You should also create the element set PRESS shown in Figure 4-16. Rememberto use the node and element numbers in your model and not those shown in the figure.

Section properties

Look up the C3D20R element in Chapter 14, "Continuum Elements," of theABAQUS/Standard User's Manual to determine the correct element section option and therequired data that must be specified for this element. You will discover that the C3D20Relement uses the *SOLID SECTION option to define the element's section properties. Becausethis is a three-dimensional element, ABAQUS needs no additional geometric data for theelement section.

Element set LUG contains all the elements, so add this element section option to your model


If you did not define an element set with the name LUG, use the name of whatever element setcontains all the elements in your model as the value of the ELSET parameter. The materialproperty definition in the model will have the name STEEL.

Materials

The connecting lug is made of a mild steel and, thus, has isotropic, linear elastic materialbehavior under the loads being applied. Assume that E = 200 GPa and that º = 0.3. These aregiven on the data line of the *ELASTIC option (remember the overhead hoist example inChapter 2, "ABAQUS Basics"). Put the following material property definition in your inputfile:


*ELASTIC

200.E9, 0.3


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The value for the NAME parameter on the *MATERIAL option must match the value of theMATERIAL parameter on the *SOLID SECTION option.

4.3.5 Reviewing the input file--the history dataThe history data portion of the input file starts at the first *STEP option. It is quite likely that thepreprocessor you used generated the following input options in your model:

*STEP, PERTURBATION

<possibly a title describing this step>

*STATIC

Many preprocessors create a linear static step in the input file by default. If these options are not inyour input file, add them at the end of the existing data. It is easier for someone else to understand yourmodel if you use the data lines following the *STEP option to add a suitable title describing the eventbeing simulated in the step.

Boundary conditions

In the model of the connecting lug, all the nodes need to be constrained in all three directionsat the left-hand end where it is attached to its parent structure (see Figure 4-19).

Figure 4-19 Boundary conditions on the connecting lug.

If you defined the boundary conditions in a preprocessor such as ABAQUS/CAE, eachconstrained degree of freedom is specified individually in the *BOUNDARY option block, asshown below:

*BOUNDARY

3241, 1,1

3241, 2,2

3241, 3,3

......

If a large number of nodes are constrained, these data can occupy a great deal of space in thecomputer's memory. Where a number of nodes all have the same boundary conditions, it ismore efficient to apply the constraints directly to a node set containing all the nodes. Thus, in

The value for the NAME parameter on the *MATERIAL option must match the value of theMATERIAL parameter on the *SOLID SECTION option.

4.3.5 Reviewing the input file--the history dataThe history data portion of the input file starts at the first *STEP option. It is quite likely that thepreprocessor you used generated the following input options in your model:

*STEP, PERTURBATION

<possibly a title describing this step>

*STATIC

Many preprocessors create a linear static step in the input file by default. If these options are not inyour input file, add them at the end of the existing data. It is easier for someone else to understand yourmodel if you use the data lines following the *STEP option to add a suitable title describing the eventbeing simulated in the step.

Boundary conditions

In the model of the connecting lug, all the nodes need to be constrained in all three directionsat the left-hand end where it is attached to its parent structure (see Figure 4-19).

Figure 4-19 Boundary conditions on the connecting lug.

If you defined the boundary conditions in a preprocessor such as ABAQUS/CAE, eachconstrained degree of freedom is specified individually in the *BOUNDARY option block, asshown below:

*BOUNDARY

3241, 1,1

3241, 2,2

3241, 3,3

......

If a large number of nodes are constrained, these data can occupy a great deal of space in thecomputer's memory. Where a number of nodes all have the same boundary conditions, it ismore efficient to apply the constraints directly to a node set containing all the nodes. Thus, in


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the lug model we prefer to create the node set LHEND to specify the constraints:

*BOUNDARY

LHEND, ENCASTRE

If you think that you defined the boundary conditions incorrectly, you can display them inABAQUS/Viewer and compare them with the boundary conditions shown in Figure 4-19. Thepostprocessing instructions given at the end of ``Postprocessing with ABAQUS/Viewer,''Section 2.7, discuss how to do this.

Loading

The lug carries a pressure of 50 MPa distributed around the bottom-half of the hole. Pressureloads can be defined conveniently using the preprocessor by selecting the element faces towhich the load is applied. In the connecting lug input file, these loads will appear as a*DLOAD option block. For example, the *DLOAD option block for the connecting lug maylook like

*DLOAD

1, P6, 5.E+07

2, P6, 5.E+07

3, P6, 5.E+07

4, P6, 5.E+07

13, P6, 5.E+07

...

1015, P6, 5.E+07

1016, P6, 5.E+07

The format of each data line is

<element or element set name>, <load ID>, <load magnitude>

In this case the "load ID" consists of the letter "P" followed by the number of the element faceto which pressure is applied. The face numbers depend on the connectivity of the element andare defined for each element type in the ABAQUS/Standard User's Manual. For thethree-dimensional hexahedral elements used in this example, the face numbers are shown inFigure 4-20.

Figure 4-20 Face numbers on hexahedral elements.

the lug model we prefer to create the node set LHEND to specify the constraints:

*BOUNDARY

LHEND, ENCASTRE

If you think that you defined the boundary conditions incorrectly, you can display them inABAQUS/Viewer and compare them with the boundary conditions shown in Figure 4-19. Thepostprocessing instructions given at the end of ``Postprocessing with ABAQUS/Viewer,''Section 2.7, discuss how to do this.

Loading

The lug carries a pressure of 50 MPa distributed around the bottom-half of the hole. Pressureloads can be defined conveniently using the preprocessor by selecting the element faces towhich the load is applied. In the connecting lug input file, these loads will appear as a*DLOAD option block. For example, the *DLOAD option block for the connecting lug maylook like

*DLOAD

1, P6, 5.E+07

2, P6, 5.E+07

3, P6, 5.E+07

4, P6, 5.E+07

13, P6, 5.E+07

...

1015, P6, 5.E+07

1016, P6, 5.E+07

The format of each data line is

<element or element set name>, <load ID>, <load magnitude>

In this case the "load ID" consists of the letter "P" followed by the number of the element faceto which pressure is applied. The face numbers depend on the connectivity of the element andare defined for each element type in the ABAQUS/Standard User's Manual. For thethree-dimensional hexahedral elements used in this example, the face numbers are shown inFigure 4-20.

Figure 4-20 Face numbers on hexahedral elements.


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In the model, as defined in ``Connecting lug,'' Section A.2, the pressure is applied to face 6 ofthe elements around the bottom of the hole, so the load ID is "P6."

For meshes generated with a preprocessor, the face numbers and, hence, the load IDs willdepend on how the mesh is generated. Some preprocessors, such as ABAQUS/CAE, candetermine the correct load ID automatically; this makes it very easy to specify pressure loadson complicated meshes. However, this method tends to produce long lists of data lines in theinput file. In models where the same load ID and load magnitude are used for each element,you can use an element set--which is more efficient--to apply the pressure loads. For example,in this model the *DLOAD option block may appear as

*DLOAD

PRESS, P6, 5.E+07

where we have made use of the element set PRESS whose members are shown in Figure 4-16.

Output requests

By default, many preprocessors create an ABAQUS input file that has a large number ofoutput request options. These requests are in addition to the output database file request that isgenerated automatically by ABAQUS. If, when you edit your input file, you find that theseadditional output options were created, delete them because they will generally generate toomuch unnecessary output.

You were asked to determine the deflection of the connecting lug when the load is applied. Asimple method for obtaining this result is to print out all the displacements in the model.However, it is likely that the location on the lug with the largest deflection is probably goingto be on the bottom of the hole, where the load is applied. Furthermore, only the displacementin the 2-direction (U2) is going to be of interest. You should have created a node set,HOLEBOT, containing those nodes. Use that set to limit the requested displacements to justthose five nodes at the bottom of the hole and to limit the output to just the verticaldisplacements.

*NODE PRINT, NSET=HOLEBOT

U2,

It is good practice to check that the reaction forces at the constraints balance the applied loads.All the reaction forces at a node can be printed by specifying the variable RF. We again usethe node set LHEND to limit the output to those nodes that are constrained.

*NODE PRINT, NSET=LHEND, TOTAL=YES, SUMMARY=NO

RF,

You can define several *NODE PRINT and *EL PRINT options. The parameter TOTAL=YEScauses the sum of the reaction forces at all the nodes in the node set to be printed. TheSUMMARY=NO parameter prevents the minimum and maximum values in the table frombeing printed.

Add the following commands to print the stress tensor (variable S) and the von Mises stress

In the model, as defined in ``Connecting lug,'' Section A.2, the pressure is applied to face 6 ofthe elements around the bottom of the hole, so the load ID is "P6."

For meshes generated with a preprocessor, the face numbers and, hence, the load IDs willdepend on how the mesh is generated. Some preprocessors, such as ABAQUS/CAE, candetermine the correct load ID automatically; this makes it very easy to specify pressure loadson complicated meshes. However, this method tends to produce long lists of data lines in theinput file. In models where the same load ID and load magnitude are used for each element,you can use an element set--which is more efficient--to apply the pressure loads. For example,in this model the *DLOAD option block may appear as

*DLOAD

PRESS, P6, 5.E+07

where we have made use of the element set PRESS whose members are shown in Figure 4-16.

Output requests

By default, many preprocessors create an ABAQUS input file that has a large number ofoutput request options. These requests are in addition to the output database file request that isgenerated automatically by ABAQUS. If, when you edit your input file, you find that theseadditional output options were created, delete them because they will generally generate toomuch unnecessary output.

You were asked to determine the deflection of the connecting lug when the load is applied. Asimple method for obtaining this result is to print out all the displacements in the model.However, it is likely that the location on the lug with the largest deflection is probably goingto be on the bottom of the hole, where the load is applied. Furthermore, only the displacementin the 2-direction (U2) is going to be of interest. You should have created a node set,HOLEBOT, containing those nodes. Use that set to limit the requested displacements to justthose five nodes at the bottom of the hole and to limit the output to just the verticaldisplacements.


U2,

It is good practice to check that the reaction forces at the constraints balance the applied loads.All the reaction forces at a node can be printed by specifying the variable RF. We again usethe node set LHEND to limit the output to those nodes that are constrained.


RF,

You can define several *NODE PRINT and *EL PRINT options. The parameter TOTAL=YEScauses the sum of the reaction forces at all the nodes in the node set to be printed. TheSUMMARY=NO parameter prevents the minimum and maximum values in the table frombeing printed.

Add the following commands to print the stress tensor (variable S) and the von Mises stress


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(variable MISES) for the elements at the constrained end (element set BUILTIN):

*EL PRINT, ELSET=BUILTIN

S, MISES

Indicate the end of a step with the option

*END STEP

Make sure that this input option is the last option in your model.

4.3.6 Running the analysisStore the input in a file called lug.inp (an example file is listed in ``Connecting lug,'' Section A.2).Then, run the simulation using the command:

abaqus job=lug interactive

When the job has completed, check the data file, lug.dat , for any errors or warnings. If there areany errors, correct the input file and run the simulation again. If you have problems correcting anyerrors, try comparing your input file to the one given in ``Connecting lug,'' Section A.2. Check that youhave the correct parameters for each input option.

4.3.7 ResultsWhen the job has completed successfully, look at the three tables of output that you requested. Theywill be found at the end of the data file. A portion of the table of element stresses is shown in Figure4-21. The maximum von Mises stress at the built-in end is 306 MPa.

Figure 4-21 Table of element stresses in the data file.

(variable MISES) for the elements at the constrained end (element set BUILTIN):


S, MISES

Indicate the end of a step with the option

*END STEP

Make sure that this input option is the last option in your model.

4.3.6 Running the analysisStore the input in a file called lug.inp (an example file is listed in ``Connecting lug,'' Section A.2).Then, run the simulation using the command:

abaqus job=lug interactive

When the job has completed, check the data file, lug.dat , for any errors or warnings. If there areany errors, correct the input file and run the simulation again. If you have problems correcting anyerrors, try comparing your input file to the one given in ``Connecting lug,'' Section A.2. Check that youhave the correct parameters for each input option.

4.3.7 ResultsWhen the job has completed successfully, look at the three tables of output that you requested. Theywill be found at the end of the data file. A portion of the table of element stresses is shown in Figure4-21. The maximum von Mises stress at the built-in end is 306 MPa.

Figure 4-21 Table of element stresses in the data file.


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The tables showing the displacements of the nodes along the bottom of the hole and the reaction forcesat the constrained nodes are shown in Figure 4-22 and Figure 4-23, respectively.

Figure 4-22 Table of nodal displacements in the data file.

Figure 4-23 Table of reaction forces in the data file.

The tables showing the displacements of the nodes along the bottom of the hole and the reaction forcesat the constrained nodes are shown in Figure 4-22 and Figure 4-23, respectively.

Figure 4-22 Table of nodal displacements in the data file.

Figure 4-23 Table of reaction forces in the data file.


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The bottom of the hole in the lug has displaced about 0.3 mm. The total reaction force in the2-direction at the constrained nodes is equal and opposite to the applied load in that direction of -30kN.

4.3.8 PostprocessingOnce you have looked at the results in the data file, start ABAQUS/Viewer by typing

abaqus viewer odb=lug


Plotting the deformed shape

To begin this exercise, plot the deformed model shape.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.Figure 4-24 displays the deformed model shape at the end of the analysis. What is thedisplacement magnification level?

The bottom of the hole in the lug has displaced about 0.3 mm. The total reaction force in the2-direction at the constrained nodes is equal and opposite to the applied load in that direction of -30kN.

4.3.8 PostprocessingOnce you have looked at the results in the data file, start ABAQUS/Viewer by typing

abaqus viewer odb=lug


Plotting the deformed shape

To begin this exercise, plot the deformed model shape.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.Figure 4-24 displays the deformed model shape at the end of the analysis. What is thedisplacement magnification level?


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Figure 4-24 Deformed model shape of connecting lug (wireframe).

Changing the view

The default view is isometric. You can change the view using the options in the View menu orthe view tools in the toolbar.

To rotate the view:

1. From the main menu bar, select View->Rotate; or use the tool from the toolbar.

2. Drag the mouse over the virtual trackball in the viewport.

The view rotates interactively.

You can also specify a view by entering values for rotation angles, viewpoint, zoom factor, orfraction of viewport to pan.

To specify the view:

1. From the main menu bar, select View->Specify.

The Specify View dialog box appears.

2. From the list of available methods, select Viewpoint.

In the Viewpoint method, you enter three values representing the X-, Y-, and Z-position ofan observer. You can also specify an up vector. ABAQUS positions your model so thatthis vector points upward.

3. Enter the X-, Y-, and Z-coordinates of the viewpoint vector as 1, 1, 3 and thecoordinates of the up vector as 0, 1, 0.

Figure 4-24 Deformed model shape of connecting lug (wireframe).

Changing the view

The default view is isometric. You can change the view using the options in the View menu orthe view tools in the toolbar.

To rotate the view:



The view rotates interactively.

You can also specify a view by entering values for rotation angles, viewpoint, zoom factor, orfraction of viewport to pan.





In the Viewpoint method, you enter three values representing the X-, Y-, and Z-position ofan observer. You can also specify an up vector. ABAQUS positions your model so thatthis vector points upward.



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4. Click OK.

ABAQUS/Viewer displays your model in the specified view, as shown in Figure 4-25.

Figure 4-25 Deformed model shape viewed from specified viewpoint.

Visible edges

Several options are available for choosing which edges will be visible in the model display.The previous plots show only feature edges of the model; Figure 4-26 displays all elementedges.

Figure 4-26 Deformed shape with all edges visible.

To display all element edges:


4. Click OK.


Figure 4-25 Deformed model shape viewed from specified viewpoint.

Visible edges

Several options are available for choosing which edges will be visible in the model display.The previous plots show only feature edges of the model; Figure 4-26 displays all elementedges.

Figure 4-26 Deformed shape with all edges visible.

To display all element edges:



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The Deformed Plot Options dialog box appears.

2. Click the Basic tab if it is not already selected.

3. From the Visible Edges options, choose All.

4. Click OK.

The deformed plot in the current viewport changes to display all element edges, as shownin Figure 4-26.

Render style

A wireframe model showing internal edges can be visually confusing for complexthree-dimensional models. Three other render styles provide additional display options: hiddenline, filled, and shaded. You can select a render style from the Options menu on the main

menu bar or from the render style tools on the toolbar: wireframe , hidden line , filled

, and shaded . To display the hidden line plot shown in Figure 4-27, select hidden

line plotting for the deformed plot mode by clicking the tool. Deformed shape plots willbe displayed in the hidden line render style until you select another type of display.

Figure 4-27 Hidden line plot.

You can use the other render style tools to select filled and shaded render styles, shown inFigure 4-28 and Figure 4-29, respectively.

Figure 4-28 Filled element plot.


2. Click the Basic tab if it is not already selected.

3. From the Visible Edges options, choose All.

4. Click OK.

The deformed plot in the current viewport changes to display all element edges, as shownin Figure 4-26.

Render style

A wireframe model showing internal edges can be visually confusing for complexthree-dimensional models. Three other render styles provide additional display options: hiddenline, filled, and shaded. You can select a render style from the Options menu on the main

menu bar or from the render style tools on the toolbar: wireframe , hidden line , filled

, and shaded . To display the hidden line plot shown in Figure 4-27, select hidden

line plotting for the deformed plot mode by clicking the tool. Deformed shape plots willbe displayed in the hidden line render style until you select another type of display.

Figure 4-27 Hidden line plot.

You can use the other render style tools to select filled and shaded render styles, shown inFigure 4-28 and Figure 4-29, respectively.

Figure 4-28 Filled element plot.


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A filled plot colors the element faces a solid color. A shaded plot is a filled plot in which alightsource appears to be directed at the model. These render styles can be very useful whenviewing complex three-dimensional models.

Figure 4-29 Shaded plot.

Contour plots

Contour plots display the variation of a variable across the surface of a model. You can createfilled or shaded contour plots of field output results from the output database.

To generate a contour plot of the von Mises stress:

1. From the main menu bar, select Plot->Contours.

The filled contour plot shown in Figure 4-30 appears.

A filled plot colors the element faces a solid color. A shaded plot is a filled plot in which alightsource appears to be directed at the model. These render styles can be very useful whenviewing complex three-dimensional models.

Figure 4-29 Shaded plot.

Contour plots

Contour plots display the variation of a variable across the surface of a model. You can createfilled or shaded contour plots of field output results from the output database.

To generate a contour plot of the von Mises stress:

1. From the main menu bar, select Plot->Contours.

The filled contour plot shown in Figure 4-30 appears.


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The von Mises stress, S Mises, indicated in the legend title is the default variablechosen by ABAQUS for this analysis. You can select a different variable to plot.

Figure 4-30 Filled contour plot of von Mises stress.

2. From the main menu bar, select Result->Field Output.

The Field Output dialog box appears; by default, the Primary Variable tab is selected.

3. From the list of available output variables, select a new variable to plot.

4. Click OK.

The contour plot in the current viewport changes to reflect your selection.

ABAQUS/Viewer offers many options to customize contour plots. To see the availableoptions, click Contour Options in the prompt area. By default, ABAQUS/Viewerautomatically computes the minimum and maximum values shown in your contour plots andevenly divides the range between these values into 12 intervals. You can control the minimumand maximum values ABAQUS/Viewer displays (for example, to examine variations within afixed set of bounds), as well as the number of intervals.

To generate a customized contour plot:

1. In the Contour Limits field of the Contour Plot Options dialog box, choose Specifybeside Max; then enter a maximum value of 300E+6.

2. Choose Specify beside Min; then enter a minimum value of 60E+6.

3. Drag the Contour Intervals slider to change the number of intervals to nine. Select

The von Mises stress, S Mises, indicated in the legend title is the default variablechosen by ABAQUS for this analysis. You can select a different variable to plot.

Figure 4-30 Filled contour plot of von Mises stress.



3. From the list of available output variables, select a new variable to plot.

4. Click OK.

The contour plot in the current viewport changes to reflect your selection.

ABAQUS/Viewer offers many options to customize contour plots. To see the availableoptions, click Contour Options in the prompt area. By default, ABAQUS/Viewerautomatically computes the minimum and maximum values shown in your contour plots andevenly divides the range between these values into 12 intervals. You can control the minimumand maximum values ABAQUS/Viewer displays (for example, to examine variations within afixed set of bounds), as well as the number of intervals.

To generate a customized contour plot:

1. In the Contour Limits field of the Contour Plot Options dialog box, choose Specifybeside Max; then enter a maximum value of 300E+6.

2. Choose Specify beside Min; then enter a minimum value of 60E+6.

3. Drag the Contour Intervals slider to change the number of intervals to nine. Select


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Exterior visible edges.

4. Click Apply.

ABAQUS/Viewer displays your model with the specified contour option settings, asshown in Figure 4-31. These settings remain in effect for all subsequent contour plots untilyou change them or reset them to their default values.

Figure 4-31 Customized plot of von Mises stress.

Maximum and minimum values

The maximum and minimum values of a variable in a model can be determined easily.

To report the minimum and maximum values of a contour variable:

1. From the main menu bar, select View->Viewport Annotations; then click the Legend tabin the dialog box that appears.

The Legend options become available.

2. Toggle on Show min/max values.

3. Click OK.

The contour legend changes to report the minimum and maximum contour values.

Based on the contour legend in your plots, what is the maximum value of von Mises stress inthe model? How does it compare to the value reported in the data (.dat) file? Do the twomaximum values correspond to the same point in the model? The von Mises stresses shown inthe contour plot have been extrapolated to the nodes, whereas the stresses written to the data

Exterior visible edges.

4. Click Apply.

ABAQUS/Viewer displays your model with the specified contour option settings, asshown in Figure 4-31. These settings remain in effect for all subsequent contour plots untilyou change them or reset them to their default values.

Figure 4-31 Customized plot of von Mises stress.

Maximum and minimum values

The maximum and minimum values of a variable in a model can be determined easily.

To report the minimum and maximum values of a contour variable:

1. From the main menu bar, select View->Viewport Annotations; then click the Legend tabin the dialog box that appears.


2. Toggle on Show min/max values.

3. Click OK.

The contour legend changes to report the minimum and maximum contour values.

Based on the contour legend in your plots, what is the maximum value of von Mises stress inthe model? How does it compare to the value reported in the data (.dat) file? Do the twomaximum values correspond to the same point in the model? The von Mises stresses shown inthe contour plot have been extrapolated to the nodes, whereas the stresses written to the data


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file for this problem correspond to the element integration points. Therefore, the location ofthe maximum von Mises stress in the data file is not exactly the same as the location of themaximum von Mises stress in the contour plot. This difference can be resolved by requestingthat stress output at the nodes (extrapolated from the element integration points and averagedover all elements containing a given node) be written to the data file. If the difference is largeenough to be of concern, this is an indication that the mesh may be too coarse.

One of the goals of this workshop was to determine the deflection of the lug in the negative2-direction. You will contour the displacement component of the lug in the 2-direction todetermine its peak displacement in the vertical direction. In the Contour Plot Options dialogbox, click Defaults to reset the minimum and maximum contour values and the number ofintervals to their default values before proceeding.

To contour the displacement of the connecting lug in the 2-direction:



2. From the list of available output variables, select U.

3. From the list of available components, select U2.

4. Click OK.

What is the maximum displacement value in the negative 2-direction? Is the value the same aswas given in the data file? Which do you think is a more appropriate value to report for thisapplication (remember what you were asked to determine from this simulation)?

Displaying a subset of the model

By default, ABAQUS/Viewer displays your entire model; however, you can choose to displaya subset of your model called a display group. This subset can contain any combination ofnodes, elements, and surfaces from the current output database. For the connecting lug modelyou will create a display group consisting of the elements to which the pressure load wasapplied.

To display a subset of the model:

1. From the main menu bar, select Tools->Display Group->Create.

The Create Display Group dialog box opens.

2. From the ODB Item list, select Elements. From the Selection Method list, selectElement sets.

Once you have selected these items, the list on the right hand side of the Create DisplayGroup dialog box shows the available selections.

3. From this list, choose element set PRESS. Toggle on Highlight items in viewport below

file for this problem correspond to the element integration points. Therefore, the location ofthe maximum von Mises stress in the data file is not exactly the same as the location of themaximum von Mises stress in the contour plot. This difference can be resolved by requestingthat stress output at the nodes (extrapolated from the element integration points and averagedover all elements containing a given node) be written to the data file. If the difference is largeenough to be of concern, this is an indication that the mesh may be too coarse.

One of the goals of this workshop was to determine the deflection of the lug in the negative2-direction. You will contour the displacement component of the lug in the 2-direction todetermine its peak displacement in the vertical direction. In the Contour Plot Options dialogbox, click Defaults to reset the minimum and maximum contour values and the number ofintervals to their default values before proceeding.

To contour the displacement of the connecting lug in the 2-direction:



2. From the list of available output variables, select U.

3. From the list of available components, select U2.

4. Click OK.

What is the maximum displacement value in the negative 2-direction? Is the value the same aswas given in the data file? Which do you think is a more appropriate value to report for thisapplication (remember what you were asked to determine from this simulation)?

Displaying a subset of the model

By default, ABAQUS/Viewer displays your entire model; however, you can choose to displaya subset of your model called a display group. This subset can contain any combination ofnodes, elements, and surfaces from the current output database. For the connecting lug modelyou will create a display group consisting of the elements to which the pressure load wasapplied.

To display a subset of the model:


The Create Display Group dialog box opens.

2. From the ODB Item list, select Elements. From the Selection Method list, selectElement sets.

Once you have selected these items, the list on the right hand side of the Create DisplayGroup dialog box shows the available selections.

3. From this list, choose element set PRESS. Toggle on Highlight items in viewport below


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the list.

The outlines of the elements in element set PRESS are highlighted in red.

4. Click Replace to replace the current model display with the element set PRESS.

ABAQUS/Viewer displays the specified subset of your model.

When creating an ABAQUS model, you may need to determine the face labels for a solidelement so that you can determine the correct load ID when applying pressure loads or whendefining surfaces for contact. If you use a preprocessor to generate the mesh, you may not besure of the connectivity or orientation of the elements. In such situations you can useABAQUS/Viewer to display the mesh after you have run a datacheck analysis that creates anoutput database file.

To display the face identification labels and element numbers on the undeformed modelshape:



2. Change the render style to filled since face labels cannot be displayed when the renderstyle is set to wireframe; all visible element edges will be displayed for convenience.

a. Toggle on Filled under Render Style.

b. Toggle on All under Visible Edges.

3. Click the Labels tab, and toggle on Show element labels and Show face labels.

4. Click OK to apply the plot options and to close the dialog box.

5. From the main menu bar, select Plot->Undeformed Shape.

ABAQUS/Viewer displays the element and face identification labels in element setPRESS.

4.4 Mesh convergenceIt is important that you use a sufficiently refined mesh to ensure that the results from your ABAQUSsimulation are adequate. Coarse meshes can yield inaccurate results. The numerical solution providedby your model will tend toward a unique value as you increase the mesh density. The computerresources required to run your simulation also increase as the mesh is refined. The mesh is said to beconverged when further mesh refinement produces a negligible change in the solution.

As you gain experience, you will learn to judge what level of refinement produces a suitable mesh togive acceptable results for most simulations. However, it is always good practice to perform a meshconvergence study, where you simulate the same problem with a finer mesh and compare the results.

the list.

The outlines of the elements in element set PRESS are highlighted in red.

4. Click Replace to replace the current model display with the element set PRESS.

ABAQUS/Viewer displays the specified subset of your model.

When creating an ABAQUS model, you may need to determine the face labels for a solidelement so that you can determine the correct load ID when applying pressure loads or whendefining surfaces for contact. If you use a preprocessor to generate the mesh, you may not besure of the connectivity or orientation of the elements. In such situations you can useABAQUS/Viewer to display the mesh after you have run a datacheck analysis that creates anoutput database file.

To display the face identification labels and element numbers on the undeformed modelshape:



2. Change the render style to filled since face labels cannot be displayed when the renderstyle is set to wireframe; all visible element edges will be displayed for convenience.

a. Toggle on Filled under Render Style.

b. Toggle on All under Visible Edges.

3. Click the Labels tab, and toggle on Show element labels and Show face labels.

4. Click OK to apply the plot options and to close the dialog box.

5. From the main menu bar, select Plot->Undeformed Shape.

ABAQUS/Viewer displays the element and face identification labels in element setPRESS.

4.4 Mesh convergenceIt is important that you use a sufficiently refined mesh to ensure that the results from your ABAQUSsimulation are adequate. Coarse meshes can yield inaccurate results. The numerical solution providedby your model will tend toward a unique value as you increase the mesh density. The computerresources required to run your simulation also increase as the mesh is refined. The mesh is said to beconverged when further mesh refinement produces a negligible change in the solution.

As you gain experience, you will learn to judge what level of refinement produces a suitable mesh togive acceptable results for most simulations. However, it is always good practice to perform a meshconvergence study, where you simulate the same problem with a finer mesh and compare the results.


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You can have confidence that your model is producing a mathematically accurate solution if the twomeshes give essentially the same result.

The connecting lug is analyzed using four different mesh densities (Figure 4-32) as an example of amesh refinement study. The number of elements used in each mesh is indicated in the figure.

Figure 4-32 Different meshes for the connecting lug problem.

We consider the influence of the mesh density on three particular results from this model:

� The displacement of the bottom of the hole.

� The peak von Mises stress at the stress concentration on the bottom surface of the hole.

� The peak von Mises stress where the lug is attached to the parent structure.

These effects are shown in Figure 4-33. The results for each of the four mesh densities are compared inTable 4-3, along with the CPU time required to run each simulation.

Figure 4-33 Annotated contour plot of von Mises stress.

You can have confidence that your model is producing a mathematically accurate solution if the twomeshes give essentially the same result.

The connecting lug is analyzed using four different mesh densities (Figure 4-32) as an example of amesh refinement study. The number of elements used in each mesh is indicated in the figure.

Figure 4-32 Different meshes for the connecting lug problem.

We consider the influence of the mesh density on three particular results from this model:

� The displacement of the bottom of the hole.

� The peak von Mises stress at the stress concentration on the bottom surface of the hole.

� The peak von Mises stress where the lug is attached to the parent structure.

These effects are shown in Figure 4-33. The results for each of the four mesh densities are compared inTable 4-3, along with the CPU time required to run each simulation.

Figure 4-33 Annotated contour plot of von Mises stress.


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Table 4-3. Results of mesh refinement study.Mesh Displacement of

bottom of holeStress at bottom

of holeStress at

attachmentRelative

CPU timeCoarse 2.01E-4 180.E6 205.E6 0.26Normal 3.13E-4 311.E6 365.E6 1.0Fine 3.14E-4 332.E6 426.E6 2.7Very fine 3.15E-4 345.E6 496.E6 22.5

The coarse mesh predicts inaccurate displacements at the bottom of hole, but the normal, fine, and veryfine meshes all predict similar results. The normal mesh is, therefore, converged as far as thedisplacements are concerned. The convergence of the results is plotted in Figure 4-34.

Figure 4-34 Convergence of results in mesh refinement study.

Table 4-3. Results of mesh refinement study.Mesh Displacement of

bottom of holeStress at bottom

of holeStress at

attachmentRelative

CPU timeCoarse 2.01E-4 180.E6 205.E6 0.26Normal 3.13E-4 311.E6 365.E6 1.0Fine 3.14E-4 332.E6 426.E6 2.7Very fine 3.15E-4 345.E6 496.E6 22.5

The coarse mesh predicts inaccurate displacements at the bottom of hole, but the normal, fine, and veryfine meshes all predict similar results. The normal mesh is, therefore, converged as far as thedisplacements are concerned. The convergence of the results is plotted in Figure 4-34.

Figure 4-34 Convergence of results in mesh refinement study.


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All the results are normalized with respect to the values predicted by the coarse mesh. The peak stresson the bottom of the hole converges much more slowly than the displacements because stress andstrain are calculated from the displacement gradients; thus, a much finer mesh is required to predictaccurate displacement gradients than is needed to calculate accurate displacements.

Mesh refinement significantly changes the stress calculated at the attachment of the connecting lug; itcontinues to increase with continued mesh refinement. A stress singularity exists at the corner of thelug where it attaches to the parent structure. Theoretically the stress is infinite at this location;therefore, increasing the mesh density will not produce a converged stress value at this location. Thissingularity occurs because of the idealizations used in the finite element model. The connectionbetween the lug and the parent structure has been modeled as a sharp corner, and the parent structurehas been modeled as rigid. These idealizations lead to the stress singularity. In reality there probablywill be a small fillet between the lug and the parent structure, and the parent structure will bedeformable, not rigid. If the exact stress in this location is required, the fillet between the componentsmust be modeled accurately (see Figure 4-35) and the stiffness of the parent structure must also beconsidered.

Figure 4-35 Idealizing a fillet as a sharp corner.

All the results are normalized with respect to the values predicted by the coarse mesh. The peak stresson the bottom of the hole converges much more slowly than the displacements because stress andstrain are calculated from the displacement gradients; thus, a much finer mesh is required to predictaccurate displacement gradients than is needed to calculate accurate displacements.

Mesh refinement significantly changes the stress calculated at the attachment of the connecting lug; itcontinues to increase with continued mesh refinement. A stress singularity exists at the corner of thelug where it attaches to the parent structure. Theoretically the stress is infinite at this location;therefore, increasing the mesh density will not produce a converged stress value at this location. Thissingularity occurs because of the idealizations used in the finite element model. The connectionbetween the lug and the parent structure has been modeled as a sharp corner, and the parent structurehas been modeled as rigid. These idealizations lead to the stress singularity. In reality there probablywill be a small fillet between the lug and the parent structure, and the parent structure will bedeformable, not rigid. If the exact stress in this location is required, the fillet between the componentsmust be modeled accurately (see Figure 4-35) and the stiffness of the parent structure must also beconsidered.

Figure 4-35 Idealizing a fillet as a sharp corner.


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It is common to omit small details like fillet radii from a finite element model to simplify the analysisand to keep the model size reasonable. However, the introduction of any sharp corner into a model willlead to a stress singularity at that location. This normally has a negligible effect on the overall responseof the model, but the predicted stresses close to the singularity will be inaccurate.

For complex, three-dimensional simulations the available computer resources often dictate a practicallimit on the mesh density that you can use. In this case you must use the results from the analysiscarefully. Coarse meshes are often adequate to predict trends and to compare how different conceptsbehave relative to each other. However, you should use the actual magnitudes of displacement andstress calculated with a coarse mesh with caution.

It is rarely necessary to use a uniformly fine mesh throughout the structure being analyzed. You shoulduse a fine mesh mainly in the areas of high stress gradients and use a coarser mesh in areas of lowstress gradients or where the magnitude of the stresses is not of interest. For example, Figure 4-36shows a mesh that is designed to give an accurate prediction of the stress concentration at the bottomof the hole.

Figure 4-36 Mesh refined around the hole.

A fine mesh is used only in the region of high stress gradients, and a coarse mesh is used elsewhere.The results from an ABAQUS simulation with this locally refined mesh are shown in Table 4-4. Thistable shows that the results are comparable to those from the very fine mesh, but the simulation withthe locally refined mesh required considerably less CPU time than the analysis with the very fine mesh.

Table 4-4. Comparison of very fine and locally refined meshes.Mesh Displacement of Stress at Relative

bottom of hole bottom of hole CPU timeVery fine 3.15E-4 345.E6 22.5Locally refined 3.14E-4 346.E6 3.44

You can often predict the locations of the highly stressed regions of a model--and, hence, the regionswhere a fine mesh is required--using your knowledge of similar components or with hand calculations.This information can also be gained by using a coarse mesh initially to identify the regions of highstress and then refining the mesh in these regions. The latter procedure is carried out easily usingpreprocessors like ABAQUS/CAE where the complete numerical model (i.e., material properties,

It is common to omit small details like fillet radii from a finite element model to simplify the analysisand to keep the model size reasonable. However, the introduction of any sharp corner into a model willlead to a stress singularity at that location. This normally has a negligible effect on the overall responseof the model, but the predicted stresses close to the singularity will be inaccurate.

For complex, three-dimensional simulations the available computer resources often dictate a practicallimit on the mesh density that you can use. In this case you must use the results from the analysiscarefully. Coarse meshes are often adequate to predict trends and to compare how different conceptsbehave relative to each other. However, you should use the actual magnitudes of displacement andstress calculated with a coarse mesh with caution.

It is rarely necessary to use a uniformly fine mesh throughout the structure being analyzed. You shoulduse a fine mesh mainly in the areas of high stress gradients and use a coarser mesh in areas of lowstress gradients or where the magnitude of the stresses is not of interest. For example, Figure 4-36shows a mesh that is designed to give an accurate prediction of the stress concentration at the bottomof the hole.

Figure 4-36 Mesh refined around the hole.

A fine mesh is used only in the region of high stress gradients, and a coarse mesh is used elsewhere.The results from an ABAQUS simulation with this locally refined mesh are shown in Table 4-4. Thistable shows that the results are comparable to those from the very fine mesh, but the simulation withthe locally refined mesh required considerably less CPU time than the analysis with the very fine mesh.

Table 4-4. Comparison of very fine and locally refined meshes.Mesh Displacement of Stress at Relative

bottom of hole bottom of hole CPU timeVery fine 3.15E-4 345.E6 22.5Locally refined 3.14E-4 346.E6 3.44

You can often predict the locations of the highly stressed regions of a model--and, hence, the regionswhere a fine mesh is required--using your knowledge of similar components or with hand calculations.This information can also be gained by using a coarse mesh initially to identify the regions of highstress and then refining the mesh in these regions. The latter procedure is carried out easily usingpreprocessors like ABAQUS/CAE where the complete numerical model (i.e., material properties,


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boundary conditions, loads, etc.) can be defined based on the geometry of the structure. It is simple tomesh the geometry coarsely for the initial simulation and then to refine the mesh in the appropriateregions, as indicated by the results from the coarse simulation.

ABAQUS provides an advanced feature, called submodeling, that allows you to obtain more detailed(and accurate) results in a region of interest in the structure. The solution from a coarse mesh of theentire structure is used to "drive" a detailed local analysis that uses a fine mesh in this region ofinterest. (This topic is beyond the scope of this guide. See the ABAQUS/Standard User's Manual forfurther details.)

4.5 Related ABAQUS examplesIf you are interested in learning more about using continuum elements in ABAQUS, you shouldexamine the following problems:

� ``Geometrically nonlinear analysis of a cantilever beam,'' Section 2.1.2 of the ABAQUSBenchmarks Manual

� ``Spherical cavity in an infinite medium,'' Section 2.2.4 of the ABAQUS Benchmarks Manual

� ``Performance of continuum and shell elements for linear analysis of bending problems, '' Section2.3.5 of the ABAQUS Benchmarks Manual

4.6 Suggested readingThe volume of literature that has been written on the finite element method and the applications offinite element analysis is enormous. In most of the remaining chapters of this guide, a list of suggestedbooks and articles is provided so that you can explore the topics in more depth if you wish. While theadvanced references will not be of interest to most users, they provide detailed theoretical informationfor the interested user.

General texts on the finite element method

� NAFEMS Ltd., A Finite Element Primer, 1986.

� Becker, E. B., G. F. Carey, J. T. Oden, Finite Elements: An Introduction, Prentice-Hall,1981.

� Cook, R. D., D. S. Malkus, and M. E. Plesha, Concepts and Applications of FiniteElement Analysis, John Wiley & Sons, 1989.

� Hughes, T. J. R., The Finite Element Method, Prentice-Hall, Inc., 1987.

� Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill, 1977.

Performance of linear solid elements

� Prathap, G., "The Poor Bending Response of the Four-Node Plane Stress Quadrilaterals, "

boundary conditions, loads, etc.) can be defined based on the geometry of the structure. It is simple tomesh the geometry coarsely for the initial simulation and then to refine the mesh in the appropriateregions, as indicated by the results from the coarse simulation.

ABAQUS provides an advanced feature, called submodeling, that allows you to obtain more detailed(and accurate) results in a region of interest in the structure. The solution from a coarse mesh of theentire structure is used to "drive" a detailed local analysis that uses a fine mesh in this region ofinterest. (This topic is beyond the scope of this guide. See the ABAQUS/Standard User's Manual forfurther details.)

4.5 Related ABAQUS examplesIf you are interested in learning more about using continuum elements in ABAQUS, you shouldexamine the following problems:


� ``Spherical cavity in an infinite medium,'' Section 2.2.4 of the ABAQUS Benchmarks Manual

� ``Performance of continuum and shell elements for linear analysis of bending problems, '' Section2.3.5 of the ABAQUS Benchmarks Manual

4.6 Suggested readingThe volume of literature that has been written on the finite element method and the applications offinite element analysis is enormous. In most of the remaining chapters of this guide, a list of suggestedbooks and articles is provided so that you can explore the topics in more depth if you wish. While theadvanced references will not be of interest to most users, they provide detailed theoretical informationfor the interested user.

General texts on the finite element method

� NAFEMS Ltd., A Finite Element Primer, 1986.

� Becker, E. B., G. F. Carey, J. T. Oden, Finite Elements: An Introduction, Prentice-Hall,1981.



� Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill, 1977.

Performance of linear solid elements

� Prathap, G., "The Poor Bending Response of the Four-Node Plane Stress Quadrilaterals, "


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International Journal for Numerical Methods in Engineering, vol. 21, 825-835, 1985.

Hourglass control in solid elements

� Belytschko, T., W. K. Liu, and J. M. Kennedy, "Hourglass Control in Linear andNonlinear Problems," Computer Methods in Applied Mechanics and Engineering, vol. 43,251-276, 1984.

� Flanagan, D. P., and T. Belytschko, "A Uniform Strain Hexahedron and Quadrilateral withHourglass Control," International Journal for Numerical Methods in Engineering, vol. 17,679-706, 1981.

Incompatible mode elements

� Simo, J. C. and M. S. Rifai, "A Class of Assumed Strain Methods and the Method ofIncompatible Modes," International Journal for Numerical Methods in Engineering, vol.29, 1595-1638, 1990.

4.7 Summary� The formulation and order of integration used in a continuum element can have a very significant

effect on the accuracy and cost of the analysis.

� First-order (linear) elements using full integration are prone to shear locking and normally shouldnot be used.

� First-order, reduced-integration elements are prone to hourglassing; sufficient mesh refinementminimizes this problem.

� When using first-order, reduced-integration elements in a simulation where bending deformationwill occur, use at least four elements through the thickness.

� Hourglassing is rarely a problem in second-order, reduced-integration elements. You shouldconsider using these elements for most general applications when there is no contact.

� The accuracy of incompatible mode elements is strongly influenced by the amount of elementdistortion.

� The numerical accuracy of the results depends on the mesh that has been used. Ideally a meshrefinement study should be carried out to ensure that the mesh provides a unique solution to theproblem. However, remember that using a converged mesh does not ensure that the results fromthe finite element simulation will match the actual behavior of the physical problem: that alsodepends on other approximations and idealizations in the model.

� In general, refine the mesh mainly in regions where you want accurate results; a finer mesh isrequired to predict accurate stresses than is needed to calculate accurate displacements.

International Journal for Numerical Methods in Engineering, vol. 21, 825-835, 1985.

Hourglass control in solid elements

� Belytschko, T., W. K. Liu, and J. M. Kennedy, "Hourglass Control in Linear andNonlinear Problems," Computer Methods in Applied Mechanics and Engineering, vol. 43,251-276, 1984.

� Flanagan, D. P., and T. Belytschko, "A Uniform Strain Hexahedron and Quadrilateral withHourglass Control," International Journal for Numerical Methods in Engineering, vol. 17,679-706, 1981.

Incompatible mode elements

� Simo, J. C. and M. S. Rifai, "A Class of Assumed Strain Methods and the Method ofIncompatible Modes," International Journal for Numerical Methods in Engineering, vol.29, 1595-1638, 1990.

4.7 Summary� The formulation and order of integration used in a continuum element can have a very significant

effect on the accuracy and cost of the analysis.

� First-order (linear) elements using full integration are prone to shear locking and normally shouldnot be used.

� First-order, reduced-integration elements are prone to hourglassing; sufficient mesh refinementminimizes this problem.

� When using first-order, reduced-integration elements in a simulation where bending deformationwill occur, use at least four elements through the thickness.

� Hourglassing is rarely a problem in second-order, reduced-integration elements. You shouldconsider using these elements for most general applications when there is no contact.

� The accuracy of incompatible mode elements is strongly influenced by the amount of elementdistortion.

� The numerical accuracy of the results depends on the mesh that has been used. Ideally a meshrefinement study should be carried out to ensure that the mesh provides a unique solution to theproblem. However, remember that using a converged mesh does not ensure that the results fromthe finite element simulation will match the actual behavior of the physical problem: that alsodepends on other approximations and idealizations in the model.

� In general, refine the mesh mainly in regions where you want accurate results; a finer mesh isrequired to predict accurate stresses than is needed to calculate accurate displacements.


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� Advanced features such as submodeling are available in ABAQUS to help you to obtain usefulresults for complex simulations.

� Advanced features such as submodeling are available in ABAQUS to help you to obtain usefulresults for complex simulations.


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5. Using Shell ElementsUse shell elements to model structures in which one dimension (the thickness) is significantly smallerthan the other dimensions and in which the stresses in the thickness direction are negligible. Astructure, such as a pressure vessel, whose thickness is less than 1/10 of a typical global structuraldimension generally can be modeled with shell elements. The following are examples of typical globaldimensions:

� the distance between supports,

� the distance between stiffeners or large changes in section thickness,

� the radius of curvature, and

� the wavelength of the highest vibration mode of interest.

ABAQUS shell elements assume that plane sections perpendicular to the plane of the shell remainplane. Do not be confused into thinking that the thickness must be less than 1/10 of the elementdimensions. A highly refined mesh may contain shell elements whose thickness is greater than theirin-plane dimensions, although this is not generally recommended--continuum elements may be moresuitable in such a case.

5.1 Element geometryThe nodes of a shell element define the element's planar dimensions, its surface normal, and its initialcurvature. The nodes do not define the shell thickness.

5.1.1 Shell thickness and section pointsYou define the shell thickness using either the *SHELL SECTION or *SHELL GENERAL SECTIONoption.

If you use the *SHELL SECTION option, ABAQUS uses numerical integration to calculate thestresses and strains independently at each section point (integration point) through the thickness of theshell, thus allowing nonlinear material behavior. For example, an elastic-plastic shell may yield at theouter section points while remaining elastic at the inner section points. The location of the singleintegration point in an S4R (4-node, reduced integration) element and the configuration of the sectionpoints through the shell thickness are shown in Figure 5-1.

Figure 5-1 Configuration of section points in a numerically integrated shell.

5. Using Shell ElementsUse shell elements to model structures in which one dimension (the thickness) is significantly smallerthan the other dimensions and in which the stresses in the thickness direction are negligible. Astructure, such as a pressure vessel, whose thickness is less than 1/10 of a typical global structuraldimension generally can be modeled with shell elements. The following are examples of typical globaldimensions:


� the distance between stiffeners or large changes in section thickness,

� the radius of curvature, and


ABAQUS shell elements assume that plane sections perpendicular to the plane of the shell remainplane. Do not be confused into thinking that the thickness must be less than 1/10 of the elementdimensions. A highly refined mesh may contain shell elements whose thickness is greater than theirin-plane dimensions, although this is not generally recommended--continuum elements may be moresuitable in such a case.

5.1 Element geometryThe nodes of a shell element define the element's planar dimensions, its surface normal, and its initialcurvature. The nodes do not define the shell thickness.

5.1.1 Shell thickness and section pointsYou define the shell thickness using either the *SHELL SECTION or *SHELL GENERAL SECTIONoption.

If you use the *SHELL SECTION option, ABAQUS uses numerical integration to calculate thestresses and strains independently at each section point (integration point) through the thickness of theshell, thus allowing nonlinear material behavior. For example, an elastic-plastic shell may yield at theouter section points while remaining elastic at the inner section points. The location of the singleintegration point in an S4R (4-node, reduced integration) element and the configuration of the sectionpoints through the shell thickness are shown in Figure 5-1.

Figure 5-1 Configuration of section points in a numerically integrated shell.

Using Shell Elements

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You can specify any odd number of section points through the shell thickness with the *SHELLSECTION option. By default, ABAQUS uses five section points through the thickness of ahomogeneous shell, which is sufficient for most nonlinear design problems. However, you should usemore section points in some complicated simulations, especially when you anticipate reversed plasticbending (nine is normally sufficient in this case). For linear problems three section points provideexact integration through the thickness. However, the *SHELL GENERAL SECTION option is moreefficient for linear elastic shells.

If you use the *SHELL GENERAL SECTION option, the material behavior must be linear elastic, asthe stiffness of the cross-section is calculated only once at the beginning of the simulation. Allcalculations are done in terms of the resultant forces and moments across the entire cross-section. Ifyou request stress or strain output, ABAQUS provides output for the bottom surface, the midplane, andthe top surface.

5.1.2 Shell normals and shell surfacesThe connectivity of the shell element defines the direction of the positive normal, as shown in Figure5-2.

Figure 5-2 Positive normals for shells.

You can specify any odd number of section points through the shell thickness with the *SHELLSECTION option. By default, ABAQUS uses five section points through the thickness of ahomogeneous shell, which is sufficient for most nonlinear design problems. However, you should usemore section points in some complicated simulations, especially when you anticipate reversed plasticbending (nine is normally sufficient in this case). For linear problems three section points provideexact integration through the thickness. However, the *SHELL GENERAL SECTION option is moreefficient for linear elastic shells.

If you use the *SHELL GENERAL SECTION option, the material behavior must be linear elastic, asthe stiffness of the cross-section is calculated only once at the beginning of the simulation. Allcalculations are done in terms of the resultant forces and moments across the entire cross-section. Ifyou request stress or strain output, ABAQUS provides output for the bottom surface, the midplane, andthe top surface.

5.1.2 Shell normals and shell surfacesThe connectivity of the shell element defines the direction of the positive normal, as shown in Figure5-2.

Figure 5-2 Positive normals for shells.


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For axisymmetric shell elements the positive normal direction is defined by a 90° counterclockwiserotation from the direction going from node 1 to node 2. For three-dimensional shell elements thepositive normal is given by the right-hand rule going around the nodes in the order in which theyappear in the element definition.

The "top" surface of a shell is the surface in the positive normal direction and is called the SPOS facefor contact definition. The "bottom" surface is in the negative direction along the normal and is calledthe SNEG face for contact definition. Normals should be consistent among adjoining shell elements.

The positive normal direction defines the convention for pressure load application and output ofquantities that vary through the shell thickness. A positive pressure load applied to a shell elementproduces load that acts in the direction of the positive normal. (The pressure load convention for shellelements is opposite that for continuum elements.)

5.1.3 Initial shell curvatureMost shell elements in ABAQUS are formulated as true curved shell elements and, therefore, requireaccurate calculation of the initial surface curvature. ABAQUS automatically calculates the surfacenormals at the nodes of every shell element to estimate the initial curvature of the shell. The surfacenormal at each node is determined using a fairly elaborate algorithm, which is discussed in detail in theABAQUS/Standard User's Manual.

With a coarse mesh as shown in Figure 5-3, ABAQUS may determine several independent surfacenormals at the same node for adjoining elements. Physically, multiple normals at a single node meanthat there is a fold line between the elements sharing the node. While it is possible that you intend tomodel such a structure, it is more likely that you intend to model a smoothly curved shell; ABAQUSwill try to smooth the shell by creating an averaged normal at a node.

Figure 5-3 Effect of mesh refinement on the nodal surface normals.

For axisymmetric shell elements the positive normal direction is defined by a 90° counterclockwiserotation from the direction going from node 1 to node 2. For three-dimensional shell elements thepositive normal is given by the right-hand rule going around the nodes in the order in which theyappear in the element definition.

The "top" surface of a shell is the surface in the positive normal direction and is called the SPOS facefor contact definition. The "bottom" surface is in the negative direction along the normal and is calledthe SNEG face for contact definition. Normals should be consistent among adjoining shell elements.

The positive normal direction defines the convention for pressure load application and output ofquantities that vary through the shell thickness. A positive pressure load applied to a shell elementproduces load that acts in the direction of the positive normal. (The pressure load convention for shellelements is opposite that for continuum elements.)

5.1.3 Initial shell curvatureMost shell elements in ABAQUS are formulated as true curved shell elements and, therefore, requireaccurate calculation of the initial surface curvature. ABAQUS automatically calculates the surfacenormals at the nodes of every shell element to estimate the initial curvature of the shell. The surfacenormal at each node is determined using a fairly elaborate algorithm, which is discussed in detail in theABAQUS/Standard User's Manual.

With a coarse mesh as shown in Figure 5-3, ABAQUS may determine several independent surfacenormals at the same node for adjoining elements. Physically, multiple normals at a single node meanthat there is a fold line between the elements sharing the node. While it is possible that you intend tomodel such a structure, it is more likely that you intend to model a smoothly curved shell; ABAQUSwill try to smooth the shell by creating an averaged normal at a node.

Figure 5-3 Effect of mesh refinement on the nodal surface normals.


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The basic smoothing algorithm used is as follows: if the normals at a node for each shell elementattached to the node are within 20° of each other, the normals will be averaged. The averaged normalwill be used at that node for all elements attached to the node. If ABAQUS cannot smooth the shell, awarning message is issued in the data (.dat) file.

You may override the default algorithm. To introduce fold lines into a curved shell or to model acurved shell with a coarse mesh, use the *NODE and *NORMAL options to define the normalsmanually. With the *NODE option you specify the surface normal at a node as the 4th, 5th, and 6thvalues on the data line, following the nodal coordinates. A normal you define with *NODE is thenormal used for all elements sharing that node, unless *NORMAL is also used. Use the *NORMALoption to specify a normal at a node for selected elements only. Normals defined with *NORMALoverride normals defined with *NODE. See the ABAQUS/Standard User's Manual for further details.

5.1.4 Reference surface offsetsThe reference surface of the shell is defined by the shell element's nodes and normal definitions. Whenmodeling with shell elements, the reference surface is typically coincident with the shell's midsurface.However, many situations arise in which it is more convenient to define the reference surface as offsetfrom the shell's midsurface. For example, surfaces created in CAD packages usually represent eitherthe top or bottom surface of the shell body. In this case it may be easier to define the reference surfaceto be coincident with the CAD surface and, therefore, offset from the shell's midsurface.

Shell offsets can also be used to define a more precise surface geometry for contact problems whereshell thickness is important. Shell offsets can be useful when modeling a shell with continuouslyvarying thickness. In this case defining the nodes at the shell midplane can be difficult. If one surfaceis smooth while the other is rough, as in some aircraft structures, it is easiest to use shell offsets todefine the nodes at the smooth surface.

Offsets can be introduced by using the OFFSET parameter on the *SHELL SECTION and *SHELLGENERAL SECTION options. The offset value is defined as a fraction of the shell thickness measuredfrom the shell's midsurface to the shell's reference surface, as shown in Figure 5-4.

The basic smoothing algorithm used is as follows: if the normals at a node for each shell elementattached to the node are within 20° of each other, the normals will be averaged. The averaged normalwill be used at that node for all elements attached to the node. If ABAQUS cannot smooth the shell, awarning message is issued in the data (.dat) file.

You may override the default algorithm. To introduce fold lines into a curved shell or to model acurved shell with a coarse mesh, use the *NODE and *NORMAL options to define the normalsmanually. With the *NODE option you specify the surface normal at a node as the 4th, 5th, and 6thvalues on the data line, following the nodal coordinates. A normal you define with *NODE is thenormal used for all elements sharing that node, unless *NORMAL is also used. Use the *NORMALoption to specify a normal at a node for selected elements only. Normals defined with *NORMALoverride normals defined with *NODE. See the ABAQUS/Standard User's Manual for further details.

5.1.4 Reference surface offsetsThe reference surface of the shell is defined by the shell element's nodes and normal definitions. Whenmodeling with shell elements, the reference surface is typically coincident with the shell's midsurface.However, many situations arise in which it is more convenient to define the reference surface as offsetfrom the shell's midsurface. For example, surfaces created in CAD packages usually represent eitherthe top or bottom surface of the shell body. In this case it may be easier to define the reference surfaceto be coincident with the CAD surface and, therefore, offset from the shell's midsurface.

Shell offsets can also be used to define a more precise surface geometry for contact problems whereshell thickness is important. Shell offsets can be useful when modeling a shell with continuouslyvarying thickness. In this case defining the nodes at the shell midplane can be difficult. If one surfaceis smooth while the other is rough, as in some aircraft structures, it is easiest to use shell offsets todefine the nodes at the smooth surface.

Offsets can be introduced by using the OFFSET parameter on the *SHELL SECTION and *SHELLGENERAL SECTION options. The offset value is defined as a fraction of the shell thickness measuredfrom the shell's midsurface to the shell's reference surface, as shown in Figure 5-4.


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Figure 5-4 Schematic of shell offset for an offset value of 0.5.

5.2 Shell formulation - thick or thinShell problems generally fall into one of two categories: thin shell problems and thick shell problems.Thick shell problems assume that the effects of transverse shear deformation are important to thesolution. Thin shell problems, on the other hand, assume that transverse shear deformation is smallenough to be neglected. Figure 5-5 illustrates the effect of transverse shear deformation.

Figure 5-5 Behavior of transverse shell sections in (a) thin shells and (b) thick shells.

There are three different classes of shell elements in ABAQUS, distinguished by the element'sapplicability to thin and thick shell problems. General-purpose shell elements are valid for use withboth thick and thin shell problems. Furthermore, all general-purpose shell elements in ABAQUSconsider finite membrane strains. In certain cases, for specific applications, enhanced performance canbe obtained by using special-purpose shell elements. All special-purpose shell elements assume finiterotations; however, they assume small strains. The special-purpose shell elements fall into twocategories: thin-only shell elements and thick-only shell elements. The thin-only shell elements enforcethe Kirchhoff constraint; that is, plane sections normal to the mid-section of the shell remain normal tothe mid-surface. The Kirchhoff constraint is enforced either analytically in the element formulation(STRI3) or numerically through the use of a penalty constraint. The thick-only shell elements are

Figure 5-4 Schematic of shell offset for an offset value of 0.5.

5.2 Shell formulation - thick or thinShell problems generally fall into one of two categories: thin shell problems and thick shell problems.Thick shell problems assume that the effects of transverse shear deformation are important to thesolution. Thin shell problems, on the other hand, assume that transverse shear deformation is smallenough to be neglected. Figure 5-5 illustrates the effect of transverse shear deformation.

Figure 5-5 Behavior of transverse shell sections in (a) thin shells and (b) thick shells.

There are three different classes of shell elements in ABAQUS, distinguished by the element'sapplicability to thin and thick shell problems. General-purpose shell elements are valid for use withboth thick and thin shell problems. Furthermore, all general-purpose shell elements in ABAQUSconsider finite membrane strains. In certain cases, for specific applications, enhanced performance canbe obtained by using special-purpose shell elements. All special-purpose shell elements assume finiterotations; however, they assume small strains. The special-purpose shell elements fall into twocategories: thin-only shell elements and thick-only shell elements. The thin-only shell elements enforcethe Kirchhoff constraint; that is, plane sections normal to the mid-section of the shell remain normal tothe mid-surface. The Kirchhoff constraint is enforced either analytically in the element formulation(STRI3) or numerically through the use of a penalty constraint. The thick-only shell elements are


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second-order quadrilaterals that may produce more accurate results than the general-purpose shellelements in small-strain applications where the loading is such that the solution is smoothly varyingover the span of the shell. Table 5-1 lists the three categories of shell elements in ABAQUS.




S8R, S8RT

To decide if a given application is a thin or thick shell problem, we can offer a few guidelines. Forthick shells transverse shear flexibility is important, while for thin shells it is negligible. Thesignificance of transverse shear in a shell can be estimated by its thickness-to-span ratio. A shell madeof a single isotropic material with a ratio greater than 1/15 is considered "thick"; if the ratio is less than1/15, the shell is considered "thin." These estimates are approximate; you should always check thetransverse shear effects in your model to verify the assumed shell behavior. Since transverse shearflexibility can be significant in laminated composite shell structures, this ratio should be much smallerfor "thin" shell theory to apply. Composite shells with very compliant interior layers (so-called"sandwich" composites) have very low transverse shear stiffness and should almost always be modeledwith "thick" shells; if the assumption of plane sections remaining plane is violated, continuumelements should be used.

Transverse shear force and strain are available for general-purpose and thick-only shell elements. Forthree-dimensional elements, estimates of transverse shear stress are provided. The calculation of thesestresses neglects coupling between bending and twisting deformation and assumes small spatialgradients of material properties and bending moments.

5.3 Shell material directionsShell elements, unlike continuum elements, use material directions local to each element. Anisotropicmaterial data, such as that for fiber reinforced composites, and element output variables, such as stressand strain, are defined in terms of these local material directions. In large-displacement analyses thelocal material axes on a shell surface rotate with the average motion of the material at each integrationpoint.

5.3.1 Default local material directionsThe local material 1- and 2-directions lie in the plane of the shell. The default local 1-direction is theprojection of the global 1-axis onto the shell surface. If the global 1-axis is normal to the shell surface,the local 1-direction is the projection of the global 3-axis onto the shell surface. The local 2-directionis perpendicular to the local 1-direction in the surface of the shell, so that the local 1-direction, local2-direction, and the positive normal to the surface form a right-handed set (see Figure 5-6).

Figure 5-6 Default local shell material directions.

second-order quadrilaterals that may produce more accurate results than the general-purpose shellelements in small-strain applications where the loading is such that the solution is smoothly varyingover the span of the shell. Table 5-1 lists the three categories of shell elements in ABAQUS.




S8R, S8RT

To decide if a given application is a thin or thick shell problem, we can offer a few guidelines. Forthick shells transverse shear flexibility is important, while for thin shells it is negligible. Thesignificance of transverse shear in a shell can be estimated by its thickness-to-span ratio. A shell madeof a single isotropic material with a ratio greater than 1/15 is considered "thick"; if the ratio is less than1/15, the shell is considered "thin." These estimates are approximate; you should always check thetransverse shear effects in your model to verify the assumed shell behavior. Since transverse shearflexibility can be significant in laminated composite shell structures, this ratio should be much smallerfor "thin" shell theory to apply. Composite shells with very compliant interior layers (so-called"sandwich" composites) have very low transverse shear stiffness and should almost always be modeledwith "thick" shells; if the assumption of plane sections remaining plane is violated, continuumelements should be used.

Transverse shear force and strain are available for general-purpose and thick-only shell elements. Forthree-dimensional elements, estimates of transverse shear stress are provided. The calculation of thesestresses neglects coupling between bending and twisting deformation and assumes small spatialgradients of material properties and bending moments.

5.3 Shell material directionsShell elements, unlike continuum elements, use material directions local to each element. Anisotropicmaterial data, such as that for fiber reinforced composites, and element output variables, such as stressand strain, are defined in terms of these local material directions. In large-displacement analyses thelocal material axes on a shell surface rotate with the average motion of the material at each integrationpoint.

5.3.1 Default local material directionsThe local material 1- and 2-directions lie in the plane of the shell. The default local 1-direction is theprojection of the global 1-axis onto the shell surface. If the global 1-axis is normal to the shell surface,the local 1-direction is the projection of the global 3-axis onto the shell surface. The local 2-directionis perpendicular to the local 1-direction in the surface of the shell, so that the local 1-direction, local2-direction, and the positive normal to the surface form a right-handed set (see Figure 5-6).

Figure 5-6 Default local shell material directions.


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The default set of local material directions can sometimes cause problems; a case in point is thecylinder shown in Figure 5-7. For most of the elements in the figure the local 1-direction iscircumferential. However, there is a line of elements that are normal to the global 1-axis. For theseelements the local 1-direction is the projection of the global 3-axis onto the shell, making the local1-direction axial instead of circumferential. A contour plot of the direct stress in the local 1-direction,¾11 , looks very strange, since for most elements ¾11 is the circumferential stress, whereas for someelements it is the axial stress. In such cases use the *ORIENTATION option to define moreappropriate local directions for the model. The *ORIENTATION option is discussed in the nextsection.

Figure 5-7 Default local material 1-direction in a cylinder.

5.3.2 Creating alternative material directionsThe *ORIENTATION option allows you to control the local material directions directly. With it youcan replace the global Cartesian coordinate system with a local rectangular, cylindrical, or sphericalcoordinate system. You define the orientation of the local (x0, y0, z0) coordinate system by specifyingthe location of two points, a and b, as shown in Figure 5-8. For example, a local rectangular system isdefined with the following option:

*ORIENTATION, SYSTEM=RECTANGULAR, NAME=LOCALR

<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>

The default set of local material directions can sometimes cause problems; a case in point is thecylinder shown in Figure 5-7. For most of the elements in the figure the local 1-direction iscircumferential. However, there is a line of elements that are normal to the global 1-axis. For theseelements the local 1-direction is the projection of the global 3-axis onto the shell, making the local1-direction axial instead of circumferential. A contour plot of the direct stress in the local 1-direction,¾11 , looks very strange, since for most elements ¾11 is the circumferential stress, whereas for someelements it is the axial stress. In such cases use the *ORIENTATION option to define moreappropriate local directions for the model. The *ORIENTATION option is discussed in the nextsection.

Figure 5-7 Default local material 1-direction in a cylinder.

5.3.2 Creating alternative material directionsThe *ORIENTATION option allows you to control the local material directions directly. With it youcan replace the global Cartesian coordinate system with a local rectangular, cylindrical, or sphericalcoordinate system. You define the orientation of the local (x0, y0, z0) coordinate system by specifyingthe location of two points, a and b, as shown in Figure 5-8. For example, a local rectangular system isdefined with the following option:


<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>


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Figure 5-8 Definition of local coordinate systems.

The parameter NAME specifies a label for this orientation, and the coordinates of point a (xa1 , xa

2 , xa3 )

and point b (xb1 , xb

2 , xb3) are given in the global Cartesian system. The local coordinate system is then

referred to by the ORIENTATION parameter on the *SHELL SECTION or *SHELL GENERALSECTION option.

You must still specify another piece of information. ABAQUS must also be told which of the localaxes corresponds to which material direction. On the second data line following *ORIENTATION,specify the local axis (1, 2, or 3) that is closest to being normal to the shell's 1 and 2 material directionsand a rotation about that axis. The two axes that you did not select are projected onto the shell andform the local 1 and 2 material directions. For example, if you choose the x0-axis, ABAQUS projectsthe y0- and z0-axes onto the shell, and the projection of the y0-axis is the material 1-direction. Thematerial 2-direction is defined by the cross product of the shell normal and the material 1-direction.Normally, the final material 2-direction and the projection of the other local axis, in this case thez0-axis, will not coincide for curved shells.

If these local axes do not create the desired material directions, you can specify a rotation about theselected axis. The other two local axes are rotated by this amount before they are projected onto theshell's surface to give the final material directions. The following option block:


<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>

1, ®

would create the local system shown in Figure 5-9. Again, it is the rotated y0- and z0-axes thatABAQUS projects onto the surface of the shell elements. For the projections to be interpreted easily,

Figure 5-8 Definition of local coordinate systems.

The parameter NAME specifies a label for this orientation, and the coordinates of point a (xa1 , xa

2 , xa3 )

and point b (xb1 , xb

2 , xb3) are given in the global Cartesian system. The local coordinate system is then

referred to by the ORIENTATION parameter on the *SHELL SECTION or *SHELL GENERALSECTION option.

You must still specify another piece of information. ABAQUS must also be told which of the localaxes corresponds to which material direction. On the second data line following *ORIENTATION,specify the local axis (1, 2, or 3) that is closest to being normal to the shell's 1 and 2 material directionsand a rotation about that axis. The two axes that you did not select are projected onto the shell andform the local 1 and 2 material directions. For example, if you choose the x0-axis, ABAQUS projectsthe y0- and z0-axes onto the shell, and the projection of the y0-axis is the material 1-direction. Thematerial 2-direction is defined by the cross product of the shell normal and the material 1-direction.Normally, the final material 2-direction and the projection of the other local axis, in this case thez0-axis, will not coincide for curved shells.

If these local axes do not create the desired material directions, you can specify a rotation about theselected axis. The other two local axes are rotated by this amount before they are projected onto theshell's surface to give the final material directions. The following option block:


<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>

1, ®

would create the local system shown in Figure 5-9. Again, it is the rotated y0- and z0-axes thatABAQUS projects onto the surface of the shell elements. For the projections to be interpreted easily,


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the selected axis should be as close as possible to the shell normal.

Figure 5-9 Rotation of the local coordinate system for shell elements.

If the centerline of the cylinder shown in Figure 5-7coincides with the global 3-axis, the followingoption block could be used to define consistent material directions:

*ORIENTATION, SYSTEM=CYLINDRICAL, NAME=CYLIND1

0., 0., 0., 0., 0., 1.

1, 0.

Points a and b lie along the centerline of the cylinder. Since the orientation of the cylinder matches theorientation of our newly defined cylindrical coordinate system, the x0-axis is radial, the y0-axis iscircumferential, and the z0-axis is axial. The x0-axis corresponds approximately to the shell normaldirection, and a zero rotation is specified; therefore, the projection of the y0-axis onto the shell'ssurface is the material 1-direction. Thus, the material 1-direction is always circumferential, and thecorresponding material 2-direction is always axial.

5.4 Selecting shell elements� The linear, finite-membrane-strain, fully integrated, quadrilateral shell element ( S4) should be

used when greater solution accuracy is desired. This element is well suited for problems prone tomembrane- or bending-mode hourglassing or for problems where in-plane bending is expected.

� The linear, finite-membrane-strain, reduced integration, quadrilateral shell element ( S4R) is robustand is suitable for a wide range of applications.

� The linear, finite-membrane-strain, triangular shell element (S3/S3R) can be used as ageneral-purpose element. A refined mesh may be needed to capture bending deformations or highstrain gradients because of the constant strain approximation in the element.

� To account for the influence of shear flexibility in laminated composite shell models, use the shellelements suitable for modeling thick shells (S4, S4R, S3/S3R, S8R); check that the assumption ofplane sections remaining plane is satisfied.

� Quadratic shell elements, either quadrilateral or triangular, are very effective for general,small-strain, thin-shell applications. These elements are not susceptible to shear or membrane

the selected axis should be as close as possible to the shell normal.

Figure 5-9 Rotation of the local coordinate system for shell elements.

If the centerline of the cylinder shown in Figure 5-7coincides with the global 3-axis, the followingoption block could be used to define consistent material directions:

*ORIENTATION, SYSTEM=CYLINDRICAL, NAME=CYLIND1

0., 0., 0., 0., 0., 1.

1, 0.

Points a and b lie along the centerline of the cylinder. Since the orientation of the cylinder matches theorientation of our newly defined cylindrical coordinate system, the x0-axis is radial, the y0-axis iscircumferential, and the z0-axis is axial. The x0-axis corresponds approximately to the shell normaldirection, and a zero rotation is specified; therefore, the projection of the y0-axis onto the shell'ssurface is the material 1-direction. Thus, the material 1-direction is always circumferential, and thecorresponding material 2-direction is always axial.

5.4 Selecting shell elements� The linear, finite-membrane-strain, fully integrated, quadrilateral shell element ( S4) should be

used when greater solution accuracy is desired. This element is well suited for problems prone tomembrane- or bending-mode hourglassing or for problems where in-plane bending is expected.

� The linear, finite-membrane-strain, reduced integration, quadrilateral shell element ( S4R) is robustand is suitable for a wide range of applications.

� The linear, finite-membrane-strain, triangular shell element (S3/S3R) can be used as ageneral-purpose element. A refined mesh may be needed to capture bending deformations or highstrain gradients because of the constant strain approximation in the element.

� To account for the influence of shear flexibility in laminated composite shell models, use the shellelements suitable for modeling thick shells (S4, S4R, S3/S3R, S8R); check that the assumption ofplane sections remaining plane is satisfied.

� Quadratic shell elements, either quadrilateral or triangular, are very effective for general,small-strain, thin-shell applications. These elements are not susceptible to shear or membrane


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locking.

� Do not use the quadratic, triangular shell element ( STRI65) in contact simulations. Use the 9-node,quadrilateral shell element (S9R5).

� For very large models that will experience only geometrically linear behavior, the linear, thin-shellelement (S4R5) will generally be more cost-effective than the general-purpose shell elements.

5.5 Example: skew plateYou have been asked to model the plate shown in Figure 5-10. It is skewed 30° to the global 1-axis, isbuilt-in at one end, and is constrained to move on rails parallel to the plate axis at the other end. Youare to determine the midspan deflection when the plate carries a uniform pressure. You are also toassess whether a linear analysis is valid for this problem.

Figure 5-10 Sketch of the skewed plate.

5.5.1 Coordinate systemThe orientation of the structure in the global coordinate system and the suggested origin of the systemare shown in Figure 5-10. The plate lies in the global 1-2 plane. Will it be easy to interpret the results

locking.

� Do not use the quadratic, triangular shell element ( STRI65) in contact simulations. Use the 9-node,quadrilateral shell element (S9R5).

� For very large models that will experience only geometrically linear behavior, the linear, thin-shellelement (S4R5) will generally be more cost-effective than the general-purpose shell elements.

5.5 Example: skew plateYou have been asked to model the plate shown in Figure 5-10. It is skewed 30° to the global 1-axis, isbuilt-in at one end, and is constrained to move on rails parallel to the plate axis at the other end. Youare to determine the midspan deflection when the plate carries a uniform pressure. You are also toassess whether a linear analysis is valid for this problem.

Figure 5-10 Sketch of the skewed plate.

5.5.1 Coordinate systemThe orientation of the structure in the global coordinate system and the suggested origin of the systemare shown in Figure 5-10. The plate lies in the global 1-2 plane. Will it be easy to interpret the results


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of the simulation if you use the default material directions for the shell elements in this model?

5.5.2 Mesh designFigure 5-11 shows the suggested mesh for this simulation. The node and element numbers shown werecreated by the input file in ``Skew plate,'' Section A.3. The numbering in your model will almostcertainly be different.

Figure 5-11 Suggested mesh design for the skewed plate simulation.

You must answer the following questions before selecting an element type: Is the plate thin or thick?Are the strains small or large? The plate is quite thin, with a thickness-to-minimum span ratio of 0.02.(The thickness is 0.8 cm and the minimum span is 40 cm.) While we cannot readily predict themagnitude of the strains in the structure, we think that the strains will be small. Based on thisinformation, you choose quadratic shell elements (S8R5), because they give accurate results for thinshells in small-strain simulations. For further details on shell element selection, consult theABAQUS/Standard User's Manual.

5.5.3 Preprocessing--creating the modelUse a preprocessor such as ABAQUS/CAE to generate the mesh shown in Figure 5-11. Create thenode sets shown in Figure 5-12, and put all of the elements in an element set called PLATE.

Figure 5-12 Node sets needed for the skew plate simulation.

of the simulation if you use the default material directions for the shell elements in this model?

5.5.2 Mesh designFigure 5-11 shows the suggested mesh for this simulation. The node and element numbers shown werecreated by the input file in ``Skew plate,'' Section A.3. The numbering in your model will almostcertainly be different.

Figure 5-11 Suggested mesh design for the skewed plate simulation.

You must answer the following questions before selecting an element type: Is the plate thin or thick?Are the strains small or large? The plate is quite thin, with a thickness-to-minimum span ratio of 0.02.(The thickness is 0.8 cm and the minimum span is 40 cm.) While we cannot readily predict themagnitude of the strains in the structure, we think that the strains will be small. Based on thisinformation, you choose quadratic shell elements (S8R5), because they give accurate results for thinshells in small-strain simulations. For further details on shell element selection, consult theABAQUS/Standard User's Manual.

5.5.3 Preprocessing--creating the modelUse a preprocessor such as ABAQUS/CAE to generate the mesh shown in Figure 5-11. Create thenode sets shown in Figure 5-12, and put all of the elements in an element set called PLATE.

Figure 5-12 Node sets needed for the skew plate simulation.


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You can define most of this model in your preprocessor, but leave out the material and historyinformation so you can add it later with an editor. This exercise will give you a better understanding ofhow the various option blocks combine to define an ABAQUS model. If you do not have apreprocessor, use the ABAQUS mesh generation options described in ``Skew plate,'' Section A.3.

Before you start to build the model, decide on a system of units. The dimensions are given in cm, butthe loading and material properties are given in MPa and GPa. Since these are not consistent units, youmust choose a consistent system to use in your model and convert the necessary input data.

5.5.4 Reviewing the input file--the model dataAt this point we assume that you have created the basic mesh using your preprocessor. In this sectionyou will review and make corrections to your input file, as well as include additional information, suchas material data.

Model description

The following would be a suitable description in the *HEADING option for this simulation:

*HEADING

Linear Elastic Skew Plate. 20 kPa Load.

S.I. Units (meters, Newtons, sec, kilograms)

It clearly explains what you are modeling and what units you are using.


Check to make sure that you are using the correct element type (S8R5). It is possible that youspecified the wrong element type in the preprocessor or that the translator made a mistakewhen generating the input file. The *ELEMENT option block in your model should begin withthe following:

*ELEMENT, TYPE=S8R5, ELSET=PLATE

The name given for the ELSET parameter in your model may not be PLATE. If necessary,change the value to PLATE. Meaningful names for node and element sets make input fileseasy to understand.

Node sets

You can define most of this model in your preprocessor, but leave out the material and historyinformation so you can add it later with an editor. This exercise will give you a better understanding ofhow the various option blocks combine to define an ABAQUS model. If you do not have apreprocessor, use the ABAQUS mesh generation options described in ``Skew plate,'' Section A.3.

Before you start to build the model, decide on a system of units. The dimensions are given in cm, butthe loading and material properties are given in MPa and GPa. Since these are not consistent units, youmust choose a consistent system to use in your model and convert the necessary input data.

5.5.4 Reviewing the input file--the model dataAt this point we assume that you have created the basic mesh using your preprocessor. In this sectionyou will review and make corrections to your input file, as well as include additional information, suchas material data.

Model description

The following would be a suitable description in the *HEADING option for this simulation:

*HEADING



It clearly explains what you are modeling and what units you are using.


Check to make sure that you are using the correct element type (S8R5). It is possible that youspecified the wrong element type in the preprocessor or that the translator made a mistakewhen generating the input file. The *ELEMENT option block in your model should begin withthe following:


The name given for the ELSET parameter in your model may not be PLATE. If necessary,change the value to PLATE. Meaningful names for node and element sets make input fileseasy to understand.

Node sets


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The three node sets shown in Figure 5-12will be useful in completing the model of the plate. Ifyou were unable to define them with your preprocessor, use a text editor and the *NSEToption to create node sets with these names.

Defining alternative material directions

If you use the default material directions, the direct stress in the material 1-direction, ¾11 , willcontain contributions from both the axial stress, produced by the bending of the plate, and thestress transverse to the axis of the plate. It will be easier to interpret the results if the materialdirections are aligned with the axis of the plate and the transverse direction. Therefore, a localrectangular coordinate system is needed in which the local x0-direction lies along the axis ofthe plate (i.e., at 30° to the global 1-axis) and the local y0-direction is also in the plane of theplate.

As you learned in ``Shell material directions,'' Section 5.3, the *ORIENTATION optiondefines such a local coordinate system. Choose point a (see Figure 5-8) to have coordinates(10.0E-2, 5.77E-2, 0.)--so that xa

2=xa1 = tan 30°--and point b to have coordinates (-5.77E-2,

10.E-2, 0.). You must also specify which axis is not projected onto the shell surface (thez0-direction in this model) as well as an additional rotation (zero using this method). Use thefollowing *ORIENTATION option block to create the proper local coordinate system, namedSKEW:

*ORIENTATION, NAME=SKEW, SYSTEM=RECTANGULAR

10.0E-2,5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

3, 0.0

Alternatively, you can define exactly the same local coordinate system by choosing point aand point b to lie on the global coordinate 1- and 2-axes and specifying an additional rotationof 30°:


1., 0., 0., 0., 1., 0.

3, 30.

Section properties

Since the structure is made of a single material with constant thickness, the section propertiesare the same for all elements. Therefore, you can use the element set PLATE (which includesall elements) to assign the physical and material properties to the elements. Since you assumethat the plate is linear elastic, the *SHELL GENERAL SECTION option is more efficient thanusing the *SHELL SECTION option. Use the following element property option block in yourmodel:

*SHELL GENERAL SECTION, ELSET=PLATE, MATERIAL=MAT1,

ORIENTATION=SKEW

0.8E-2,

The ORIENTATION parameter tells ABAQUS to use the local coordinate system namedSKEW to define the material directions for the shells in element set PLATE. All element

The three node sets shown in Figure 5-12will be useful in completing the model of the plate. Ifyou were unable to define them with your preprocessor, use a text editor and the *NSEToption to create node sets with these names.

Defining alternative material directions

If you use the default material directions, the direct stress in the material 1-direction, ¾11 , willcontain contributions from both the axial stress, produced by the bending of the plate, and thestress transverse to the axis of the plate. It will be easier to interpret the results if the materialdirections are aligned with the axis of the plate and the transverse direction. Therefore, a localrectangular coordinate system is needed in which the local x0-direction lies along the axis ofthe plate (i.e., at 30° to the global 1-axis) and the local y0-direction is also in the plane of theplate.

As you learned in ``Shell material directions,'' Section 5.3, the *ORIENTATION optiondefines such a local coordinate system. Choose point a (see Figure 5-8) to have coordinates(10.0E-2, 5.77E-2, 0.)--so that xa

2=xa1 = tan 30°--and point b to have coordinates (-5.77E-2,

10.E-2, 0.). You must also specify which axis is not projected onto the shell surface (thez0-direction in this model) as well as an additional rotation (zero using this method). Use thefollowing *ORIENTATION option block to create the proper local coordinate system, namedSKEW:


10.0E-2,5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

3, 0.0

Alternatively, you can define exactly the same local coordinate system by choosing point aand point b to lie on the global coordinate 1- and 2-axes and specifying an additional rotationof 30°:


1., 0., 0., 0., 1., 0.

3, 30.

Section properties

Since the structure is made of a single material with constant thickness, the section propertiesare the same for all elements. Therefore, you can use the element set PLATE (which includesall elements) to assign the physical and material properties to the elements. Since you assumethat the plate is linear elastic, the *SHELL GENERAL SECTION option is more efficient thanusing the *SHELL SECTION option. Use the following element property option block in yourmodel:


ORIENTATION=SKEW

0.8E-2,

The ORIENTATION parameter tells ABAQUS to use the local coordinate system namedSKEW to define the material directions for the shells in element set PLATE. All element


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variables will be defined in the SKEW coordinate system.

Material properties

The plate is made of an isotropic, linear elastic material that has a Young's modulus of 30.0GPa and a Poisson's ratio of 0.3. Use the following material option block in your model:

*MATERIAL, NAME=MAT1

*ELASTIC

30.0E9, 0.3

Local directions at the nodes

While the *ORIENTATION option defines a local coordinate system for elements, you mustuse the *TRANSFORM option to define a local coordinate system for nodes. The two optionsare completely independent of each other. If a node refers to a local coordinate system definedwith *TRANSFORM, all data pertaining to the node--such as boundary conditions,concentrated loads, or nodal output variables (displacements, velocities, reaction forces,etc.)--are defined in the transformed coordinate system.

The *TRANSFORM option has the following format:

*TRANSFORM, NSET=<node set name>, TYPE=<axis type>

<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>

The data line specifies the coordinates of two points, a and b, in much the same way as the*ORIENTATION option. The coordinate system defined with *TRANSFORM does not rotateas the body deforms; it is fixed in the original directions defined at the beginning of thesimulation. Rectangular (TYPE=R), cylindrical (TYPE=C), and spherical (TYPE=S)coordinate systems can be specified. Use the NSET parameter to specify the node sets that usethis local coordinate system.

As shown in Figure 5-10, one end of the plate is constrained to move on rails that are parallelto the axis of the plate. Since this boundary condition does not coincide with the global axes,you must transform the nodes on this end of the plate into a local coordinate system that has anaxis aligned with the plate. Use the following *TRANSFORM option:

*TRANSFORM, NSET=ENDB, TYPE=R

10.0E-2,5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

This option block defines the degrees of freedom for node set ENDB in a local coordinatesystem whose x0-axis is aligned with the long axis of the plate (i.e., the local system is rotated30° about the global 3-axis).

5.5.5 Reviewing the input file--the history dataWe now review the history definition portion of the input file. A single step is needed to define thissimulation.

Step definition

variables will be defined in the SKEW coordinate system.

Material properties

The plate is made of an isotropic, linear elastic material that has a Young's modulus of 30.0GPa and a Poisson's ratio of 0.3. Use the following material option block in your model:


*ELASTIC

30.0E9, 0.3

Local directions at the nodes

While the *ORIENTATION option defines a local coordinate system for elements, you mustuse the *TRANSFORM option to define a local coordinate system for nodes. The two optionsare completely independent of each other. If a node refers to a local coordinate system definedwith *TRANSFORM, all data pertaining to the node--such as boundary conditions,concentrated loads, or nodal output variables (displacements, velocities, reaction forces,etc.)--are defined in the transformed coordinate system.

The *TRANSFORM option has the following format:

*TRANSFORM, NSET=<node set name>, TYPE=<axis type>

<xa1>, <x

a2>, <x

a3>, <x

b1>, <x

b2>, <x

b3>

The data line specifies the coordinates of two points, a and b, in much the same way as the*ORIENTATION option. The coordinate system defined with *TRANSFORM does not rotateas the body deforms; it is fixed in the original directions defined at the beginning of thesimulation. Rectangular (TYPE=R), cylindrical (TYPE=C), and spherical (TYPE=S)coordinate systems can be specified. Use the NSET parameter to specify the node sets that usethis local coordinate system.

As shown in Figure 5-10, one end of the plate is constrained to move on rails that are parallelto the axis of the plate. Since this boundary condition does not coincide with the global axes,you must transform the nodes on this end of the plate into a local coordinate system that has anaxis aligned with the plate. Use the following *TRANSFORM option:


10.0E-2,5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

This option block defines the degrees of freedom for node set ENDB in a local coordinatesystem whose x0-axis is aligned with the long axis of the plate (i.e., the local system is rotated30° about the global 3-axis).

5.5.5 Reviewing the input file--the history dataWe now review the history definition portion of the input file. A single step is needed to define thissimulation.

Step definition


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Your choice of *STEP definition must reflect the fact that this is a linear, static simulation. Ifthe preprocessor did not create the option blocks to define the step, add the following:

*STEP, PERTURBATION

Uniform pressure (20.0 kPa) load

*STATIC

The line following *STEP, PERTURBATION contains a clear description of the loadingapplied in this step.

Boundary conditions

The nodes at the left-hand end of the plate (node set ENDA) need to be constrained completely.If necessary, add the following to your input file:

*BOUNDARY

ENDA, ENCASTRE

The nodes at the right-hand end of the plate need to be constrained to model their "rail"boundary condition. Since you have transformed the nodes at this end using *TRANSFORM,you must apply the boundary conditions in the local coordinate system. To allow these nodesto move in the local 1-direction (along the axis of the plate) only, all other degrees of freedommust be constrained as follows:

ENDB, 2,6

Had you not defined node sets ENDA and ENDB, you would have had to create a data line foreach node.

Loading

A distributed pressure load of 20.0 kPa is applied to the plate in this simulation. As shown inFigure 5-10, the pressure acts in the negative global 3-direction. Pressure loads are applied tothe faces of elements with the *DLOAD option (*DLOAD is described in Chapter 4, "UsingContinuum Elements," for the lug model example). Shell elements have only one face;therefore, the load identifier for pressure is just "P." A positive pressure on a shell acts in thedirection of the positive element normal. The shell elements in the input file from ``Skewplate,'' Section A.3, have normals that align with the positive global 3-axis. Thus, thefollowing input defines the correct pressure loading in that model:

*DLOAD

PLATE, P, -20000.0

Since element set PLATE contains all elements in the model, this option block applies apressure load to all elements in the model.

Output requests

If the preprocessor has generated default output request options, you should delete them. Tocreate an output database ( .odb) file for use with ABAQUS/Viewer and printed tables of theelement stresses, nodal reaction forces, and displacements at the midspan of the plate, add the

Your choice of *STEP definition must reflect the fact that this is a linear, static simulation. Ifthe preprocessor did not create the option blocks to define the step, add the following:

*STEP, PERTURBATION


*STATIC

The line following *STEP, PERTURBATION contains a clear description of the loadingapplied in this step.

Boundary conditions

The nodes at the left-hand end of the plate (node set ENDA) need to be constrained completely.If necessary, add the following to your input file:

*BOUNDARY

ENDA, ENCASTRE

The nodes at the right-hand end of the plate need to be constrained to model their "rail"boundary condition. Since you have transformed the nodes at this end using *TRANSFORM,you must apply the boundary conditions in the local coordinate system. To allow these nodesto move in the local 1-direction (along the axis of the plate) only, all other degrees of freedommust be constrained as follows:

ENDB, 2,6

Had you not defined node sets ENDA and ENDB, you would have had to create a data line foreach node.

Loading

A distributed pressure load of 20.0 kPa is applied to the plate in this simulation. As shown inFigure 5-10, the pressure acts in the negative global 3-direction. Pressure loads are applied tothe faces of elements with the *DLOAD option (*DLOAD is described in Chapter 4, "UsingContinuum Elements," for the lug model example). Shell elements have only one face;therefore, the load identifier for pressure is just "P." A positive pressure on a shell acts in thedirection of the positive element normal. The shell elements in the input file from ``Skewplate,'' Section A.3, have normals that align with the positive global 3-axis. Thus, thefollowing input defines the correct pressure loading in that model:

*DLOAD

PLATE, P, -20000.0

Since element set PLATE contains all elements in the model, this option block applies apressure load to all elements in the model.

Output requests

If the preprocessor has generated default output request options, you should delete them. Tocreate an output database ( .odb) file for use with ABAQUS/Viewer and printed tables of theelement stresses, nodal reaction forces, and displacements at the midspan of the plate, add the


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following output requests to your input file:

*OUTPUT, FIELD, OP=NEW

*NODE OUTPUT

U,

*ELEMENT OUTPUT

S,

*OUTPUT, HISTORY, OP=NEW

*NODE OUTPUT, NSET=MIDSPAN

U,

*EL PRINT

S,

*NODE PRINT, SUMMARY=NO, TOTALS=YES

RF,

*NODE PRINT, NSET=MIDSPAN

U,

Specifying the *OUTPUT option overrides the default output selections noted in the previouschapters. The option is used with the FIELD and HISTORY parameters to request field andhistory output to the output database file. In general, field output is used to generate contourplots, symbol plots, and deformed shape plots; history output is used for X-Y plotting. Inconjunction with the *OUTPUT option, the *NODE OUTPUT option is used to request outputof nodal variables and the *ELEMENT OUTPUT option is used for output of elementvariables.

You may also want output to be written to a results file, skew.fil, for use with a third-partypostprocessor. Use the *NODE FILE and *EL FILE options, which are very similar in formatto the *NODE PRINT and *EL PRINT options, to specify which results are to be stored in theresults file. The following input stores the displacements at all nodes and the stresses in allelements in the results file:

*NODE FILE

U,

*EL FILE

S,

*END STEP

5.5.6 Running the analysisAfter storing your input in a file called skew.inp, run the analysis interactively. If you do notremember how to run the analysis, see ``Running the analysis,'' Section 4.3.6. If your analysis does notcomplete, check the data file, skew.dat, for error messages. Modify your input file to remove theerrors; if you still have trouble running your model, compare your input file to the one given in ``Skewplate,'' Section A.3.

5.5.7 Results

following output requests to your input file:

*OUTPUT, FIELD, OP=NEW

*NODE OUTPUT

U,

*ELEMENT OUTPUT

S,

*OUTPUT, HISTORY, OP=NEW


U,

*EL PRINT

S,


RF,


U,

Specifying the *OUTPUT option overrides the default output selections noted in the previouschapters. The option is used with the FIELD and HISTORY parameters to request field andhistory output to the output database file. In general, field output is used to generate contourplots, symbol plots, and deformed shape plots; history output is used for X-Y plotting. Inconjunction with the *OUTPUT option, the *NODE OUTPUT option is used to request outputof nodal variables and the *ELEMENT OUTPUT option is used for output of elementvariables.

You may also want output to be written to a results file, skew.fil, for use with a third-partypostprocessor. Use the *NODE FILE and *EL FILE options, which are very similar in formatto the *NODE PRINT and *EL PRINT options, to specify which results are to be stored in theresults file. The following input stores the displacements at all nodes and the stresses in allelements in the results file:

*NODE FILE

U,

*EL FILE

S,

*END STEP

5.5.6 Running the analysisAfter storing your input in a file called skew.inp, run the analysis interactively. If you do notremember how to run the analysis, see ``Running the analysis,'' Section 4.3.6. If your analysis does notcomplete, check the data file, skew.dat, for error messages. Modify your input file to remove theerrors; if you still have trouble running your model, compare your input file to the one given in ``Skewplate,'' Section A.3.

5.5.7 Results


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After running the simulation successfully, look at the table of stresses in the data file, skew.dat. Anexcerpt from the table is shown below.

The second column (SEC PT--section point) identifies the location in the element where the stress wascalculated. Section point 1 lies on the SNEG surface of the shell, and section point 3 lies on the SPOSsurface. The letters OR appear in the FOOTNOTE column, indicating that an *ORIENTATION optionhas been used for the element: the stresses refer to a local coordinate system.

Check that the small-strain assumption was valid for this simulation. The axial strain corresponding tothe peak stress is "11 ¼ 0.0066. Since the strain is typically considered small if it is less than 4 or 5%,a strain of 0.0066 is well within the appropriate range to be modeled with S8R5 elements.

Look at the reaction forces and moments in the following table:

After running the simulation successfully, look at the table of stresses in the data file, skew.dat. Anexcerpt from the table is shown below.

The second column (SEC PT--section point) identifies the location in the element where the stress wascalculated. Section point 1 lies on the SNEG surface of the shell, and section point 3 lies on the SPOSsurface. The letters OR appear in the FOOTNOTE column, indicating that an *ORIENTATION optionhas been used for the element: the stresses refer to a local coordinate system.

Check that the small-strain assumption was valid for this simulation. The axial strain corresponding tothe peak stress is "11 ¼ 0.0066. Since the strain is typically considered small if it is less than 4 or 5%,a strain of 0.0066 is well within the appropriate range to be modeled with S8R5 elements.

Look at the reaction forces and moments in the following table:


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To create the output shown, we included the additional parameter GLOBAL=YES on the *NODEPRINT option to obtain reaction forces in the global coordinate system. Check that the sum of thereaction forces and reaction moments with the corresponding applied loads is zero. The nonzeroreaction force in the 3-direction equilibrates the vertical force of the pressure load (20 kPa � 1.0 m �0.4 m). In addition to the reaction forces, the pressure load causes self-equilibrating reaction momentsat the constrained rotational degrees of freedom.

The table of displacements (which is not shown here) shows that the mid-span deflection across theplate is 5.3 cm, which is approximately 5% of the plate's length. By running this as a linear analysis,we assume the displacements to be small. It is questionable whether these displacements are trulysmall relative to the dimensions of the structure; nonlinear effects may be important, requiring furtherinvestigation. In this case we need to perform a geometrically nonlinear analysis, which is discussed inChapter 7, "Nonlinearity."

5.5.8 PostprocessingWhile you have been able to answer this simulation's important questions using the printed output inthe data file, more complicated simulations often require graphical postprocessing. This sectiondiscusses postprocessing with ABAQUS/Viewer. Both contour and symbol plots are useful forvisualizing shell analysis results. Since contour plotting was discussed in detail in Chapter 4, "UsingContinuum Elements," we use symbol plots here.


abaqus viewer odb=skew


By default, ABAQUS/Viewer plots the fast representation of the model. Plot the undeformed model

To create the output shown, we included the additional parameter GLOBAL=YES on the *NODEPRINT option to obtain reaction forces in the global coordinate system. Check that the sum of thereaction forces and reaction moments with the corresponding applied loads is zero. The nonzeroreaction force in the 3-direction equilibrates the vertical force of the pressure load (20 kPa � 1.0 m �0.4 m). In addition to the reaction forces, the pressure load causes self-equilibrating reaction momentsat the constrained rotational degrees of freedom.

The table of displacements (which is not shown here) shows that the mid-span deflection across theplate is 5.3 cm, which is approximately 5% of the plate's length. By running this as a linear analysis,we assume the displacements to be small. It is questionable whether these displacements are trulysmall relative to the dimensions of the structure; nonlinear effects may be important, requiring furtherinvestigation. In this case we need to perform a geometrically nonlinear analysis, which is discussed inChapter 7, "Nonlinearity."

5.5.8 PostprocessingWhile you have been able to answer this simulation's important questions using the printed output inthe data file, more complicated simulations often require graphical postprocessing. This sectiondiscusses postprocessing with ABAQUS/Viewer. Both contour and symbol plots are useful forvisualizing shell analysis results. Since contour plotting was discussed in detail in Chapter 4, "UsingContinuum Elements," we use symbol plots here.


abaqus viewer odb=skew


By default, ABAQUS/Viewer plots the fast representation of the model. Plot the undeformed model


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shape by selecting Plot->Undeformed Shape from the main menu bar or by clicking the tool inthe toolbox.

Element normals

Use the undeformed shape plot to check the model definition. Check that the element normalsfor the skew-plate model were defined correctly and point in the positive 3-direction.

To display the element normals:

1. In the prompt area, click Undeformed Options.


2. Choose Exterior visible edges and set the render style to Shaded.

3. Click the Normals tab.

4. Toggle on Show Normals.

5. Click OK to apply the settings and to close the dialog box.

The default view is isometric. You can change the view using the options in the view menu or

the view tools (such as ) from the toolbar.

To change the view:




3. Enter the X-, Y - and Z-coordinates of the viewpoint vector as -0.2, -1, 0.8 and thecoordinates of the up vector as 0, 0, 1.

4. Click OK.


Figure 5-13 Shell element normals in skewed plate model.

shape by selecting Plot->Undeformed Shape from the main menu bar or by clicking the tool inthe toolbox.

Element normals

Use the undeformed shape plot to check the model definition. Check that the element normalsfor the skew-plate model were defined correctly and point in the positive 3-direction.

To display the element normals:



2. Choose Exterior visible edges and set the render style to Shaded.

3. Click the Normals tab.

4. Toggle on Show Normals.


The default view is isometric. You can change the view using the options in the view menu or

the view tools (such as ) from the toolbar.

To change the view:




3. Enter the X-, Y - and Z-coordinates of the viewpoint vector as -0.2, -1, 0.8 and thecoordinates of the up vector as 0, 0, 1.

4. Click OK.


Figure 5-13 Shell element normals in skewed plate model.


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Symbol plots

Symbol plots display the specified variable as a vector originating from the node. You canproduce symbol plots of most tensor- and vector-valued variables. The exceptions are mainlynon-mechanical output variables and element results stored at nodes, such as nodal forces. Therelative size of the arrows indicate the relative magnitude of the results, and the vectors areoriented along the global direction of the results. You can plot results for the resultant ofvariables such as displacement (U), reaction force (RF), etc., or you can plot individualcomponents of these variables.

To generate a symbol plot of the displacement:



2. From the list of output variables, select U.

3. From the list of components, select U3.

4. Click OK.

The Select Plot Mode dialog box appears.

5. Toggle on Symbol, and click OK.

ABAQUS/Viewer displays a symbol plot of the displacements in the 3-direction.

6. To modify the attributes of the symbol plot, click Symbol Options in the prompt area.

The Symbol Plot Options dialog box appears; by default, the Basic tab is selected.

7. Choose Exterior visible edges.

8. To plot the symbols on the undeformed shape, click the Shape tab and toggle onUndeformed shape.


Symbol plots

Symbol plots display the specified variable as a vector originating from the node. You canproduce symbol plots of most tensor- and vector-valued variables. The exceptions are mainlynon-mechanical output variables and element results stored at nodes, such as nodal forces. Therelative size of the arrows indicate the relative magnitude of the results, and the vectors areoriented along the global direction of the results. You can plot results for the resultant ofvariables such as displacement (U), reaction force (RF), etc., or you can plot individualcomponents of these variables.

To generate a symbol plot of the displacement:



2. From the list of output variables, select U.

3. From the list of components, select U3.

4. Click OK.


5. Toggle on Symbol, and click OK.

ABAQUS/Viewer displays a symbol plot of the displacements in the 3-direction.

6. To modify the attributes of the symbol plot, click Symbol Options in the prompt area.

The Symbol Plot Options dialog box appears; by default, the Basic tab is selected.


8. To plot the symbols on the undeformed shape, click the Shape tab and toggle onUndeformed shape.



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A symbol plot on the undeformed shape appears, as shown in Figure 5-14.

Figure 5-14 Symbol plot of displacement.

You can plot principal values of tensor variables such as stress using symbol plots. A symbolplot of the principal values of stress yields three vectors at every integration point, eachcorresponding to a principal value oriented along the corresponding principal direction.Compressive values are indicated by arrows pointing toward the node, and tensile values areindicated by arrows pointing away from the node. You can also plot individual principalvalues.

To generate a symbol plot of the principal stresses:


The Field Output dialog box appears.

2. From the list of output variables, select S; and from the list of invariants, select Max.Principal.


ABAQUS/Viewer displays a symbol plot of principal stresses.

4. To change the arrow length, click Symbol Options in the prompt area.

The Symbol Plot Options dialog box appears.

5. Click the Color & Style tab; then click the Tensor tab.

6. Set the Max length option to Short.

A symbol plot on the undeformed shape appears, as shown in Figure 5-14.

Figure 5-14 Symbol plot of displacement.

You can plot principal values of tensor variables such as stress using symbol plots. A symbolplot of the principal values of stress yields three vectors at every integration point, eachcorresponding to a principal value oriented along the corresponding principal direction.Compressive values are indicated by arrows pointing toward the node, and tensile values areindicated by arrows pointing away from the node. You can also plot individual principalvalues.

To generate a symbol plot of the principal stresses:



2. From the list of output variables, select S; and from the list of invariants, select Max.Principal.


ABAQUS/Viewer displays a symbol plot of principal stresses.

4. To change the arrow length, click Symbol Options in the prompt area.

The Symbol Plot Options dialog box appears.

5. Click the Color & Style tab; then click the Tensor tab.

6. Set the Max length option to Short.


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The symbol plot shown in Figure 5-15 appears.

Figure 5-15 Symbol plot of principal stresses on the bottom surface of the plate.

8. The principal stresses are displayed at section point 1 by default. To plot stresses atnon-default section points, select Result->Section Points from the main menu bar to bring upthe Section Points dialog box.

9. Select the desired non-default section point for plotting.

10. In a complex model, the element edges can obscure the symbol plots. To suppress the displayof the element edges, choose Feature visible edges on the Basic page in the Symbol PlotOptions dialog box. Figure 5-16shows a symbol plot of the principal stresses with onlyfeature edges visible.

Figure 5-16 Feature edge plot showing vectors of principal stress.


The symbol plot shown in Figure 5-15 appears.

Figure 5-15 Symbol plot of principal stresses on the bottom surface of the plate.

8. The principal stresses are displayed at section point 1 by default. To plot stresses atnon-default section points, select Result->Section Points from the main menu bar to bring upthe Section Points dialog box.

9. Select the desired non-default section point for plotting.

10. In a complex model, the element edges can obscure the symbol plots. To suppress the displayof the element edges, choose Feature visible edges on the Basic page in the Symbol PlotOptions dialog box. Figure 5-16shows a symbol plot of the principal stresses with onlyfeature edges visible.

Figure 5-16 Feature edge plot showing vectors of principal stress.


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Material directions

ABAQUS/Viewer also lets you visualize the element material directions. This feature isparticularly helpful, allowing you to ensure the correctness of the material directions.

To plot the material directions:

1. From the main menu bar, select Plot->Material Orientation; or click the tool in thetoolbox.

The material orientation directions are plotted on the deformed shape. By default, the triadsthat represent the material orientation directions are plotted without arrowheads.

2. To display the triads with arrowheads, click Material Orientation Options in the prompt area.

The Material Orientation Plot Options dialog box appears.

3. Click the Color & Style tab; then click the Triad tab.

4. Set the Arrowhead option to use filled arrowheads in the triad.


6. From the main menu bar, select View->Views Toolbox; or click the tool in the toolbar.

The Views toolbox appears.

7. Use the predefined views available in the toolbox to display the plate as shown in Figure 5-17.

In this figure, perspective is turned off. To turn off perspective, click the tool in thetoolbar.

By default, the material 1-direction is colored blue, the material 2-direction is colored yellow,

Material directions

ABAQUS/Viewer also lets you visualize the element material directions. This feature isparticularly helpful, allowing you to ensure the correctness of the material directions.

To plot the material directions:

1. From the main menu bar, select Plot->Material Orientation; or click the tool in thetoolbox.

The material orientation directions are plotted on the deformed shape. By default, the triadsthat represent the material orientation directions are plotted without arrowheads.

2. To display the triads with arrowheads, click Material Orientation Options in the prompt area.

The Material Orientation Plot Options dialog box appears.

3. Click the Color & Style tab; then click the Triad tab.

4. Set the Arrowhead option to use filled arrowheads in the triad.


6. From the main menu bar, select View->Views Toolbox; or click the tool in the toolbar.

The Views toolbox appears.

7. Use the predefined views available in the toolbox to display the plate as shown in Figure 5-17.

In this figure, perspective is turned off. To turn off perspective, click the tool in thetoolbar.

By default, the material 1-direction is colored blue, the material 2-direction is colored yellow,


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and, if it is present, the material 3-direction is colored red.

Figure 5-17 Plot of material orientation directions in the plate.

5.6 Related ABAQUS examples� ``Pressurized fuel tank with variable shell thickness, '' Section 2.1.5 of the ABAQUS Example

Problems Manual

� ``Analysis of an anisotropic layered plate,'' Section 1.1.2 of the ABAQUS Benchmarks Manual

� ``Buckling of a simply supported square plate, '' Section 1.2.4 of the ABAQUS BenchmarksManual

� ``The barrel vault roof problem,'' Section 2.3.1 of the ABAQUS Benchmarks Manual

5.7 Suggested readingThe following references provide a more in-depth treatment of the theoretical and computationalaspects of shell theory.

Basic shell theory

� Timoshenko, S., Strength of Materials: Part II, Krieger Publishing Co., 1958.

� Timoshenko, S. and S. W. Krieger, Theory of Plates and Shells , McGraw-Hill, Inc., 1959.

and, if it is present, the material 3-direction is colored red.

Figure 5-17 Plot of material orientation directions in the plate.

5.6 Related ABAQUS examples� ``Pressurized fuel tank with variable shell thickness, '' Section 2.1.5 of the ABAQUS Example

Problems Manual

� ``Analysis of an anisotropic layered plate,'' Section 1.1.2 of the ABAQUS Benchmarks Manual

� ``Buckling of a simply supported square plate, '' Section 1.2.4 of the ABAQUS BenchmarksManual

� ``The barrel vault roof problem,'' Section 2.3.1 of the ABAQUS Benchmarks Manual

5.7 Suggested readingThe following references provide a more in-depth treatment of the theoretical and computationalaspects of shell theory.

Basic shell theory


� Timoshenko, S. and S. W. Krieger, Theory of Plates and Shells , McGraw-Hill, Inc., 1959.


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� Ugural, A. C., Stresses in Plates and Shells , McGraw-Hill, Inc., 1981.

Basic computational shell theory



Advanced shell theory

� Budiansky, B., and J. L. Sanders, "On the `Best' First-Order Linear Shell Theory,"Progress in Applied Mechanics, The Prager Anniversary Volume, 129-140, 1963.

Advanced computational shell theory

� Ashwell, D. G., and R. H. Gallagher, Finite Elements for Thin Shells and CurvedMembers, John Wiley & Sons, 1976.

� Hughes, T. J. R., T. E. Tezduyar, "Finite Elements Based upon Mindlin Plate Theory withParticular Reference to the Four-Node Bilinear Isoparametric Element," Journal ofApplied Mechanics, 587-596, 1981.

� Simo, J. C., D. D. Fox, and M. S. Rifai, "On a Stress Resultant Geometrically Exact ShellModel. Part III: Computational Aspects of the Nonlinear Theory," Computer Methods inApplied Mechanics and Engineering, vol. 79, 21-70, 1990.

5.8 Summary� The cross-section behavior of shell elements can be determined using numerical integration

through the shell thickness (*SHELL SECTION) or using a cross-section stiffness calculated at thebeginning of the analysis (*SHELL GENERAL SECTION).

� *SHELL GENERAL SECTION is efficient, but it can be used only with linear materials. *SHELLSECTION can be used with both linear and nonlinear materials.

� Numerical integration is performed at a number of section points through the shell thickness.These section points are the locations at which element variables can be output. The defaultoutermost section points lie on the surfaces of the shell.

� The direction of a shell element's normal determines the positive and negative surfaces of theelement. To define contact and interpret element output correctly, you must know which surface iswhich. The shell normal also defines the direction of positive pressure loads applied to the elementand can be plotted in ABAQUS/Viewer.

� Ugural, A. C., Stresses in Plates and Shells , McGraw-Hill, Inc., 1981.

Basic computational shell theory



Advanced shell theory

� Budiansky, B., and J. L. Sanders, "On the `Best' First-Order Linear Shell Theory,"Progress in Applied Mechanics, The Prager Anniversary Volume, 129-140, 1963.

Advanced computational shell theory

� Ashwell, D. G., and R. H. Gallagher, Finite Elements for Thin Shells and CurvedMembers, John Wiley & Sons, 1976.

� Hughes, T. J. R., T. E. Tezduyar, "Finite Elements Based upon Mindlin Plate Theory withParticular Reference to the Four-Node Bilinear Isoparametric Element," Journal ofApplied Mechanics, 587-596, 1981.

� Simo, J. C., D. D. Fox, and M. S. Rifai, "On a Stress Resultant Geometrically Exact ShellModel. Part III: Computational Aspects of the Nonlinear Theory," Computer Methods inApplied Mechanics and Engineering, vol. 79, 21-70, 1990.

5.8 Summary� The cross-section behavior of shell elements can be determined using numerical integration

through the shell thickness (*SHELL SECTION) or using a cross-section stiffness calculated at thebeginning of the analysis (*SHELL GENERAL SECTION).

� *SHELL GENERAL SECTION is efficient, but it can be used only with linear materials. *SHELLSECTION can be used with both linear and nonlinear materials.

� Numerical integration is performed at a number of section points through the shell thickness.These section points are the locations at which element variables can be output. The defaultoutermost section points lie on the surfaces of the shell.

� The direction of a shell element's normal determines the positive and negative surfaces of theelement. To define contact and interpret element output correctly, you must know which surface iswhich. The shell normal also defines the direction of positive pressure loads applied to the elementand can be plotted in ABAQUS/Viewer.


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� Shell elements use material directions local to each element. In large-displacement analyses thelocal material axes rotate with the element. *ORIENTATION can be used to define non-defaultlocal coordinate systems. The element variables, such as stress and strain, are output in the localdirections.

� *TRANSFORM defines local coordinate systems for nodes. Concentrated loads and boundaryconditions are applied in the local coordinate system. All nodal output, such as displacements, alsorefer to the local system by default.

� Symbol plots can help you visualize the results from a simulation. They are especially useful forvisualizing the motion and load paths of a structure.

� Shell elements use material directions local to each element. In large-displacement analyses thelocal material axes rotate with the element. *ORIENTATION can be used to define non-defaultlocal coordinate systems. The element variables, such as stress and strain, are output in the localdirections.

� *TRANSFORM defines local coordinate systems for nodes. Concentrated loads and boundaryconditions are applied in the local coordinate system. All nodal output, such as displacements, alsorefer to the local system by default.

� Symbol plots can help you visualize the results from a simulation. They are especially useful forvisualizing the motion and load paths of a structure.


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6. Using Beam ElementsUse beam elements to model structures in which one dimension (the length) is significantly greaterthan the other two dimensions and in which the longitudinal stress is most important. Beam theory isbased on the assumption that the deformation of the structure can be determined entirely fromvariables that are functions of position along the structure's length. For beam theory to produceacceptable results, the cross-section dimensions should be less than 1/10 of the structure's typical axialdimension. The following are examples of typical axial dimensions:


� the distance between gross changes in cross-section, and


ABAQUS beam elements assume that plane sections perpendicular to the axis of the beam remainplane. Do not be confused into thinking that the cross-section dimensions should be less than 1/10 of atypical element length. A highly refined mesh may contain beam elements whose length is less thantheir cross-section dimensions, although this is not generally recommended--continuum elements maybe more suitable in such a case.

6.1 Element cross-section geometryYou can use either the *BEAM SECTION option or the *BEAM GENERAL SECTION option todefine the beam section. With either option you can define the beam cross-section geometrically byspecifying the shape and dimensions of the section. Alternatively, the *BEAM GENERAL SECTIONoption can be used to define the beam section through section engineering properties, such as area andmoment of inertia.

ABAQUS offers a variety of common section shapes, as shown in Figure 6-1, should you decide todefine the beam section geometrically.

Figure 6-1 Beam cross-sections.

6. Using Beam ElementsUse beam elements to model structures in which one dimension (the length) is significantly greaterthan the other two dimensions and in which the longitudinal stress is most important. Beam theory isbased on the assumption that the deformation of the structure can be determined entirely fromvariables that are functions of position along the structure's length. For beam theory to produceacceptable results, the cross-section dimensions should be less than 1/10 of the structure's typical axialdimension. The following are examples of typical axial dimensions:


� the distance between gross changes in cross-section, and


ABAQUS beam elements assume that plane sections perpendicular to the axis of the beam remainplane. Do not be confused into thinking that the cross-section dimensions should be less than 1/10 of atypical element length. A highly refined mesh may contain beam elements whose length is less thantheir cross-section dimensions, although this is not generally recommended--continuum elements maybe more suitable in such a case.

6.1 Element cross-section geometryYou can use either the *BEAM SECTION option or the *BEAM GENERAL SECTION option todefine the beam section. With either option you can define the beam cross-section geometrically byspecifying the shape and dimensions of the section. Alternatively, the *BEAM GENERAL SECTIONoption can be used to define the beam section through section engineering properties, such as area andmoment of inertia.

ABAQUS offers a variety of common section shapes, as shown in Figure 6-1, should you decide todefine the beam section geometrically.

Figure 6-1 Beam cross-sections.

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SECTION=ARBITRARY allows you to define almost any thin-walled cross-section.

The basic format of the *BEAM SECTION option is

*BEAM SECTION, ELSET=<element set name>, SECTION=<section type>,

MATERIAL=<material name>

<cross-section dimensions>

<n11>,<n

21>,<n

31>

Set the SECTION parameter to one of the cross-sections shown in Figure 6-1. Provide the requiredcross-section dimensions, which are different for each type of cross-section, as specified in theABAQUS/Standard User's Manual. The vector on the second data line defines the approximate normal,n1 , which is explained later in this section.

The basic format of the *BEAM GENERAL SECTION option is

*BEAM GENERAL SECTION, ELSET=<element set name>, SECTION=<section type>

<cross-section dimensions> or <section engineering properties>

<n11>,<n

21>,<n

31>

<Young's modulus(E)>,<torsional shear modulus(G)>

To define the section's properties geometrically, set the SECTION parameter to one of thecross-sections shown in Figure 6-1. In this case you provide the required cross-section dimensions thesame way as you would with *BEAM SECTION. The vector on the second data line again defines theapproximate normal, n1 . On the third line you enter the elastic material constants, because *BEAMGENERAL SECTION does not refer to any material option block. If you define the section's propertiesgeometrically with this option, the material behavior must be linear elastic.

The alternative is to set the SECTION parameter to GENERAL or NONLINEAR GENERAL, in whichcase you provide the section engineering properties (area, moments of inertia, and torsional constants)instead of the cross-section dimensions. These parameters allow you to combine the beam's geometryand material behavior to define its response to loads. This response may be linear or nonlinear. See theABAQUS/Standard User's Manual for further details.

6.1.1 Section points

SECTION=ARBITRARY allows you to define almost any thin-walled cross-section.

The basic format of the *BEAM SECTION option is

*BEAM SECTION, ELSET=<element set name>, SECTION=<section type>,

MATERIAL=<material name>

<cross-section dimensions>

<n11>,<n

21>,<n

31>

Set the SECTION parameter to one of the cross-sections shown in Figure 6-1. Provide the requiredcross-section dimensions, which are different for each type of cross-section, as specified in theABAQUS/Standard User's Manual. The vector on the second data line defines the approximate normal,n1 , which is explained later in this section.

The basic format of the *BEAM GENERAL SECTION option is

*BEAM GENERAL SECTION, ELSET=<element set name>, SECTION=<section type>

<cross-section dimensions> or <section engineering properties>

<n11>,<n

21>,<n

31>

<Young's modulus(E)>,<torsional shear modulus(G)>

To define the section's properties geometrically, set the SECTION parameter to one of thecross-sections shown in Figure 6-1. In this case you provide the required cross-section dimensions thesame way as you would with *BEAM SECTION. The vector on the second data line again defines theapproximate normal, n1 . On the third line you enter the elastic material constants, because *BEAMGENERAL SECTION does not refer to any material option block. If you define the section's propertiesgeometrically with this option, the material behavior must be linear elastic.

The alternative is to set the SECTION parameter to GENERAL or NONLINEAR GENERAL, in whichcase you provide the section engineering properties (area, moments of inertia, and torsional constants)instead of the cross-section dimensions. These parameters allow you to combine the beam's geometryand material behavior to define its response to loads. This response may be linear or nonlinear. See theABAQUS/Standard User's Manual for further details.

6.1.1 Section points

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When you specify *BEAM SECTION, ABAQUS calculates the beam element's response at an array ofsection points throughout the beam cross-section. The number of section points, as well as the sectionpoint locations, are shown in the ABAQUS/Standard User's Manual. Element output variables, such asstress and strain, are available at any of the section points; however, by default output is provided atonly a select number of section points, as listed in the ABAQUS/Standard User's Manual. All thesection points for a rectangular cross-section (SECTION=RECT) are shown in Figure 6-2.

Figure 6-2 Integration and default section points in a B32 rectangular beam element.

For this cross-section output is provided at points 1, 5, 21, and 25 by default. The beam element shownin Figure 6-2 uses a total of 50 section points, 25 at each of the two integration points, to calculate itsstiffness.

When you specify *BEAM GENERAL SECTION, ABAQUS does not calculate the beam's response atthe section points. Instead, it uses the section engineering properties to determine the section response.Therefore, ABAQUS uses section points only as locations for output, and you need to specify thesection points at which you desire output. Use the *SECTION POINTS option, which must follow the*BEAM GENERAL SECTION option, to specify the location of the section points:

*SECTION POINTS

<x1>,<x2>

The x1- and x2-coordinates are given in the local 1-2 coordinate system of the beam cross-section. Forexample, if we require stresses at the corners of an element with the rectangular beam cross-sectionshown in Figure 6-3, we would use the following option block:

*SECTION POINTS

-0.01, -0.005

0.01, -0.005

0.01, 0.005

-0.01, 0.005

Figure 6-3 Section points at the corners of a rectangular beam.

When you specify *BEAM SECTION, ABAQUS calculates the beam element's response at an array ofsection points throughout the beam cross-section. The number of section points, as well as the sectionpoint locations, are shown in the ABAQUS/Standard User's Manual. Element output variables, such asstress and strain, are available at any of the section points; however, by default output is provided atonly a select number of section points, as listed in the ABAQUS/Standard User's Manual. All thesection points for a rectangular cross-section (SECTION=RECT) are shown in Figure 6-2.

Figure 6-2 Integration and default section points in a B32 rectangular beam element.

For this cross-section output is provided at points 1, 5, 21, and 25 by default. The beam element shownin Figure 6-2 uses a total of 50 section points, 25 at each of the two integration points, to calculate itsstiffness.

When you specify *BEAM GENERAL SECTION, ABAQUS does not calculate the beam's response atthe section points. Instead, it uses the section engineering properties to determine the section response.Therefore, ABAQUS uses section points only as locations for output, and you need to specify thesection points at which you desire output. Use the *SECTION POINTS option, which must follow the*BEAM GENERAL SECTION option, to specify the location of the section points:

*SECTION POINTS

<x1>,<x2>

The x1- and x2-coordinates are given in the local 1-2 coordinate system of the beam cross-section. Forexample, if we require stresses at the corners of an element with the rectangular beam cross-sectionshown in Figure 6-3, we would use the following option block:

*SECTION POINTS

-0.01, -0.005

0.01, -0.005

0.01, 0.005

-0.01, 0.005

Figure 6-3 Section points at the corners of a rectangular beam.

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The points you specify are assigned identifying numbers based on the order they are given; i.e., thefirst point is section point 1, the second is section point 2, etc.

6.1.2 Cross-section orientationYou must define the orientation of a beam's cross-section in global Cartesian space. The local tangentalong the beam element, t, is defined as the vector from the first node of the element to the next node.The beam cross-section is perpendicular to this local tangent. The local (1-2) beam section axes arerepresented by the vectors n1 and n2 . The three vectors t, n1 , and n2 form a local, right-handed,coordinate system (see Figure 6-4).

Figure 6-4 Orientation of the beam element tangent, t, and beam section axes, n1 and n2 .

The n1-direction is always (0.0, 0.0, -1.0) for two-dimensional beam elements.

For three-dimensional beam elements there are several ways to define the orientation of the local beamsection axes. The simplest is to specify an extra node on the data line defining the element in the*ELEMENT option. The vector, v, from the first node in the beam element to this additional node (seeFigure 6-4), is used initially as an approximate n1-direction. ABAQUS then defines the beam'sn2-direction as t£ v. Having determined n2 , ABAQUS defines the actual n1-direction as n2 £ t.This procedure ensures that the local tangent and local beam section axes form an orthogonal system.

Alternatively, you can give an approximate n1-direction on the element section option (either *BEAMSECTION or *BEAM GENERAL SECTION). ABAQUS then uses the procedure described above tocalculate the actual beam section axes. If you specify both an extra node and an approximaten1-direction, the additional node method takes precedence. ABAQUS uses the vector from the originto the point (0.0, 0.0, -1.0) as the default n1-direction if you provide no approximate n1-direction.

There are two methods that can be used to override the n2-direction defined by ABAQUS. One is togive the components of n2 as the 4th, 5th, and 6th data values following the nodal coordinates on the

The points you specify are assigned identifying numbers based on the order they are given; i.e., thefirst point is section point 1, the second is section point 2, etc.

6.1.2 Cross-section orientationYou must define the orientation of a beam's cross-section in global Cartesian space. The local tangentalong the beam element, t, is defined as the vector from the first node of the element to the next node.The beam cross-section is perpendicular to this local tangent. The local (1-2) beam section axes arerepresented by the vectors n1 and n2 . The three vectors t, n1 , and n2 form a local, right-handed,coordinate system (see Figure 6-4).

Figure 6-4 Orientation of the beam element tangent, t, and beam section axes, n1 and n2 .

The n1-direction is always (0.0, 0.0, -1.0) for two-dimensional beam elements.

For three-dimensional beam elements there are several ways to define the orientation of the local beamsection axes. The simplest is to specify an extra node on the data line defining the element in the*ELEMENT option. The vector, v, from the first node in the beam element to this additional node (seeFigure 6-4), is used initially as an approximate n1-direction. ABAQUS then defines the beam'sn2-direction as t£ v. Having determined n2 , ABAQUS defines the actual n1-direction as n2 £ t.This procedure ensures that the local tangent and local beam section axes form an orthogonal system.

Alternatively, you can give an approximate n1-direction on the element section option (either *BEAMSECTION or *BEAM GENERAL SECTION). ABAQUS then uses the procedure described above tocalculate the actual beam section axes. If you specify both an extra node and an approximaten1-direction, the additional node method takes precedence. ABAQUS uses the vector from the originto the point (0.0, 0.0, -1.0) as the default n1-direction if you provide no approximate n1-direction.

There are two methods that can be used to override the n2-direction defined by ABAQUS. One is togive the components of n2 as the 4th, 5th, and 6th data values following the nodal coordinates on the

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data lines of the *NODE option. The alternative is to use the *NORMAL option. If both methods areused, the *NORMAL option takes precedence. ABAQUS again defines the n1-direction as n2 £ t.

The n2-direction that you provide need not be orthogonal to the beam element tangent, t. When youprovide the n2-direction, the local beam element tangent is redefined as the value of the cross-productn1 £ n2 . It is quite possible in this situation that the redefined local beam tangent, t, will not alignwith the beam axis, as defined by the vector from the first to the second node. If the n2-directionsubtends an angle greater than 20° with the plane perpendicular to the element axis, ABAQUS issues awarning message in the data file.

6.1.3 Beam element curvatureThe curvature of beam elements is based on the orientation of the beam's n2-direction relative to thebeam axis. If the n2-direction and the beam axis are not orthogonal (i.e., the beam axis and the tangent,t, do not coincide), the beam element is considered to be curved initially. Since the behavior of curvedbeams is different from the behavior of straight beams, you should always check your model to ensurethat the correct normals and, hence, the correct curvatures are used. For beams and shells ABAQUSuses the same algorithm to determine the normals at nodes shared by several elements. A description isgiven in Part V, "Elements," of the ABAQUS/Standard User's Manual.

If you intend to model curved beam structures, you should probably use one of the two methodsdescribed earlier to define the n2-direction directly, allowing you great control in modeling thecurvature. Even if you intend to model a structure made up of straight beams, curvature may beintroduced as normals are averaged at shared nodes. You can rectify this problem by defining the beamnormals directly as explained previously.

6.1.4 Nodal offsets in beam sectionsWhen beam elements are used as stiffeners for shell models, it is convenient to have the beam andshell elements share the same nodes. Shell element nodes are located at the mid-plane of the shell, andbeam element nodes are located somewhere in the cross-section of the beam. Hence, if the shell andbeam elements share the same nodes, the shell and the beam stiffener will overlap unless the beamcross-section is offset from the location of the node (see Figure 6-5).

Figure 6-5 Using beams as stiffeners for shell models: (a) without offset of beam sections; (b) with

offset of beam sections.

data lines of the *NODE option. The alternative is to use the *NORMAL option. If both methods areused, the *NORMAL option takes precedence. ABAQUS again defines the n1-direction as n2 £ t.

The n2-direction that you provide need not be orthogonal to the beam element tangent, t. When youprovide the n2-direction, the local beam element tangent is redefined as the value of the cross-productn1 £ n2 . It is quite possible in this situation that the redefined local beam tangent, t, will not alignwith the beam axis, as defined by the vector from the first to the second node. If the n2-directionsubtends an angle greater than 20° with the plane perpendicular to the element axis, ABAQUS issues awarning message in the data file.

6.1.3 Beam element curvatureThe curvature of beam elements is based on the orientation of the beam's n2-direction relative to thebeam axis. If the n2-direction and the beam axis are not orthogonal (i.e., the beam axis and the tangent,t, do not coincide), the beam element is considered to be curved initially. Since the behavior of curvedbeams is different from the behavior of straight beams, you should always check your model to ensurethat the correct normals and, hence, the correct curvatures are used. For beams and shells ABAQUSuses the same algorithm to determine the normals at nodes shared by several elements. A description isgiven in Part V, "Elements," of the ABAQUS/Standard User's Manual.

If you intend to model curved beam structures, you should probably use one of the two methodsdescribed earlier to define the n2-direction directly, allowing you great control in modeling thecurvature. Even if you intend to model a structure made up of straight beams, curvature may beintroduced as normals are averaged at shared nodes. You can rectify this problem by defining the beamnormals directly as explained previously.

6.1.4 Nodal offsets in beam sectionsWhen beam elements are used as stiffeners for shell models, it is convenient to have the beam andshell elements share the same nodes. Shell element nodes are located at the mid-plane of the shell, andbeam element nodes are located somewhere in the cross-section of the beam. Hence, if the shell andbeam elements share the same nodes, the shell and the beam stiffener will overlap unless the beamcross-section is offset from the location of the node (see Figure 6-5).

Figure 6-5 Using beams as stiffeners for shell models: (a) without offset of beam sections; (b) with

offset of beam sections.

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With beam section types I, TRAPEZOID, and ARBITRARY it is possible to specify that the sectiongeometry is located at some distance from the origin of the section's local coordinate system, which islocated at the element's nodes. Since it is easy to offset beams with such cross-sections from theirnodes, they can be used readily as stiffeners as shown in Figure 6-5(b). (If flange or web buckling ofthe stiffeners is important, shells should be used to model the stiffeners.)

The I-beam shown in Figure 6-6 is attached to a shell 1.2 units thick. The following input is used toorient the beam section as it is shown in the figure:

*BEAM SECTION, SECTION=I, ELSET=<element set name>, MATERIAL=<material>

-0.6, 2.4, 3.0, 2.0, 0.2, 0.2, 0.2

<n11>, <n

21>,<n

31>

Figure 6-6 I-beam used as stiffener for a shell element.

The first item on the first data line defines the offset of the beam node from the bottom of the I-section.The offset is one half of the shell thickness or 0.6. The remaining data items are the beam depth, thewidth of the bottom and top flanges, the thickness of the bottom and top flanges, and the thickness ofthe web.

You can give the location of the centroid and shear center if you specify the *BEAM GENERALSECTION option with the parameter SECTION=GENERAL. The *SHEAR CENTER and*CENTROID options allow these locations to be offset from the node, enabling you to model stiffenersreadily. For example, the input for the I-beam attached to the shell as shown in Figure 6-6 is:

6.2 Formulation and integrationAll beam elements in ABAQUS are "beam-column" elements--meaning they allow axial, bending, andtorsional deformation. The Timoshenko beam elements also consider the effects of transverse shear

With beam section types I, TRAPEZOID, and ARBITRARY it is possible to specify that the sectiongeometry is located at some distance from the origin of the section's local coordinate system, which islocated at the element's nodes. Since it is easy to offset beams with such cross-sections from theirnodes, they can be used readily as stiffeners as shown in Figure 6-5(b). (If flange or web buckling ofthe stiffeners is important, shells should be used to model the stiffeners.)

The I-beam shown in Figure 6-6 is attached to a shell 1.2 units thick. The following input is used toorient the beam section as it is shown in the figure:

*BEAM SECTION, SECTION=I, ELSET=<element set name>, MATERIAL=<material>

-0.6, 2.4, 3.0, 2.0, 0.2, 0.2, 0.2

<n11>, <n

21>,<n

31>

Figure 6-6 I-beam used as stiffener for a shell element.

The first item on the first data line defines the offset of the beam node from the bottom of the I-section.The offset is one half of the shell thickness or 0.6. The remaining data items are the beam depth, thewidth of the bottom and top flanges, the thickness of the bottom and top flanges, and the thickness ofthe web.

You can give the location of the centroid and shear center if you specify the *BEAM GENERALSECTION option with the parameter SECTION=GENERAL. The *SHEAR CENTER and*CENTROID options allow these locations to be offset from the node, enabling you to model stiffenersreadily. For example, the input for the I-beam attached to the shell as shown in Figure 6-6 is:

6.2 Formulation and integrationAll beam elements in ABAQUS are "beam-column" elements--meaning they allow axial, bending, andtorsional deformation. The Timoshenko beam elements also consider the effects of transverse shear

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deformation.

6.2.1 Shear deformationThe linear elements (B21 and B31) and the quadratic elements (B22 and B32) are shear deformable,Timoshenko beams; thus, they are suitable for modeling both thick members, in which sheardeformation is important, and slender beams, in which shear deformation is not important. Thecross-sections of these elements behave in the same manner as the cross-sections of the thick-shellelements, as illustrated in Figure 6-5(b) and discussed in ``Shell formulation - thick or thin, '' Section5.2.

Figure 6-7 Behavior of transverse beam sections in (a) slender beams and (b) thick beams.

ABAQUS assumes the transverse shear stiffness of these beam elements to be linear elastic andconstant. In addition, these beams are formulated so that their cross-sectional area can change as afunction of the axial deformation, an effect that is considered only in geometrically nonlinearsimulations (see Chapter 7, "Nonlinearity") in which the POISSON parameter on the beam sectionproperty option has a nonzero value. These elements can provide useful results as long as thecross-section dimensions are less than 1/10 of the typical axial dimensions of the structure, which isgenerally considered to be the limit of the applicability of beam theory; if the beam cross-section doesnot remain plane under bending deformation, beam theory is not adequate to model the deformation.

The cubic elements--the so-called Euler-Bernoulli beam elements ( B23 and B33)--do not model shearflexibility. The cross-sections of these elements remain perpendicular to the beam axis (see Figure6-5(a)). Therefore, the cubic beam elements are most effective for modeling structures with relativelyslender members. Since cubic elements model a cubic variation of displacement along their lengths, astructural member often can be modeled with a single cubic element for a static analysis and with asmall number of elements for a dynamic analysis. These elements assume that shear deformations arenegligible. Generally, if the cross-section dimensions are less than 1/15 of the typical axial dimensionsof the structure, this assumption is valid.

6.2.2 Torsional response--warping

deformation.

6.2.1 Shear deformationThe linear elements (B21 and B31) and the quadratic elements (B22 and B32) are shear deformable,Timoshenko beams; thus, they are suitable for modeling both thick members, in which sheardeformation is important, and slender beams, in which shear deformation is not important. Thecross-sections of these elements behave in the same manner as the cross-sections of the thick-shellelements, as illustrated in Figure 6-5(b) and discussed in ``Shell formulation - thick or thin, '' Section5.2.

Figure 6-7 Behavior of transverse beam sections in (a) slender beams and (b) thick beams.

ABAQUS assumes the transverse shear stiffness of these beam elements to be linear elastic andconstant. In addition, these beams are formulated so that their cross-sectional area can change as afunction of the axial deformation, an effect that is considered only in geometrically nonlinearsimulations (see Chapter 7, "Nonlinearity") in which the POISSON parameter on the beam sectionproperty option has a nonzero value. These elements can provide useful results as long as thecross-section dimensions are less than 1/10 of the typical axial dimensions of the structure, which isgenerally considered to be the limit of the applicability of beam theory; if the beam cross-section doesnot remain plane under bending deformation, beam theory is not adequate to model the deformation.

The cubic elements--the so-called Euler-Bernoulli beam elements ( B23 and B33)--do not model shearflexibility. The cross-sections of these elements remain perpendicular to the beam axis (see Figure6-5(a)). Therefore, the cubic beam elements are most effective for modeling structures with relativelyslender members. Since cubic elements model a cubic variation of displacement along their lengths, astructural member often can be modeled with a single cubic element for a static analysis and with asmall number of elements for a dynamic analysis. These elements assume that shear deformations arenegligible. Generally, if the cross-section dimensions are less than 1/15 of the typical axial dimensionsof the structure, this assumption is valid.

6.2.2 Torsional response--warping

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Structural members are often subjected to torsional moments, which occur in almost anythree-dimensional frame structure. Loads that cause bending in one member may cause twisting inanother, as shown in Figure 6-8.

Figure 6-8 Torsion induced in a frame structure.

The response of a beam to torsion depends on the shape of its cross-section. Generally, torsion in abeam produces warping or non-uniform out-of-plane displacements in the cross-section. ABAQUSonly considers the effects of torsion and warping in the three-dimensional elements. The warpingcalculation assumes that the warping displacements are small. The following cross-sections behavedifferently under torsion: solid cross-sections; closed, thin-walled cross-sections; and open,thin-walled cross-sections.

Solid cross-sections

A solid, non-circular cross-section does not remain plane under torsion; instead, the sectionwarps. ABAQUS uses St. Venant warping theory to calculate a single component of shearstrain caused by the warping at each section point in the cross-section. The warping in suchsolid cross-sections is considered unconstrained and creates negligible axial stresses. (Warpingconstraints would affect the solution only in the immediate vicinity of the constrained end.)The torsional stiffness of a beam with a solid cross-section depends on the shear modulus, G,of the material and the torsion constant, J, of the beam section. The torsion constant dependson the shape and the warping characteristics of the beam cross-section. Torsional loads thatproduce large amounts of inelastic deformation in the cross-section cannot be modeledaccurately with this approach.

Closed, thin-walled cross-sections

Beams that have closed, thin-walled, non-circular cross-sections ( BOX or HEX) havesignificant torsional stiffness and, thus, behave in a manner similar to solid sections.ABAQUS assumes that warping in these sections is also unconstrained. The thin-wallednature of the cross-section allows ABAQUS to consider the shear strains to be constantthrough the wall thickness. The thin-walled assumption is generally valid provided that thewall thickness is 1/10 a typical beam cross-section dimension. Examples of typicalcross-section dimensions for thin-walled cross-sections include:

� The diameter of a pipe section.

Structural members are often subjected to torsional moments, which occur in almost anythree-dimensional frame structure. Loads that cause bending in one member may cause twisting inanother, as shown in Figure 6-8.

Figure 6-8 Torsion induced in a frame structure.

The response of a beam to torsion depends on the shape of its cross-section. Generally, torsion in abeam produces warping or non-uniform out-of-plane displacements in the cross-section. ABAQUSonly considers the effects of torsion and warping in the three-dimensional elements. The warpingcalculation assumes that the warping displacements are small. The following cross-sections behavedifferently under torsion: solid cross-sections; closed, thin-walled cross-sections; and open,thin-walled cross-sections.

Solid cross-sections

A solid, non-circular cross-section does not remain plane under torsion; instead, the sectionwarps. ABAQUS uses St. Venant warping theory to calculate a single component of shearstrain caused by the warping at each section point in the cross-section. The warping in suchsolid cross-sections is considered unconstrained and creates negligible axial stresses. (Warpingconstraints would affect the solution only in the immediate vicinity of the constrained end.)The torsional stiffness of a beam with a solid cross-section depends on the shear modulus, G,of the material and the torsion constant, J, of the beam section. The torsion constant dependson the shape and the warping characteristics of the beam cross-section. Torsional loads thatproduce large amounts of inelastic deformation in the cross-section cannot be modeledaccurately with this approach.

Closed, thin-walled cross-sections

Beams that have closed, thin-walled, non-circular cross-sections ( BOX or HEX) havesignificant torsional stiffness and, thus, behave in a manner similar to solid sections.ABAQUS assumes that warping in these sections is also unconstrained. The thin-wallednature of the cross-section allows ABAQUS to consider the shear strains to be constantthrough the wall thickness. The thin-walled assumption is generally valid provided that thewall thickness is 1/10 a typical beam cross-section dimension. Examples of typicalcross-section dimensions for thin-walled cross-sections include:

� The diameter of a pipe section.

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� The length of an edge of a box section.

� The typical edge length of an arbitrary section.

Open, thin-walled cross-sections

Open, thin-walled cross-sections are very flexible in torsion when warping is unconstrained,and the primary source of torsional stiffness in such structures is the constraint of the axialwarping strains. Constraining the warping of open, thin-walled beams introduces axial stressesthat can affect the beam's response to other loading types. ABAQUS has shear deformablebeam elements, B31OS and B32OS, which include the warping effects in open, thin-walledsections. These elements must be used when modeling structures with open, thin-walledcross-sections, such as a channel (defined as an ARBITRARY section) or an I-section, whichare subjected to significant torsional loading.

Warping function

The variation of the warping-induced axial deformation over the beam's cross-section isdefined by the section's warping function. The magnitude of this function is treated as an extradegree of freedom, 7, in the open-section beam elements. Constraining this degree of freedomprevents warping at the nodes at which the constraints are applied.

So that the warping amplitude can be different in each branch, the junction betweenopen-section beams in a frame structure generally should be modeled with separate nodes foreach branch (see Figure 6-9).

Figure 6-9 Connecting open-section beams.

However, if the connection is designed to prevent warping, all branches should share acommon node, and the warping degree of freedom should be constrained using the*BOUNDARY option.

A shear force that does not act through the beam's shear center produces torsion. The twistingmoment is equal to the shear force multiplied by its eccentricity with respect to the shearcenter. Often, the centroid and the shear center do not coincide in open, thin-walled beamsections (see Figure 6-10). If the nodes are not located at the shear center of the cross-section,the section may twist under loading.

Figure 6-10 Approximate locations of shear centers, s, and centroids, c, for a number of beam

cross-sections.

� The length of an edge of a box section.

� The typical edge length of an arbitrary section.

Open, thin-walled cross-sections

Open, thin-walled cross-sections are very flexible in torsion when warping is unconstrained,and the primary source of torsional stiffness in such structures is the constraint of the axialwarping strains. Constraining the warping of open, thin-walled beams introduces axial stressesthat can affect the beam's response to other loading types. ABAQUS has shear deformablebeam elements, B31OS and B32OS, which include the warping effects in open, thin-walledsections. These elements must be used when modeling structures with open, thin-walledcross-sections, such as a channel (defined as an ARBITRARY section) or an I-section, whichare subjected to significant torsional loading.

Warping function

The variation of the warping-induced axial deformation over the beam's cross-section isdefined by the section's warping function. The magnitude of this function is treated as an extradegree of freedom, 7, in the open-section beam elements. Constraining this degree of freedomprevents warping at the nodes at which the constraints are applied.

So that the warping amplitude can be different in each branch, the junction betweenopen-section beams in a frame structure generally should be modeled with separate nodes foreach branch (see Figure 6-9).

Figure 6-9 Connecting open-section beams.

However, if the connection is designed to prevent warping, all branches should share acommon node, and the warping degree of freedom should be constrained using the*BOUNDARY option.

A shear force that does not act through the beam's shear center produces torsion. The twistingmoment is equal to the shear force multiplied by its eccentricity with respect to the shearcenter. Often, the centroid and the shear center do not coincide in open, thin-walled beamsections (see Figure 6-10). If the nodes are not located at the shear center of the cross-section,the section may twist under loading.

Figure 6-10 Approximate locations of shear centers, s, and centroids, c, for a number of beam

cross-sections.

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6.3 Selecting beam elements� First-order, shear-deformable beam elements (B21, B31) should be used in any simulation that

includes contact.

� If the structure is either very rigid or very flexible, hybrid beam elements (B21H, B32H, etc.)should be used in geometrically nonlinear simulations.

� Euler-Bernoulli (cubic) beams (B23, B33) are very accurate for simulations that includedistributed loading, such as dynamic vibration analyses. If the transverse shear deformation is alsoimportant, use Timoshenko (quadratic) beam elements instead ( B22, B32).

� Structures with open, thin-walled cross-sections should be modeled with elements that useopen-section warping theory (B31OS, B32OS).

6.4 Example: cargo craneA light-service, cargo crane is shown in Figure 6-11. You have been asked to determine the deflectionsof the crane when it carries a load of 10 kN. You should also identify the critical members and joints inthe structure: i.e., those with the highest stresses and loads.

Figure 6-11 Sketch of a light-service cargo crane.

6.3 Selecting beam elements� First-order, shear-deformable beam elements (B21, B31) should be used in any simulation that

includes contact.

� If the structure is either very rigid or very flexible, hybrid beam elements (B21H, B32H, etc.)should be used in geometrically nonlinear simulations.

� Euler-Bernoulli (cubic) beams (B23, B33) are very accurate for simulations that includedistributed loading, such as dynamic vibration analyses. If the transverse shear deformation is alsoimportant, use Timoshenko (quadratic) beam elements instead ( B22, B32).

� Structures with open, thin-walled cross-sections should be modeled with elements that useopen-section warping theory (B31OS, B32OS).

6.4 Example: cargo craneA light-service, cargo crane is shown in Figure 6-11. You have been asked to determine the deflectionsof the crane when it carries a load of 10 kN. You should also identify the critical members and joints inthe structure: i.e., those with the highest stresses and loads.

Figure 6-11 Sketch of a light-service cargo crane.

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The crane consists of two truss structures joined together by cross bracing. The two main members ineach truss structure are steel box beams (box cross-sections). Each truss structure is stiffened byinternal bracing, which is welded to the main members. The cross bracing connecting the two trussstructures is bolted to the truss structures. These connections can transmit little, if any, moment and,therefore, are treated as pinned joints. Both the internal bracing and cross bracing use steel box beamswith smaller cross-sections than the main members of the truss structures. The two truss structures areconnected at their ends (at point E) in such a way that allows independent movement in the 3-directionand all of the rotations, while constraining the displacements in the 1- and 2-directions to be the same.The crane is welded firmly to a massive structure at points A, B, C, and D. The dimensions of thecrane are shown in Figure 6-12. In the following figures, truss A is the structure consisting of membersAE, BE, and their internal bracing; and truss B consists of members CE, DE, and their internalbracing.

Figure 6-12 Dimensions (in m) of the cargo crane.

The crane consists of two truss structures joined together by cross bracing. The two main members ineach truss structure are steel box beams (box cross-sections). Each truss structure is stiffened byinternal bracing, which is welded to the main members. The cross bracing connecting the two trussstructures is bolted to the truss structures. These connections can transmit little, if any, moment and,therefore, are treated as pinned joints. Both the internal bracing and cross bracing use steel box beamswith smaller cross-sections than the main members of the truss structures. The two truss structures areconnected at their ends (at point E) in such a way that allows independent movement in the 3-directionand all of the rotations, while constraining the displacements in the 1- and 2-directions to be the same.The crane is welded firmly to a massive structure at points A, B, C, and D. The dimensions of thecrane are shown in Figure 6-12. In the following figures, truss A is the structure consisting of membersAE, BE, and their internal bracing; and truss B consists of members CE, DE, and their internalbracing.

Figure 6-12 Dimensions (in m) of the cargo crane.

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The ratio of the typical cross-section dimension to global axial length in the main members of thecrane is much less than 1/15. The ratio is approximately 1/15 in the shortest member used for internalbracing. Therefore, it is valid to use beam elements to model the crane.

6.4.1 Coordinate systemYou should use the default global rectangular Cartesian coordinate system shown in Figure 6-11 andFigure 6-12. Locate the origin of the coordinate system midway between points A and D. If you buildyour model with a different origin or orientation of the coordinate system, ensure that the input data inyour model reflect your coordinate system and not the one shown here.

6.4.2 Mesh designThe cargo crane will be modeled with three-dimensional, slender, cubic beam elements ( B33). Thecubic interpolation in these elements allows us to use a single element for each member and still obtainaccurate results under the applied bending load. The mesh that you should use in the simulation isshown in Figure 6-13.

Figure 6-13 Mesh for cargo crane.

The ratio of the typical cross-section dimension to global axial length in the main members of thecrane is much less than 1/15. The ratio is approximately 1/15 in the shortest member used for internalbracing. Therefore, it is valid to use beam elements to model the crane.

6.4.1 Coordinate systemYou should use the default global rectangular Cartesian coordinate system shown in Figure 6-11 andFigure 6-12. Locate the origin of the coordinate system midway between points A and D. If you buildyour model with a different origin or orientation of the coordinate system, ensure that the input data inyour model reflect your coordinate system and not the one shown here.

6.4.2 Mesh designThe cargo crane will be modeled with three-dimensional, slender, cubic beam elements ( B33). Thecubic interpolation in these elements allows us to use a single element for each member and still obtainaccurate results under the applied bending load. The mesh that you should use in the simulation isshown in Figure 6-13.

Figure 6-13 Mesh for cargo crane.

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The welded joints in the crane provide complete continuity of the translations and rotations from oneelement to the next. You, therefore, need only a single node at each welded joint in the model. Thebolted joints, which connect the cross bracing to the truss structures, and the connection at the tip ofthe truss structures are different. Since these joints do not provide complete continuity for all nodaldegrees of freedom, separate nodes are needed for each element at the connection. Appropriateconstraints between these separate nodes must then be given by using the *MPC, *BOUNDARY, or*EQUATION options. The *MPC and *EQUATION options are discussed in more detail later.

The node numbers for the various members of the cargo crane model are shown in Figure 6-14.

Figure 6-14 Node numbers in crane model.

These are the node numbers from the input file given in ``Cargo crane,'' Section A.4. Separate nodes

The welded joints in the crane provide complete continuity of the translations and rotations from oneelement to the next. You, therefore, need only a single node at each welded joint in the model. Thebolted joints, which connect the cross bracing to the truss structures, and the connection at the tip ofthe truss structures are different. Since these joints do not provide complete continuity for all nodaldegrees of freedom, separate nodes are needed for each element at the connection. Appropriateconstraints between these separate nodes must then be given by using the *MPC, *BOUNDARY, or*EQUATION options. The *MPC and *EQUATION options are discussed in more detail later.

The node numbers for the various members of the cargo crane model are shown in Figure 6-14.

Figure 6-14 Node numbers in crane model.

These are the node numbers from the input file given in ``Cargo crane,'' Section A.4. Separate nodes

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have been defined on the cross-bracing elements and the truss structures that they connect. Separatenodes are also needed at the end of each truss structure, point E in Figure 6-11. The node numbers inyour model may be different from those shown here.

The element numbers for the various members of the cargo crane model are shown in Figure 6-15.

Figure 6-15 Element numbers in crane model.

These are the element numbers from the input file given in ``Cargo crane,'' Section A.4. The elementnumbers in your model may be different from those shown here.

6.4.3 Preprocessing--creating the modelWhile you can generate the mesh for this simulation using a preprocessor, you may find it easier tobuild this model with an editor. The ABAQUS input options used to create the nodes and elementsshown on the preceding pages can be found in ``Cargo crane,'' Section A.4. If you do use apreprocessor, most likely you will have to modify the input data to ensure correct modeling of thejoints between members and correct orientation of the beam element cross-sections.

6.4.4 Reviewing the input file--the model dataAt this point we assume that you have created the basic mesh either by using a preprocessor or with aneditor. In this section you will review and make corrections to your input file, as well as includeadditional information, such as beam section properties and orientations.

Heading

have been defined on the cross-bracing elements and the truss structures that they connect. Separatenodes are also needed at the end of each truss structure, point E in Figure 6-11. The node numbers inyour model may be different from those shown here.

The element numbers for the various members of the cargo crane model are shown in Figure 6-15.

Figure 6-15 Element numbers in crane model.

These are the element numbers from the input file given in ``Cargo crane,'' Section A.4. The elementnumbers in your model may be different from those shown here.

6.4.3 Preprocessing--creating the modelWhile you can generate the mesh for this simulation using a preprocessor, you may find it easier tobuild this model with an editor. The ABAQUS input options used to create the nodes and elementsshown on the preceding pages can be found in ``Cargo crane,'' Section A.4. If you do use apreprocessor, most likely you will have to modify the input data to ensure correct modeling of thejoints between members and correct orientation of the beam element cross-sections.

6.4.4 Reviewing the input file--the model dataAt this point we assume that you have created the basic mesh either by using a preprocessor or with aneditor. In this section you will review and make corrections to your input file, as well as includeadditional information, such as beam section properties and orientations.

Heading

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Use a suitable, descriptive heading in your input file, such as the following:

*HEADING

3-D model of light-service cargo crane

S.I. Units (m, kg, N, sec)


Define the nodal coordinates in a *NODE option block. If you decide to do this with an editor,you may want to use the mesh generation commands found in ``Cargo crane,'' Section A.4.Create a node set called ATTACH containing the nodes at points A, B, C, and D, the points atwhich the crane is attached to the parent structure.

*NSET, NSET=ATTACH

100,107,200,207

Create elements in your model that correspond to the elements shown in Figure 6-15, butremember that your numbering may be different. (Having the same numbering will makedefining some modeling features easier, however.) There are several element sets that you willneed in this simulation. As you define the elements, group them into the following elementsets:

OUTA 100, 101, 102, 103, 104, 105, 106

BRACEA 110, 111, 112, 113, 114

OUTB 200, 201, 202, 203, 204, 205, 206

BRACEB 210, 211, 212, 213, 214

CROSSEL 300, 301, 302, 303, 304, 305, 306, 307

where these element numbers refer to those elements shown in Figure 6-15. The element setsOUTA and OUTB contain the main outer members for the two truss structures. Element setsBRACEA and BRACEB contain the elements modeling the internal bracing within each trussstructure. Element set CROSSEL contains the cross bracing that connects the two trussstructures.

Beam element properties

Since the material behavior in this simulation is assumed to be linear elastic, use the *BEAMGENERAL SECTION option to define the section properties. All of the beams in thisstructure have a box-shaped cross-section.

A box-section is specified using the parameter SECTION=BOX. The first data line containsthe section dimensions, which are the dimensions a, b, t1 , t2 , t3 , and t4 shown in Figure 6-16for a box section. The dimensions shown in Figure 6-16 are for the main members of the twotrusses in the crane.

Figure 6-16 Cross-section geometry and dimensions (in m) of the main members.

Use a suitable, descriptive heading in your input file, such as the following:

*HEADING




Define the nodal coordinates in a *NODE option block. If you decide to do this with an editor,you may want to use the mesh generation commands found in ``Cargo crane,'' Section A.4.Create a node set called ATTACH containing the nodes at points A, B, C, and D, the points atwhich the crane is attached to the parent structure.

*NSET, NSET=ATTACH

100,107,200,207

Create elements in your model that correspond to the elements shown in Figure 6-15, butremember that your numbering may be different. (Having the same numbering will makedefining some modeling features easier, however.) There are several element sets that you willneed in this simulation. As you define the elements, group them into the following elementsets:

OUTA 100, 101, 102, 103, 104, 105, 106

BRACEA 110, 111, 112, 113, 114

OUTB 200, 201, 202, 203, 204, 205, 206

BRACEB 210, 211, 212, 213, 214

CROSSEL 300, 301, 302, 303, 304, 305, 306, 307

where these element numbers refer to those elements shown in Figure 6-15. The element setsOUTA and OUTB contain the main outer members for the two truss structures. Element setsBRACEA and BRACEB contain the elements modeling the internal bracing within each trussstructure. Element set CROSSEL contains the cross bracing that connects the two trussstructures.

Beam element properties

Since the material behavior in this simulation is assumed to be linear elastic, use the *BEAMGENERAL SECTION option to define the section properties. All of the beams in thisstructure have a box-shaped cross-section.

A box-section is specified using the parameter SECTION=BOX. The first data line containsthe section dimensions, which are the dimensions a, b, t1 , t2 , t3 , and t4 shown in Figure 6-16for a box section. The dimensions shown in Figure 6-16 are for the main members of the twotrusses in the crane.

Figure 6-16 Cross-section geometry and dimensions (in m) of the main members.

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The beam section axes for the main members should be oriented such that the beam 1-axis isorthogonal to the plane of the truss structures shown in the elevation view ( Figure 6-12) andthe beam 2-axis is orthogonal to the elements in that plane. Specify this orientation by givingthe approximate direction of the beam 1-axis (the n1-vector) on the second data line of the*BEAM GENERAL SECTION option. To get the correct normal, n2 , in this case, you need toprovide a very accurate n1 . It is somewhat easier to provide an approximate direction, whichwould be the negative 3-direction. However, given the logic that ABAQUS uses to determinen2 given t and n1 , the normal n2 is rotated slightly from its proper orientation if we use thisapproximate n1 . You can specify the same n1-direction for all elements in each of the twotruss structures. The third data line contains the elastic and shear moduli, assuming a mildstrength steel with E = 200.0 GPa, º = 0.25, and G = 80.0 GPa. Add the following optionblocks to your input file:

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=OUTA

0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=OUTB

0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9

The dimensions of the beam sections for the bracing members are shown in Figure 6-17. Boththe cross bracing and the bracing within each truss structure have the same beam sectiongeometry, but they do not share the same orientation of the beam section axes. Therefore,separate *BEAM GENERAL SECTION options must be used. The bracing is made from thesame steel as the main members.

Figure 6-17 Cross-section geometry and dimensions (in m) of the bracing members.

The beam section axes for the main members should be oriented such that the beam 1-axis isorthogonal to the plane of the truss structures shown in the elevation view ( Figure 6-12) andthe beam 2-axis is orthogonal to the elements in that plane. Specify this orientation by givingthe approximate direction of the beam 1-axis (the n1-vector) on the second data line of the*BEAM GENERAL SECTION option. To get the correct normal, n2 , in this case, you need toprovide a very accurate n1 . It is somewhat easier to provide an approximate direction, whichwould be the negative 3-direction. However, given the logic that ABAQUS uses to determinen2 given t and n1 , the normal n2 is rotated slightly from its proper orientation if we use thisapproximate n1 . You can specify the same n1-direction for all elements in each of the twotruss structures. The third data line contains the elastic and shear moduli, assuming a mildstrength steel with E = 200.0 GPa, º = 0.25, and G = 80.0 GPa. Add the following optionblocks to your input file:


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9

The dimensions of the beam sections for the bracing members are shown in Figure 6-17. Boththe cross bracing and the bracing within each truss structure have the same beam sectiongeometry, but they do not share the same orientation of the beam section axes. Therefore,separate *BEAM GENERAL SECTION options must be used. The bracing is made from thesame steel as the main members.

Figure 6-17 Cross-section geometry and dimensions (in m) of the bracing members.

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The approximate n1-vector for the internal truss bracing is the same as for the main membersof the respective truss structures. Use the following input to define the element properties ofthis bracing:

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEA

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEB

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

We make some assumptions so that the orientation of the cross bracing is somewhat easier tospecify. All of the beam normals (n2-vectors) should lie approximately in the plane of the planview of the cargo crane (see Figure 6-17). This plane is skewed slightly from the global 1-3plane. Again, a simple method for defining such an orientation is to provide an approximaten1-vector that is orthogonal to this plane on the element property options. The vector shouldbe nearly parallel to the global 2-direction. Since the angle between the normals from oneelement to the next is always greater than 20°, the normals are not averaged at the nodes.

Depending on the exact orientation of the members in the cross bracing, it is possible that wewould have to define the normals individually for each of the cross-bracing elements. Such anexercise would be very similar to what you have already done by defining the normals for thetwo truss structures. Since the square cross-bracing members are subjected to primarily axialloading, their deformation is not sensitive to cross-section orientation. Therefore, we acceptthe default normals that ABAQUS calculates to be correct. The approximate n1-vector for thecross bracing is aligned with the y-axis. Add the following option block to your model:

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=CROSSEL

0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

Beam section orientation

In this model you will have a modeling error if you provide data that only define theorientation of the approximate n1-vector. The averaging of beam normals at the nodes (see

The approximate n1-vector for the internal truss bracing is the same as for the main membersof the respective truss structures. Use the following input to define the element properties ofthis bracing:


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

We make some assumptions so that the orientation of the cross bracing is somewhat easier tospecify. All of the beam normals (n2-vectors) should lie approximately in the plane of the planview of the cargo crane (see Figure 6-17). This plane is skewed slightly from the global 1-3plane. Again, a simple method for defining such an orientation is to provide an approximaten1-vector that is orthogonal to this plane on the element property options. The vector shouldbe nearly parallel to the global 2-direction. Since the angle between the normals from oneelement to the next is always greater than 20°, the normals are not averaged at the nodes.

Depending on the exact orientation of the members in the cross bracing, it is possible that wewould have to define the normals individually for each of the cross-bracing elements. Such anexercise would be very similar to what you have already done by defining the normals for thetwo truss structures. Since the square cross-bracing members are subjected to primarily axialloading, their deformation is not sensitive to cross-section orientation. Therefore, we acceptthe default normals that ABAQUS calculates to be correct. The approximate n1-vector for thecross bracing is aligned with the y-axis. Add the following option block to your model:


0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

Beam section orientation

In this model you will have a modeling error if you provide data that only define theorientation of the approximate n1-vector. The averaging of beam normals at the nodes (see

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``Beam element curvature,'' Section 6.1.3) causes ABAQUS to use incorrect geometry for thecargo crane model. To detect this type of modeling error, you can use ABAQUS/Viewer todisplay the beam section axes and beam tangent vectors (see ``Postprocessing,'' Section 6.4.7).However, the normals in the crane model appear to be correct in ABAQUS/Viewer; yet, theyare, in fact, slightly incorrect. You can also find such modeling mistakes by examining theaveraged nodal normals that are printed in the data (.dat) file. Some of the normals in theincorrect model of the cargo crane are shown in the following output:

The problem is the normal vector for node 102, which does not match those at the other nodesdefining the lower, main member in truss A (see Figure 6-14). Four elements (101, 102, 112,and 113) contain node 102. When the averaging of beam normals at nodes produces multipleindependent normals, the additional normals at the node are also printed in the data file (see``Beam element curvature,'' Section 6.1.3, for details). The correct geometry for the cranemodel requires three independent beam normals at node 102: one each for the bracingelements 112 and 113 and a single normal for elements 101 and 102. The normal shown abovefor node 102 is not the normal needed for elements 101 and 102. If it were, it would match thenormals shown for nodes 100, 101, or 103. Nor is it the correct normal for either of the othertwo elements, whose normals are printed in the data file, as shown below.

N O R M A L D E F I N I T I O N S

ELEMENT NODE NORMAL ELEMENT NODE N

110 101 -0.2947 -0.9550 3.3161E-02 110 107 -0.2947 -0

111 101 -0.7581 0.6465 8.5304E-02 111 106 -0.7581 0

112 102 -0.2947 -0.9550 3.3161E-02 112 106 -0.2947 -0

114 103 -0.2947 -0.9550 3.3161E-02 114 105 -0.2947 -0

In this table element 113 shows no normal for either of its nodes, node 102 and node 105.Thus, the normal shown for node 102 above in the NODE DEFINITIONS table was theaverage of the normals from elements 101, 102, and 113. Using the ABAQUS logic foraveraging normals, we could have predicted that the normals at the nodes of element 113would be averaged with the normals for the adjacent elements. For this problem the importantpart of the averaging logic is that normals that subtend an angle less than 20° with thereference normal are averaged with the reference normal to define a new reference normal. Inthe case of the normals at node 102, the original reference normal is the normal for elements101 and 102. Since the normal for element 113 at node 102 subtends an angle less than 20°with the original reference normal, it is averaged with the normals for elements 101 and 102 atnode 102 to define the new reference normal at that node. On the other hand, since the normalfor element 112 subtends an angle of approximately 30° with the original reference normal, ithas an independent normal at node 102, as shown in the data file.

``Beam element curvature,'' Section 6.1.3) causes ABAQUS to use incorrect geometry for thecargo crane model. To detect this type of modeling error, you can use ABAQUS/Viewer todisplay the beam section axes and beam tangent vectors (see ``Postprocessing,'' Section 6.4.7).However, the normals in the crane model appear to be correct in ABAQUS/Viewer; yet, theyare, in fact, slightly incorrect. You can also find such modeling mistakes by examining theaveraged nodal normals that are printed in the data (.dat) file. Some of the normals in theincorrect model of the cargo crane are shown in the following output:

The problem is the normal vector for node 102, which does not match those at the other nodesdefining the lower, main member in truss A (see Figure 6-14). Four elements (101, 102, 112,and 113) contain node 102. When the averaging of beam normals at nodes produces multipleindependent normals, the additional normals at the node are also printed in the data file (see``Beam element curvature,'' Section 6.1.3, for details). The correct geometry for the cranemodel requires three independent beam normals at node 102: one each for the bracingelements 112 and 113 and a single normal for elements 101 and 102. The normal shown abovefor node 102 is not the normal needed for elements 101 and 102. If it were, it would match thenormals shown for nodes 100, 101, or 103. Nor is it the correct normal for either of the othertwo elements, whose normals are printed in the data file, as shown below.

N O R M A L D E F I N I T I O N S

ELEMENT NODE NORMAL ELEMENT NODE N

110 101 -0.2947 -0.9550 3.3161E-02 110 107 -0.2947 -0

111 101 -0.7581 0.6465 8.5304E-02 111 106 -0.7581 0

112 102 -0.2947 -0.9550 3.3161E-02 112 106 -0.2947 -0

114 103 -0.2947 -0.9550 3.3161E-02 114 105 -0.2947 -0

In this table element 113 shows no normal for either of its nodes, node 102 and node 105.Thus, the normal shown for node 102 above in the NODE DEFINITIONS table was theaverage of the normals from elements 101, 102, and 113. Using the ABAQUS logic foraveraging normals, we could have predicted that the normals at the nodes of element 113would be averaged with the normals for the adjacent elements. For this problem the importantpart of the averaging logic is that normals that subtend an angle less than 20° with thereference normal are averaged with the reference normal to define a new reference normal. Inthe case of the normals at node 102, the original reference normal is the normal for elements101 and 102. Since the normal for element 113 at node 102 subtends an angle less than 20°with the original reference normal, it is averaged with the normals for elements 101 and 102 atnode 102 to define the new reference normal at that node. On the other hand, since the normalfor element 112 subtends an angle of approximately 30° with the original reference normal, ithas an independent normal at node 102, as shown in the data file.

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This incorrect average normal means that elements 101, 102, and 113 have a section geometrythat twists about the beam axis from one end of the element to the other, which is not theintended geometry. You must use the *NORMAL option to define the normals at node 102 forelement 113 explicitly. Explicitly specifying the normal directions prevents ABAQUS fromapplying its averaging algorithm. You must also use *NORMAL for the correspondingelements on the opposite side of the crane: element 213, node 202 of truss B.

There is also a problem with the normals at nodes 104 and 204 at the tip of each trussstructure, again because the angle between elements 103 and 104 is less than 20°. Since we aremodeling straight beam elements, the normals are constant at both nodes in each element.Thus, the *NORMAL option block that you should put in your input file has six data lines. Ifyou used the numbering scheme shown in Figure 6-14 and Figure 6-15, the following optionblock should be added to your input file:

Multi-point constraints

The cross bracing, unlike the internal truss bracing, is bolted to the truss members. You canassume that these bolted connections are unable to transmit rotations or torsion. The duplicatenodes that were defined at these locations are needed to define this constraint. In ABAQUSsuch constraints between nodes can be defined using multi-point constraints ( MPCs) orconstraint equations.

Multi-point constraints allow constraints to be imposed between different degrees of freedomof the model. A large library of MPCs is available in ABAQUS. (See the ABAQUS/StandardUser's Manual for a complete list and a description of each one.) The format of the *MPCoption is

*MPC

<type of MPC>,<node 1 or node set 1>,<node 2 or node set 2>, ......

You can define multiple constraints of the same type with just a single data line by using nodesets. The MPC type needed to model the bolted connection is PIN. The pinned joint created bythis MPC constrains the displacements at two nodes to be equal, but the rotations, if they existat the nodes, remain independent.

There are many bolted joints in the crane model. The following is the complete *MPC optionblock for the model from ``Cargo crane,'' Section A.4:

*MPC

PIN,101,301

PIN,102,302

PIN,103,303

This incorrect average normal means that elements 101, 102, and 113 have a section geometrythat twists about the beam axis from one end of the element to the other, which is not theintended geometry. You must use the *NORMAL option to define the normals at node 102 forelement 113 explicitly. Explicitly specifying the normal directions prevents ABAQUS fromapplying its averaging algorithm. You must also use *NORMAL for the correspondingelements on the opposite side of the crane: element 213, node 202 of truss B.

There is also a problem with the normals at nodes 104 and 204 at the tip of each trussstructure, again because the angle between elements 103 and 104 is less than 20°. Since we aremodeling straight beam elements, the normals are constant at both nodes in each element.Thus, the *NORMAL option block that you should put in your input file has six data lines. Ifyou used the numbering scheme shown in Figure 6-14 and Figure 6-15, the following optionblock should be added to your input file:


The cross bracing, unlike the internal truss bracing, is bolted to the truss members. You canassume that these bolted connections are unable to transmit rotations or torsion. The duplicatenodes that were defined at these locations are needed to define this constraint. In ABAQUSsuch constraints between nodes can be defined using multi-point constraints ( MPCs) orconstraint equations.

Multi-point constraints allow constraints to be imposed between different degrees of freedomof the model. A large library of MPCs is available in ABAQUS. (See the ABAQUS/StandardUser's Manual for a complete list and a description of each one.) The format of the *MPCoption is

*MPC

<type of MPC>,<node 1 or node set 1>,<node 2 or node set 2>, ......

You can define multiple constraints of the same type with just a single data line by using nodesets. The MPC type needed to model the bolted connection is PIN. The pinned joint created bythis MPC constrains the displacements at two nodes to be equal, but the rotations, if they existat the nodes, remain independent.

There are many bolted joints in the crane model. The following is the complete *MPC optionblock for the model from ``Cargo crane,'' Section A.4:

*MPC

PIN,101,301

PIN,102,302

PIN,103,303

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PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

Add a similar option block to your model, changing the node numbers to correspond to thosein your model. If all of the nodes on the two truss structures had been grouped into a node setcalled TRUSNODE and all of the nodes on the cross bracing had been grouped into a node setcalled CROSNODE, the option block could have been shortened to the following:

*NSET, NSET=TRUSNODE, UNSORTED

101,102,103,104,105,201,202,203,204,205

*NSET, NSET=CROSNODE, UNSORTED

301,302,303,304,305,401,402,403,404,405

*MPC

PIN, TRUSNODE, CROSNODE

If a node set is provided as the first item after the MPC type, the second item can be eitheranother node set or a single node. When the data line of an *MPC option contains a node setand then a single node, as shown below, ABAQUS creates an MPC constraint between eachnode in the set and the individual node specified. For example, the following option blockwould create a pinned joint between node 301 and each node in node set TRUSNODE.

*MPC

PIN, TRUSNODE, 301

Constraint equations

Constraints between nodal degrees of freedom can also be specified with linear equations byusing the *EQUATION option. The form of each equation is,

A1u1 +A2u2 + :::: +Anun = 0;

where Ai is the coefficient associated with degree of freedom ui. Each linear constraintequation requires at least two data lines. The number of terms, n, involved in the equation isgiven on the first data line beneath the *EQUATION option. On the subsequent data lines theformat is

<node1>,<dof1>,<A1>,<node2>,<dof2>,<A2>,: : : ,<noden>,<dofn>,<An>

Exactly four terms must be given on each data line except the final one, which can have fewerterms.

In the crane model the tips of the two trusses are connected together such that degrees offreedom 1 and 2 (the translations in the 1- and 2-directions) are equal, while the other degrees

PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

Add a similar option block to your model, changing the node numbers to correspond to thosein your model. If all of the nodes on the two truss structures had been grouped into a node setcalled TRUSNODE and all of the nodes on the cross bracing had been grouped into a node setcalled CROSNODE, the option block could have been shortened to the following:

*NSET, NSET=TRUSNODE, UNSORTED

101,102,103,104,105,201,202,203,204,205

*NSET, NSET=CROSNODE, UNSORTED

301,302,303,304,305,401,402,403,404,405

*MPC

PIN, TRUSNODE, CROSNODE

If a node set is provided as the first item after the MPC type, the second item can be eitheranother node set or a single node. When the data line of an *MPC option contains a node setand then a single node, as shown below, ABAQUS creates an MPC constraint between eachnode in the set and the individual node specified. For example, the following option blockwould create a pinned joint between node 301 and each node in node set TRUSNODE.

*MPC

PIN, TRUSNODE, 301


Constraints between nodal degrees of freedom can also be specified with linear equations byusing the *EQUATION option. The form of each equation is,

A1u1 +A2u2 + :::: +Anun = 0;

where Ai is the coefficient associated with degree of freedom ui. Each linear constraintequation requires at least two data lines. The number of terms, n, involved in the equation isgiven on the first data line beneath the *EQUATION option. On the subsequent data lines theformat is

<node1>,<dof1>,<A1>,<node2>,<dof2>,<A2>,: : : ,<noden>,<dofn>,<An>

Exactly four terms must be given on each data line except the final one, which can have fewerterms.

In the crane model the tips of the two trusses are connected together such that degrees offreedom 1 and 2 (the translations in the 1- and 2-directions) are equal, while the other degrees

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of freedom (3-6) at the nodes are independent. We need two linear constraints, one equatingdegree of freedom 1 at node 104 to degree of freedom 1 at node 204:

(1)u1041 + (¡1)u204

1 = 0

and the other equating degree of freedom 2 at node 104 to degree of freedom 2 at node 204:

(1)u1042 + (¡1)u204

2 = 0:

Remember to change the node numbers if necessary for your model. The following optionblock defines the appropriate constraints at point E in the crane model (see Figure 6-11):

*EQUATION

2

104,1,1.0, 204,1,-1.0

2

104,2,1.0, 204,2,-1.0

The degrees of freedom at the first node defined in an *MPC or *EQUATION are eliminatedfrom the stiffness matrix. Therefore, these nodes should not appear in other MPCs orconstraint equations. Nor should boundary conditions be applied to the eliminated degrees offreedom.

6.4.5 Reviewing the input file--the history dataSpecify a static, linear perturbation simulation using the following options:

*STEP, PERTURBATION

Static tip load on crane

*STATIC

Boundary conditions

The crane is attached firmly to the parent structure. Use the following *BOUNDARY optionblock to constrain all of the nodes at the attachment points, which should have been groupedinto node set ATTACH:

*BOUNDARY

ATTACH, ENCASTRE

Loading

A concentrated load of 10 kN is applied in the negative y-direction to node 104. Since there isa constraint equation connecting the y-displacement of nodes 104 and 204, the load is carriedequally by both nodes. The *CLOAD option block that you should use is:

*CLOAD

of freedom (3-6) at the nodes are independent. We need two linear constraints, one equatingdegree of freedom 1 at node 104 to degree of freedom 1 at node 204:

(1)u1041 + (¡1)u204

1 = 0

and the other equating degree of freedom 2 at node 104 to degree of freedom 2 at node 204:

(1)u1042 + (¡1)u204

2 = 0:

Remember to change the node numbers if necessary for your model. The following optionblock defines the appropriate constraints at point E in the crane model (see Figure 6-11):

*EQUATION

2

104,1,1.0, 204,1,-1.0

2

104,2,1.0, 204,2,-1.0

The degrees of freedom at the first node defined in an *MPC or *EQUATION are eliminatedfrom the stiffness matrix. Therefore, these nodes should not appear in other MPCs orconstraint equations. Nor should boundary conditions be applied to the eliminated degrees offreedom.

6.4.5 Reviewing the input file--the history dataSpecify a static, linear perturbation simulation using the following options:

*STEP, PERTURBATION

Static tip load on crane

*STATIC

Boundary conditions

The crane is attached firmly to the parent structure. Use the following *BOUNDARY optionblock to constrain all of the nodes at the attachment points, which should have been groupedinto node set ATTACH:

*BOUNDARY

ATTACH, ENCASTRE

Loading

A concentrated load of 10 kN is applied in the negative y-direction to node 104. Since there isa constraint equation connecting the y-displacement of nodes 104 and 204, the load is carriedequally by both nodes. The *CLOAD option block that you should use is:

*CLOAD

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104,2,-1.0E4

Output requests

Write the displacements (U) and reaction forces (RF) at the nodes, and the section forces (SF)in the elements to the output database for postprocessing with ABAQUS/Viewer. Also, printU, RF, and SF to the data file. Add the following option blocks to your input file:

*OUTPUT, FIELD

*NODE OUTPUT

U, RF

*ELEMENT OUTPUT

SF,

*NODE PRINT

U,

RF,

*EL PRINT

SF,

*END STEP

6.4.6 Running the analysisStore the input in a file called crane.inp. Run the analysis using the command

abaqus job=crane

6.4.7 PostprocessingStart ABAQUS/Viewer by typing

abaqus viewer odb=crane


ABAQUS/Viewer displays a fast plot of the crane model.

Plotting the deformed model shape

To begin this exercise, plot the deformed shape.

To plot the deformed model shape:

1. From the main menu bar, select Plot->Deformed Shape; or use the tool in thetoolbox to display the deformed model shape at the end of the analysis.


The Deformed Plot Options dialog box appears; by default, the Basic tab is selected.

3. Toggle on Superimpose undeformed plot.

104,2,-1.0E4

Output requests

Write the displacements (U) and reaction forces (RF) at the nodes, and the section forces (SF)in the elements to the output database for postprocessing with ABAQUS/Viewer. Also, printU, RF, and SF to the data file. Add the following option blocks to your input file:

*OUTPUT, FIELD

*NODE OUTPUT

U, RF

*ELEMENT OUTPUT

SF,

*NODE PRINT

U,

RF,

*EL PRINT

SF,

*END STEP

6.4.6 Running the analysisStore the input in a file called crane.inp. Run the analysis using the command

abaqus job=crane

6.4.7 PostprocessingStart ABAQUS/Viewer by typing

abaqus viewer odb=crane


ABAQUS/Viewer displays a fast plot of the crane model.


To begin this exercise, plot the deformed shape.


1. From the main menu bar, select Plot->Deformed Shape; or use the tool in thetoolbox to display the deformed model shape at the end of the analysis.


The Deformed Plot Options dialog box appears; by default, the Basic tab is selected.


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4. Click OK.

ABAQUS/Viewer displays an image of the deformed crane superimposed upon theundeformed crane.

Changing the view

The default view is isometric. You can change the view using the options in the View menu orthe view tools on the toolbar.






4. Click OK.

ABAQUS/Viewer displays your model in the specified view.

The deformed shape of the crane superimposed upon the undeformed shape is shown in Figure6-18.

Figure 6-18 Deformed shape of cargo crane.

Plotting individual element and node sets

You can use display groups in ABAQUS/Viewer to plot existing node and element sets as wellas surfaces. You will display only the elements in element set OUTA, the main members intruss structure A.

To display the elements in the element set OUTA:

1. From the main menu bar, select Tools->Display Group->Create; or use the tool in

4. Click OK.

ABAQUS/Viewer displays an image of the deformed crane superimposed upon theundeformed crane.

Changing the view

The default view is isometric. You can change the view using the options in the View menu orthe view tools on the toolbar.






4. Click OK.

ABAQUS/Viewer displays your model in the specified view.

The deformed shape of the crane superimposed upon the undeformed shape is shown in Figure6-18.

Figure 6-18 Deformed shape of cargo crane.

Plotting individual element and node sets

You can use display groups in ABAQUS/Viewer to plot existing node and element sets as wellas surfaces. You will display only the elements in element set OUTA, the main members intruss structure A.

To display the elements in the element set OUTA:

1. From the main menu bar, select Tools->Display Group->Create; or use the tool in

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the toolbox.

The Create Display Group dialog box appears.

2. From the ODB Item list in the top-left portion of this dialog box, select Elements.

3. From the Selection Method list, select Element sets. A list of the available element setsappears in the right hand side of the dialog box. Select the element set OUTA.

4. From the icons in the bottom portion of the dialog box, click to activate theBoolean Replace operation.

5. Click Dismiss to close the dialog box.

ABAQUS/Viewer now displays only the elements in element set OUTA.

Beam cross-section orientation

You can plot the section axes and beam tangents on the undeformed model shape.

To plot the beam section axes:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox to display the undeformed model shape.

2. From the toolbar, click and select the isometric view from the Views dialog box thatappears.

3. From the main menu bar, select Options->Undeformed; then, click the Normals tab inthe dialog box that appears.

4. Toggle on Show normals.

5. In the Style area at the bottom of the Normals page, specify the Length to be Long.

6. Click OK.

The section axes and beam tangents are displayed on the undeformed shape.

The resulting plot is shown in Figure 6-19. The text annotations in Figure 6-19 that identifythe section axes and beam tangent will not appear in your image. The vector showing the localbeam 1-axis, n1 , is blue; the vector showing the beam 2-axis, n2 , is red; and the vectorshowing the beam tangent, t, is white.

Figure 6-19 Plot of beam section axes and tangents for element set OUTA.

the toolbox.


2. From the ODB Item list in the top-left portion of this dialog box, select Elements.

3. From the Selection Method list, select Element sets. A list of the available element setsappears in the right hand side of the dialog box. Select the element set OUTA.

4. From the icons in the bottom portion of the dialog box, click to activate theBoolean Replace operation.


ABAQUS/Viewer now displays only the elements in element set OUTA.

Beam cross-section orientation

You can plot the section axes and beam tangents on the undeformed model shape.

To plot the beam section axes:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool in thetoolbox to display the undeformed model shape.

2. From the toolbar, click and select the isometric view from the Views dialog box thatappears.

3. From the main menu bar, select Options->Undeformed; then, click the Normals tab inthe dialog box that appears.

4. Toggle on Show normals.

5. In the Style area at the bottom of the Normals page, specify the Length to be Long.

6. Click OK.

The section axes and beam tangents are displayed on the undeformed shape.

The resulting plot is shown in Figure 6-19. The text annotations in Figure 6-19 that identifythe section axes and beam tangent will not appear in your image. The vector showing the localbeam 1-axis, n1 , is blue; the vector showing the beam 2-axis, n2 , is red; and the vectorshowing the beam tangent, t, is white.

Figure 6-19 Plot of beam section axes and tangents for element set OUTA.

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Creating a hard copy

You can save the image of the beam normals to a file for hardcopy output.

To create a PostScript file of the beam normals image:

1. From the main menu bar, select File->Print.

The Print dialog box appears.

2. From the Settings area in the Print dialog box, select Black&White as the Renditiontype; and toggle on File as the Destination.

3. Select PS as the Format, and enter beamsectaxes.ps as the File name.

4. Click PS Options.

The PostScript Options dialog box appears.

5. From the PostScript Options dialog box, select 600 dpi as the Resolution; and toggleoff Print date.

6. In the PostScript Options dialog box, click OK to apply your selections and to close thedialog box.

7. In the Print dialog box, click OK.

ABAQUS/Viewer creates a PostScript file of the beam normals image and saves it in yourworking directory as beamsectaxes.ps. You can print this file using your system'scommand for printing PostScript files.

Displacement summary

You can write a summary of the displacements of all nodes in element set OUTA to a file.

To write the displacement summary to a report file:

1. From the main menu bar, select Report->Field Output.

The Report Field Output dialog box appears; by default, the Variables tab is selected.

2. Select Unique Nodal from the Position menu, and toggle on U Spatial displacement.

Creating a hard copy

You can save the image of the beam normals to a file for hardcopy output.

To create a PostScript file of the beam normals image:

1. From the main menu bar, select File->Print.

The Print dialog box appears.

2. From the Settings area in the Print dialog box, select Black&White as the Renditiontype; and toggle on File as the Destination.

3. Select PS as the Format, and enter beamsectaxes.ps as the File name.

4. Click PS Options.

The PostScript Options dialog box appears.

5. From the PostScript Options dialog box, select 600 dpi as the Resolution; and toggleoff Print date.

6. In the PostScript Options dialog box, click OK to apply your selections and to close thedialog box.

7. In the Print dialog box, click OK.

ABAQUS/Viewer creates a PostScript file of the beam normals image and saves it in yourworking directory as beamsectaxes.ps. You can print this file using your system'scommand for printing PostScript files.

Displacement summary

You can write a summary of the displacements of all nodes in element set OUTA to a file.

To write the displacement summary to a report file:

1. From the main menu bar, select Report->Field Output.

The Report Field Output dialog box appears; by default, the Variables tab is selected.

2. Select Unique Nodal from the Position menu, and toggle on U Spatial displacement.

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3. From the Report Field Output dialog box, click the Setup tab.

4. Enter the file name crane.rpt.

5. Click OK.

The displacements of all the nodes in element set OUTA are written to a file namedcrane.rpt.

The peak displacement at the tip of the crane in the 2-direction is 0.0192 m.

Section forces and moments

ABAQUS can provide output for structural elements in terms of forces and moments acting onthe cross-section at a given point. These section forces and moments are defined in the localbeam coordinate system. Contour the section moment about the beam 1-axis in the membersof element set OUTA. For clarity, reset the view so that the elements are displayed in the 1-2plane.

To create a "bending moment"-type contour plot:



2. Select SM from the list of available output variables; then select SM1 from the componentfield.

3. Click OK.


4. Toggle on Contour, and click OK.

ABAQUS/Viewer displays a contour plot of the bending moment about the beam 1-axis. Thecontour is plotted on the deformed model shape. The deformation scale factor is chosenautomatically since geometric nonlinearity was not considered in the analysis.

Color contour plots of this type typically are not very useful for one-dimensional elementssuch as beams. A more useful plot is a "bending moment"-type plot, which you can produceusing the contour options.

5. From the prompt area, click Contour Options.

The Contour Plot Options dialog box appears; by default, the Basic tab is selected.

6. In the Contour Type field, toggle on Show tick marks for line elements.

7. Click the Shape tab, and select a Uniform deformation scale factor of 1.0.

8. Click OK.

3. From the Report Field Output dialog box, click the Setup tab.

4. Enter the file name crane.rpt.

5. Click OK.

The displacements of all the nodes in element set OUTA are written to a file namedcrane.rpt.

The peak displacement at the tip of the crane in the 2-direction is 0.0192 m.

Section forces and moments

ABAQUS can provide output for structural elements in terms of forces and moments acting onthe cross-section at a given point. These section forces and moments are defined in the localbeam coordinate system. Contour the section moment about the beam 1-axis in the membersof element set OUTA. For clarity, reset the view so that the elements are displayed in the 1-2plane.

To create a "bending moment"-type contour plot:



2. Select SM from the list of available output variables; then select SM1 from the componentfield.

3. Click OK.



ABAQUS/Viewer displays a contour plot of the bending moment about the beam 1-axis. Thecontour is plotted on the deformed model shape. The deformation scale factor is chosenautomatically since geometric nonlinearity was not considered in the analysis.

Color contour plots of this type typically are not very useful for one-dimensional elementssuch as beams. A more useful plot is a "bending moment"-type plot, which you can produceusing the contour options.

5. From the prompt area, click Contour Options.

The Contour Plot Options dialog box appears; by default, the Basic tab is selected.

6. In the Contour Type field, toggle on Show tick marks for line elements.

7. Click the Shape tab, and select a Uniform deformation scale factor of 1.0.

8. Click OK.

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The plot shown in Figure 6-20 appears. The magnitude of the variable at each node is nowindicated by the position at which the contour curve intersects a "tick mark" drawnperpendicular to the element. This "bending moment"-type plot can be used for any variable(not just bending moments) for any one-dimensional element, including trusses andaxisymmetric shells as well as beams.

Figure 6-20 Bending moment diagram (moment about beam 1-axis) for element set OUTA. The

locations with the highest stress (created by the bending of the elements) are indicated.

6.5 Related ABAQUS examples� ``Detroit Edison pipe whip experiment, '' Section 2.1.2 of the ABAQUS Example Problems Manual

� ``Buckling analysis of beams,'' Section 1.2.1 of the ABAQUS Benchmarks Manual

� ``Crash simulation of a motor vehicle, '' Section 1.3.14 of the ABAQUS Benchmarks Manual


6.6 Suggested readingBasic beam theory


� Oden, J. T. and E. A. Ripperger, Mechanics of Elastic Structures , McGraw-Hill, 1981.

Basic computational beam theory

The plot shown in Figure 6-20 appears. The magnitude of the variable at each node is nowindicated by the position at which the contour curve intersects a "tick mark" drawnperpendicular to the element. This "bending moment"-type plot can be used for any variable(not just bending moments) for any one-dimensional element, including trusses andaxisymmetric shells as well as beams.

Figure 6-20 Bending moment diagram (moment about beam 1-axis) for element set OUTA. The

locations with the highest stress (created by the bending of the elements) are indicated.

6.5 Related ABAQUS examples� ``Detroit Edison pipe whip experiment, '' Section 2.1.2 of the ABAQUS Example Problems Manual

� ``Buckling analysis of beams,'' Section 1.2.1 of the ABAQUS Benchmarks Manual

� ``Crash simulation of a motor vehicle, '' Section 1.3.14 of the ABAQUS Benchmarks Manual


6.6 Suggested readingBasic beam theory


� Oden, J. T. and E. A. Ripperger, Mechanics of Elastic Structures , McGraw-Hill, 1981.

Basic computational beam theory

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� Hughes, T. J. R., The Finite Element Method, Prentice-Hall Inc., 1987.

6.7 Summary� The behavior of beam elements can be determined by numerical integration of the section (either

*BEAM SECTION or *BEAM GENERAL SECTION) or can be given directly in terms of area,moments of inertia, and torsional constants (*BEAM GENERAL SECTION).

� When *BEAM GENERAL SECTION is used with numerical integration, the calculations are doneonce at the start of the simulation, and elastic behavior is assumed.

� ABAQUS includes a number of standard cross-section shapes. Other shapes, provided they are"thin-walled," can be modeled using SECTION=ARBITRARY.

� The orientation of the cross-section must be defined either by specifying a third node or bydefining a vector as part of the element property definition. The orientations can be plotted inABAQUS/Viewer.

� The beam cross-section can be offset from the nodes that define the beam. This procedure is usefulin modeling stiffeners on shells.

� The linear and quadratic beams include the effects of shear deformation. The cubic beams do notaccount for shear flexibility. The open-section beam elements correctly model the effects oftorsion and warping (including warping constraints) in thin-walled, open sections.

� Multi-point constraints and constraint equations can be used to connect degrees of freedom atnodes to model pinned connections, rigid links, etc.

� "Bending moment"-type plots allow the results of one-dimensional elements, such as beams, to bevisualized easily.

� Hard copies of ABAQUS/Viewer plots can be obtained in PostScript ( PS), EncapsulatedPostScript (EPS), Tag Image File Format (TIFF), and Portable Network Graphics (PNG) formats.


� Hughes, T. J. R., The Finite Element Method, Prentice-Hall Inc., 1987.

6.7 Summary� The behavior of beam elements can be determined by numerical integration of the section (either

*BEAM SECTION or *BEAM GENERAL SECTION) or can be given directly in terms of area,moments of inertia, and torsional constants (*BEAM GENERAL SECTION).

� When *BEAM GENERAL SECTION is used with numerical integration, the calculations are doneonce at the start of the simulation, and elastic behavior is assumed.

� ABAQUS includes a number of standard cross-section shapes. Other shapes, provided they are"thin-walled," can be modeled using SECTION=ARBITRARY.

� The orientation of the cross-section must be defined either by specifying a third node or bydefining a vector as part of the element property definition. The orientations can be plotted inABAQUS/Viewer.

� The beam cross-section can be offset from the nodes that define the beam. This procedure is usefulin modeling stiffeners on shells.

� The linear and quadratic beams include the effects of shear deformation. The cubic beams do notaccount for shear flexibility. The open-section beam elements correctly model the effects oftorsion and warping (including warping constraints) in thin-walled, open sections.

� Multi-point constraints and constraint equations can be used to connect degrees of freedom atnodes to model pinned connections, rigid links, etc.

� "Bending moment"-type plots allow the results of one-dimensional elements, such as beams, to bevisualized easily.

� Hard copies of ABAQUS/Viewer plots can be obtained in PostScript ( PS), EncapsulatedPostScript (EPS), Tag Image File Format (TIFF), and Portable Network Graphics (PNG) formats.

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7. NonlinearityThis chapter discusses nonlinear structural analysis in ABAQUS. The differences between linear andnonlinear analyses are summarized below.

Linear analysis

All the analyses discussed so far have been linear: there is a linear relationship between theapplied loads and the response of the system. For example, if a linear spring extends by 1 munder a load of 10 N, it will extend by 2 m when a load of 20 N is applied. This means that, ina linear analysis, the flexibility of the structure need only be calculated once (by assemblingthe stiffness matrix and inverting it). The linear response of the structure to other load casescan be found by multiplying the new vector of loads by the inverted stiffness matrix.Furthermore, the structure's response to various load cases can be scaled by constants and/orsuperimposed on one another to determine its response to a completely new load case,provided that the new load case is the sum (or multiple) of previous ones. This principle ofsuperposition of load cases assumes that the same boundary conditions are used for all theload cases.

ABAQUS uses the principle of superposition of load cases in linear dynamics simulations,which are discussed in Chapter 9, "Dynamics."

Nonlinear analysis

A nonlinear structural problem is one in which the structure's stiffness changes as it deforms.All physical structures are nonlinear. Linear analysis is a convenient approximation that isoften adequate for design purposes. It is obviously inadequate for many structural simulationsincluding manufacturing processes, such as forging or stamping; crash analyses; and analysesof rubber components, such as tires or engine mounts. A simple example is a spring with anonlinear stiffening response (see Figure 7-1).

Figure 7-1 Linear and nonlinear spring characteristics.

Since the stiffness is now dependent on the displacement, the initial flexibility can no longerbe multiplied by the applied load to calculate the spring's displacement for any load. In anonlinear analysis the stiffness matrix of the structure has to be assembled and inverted manytimes during the course of the analysis, making it much more expensive to solve than a linearanalysis.

7. NonlinearityThis chapter discusses nonlinear structural analysis in ABAQUS. The differences between linear andnonlinear analyses are summarized below.

Linear analysis

All the analyses discussed so far have been linear: there is a linear relationship between theapplied loads and the response of the system. For example, if a linear spring extends by 1 munder a load of 10 N, it will extend by 2 m when a load of 20 N is applied. This means that, ina linear analysis, the flexibility of the structure need only be calculated once (by assemblingthe stiffness matrix and inverting it). The linear response of the structure to other load casescan be found by multiplying the new vector of loads by the inverted stiffness matrix.Furthermore, the structure's response to various load cases can be scaled by constants and/orsuperimposed on one another to determine its response to a completely new load case,provided that the new load case is the sum (or multiple) of previous ones. This principle ofsuperposition of load cases assumes that the same boundary conditions are used for all theload cases.

ABAQUS uses the principle of superposition of load cases in linear dynamics simulations,which are discussed in Chapter 9, "Dynamics."

Nonlinear analysis

A nonlinear structural problem is one in which the structure's stiffness changes as it deforms.All physical structures are nonlinear. Linear analysis is a convenient approximation that isoften adequate for design purposes. It is obviously inadequate for many structural simulationsincluding manufacturing processes, such as forging or stamping; crash analyses; and analysesof rubber components, such as tires or engine mounts. A simple example is a spring with anonlinear stiffening response (see Figure 7-1).

Figure 7-1 Linear and nonlinear spring characteristics.

Since the stiffness is now dependent on the displacement, the initial flexibility can no longerbe multiplied by the applied load to calculate the spring's displacement for any load. In anonlinear analysis the stiffness matrix of the structure has to be assembled and inverted manytimes during the course of the analysis, making it much more expensive to solve than a linearanalysis.

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Since the response of a nonlinear system is not a linear function of the magnitude of theapplied load, it is not possible to create solutions for different load cases by superposition.Each load case must be defined and solved as a separate analysis.

7.1 Sources of nonlinearityThere are three sources of nonlinearity in structural mechanics simulations:

� Material nonlinearity.

� Boundary nonlinearity.

� Geometric nonlinearity.

7.1.1 Material nonlinearityThis type of nonlinearity is probably the one that you are most familiar with and is covered in moredepth in Chapter 8, "Materials." Most metals have a fairly linear stress/strain relationship at low strainvalues, but at higher strains the material yields, at which point the response becomes nonlinear andirreversible (Figure 7-2).

Figure 7-2 Stress-strain curve for an elastic-plastic material under uniaxial tension.

Rubber materials have a nonlinear, reversible (elastic) response ( Figure 7-3).

Figure 7-3 Stress-strain curve for a rubber-type material.

Since the response of a nonlinear system is not a linear function of the magnitude of theapplied load, it is not possible to create solutions for different load cases by superposition.Each load case must be defined and solved as a separate analysis.

7.1 Sources of nonlinearityThere are three sources of nonlinearity in structural mechanics simulations:

� Material nonlinearity.

� Boundary nonlinearity.

� Geometric nonlinearity.

7.1.1 Material nonlinearityThis type of nonlinearity is probably the one that you are most familiar with and is covered in moredepth in Chapter 8, "Materials." Most metals have a fairly linear stress/strain relationship at low strainvalues, but at higher strains the material yields, at which point the response becomes nonlinear andirreversible (Figure 7-2).

Figure 7-2 Stress-strain curve for an elastic-plastic material under uniaxial tension.

Rubber materials have a nonlinear, reversible (elastic) response ( Figure 7-3).

Figure 7-3 Stress-strain curve for a rubber-type material.

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Material nonlinearity may be related to factors other than strain. Strain-rate-dependent material dataand material failure are both forms of material nonlinearity. Material properties can also be a functionof temperature and other predefined fields.

7.1.2 Boundary nonlinearityBoundary nonlinearity occurs if the boundary conditions change during the analysis. Consider thecantilever beam, shown in Figure 7-4, that deflects under an applied load until it hits a "stop."

Figure 7-4 Cantilever beam hitting a stop.

The vertical deflection of the tip is linearly related to the load (if the deflection is small) until itcontacts the stop. There is then a sudden change in the boundary condition at the tip of the beam,preventing any further vertical deflection, and so the response of the beam is no longer linear.Boundary nonlinearities are extremely discontinuous: when contact occurs during a simulation, there isa large and instantaneous change in the response of the structure.

Another example of boundary nonlinearity is blowing a sheet of material into a mold. The sheetexpands relatively easily under the applied pressure until it begins to contact the mold. From then onthe pressure must be increased to continue forming the sheet because of the change in boundaryconditions.

Boundary nonlinearity is covered in Chapter 11, "Contact."

7.1.3 Geometric nonlinearityThe third source of nonlinearity is related to changes in the geometry of the model during the analysis.

Material nonlinearity may be related to factors other than strain. Strain-rate-dependent material dataand material failure are both forms of material nonlinearity. Material properties can also be a functionof temperature and other predefined fields.

7.1.2 Boundary nonlinearityBoundary nonlinearity occurs if the boundary conditions change during the analysis. Consider thecantilever beam, shown in Figure 7-4, that deflects under an applied load until it hits a "stop."

Figure 7-4 Cantilever beam hitting a stop.

The vertical deflection of the tip is linearly related to the load (if the deflection is small) until itcontacts the stop. There is then a sudden change in the boundary condition at the tip of the beam,preventing any further vertical deflection, and so the response of the beam is no longer linear.Boundary nonlinearities are extremely discontinuous: when contact occurs during a simulation, there isa large and instantaneous change in the response of the structure.

Another example of boundary nonlinearity is blowing a sheet of material into a mold. The sheetexpands relatively easily under the applied pressure until it begins to contact the mold. From then onthe pressure must be increased to continue forming the sheet because of the change in boundaryconditions.

Boundary nonlinearity is covered in Chapter 11, "Contact."

7.1.3 Geometric nonlinearityThe third source of nonlinearity is related to changes in the geometry of the model during the analysis.

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Geometric nonlinearity occurs whenever the magnitude of the displacements affects the response of thestructure. This may be caused by:

� Large deflections or rotations.

� "Snap through."

� Initial stresses or load stiffening.

For example, consider a cantilever beam loaded vertically at the tip ( Figure 7-5).

Figure 7-5 Large deflection of a cantilever beam.

If the tip deflection is small, the analysis can be considered as being approximately linear. However, ifthe tip deflections are large, the shape of the structure and, hence, its stiffness changes. In addition, ifthe load does not remain perpendicular to the beam, the action of the load on the structure changessignificantly. As the cantilever beam deflects, the load can be resolved into a component perpendicularto the beam and a component acting along the length of the beam. Both of these effects contribute tothe nonlinear response of the cantilever beam (i.e., the changing of the beam's stiffness as the load itcarries increases).

One would expect large deflections and rotations to have a significant effect on the way that structurescarry loads. However, displacements do not necessarily have to be large relative to the dimensions ofthe structure for geometric nonlinearity to be important. Consider the "snap through" under appliedpressure of a large panel with a shallow curve, as shown in Figure 7-6.

Figure 7-6 Snap-through of a large panel.

Geometric nonlinearity occurs whenever the magnitude of the displacements affects the response of thestructure. This may be caused by:

� Large deflections or rotations.

� "Snap through."

� Initial stresses or load stiffening.

For example, consider a cantilever beam loaded vertically at the tip ( Figure 7-5).

Figure 7-5 Large deflection of a cantilever beam.

If the tip deflection is small, the analysis can be considered as being approximately linear. However, ifthe tip deflections are large, the shape of the structure and, hence, its stiffness changes. In addition, ifthe load does not remain perpendicular to the beam, the action of the load on the structure changessignificantly. As the cantilever beam deflects, the load can be resolved into a component perpendicularto the beam and a component acting along the length of the beam. Both of these effects contribute tothe nonlinear response of the cantilever beam (i.e., the changing of the beam's stiffness as the load itcarries increases).

One would expect large deflections and rotations to have a significant effect on the way that structurescarry loads. However, displacements do not necessarily have to be large relative to the dimensions ofthe structure for geometric nonlinearity to be important. Consider the "snap through" under appliedpressure of a large panel with a shallow curve, as shown in Figure 7-6.

Figure 7-6 Snap-through of a large panel.

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In this example there is a dramatic change in the stiffness of the panel as it deforms. As the panel"snaps through," the stiffness becomes negative. Thus, although the magnitude of the displacements,relative to the panel's dimensions, is quite small, there is significant geometric nonlinearity in thesimulation, which must be taken into consideration.

7.2 The solution of nonlinear problemsThe nonlinear load-displacement curve for a structure is shown in Figure 7-7.

Figure 7-7 Nonlinear load-displacement curve.

The objective of the analysis is to determine this response. ABAQUS uses the Newton-Raphsonmethod to obtain solutions for nonlinear problems. In a nonlinear analysis the solution cannot becalculated by solving a single system of equations, as would be done in a linear problem. Instead, thesolution is found by applying the specified loads gradually and incrementally working toward the finalsolution. Therefore, ABAQUS breaks the simulation into a number of load increments and finds theapproximate equilibrium configuration at the end of each load increment. It often takes ABAQUSseveral iterations to determine an acceptable solution to a given load increment. The sum of all of theincremental responses is the approximate solution for the nonlinear analysis.

In this example there is a dramatic change in the stiffness of the panel as it deforms. As the panel"snaps through," the stiffness becomes negative. Thus, although the magnitude of the displacements,relative to the panel's dimensions, is quite small, there is significant geometric nonlinearity in thesimulation, which must be taken into consideration.

7.2 The solution of nonlinear problemsThe nonlinear load-displacement curve for a structure is shown in Figure 7-7.

Figure 7-7 Nonlinear load-displacement curve.

The objective of the analysis is to determine this response. ABAQUS uses the Newton-Raphsonmethod to obtain solutions for nonlinear problems. In a nonlinear analysis the solution cannot becalculated by solving a single system of equations, as would be done in a linear problem. Instead, thesolution is found by applying the specified loads gradually and incrementally working toward the finalsolution. Therefore, ABAQUS breaks the simulation into a number of load increments and finds theapproximate equilibrium configuration at the end of each load increment. It often takes ABAQUSseveral iterations to determine an acceptable solution to a given load increment. The sum of all of theincremental responses is the approximate solution for the nonlinear analysis.

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Consider the internal (nodal) forces, I , and the external forces, P , acting on a body (see Figure 7-8(b)and Figure 7-8(a), respectively). The internal loads acting on a node are caused by the stresses in theelements that contain that node.

Figure 7-8 Internal and external loads on a body.

For the body to be in equilibrium, the net force acting at every node must be zero. Therefore, the basicstatement of equilibrium is that the internal forces, I , and the external forces, P , must balance eachother:

P ¡ I = 0:

7.2.1 Steps, increments, and iterationsThis section introduces some new vocabulary for describing the various parts of an analysis. It isimportant that you clearly understand the difference between an analysis step, a load increment, and aniteration.

� The load history for a simulation consists of one or more steps. You define the steps, whichgenerally consist of an analysis procedure option, loading options, and output request options.Different loads, boundary conditions, analysis procedure options, and output requests can be usedin each step. For example:

� Step 1: Hold a plate between rigid jaws.

� Step 2: Add loads to deform the plate.

� Step 3: Find the natural frequencies of the deformed plate.

� An increment is part of a step. In nonlinear analyses the total load applied in a step is broken intosmaller increments so that the nonlinear solution path can be followed. You suggest the size of thefirst increment, and ABAQUS automatically chooses the size of the subsequent increments. At theend of each increment the structure is in (approximate) equilibrium and results are available forwriting to the output database, restart, data, or results files. The increments at which you selectresults to be written to the output database file are called frames.

� An iteration is an attempt at finding an equilibrium solution in an increment. If the model is not inequilibrium at the end of the iteration, ABAQUS tries another iteration. With every iteration the

Consider the internal (nodal) forces, I , and the external forces, P , acting on a body (see Figure 7-8(b)and Figure 7-8(a), respectively). The internal loads acting on a node are caused by the stresses in theelements that contain that node.

Figure 7-8 Internal and external loads on a body.

For the body to be in equilibrium, the net force acting at every node must be zero. Therefore, the basicstatement of equilibrium is that the internal forces, I , and the external forces, P , must balance eachother:

P ¡ I = 0:

7.2.1 Steps, increments, and iterationsThis section introduces some new vocabulary for describing the various parts of an analysis. It isimportant that you clearly understand the difference between an analysis step, a load increment, and aniteration.

� The load history for a simulation consists of one or more steps. You define the steps, whichgenerally consist of an analysis procedure option, loading options, and output request options.Different loads, boundary conditions, analysis procedure options, and output requests can be usedin each step. For example:

� Step 1: Hold a plate between rigid jaws.

� Step 2: Add loads to deform the plate.

� Step 3: Find the natural frequencies of the deformed plate.

� An increment is part of a step. In nonlinear analyses the total load applied in a step is broken intosmaller increments so that the nonlinear solution path can be followed. You suggest the size of thefirst increment, and ABAQUS automatically chooses the size of the subsequent increments. At theend of each increment the structure is in (approximate) equilibrium and results are available forwriting to the output database, restart, data, or results files. The increments at which you selectresults to be written to the output database file are called frames.

� An iteration is an attempt at finding an equilibrium solution in an increment. If the model is not inequilibrium at the end of the iteration, ABAQUS tries another iteration. With every iteration the

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solution ABAQUS obtains should be closer to equilibrium; sometimes ABAQUS may need manyiterations to obtain an equilibrium solution. Because the structure is not necessarily in equilibriumat the end of a given iteration, results can be requested only for the iteration that produces anequilibrium solution.

7.2.2 Equilibrium iterations and convergenceThe nonlinear response of a structure to a small load increment, ¢P , is shown in Figure 7-9.ABAQUS uses the structure's initial stiffness, K0 , which is based on its configuration at u0 , and ¢P

to calculate a displacement correction, ca, for the structure. Using ca, the structure's configuration isupdated to ua.

Figure 7-9 First iteration in an increment.

Convergence

ABAQUS forms a new stiffness, Ka , for the structure, based on its updated configuration, ua.ABAQUS uses Ka to calculate the structure's internal forces, Ia, in this updated configuration.The difference between the total applied load, P , and Ia can now be calculated as:

Ra = P ¡ Ia ;

where Ra is the force residual for the iteration.

If Ra is zero at every degree of freedom in the model, point a in Figure 7-9 would lie on theload-deflection curve, and the structure would be in equilibrium. In a nonlinear problem it isalmost impossible to have Ra equal zero, so ABAQUS compares it to a tolerance value. If Ra

is less than this force residual tolerance, ABAQUS accepts the structure's updatedconfiguration as the equilibrium solution. By default, this tolerance value is set to 0.5% of anaverage force in the structure, averaged over time. ABAQUS automatically calculates thisspatially and time-averaged force throughout the simulation.

If Ra is less than the current tolerance value, P and Ia are in equilibrium, and ua is a validequilibrium configuration for the structure under the applied load. However, before ABAQUSaccepts the solution, it also checks that the displacement correction, ca, is small relative to the

solution ABAQUS obtains should be closer to equilibrium; sometimes ABAQUS may need manyiterations to obtain an equilibrium solution. Because the structure is not necessarily in equilibriumat the end of a given iteration, results can be requested only for the iteration that produces anequilibrium solution.

7.2.2 Equilibrium iterations and convergenceThe nonlinear response of a structure to a small load increment, ¢P , is shown in Figure 7-9.ABAQUS uses the structure's initial stiffness, K0 , which is based on its configuration at u0 , and ¢P

to calculate a displacement correction, ca, for the structure. Using ca, the structure's configuration isupdated to ua.

Figure 7-9 First iteration in an increment.

Convergence

ABAQUS forms a new stiffness, Ka , for the structure, based on its updated configuration, ua.ABAQUS uses Ka to calculate the structure's internal forces, Ia, in this updated configuration.The difference between the total applied load, P , and Ia can now be calculated as:

Ra = P ¡ Ia ;

where Ra is the force residual for the iteration.

If Ra is zero at every degree of freedom in the model, point a in Figure 7-9 would lie on theload-deflection curve, and the structure would be in equilibrium. In a nonlinear problem it isalmost impossible to have Ra equal zero, so ABAQUS compares it to a tolerance value. If Ra

is less than this force residual tolerance, ABAQUS accepts the structure's updatedconfiguration as the equilibrium solution. By default, this tolerance value is set to 0.5% of anaverage force in the structure, averaged over time. ABAQUS automatically calculates thisspatially and time-averaged force throughout the simulation.

If Ra is less than the current tolerance value, P and Ia are in equilibrium, and ua is a validequilibrium configuration for the structure under the applied load. However, before ABAQUSaccepts the solution, it also checks that the displacement correction, ca, is small relative to the

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total incremental displacement, ¢ua = ua ¡ u0 . If ca is greater than 1% of the incrementaldisplacement, ABAQUS performs another iteration. Both convergence checks must besatisfied before a solution is said to have converged for that load increment. The exception tothis rule is the case of a linear increment, which is defined as any increment in which thelargest force residual is less than 10-8 times the time-averaged force. Any case that passes sucha stringent comparison of the largest force residual with the time-averaged force is consideredlinear and does not require further iteration. The solution is accepted without any check on thesize of the displacement correction.

If the solution from an iteration is not converged, ABAQUS performs another iteration to tryto bring the internal and external forces into balance. This second iteration uses the stiffness,Ka , calculated at the end of the previous iteration together with Ra to determine anotherdisplacement correction, cb, that brings the system closer to equilibrium (point b in Figure7-10).

Figure 7-10 Second iteration.

ABAQUS calculates a new force residual, Rb , using the internal forces from the structure'snew configuration, ub. Again, the largest force residual at any degree of freedom, Rb , iscompared against the force residual tolerance, and the displacement correction for the seconditeration, cb, is compared to the increment of displacement, ¢ub = ub ¡ u0 . If necessaryABAQUS performs further iterations.

For each iteration in a nonlinear analysis ABAQUS forms the model's stiffness matrix andsolves a system of equations. This means that each iteration is equivalent, in computationalcost, to conducting a complete linear analysis. It should now be clear that the computationalexpense of a nonlinear analysis can be many times greater than for a linear one.

It is possible with ABAQUS to save results at each converged increment. Thus, the amount ofoutput data available from a nonlinear simulation is many times that available from a linearanalysis of the same geometry. Consider both of these factors and the types of nonlinearsimulations that you want to perform when planning your computer resources.

total incremental displacement, ¢ua = ua ¡ u0 . If ca is greater than 1% of the incrementaldisplacement, ABAQUS performs another iteration. Both convergence checks must besatisfied before a solution is said to have converged for that load increment. The exception tothis rule is the case of a linear increment, which is defined as any increment in which thelargest force residual is less than 10-8 times the time-averaged force. Any case that passes sucha stringent comparison of the largest force residual with the time-averaged force is consideredlinear and does not require further iteration. The solution is accepted without any check on thesize of the displacement correction.

If the solution from an iteration is not converged, ABAQUS performs another iteration to tryto bring the internal and external forces into balance. This second iteration uses the stiffness,Ka , calculated at the end of the previous iteration together with Ra to determine anotherdisplacement correction, cb, that brings the system closer to equilibrium (point b in Figure7-10).

Figure 7-10 Second iteration.

ABAQUS calculates a new force residual, Rb , using the internal forces from the structure'snew configuration, ub. Again, the largest force residual at any degree of freedom, Rb , iscompared against the force residual tolerance, and the displacement correction for the seconditeration, cb, is compared to the increment of displacement, ¢ub = ub ¡ u0 . If necessaryABAQUS performs further iterations.

For each iteration in a nonlinear analysis ABAQUS forms the model's stiffness matrix andsolves a system of equations. This means that each iteration is equivalent, in computationalcost, to conducting a complete linear analysis. It should now be clear that the computationalexpense of a nonlinear analysis can be many times greater than for a linear one.

It is possible with ABAQUS to save results at each converged increment. Thus, the amount ofoutput data available from a nonlinear simulation is many times that available from a linearanalysis of the same geometry. Consider both of these factors and the types of nonlinearsimulations that you want to perform when planning your computer resources.

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7.2.3 Automatic incrementation controlABAQUS automatically adjusts the size of the load increments so that it solves nonlinear problemseasily and efficiently. You only need to suggest the size of the first increment in each step of yoursimulation. Thereafter, ABAQUS automatically adjusts the size of the increments. If you do notprovide a suggested initial increment size, ABAQUS will try to apply all of the loads defined in thestep in the first increment. In highly nonlinear problems ABAQUS will have to reduce the incrementsize repeatedly, resulting in wasted CPU time. Generally it is to your advantage to provide a reasonableinitial increment size (see ``Modifications to the input file--the history data, '' Section 7.4.1, for anexample); only in very mildly nonlinear problems can all of the loads in a step be applied in a singleincrement.

The number of iterations needed to find a converged solution for a load increment will vary dependingon the degree of nonlinearity in the system. By default, if the solution has not converged within 16iterations or if the solution appears to diverge, ABAQUS abandons the increment and starts again withthe increment size set to 25% of its previous value. An attempt is then made at finding a convergedsolution with this smaller load increment. If the increment still fails to converge, ABAQUS reduces theincrement size again. By default, ABAQUS allows a maximum of five cutbacks in an increment beforestopping the analysis.

If the increment converges in fewer than five iterations, this indicates that the solution is being foundfairly easily. Therefore, ABAQUS automatically increases the increment size by 50% if twoconsecutive increments require fewer than five iterations to obtain a converged solution.

Details of the automatic load incrementation scheme are given in the message file, as shown in moredetail in ``Results,'' Section 7.4.3.

7.3 Including nonlinearity in an ABAQUS analysisWe now discuss how to account for nonlinearity in an ABAQUS analysis. The main focus is ongeometric nonlinearity.

7.3.1 Geometric nonlinearityIncorporating the effects of geometric nonlinearity in an analysis requires only minor changes to theinput file. You need to add the NLGEOM parameter to the *STEP option and remove thePERTURBATION parameter. You may also add the INC parameter to specify the maximum numberof increments allowed during the step. ABAQUS terminates the analysis with an error message if itneeds more increments than this limit to complete the step. The default number of increments for astep is 10; if significant nonlinearity is present in the simulation, the analysis may require many moreincrements. The INC parameter specifies an upper limit on the number of increments that ABAQUScan use, rather than the number of increments it must use. For example, a step involving nonlineargeometry with a maximum of 25 increments would be specified as:

*STEP, NLGEOM, INC=25

In a nonlinear analysis a step takes place over a finite period of "time," although this "time" has no

7.2.3 Automatic incrementation controlABAQUS automatically adjusts the size of the load increments so that it solves nonlinear problemseasily and efficiently. You only need to suggest the size of the first increment in each step of yoursimulation. Thereafter, ABAQUS automatically adjusts the size of the increments. If you do notprovide a suggested initial increment size, ABAQUS will try to apply all of the loads defined in thestep in the first increment. In highly nonlinear problems ABAQUS will have to reduce the incrementsize repeatedly, resulting in wasted CPU time. Generally it is to your advantage to provide a reasonableinitial increment size (see ``Modifications to the input file--the history data, '' Section 7.4.1, for anexample); only in very mildly nonlinear problems can all of the loads in a step be applied in a singleincrement.

The number of iterations needed to find a converged solution for a load increment will vary dependingon the degree of nonlinearity in the system. By default, if the solution has not converged within 16iterations or if the solution appears to diverge, ABAQUS abandons the increment and starts again withthe increment size set to 25% of its previous value. An attempt is then made at finding a convergedsolution with this smaller load increment. If the increment still fails to converge, ABAQUS reduces theincrement size again. By default, ABAQUS allows a maximum of five cutbacks in an increment beforestopping the analysis.

If the increment converges in fewer than five iterations, this indicates that the solution is being foundfairly easily. Therefore, ABAQUS automatically increases the increment size by 50% if twoconsecutive increments require fewer than five iterations to obtain a converged solution.

Details of the automatic load incrementation scheme are given in the message file, as shown in moredetail in ``Results,'' Section 7.4.3.

7.3 Including nonlinearity in an ABAQUS analysisWe now discuss how to account for nonlinearity in an ABAQUS analysis. The main focus is ongeometric nonlinearity.

7.3.1 Geometric nonlinearityIncorporating the effects of geometric nonlinearity in an analysis requires only minor changes to theinput file. You need to add the NLGEOM parameter to the *STEP option and remove thePERTURBATION parameter. You may also add the INC parameter to specify the maximum numberof increments allowed during the step. ABAQUS terminates the analysis with an error message if itneeds more increments than this limit to complete the step. The default number of increments for astep is 10; if significant nonlinearity is present in the simulation, the analysis may require many moreincrements. The INC parameter specifies an upper limit on the number of increments that ABAQUScan use, rather than the number of increments it must use. For example, a step involving nonlineargeometry with a maximum of 25 increments would be specified as:


In a nonlinear analysis a step takes place over a finite period of "time," although this "time" has no

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physical meaning unless inertial effects or rate-dependent behavior are important. The data line on theprocedure option used in the step specifies the initial time increment and the total time for the step. Forexample,

defines a static analysis that occurs over 1.0 units of time and has an initial increment of 0.1. Thesedata also specify the proportion of load applied in the first increment. The initial load increment isgiven by:

¢Tinitial

Ttotal£ Load magnitude:

The choice of initial time increment can be critical in certain nonlinear simulations, but for mostanalyses an initial increment size that is 5% to 10% of the total step time is usually sufficient. In staticsimulations the total step time is usually set to 1.0, for convenience, unless, for example,rate-dependent material effects or dashpots are included in the model. With a total step time of 1.0 theproportion of load applied is always equal to the current step time; i.e., 50% of the total load is appliedwhen the step time is 0.5.

Although you must specify the initial increment size, ABAQUS automatically controls the size of thesubsequent increments. This automatic control of the increment size is suitable for the majority ofnonlinear simulations performed with ABAQUS, although further controls on the increment size areavailable. ABAQUS will terminate an analysis if excessive cutbacks caused by convergence problemsreduce the increment size below the minimum value. The default minimum allowable time increment,¢Tmin , is 10-5 times the total step time. By default, ABAQUS has no upper limit on the incrementsize, ¢Tmax , other than the total step time. Depending on your ABAQUS simulation, you may want tospecify different minimum and/or maximum allowable increment sizes. For example, if you know thatyour simulation may have trouble obtaining a solution if too large a load increment is applied, perhapsbecause the model may undergo plastic deformation, you may want to decrease ¢Tmax .

The following input describes a static analysis in which the load is applied over 5 units of time and theinitial time increment is 1 unit of time; the minimum and maximum time increments are set to 0.0001and 1.5, respectively:

ABAQUS will apply 20% (1.0/5.0) of the total load in the first increment, and it will terminate theanalysis if it has problems converging and requires an increment smaller than 0.0001. If the timeincrement grows because the solution is converging easily, the maximum time increment ABAQUS

physical meaning unless inertial effects or rate-dependent behavior are important. The data line on theprocedure option used in the step specifies the initial time increment and the total time for the step. Forexample,

defines a static analysis that occurs over 1.0 units of time and has an initial increment of 0.1. Thesedata also specify the proportion of load applied in the first increment. The initial load increment isgiven by:

¢Tinitial

Ttotal£ Load magnitude:

The choice of initial time increment can be critical in certain nonlinear simulations, but for mostanalyses an initial increment size that is 5% to 10% of the total step time is usually sufficient. In staticsimulations the total step time is usually set to 1.0, for convenience, unless, for example,rate-dependent material effects or dashpots are included in the model. With a total step time of 1.0 theproportion of load applied is always equal to the current step time; i.e., 50% of the total load is appliedwhen the step time is 0.5.

Although you must specify the initial increment size, ABAQUS automatically controls the size of thesubsequent increments. This automatic control of the increment size is suitable for the majority ofnonlinear simulations performed with ABAQUS, although further controls on the increment size areavailable. ABAQUS will terminate an analysis if excessive cutbacks caused by convergence problemsreduce the increment size below the minimum value. The default minimum allowable time increment,¢Tmin , is 10-5 times the total step time. By default, ABAQUS has no upper limit on the incrementsize, ¢Tmax , other than the total step time. Depending on your ABAQUS simulation, you may want tospecify different minimum and/or maximum allowable increment sizes. For example, if you know thatyour simulation may have trouble obtaining a solution if too large a load increment is applied, perhapsbecause the model may undergo plastic deformation, you may want to decrease ¢Tmax .

The following input describes a static analysis in which the load is applied over 5 units of time and theinitial time increment is 1 unit of time; the minimum and maximum time increments are set to 0.0001and 1.5, respectively:

ABAQUS will apply 20% (1.0/5.0) of the total load in the first increment, and it will terminate theanalysis if it has problems converging and requires an increment smaller than 0.0001. If the timeincrement grows because the solution is converging easily, the maximum time increment ABAQUS

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can use is 1.5.

Local directions

In a geometrically nonlinear analysis the local material directions may rotate with thedeformation in each element. For shell, beam, and truss elements the local material directionsalways rotate with the deformation. For solid elements the local material directions rotate withthe deformation only if the elements refer to an *ORIENTATION option; otherwise, the localmaterial directions remain constant throughout the analysis.

Local directions defined at nodes by using the *TRANSFORM option remain fixed throughoutthe analysis; they do not rotate with the deformation.

Effect on subsequent steps

Once you include the NLGEOM parameter on the *STEP option, geometric nonlinearity isconsidered in all subsequent steps. If the NLGEOM parameter is omitted in a subsequent step,ABAQUS will issue a warning stating that nonlinear geometric effects are being included inthe step anyway.

Other geometrically nonlinear effects

The large deformations in a model are not the only important nonlinear geometric effects thatare considered when the NLGEOM parameter is used. Using this parameter also causesABAQUS to include terms in the element stiffness calculations that are caused by the appliedloads, the so-called load stiffness. These terms improve convergence behavior. In addition, themembrane loads in shells and the axial loads in cables and beams contribute much of thestiffness of these structures in response to transverse loads. By including geometricnonlinearity, the membrane stiffness in response to transverse loads is considered as well.

7.3.2 Material nonlinearityThe addition of material nonlinearity to ABAQUS input files is discussed in Chapter 8, "Materials."

7.3.3 Boundary nonlinearityThe introduction of boundary nonlinearity is discussed in Chapter 11, "Contact."

7.4 Example: nonlinear skew plateThis example is a continuation of the linear skew plate simulation described in Chapter 5, "Using ShellElements," and shown in Figure 7-11.

Figure 7-11 Skew plate.

can use is 1.5.

Local directions

In a geometrically nonlinear analysis the local material directions may rotate with thedeformation in each element. For shell, beam, and truss elements the local material directionsalways rotate with the deformation. For solid elements the local material directions rotate withthe deformation only if the elements refer to an *ORIENTATION option; otherwise, the localmaterial directions remain constant throughout the analysis.

Local directions defined at nodes by using the *TRANSFORM option remain fixed throughoutthe analysis; they do not rotate with the deformation.

Effect on subsequent steps

Once you include the NLGEOM parameter on the *STEP option, geometric nonlinearity isconsidered in all subsequent steps. If the NLGEOM parameter is omitted in a subsequent step,ABAQUS will issue a warning stating that nonlinear geometric effects are being included inthe step anyway.

Other geometrically nonlinear effects

The large deformations in a model are not the only important nonlinear geometric effects thatare considered when the NLGEOM parameter is used. Using this parameter also causesABAQUS to include terms in the element stiffness calculations that are caused by the appliedloads, the so-called load stiffness. These terms improve convergence behavior. In addition, themembrane loads in shells and the axial loads in cables and beams contribute much of thestiffness of these structures in response to transverse loads. By including geometricnonlinearity, the membrane stiffness in response to transverse loads is considered as well.

7.3.2 Material nonlinearityThe addition of material nonlinearity to ABAQUS input files is discussed in Chapter 8, "Materials."

7.3.3 Boundary nonlinearityThe introduction of boundary nonlinearity is discussed in Chapter 11, "Contact."

7.4 Example: nonlinear skew plateThis example is a continuation of the linear skew plate simulation described in Chapter 5, "Using ShellElements," and shown in Figure 7-11.

Figure 7-11 Skew plate.

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Having already simulated the linear response of the plate, you will now reanalyze it including theeffects of geometric nonlinearity. The results from the linear simulation indicated that nonlinear effectsmay be important for this problem. With the results of this analysis, you will now be able to judgewhether this is true or not.

You only need to modify the history data to convert this model from a linear simulation to a nonlinearsimulation.

Copy the input file skew.inp to skew_nl.inp, and incorporate the changes described below.

7.4.1 Modifications to the input file--the history dataAdd the NLGEOM parameter to the *STEP option, and remove the PERTURBATION parameter. Thisindicates that the analysis now includes nonlinear geometric effects. In addition, set the maximumnumber of increments to 25. ABAQUS may use fewer increments than this upper limit, but it will stopthe analysis if it needs more.

Your modified *STEP option should look like:


You may also wish to modify the description of the step to reflect that this is now a nonlinearanalysis.

Defining the step time

You need to add a data line to the *STATIC option to specify the size of the initial timeincrement, ¢Tinitial , for the analysis and the total step time for the simulation. Use a total steptime of 1.0, and specify ¢Tinitial such that ABAQUS applies 10% of the load in the firstincrement. The completed *STATIC option block will, therefore, be

*STATIC

0.1,1.0

Output control

In a linear analysis ABAQUS solves the equilibrium equations once and calculates the resultsfor this one solution. A nonlinear analysis can produce much more output because results canbe requested at the end of each converged increment. If you do not select the output requests

Having already simulated the linear response of the plate, you will now reanalyze it including theeffects of geometric nonlinearity. The results from the linear simulation indicated that nonlinear effectsmay be important for this problem. With the results of this analysis, you will now be able to judgewhether this is true or not.

You only need to modify the history data to convert this model from a linear simulation to a nonlinearsimulation.

Copy the input file skew.inp to skew_nl.inp, and incorporate the changes described below.

7.4.1 Modifications to the input file--the history dataAdd the NLGEOM parameter to the *STEP option, and remove the PERTURBATION parameter. Thisindicates that the analysis now includes nonlinear geometric effects. In addition, set the maximumnumber of increments to 25. ABAQUS may use fewer increments than this upper limit, but it will stopthe analysis if it needs more.

Your modified *STEP option should look like:


You may also wish to modify the description of the step to reflect that this is now a nonlinearanalysis.

Defining the step time

You need to add a data line to the *STATIC option to specify the size of the initial timeincrement, ¢Tinitial , for the analysis and the total step time for the simulation. Use a total steptime of 1.0, and specify ¢Tinitial such that ABAQUS applies 10% of the load in the firstincrement. The completed *STATIC option block will, therefore, be

*STATIC

0.1,1.0

Output control

In a linear analysis ABAQUS solves the equilibrium equations once and calculates the resultsfor this one solution. A nonlinear analysis can produce much more output because results canbe requested at the end of each converged increment. If you do not select the output requests

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carefully, the output files become very large, potentially filling the disk space on yourcomputer.

As noted earlier, output is available in four different files:

� the output database (.odb) file, which contains data in a neutral binary format necessary topostprocess the results with ABAQUS/Viewer;

� the data (.dat) file, which contains printed tables of selected results;

� the restart file (.res) file, which is used to continue the analysis; and

� the results (.fil) file, which is used with third-party postprocessors.

The options for restart and results file output will not be discussed here. If selected carefully,data can be saved frequently during the simulation without using excessive disk space.

Remove the existing output requests from your input file; and add the following outputoptions, which ensure that only selected output is saved during the nonlinear analysis.

To reduce the size of the output database file, use the FREQUENCY parameter on the*OUTPUT, FIELD option; in this simulation you should write field output every secondincrement. Thus, set the FREQUENCY parameter equal to 2 on the *OUTPUT option:

*OUTPUT, FIELD, FREQUENCY=2, VARIABLE=PRESELECT

The parameter VARIABLE=PRESELECT indicates that a preselected set of the mostcommonly used field variables for a given type of analysis will be written to the outputdatabase (.odb) file. If you are simply interested in the final results, set the FREQUENCYparameter equal to a large number. Results are always stored at the end of each step, regardlessof the value of the FREQUENCY parameter; therefore, using a large value causes only thefinal results to be saved.

Request that the displacements of the nodes at the midspan be saved to the output databasefile. These results will be used later to demonstrate the X-Y plotting capability inABAQUS/Viewer. Use the default FREQUENCY value (FREQUENCY=1) for this historyoutput request for the output database file. Here the *NODE OUTPUT option must appearimmediately after the history output request. Remember to use the NSET or ELSETparameters when limiting the output being requested to a subset of the model; otherwise, thedefault subset, which is the entire model, will be used.

Finally, request that reaction forces (RF) be printed for all the nodes in the model and that thedisplacements (U) be printed for the nodes at the midspan (node set MIDSPAN). You will needtwo *NODE PRINT options because you are requesting results for two different subsets of themodel. Again, use the FREQUENCY parameter to reduce the amount of output; print the dataevery second increment. Suppress the printing of all element results by using *EL PRINT,FREQUENCY=0. To suppress output on any output request option, set FREQUENCY=0 onthe output request.

The new list of output request option blocks in your input file should look like:

carefully, the output files become very large, potentially filling the disk space on yourcomputer.

As noted earlier, output is available in four different files:

� the output database (.odb) file, which contains data in a neutral binary format necessary topostprocess the results with ABAQUS/Viewer;

� the data (.dat) file, which contains printed tables of selected results;

� the restart file (.res) file, which is used to continue the analysis; and

� the results (.fil) file, which is used with third-party postprocessors.

The options for restart and results file output will not be discussed here. If selected carefully,data can be saved frequently during the simulation without using excessive disk space.

Remove the existing output requests from your input file; and add the following outputoptions, which ensure that only selected output is saved during the nonlinear analysis.

To reduce the size of the output database file, use the FREQUENCY parameter on the*OUTPUT, FIELD option; in this simulation you should write field output every secondincrement. Thus, set the FREQUENCY parameter equal to 2 on the *OUTPUT option:


The parameter VARIABLE=PRESELECT indicates that a preselected set of the mostcommonly used field variables for a given type of analysis will be written to the outputdatabase (.odb) file. If you are simply interested in the final results, set the FREQUENCYparameter equal to a large number. Results are always stored at the end of each step, regardlessof the value of the FREQUENCY parameter; therefore, using a large value causes only thefinal results to be saved.

Request that the displacements of the nodes at the midspan be saved to the output databasefile. These results will be used later to demonstrate the X-Y plotting capability inABAQUS/Viewer. Use the default FREQUENCY value (FREQUENCY=1) for this historyoutput request for the output database file. Here the *NODE OUTPUT option must appearimmediately after the history output request. Remember to use the NSET or ELSETparameters when limiting the output being requested to a subset of the model; otherwise, thedefault subset, which is the entire model, will be used.

Finally, request that reaction forces (RF) be printed for all the nodes in the model and that thedisplacements (U) be printed for the nodes at the midspan (node set MIDSPAN). You will needtwo *NODE PRINT options because you are requesting results for two different subsets of themodel. Again, use the FREQUENCY parameter to reduce the amount of output; print the dataevery second increment. Suppress the printing of all element results by using *EL PRINT,FREQUENCY=0. To suppress output on any output request option, set FREQUENCY=0 onthe output request.

The new list of output request option blocks in your input file should look like:

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*OUTPUT, HISTORY, FREQUENCY=1


U,

*NODE PRINT, NSET=MIDSPAN, FREQUENCY=2

U,

*NODE PRINT, SUMMARY=NO, TOTALS=YES, FREQUENCY=2

RF,

*EL PRINT, FREQUENCY=0

Finally, complete the step definition by using the *END STEP option.

*END STEP

7.4.2 Running the analysisStore the modified input in a file called skew_nl.inp (an example input file is listed in ``Nonlinearskew plate,'' Section A.5). Run the analysis using the following command:

abaqus job=skew_nl interactive

7.4.3 ResultsDuring a nonlinear analysis, two additional output files become very important. They are the status file(skew_nl.sta) and the message file (skew_nl.msg). As the analysis progresses, ABAQUS willwrite data to both of these files. You can view the data even as ABAQUS continues your analysis. Youwill need to learn how to use the data in these files to assess ABAQUS's progress on your simulations.There may be situations where you decide, based on information in these files, to terminate youranalysis early. A more likely scenario is that you may need to use these files to learn what causedABAQUS to terminate your analysis prematurely; i.e., what caused the convergence problems.

Status file

The status file is particularly useful for monitoring the progress of a nonlinear simulationwhile the job is running. The output below shows the status file for this nonlinear skewedplate example.

SUMMARY OF JOB INFORMATION:

STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF

DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF

ITERS FREQ

1 1 1 0 4 4 0.100 0.100 0.1000

1 2 1 0 2 2 0.200 0.200 0.1000

1 3 1 0 2 2 0.350 0.350 0.1500

1 4 1 0 3 3 0.575 0.575 0.2250

1 5 1 0 3 3 0.913 0.913 0.3375

1 6 1 0 2 2 1.00 1.00 0.08750




U,


U,


RF,


Finally, complete the step definition by using the *END STEP option.

*END STEP

7.4.2 Running the analysisStore the modified input in a file called skew_nl.inp (an example input file is listed in ``Nonlinearskew plate,'' Section A.5). Run the analysis using the following command:

abaqus job=skew_nl interactive

7.4.3 ResultsDuring a nonlinear analysis, two additional output files become very important. They are the status file(skew_nl.sta) and the message file (skew_nl.msg). As the analysis progresses, ABAQUS willwrite data to both of these files. You can view the data even as ABAQUS continues your analysis. Youwill need to learn how to use the data in these files to assess ABAQUS's progress on your simulations.There may be situations where you decide, based on information in these files, to terminate youranalysis early. A more likely scenario is that you may need to use these files to learn what causedABAQUS to terminate your analysis prematurely; i.e., what caused the convergence problems.

Status file

The status file is particularly useful for monitoring the progress of a nonlinear simulationwhile the job is running. The output below shows the status file for this nonlinear skewedplate example.




ITERS FREQ

1 1 1 0 4 4 0.100 0.100 0.1000

1 2 1 0 2 2 0.200 0.200 0.1000

1 3 1 0 2 2 0.350 0.350 0.1500

1 4 1 0 3 3 0.575 0.575 0.2250

1 5 1 0 3 3 0.913 0.913 0.3375

1 6 1 0 2 2 1.00 1.00 0.08750

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The status file contains a separate line for every converged increment in the simulation. Thefirst column shows the step number--in this case there is only one step. The second columngives the increment number. The sixth column shows the number of iterations ABAQUSneeded to obtain a converged solution in each increment; for example, ABAQUS needed 4iterations in increment 1. The eighth column shows the total step time completed, and theninth column shows the increment size (¢T ).

This example shows how ABAQUS automatically controls the increment size and, therefore,the proportion of load applied in each increment. In this analysis ABAQUS applied 10% of thetotal load in the first increment: you specified ¢Tinitial to be 0.1 and the step time to be 1.0.ABAQUS needed four iterations to converge to a solution in the first increment. ABAQUSonly needed two iterations in the second increment, so it automatically increased the size ofthe next increment by 50% to ¢T = 0.15. ABAQUS also increased ¢T in both the fourth andfifth increments. It adjusted the final increment size to be just enough to complete the analysis;in this case the final increment size was 0.0875.

Message file

The message file contains more detailed information about the progress of the analysis thanthe status file. ABAQUS lists all of the tolerances and parameters to control the analysis at thestart of each step in the message file. This is done for each step because these controls can bemodified from step to step. The default values of these controls are appropriate for mostanalyses, so normally it is not necessary for you to modify them. The modification of controltolerances and parameters is beyond the scope of this guide (it is discussed in theABAQUS/Standard User's Manual).

The status file contains a separate line for every converged increment in the simulation. Thefirst column shows the step number--in this case there is only one step. The second columngives the increment number. The sixth column shows the number of iterations ABAQUSneeded to obtain a converged solution in each increment; for example, ABAQUS needed 4iterations in increment 1. The eighth column shows the total step time completed, and theninth column shows the increment size (¢T ).

This example shows how ABAQUS automatically controls the increment size and, therefore,the proportion of load applied in each increment. In this analysis ABAQUS applied 10% of thetotal load in the first increment: you specified ¢Tinitial to be 0.1 and the step time to be 1.0.ABAQUS needed four iterations to converge to a solution in the first increment. ABAQUSonly needed two iterations in the second increment, so it automatically increased the size ofthe next increment by 50% to ¢T = 0.15. ABAQUS also increased ¢T in both the fourth andfifth increments. It adjusted the final increment size to be just enough to complete the analysis;in this case the final increment size was 0.0875.

Message file

The message file contains more detailed information about the progress of the analysis thanthe status file. ABAQUS lists all of the tolerances and parameters to control the analysis at thestart of each step in the message file. This is done for each step because these controls can bemodified from step to step. The default values of these controls are appropriate for mostanalyses, so normally it is not necessary for you to modify them. The modification of controltolerances and parameters is beyond the scope of this guide (it is discussed in theABAQUS/Standard User's Manual).

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ABAQUS lists a summary of each iteration in the message file after the lists of tolerances andcontrols. It prints the values of the largest residual force, r®max ; largest increment ofdisplacement, ¢u® ; the largest correction to displacement, c®; and the time averaged force,~q® . It also prints the nodes and degrees of freedom (dof) at which r®max , ¢u® , and c® occur. A

ABAQUS lists a summary of each iteration in the message file after the lists of tolerances andcontrols. It prints the values of the largest residual force, r®max ; largest increment ofdisplacement, ¢u® ; the largest correction to displacement, c®; and the time averaged force,~q® . It also prints the nodes and degrees of freedom (dof) at which r®max , ¢u® , and c® occur. A

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similar summary is printed for rotational degrees of freedom.

In this example the initial time increment is 0.1 seconds, as specified in the input file. Theaverage force for the increment is 18.6 N, and ~q® has the same value since this is the firstincrement. The largest residual force in the model, r®max , is 1461 N, which is clearly greaterthan 0.005 � ~q® . r®max occurred at node 57 in degree of freedom 1. ABAQUS must also checkfor equilibrium of the moments in the model since this model includes shell elements. Themoment/rotation field also failed to satisfy the equilibrium check.

Although failure to satisfy the equilibrium check is enough to cause ABAQUS to try anotheriteration, you should also examine the displacement correction. In the first iteration the largestincrement of displacement, ¢u®

max , and the largest correction to displacement, c®max , are both-5.587 � 10-3 m; and the largest increment of rotation and correction to rotation are both 1.546� 10-2 radians. Since the incremental values and the corrections are always equal in the firstiteration, the check that the largest corrections to nodal variables are less than 1% of thelargest incremental values will always fail. However, if ABAQUS judges the solution to belinear (a judgement based on the magnitude of the residuals, r®max < 10-8~q®), it will ignore thiscriterion.

Since ABAQUS did not find an equilibrium solution in the first iteration, it tries a seconditeration.

In the second iteration r®max has fallen to -0.317 N at node 651 in degree of freedom 1.However, equilibrium is not satisfied in this iteration because 0.005 £~q® , where ~q® = 1.52 N,is still less than r®max . The displacement correction criterion also failed again because c®max =-7.011 � 10-5, which occurred at node 701 in degree of freedom 1, is more than 1% of ¢u®

max

= -5.592 � 10-3, the maximum displacement increment.

Both the moment residual check and the largest correction to rotation check were satisfied in

similar summary is printed for rotational degrees of freedom.

In this example the initial time increment is 0.1 seconds, as specified in the input file. Theaverage force for the increment is 18.6 N, and ~q® has the same value since this is the firstincrement. The largest residual force in the model, r®max , is 1461 N, which is clearly greaterthan 0.005 � ~q® . r®max occurred at node 57 in degree of freedom 1. ABAQUS must also checkfor equilibrium of the moments in the model since this model includes shell elements. Themoment/rotation field also failed to satisfy the equilibrium check.

Although failure to satisfy the equilibrium check is enough to cause ABAQUS to try anotheriteration, you should also examine the displacement correction. In the first iteration the largestincrement of displacement, ¢u®

max , and the largest correction to displacement, c®max , are both-5.587 � 10-3 m; and the largest increment of rotation and correction to rotation are both 1.546� 10-2 radians. Since the incremental values and the corrections are always equal in the firstiteration, the check that the largest corrections to nodal variables are less than 1% of thelargest incremental values will always fail. However, if ABAQUS judges the solution to belinear (a judgement based on the magnitude of the residuals, r®max < 10-8~q®), it will ignore thiscriterion.

Since ABAQUS did not find an equilibrium solution in the first iteration, it tries a seconditeration.

In the second iteration r®max has fallen to -0.317 N at node 651 in degree of freedom 1.However, equilibrium is not satisfied in this iteration because 0.005 £~q® , where ~q® = 1.52 N,is still less than r®max . The displacement correction criterion also failed again because c®max =-7.011 � 10-5, which occurred at node 701 in degree of freedom 1, is more than 1% of ¢u®

max

= -5.592 � 10-3, the maximum displacement increment.

Both the moment residual check and the largest correction to rotation check were satisfied in

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this second iteration; however, ABAQUS must perform another iteration because the solutiondid not pass the force residual check (or the largest correction to displacement criterion). Themessage file summaries for the additional iterations necessary to obtain a solution in the firstincrement are shown below.

After four iterations ~q® = 1.51 N and r®max = 2.004 � 10-7 N at node 357 in degree offreedom 2. These values satisfy r®max< 0.005 � ~q® , so the force residual check is satisfied.Comparing c®max to the largest increment of displacement shows that the displacementcorrection is below the required tolerance. The solution for the forces and displacements has,therefore, converged. The checks for both the moment residual and the rotation correctioncontinue to be satisfied, as they have been since the second iteration. With a solution thatsatisfies equilibrium for all variables (displacement and rotation in this case), the first loadincrement is complete. The increment summary shows the number of iterations that wererequired for this increment, the size of the increment, and the fraction of the step that has beencompleted.

The second increment requires two iterations to converge.

this second iteration; however, ABAQUS must perform another iteration because the solutiondid not pass the force residual check (or the largest correction to displacement criterion). Themessage file summaries for the additional iterations necessary to obtain a solution in the firstincrement are shown below.

After four iterations ~q® = 1.51 N and r®max = 2.004 � 10-7 N at node 357 in degree offreedom 2. These values satisfy r®max< 0.005 � ~q® , so the force residual check is satisfied.Comparing c®max to the largest increment of displacement shows that the displacementcorrection is below the required tolerance. The solution for the forces and displacements has,therefore, converged. The checks for both the moment residual and the rotation correctioncontinue to be satisfied, as they have been since the second iteration. With a solution thatsatisfies equilibrium for all variables (displacement and rotation in this case), the first loadincrement is complete. The increment summary shows the number of iterations that wererequired for this increment, the size of the increment, and the fraction of the step that has beencompleted.

The second increment requires two iterations to converge.

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ABAQUS continues this process of applying an increment of load then iterating to find asolution until it completes the whole analysis (or reaches the increment specified as the valueof the INC parameter). In this analysis it required four more increments. ABAQUS gives asummary at the end of the message file of how the analysis progressed and how many errorand warning messages it issued. The summary for this analysis is shown below. An importantitem to check is how many iterations ABAQUS uses. In this analysis it performed 16 iterationsover the six increments: the model's system of equations was solved 16 times (i.e., 16 matrixdecompositions), illustrating the increased computational expense of nonlinear analysescompared with linear simulations.

ABAQUS continues this process of applying an increment of load then iterating to find asolution until it completes the whole analysis (or reaches the increment specified as the valueof the INC parameter). In this analysis it required four more increments. ABAQUS gives asummary at the end of the message file of how the analysis progressed and how many errorand warning messages it issued. The summary for this analysis is shown below. An importantitem to check is how many iterations ABAQUS uses. In this analysis it performed 16 iterationsover the six increments: the model's system of equations was solved 16 times (i.e., 16 matrixdecompositions), illustrating the increased computational expense of nonlinear analysescompared with linear simulations.

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You should always check this summary at the end of every ABAQUS simulation. It tells you ifthe analysis was successful or not, and it gives the number of error and warning messagesABAQUS issued during the simulation. Always investigate any errors or warnings. Allwarnings and errors generated during the analysis are found in the message (.msg) file.Warnings issued "during user input processing" are found in the data (.dat) file.

Data file

The tables of displacements and reaction forces that you requested are in the data (.dat) file.The midspan deflections at the end of the step can be found near the end of the file.

THE FOLLOWING TABLE IS PRINTED FOR NODES BELONGING TO NODE SET M

NODE FOOT- U1 U2 U3 UR1 UR

NOTE

351 -2.6632E-03 -7.4371E-04 -4.9226E-02

353 -2.5267E-03 -7.5635E-04 -4.8088E-02

355 -2.4189E-03 -7.7812E-04 -4.6726E-02

357 -2.2649E-03 -7.9809E-04 -4.5314E-02

MAXIMUM -2.2649E-03 -7.4371E-04 -4.5314E-02 0.0000E+00 0.00

AT NODE 357 351 357 0

MINIMUM -2.6632E-03 -7.9809E-04 -4.9226E-02 0.0000E+00 0.00

AT NODE 351 357 351 0

You should always check this summary at the end of every ABAQUS simulation. It tells you ifthe analysis was successful or not, and it gives the number of error and warning messagesABAQUS issued during the simulation. Always investigate any errors or warnings. Allwarnings and errors generated during the analysis are found in the message (.msg) file.Warnings issued "during user input processing" are found in the data (.dat) file.

Data file

The tables of displacements and reaction forces that you requested are in the data (.dat) file.The midspan deflections at the end of the step can be found near the end of the file.

THE FOLLOWING TABLE IS PRINTED FOR NODES BELONGING TO NODE SET M

NODE FOOT- U1 U2 U3 UR1 UR

NOTE

351 -2.6632E-03 -7.4371E-04 -4.9226E-02

353 -2.5267E-03 -7.5635E-04 -4.8088E-02

355 -2.4189E-03 -7.7812E-04 -4.6726E-02

357 -2.2649E-03 -7.9809E-04 -4.5314E-02

MAXIMUM -2.2649E-03 -7.4371E-04 -4.5314E-02 0.0000E+00 0.00

AT NODE 357 351 357 0

MINIMUM -2.6632E-03 -7.9809E-04 -4.9226E-02 0.0000E+00 0.00

AT NODE 351 357 351 0

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Compare these with the displacements from the linear analysis in Chapter 5, "Using ShellElements." The maximum displacement at the midspan in this simulation is about 9% less thanthat predicted from the linear analysis. Including the nonlinear geometric effects in thesimulation reduces the vertical deflection (U3) of the midspan of the plate.

Another difference between the two analyses is that in the nonlinear simulation there arenonzero deflections in the 1- and 2-directions. What effects make the in-plane displacements,U1 and U2, nonzero in the nonlinear analysis? Why is the vertical deflection of the plate lessin the nonlinear analysis?

The plate deforms into a curved shape: a geometry change that is taken into account in thenonlinear simulation. As a consequence, membrane effects cause some of the load to becarried by membrane action rather than by bending alone. This makes the plate stiffer. Inaddition, the pressure loading, which is always normal to the plate's surface, starts to have acomponent in the 1- and 2-directions as the plate deforms. The nonlinear analysis includes theeffects of this stiffening and the changing direction of the pressure. Neither of these effects isincluded in the linear simulation.

The differences between the linear and nonlinear simulations are sufficiently large to indicatethat a linear simulation is not adequate for this plate under this particular loadingcondition.

For five degree of freedom shells, such as the S8R5 element used in this analysis, ABAQUSdoes not output total rotations at the nodes.

7.4.4 PostprocessingWhen you are in the directory containing the output database file skew_nl.odb, type the command

abaqus viewer odb=skew_nl


Showing the available frames

To begin this exercise, determine the available output frames (the increment intervals at whichresults were written to the output database).

To show the available frames:

1. From the main menu bar, select Result->Frame.

The Frame Selector dialog box appears.

During the analysis ABAQUS wrote field output results to the output database file everysecond increment (since the command *OUTPUT, FIELD, FREQUENCY=2,VARIABLE=PRESELECT was in the input file). ABAQUS/Viewer displays the list of theavailable frames, as shown in Figure 7-12.

Compare these with the displacements from the linear analysis in Chapter 5, "Using ShellElements." The maximum displacement at the midspan in this simulation is about 9% less thanthat predicted from the linear analysis. Including the nonlinear geometric effects in thesimulation reduces the vertical deflection (U3) of the midspan of the plate.

Another difference between the two analyses is that in the nonlinear simulation there arenonzero deflections in the 1- and 2-directions. What effects make the in-plane displacements,U1 and U2, nonzero in the nonlinear analysis? Why is the vertical deflection of the plate lessin the nonlinear analysis?

The plate deforms into a curved shape: a geometry change that is taken into account in thenonlinear simulation. As a consequence, membrane effects cause some of the load to becarried by membrane action rather than by bending alone. This makes the plate stiffer. Inaddition, the pressure loading, which is always normal to the plate's surface, starts to have acomponent in the 1- and 2-directions as the plate deforms. The nonlinear analysis includes theeffects of this stiffening and the changing direction of the pressure. Neither of these effects isincluded in the linear simulation.

The differences between the linear and nonlinear simulations are sufficiently large to indicatethat a linear simulation is not adequate for this plate under this particular loadingcondition.

For five degree of freedom shells, such as the S8R5 element used in this analysis, ABAQUSdoes not output total rotations at the nodes.

7.4.4 PostprocessingWhen you are in the directory containing the output database file skew_nl.odb, type the command

abaqus viewer odb=skew_nl


Showing the available frames

To begin this exercise, determine the available output frames (the increment intervals at whichresults were written to the output database).

To show the available frames:



During the analysis ABAQUS wrote field output results to the output database file everysecond increment (since the command *OUTPUT, FIELD, FREQUENCY=2,VARIABLE=PRESELECT was in the input file). ABAQUS/Viewer displays the list of theavailable frames, as shown in Figure 7-12.

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Figure 7-12 Available frames.

The list tabulates the steps and increments for which field variables are stored. This analysisconsisted of a single step with six increments, and you saved data for increments 2, 4, and 6.By default, ABAQUS/Viewer always uses the data for the last available increment saved onthe output database file.

2. Click OK to dismiss the Frame Selector dialog box.

Displaying the deformed and undeformed model shapes with visible exterior edges

You will now display the deformed and undeformed model shapes and make visible theexterior edges.

To display the deformed and undeformed model shapes with visible exterior edges:

1. From the main menu bar, select Plot->Deformed Shape; or use the tool in thetoolbox.

ABAQUS/Viewer displays the deformed model shape at the end of the analysis.


ABAQUS/Viewer displays the Deformed Plot Options dialog box.

3. Click the Basic tab if it is not already selected, and do the following:

a. Toggle on Superimpose undeformed plot.

b. Choose Exterior visible edges.

c. Click OK to apply your changes and to close the dialog box.

4. To turn on the exterior visible edges for the undeformed model shape as well, selectOptions->Undeformed from the main menu bar.

Figure 7-12 Available frames.

The list tabulates the steps and increments for which field variables are stored. This analysisconsisted of a single step with six increments, and you saved data for increments 2, 4, and 6.By default, ABAQUS/Viewer always uses the data for the last available increment saved onthe output database file.

2. Click OK to dismiss the Frame Selector dialog box.

Displaying the deformed and undeformed model shapes with visible exterior edges

You will now display the deformed and undeformed model shapes and make visible theexterior edges.

To display the deformed and undeformed model shapes with visible exterior edges:


ABAQUS/Viewer displays the deformed model shape at the end of the analysis.


ABAQUS/Viewer displays the Deformed Plot Options dialog box.


a. Toggle on Superimpose undeformed plot.

b. Choose Exterior visible edges.

c. Click OK to apply your changes and to close the dialog box.

4. To turn on the exterior visible edges for the undeformed model shape as well, selectOptions->Undeformed from the main menu bar.

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ABAQUS/Viewer displays the Undeformed Plot Options dialog box.


a. Choose Exterior visible edges.

b. Click OK to apply your changes and to close the dialog box.

Changing the view

You can change the default isometric view using the options in the View menu or the viewtools in the toolbar.

To rotate the view:



The view rotates interactively. Move the mouse, with mouse button 1 depressed, to obtain aview similar to that shown in Figure 7-13.

Figure 7-13 Deformed and undeformed model shapes of the skewed plate.

3. Click the tool from the toolbar again to exit the rotation mode.

Using results from other frames

You can evaluate the results from other increments saved to the output database file byselecting the appropriate frame.

To select a new frame:



ABAQUS/Viewer displays the Undeformed Plot Options dialog box.


a. Choose Exterior visible edges.

b. Click OK to apply your changes and to close the dialog box.

Changing the view

You can change the default isometric view using the options in the View menu or the viewtools in the toolbar.

To rotate the view:



The view rotates interactively. Move the mouse, with mouse button 1 depressed, to obtain aview similar to that shown in Figure 7-13.

Figure 7-13 Deformed and undeformed model shapes of the skewed plate.

3. Click the tool from the toolbar again to exit the rotation mode.

Using results from other frames

You can evaluate the results from other increments saved to the output database file byselecting the appropriate frame.

To select a new frame:



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2. Select Increment 4 from the Frame menu.

3. Click OK to apply your changes and to close the Frame Selector dialog box.

Any plots now requested will use results from increment 4. Repeat this procedure, substitutingthe increment number of interest, to move through the output database file.

X-Y plotting

You saved the displacements of the midspan nodes (node set MIDSPAN) in the history portionof the output database file skew_nl.odb for each increment of the simulation. You can usethese results to create X-Y plots.

To create X-Y plots of the midspan displacements:

1. From the main menu bar, select Result->History Output.

ABAQUS/Viewer displays the ODB History Output dialog box. To see the completedescription of the variable choices, increase the width of the ODB History Output dialog boxby dragging the right or left edge.

2. The Output Variables field contains a list of all the variables in the history portion of theoutput database; these are the only history output variables you can plot. Select (using[Ctrl]+Click) the vertical motion of both node 351 (Spatial displacement: U3 atNode 351 in NSET MIDSPAN) and node 357 (Spatial displacement: U3 atNode 357 in NSET MIDSPAN).

3. Click Plot.

ABAQUS reads the data for both curves from the output database file and plots a graphsimilar to the one shown in Figure 7-14. (For clarity, the second curve has been changed to adashed line.)

Figure 7-14 History of the displacements at the edges of the plate's midspan.

2. Select Increment 4 from the Frame menu.

3. Click OK to apply your changes and to close the Frame Selector dialog box.

Any plots now requested will use results from increment 4. Repeat this procedure, substitutingthe increment number of interest, to move through the output database file.

X-Y plotting

You saved the displacements of the midspan nodes (node set MIDSPAN) in the history portionof the output database file skew_nl.odb for each increment of the simulation. You can usethese results to create X-Y plots.

To create X-Y plots of the midspan displacements:


ABAQUS/Viewer displays the ODB History Output dialog box. To see the completedescription of the variable choices, increase the width of the ODB History Output dialog boxby dragging the right or left edge.

2. The Output Variables field contains a list of all the variables in the history portion of theoutput database; these are the only history output variables you can plot. Select (using[Ctrl]+Click) the vertical motion of both node 351 (Spatial displacement: U3 atNode 351 in NSET MIDSPAN) and node 357 (Spatial displacement: U3 atNode 357 in NSET MIDSPAN).

3. Click Plot.

ABAQUS reads the data for both curves from the output database file and plots a graphsimilar to the one shown in Figure 7-14. (For clarity, the second curve has been changed to adashed line.)

Figure 7-14 History of the displacements at the edges of the plate's midspan.

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The nonlinear nature of this simulation is clearly seen in these curves: as the analysisprogresses, the plate stiffens. The effects of geometric nonlinearity mean that a structure'sstiffness changes as it deforms. In this simulation the plate deforms as it gets stiffer due tomembrane effects. Therefore, the resulting peak displacement is less than that predicted by thelinear analysis, which did not include this effect.

You can create X-Y curves from data stored in the output database (.odb) file. X-Y curves canalso be read from an external file or they can be typed into ABAQUS/Viewer interactively.Once curves have been created, their data can be further manipulated and plotted to the screenin graphical form.

The X-Y plotting capability of ABAQUS/Viewer is discussed further in Chapter 8, "Materials."

7.5 Related ABAQUS examples� `Èlastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure, ''

Section 1.1.2 of the ABAQUS Example Problems Manual

� ``Laminated composite shells: buckling of a cylindrical panel with a circular hole, '' Section 1.2.2of the ABAQUS Example Problems Manual

� `Ùnstable static problem: reinforced plate under compressive loads, '' Section 1.2.5 of theABAQUS Example Problems Manual

� ``Large rotation of a one degree of freedom system,'' Section 1.3.5 of the ABAQUS BenchmarksManual

� ``Vibration of a cable under tension, '' Section 1.4.3 of the ABAQUS Benchmarks Manual

7.6 Suggested readingThe following references provide additional information on nonlinear finite element methods. They

The nonlinear nature of this simulation is clearly seen in these curves: as the analysisprogresses, the plate stiffens. The effects of geometric nonlinearity mean that a structure'sstiffness changes as it deforms. In this simulation the plate deforms as it gets stiffer due tomembrane effects. Therefore, the resulting peak displacement is less than that predicted by thelinear analysis, which did not include this effect.

You can create X-Y curves from data stored in the output database (.odb) file. X-Y curves canalso be read from an external file or they can be typed into ABAQUS/Viewer interactively.Once curves have been created, their data can be further manipulated and plotted to the screenin graphical form.

The X-Y plotting capability of ABAQUS/Viewer is discussed further in Chapter 8, "Materials."

7.5 Related ABAQUS examples� `Èlastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure, ''

Section 1.1.2 of the ABAQUS Example Problems Manual

� ``Laminated composite shells: buckling of a cylindrical panel with a circular hole, '' Section 1.2.2of the ABAQUS Example Problems Manual

� `Ùnstable static problem: reinforced plate under compressive loads, '' Section 1.2.5 of theABAQUS Example Problems Manual

� ``Large rotation of a one degree of freedom system,'' Section 1.3.5 of the ABAQUS BenchmarksManual


7.6 Suggested readingThe following references provide additional information on nonlinear finite element methods. They

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allow the interested user to explore the topic in more depth.

General texts on nonlinear analysis

� Cook, R. D., D. S. Malkus, and M. E. Plesha, Concepts and Applications of FiniteElement Analysis, Wiley & Sons, 1989.

� E. Hinton (editor), NAFEMS Introduction to Nonlinear Finite Element Analysis ,NAFEMS Ltd., 1992.

7.7 Summary� There are three sources of nonlinearity in structural problems: material, geometric, and boundary

(contact). Any combination of these may be present in an ABAQUS analysis.

� Geometric nonlinearity occurs whenever the magnitude of the displacements affects the responseof the structure. It includes the effects of large displacements and rotations, snap through, and loadstiffening.

� Nonlinear problems are solved iteratively using the Newton-Raphson method. A nonlinearproblem will require many times the computer resources required by a linear problem.

� The nonlinear analysis step is split into a number of increments. ABAQUS iterates to find theapproximate static equilibrium obtained at the end of each new load increment. ABAQUS controlsthe load incrementation by using convergence controls throughout the simulation.

� The status file allows the progress of an analysis to be monitored while it is running. The messagefile contains the details of the load incrementation and iterations.

� Results can be saved at the end of each increment so that the evolution of the structure's responsecan be visualized in ABAQUS/Viewer. Results can also be plotted in the form of X-Y graphs.

allow the interested user to explore the topic in more depth.

General texts on nonlinear analysis

� Cook, R. D., D. S. Malkus, and M. E. Plesha, Concepts and Applications of FiniteElement Analysis, Wiley & Sons, 1989.

� E. Hinton (editor), NAFEMS Introduction to Nonlinear Finite Element Analysis ,NAFEMS Ltd., 1992.

7.7 Summary� There are three sources of nonlinearity in structural problems: material, geometric, and boundary

(contact). Any combination of these may be present in an ABAQUS analysis.

� Geometric nonlinearity occurs whenever the magnitude of the displacements affects the responseof the structure. It includes the effects of large displacements and rotations, snap through, and loadstiffening.

� Nonlinear problems are solved iteratively using the Newton-Raphson method. A nonlinearproblem will require many times the computer resources required by a linear problem.

� The nonlinear analysis step is split into a number of increments. ABAQUS iterates to find theapproximate static equilibrium obtained at the end of each new load increment. ABAQUS controlsthe load incrementation by using convergence controls throughout the simulation.

� The status file allows the progress of an analysis to be monitored while it is running. The messagefile contains the details of the load incrementation and iterations.

� Results can be saved at the end of each increment so that the evolution of the structure's responsecan be visualized in ABAQUS/Viewer. Results can also be plotted in the form of X-Y graphs.

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8. MaterialsThe material library in ABAQUS allows most engineering materials to be modeled, including metals,plastics, rubbers, foams, composites, granular soils, rocks, and plain and reinforced concrete. Thisguide only discusses three of the most commonly used material models: linear elasticity, metalplasticity, and rubber elasticity. All of the material models are discussed in detail in theABAQUS/Standard User's Manual.

8.1 Defining materials in ABAQUSYou can use any number of different materials in your simulation. Each material definition starts witha *MATERIAL option. The NAME parameter identifies the name associated with the material beingdefined. This name is used to assign the material definition to specific elements in the model.

This is one of the few situations where the position of option blocks in the ABAQUS input file isimportant. All of the option blocks defining specific aspects of a material's behavior, such as elasticmodulus or density, must follow the *MATERIAL option. Furthermore, the material option blocksdefining the behavior of a particular material cannot be interrupted by other non-material options.ABAQUS issues an error message if it cannot associate a material behavior option block, such as*ELASTIC, with a prior *MATERIAL option.

For example, consider a material description, such as an elastic-plastic metal subjected to gravitationalloads, that requires several material behavior option blocks to supply ABAQUS with the necessarydata. In addition to the elastic and plastic property option blocks, ABAQUS needs the material'sdensity to calculate the gravitational loads. Thus, the complete material description would be

A non-material option block between the *PLASTIC and *DENSITY options, as shown in thefollowing input, would cause ABAQUS to terminate the analysis with an error message.

8.2 Plasticity in ductile metalsMany metals have approximately linear elastic behavior at low strain magnitudes (see Figure 8-1), and

8. MaterialsThe material library in ABAQUS allows most engineering materials to be modeled, including metals,plastics, rubbers, foams, composites, granular soils, rocks, and plain and reinforced concrete. Thisguide only discusses three of the most commonly used material models: linear elasticity, metalplasticity, and rubber elasticity. All of the material models are discussed in detail in theABAQUS/Standard User's Manual.

8.1 Defining materials in ABAQUSYou can use any number of different materials in your simulation. Each material definition starts witha *MATERIAL option. The NAME parameter identifies the name associated with the material beingdefined. This name is used to assign the material definition to specific elements in the model.

This is one of the few situations where the position of option blocks in the ABAQUS input file isimportant. All of the option blocks defining specific aspects of a material's behavior, such as elasticmodulus or density, must follow the *MATERIAL option. Furthermore, the material option blocksdefining the behavior of a particular material cannot be interrupted by other non-material options.ABAQUS issues an error message if it cannot associate a material behavior option block, such as*ELASTIC, with a prior *MATERIAL option.

For example, consider a material description, such as an elastic-plastic metal subjected to gravitationalloads, that requires several material behavior option blocks to supply ABAQUS with the necessarydata. In addition to the elastic and plastic property option blocks, ABAQUS needs the material'sdensity to calculate the gravitational loads. Thus, the complete material description would be

A non-material option block between the *PLASTIC and *DENSITY options, as shown in thefollowing input, would cause ABAQUS to terminate the analysis with an error message.

8.2 Plasticity in ductile metalsMany metals have approximately linear elastic behavior at low strain magnitudes (see Figure 8-1), and

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the stiffness of the material, known as the Young's or elastic modulus, is constant.

Figure 8-1 Stress-strain behavior for a linear elastic material, such as steel, at small strains.

At higher stress (and strain) magnitudes, metals begin to have nonlinear, inelastic behavior (see Figure8-2), which is referred to as plasticity.

Figure 8-2 Nominal stress-strain behavior of an elastic-plastic material in a tensile test.

8.2.1 Characteristics of plasticity in ductile metalsThe plastic behavior of a material is described by its yield point and its post-yield hardening. The shiftfrom elastic to plastic behavior occurs at a certain point, known as the elastic limit or yield point, on amaterial's stress-strain curve (see Figure 8-2). The stress at the yield point is called the yield stress. Inmost metals the initial yield stress is 0.05 to 0.1% of the material's elastic modulus.

The deformation of the metal prior to reaching the yield point creates only elastic strains, which arefully recovered if the applied load is removed. However, once the stress in the metal exceeds the yieldstress, permanent (plastic) deformation begins to occur. The strains associated with this permanentdeformation are called plastic strains. Both elastic and plastic strains accumulate as the metal deformsin the post-yield region.

the stiffness of the material, known as the Young's or elastic modulus, is constant.

Figure 8-1 Stress-strain behavior for a linear elastic material, such as steel, at small strains.

At higher stress (and strain) magnitudes, metals begin to have nonlinear, inelastic behavior (see Figure8-2), which is referred to as plasticity.

Figure 8-2 Nominal stress-strain behavior of an elastic-plastic material in a tensile test.

8.2.1 Characteristics of plasticity in ductile metalsThe plastic behavior of a material is described by its yield point and its post-yield hardening. The shiftfrom elastic to plastic behavior occurs at a certain point, known as the elastic limit or yield point, on amaterial's stress-strain curve (see Figure 8-2). The stress at the yield point is called the yield stress. Inmost metals the initial yield stress is 0.05 to 0.1% of the material's elastic modulus.

The deformation of the metal prior to reaching the yield point creates only elastic strains, which arefully recovered if the applied load is removed. However, once the stress in the metal exceeds the yieldstress, permanent (plastic) deformation begins to occur. The strains associated with this permanentdeformation are called plastic strains. Both elastic and plastic strains accumulate as the metal deformsin the post-yield region.

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The stiffness of a metal typically decreases dramatically once the material yields (see Figure 8-2). Aductile metal that has yielded will recover its initial elastic stiffness when the applied load is removed(see Figure 8-2). Often the plastic deformation of the material increases its yield stress for subsequentloadings: this behavior is called work hardening.

Another important feature of metal plasticity is that the inelastic deformation is associated with nearlyincompressible material behavior. Modeling this effect places some severe restrictions on the type ofelements that can be used in elastic-plastic simulations.

A metal deforming plastically under a tensile load may experience highly localized extension andthinning, called necking, as the material fails (see Figure 8-2). The engineering stress (force per unitundeformed area) in the metal is known as the nominal stress, with the conjugate nominal strain(length change per unit undeformed length). The nominal stress in the metal as it is necking is muchlower than the material's ultimate strength. This material behavior is caused by the geometry of the testspecimen, the nature of the test itself, and the stress and strain measures used. For example, testing thesame material in compression produces a stress-strain plot that does not have a necking region becausethe specimen is not going to thin as it deforms under compressive loads. A mathematical modeldescribing the plastic behavior of metals should be able to account for differences in the compressiveand tensile behavior independent of the structure's geometry or the nature of the applied loads. Thisgoal can be accomplished if the familiar definitions of nominal stress, F=A0 , and nominal strain,¢l=l0 , where the subscript 0 indicates a value from the undeformed state of the material, are replacedby new measures of stress and strain that account for the change in area during the finitedeformations.

8.2.2 Stress and strain measures for finite deformationsStrains in compression and tension are the same only if considered in the limit as ¢l ! dl ! 0; i.e.,

d" =dl

l

and

" =

Z l

l0

dl

l= ln

µl

l0

¶;

where l is the current length, l0 is the original length, and " is the true strain or logarithmic strain.

The stress measure that is the conjugate to the true strain is called the true stress and is defined as

¾ =F

A;

where F is the force in the material and A is the current area. A ductile metal subjected to finitedeformations will have identical stress-strain behavior in tension and compression if true stress isplotted against true strain.

The stiffness of a metal typically decreases dramatically once the material yields (see Figure 8-2). Aductile metal that has yielded will recover its initial elastic stiffness when the applied load is removed(see Figure 8-2). Often the plastic deformation of the material increases its yield stress for subsequentloadings: this behavior is called work hardening.

Another important feature of metal plasticity is that the inelastic deformation is associated with nearlyincompressible material behavior. Modeling this effect places some severe restrictions on the type ofelements that can be used in elastic-plastic simulations.

A metal deforming plastically under a tensile load may experience highly localized extension andthinning, called necking, as the material fails (see Figure 8-2). The engineering stress (force per unitundeformed area) in the metal is known as the nominal stress, with the conjugate nominal strain(length change per unit undeformed length). The nominal stress in the metal as it is necking is muchlower than the material's ultimate strength. This material behavior is caused by the geometry of the testspecimen, the nature of the test itself, and the stress and strain measures used. For example, testing thesame material in compression produces a stress-strain plot that does not have a necking region becausethe specimen is not going to thin as it deforms under compressive loads. A mathematical modeldescribing the plastic behavior of metals should be able to account for differences in the compressiveand tensile behavior independent of the structure's geometry or the nature of the applied loads. Thisgoal can be accomplished if the familiar definitions of nominal stress, F=A0 , and nominal strain,¢l=l0 , where the subscript 0 indicates a value from the undeformed state of the material, are replacedby new measures of stress and strain that account for the change in area during the finitedeformations.

8.2.2 Stress and strain measures for finite deformationsStrains in compression and tension are the same only if considered in the limit as ¢l ! dl ! 0; i.e.,

d" =dl

l

and

" =

Z l

l0

dl

l= ln

µl

l0

¶;

where l is the current length, l0 is the original length, and " is the true strain or logarithmic strain.

The stress measure that is the conjugate to the true strain is called the true stress and is defined as

¾ =F

A;

where F is the force in the material and A is the current area. A ductile metal subjected to finitedeformations will have identical stress-strain behavior in tension and compression if true stress isplotted against true strain.

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8.2.3 Defining plasticity in ABAQUSWhen defining plasticity data in ABAQUS, you must use true stress and true strain. ABAQUSrequires these values to interpret the data in the input file correctly.

Quite often material test data are supplied using values of nominal stress and strain. In such situationsyou must use the expressions presented below to convert the plastic material data from nominalstress/strain values to true stress/strain values.

The relationship between true strain and nominal strain is established by expressing the nominal strainas

"nom =l ¡ l0l0

=l

l0¡ l0

l0=

l

l0¡ 1:

Adding unity to both sides of this expression and taking the natural log of both sides provides therelationship between the true strain and the nominal strain:

" = ln (1 + "nom ):

The relationship between true stress and nominal stress is formed by considering the incompressiblenature of the plastic deformation and assuming the elasticity is also incompressible, so

l0A0 = lA:

The current area is related to the original area by

A = A0l0l:

Substituting this definition of A into the definition of true stress gives

¾ =F

A=

F

A0

l

l0= ¾nom

µl

l0

¶;

where

l

l0

can also be written as

1 + "nom :

Making this final substitution provides the relationship between true stress and nominal stress andstrain:

8.2.3 Defining plasticity in ABAQUSWhen defining plasticity data in ABAQUS, you must use true stress and true strain. ABAQUSrequires these values to interpret the data in the input file correctly.

Quite often material test data are supplied using values of nominal stress and strain. In such situationsyou must use the expressions presented below to convert the plastic material data from nominalstress/strain values to true stress/strain values.

The relationship between true strain and nominal strain is established by expressing the nominal strainas

"nom =l ¡ l0l0

=l

l0¡ l0

l0=

l

l0¡ 1:

Adding unity to both sides of this expression and taking the natural log of both sides provides therelationship between the true strain and the nominal strain:

" = ln (1 + "nom ):

The relationship between true stress and nominal stress is formed by considering the incompressiblenature of the plastic deformation and assuming the elasticity is also incompressible, so

l0A0 = lA:

The current area is related to the original area by

A = A0l0l:

Substituting this definition of A into the definition of true stress gives

¾ =F

A=

F

A0

l

l0= ¾nom

µl

l0

¶;

where

l

l0

can also be written as

1 + "nom :

Making this final substitution provides the relationship between true stress and nominal stress andstrain:

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¾ = ¾nom (1 + "nom ):

The *PLASTIC option in ABAQUS defines the post-yield behavior for most metals. ABAQUSapproximates the smooth stress-strain behavior of the material with a series of straight lines joining thegiven data points. Any number of points can be used to approximate the actual material behavior;therefore, it is possible to use a very close approximation of the actual material behavior. The data onthe *PLASTIC option define the true yield stress of the material as a function of true plastic strain. Thefirst piece of data given defines the initial yield stress of the material and, therefore, should have aplastic strain value of zero.

The strains provided in material test data used to define the plastic behavior are not likely to be theplastic strains in the material. Instead, they will probably be the total strains in the material. You mustdecompose these total strain values into the elastic and plastic strain components. The plastic strain isobtained by subtracting the elastic strain, defined as the value of true stress divided by the Young'smodulus, from the value of total strain (see Figure 8-3).

Figure 8-3 Decomposition of the total strain into elastic and plastic components.

This relationship is written

"pl = "t ¡ "el = "t ¡ ¾=E;

where

"pl

is true plastic strain,

"t

is true total strain,

"el

is true elastic strain,¾

¾ = ¾nom (1 + "nom ):

The *PLASTIC option in ABAQUS defines the post-yield behavior for most metals. ABAQUSapproximates the smooth stress-strain behavior of the material with a series of straight lines joining thegiven data points. Any number of points can be used to approximate the actual material behavior;therefore, it is possible to use a very close approximation of the actual material behavior. The data onthe *PLASTIC option define the true yield stress of the material as a function of true plastic strain. Thefirst piece of data given defines the initial yield stress of the material and, therefore, should have aplastic strain value of zero.

The strains provided in material test data used to define the plastic behavior are not likely to be theplastic strains in the material. Instead, they will probably be the total strains in the material. You mustdecompose these total strain values into the elastic and plastic strain components. The plastic strain isobtained by subtracting the elastic strain, defined as the value of true stress divided by the Young'smodulus, from the value of total strain (see Figure 8-3).

Figure 8-3 Decomposition of the total strain into elastic and plastic components.

This relationship is written

"pl = "t ¡ "el = "t ¡ ¾=E;

where

"pl

is true plastic strain,

"t

is true total strain,

"el

is true elastic strain,¾

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is true stress, and

E

is Young's modulus.

Example of converting material test data to ABAQUS input

The stress-strain curve in Figure 8-4 will be used as an example of how to convert the test datadefining a material's plastic behavior into the appropriate input format for ABAQUS. The sixpoints shown on the nominal stress-strain curve will be used as the data for the *PLASTICoption.

Figure 8-4 Elastic-plastic material behavior and correspondng ABAQUS input data.

The first step is to use the equations relating the true stress to the nominal stress and strain andthe true strain to the nominal strain (shown earlier) to convert the nominal stress and nominalstrain to true stress and true strain. Once these values are known, the equation relating theplastic strain to the total and elastic strains (shown earlier) can be used to determine the plasticstrains associated with each yield stress value. The converted data are shown in Table 8-1.

Table 8-1. Stress and strain conversions.Nominal Nominal True True Plastic

Stress Strain Stress Strain Strain200E6 0.00095 200.2E6 0.00095 0.0240E6 0.025 246E6 0.0247 0.0235280E6 0.050 294E6 0.0488 0.0474340E6 0.100 374E6 0.0953 0.0935380E6 0.150 437E6 0.1398 0.1377400E6 0.200 480E6 0.1823 0.1800

While there are few differences between the nominal and true values at small strains, there arevery significant differences at larger strain values; therefore, it is extremely important toprovide the proper stress-strain data to ABAQUS if the strains in the simulation will be large.The format of the input data defining this material behavior is shown in Figure 8-4.

is true stress, and

E

is Young's modulus.

Example of converting material test data to ABAQUS input

The stress-strain curve in Figure 8-4 will be used as an example of how to convert the test datadefining a material's plastic behavior into the appropriate input format for ABAQUS. The sixpoints shown on the nominal stress-strain curve will be used as the data for the *PLASTICoption.

Figure 8-4 Elastic-plastic material behavior and correspondng ABAQUS input data.

The first step is to use the equations relating the true stress to the nominal stress and strain andthe true strain to the nominal strain (shown earlier) to convert the nominal stress and nominalstrain to true stress and true strain. Once these values are known, the equation relating theplastic strain to the total and elastic strains (shown earlier) can be used to determine the plasticstrains associated with each yield stress value. The converted data are shown in Table 8-1.

Table 8-1. Stress and strain conversions.Nominal Nominal True True Plastic

Stress Strain Stress Strain Strain200E6 0.00095 200.2E6 0.00095 0.0240E6 0.025 246E6 0.0247 0.0235280E6 0.050 294E6 0.0488 0.0474340E6 0.100 374E6 0.0953 0.0935380E6 0.150 437E6 0.1398 0.1377400E6 0.200 480E6 0.1823 0.1800

While there are few differences between the nominal and true values at small strains, there arevery significant differences at larger strain values; therefore, it is extremely important toprovide the proper stress-strain data to ABAQUS if the strains in the simulation will be large.The format of the input data defining this material behavior is shown in Figure 8-4.

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ABAQUS interpolates linearly between the data points provided to obtain the material'sresponse and assumes that the response is constant outside the range defined by the input data.Thus, the stress in this material will never exceed 480 MPa; when the stress in the materialreaches 480 MPa, the material will deform continuously until the stress is reduced below thisvalue.

8.3 Selecting elements for elastic-plastic problemsThe incompressible nature of plastic deformation in metals places limitations on the types of elementsthat can be used for an elastic-plastic simulation. The limitations arise because modelingincompressible material behavior adds kinematic constraints to an element; in this case the limitationsconstrain the volume at the element's integration points to remain constant. In certain classes ofelements the addition of these incompressibility constraints makes the element overconstrained. Whenthese elements cannot resolve all of these constraints, they suffer from volumetric locking, whichcauses their response to be too stiff. Volumetric locking is indicated by a rapid variation of hydrostaticpressure stress from element to element or integration point to integration point.

Fully integrated, second-order, solid elements are very susceptible to volumetric locking whenmodeling incompressible material behavior and, therefore, should not be used in elastic-plasticsimulations. The ABAQUS fully integrated, first-order, solid elements do not suffer from volumetriclocking because ABAQUS actually uses a constant volume strain in these elements. Thus, they can beused safely in plasticity problems.

Reduced-integration, solid elements have fewer integration points at which the incompressibilityconstraints must be satisfied. Therefore, they are not overconstrained and can be used for mostelastic-plastic simulations. Second-order, reduced-integration elements should be used with caution ifthe strains exceed 20-40% because at this magnitude they can suffer from volumetric locking. Thiseffect can be reduced with mesh refinement.

If you have to use fully-integrated, second-order elements, use the hybrid versions, which are designedto model incompressible behavior; however, remember that the additional degrees of freedom in theseelements will make the analysis more computationally expensive.

8.4 Example: connecting lug with plasticityYou have been asked to investigate what happens if the steel connecting lug from Chapter 4, "UsingContinuum Elements," is subjected to an extreme load (60 kN) caused by an accident. The results fromthe linear analysis indicate that the lug will yield. You need to determine the extent of the plasticdeformation in the lug and the magnitude of the plastic strains, so that you can assess whether or notthe lug will fail.

The only inelastic material data available for the steel are its yield stress (380 MPa) and its strain atfailure (0.15). You decide to assume that the steel is perfectly plastic: the material does not harden, andthe stress can never exceed 380 MPa (see Figure 8-5).

Figure 8-5 Stress-strain behavior for the steel.

ABAQUS interpolates linearly between the data points provided to obtain the material'sresponse and assumes that the response is constant outside the range defined by the input data.Thus, the stress in this material will never exceed 480 MPa; when the stress in the materialreaches 480 MPa, the material will deform continuously until the stress is reduced below thisvalue.

8.3 Selecting elements for elastic-plastic problemsThe incompressible nature of plastic deformation in metals places limitations on the types of elementsthat can be used for an elastic-plastic simulation. The limitations arise because modelingincompressible material behavior adds kinematic constraints to an element; in this case the limitationsconstrain the volume at the element's integration points to remain constant. In certain classes ofelements the addition of these incompressibility constraints makes the element overconstrained. Whenthese elements cannot resolve all of these constraints, they suffer from volumetric locking, whichcauses their response to be too stiff. Volumetric locking is indicated by a rapid variation of hydrostaticpressure stress from element to element or integration point to integration point.

Fully integrated, second-order, solid elements are very susceptible to volumetric locking whenmodeling incompressible material behavior and, therefore, should not be used in elastic-plasticsimulations. The ABAQUS fully integrated, first-order, solid elements do not suffer from volumetriclocking because ABAQUS actually uses a constant volume strain in these elements. Thus, they can beused safely in plasticity problems.

Reduced-integration, solid elements have fewer integration points at which the incompressibilityconstraints must be satisfied. Therefore, they are not overconstrained and can be used for mostelastic-plastic simulations. Second-order, reduced-integration elements should be used with caution ifthe strains exceed 20-40% because at this magnitude they can suffer from volumetric locking. Thiseffect can be reduced with mesh refinement.

If you have to use fully-integrated, second-order elements, use the hybrid versions, which are designedto model incompressible behavior; however, remember that the additional degrees of freedom in theseelements will make the analysis more computationally expensive.

8.4 Example: connecting lug with plasticityYou have been asked to investigate what happens if the steel connecting lug from Chapter 4, "UsingContinuum Elements," is subjected to an extreme load (60 kN) caused by an accident. The results fromthe linear analysis indicate that the lug will yield. You need to determine the extent of the plasticdeformation in the lug and the magnitude of the plastic strains, so that you can assess whether or notthe lug will fail.

The only inelastic material data available for the steel are its yield stress (380 MPa) and its strain atfailure (0.15). You decide to assume that the steel is perfectly plastic: the material does not harden, andthe stress can never exceed 380 MPa (see Figure 8-5).

Figure 8-5 Stress-strain behavior for the steel.

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In reality some hardening will probably occur, but this assumption is conservative; if the materialhardens, the plastic strains will be less than those predicted by the simulation.

Copy the existing input file, lug.inp, to a new file, lug_plas.inp. Modify this file to include thechanges described below.

8.4.1 Modifications to the input file--the model dataSpecify the post-yield behavior of the material using the *PLASTIC option. The Young's modulus forthe material is 200 GPa, and the initial yield stress at zero plastic strain is 380 MPa. Since you aremodeling the steel as perfectly plastic, no other yield stresses are given on the *PLASTIC option.

The complete material definition is, therefore:


*ELASTIC

200.E9, 0.3

*PLASTIC

380.E6,0.0

All other option blocks in the model definition portion of the input file remain unchanged.

8.4.2 Modifications to the input file--the history dataYou must perform a general, nonlinear simulation because of the nonlinear material behavior in themodel. Therefore, you must remove the PERTURBATION parameter from the *STEP option andspecify the maximum number of increments allowed in the analysis. Since you do not know how manyincrements this simulation may require, pick a reasonably large number: in this case specify 25increments. This simulation is a static analysis of the lug under the extreme loads. Set the total steptime in the *STATIC procedure option block to 1.0. Use an initial increment size that is 20% of thetotal step time. These modifications to your input file should look like:

*STEP, INC=25

*STATIC

0.2,1.0

You assume that the effects of geometric nonlinearity will not be important in this simulation, so youomit the NLGEOM parameter from the *STEP option.

In reality some hardening will probably occur, but this assumption is conservative; if the materialhardens, the plastic strains will be less than those predicted by the simulation.

Copy the existing input file, lug.inp, to a new file, lug_plas.inp. Modify this file to include thechanges described below.

8.4.1 Modifications to the input file--the model dataSpecify the post-yield behavior of the material using the *PLASTIC option. The Young's modulus forthe material is 200 GPa, and the initial yield stress at zero plastic strain is 380 MPa. Since you aremodeling the steel as perfectly plastic, no other yield stresses are given on the *PLASTIC option.

The complete material definition is, therefore:


*ELASTIC

200.E9, 0.3

*PLASTIC

380.E6,0.0

All other option blocks in the model definition portion of the input file remain unchanged.

8.4.2 Modifications to the input file--the history dataYou must perform a general, nonlinear simulation because of the nonlinear material behavior in themodel. Therefore, you must remove the PERTURBATION parameter from the *STEP option andspecify the maximum number of increments allowed in the analysis. Since you do not know how manyincrements this simulation may require, pick a reasonably large number: in this case specify 25increments. This simulation is a static analysis of the lug under the extreme loads. Set the total steptime in the *STATIC procedure option block to 1.0. Use an initial increment size that is 20% of thetotal step time. These modifications to your input file should look like:

*STEP, INC=25

*STATIC

0.2,1.0

You assume that the effects of geometric nonlinearity will not be important in this simulation, so youomit the NLGEOM parameter from the *STEP option.

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Loading

The load applied in this simulation is twice what was applied in the linear elastic simulation ofthe lug (60 kN vs. 30 kN). Therefore, double the magnitude of the pressures applied to the lug.The modified *DLOAD option block in your model will look similar to the following:

*DLOAD

PRESS, P6, 1.E+08

Output requests

You will use ABAQUS/Viewer to review all of the results from this simulation, so suppressall printed output requests by setting the FREQUENCY parameter equal to zero. In addition,in the output database (.odb) file you only need results from the end of the simulation. Youcan set the FREQUENCY parameter on the *OUTPUT, FIELD option to a large number tolimit the output to the output database file. The output request options in your input file shouldlook similar to those shown below:

*NODE PRINT, FREQUENCY=0



You will need to save some history data in the output database file to use with the X-Y plottingcapability in ABAQUS/Viewer. Store the displacements for node set HOLEBOT, which shouldalready exist, using the following option:

*OUTPUT, HISTORY

*NODE OUTPUT, NSET=HOLEBOT

U,

You also want detailed results for one of the elements along the built-in end of the lug (seeFigure 8-6).

Figure 8-6 Element 206.

This is element 206 in the mesh generated by the commands found in ``Connecting lug with

Loading

The load applied in this simulation is twice what was applied in the linear elastic simulation ofthe lug (60 kN vs. 30 kN). Therefore, double the magnitude of the pressures applied to the lug.The modified *DLOAD option block in your model will look similar to the following:

*DLOAD

PRESS, P6, 1.E+08

Output requests

You will use ABAQUS/Viewer to review all of the results from this simulation, so suppressall printed output requests by setting the FREQUENCY parameter equal to zero. In addition,in the output database (.odb) file you only need results from the end of the simulation. Youcan set the FREQUENCY parameter on the *OUTPUT, FIELD option to a large number tolimit the output to the output database file. The output request options in your input file shouldlook similar to those shown below:




You will need to save some history data in the output database file to use with the X-Y plottingcapability in ABAQUS/Viewer. Store the displacements for node set HOLEBOT, which shouldalready exist, using the following option:

*OUTPUT, HISTORY


U,

You also want detailed results for one of the elements along the built-in end of the lug (seeFigure 8-6).

Figure 8-6 Element 206.

This is element 206 in the mesh generated by the commands found in ``Connecting lug with

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plasticity,'' Section A.6; the element in this location may have a different number in yourmodel. This element is chosen because it is the element for which the stresses are most likelyto be largest in magnitude. Create an element set that contains the element, and save thestresses (S), stress invariants (SINV), plastic strains (PE), and strains (E) for that element setin the output database file. The necessary option blocks are shown below:

*ELEMENT OUTPUT, ELSET=EL206

S, SINV, PE, E

*ELSET, ELSET=EL206

206,

Note: The *ELSET option used to define the element set appears after the output requests soas to not break up the block of suboptions associated with the *OUTPUT option.

8.4.3 Running the analysisSave these changes to your model, and run the analysis with the following command:

abaqus job=lug_plas

Status file

Monitor the simulation while it is running by looking at the status file, lug_plas.sta.When ABAQUS has finished the simulation, your status file will contain information similarto the following:




ITERS FREQ

1 1 1 0 1 1 0.200 0.200 0.2000

1 2 1 0 1 1 0.400 0.400 0.2000

1 3 1 0 3 3 0.700 0.700 0.3000

1 4 2 0 2 2 0.775 0.775 0.07500

1 5 1 0 4 4 0.887 0.887 0.1125

1 6 2 0 3 3 0.916 0.916 0.02813

1 7 2 0 2 2 0.926 0.926 0.01055

1 8 1 0 4 4 0.942 0.942 0.01582

1 9 2 0 5 5 0.948 0.948 0.005933

1 10 2 0 4 4 0.949 0.949 0.001483

1 11 1 0 4 4 0.951 0.951 0.001483

1 12 2 0 3 3 0.951 0.951 0.0005562

1 13 1 0 4 4 0.952 0.952 0.0008343

1 14 2 0 4 4 0.953 0.953 0.0003129

1 15 2 0 3 3 0.953 0.953 0.0001173

1 16 2 0 3 3 0.953 0.953 4.399e-05

plasticity,'' Section A.6; the element in this location may have a different number in yourmodel. This element is chosen because it is the element for which the stresses are most likelyto be largest in magnitude. Create an element set that contains the element, and save thestresses (S), stress invariants (SINV), plastic strains (PE), and strains (E) for that element setin the output database file. The necessary option blocks are shown below:


S, SINV, PE, E

*ELSET, ELSET=EL206

206,

Note: The *ELSET option used to define the element set appears after the output requests soas to not break up the block of suboptions associated with the *OUTPUT option.

8.4.3 Running the analysisSave these changes to your model, and run the analysis with the following command:

abaqus job=lug_plas

Status file

Monitor the simulation while it is running by looking at the status file, lug_plas.sta.When ABAQUS has finished the simulation, your status file will contain information similarto the following:




ITERS FREQ

1 1 1 0 1 1 0.200 0.200 0.2000

1 2 1 0 1 1 0.400 0.400 0.2000

1 3 1 0 3 3 0.700 0.700 0.3000

1 4 2 0 2 2 0.775 0.775 0.07500

1 5 1 0 4 4 0.887 0.887 0.1125

1 6 2 0 3 3 0.916 0.916 0.02813

1 7 2 0 2 2 0.926 0.926 0.01055

1 8 1 0 4 4 0.942 0.942 0.01582

1 9 2 0 5 5 0.948 0.948 0.005933

1 10 2 0 4 4 0.949 0.949 0.001483

1 11 1 0 4 4 0.951 0.951 0.001483

1 12 2 0 3 3 0.951 0.951 0.0005562

1 13 1 0 4 4 0.952 0.952 0.0008343

1 14 2 0 4 4 0.953 0.953 0.0003129

1 15 2 0 3 3 0.953 0.953 0.0001173

1 16 2 0 3 3 0.953 0.953 4.399e-05

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1 17 1 0 3 3 0.953 0.953 6.599e-05

1 18 2 0 2 2 0.953 0.953 2.475e-05

1 19 1 0 3 3 0.953 0.953 3.712e-05

1 20 2 0 3 3 0.953 0.953 1.392e-05

1 21 1 0 3 3 0.953 0.953 2.088e-05

1 22 2 0 2 2 0.953 0.953 1.000e-05

1 23 1 0 3 3 0.953 0.953 1.500e-05

1 24 2 0 2 2 0.953 0.953 1.000e-05

ABAQUS was able to apply only 95% of the prescribed load to the model and still obtain aconverged solution. The status file shows that ABAQUS reduced the size of the timeincrement, which is shown in the last (right-hand) column, many times during the simulationand stopped the analysis in the 24th increment. You will have to look at the information in themessage file to understand why ABAQUS terminated the simulation early.

Message file

The message file, lug_plas.msg, contains detailed information about the simulation'sprogress (see ``Results,'' Section 7.4.3, for more information about the format of the messagefile).

Look at the information for the first increment in the analysis (it is also shown below); youwill discover that the model's initial behavior is linear. The model's behavior was also linear inthe second increment.

ABAQUS requires several iterations to obtain a converged solution in the third increment,which indicates that nonlinear behavior occurred in the model during this increment. The onlynonlinearity in the model is the plastic material behavior, so the steel must have started toyield somewhere in the lug at this applied load magnitude. The summaries of the iterations forthe third increment are shown below.

INCREMENT 3 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT 0.300

EQUILIBRIUM ITERATION 1

AVERAGE FORCE 450. TIME AVG. FORCE

LARGEST RESIDUAL FORCE -524. AT NODE 13041

LARGEST INCREMENT OF DISP. -2.545E-04 AT NODE 815

LARGEST CORRECTION TO DISP. -1.800E-06 AT NODE 10817

1 17 1 0 3 3 0.953 0.953 6.599e-05

1 18 2 0 2 2 0.953 0.953 2.475e-05

1 19 1 0 3 3 0.953 0.953 3.712e-05

1 20 2 0 3 3 0.953 0.953 1.392e-05

1 21 1 0 3 3 0.953 0.953 2.088e-05

1 22 2 0 2 2 0.953 0.953 1.000e-05

1 23 1 0 3 3 0.953 0.953 1.500e-05

1 24 2 0 2 2 0.953 0.953 1.000e-05

ABAQUS was able to apply only 95% of the prescribed load to the model and still obtain aconverged solution. The status file shows that ABAQUS reduced the size of the timeincrement, which is shown in the last (right-hand) column, many times during the simulationand stopped the analysis in the 24th increment. You will have to look at the information in themessage file to understand why ABAQUS terminated the simulation early.

Message file

The message file, lug_plas.msg, contains detailed information about the simulation'sprogress (see ``Results,'' Section 7.4.3, for more information about the format of the messagefile).

Look at the information for the first increment in the analysis (it is also shown below); youwill discover that the model's initial behavior is linear. The model's behavior was also linear inthe second increment.

ABAQUS requires several iterations to obtain a converged solution in the third increment,which indicates that nonlinear behavior occurred in the model during this increment. The onlynonlinearity in the model is the plastic material behavior, so the steel must have started toyield somewhere in the lug at this applied load magnitude. The summaries of the iterations forthe third increment are shown below.




LARGEST RESIDUAL FORCE -524. AT NODE 13041



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FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE.



LARGEST RESIDUAL FORCE -1.71 AT NODE 23041






LARGEST RESIDUAL FORCE 1.174E-04 AT NODE 3041



THE FORCE EQUILIBRIUM EQUATIONS HAVE CONVERGED

ITERATION SUMMARY FOR THE INCREMENT: 3 TOTAL ITERATIONS, OF WHIC

0 ARE SEVERE DISCONTINUITY ITERATIONS AND 3 ARE EQUILIBRIUM ITE

TIME INCREMENT COMPLETED 0.300 , FRACTION OF STEP COMPLETED

STEP TIME COMPLETED 0.700 , TOTAL TIME COMPLETED

ABAQUS attempts to find a solution in the fourth increment using an increment size of 0.3,which means it is applying 30% of the total load, or 30 MPa, during this increment. Afterseveral iterations, ABAQUS issues warning messages that the strain increments it calculatedexceed the strain at initial yield by 50 times. After a few more iterations ABAQUS determinesthat the solution in this increment is not going to converge; instead, it is diverging. Therefore,ABAQUS abandons this attempt at finding a solution, reduces the increment size to 25% ofthe value used in the first attempt, and tries a second attempt at finding a solution. Thisreduction in increment size is called a cut-back. With the smaller increment size, ABAQUSfinds a converged solution in just a few iterations. Some of the iteration summaries from thefirst attempt of the fourth increment are shown below.




LARGEST RESIDUAL FORCE -1.71 AT NODE 23041














ABAQUS attempts to find a solution in the fourth increment using an increment size of 0.3,which means it is applying 30% of the total load, or 30 MPa, during this increment. Afterseveral iterations, ABAQUS issues warning messages that the strain increments it calculatedexceed the strain at initial yield by 50 times. After a few more iterations ABAQUS determinesthat the solution in this increment is not going to converge; instead, it is diverging. Therefore,ABAQUS abandons this attempt at finding a solution, reduces the increment size to 25% ofthe value used in the first attempt, and tries a second attempt at finding a solution. Thisreduction in increment size is called a cut-back. With the smaller increment size, ABAQUSfinds a converged solution in just a few iterations. Some of the iteration summaries from thefirst attempt of the fourth increment are shown below.

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Now that you have reviewed the early increments of the simulation, move to the end of themessage file and review the last increment ABAQUS attempted. You will see that ABAQUSis using a very small increment size, on the order of 1.0 � 10-5, in this final increment becauseof the many cut-backs. The iteration summaries for the last increment are shown below.ABAQUS makes two attempts to find a solution in this final increment, but it must cut backthe time increment in each attempt because the strain increments are so large that it does noteven try to perform the plasticity calculations. This check on the magnitude of the strainincrement is another example of the many automatic solution controls ABAQUS uses toensure that the solution obtained for your simulation is both accurate and efficient. Theautomatic solution controls are suitable for almost all simulations. Therefore, you do not haveto worry about providing parameters to control the solution algorithm: you only have to beconcerned with the input data for your model.

Now that you have reviewed the early increments of the simulation, move to the end of themessage file and review the last increment ABAQUS attempted. You will see that ABAQUSis using a very small increment size, on the order of 1.0 � 10-5, in this final increment becauseof the many cut-backs. The iteration summaries for the last increment are shown below.ABAQUS makes two attempts to find a solution in this final increment, but it must cut backthe time increment in each attempt because the strain increments are so large that it does noteven try to perform the plasticity calculations. This check on the magnitude of the strainincrement is another example of the many automatic solution controls ABAQUS uses toensure that the solution obtained for your simulation is both accurate and efficient. Theautomatic solution controls are suitable for almost all simulations. Therefore, you do not haveto worry about providing parameters to control the solution algorithm: you only have to beconcerned with the input data for your model.

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If you look at the summary at the end of the message file, you will find that ABAQUS issuedmany warning messages during the analysis. Reviewing the message file will show that mostof these warnings were the result of numerical problems with the plasticity calculations. Youknow that ABAQUS terminated the analysis early because these numerical problems forced itto cut back the time increment until it was below the minimum allowable time increment.

In the third column of the status file you will see the number of attempts ABAQUS made tosolve an increment. In the sixth column the number of iterations needed for the last attempt atan increment is printed. Now, you should look at the results in ABAQUS/Viewer tounderstand what caused this excessive plasticity.

8.4.4 Postprocessing the resultsRun ABAQUS/Viewer by entering the following command:

abaqus viewer odb=lug_plas



Create a plot of the model's deformed shape, and check that this shape is realistic. Do notinclude the undeformed model shape in the plot.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.

If you look at the summary at the end of the message file, you will find that ABAQUS issuedmany warning messages during the analysis. Reviewing the message file will show that mostof these warnings were the result of numerical problems with the plasticity calculations. Youknow that ABAQUS terminated the analysis early because these numerical problems forced itto cut back the time increment until it was below the minimum allowable time increment.

In the third column of the status file you will see the number of attempts ABAQUS made tosolve an increment. In the sixth column the number of iterations needed for the last attempt atan increment is printed. Now, you should look at the results in ABAQUS/Viewer tounderstand what caused this excessive plasticity.

8.4.4 Postprocessing the resultsRun ABAQUS/Viewer by entering the following command:

abaqus viewer odb=lug_plas



Create a plot of the model's deformed shape, and check that this shape is realistic. Do notinclude the undeformed model shape in the plot.

From the main menu bar, select Plot->Deformed Shape; or use the tool in the toolbox.

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ABAQUS/Viewer displays the plot shown in Figure 8-7.

Figure 8-7 Deformed model shape using results for the simulation without hardening.

The default view is isometric. You can set the view shown in Figure 8-7 by using the optionsin the View menu or the view tools in the toolbar. The displacements and, particularly, therotations of the lug shown in the plot are large. Yet they do not seem large enough to havecaused all of the numerical problems seen in the simulation. Look closely at the information inthe plot's title for an explanation. The deformation scale factor ABAQUS/Viewer used in thisplot was 0.02; the displacements are scaled to 2% of their actual values.

ABAQUS/Viewer always scales the displacements in a geometrically linear simulation suchthat the deformed shape of the model fits into the viewport. (This is in contrast to ageometrically nonlinear simulation, where ABAQUS/Viewer does not scale the displacementsand, instead, adjusts the view by zooming in or out to fit the deformed shape in the plot.) Ifyou want to plot the actual displacements, you must set the deformation scale factor to 1.0.

To set the deformation scale factor:



2. In the Deformation Scale Factor area, toggle on Uniform and enter 1.0 in the Valuetext field that appears.

3. Click OK to apply your selection and to close the dialog box.

This will produce a plot of the model in which the lug has rotated until it is almost parallel tothe vertical (global Y) axis.

The applied load of 60 kN exceeds the limit load of the lug, and the lug collapses when thematerial yields at all of the integration points through its thickness. The lug then has nostiffness to resist further deformation because of the perfectly plastic post-yield behavior of thesteel.

ABAQUS/Viewer displays the plot shown in Figure 8-7.

Figure 8-7 Deformed model shape using results for the simulation without hardening.

The default view is isometric. You can set the view shown in Figure 8-7 by using the optionsin the View menu or the view tools in the toolbar. The displacements and, particularly, therotations of the lug shown in the plot are large. Yet they do not seem large enough to havecaused all of the numerical problems seen in the simulation. Look closely at the information inthe plot's title for an explanation. The deformation scale factor ABAQUS/Viewer used in thisplot was 0.02; the displacements are scaled to 2% of their actual values.

ABAQUS/Viewer always scales the displacements in a geometrically linear simulation suchthat the deformed shape of the model fits into the viewport. (This is in contrast to ageometrically nonlinear simulation, where ABAQUS/Viewer does not scale the displacementsand, instead, adjusts the view by zooming in or out to fit the deformed shape in the plot.) Ifyou want to plot the actual displacements, you must set the deformation scale factor to 1.0.

To set the deformation scale factor:



2. In the Deformation Scale Factor area, toggle on Uniform and enter 1.0 in the Valuetext field that appears.

3. Click OK to apply your selection and to close the dialog box.

This will produce a plot of the model in which the lug has rotated until it is almost parallel tothe vertical (global Y) axis.

The applied load of 60 kN exceeds the limit load of the lug, and the lug collapses when thematerial yields at all of the integration points through its thickness. The lug then has nostiffness to resist further deformation because of the perfectly plastic post-yield behavior of thesteel.

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8.4.5 Adding hardening to the material modelThe ABAQUS connecting lug simulation with perfectly plastic material behavior predicts that the lugwill suffer catastrophic failure caused by the collapse of the structure. We have already mentioned thatthe steel would probably exhibit some hardening after it has yielded. You suspect that includinghardening behavior would allow the lug to withstand this 60 kN load because of the additionalstiffness it would provide. Therefore, you decide to add some hardening to the steel's material propertydefinition. You assume that the yield stress increases to 580 MPa at a plastic strain of 0.35, whichrepresents typical hardening for this class of steel. The stress-strain curve for the modified materialmodel is shown in Figure 8-8. Modify your *PLASTIC option block as follows so that it includes thehardening data:

*PLASTIC

380.E6, 0.00

580.E6, 0.35

Figure 8-8 Modifed stress-strain behavior of the steel.

8.4.6 Running the analysis with plastic hardeningSave the model with plastic hardening to a new input file named lug_plas_hard.inp, and run theanalysis using the command

abaqus job=lug_plas_hard

Status file

The summary of the analysis in the status file, which is shown below, shows that ABAQUSfound a converged solution when the full 60 kN load was applied. The hardening data addedenough stiffness to the lug to prevent it from collapsing under the 60 kN load.




8.4.5 Adding hardening to the material modelThe ABAQUS connecting lug simulation with perfectly plastic material behavior predicts that the lugwill suffer catastrophic failure caused by the collapse of the structure. We have already mentioned thatthe steel would probably exhibit some hardening after it has yielded. You suspect that includinghardening behavior would allow the lug to withstand this 60 kN load because of the additionalstiffness it would provide. Therefore, you decide to add some hardening to the steel's material propertydefinition. You assume that the yield stress increases to 580 MPa at a plastic strain of 0.35, whichrepresents typical hardening for this class of steel. The stress-strain curve for the modified materialmodel is shown in Figure 8-8. Modify your *PLASTIC option block as follows so that it includes thehardening data:

*PLASTIC

380.E6, 0.00

580.E6, 0.35

Figure 8-8 Modifed stress-strain behavior of the steel.

8.4.6 Running the analysis with plastic hardeningSave the model with plastic hardening to a new input file named lug_plas_hard.inp, and run theanalysis using the command

abaqus job=lug_plas_hard

Status file

The summary of the analysis in the status file, which is shown below, shows that ABAQUSfound a converged solution when the full 60 kN load was applied. The hardening data addedenough stiffness to the lug to prevent it from collapsing under the 60 kN load.




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ITERS FREQ

1 1 1 0 1 1 0.200 0.200 0.2000

1 2 1 0 1 1 0.400 0.400 0.2000

1 3 1 0 3 3 0.700 0.700 0.3000

1 4 1 0 7 7 1.00 1.00 0.3000

In this simulation there is very little information of interest in the message file. There are nowarnings issued during the analysis, so you can proceed directly to postprocessing the resultswith ABAQUS/Viewer.

8.4.7 Postprocessing the resultsStart ABAQUS/Viewer, and use the following command to review the results of the second analysis:

abaqus viewer odb=lug_plas_hard

Deformed model shape and peak displacements

Plot the deformed model shape with these new results. It should look similar to the plot shownin Figure 8-9. The default deformation scale factor in this plot is 2.04, making the displayeddeformations roughly double the actual deformations.

Figure 8-9 Deformed model shape for the simulation with plastic hardening.

Contour plot of von Mises stress

Contour the von Mises stress in the model. Create a filled contour plot on the actual deformedshape of the lug with the plot title suppressed.

To contour the von Mises stress:



2. From the list of output variables, select S.

3. From the list of invariants, select Mises.

ITERS FREQ

1 1 1 0 1 1 0.200 0.200 0.2000

1 2 1 0 1 1 0.400 0.400 0.2000

1 3 1 0 3 3 0.700 0.700 0.3000

1 4 1 0 7 7 1.00 1.00 0.3000

In this simulation there is very little information of interest in the message file. There are nowarnings issued during the analysis, so you can proceed directly to postprocessing the resultswith ABAQUS/Viewer.

8.4.7 Postprocessing the resultsStart ABAQUS/Viewer, and use the following command to review the results of the second analysis:

abaqus viewer odb=lug_plas_hard

Deformed model shape and peak displacements

Plot the deformed model shape with these new results. It should look similar to the plot shownin Figure 8-9. The default deformation scale factor in this plot is 2.04, making the displayeddeformations roughly double the actual deformations.

Figure 8-9 Deformed model shape for the simulation with plastic hardening.

Contour plot of von Mises stress

Contour the von Mises stress in the model. Create a filled contour plot on the actual deformedshape of the lug with the plot title suppressed.





3. From the list of invariants, select Mises.

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4. Click OK.



ABAQUS/Viewer displays a contour plot of the von Mises stress.

You will add the visible edges and set the deformation scale factor to produce the plotshown in Figure 8-10.

6. Click Contour Options in the prompt area.

ABAQUS/Viewer displays the Contour Plot Options dialog box.

7. Click the Shape tab, select the Uniform deformation scale factor, and set its value to1.0.

8. Click the Basic tab, drag the uniform contour intervals slider to 10, and choose Exteriorvisible edges.

9. Click OK to view the contour plot and to close the Contour Plot Options dialog box.

The contour plot title is displayed by default but can be turned off.

10. To suppress the plot title and state blocks, select View->Viewport Annotations.

The Viewport Annotations dialog box appears.

11. Click the Title Block tab, and toggle off Show Title Block.

12. Similarly, click the State Block tab and toggle off Show State Block to suppress displayof the state block.

13. Click OK to apply your selections and to close the dialog box.

The customized contour plot appears.

Use the view manipulation tools to position and size the model to obtain a plot similar to thatshown in Figure 8-10.

Figure 8-10 Contour of von Mises stress.

4. Click OK.



ABAQUS/Viewer displays a contour plot of the von Mises stress.

You will add the visible edges and set the deformation scale factor to produce the plotshown in Figure 8-10.


ABAQUS/Viewer displays the Contour Plot Options dialog box.

7. Click the Shape tab, select the Uniform deformation scale factor, and set its value to1.0.

8. Click the Basic tab, drag the uniform contour intervals slider to 10, and choose Exteriorvisible edges.

9. Click OK to view the contour plot and to close the Contour Plot Options dialog box.

The contour plot title is displayed by default but can be turned off.

10. To suppress the plot title and state blocks, select View->Viewport Annotations.


11. Click the Title Block tab, and toggle off Show Title Block.

12. Similarly, click the State Block tab and toggle off Show State Block to suppress displayof the state block.

13. Click OK to apply your selections and to close the dialog box.

The customized contour plot appears.

Use the view manipulation tools to position and size the model to obtain a plot similar to thatshown in Figure 8-10.

Figure 8-10 Contour of von Mises stress.

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Do the values listed in the contour legend surprise you? The maximum stress is greater than580 MPa, which should not be possible since the material was assumed to be perfectly plasticat this stress magnitude. This misleading result occurs because of the algorithm thatABAQUS/Viewer uses to create contour plots for element variables, such as stress. Thecontouring algorithm requires data at the nodes; however, ABAQUS/Standard calculateselement variables at the integration points. ABAQUS/Viewer calculates nodal values ofelement variables by extrapolating the data from the integration points to the nodes. Theextrapolation order depends on the element type; for second-order, reduced-integrationelements ABAQUS/Viewer uses linear extrapolation to calculate the nodal values of elementvariables. To display a contour plot of von Mises stress, ABAQUS/Viewer extrapolates thestress components from the integration points to the nodal locations within each element andcalculates the von Mises stress. If the differences in von Mises stress values fall within thespecified averaging threshold, nodal averaged von Mises stresses are calculated from eachsurrounding element's invariant stress value. Invariant values exceeding the elastic limit can beproduced by the extrapolation process.

Try plotting contours of each component of the stress tensor (variables S11, S22, S33, S12,S23, and S13). You will see that there are significant variations in these stresses across theelements at the built-in end. This causes the extrapolated nodal stresses to be higher than thevalues at the integration points. The von Mises stress calculated from these values will,therefore, also be higher.

The von Mises stress at an integration point can never exceed the current yield stress of theelement's material; however, the extrapolated nodal values reported in a contour plot may doso.

You can use the query tools in ABAQUS/Viewer to check the von Mises stress at theintegration points.

To query the von Mises stress:

1. From the main menu bar, select Tools->Query.

Do the values listed in the contour legend surprise you? The maximum stress is greater than580 MPa, which should not be possible since the material was assumed to be perfectly plasticat this stress magnitude. This misleading result occurs because of the algorithm thatABAQUS/Viewer uses to create contour plots for element variables, such as stress. Thecontouring algorithm requires data at the nodes; however, ABAQUS/Standard calculateselement variables at the integration points. ABAQUS/Viewer calculates nodal values ofelement variables by extrapolating the data from the integration points to the nodes. Theextrapolation order depends on the element type; for second-order, reduced-integrationelements ABAQUS/Viewer uses linear extrapolation to calculate the nodal values of elementvariables. To display a contour plot of von Mises stress, ABAQUS/Viewer extrapolates thestress components from the integration points to the nodal locations within each element andcalculates the von Mises stress. If the differences in von Mises stress values fall within thespecified averaging threshold, nodal averaged von Mises stresses are calculated from eachsurrounding element's invariant stress value. Invariant values exceeding the elastic limit can beproduced by the extrapolation process.

Try plotting contours of each component of the stress tensor (variables S11, S22, S33, S12,S23, and S13). You will see that there are significant variations in these stresses across theelements at the built-in end. This causes the extrapolated nodal stresses to be higher than thevalues at the integration points. The von Mises stress calculated from these values will,therefore, also be higher.

The von Mises stress at an integration point can never exceed the current yield stress of theelement's material; however, the extrapolated nodal values reported in a contour plot may doso.

You can use the query tools in ABAQUS/Viewer to check the von Mises stress at theintegration points.

To query the von Mises stress:


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The Query dialog box appears.

2. In the Visualization Queries field, select Probe values.

3. Click OK.

The Probe Values dialog box appears.

4. Select the von Mises stress output by clicking in the column to the left of S:Mises.

A check mark appears in the S:Mises row.

5. Select Elements, and select the output position Integration Pt.

6. Use the cursor to select elements near the constrained end of the lug.

ABAQUS/Viewer reports the element ID and type by default and the value of the vonMises stress at each integration point starting with the first integration point. The vonMises stress values at the integration points are all lower than the values reported in thecontour legend and also below the yield stress of 580 MPa. You can click mouse button 1to store probed values.

7. Click Cancel when you have finished probing the results.

The fact that the extrapolated values are so different from the integration point values indicatesthat there is a rapid variation of stress across the elements and that the mesh is too coarse foraccurate stress calculations. This extrapolation error will be less significant if the mesh isrefined but will always be present to some extent. Therefore, always use nodal values ofelement variables with caution.

Contour plot of equivalent plastic strain

The equivalent plastic strain in a material (PEEQ) is a scalar variable that is used to representthe material's inelastic deformation. If this variable is greater than zero, the material hasyielded. Those parts of the lug that have yielded can be identified in a contour plot of PEEQby selecting Result->Field Output from the main menu bar and selecting PEEQ from the listof output variables in the dialog box that appears. Set the minimum contour limit to a verysmall magnitude (1.E-6) of equivalent plastic strain by bringing up the Contour PlotOptions dialog box. Thus, any areas in the model plotted in dark blue in ABAQUS/Viewerstill have elastic material behavior (see Figure 8-11).

Figure 8-11 Contour of equivalent plastic strain (PEEQ).


2. In the Visualization Queries field, select Probe values.

3. Click OK.


4. Select the von Mises stress output by clicking in the column to the left of S:Mises.

A check mark appears in the S:Mises row.

5. Select Elements, and select the output position Integration Pt.

6. Use the cursor to select elements near the constrained end of the lug.

ABAQUS/Viewer reports the element ID and type by default and the value of the vonMises stress at each integration point starting with the first integration point. The vonMises stress values at the integration points are all lower than the values reported in thecontour legend and also below the yield stress of 580 MPa. You can click mouse button 1to store probed values.

7. Click Cancel when you have finished probing the results.

The fact that the extrapolated values are so different from the integration point values indicatesthat there is a rapid variation of stress across the elements and that the mesh is too coarse foraccurate stress calculations. This extrapolation error will be less significant if the mesh isrefined but will always be present to some extent. Therefore, always use nodal values ofelement variables with caution.

Contour plot of equivalent plastic strain

The equivalent plastic strain in a material (PEEQ) is a scalar variable that is used to representthe material's inelastic deformation. If this variable is greater than zero, the material hasyielded. Those parts of the lug that have yielded can be identified in a contour plot of PEEQby selecting Result->Field Output from the main menu bar and selecting PEEQ from the listof output variables in the dialog box that appears. Set the minimum contour limit to a verysmall magnitude (1.E-6) of equivalent plastic strain by bringing up the Contour PlotOptions dialog box. Thus, any areas in the model plotted in dark blue in ABAQUS/Viewerstill have elastic material behavior (see Figure 8-11).

Figure 8-11 Contour of equivalent plastic strain (PEEQ).

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It is clear from the plot that there is gross yielding in the lug where it is attached to its parentstructure. The maximum plastic strain reported in the contour legend is 0.0859. However, this

value may contain errors from the extrapolation process. Use the visualization query tool in ABAQUS/Viewer to check the integration point values of PEEQ in the elements with thelargest plastic strains. You will find that the largest equivalent plastic strains in the model areabout 0.0666 at the integration points. This does not necessarily indicate a large extrapolationerror since there are strain gradients present in the vicinity of the peak plastic deformation.

Creating a variable-variable (stress-strain) plot

The X-Y plotting capability in ABAQUS/Viewer was introduced in Chapter 7, "Nonlinearity."In this section you will learn how to create X-Y plots showing the variation of one variable as afunction of another. You will create a stress-strain plot for one of the integration points in anelement adjacent to the constrained end of the lug. You saved the stress and strain data neededfor this plot in the output database (.odb) file; therefore, use the following procedure tocreate X-Y plots in ABAQUS/Viewer.

In the model used in this discussion, the stress-strain data were saved for element 206. Youmay have specified a different element number in your model; if you did, use that elementnumber in place of 206 in the input examples that follow. Also, use data from an integrationpoint that is closest to the top surface of the lug but not adjacent to the constrained nodes. Inthe model discussed here, integration point 1 satisfies these requirements, as shown in Figure8-12.

Figure 8-12 Location of integration point 1 in element 206.

It is clear from the plot that there is gross yielding in the lug where it is attached to its parentstructure. The maximum plastic strain reported in the contour legend is 0.0859. However, this

value may contain errors from the extrapolation process. Use the visualization query tool in ABAQUS/Viewer to check the integration point values of PEEQ in the elements with thelargest plastic strains. You will find that the largest equivalent plastic strains in the model areabout 0.0666 at the integration points. This does not necessarily indicate a large extrapolationerror since there are strain gradients present in the vicinity of the peak plastic deformation.

Creating a variable-variable (stress-strain) plot

The X-Y plotting capability in ABAQUS/Viewer was introduced in Chapter 7, "Nonlinearity."In this section you will learn how to create X-Y plots showing the variation of one variable as afunction of another. You will create a stress-strain plot for one of the integration points in anelement adjacent to the constrained end of the lug. You saved the stress and strain data neededfor this plot in the output database (.odb) file; therefore, use the following procedure tocreate X-Y plots in ABAQUS/Viewer.

In the model used in this discussion, the stress-strain data were saved for element 206. Youmay have specified a different element number in your model; if you did, use that elementnumber in place of 206 in the input examples that follow. Also, use data from an integrationpoint that is closest to the top surface of the lug but not adjacent to the constrained nodes. Inthe model discussed here, integration point 1 satisfies these requirements, as shown in Figure8-12.

Figure 8-12 Location of integration point 1 in element 206.

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To create history curves of stress and direct strain along the lug in element 206:


The ODB History Output dialog box appears.

2. Use the scroll bar in the Output Variables list to locate and select the MISES stress atelement 206, integration point 1.

The von Mises stress, rather than the component of the true stress tensor, is used because theplasticity model defines plastic yield in terms of von Mises stress.

3. Click Save As to save the X-Y data.

The Save XYData As dialog box appears.

4. Enter the name MISES, and click OK.

5. Use the scroll bar in the Output Variables list to locate and select the direct strain ( E11) at thesame integration point, and save it under the name E11.

This strain component is used because it is the largest component of the total strain tensor atthis point; using it clearly shows the elastic, as well as the plastic, behavior of the material atthis integration point.

6. Click Dismiss to close the ODB History Output dialog box.

Each of the curves that you have created is a history (variable versus time) plot. You mustcombine these two plots, eliminating the time dependence, to produce the desired stress-strainplot.

To combine history curves to produce a stress-strain plot:

1. From the main menu bar, select Tools->XY Data->Create.

The Create XY Data dialog box appears.

2. Select Operate on XY data, and click Continue.

To create history curves of stress and direct strain along the lug in element 206:



2. Use the scroll bar in the Output Variables list to locate and select the MISES stress atelement 206, integration point 1.

The von Mises stress, rather than the component of the true stress tensor, is used because theplasticity model defines plastic yield in terms of von Mises stress.



4. Enter the name MISES, and click OK.

5. Use the scroll bar in the Output Variables list to locate and select the direct strain ( E11) at thesame integration point, and save it under the name E11.

This strain component is used because it is the largest component of the total strain tensor atthis point; using it clearly shows the elastic, as well as the plastic, behavior of the material atthis integration point.


Each of the curves that you have created is a history (variable versus time) plot. You mustcombine these two plots, eliminating the time dependence, to produce the desired stress-strainplot.

To combine history curves to produce a stress-strain plot:




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The Operate on XY data dialog box appears.

3. From the Operators listed, select combine(X,X).

combine( ) appears in the text field at the top of the dialog box.

4. In the XY Data field, drag the cursor across E11 and MISES to select both data objects.

When you release the mouse button, the expression combine("E11", "MISES")appears in the text field. In this expression "E11" will determine the X-values and"MISES" will determine the Y-values in the combined plot.

Warning: Note that if you select the data objects individually, you must enter thecomma in the text field yourself.

5. Save the combined data object by clicking Save As at the bottom of the dialog box.

The Save XYData As dialog box appears. In the Name text field, type SVE11; and clickOK to close the dialog box.

6. To view the combined stress-strain plot, click Plot Expression at the bottom of the dialogbox.


This X-Y plot would be clearer if the limits on the X- and Y-axes were changed. Use the XYPlot Options in ABAQUS/Viewer to do this.

To customize the stress-strain curve:

1. Click XY Plot Options in the prompt area.

The XY Plot Options dialog box appears.

2. Set the maximum range of the X-axis (E11) to 0.07 strain, the maximum range of theY-axis (MISES) to 500 MPa stress, and the minimum stress to 0.0 MPa.

3. Click OK to close the dialog box.

4. It will also be helpful to display a symbol at each data point of the curve. Click XY CurveOptions in the prompt area.

The XY Curve Options dialog box appears.

5. From the XY Data field, select the stress-strain curve (SVE11).

The SVE11 data object is highlighted.

6. Toggle on Show symbol. Accept the defaults, and click Dismiss at the bottom of thedialog box.

The Operate on XY data dialog box appears.

3. From the Operators listed, select combine(X,X).

combine( ) appears in the text field at the top of the dialog box.

4. In the XY Data field, drag the cursor across E11 and MISES to select both data objects.

When you release the mouse button, the expression combine("E11", "MISES")appears in the text field. In this expression "E11" will determine the X-values and"MISES" will determine the Y-values in the combined plot.

Warning: Note that if you select the data objects individually, you must enter thecomma in the text field yourself.

5. Save the combined data object by clicking Save As at the bottom of the dialog box.

The Save XYData As dialog box appears. In the Name text field, type SVE11; and clickOK to close the dialog box.

6. To view the combined stress-strain plot, click Plot Expression at the bottom of the dialogbox.


This X-Y plot would be clearer if the limits on the X- and Y-axes were changed. Use the XYPlot Options in ABAQUS/Viewer to do this.

To customize the stress-strain curve:

1. Click XY Plot Options in the prompt area.


2. Set the maximum range of the X-axis (E11) to 0.07 strain, the maximum range of theY-axis (MISES) to 500 MPa stress, and the minimum stress to 0.0 MPa.

3. Click OK to close the dialog box.

4. It will also be helpful to display a symbol at each data point of the curve. Click XY CurveOptions in the prompt area.


5. From the XY Data field, select the stress-strain curve (SVE11).

The SVE11 data object is highlighted.

6. Toggle on Show symbol. Accept the defaults, and click Dismiss at the bottom of thedialog box.

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The stress-strain plot appears with a symbol at each data point of the curve.

You should now have a plot similar to the one shown in Figure 8-13.

Figure 8-13 Von Mises stress vs. direct strain (E11) along the lug in element 206.

The stress-strain curve shows that the material behavior was linear elastic for this integrationpoint during the first two increments of the simulation. In this plot it appears that the materialremains linear during the third increment of the analysis; however, it does yield during thisincrement. This illusion is created by the extent of strain shown in the plot. If you limit themaximum strain displayed to 0.01 and set the minimum value to 0.0, the nonlinear materialbehavior in the third increment can be seen more clearly (see Figure 8-14). This stress-straincurve contains another apparent error. It appears that the material yields at 250 MPa, which iswell below the initial yield stress. However, this error is caused by the fact thatABAQUS/Viewer connects the data points on the curve with straight lines. If you limit theincrement size, the additional points on the graph will provide a better display of the materialresponse and show yield occurring at exactly 380 MPa.

Figure 8-14 Von Mises stress vs. direct strain (E11) along the lug in element 206. Maximum

strain 0.01.

The stress-strain plot appears with a symbol at each data point of the curve.

You should now have a plot similar to the one shown in Figure 8-13.

Figure 8-13 Von Mises stress vs. direct strain (E11) along the lug in element 206.

The stress-strain curve shows that the material behavior was linear elastic for this integrationpoint during the first two increments of the simulation. In this plot it appears that the materialremains linear during the third increment of the analysis; however, it does yield during thisincrement. This illusion is created by the extent of strain shown in the plot. If you limit themaximum strain displayed to 0.01 and set the minimum value to 0.0, the nonlinear materialbehavior in the third increment can be seen more clearly (see Figure 8-14). This stress-straincurve contains another apparent error. It appears that the material yields at 250 MPa, which iswell below the initial yield stress. However, this error is caused by the fact thatABAQUS/Viewer connects the data points on the curve with straight lines. If you limit theincrement size, the additional points on the graph will provide a better display of the materialresponse and show yield occurring at exactly 380 MPa.

Figure 8-14 Von Mises stress vs. direct strain (E11) along the lug in element 206. Maximum

strain 0.01.

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The results from this second simulation indicate that the lug will withstand this 60 kN load ifthe steel hardens after it yields. Taken together, the results of the two simulations demonstratethat it is very important to determine the actual post-yield hardening behavior of the steel. Ifthe steel has very little hardening, the lug may collapse under the 60 kN load. Whereas if it hasmoderate hardening, the lug will probably withstand the load although there will be extensiveplastic yielding in the lug (see Figure 8-11). However, even with plastic hardening, the factorof safety for this loading will probably be very small.

8.5 HyperelasticityWe now turn our attention to another class of material nonlinearity, namely, the nonlinear elasticresponse exhibited by rubber materials.

8.5.1 IntroductionThe stress-strain behavior of typical rubber materials, shown in Figure 8-15, is elastic but highlynonlinear. This type of material behavior is called hyperelasticity. The deformation of hyperelasticmaterials, such as rubber, remains elastic up to large strain values (often well over 100%). ABAQUSmakes the following assumptions when modeling a hyperelastic material:

� The material behavior is elastic.

� The material behavior is isotropic.

� The material is incompressible by default.

� The simulation will include nonlinear geometric effects (NLGEOM will be used).

Figure 8-15 Typical stress-strain curve for rubber.

The results from this second simulation indicate that the lug will withstand this 60 kN load ifthe steel hardens after it yields. Taken together, the results of the two simulations demonstratethat it is very important to determine the actual post-yield hardening behavior of the steel. Ifthe steel has very little hardening, the lug may collapse under the 60 kN load. Whereas if it hasmoderate hardening, the lug will probably withstand the load although there will be extensiveplastic yielding in the lug (see Figure 8-11). However, even with plastic hardening, the factorof safety for this loading will probably be very small.

8.5 HyperelasticityWe now turn our attention to another class of material nonlinearity, namely, the nonlinear elasticresponse exhibited by rubber materials.

8.5.1 IntroductionThe stress-strain behavior of typical rubber materials, shown in Figure 8-15, is elastic but highlynonlinear. This type of material behavior is called hyperelasticity. The deformation of hyperelasticmaterials, such as rubber, remains elastic up to large strain values (often well over 100%). ABAQUSmakes the following assumptions when modeling a hyperelastic material:

� The material behavior is elastic.

� The material behavior is isotropic.

� The material is incompressible by default.

� The simulation will include nonlinear geometric effects (NLGEOM will be used).

Figure 8-15 Typical stress-strain curve for rubber.

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ABAQUS has a special family of "hybrid" elements that must be used to model the fullyincompressible behavior seen in hyperelastic materials. These "hybrid" elements are identified by theletter `H' in their name; for example, the hybrid form of the 8-node brick, C3D8, is called C3D8H.

Elastomeric foams are another class of highly nonlinear, elastic materials. They differ from rubbermaterials in that they have very compressible behavior when subjected to compressive loads. They aremodeled with a separate material model in ABAQUS and are not discussed in detail in this guide.

8.5.2 Strain energy potentialABAQUS uses a strain energy potential (U), rather than a Young's modulus and Poisson's ratio, torelate stresses to strains in hyperelastic materials. Several different strain energy potentials areavailable: a polynomial model, the Ogden model, the Arruda-Boyce model, and the van der Waalsmodel. Simpler forms of the polynomial model are also available, including the Mooney-Rivlin,neo-Hookean, reduced polynomial, and Yeoh models.

The polynomial form of the strain energy potential is the one that is most commonly used. Its form is

U =

NXi+j=1

Cij (¹I1 ¡ 3)i(¹I2 ¡ 3)

j+

NXi=1

1

Di(Jel ¡ 1)

2i;

where U is the strain energy potential; Jel is the elastic volume ratio; ¹I1 and ¹I2 are measures of thedistortion in the material; and N , Cij , and Di are material parameters, which may be functions oftemperature. The Cij parameters describe the shear behavior of the material, and the Di parametersintroduce compressibility. If all the Di are set to zero, ABAQUS will recognize that the material isfully incompressible and will ignore the second part of the equation shown above. If the number ofterms, N , is one, the initial shear modulus, ¹0 , and bulk modulus, K0 , are given by

¹0 = 2(C01 + C10);

and

ABAQUS has a special family of "hybrid" elements that must be used to model the fullyincompressible behavior seen in hyperelastic materials. These "hybrid" elements are identified by theletter `H' in their name; for example, the hybrid form of the 8-node brick, C3D8, is called C3D8H.

Elastomeric foams are another class of highly nonlinear, elastic materials. They differ from rubbermaterials in that they have very compressible behavior when subjected to compressive loads. They aremodeled with a separate material model in ABAQUS and are not discussed in detail in this guide.

8.5.2 Strain energy potentialABAQUS uses a strain energy potential (U), rather than a Young's modulus and Poisson's ratio, torelate stresses to strains in hyperelastic materials. Several different strain energy potentials areavailable: a polynomial model, the Ogden model, the Arruda-Boyce model, and the van der Waalsmodel. Simpler forms of the polynomial model are also available, including the Mooney-Rivlin,neo-Hookean, reduced polynomial, and Yeoh models.

The polynomial form of the strain energy potential is the one that is most commonly used. Its form is

U =

NXi+j=1

Cij (¹I1 ¡ 3)i(¹I2 ¡ 3)

j+

NXi=1

1

Di(Jel ¡ 1)

2i;

where U is the strain energy potential; Jel is the elastic volume ratio; ¹I1 and ¹I2 are measures of thedistortion in the material; and N , Cij , and Di are material parameters, which may be functions oftemperature. The Cij parameters describe the shear behavior of the material, and the Di parametersintroduce compressibility. If all the Di are set to zero, ABAQUS will recognize that the material isfully incompressible and will ignore the second part of the equation shown above. If the number ofterms, N , is one, the initial shear modulus, ¹0 , and bulk modulus, K0 , are given by

¹0 = 2(C01 + C10);

and

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K0 =2

D1:

If the material is incompressible, the equation for the strain energy density is

U = C10(¹I1 ¡ 3) + C01(¹I2 ¡ 3):

This expression is commonly referred to as the Mooney-Rivlin material model. If C01 is also zero, thematerial is called neo-Hookean.

The other hyperelastic models are similar in concept and are described in the ABAQUS/StandardUser's Manual.

You must provide ABAQUS with the relevant material parameters to use a hyperelastic material. Forthe polynomial form these are N , Cij , and Di. It is possible that you will be supplied with theseparameters when modeling hyperelastic materials; however, more likely you will be given test data forthe materials that you must model. Fortunately, ABAQUS can accept test data directly and calculatethe material parameters for you (using a least squares fit).

8.5.3 Defining hyperelastic behavior using test dataA convenient way of defining a hyperelastic material is to supply ABAQUS with experimental testdata. ABAQUS then calculates the constants using a least squares method. The experimental tests forwhich ABAQUS can fit data are:

� Uniaxial tension and compression.

� Equibiaxial tension and compression.

� Planar tension and compression (pure shear).

� Volumetric tension and compression.

The deformation modes seen in these tests and the ABAQUS input options used to define the data foreach are shown in Figure 8-16. Unlike plasticity data, the test data for hyperelastic materials must begiven to ABAQUS as nominal stress and nominal strain values.

Figure 8-16 Deformation modes and ABAQUS input options for the various experimental tests for

defining hyperelastic material behavior.

K0 =2

D1:

If the material is incompressible, the equation for the strain energy density is

U = C10(¹I1 ¡ 3) + C01(¹I2 ¡ 3):

This expression is commonly referred to as the Mooney-Rivlin material model. If C01 is also zero, thematerial is called neo-Hookean.

The other hyperelastic models are similar in concept and are described in the ABAQUS/StandardUser's Manual.

You must provide ABAQUS with the relevant material parameters to use a hyperelastic material. Forthe polynomial form these are N , Cij , and Di. It is possible that you will be supplied with theseparameters when modeling hyperelastic materials; however, more likely you will be given test data forthe materials that you must model. Fortunately, ABAQUS can accept test data directly and calculatethe material parameters for you (using a least squares fit).

8.5.3 Defining hyperelastic behavior using test dataA convenient way of defining a hyperelastic material is to supply ABAQUS with experimental testdata. ABAQUS then calculates the constants using a least squares method. The experimental tests forwhich ABAQUS can fit data are:

� Uniaxial tension and compression.

� Equibiaxial tension and compression.

� Planar tension and compression (pure shear).

� Volumetric tension and compression.

The deformation modes seen in these tests and the ABAQUS input options used to define the data foreach are shown in Figure 8-16. Unlike plasticity data, the test data for hyperelastic materials must begiven to ABAQUS as nominal stress and nominal strain values.

Figure 8-16 Deformation modes and ABAQUS input options for the various experimental tests for

defining hyperelastic material behavior.

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Volumetric compression data only need to be given if the material's compressibility is important.Normally it is not important, and the default fully incompressible behavior is used.

Achieving the best material model from your data

The quality of the results from a simulation using hyperelastic materials strongly depends onthe material test data that you provide ABAQUS. There are several things that you can do tohelp ABAQUS calculate the best possible material parameters.

Wherever possible, try to obtain experimental test data from more than one deformationstate--this allows ABAQUS to form a much more accurate and stable material model.However, some of the tests shown in Figure 8-16 produce equivalent deformation modes forincompressible materials. The following are equivalent tests for incompressible materials:

Volumetric compression data only need to be given if the material's compressibility is important.Normally it is not important, and the default fully incompressible behavior is used.

Achieving the best material model from your data

The quality of the results from a simulation using hyperelastic materials strongly depends onthe material test data that you provide ABAQUS. There are several things that you can do tohelp ABAQUS calculate the best possible material parameters.

Wherever possible, try to obtain experimental test data from more than one deformationstate--this allows ABAQUS to form a much more accurate and stable material model.However, some of the tests shown in Figure 8-16 produce equivalent deformation modes forincompressible materials. The following are equivalent tests for incompressible materials:

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� Uniaxial tension � Equibiaxial compression

� Uniaxial compression � Equibiaxial tension

� Planar tension � Planar compression

You do not need to include data from a particular test if you already have data from anothertest that models a particular deformation mode.

In addition, the following may improve your hyperelastic material model:

� Obtain test data for the deformation modes that are likely to occur in your simulation. Forexample, if your component is loaded in compression, make sure that your test datainclude compressive, rather than tensile, loading.

� Both tension and compression data are allowed, with compressive stresses and strainsentered as negative values. If possible, use compression or tension data depending on theapplication, since the fit of a single material model to both tensile and compressive datawill normally be less accurate for each individual test.

� Try to include test data from the planar test. This test measures shear behavior, which canbe very important.

� Provide more data at the strain magnitudes that you expect the material will be subjectedto during the simulation. For example, if the material will only have small tensile strains,say under 50%, do not provide much, if any, test data at high strain values (over 100%).

� Perform one-element simulations of the experimental tests and compare the resultsABAQUS calculates to the experimental data. If the computational results are poor for aparticular deformation mode that is important to you, try to obtain more experimental datafor that deformation mode. These one-element simulations are very easy to perform inABAQUS/CAE. Please consult the ABAQUS/CAE User's Manual for details.

Stability of the material model

It is common for the material model determined from the test data to be unstable at certainstrain magnitudes. ABAQUS performs a stability check to determine the strain magnitudeswhere unstable behavior will occur and prints a warning message in the data (.dat) file. Youshould check this information carefully since your simulation may not converge if any part ofthe model experiences strains beyond the stability limits. The stability checks are done forspecific deformations, so it is possible for the material to be unstable at the strain levelsindicated if the deformation is more complex. Likewise, it is possible for the material to beunstable at lower strain levels if the deformation is more complex.

8.6 Example: axisymmetric mountYou have been asked to find the axial stiffness of the rubber mount shown in Figure 8-17 and toidentify any areas of high maximum principal stress that might limit the fatigue life of the mount. The

� Uniaxial tension � Equibiaxial compression

� Uniaxial compression � Equibiaxial tension

� Planar tension � Planar compression

You do not need to include data from a particular test if you already have data from anothertest that models a particular deformation mode.

In addition, the following may improve your hyperelastic material model:

� Obtain test data for the deformation modes that are likely to occur in your simulation. Forexample, if your component is loaded in compression, make sure that your test datainclude compressive, rather than tensile, loading.

� Both tension and compression data are allowed, with compressive stresses and strainsentered as negative values. If possible, use compression or tension data depending on theapplication, since the fit of a single material model to both tensile and compressive datawill normally be less accurate for each individual test.

� Try to include test data from the planar test. This test measures shear behavior, which canbe very important.

� Provide more data at the strain magnitudes that you expect the material will be subjectedto during the simulation. For example, if the material will only have small tensile strains,say under 50%, do not provide much, if any, test data at high strain values (over 100%).

� Perform one-element simulations of the experimental tests and compare the resultsABAQUS calculates to the experimental data. If the computational results are poor for aparticular deformation mode that is important to you, try to obtain more experimental datafor that deformation mode. These one-element simulations are very easy to perform inABAQUS/CAE. Please consult the ABAQUS/CAE User's Manual for details.

Stability of the material model

It is common for the material model determined from the test data to be unstable at certainstrain magnitudes. ABAQUS performs a stability check to determine the strain magnitudeswhere unstable behavior will occur and prints a warning message in the data (.dat) file. Youshould check this information carefully since your simulation may not converge if any part ofthe model experiences strains beyond the stability limits. The stability checks are done forspecific deformations, so it is possible for the material to be unstable at the strain levelsindicated if the deformation is more complex. Likewise, it is possible for the material to beunstable at lower strain levels if the deformation is more complex.

8.6 Example: axisymmetric mountYou have been asked to find the axial stiffness of the rubber mount shown in Figure 8-17 and toidentify any areas of high maximum principal stress that might limit the fatigue life of the mount. The

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mount is bonded at both ends to steel plates. It will experience axial loads up to 5.5 kN distributeduniformly across the plates. The cross-section geometry and dimensions are given in Figure 8-17.

Figure 8-17 Axisymmetric mount.

You can use axisymmetric elements for this simulation since both the geometry of the model and theloading are axisymmetric. Therefore, you only need to model a plane through the component: eachelement represents a complete 360° ring.

8.6.1 SymmetryYou do not need to model the whole section of this axisymmetric component because the problem issymmetric about a horizontal line through the center of the mount. By modeling only half of thesection, you can use half as many elements and, hence, half the number of degrees of freedom. Thissignificantly reduces the run time and storage requirements for the analysis or, alternatively, allowsyou to use a more refined mesh.

Many problems contain some degree of symmetry. For example, mirror symmetry, cyclic symmetry,axisymmetry, or repetitive symmetry (shown in Figure 8-18) are common. More than one type ofsymmetry may be present in the structure or component that you want to model.

Figure 8-18 Various forms of symmetry.

mount is bonded at both ends to steel plates. It will experience axial loads up to 5.5 kN distributeduniformly across the plates. The cross-section geometry and dimensions are given in Figure 8-17.

Figure 8-17 Axisymmetric mount.

You can use axisymmetric elements for this simulation since both the geometry of the model and theloading are axisymmetric. Therefore, you only need to model a plane through the component: eachelement represents a complete 360° ring.

8.6.1 SymmetryYou do not need to model the whole section of this axisymmetric component because the problem issymmetric about a horizontal line through the center of the mount. By modeling only half of thesection, you can use half as many elements and, hence, half the number of degrees of freedom. Thissignificantly reduces the run time and storage requirements for the analysis or, alternatively, allowsyou to use a more refined mesh.

Many problems contain some degree of symmetry. For example, mirror symmetry, cyclic symmetry,axisymmetry, or repetitive symmetry (shown in Figure 8-18) are common. More than one type ofsymmetry may be present in the structure or component that you want to model.

Figure 8-18 Various forms of symmetry.

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When modeling just a portion of a symmetric component, you have to add boundary conditions tomake the model behave as if the whole component were being modeled. You may also have to adjustthe applied loads to reflect the portion of the structure actually being modeled. Consider the portalframe in Figure 8-19.

Figure 8-19 Symmetric portal frame.

The frame is symmetric about the vertical line shown in the figure. To maintain symmetry in themodel, any nodes on the symmetry line must be constrained from translating in the 1-direction andfrom rotating about the 2- or 3-axes (degrees of freedom 5 and 6). Therefore, the symmetry constraintswould be

*BOUNDARY

<node>,1

<node>,5,6

In this problem the load is applied along the model's symmetry plane; therefore, only half of the total

When modeling just a portion of a symmetric component, you have to add boundary conditions tomake the model behave as if the whole component were being modeled. You may also have to adjustthe applied loads to reflect the portion of the structure actually being modeled. Consider the portalframe in Figure 8-19.

Figure 8-19 Symmetric portal frame.

The frame is symmetric about the vertical line shown in the figure. To maintain symmetry in themodel, any nodes on the symmetry line must be constrained from translating in the 1-direction andfrom rotating about the 2- or 3-axes (degrees of freedom 5 and 6). Therefore, the symmetry constraintswould be

*BOUNDARY

<node>,1

<node>,5,6

In this problem the load is applied along the model's symmetry plane; therefore, only half of the total

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value should be applied to the portion you are modeling.

In axisymmetric analyses using axisymmetric elements, such as this rubber mount example, we needmodel only the cross-section of the component. The element formulation automatically includes theeffects of axial symmetry.

8.6.2 Coordinate systemUse the default r-z (1-2) axisymmetric coordinate system in this simulation. Place the origin at thelevel of the bottom of the plate, as shown in Figure 8-17.

8.6.3 Mesh designUse a 30 � 15 mesh of first-order, axisymmetric, hybrid solid elements (CAX4H) for the rubber mount.You should include only the bottom half of the mount in your model, as shown in Figure 8-20.

Figure 8-20 Mesh for the rubber mount.

You must use hybrid elements because the material is fully incompressible. The elements are notexpected to be subjected to bending, so shear locking in these fully integrated elements should not be aconcern. Model the steel plates with a single layer of incompatible mode elements ( CAX4I) because itis possible that the plates may bend as the rubber underneath them deforms.

The node and element numbers from the input file given in ``Axisymmetric mount,'' Section A.7, areshown in Figure 8-21 and Figure 8-22. These will be used in the discussion of this example. If youbuild the model yourself, it will probably have different node and element numbers.

Figure 8-21 Node numbers.

value should be applied to the portion you are modeling.

In axisymmetric analyses using axisymmetric elements, such as this rubber mount example, we needmodel only the cross-section of the component. The element formulation automatically includes theeffects of axial symmetry.

8.6.2 Coordinate systemUse the default r-z (1-2) axisymmetric coordinate system in this simulation. Place the origin at thelevel of the bottom of the plate, as shown in Figure 8-17.

8.6.3 Mesh designUse a 30 � 15 mesh of first-order, axisymmetric, hybrid solid elements (CAX4H) for the rubber mount.You should include only the bottom half of the mount in your model, as shown in Figure 8-20.

Figure 8-20 Mesh for the rubber mount.

You must use hybrid elements because the material is fully incompressible. The elements are notexpected to be subjected to bending, so shear locking in these fully integrated elements should not be aconcern. Model the steel plates with a single layer of incompatible mode elements ( CAX4I) because itis possible that the plates may bend as the rubber underneath them deforms.

The node and element numbers from the input file given in ``Axisymmetric mount,'' Section A.7, areshown in Figure 8-21 and Figure 8-22. These will be used in the discussion of this example. If youbuild the model yourself, it will probably have different node and element numbers.

Figure 8-21 Node numbers.

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Figure 8-22 Element numbers.

8.6.4 Preprocessing--creating the modelIf you use ABAQUS/CAE or another preprocessor to create the mesh for this model, try to create anode set MIDDLE containing all the nodes on the symmetry plane, and apply a pressure load of 0.50MPa to the bottom of the plate (Figure 8-23). Check that this pressure results in a total applied load of5.5 kN (5.5 kN = 0.50 MPa � ¼

¡r2o ¡ r2i

¢). If you can't create the mesh, the ABAQUS input options

used to create the model can be found in ``Axisymmetric mount,'' Section A.7.

Figure 8-23 Node set MIDDLE and pressure loading.

Figure 8-22 Element numbers.

8.6.4 Preprocessing--creating the modelIf you use ABAQUS/CAE or another preprocessor to create the mesh for this model, try to create anode set MIDDLE containing all the nodes on the symmetry plane, and apply a pressure load of 0.50MPa to the bottom of the plate (Figure 8-23). Check that this pressure results in a total applied load of5.5 kN (5.5 kN = 0.50 MPa � ¼

¡r2o ¡ r2i

¢). If you can't create the mesh, the ABAQUS input options

used to create the model can be found in ``Axisymmetric mount,'' Section A.7.

Figure 8-23 Node set MIDDLE and pressure loading.

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8.6.5 Reviewing the input file--the model dataWe review the model data, including the geometry definition (nodes and elements) and the materialproperties.

Model description

Ensure that the input file contains a suitable description of the analysis.

*HEADING

Axisymmetric mount analysis under axial loading.



There will be at least two *ELEMENT option blocks in the input file since two different typesof elements are used in the simulation. Check that the element types are correct and that theelement sets containing the elements have descriptive names. The *ELEMENT options in yourinput file should look like

*ELEMENT, TYPE=CAX4I, ELSET=PLATE

*ELEMENT, TYPE=CAX4H, ELSET=RUBBER

Node sets

Check that the node set MIDDLE has been created. If it has not, add it using an editor.

Property definition

Two element property definitions are required: one for the elements modeling the rubber andone for those modeling the plates. The following element property definitions should be inyour model:

*SOLID SECTION, MATERIAL=RUBBER, ELSET=RUBBER

*SOLID SECTION, MATERIAL=STEEL, ELSET=PLATE

Material properties: hyperelastic model for the rubber

8.6.5 Reviewing the input file--the model dataWe review the model data, including the geometry definition (nodes and elements) and the materialproperties.

Model description

Ensure that the input file contains a suitable description of the analysis.

*HEADING




There will be at least two *ELEMENT option blocks in the input file since two different typesof elements are used in the simulation. Check that the element types are correct and that theelement sets containing the elements have descriptive names. The *ELEMENT options in yourinput file should look like

*ELEMENT, TYPE=CAX4I, ELSET=PLATE

*ELEMENT, TYPE=CAX4H, ELSET=RUBBER

Node sets

Check that the node set MIDDLE has been created. If it has not, add it using an editor.

Property definition

Two element property definitions are required: one for the elements modeling the rubber andone for those modeling the plates. The following element property definitions should be inyour model:



Material properties: hyperelastic model for the rubber

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You have been given some experimental test data, shown in Figure 8-24, for the rubbermaterial used in the mount. Three different sets of test data--a uniaxial test, a biaxial test, anda planar (shear) test--are available. You decide to have ABAQUS calculate the appropriatehyperelastic material constants from the test data. You are not sure how large the strains willbe in the rubber mount, but you suspect that they will be under 2.0.

Figure 8-24 Material test data for the rubber material.

The test data for the biaxial and planar tests go well beyond this magnitude, so you decide toperform a one-element simulation of the experimental tests to confirm that the coefficients thatABAQUS calculates from the test data are adequate.

Use a first-order, polynomial strain energy function to model the rubber material. Indicatethese choices by using the N=1 and POLYNOMIAL parameters on the *HYPERELASTICoption. Use the TEST DATA INPUT parameter to indicate that ABAQUS should find thematerial constants from the test data you will provide. The test data are given on options thatimmediately follow the *HYPERELASTIC option. The data should be entered as nominalstress and the corresponding nominal strain, with negative values indicating compression. Youmay be able to enter the data directly using your preprocessor (for instance, if you are usingABAQUS/CAE); otherwise, you will have to add it to your input file with an editor. Thematerial definition for the rubber will look like

*MATERIAL, NAME=RUBBER

*HYPERELASTIC, N=1, POLYNOMIAL, TEST DATA INPUT

*UNIAXIAL TEST DATA

0.054E6, 0.0380

0.152E6, 0.1338

0.254E6, 0.2210

0.362E6, 0.3450

0.459E6, 0.4600

0.583E6, 0.6242

0.656E6, 0.8510

You have been given some experimental test data, shown in Figure 8-24, for the rubbermaterial used in the mount. Three different sets of test data--a uniaxial test, a biaxial test, anda planar (shear) test--are available. You decide to have ABAQUS calculate the appropriatehyperelastic material constants from the test data. You are not sure how large the strains willbe in the rubber mount, but you suspect that they will be under 2.0.

Figure 8-24 Material test data for the rubber material.

The test data for the biaxial and planar tests go well beyond this magnitude, so you decide toperform a one-element simulation of the experimental tests to confirm that the coefficients thatABAQUS calculates from the test data are adequate.

Use a first-order, polynomial strain energy function to model the rubber material. Indicatethese choices by using the N=1 and POLYNOMIAL parameters on the *HYPERELASTICoption. Use the TEST DATA INPUT parameter to indicate that ABAQUS should find thematerial constants from the test data you will provide. The test data are given on options thatimmediately follow the *HYPERELASTIC option. The data should be entered as nominalstress and the corresponding nominal strain, with negative values indicating compression. Youmay be able to enter the data directly using your preprocessor (for instance, if you are usingABAQUS/CAE); otherwise, you will have to add it to your input file with an editor. Thematerial definition for the rubber will look like

*MATERIAL, NAME=RUBBER


*UNIAXIAL TEST DATA

0.054E6, 0.0380

0.152E6, 0.1338

0.254E6, 0.2210

0.362E6, 0.3450

0.459E6, 0.4600

0.583E6, 0.6242

0.656E6, 0.8510

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0.730E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621

The input file for the single-element simulation of the three experimental tests is shown in``Test fit of hyperelastic material data,'' Section A.8. The computational and experimentalresults for the various types of tests are compared in Figure 8-25, Figure 8-26, and Figure8-27.

Figure 8-25 Comparison of experimental data (BIAXIAL) and ABAQUS results

(BIAX_COMP): biaxial tension.

Figure 8-26 Comparison of experimental data (UNIAXIAL) and ABAQUS results

(UNI_COMP): uniaxial tension.

0.730E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621

The input file for the single-element simulation of the three experimental tests is shown in``Test fit of hyperelastic material data,'' Section A.8. The computational and experimentalresults for the various types of tests are compared in Figure 8-25, Figure 8-26, and Figure8-27.

Figure 8-25 Comparison of experimental data (BIAXIAL) and ABAQUS results

(BIAX_COMP): biaxial tension.

Figure 8-26 Comparison of experimental data (UNIAXIAL) and ABAQUS results

(UNI_COMP): uniaxial tension.

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The ABAQUS and experimental results for the biaxial tension test match very well. Thecomputational and experimental results for the uniaxial tension and planar tests match well atstrains less than 100%. The hyperelastic material model created from these material test data isprobably not suitable for use in general simulations where the strains may be larger than100%. However, the model will be adequate for this simulation if the principal strains remainwithin the

Figure 8-27 Comparison of experimental data (PLANAR) and ABAQUS results

(PLANAR_COMP): planar shear.

strain magnitudes where the data and the hyperelastic model fit well. If you find that theresults are beyond these magnitudes or if you are asked to perform a different simulation, youwill have to insist on getting better material data. Otherwise, you will not be able to havemuch confidence in your results.

Material properties: elastic properties for the steel

The steel is modeled with linear elastic properties only (E = 200 GPa, º = 0.3) because theloads should not be large enough to cause inelastic deformations. Thus, the material optionblocks for the steel are

The ABAQUS and experimental results for the biaxial tension test match very well. Thecomputational and experimental results for the uniaxial tension and planar tests match well atstrains less than 100%. The hyperelastic material model created from these material test data isprobably not suitable for use in general simulations where the strains may be larger than100%. However, the model will be adequate for this simulation if the principal strains remainwithin the

Figure 8-27 Comparison of experimental data (PLANAR) and ABAQUS results

(PLANAR_COMP): planar shear.

strain magnitudes where the data and the hyperelastic model fit well. If you find that theresults are beyond these magnitudes or if you are asked to perform a different simulation, youwill have to insist on getting better material data. Otherwise, you will not be able to havemuch confidence in your results.

Material properties: elastic properties for the steel

The steel is modeled with linear elastic properties only (E = 200 GPa, º = 0.3) because theloads should not be large enough to cause inelastic deformations. Thus, the material optionblocks for the steel are

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*ELASTIC

2.0E11,0.3

8.6.6 Reviewing the input--the history dataWe now discuss the history data associated with this problem, including the time incrementationparameters, boundary conditions, loading, and output requests.

Including nonlinear geometry and specifying the initial increment size

When hyperelastic materials are used in a model, ABAQUS assumes that it may undergo largedeformations. But large deformations and other nonlinear geometric effects are included onlyif the NLGEOM parameter is used on the *STEP option. Therefore, you must include it in thissimulation or ABAQUS will terminate the analysis with an input error. Use the INC parameterto specify that up to 20 increments may be carried out in this analysis. The *STEP optionshould look like


The simulation will be a static analysis with a total step time of 1.0. Specify the initial timeincrement to be 1/100th of the total step time. The procedure option block should look like

*STATIC

.01,1.0

Boundary conditions

Specify symmetry boundary conditions on the nodes lying on the symmetry plane. In thismodel the symmetry conditions prevent the nodes from moving in degree of freedom 2(axially), as shown in Figure 8-28.

Figure 8-28 Boundary conditions on the rubber mount.

Symmetry conditions that constrain motion in the global 2-direction can be applied using theYSYMM type boundary condition, or you can simply constrain the 2-direction. In this case the*BOUNDARY option block has the following format:


*ELASTIC

2.0E11,0.3

8.6.6 Reviewing the input--the history dataWe now discuss the history data associated with this problem, including the time incrementationparameters, boundary conditions, loading, and output requests.

Including nonlinear geometry and specifying the initial increment size

When hyperelastic materials are used in a model, ABAQUS assumes that it may undergo largedeformations. But large deformations and other nonlinear geometric effects are included onlyif the NLGEOM parameter is used on the *STEP option. Therefore, you must include it in thissimulation or ABAQUS will terminate the analysis with an input error. Use the INC parameterto specify that up to 20 increments may be carried out in this analysis. The *STEP optionshould look like


The simulation will be a static analysis with a total step time of 1.0. Specify the initial timeincrement to be 1/100th of the total step time. The procedure option block should look like

*STATIC

.01,1.0

Boundary conditions

Specify symmetry boundary conditions on the nodes lying on the symmetry plane. In thismodel the symmetry conditions prevent the nodes from moving in degree of freedom 2(axially), as shown in Figure 8-28.

Figure 8-28 Boundary conditions on the rubber mount.

Symmetry conditions that constrain motion in the global 2-direction can be applied using theYSYMM type boundary condition, or you can simply constrain the 2-direction. In this case the*BOUNDARY option block has the following format:

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*BOUNDARY

MIDDLE, 2, 2, 0.0

No boundary constraints are needed in the radial direction (global 1-direction) because theaxisymmetric nature of the model does not allow the structure to move as a rigid body in theradial direction. ABAQUS will allow nodes to move in the radial direction, even thoseinitially on the axis of symmetry (i.e., those with a radial coordinate of 0.0), if no boundaryconditions are applied to their radial displacements (degree of freedom 1). Since you want tolet the mount deform radially in this analysis, do not apply any boundary conditions; again,ABAQUS will prevent rigid body motions automatically.

Loading

The mount must carry a maximum axial load of 5.5 kN, spread uniformly over the steel plates.A distributed load is, therefore, applied to the bottom of the steel plate. The magnitude of thepressure is given by

p = 5500=¼(0:062 ¡ 0:012) »= 0:50MPa

If you generated the pressure loading using a preprocessor, a *DLOAD option block withmany data lines may be present in the input file.

*DLOAD

1, P1, 0.50E6

2, P1, 0.50E6

...

29, P1, 0.50E6

30, P1, 0.50E6

For the element and node numbering discussed here, the pressure is applied to face 1 of all theelements in element set PLATE. This allows us to use a much more compact format for thedata lines of the *DLOAD option block.

*DLOAD

PLATE, P1, 0.50E6

Output requests

Use the FREQUENCY parameter to suppress all printed output. Write the preselectedvariables as field output to the output database file. In addition, write the displacement of oneof the nodes on the bottom of the steel plate to the output database file so that the stiffness ofthe mount can be calculated. You will need to create a node set containing the node. Theoutput option blocks in your model should be similar to the following:



*BOUNDARY

MIDDLE, 2, 2, 0.0

No boundary constraints are needed in the radial direction (global 1-direction) because theaxisymmetric nature of the model does not allow the structure to move as a rigid body in theradial direction. ABAQUS will allow nodes to move in the radial direction, even thoseinitially on the axis of symmetry (i.e., those with a radial coordinate of 0.0), if no boundaryconditions are applied to their radial displacements (degree of freedom 1). Since you want tolet the mount deform radially in this analysis, do not apply any boundary conditions; again,ABAQUS will prevent rigid body motions automatically.

Loading

The mount must carry a maximum axial load of 5.5 kN, spread uniformly over the steel plates.A distributed load is, therefore, applied to the bottom of the steel plate. The magnitude of thepressure is given by

p = 5500=¼(0:062 ¡ 0:012) »= 0:50MPa

If you generated the pressure loading using a preprocessor, a *DLOAD option block withmany data lines may be present in the input file.

*DLOAD

1, P1, 0.50E6

2, P1, 0.50E6

...

29, P1, 0.50E6

30, P1, 0.50E6

For the element and node numbering discussed here, the pressure is applied to face 1 of all theelements in element set PLATE. This allows us to use a much more compact format for thedata lines of the *DLOAD option block.

*DLOAD

PLATE, P1, 0.50E6

Output requests

Use the FREQUENCY parameter to suppress all printed output. Write the preselectedvariables as field output to the output database file. In addition, write the displacement of oneof the nodes on the bottom of the steel plate to the output database file so that the stiffness ofthe mount can be calculated. You will need to create a node set containing the node. Theoutput option blocks in your model should be similar to the following:



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*NSET, NSET=NODEOUT

1,

*OUTPUT, FIELD, VARIABLE=PRESELECT, FREQUENCY=999

*OUTPUT, HISTORY

*NODE OUTPUT, NSET=NODEOUT

U,

Ensure that the end of the step definition is clearly marked with an *END STEP option.

8.6.7 Running the analysisStore your input options in a file called mount.inp. The input options for the model discussed in theabove sections can be found in `Àxisymmetric mount,'' Section A.7. Since the nonlinear nature of thesimulation means that it may take some time to complete, use the following command to run theanalysis in the background:

abaqus job=mount

When the job has completed, check the data file, mount.dat, for errors. If there are any, correct theinput file and rerun the analysis. If necessary, compare your input with that shown in `Àxisymmetricmount,'' Section A.7.

8.6.8 ResultsWe briefly review the results associated with the polynomial fit of the test data.

The hyperelastic material parameters

In this simulation you specified that the material is incompressible (D1=0). Theincompressibility is assumed since no volumetric test data were provided. To simulatecompressible behavior, you must provide volumetric test data in addition to the other test data.You also specified that ABAQUS should use a first-order, polynomial strain energy function.This form of the hyperelasticity model is known as the Mooney-Rivlin material model.

The hyperelastic material coefficients--C10 , C01 , and D1--that ABAQUS calculates from thematerial test data are given in the data file, mount.dat, provided that you used the*PREPRINT, MODEL=YES option in the model data section of the input file. The materialtest data are also written in the file so that you can ensure that ABAQUS used the correct data,as shown below.

M A T E R I A L D E S C R

MATERIAL NAME: RUBBER

UNIAXIAL TEST DATA

NOMINAL STRESS NOMINAL STRAIN TEMP.

5.4000E+04 3.8000E-02

*NSET, NSET=NODEOUT

1,

*OUTPUT, FIELD, VARIABLE=PRESELECT, FREQUENCY=999

*OUTPUT, HISTORY


U,

Ensure that the end of the step definition is clearly marked with an *END STEP option.

8.6.7 Running the analysisStore your input options in a file called mount.inp. The input options for the model discussed in theabove sections can be found in `Àxisymmetric mount,'' Section A.7. Since the nonlinear nature of thesimulation means that it may take some time to complete, use the following command to run theanalysis in the background:

abaqus job=mount

When the job has completed, check the data file, mount.dat, for errors. If there are any, correct theinput file and rerun the analysis. If necessary, compare your input with that shown in `Àxisymmetricmount,'' Section A.7.

8.6.8 ResultsWe briefly review the results associated with the polynomial fit of the test data.

The hyperelastic material parameters

In this simulation you specified that the material is incompressible (D1=0). Theincompressibility is assumed since no volumetric test data were provided. To simulatecompressible behavior, you must provide volumetric test data in addition to the other test data.You also specified that ABAQUS should use a first-order, polynomial strain energy function.This form of the hyperelasticity model is known as the Mooney-Rivlin material model.

The hyperelastic material coefficients--C10 , C01 , and D1--that ABAQUS calculates from thematerial test data are given in the data file, mount.dat, provided that you used the*PREPRINT, MODEL=YES option in the model data section of the input file. The materialtest data are also written in the file so that you can ensure that ABAQUS used the correct data,as shown below.

M A T E R I A L D E S C R

MATERIAL NAME: RUBBER

UNIAXIAL TEST DATA


5.4000E+04 3.8000E-02

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1.5200E+05 0.1338

2.5400E+05 0.2210

3.6200E+05 0.3450

4.5900E+05 0.4600

5.8300E+05 0.6242

6.5600E+05 0.8510

7.3000E+05 1.427

BIAXIAL TEST DATA


8.9000E+04 2.0000E-02

2.5500E+05 0.1400

5.0300E+05 0.4200

9.5800E+05 1.490

1.7030E+06 2.750

2.4130E+06 3.450

PLANAR TEST DATA


5.5000E+04 6.9000E-02

3.2400E+05 0.2828

7.5800E+05 1.386

1.2690E+06 3.034

1.7790E+06 4.062

HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY

D1 C10 C01

0.000000E+00 176051. 4332.63

If there were any problems with the stability of the hyperelastic material model, warningmessages would be given before the material parameters. The material model is stable at allstrains with these material test data and this strain energy function. However, if you specifiedthat a second-order (N=2), polynomial strain energy function be used, you would see thefollowing warnings in the data file:


***WARNING: UNSTABLE HYPERELASTIC MATERIAL

FOR UNIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR UNIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN

FOR BIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR BIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN

1.5200E+05 0.1338

2.5400E+05 0.2210

3.6200E+05 0.3450

4.5900E+05 0.4600

5.8300E+05 0.6242

6.5600E+05 0.8510

7.3000E+05 1.427

BIAXIAL TEST DATA


8.9000E+04 2.0000E-02

2.5500E+05 0.1400

5.0300E+05 0.4200

9.5800E+05 1.490

1.7030E+06 2.750

2.4130E+06 3.450

PLANAR TEST DATA


5.5000E+04 6.9000E-02

3.2400E+05 0.2828

7.5800E+05 1.386

1.2690E+06 3.034

1.7790E+06 4.062

HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY

D1 C10 C01

0.000000E+00 176051. 4332.63

If there were any problems with the stability of the hyperelastic material model, warningmessages would be given before the material parameters. The material model is stable at allstrains with these material test data and this strain energy function. However, if you specifiedthat a second-order (N=2), polynomial strain energy function be used, you would see thefollowing warnings in the data file:


***WARNING: UNSTABLE HYPERELASTIC MATERIAL

FOR UNIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR UNIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN

FOR BIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR BIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN

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FOR PLANE TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR PLANE COMPRESSION WITH A NOMINAL STRAIN LESS THAN

POLYNOMIAL STRAIN ENERGY FUNCTION WITH N = 2

D1 C10 C01

D2 C20 C11 C02

0.0000E+00 0.1934E+06 -148.2

0.0000E+00 -805.7 180.0 -3.967

If you only had the uniaxial test data available for this problem, you would find that theMooney-Rivlin material model ABAQUS creates would have unstable material behaviorabove certain strain magnitudes.

8.6.9 PostprocessingUse ABAQUS/Viewer to look at the analysis results by issuing the following command:

abaqus viewer odb=mount


Calculating the stiffness of the mount

Determine the stiffness of the mount by creating an X-Y plot of the displacement of the steelplate as a function of the applied load. Create a plot of the vertical displacement of the node onthe steel plate for which you wrote data to the output database file. Data were written fornode 1 in this model. Swap the X- and Y-axes, and name the plot SWAPPED as shown in theprocedure below.

To create a history curve of vertical displacement at node 1 and swap the X- and Y-axes:



2. In the Output Variables field, use the scroll bar to locate and select the verticaldisplacement U2 at node 1.



4. Type the name DISP, and click OK.





FOR PLANE TENSION WITH A NOMINAL STRAIN LARGER THAN

FOR PLANE COMPRESSION WITH A NOMINAL STRAIN LESS THAN

POLYNOMIAL STRAIN ENERGY FUNCTION WITH N = 2

D1 C10 C01

D2 C20 C11 C02

0.0000E+00 0.1934E+06 -148.2

0.0000E+00 -805.7 180.0 -3.967

If you only had the uniaxial test data available for this problem, you would find that theMooney-Rivlin material model ABAQUS creates would have unstable material behaviorabove certain strain magnitudes.

8.6.9 PostprocessingUse ABAQUS/Viewer to look at the analysis results by issuing the following command:

abaqus viewer odb=mount


Calculating the stiffness of the mount

Determine the stiffness of the mount by creating an X-Y plot of the displacement of the steelplate as a function of the applied load. Create a plot of the vertical displacement of the node onthe steel plate for which you wrote data to the output database file. Data were written fornode 1 in this model. Swap the X- and Y-axes, and name the plot SWAPPED as shown in theprocedure below.

To create a history curve of vertical displacement at node 1 and swap the X- and Y-axes:



2. In the Output Variables field, use the scroll bar to locate and select the verticaldisplacement U2 at node 1.



4. Type the name DISP, and click OK.





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The Operate on XY Data dialog box appears.

8. From the Operators listed, click swap(X).

swap( ) appears in the text field at the top of the dialog box.

9. In the XY Data field, select DISP.

The expression swap( "DISP" ) appears in the text field.

10. Save the swapped data object by clicking Save As at the bottom of the dialog box.


11. In the Name text field, type SWAPPED; and click OK to close the dialog box.

12. To view the swapped plot of time-displacement, click Plot Expression at the bottom ofthe dialog box.

You now have a curve of time-displacement. What you need is a curve showingforce-displacement. This is easy to create because in this simulation the force applied to themount is directly proportional to the total time in the analysis. All you have to do to plot aforce-displacement curve, called FORCEDEF, is multiply the curve SWAPPED by themagnitude of the load (5.5 kN).

To multiply a curve by a constant value:

1. In the Operate on XY Data dialog box, click Clear Expression.

2. In the XY Data field, select SWAPPED.

The expression "SWAPPED" appears in the text field at the top of the dialog box.

3. Click the text field.

The text field becomes highlighted, and the text cursor becomes visible. Place the text cursorat the end of the text field by using either the keyboard arrow buttons or the mouse.

4. Multiply the data object in the text field by the magnitude of the applied load by entering*5500.

The expression "SWAPPED"*5500 appears in the text field.

5. Save the multiplied data object by clicking Save As at the bottom of the dialog box.


6. In the Name text field, type FORCEDEF; and click OK to close the dialog box.

7. To view the force-displacement plot, click Plot Expression at the bottom of the dialog box.


8. From the Operators listed, click swap(X).

swap( ) appears in the text field at the top of the dialog box.


The expression swap( "DISP" ) appears in the text field.

10. Save the swapped data object by clicking Save As at the bottom of the dialog box.


11. In the Name text field, type SWAPPED; and click OK to close the dialog box.

12. To view the swapped plot of time-displacement, click Plot Expression at the bottom ofthe dialog box.

You now have a curve of time-displacement. What you need is a curve showingforce-displacement. This is easy to create because in this simulation the force applied to themount is directly proportional to the total time in the analysis. All you have to do to plot aforce-displacement curve, called FORCEDEF, is multiply the curve SWAPPED by themagnitude of the load (5.5 kN).

To multiply a curve by a constant value:

1. In the Operate on XY Data dialog box, click Clear Expression.

2. In the XY Data field, select SWAPPED.

The expression "SWAPPED" appears in the text field at the top of the dialog box.

3. Click the text field.

The text field becomes highlighted, and the text cursor becomes visible. Place the text cursorat the end of the text field by using either the keyboard arrow buttons or the mouse.

4. Multiply the data object in the text field by the magnitude of the applied load by entering*5500.

The expression "SWAPPED"*5500 appears in the text field.

5. Save the multiplied data object by clicking Save As at the bottom of the dialog box.


6. In the Name text field, type FORCEDEF; and click OK to close the dialog box.

7. To view the force-displacement plot, click Plot Expression at the bottom of the dialog box.

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You have now created a curve with the force-deflection characteristic of the mount. To get thestiffness, you need to differentiate the curve FORCEDEF. You can do this by using thedifferentiate( ) operator in the Operate on XY data dialog box.

To obtain the stiffness:

1. In the Operate on XY Data dialog box, clear the current expression.

2. From the Operators listed, click differentiate(X).

differentiate( ) appears in the text field at the top of the dialog box.

3. In the XY Data field, select FORCEDEF.

The expression differentiate( "FORCEDEF" ) appears in the text field.

4. Save the differentiated data object by clicking Save As at the bottom of the dialog box.


5. In the Name text field, type STIFF; and click OK to close the dialog box.

6. To plot the stiffness-displacement curve, click Plot Expression at the bottom of thedialog box.


8. Click XY Plot Options in the prompt area of the main window.


9. Click the Titles tab, and select User-specified as the Title Source.

10. Customize the axis labels as they appear in Figure 8-29 by filling in the Title text fieldsfor the X- and Y-axes.

11. Click OK to close the XY Plot Options dialog box.

12. From the main menu bar, select Tools->XY Data->Plot. From the list of curves selectSTIFF to view the plot in Figure 8-29 that shows the variation of the mount's axialstiffness as it deforms.

Figure 8-29 Stiffness characteristic of the mount.

You have now created a curve with the force-deflection characteristic of the mount. To get thestiffness, you need to differentiate the curve FORCEDEF. You can do this by using thedifferentiate( ) operator in the Operate on XY data dialog box.

To obtain the stiffness:

1. In the Operate on XY Data dialog box, clear the current expression.

2. From the Operators listed, click differentiate(X).


3. In the XY Data field, select FORCEDEF.

The expression differentiate( "FORCEDEF" ) appears in the text field.




6. To plot the stiffness-displacement curve, click Plot Expression at the bottom of thedialog box.





10. Customize the axis labels as they appear in Figure 8-29 by filling in the Title text fieldsfor the X- and Y-axes.


12. From the main menu bar, select Tools->XY Data->Plot. From the list of curves selectSTIFF to view the plot in Figure 8-29 that shows the variation of the mount's axialstiffness as it deforms.

Figure 8-29 Stiffness characteristic of the mount.

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The stiffness of the mount increases by almost 40% as the mount deforms. This is a result ofthe nonlinear nature of the rubber and the change in shape of the mount as it deforms.Alternatively, you could have created the stiffness-displacement curve directly by combiningall the operators above into one expression.

To define the stiffness curve directly:





3. Clear the current expression; and from the Operators listed, click differentiate(X).


4. From the Operators listed, click swap( X ).

differentiate( swap( ) ) appears in the text field.


The expression differentiate( swap( "DISP" ) ) appears in the text field.

6. Place the cursor in the text field directly after the swap( "DISP" ) data object, andtype *5500 to multiply the swapped data by the constant total force value.

differentiate( swap( "DISP" )*5500 ) appears in the text field.




The stiffness of the mount increases by almost 40% as the mount deforms. This is a result ofthe nonlinear nature of the rubber and the change in shape of the mount as it deforms.Alternatively, you could have created the stiffness-displacement curve directly by combiningall the operators above into one expression.

To define the stiffness curve directly:





3. Clear the current expression; and from the Operators listed, click differentiate(X).


4. From the Operators listed, click swap( X ).

differentiate( swap( ) ) appears in the text field.


The expression differentiate( swap( "DISP" ) ) appears in the text field.

6. Place the cursor in the text field directly after the swap( "DISP" ) data object, andtype *5500 to multiply the swapped data by the constant total force value.

differentiate( swap( "DISP" )*5500 ) appears in the text field.




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12. Customize the axis labels as they appear in Figure 8-29 by filling in the Title text fieldsfor the X- and Y- axes.


14. From the main menu bar, select Tools->XY Data->Plot.

15. From the list of curves, select STIFF to view the plot in Figure 8-29 that shows thevariation of the mount's axial stiffness as the mount deforms.

Model shape plots

You will now plot the undeformed and deformed model shapes of the mount. The latter plotwill allow you to evaluate the quality of the deformed mesh and to assess the need for meshrefinement.

To plot the undeformed and deformed model shapes:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool on thetoolbar to plot the undeformed model shape (Figure 8-30).

Figure 8-30 Undeformed model shape of the rubber mount.

2. Select Options->Undeformed, or click Undeformed Options in the prompt area.






12. Customize the axis labels as they appear in Figure 8-29 by filling in the Title text fieldsfor the X- and Y- axes.


14. From the main menu bar, select Tools->XY Data->Plot.

15. From the list of curves, select STIFF to view the plot in Figure 8-29 that shows thevariation of the mount's axial stiffness as the mount deforms.

Model shape plots

You will now plot the undeformed and deformed model shapes of the mount. The latter plotwill allow you to evaluate the quality of the deformed mesh and to assess the need for meshrefinement.

To plot the undeformed and deformed model shapes:

1. From the main menu bar, select Plot->Undeformed Shape; or use the tool on thetoolbar to plot the undeformed model shape (Figure 8-30).

Figure 8-30 Undeformed model shape of the rubber mount.

2. Select Options->Undeformed, or click Undeformed Options in the prompt area.


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3. In the Undeformed Plot Options dialog box, set Visible Edges to All to view all elementedges.

4. Click OK to see the modified undeformed plot.

Now plot the deformed model shape (Figure 8-31).

Figure 8-31 Deformed model shape of the rubber under an applied load of 5500 N.

5. Select Plot->Deformed Shape, or use the tool to plot the deformed model shape of themount.

6. Set the same edge options for the deformed shape plot as you did for the undeformed shapeplot by selecting Options->Deformed or by clicking Deformed Options in the prompt area.

If the figure obscures the plot title, you can move the plot by clicking the tool andholding down mouse button 1 to pan the deformed shape to the desired location. Alternatively,you can turn the plot title off.

To turn off the title and state blocks:

1. From the main menu bar, select View->Viewport Annotations.


2. Click the Title Block tab, and toggle off Show title block.

3. Similarly, click the State Block tab, and toggle off Show state block.

4. Click OK to apply the changes and to close the Viewport Annotations dialog box.

The plate has been pushed up, causing the rubber to bulge at the sides.

3. In the Undeformed Plot Options dialog box, set Visible Edges to All to view all elementedges.

4. Click OK to see the modified undeformed plot.

Now plot the deformed model shape (Figure 8-31).

Figure 8-31 Deformed model shape of the rubber under an applied load of 5500 N.

5. Select Plot->Deformed Shape, or use the tool to plot the deformed model shape of themount.

6. Set the same edge options for the deformed shape plot as you did for the undeformed shapeplot by selecting Options->Deformed or by clicking Deformed Options in the prompt area.

If the figure obscures the plot title, you can move the plot by clicking the tool andholding down mouse button 1 to pan the deformed shape to the desired location. Alternatively,you can turn the plot title off.

To turn off the title and state blocks:



2. Click the Title Block tab, and toggle off Show title block.

3. Similarly, click the State Block tab, and toggle off Show state block.

4. Click OK to apply the changes and to close the Viewport Annotations dialog box.

The plate has been pushed up, causing the rubber to bulge at the sides.

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5. Zoom in on the bottom left corner of the mesh using the tool from the toolbar.

Click mouse button 1, and hold it down to define the first corner of the new view; movethe mouse to create a box enclosing the viewing area that you want (Figure 8-32); andrelease the mouse button. Alternatively, you can zoom and pan the plot by selectingView->Specify from the main menu bar.

You should have a plot similar to the one shown in Figure 8-32.

Figure 8-32 Distortion in the corner of the rubber mount model.

Some elements in this corner of the model are becoming badly distorted because the meshdesign in this area was inadequate for the type of deformation that occurs there. Although theshape of the elements is fine at the start of the analysis, they become badly distorted as therubber bulges outward, especially the element in the corner. If the loading were increasedfurther, the element distortion may become so excessive that the analysis may abort. ``Meshdesign for large distortions,'' Section 8.7, discusses how to improve the mesh design for thisproblem.

The keystoning pattern exhibited by the distorted elements in the bottom right-hand corner ofthe model indicates that they are locking. A contour plot of the hydrostatic pressure stress inthese elements (without averaging across elements sharing common nodes) shows rapidvariation in the pressure stress between adjacent elements. This indicates that these elementsare suffering from volumetric locking, which was discussed earlier in ``Selecting elements forelastic-plastic problems,'' Section 8.3, in the context of plastic incompressibility. Volumetriclocking arises in this problem from overconstraint. The steel is very stiff compared to therubber. Thus, along the bond line the rubber elements cannot deform laterally. Since theseelements must also satisfy incompressibility requirements, they are highly constrained andlocking occurs. Analysis techniques that address volumetric locking are discussed in``Techniques for reducing volumetric locking,'' Section 8.8.

Contouring the maximum principal stress

Plot the maximum in-plane principal stress in the model. Follow the procedure given below tocreate a filled contour plot on the actual deformed shape of the lug with the plot title

5. Zoom in on the bottom left corner of the mesh using the tool from the toolbar.

Click mouse button 1, and hold it down to define the first corner of the new view; movethe mouse to create a box enclosing the viewing area that you want (Figure 8-32); andrelease the mouse button. Alternatively, you can zoom and pan the plot by selectingView->Specify from the main menu bar.

You should have a plot similar to the one shown in Figure 8-32.

Figure 8-32 Distortion in the corner of the rubber mount model.

Some elements in this corner of the model are becoming badly distorted because the meshdesign in this area was inadequate for the type of deformation that occurs there. Although theshape of the elements is fine at the start of the analysis, they become badly distorted as therubber bulges outward, especially the element in the corner. If the loading were increasedfurther, the element distortion may become so excessive that the analysis may abort. ``Meshdesign for large distortions,'' Section 8.7, discusses how to improve the mesh design for thisproblem.

The keystoning pattern exhibited by the distorted elements in the bottom right-hand corner ofthe model indicates that they are locking. A contour plot of the hydrostatic pressure stress inthese elements (without averaging across elements sharing common nodes) shows rapidvariation in the pressure stress between adjacent elements. This indicates that these elementsare suffering from volumetric locking, which was discussed earlier in ``Selecting elements forelastic-plastic problems,'' Section 8.3, in the context of plastic incompressibility. Volumetriclocking arises in this problem from overconstraint. The steel is very stiff compared to therubber. Thus, along the bond line the rubber elements cannot deform laterally. Since theseelements must also satisfy incompressibility requirements, they are highly constrained andlocking occurs. Analysis techniques that address volumetric locking are discussed in``Techniques for reducing volumetric locking,'' Section 8.8.

Contouring the maximum principal stress

Plot the maximum in-plane principal stress in the model. Follow the procedure given below tocreate a filled contour plot on the actual deformed shape of the lug with the plot title

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suppressed.

To contour the maximum principal stress:




3. From the list of invariants, select Max. Principal.

4. Click OK.



ABAQUS/Viewer displays a contour plot of the maximum in-plane principal stresses.Youmust add the visible edges and set the deformation scale factor to produce the plot shownin Figure 8-33.

Figure 8-33 Contours of maximum principal stress in the rubber mount.


The Contour Plot Options dialog box appears.

7. Drag the uniform contour intervals slider to 8 and choose Exterior visible edges.

8. Click OK to view the contour plot and to close the dialog box.

Create a display group showing only the elements in the rubber mount.

9. Select Tools->Display Group->Create.

suppressed.

To contour the maximum principal stress:





4. Click OK.



ABAQUS/Viewer displays a contour plot of the maximum in-plane principal stresses.Youmust add the visible edges and set the deformation scale factor to produce the plot shownin Figure 8-33.

Figure 8-33 Contours of maximum principal stress in the rubber mount.



7. Drag the uniform contour intervals slider to 8 and choose Exterior visible edges.

8. Click OK to view the contour plot and to close the dialog box.

Create a display group showing only the elements in the rubber mount.

9. Select Tools->Display Group->Create.

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10. From the ODB Item list, select Elements and the RUBBER element set. Click toreplace the current display group with the RUBBER element set.

The viewport display changes and displays only the rubber mount elements.

11. Click Dismiss to close the Create Display Group dialog box.

The maximum principal stress in the model, reported in the contour legend, is 135 kPa.Although the mesh in this model is fairly refined and, thus, the extrapolation error should be

minimal, you may want to use the query tool in ABAQUS/Viewer to determine the moreaccurate integration point values of the maximum principal stress.

When you look at the integration point values, you will discover that the peak value ofmaximum principal stress occurs in one of the distorted elements in the bottom right-hand partof the model. This value is likely to be unreliable because of the levels of element distortionand volumetric locking. If this value is ignored, there is an area near the plane of symmetrywhere the maximum principal stress is around 88.3 kPa.

The easiest way to check the range of the principal strains in the model is to display themaximum and minimum values in the contour legend.

To check the principal strain magnitude:



2. Click the Legend tab, and toggle on Show min/max values.

3. Click OK.

The maximum and minimum values appear at the bottom of the contour legend in theviewport.



5. From the list of output variables, select E.


7. Click Apply.

The contour plot changes to display values for maximum principal strain. Note the valueof the maximum principal strain from the contour legend.


10. From the ODB Item list, select Elements and the RUBBER element set. Click toreplace the current display group with the RUBBER element set.

The viewport display changes and displays only the rubber mount elements.

11. Click Dismiss to close the Create Display Group dialog box.

The maximum principal stress in the model, reported in the contour legend, is 135 kPa.Although the mesh in this model is fairly refined and, thus, the extrapolation error should be

minimal, you may want to use the query tool in ABAQUS/Viewer to determine the moreaccurate integration point values of the maximum principal stress.

When you look at the integration point values, you will discover that the peak value ofmaximum principal stress occurs in one of the distorted elements in the bottom right-hand partof the model. This value is likely to be unreliable because of the levels of element distortionand volumetric locking. If this value is ignored, there is an area near the plane of symmetrywhere the maximum principal stress is around 88.3 kPa.

The easiest way to check the range of the principal strains in the model is to display themaximum and minimum values in the contour legend.

To check the principal strain magnitude:




3. Click OK.

The maximum and minimum values appear at the bottom of the contour legend in theviewport.



5. From the list of output variables, select E.


7. Click Apply.

The contour plot changes to display values for maximum principal strain. Note the valueof the maximum principal strain from the contour legend.

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8. From the list of invariants in the Field Output dialog box, select Min. Principal.

9. Click OK to dismiss the Field Output dialog box.

The contour plot changes to display values for minimum principal strain. Note the valueof the minimum principal strain from the contour legend.

The maximum and minimum principal strain values indicate that the maximum tensile strainin the model is about 62% and the maximum compressive strain is about 64%. Because thestrains in the model remained within the range where the ABAQUS hyperelasticity model hasa good fit to the material data, you can be fairly confident that the response predicted by themount is reasonable from a material modeling viewpoint.

8.7 Mesh design for large distortionsWe know that the element distortions in the corners of the rubber mount are undesirable. The results inthese areas are unreliable; and if the load were increased, the analysis might fail. These problems canbe corrected by using a better mesh design. The mesh shown in Figure 8-34 is an example of analternate mesh design that might be used to reduce element distortion in the bottom left corner of therubber model. The issues surrounding the mesh distortion in the opposite corner are addressed in``Techniques for reducing volumetric locking,'' Section 8.8.

Figure 8-34 Modified mesh to minimize element distortions in the bottom left corner of the rubber

model during the simulation.

The elements in the bottom left-hand corner region are now much more distorted in the initial,undeformed configuration. However, as the analysis progresses and the elements deform, their shapeactually improves. The displaced shape plot, shown in Figure 8-35, illustrates that the amount ofelement distortion in this region is reduced; the level of mesh distortion in the bottom right-handcorner of the rubber model is still significant. The contours of maximum principal stress ( Figure 8-36)show that the very localized stress in that corner has been reduced only slightly.

8. From the list of invariants in the Field Output dialog box, select Min. Principal.

9. Click OK to dismiss the Field Output dialog box.

The contour plot changes to display values for minimum principal strain. Note the valueof the minimum principal strain from the contour legend.

The maximum and minimum principal strain values indicate that the maximum tensile strainin the model is about 62% and the maximum compressive strain is about 64%. Because thestrains in the model remained within the range where the ABAQUS hyperelasticity model hasa good fit to the material data, you can be fairly confident that the response predicted by themount is reasonable from a material modeling viewpoint.

8.7 Mesh design for large distortionsWe know that the element distortions in the corners of the rubber mount are undesirable. The results inthese areas are unreliable; and if the load were increased, the analysis might fail. These problems canbe corrected by using a better mesh design. The mesh shown in Figure 8-34 is an example of analternate mesh design that might be used to reduce element distortion in the bottom left corner of therubber model. The issues surrounding the mesh distortion in the opposite corner are addressed in``Techniques for reducing volumetric locking,'' Section 8.8.

Figure 8-34 Modified mesh to minimize element distortions in the bottom left corner of the rubber

model during the simulation.

The elements in the bottom left-hand corner region are now much more distorted in the initial,undeformed configuration. However, as the analysis progresses and the elements deform, their shapeactually improves. The displaced shape plot, shown in Figure 8-35, illustrates that the amount ofelement distortion in this region is reduced; the level of mesh distortion in the bottom right-handcorner of the rubber model is still significant. The contours of maximum principal stress ( Figure 8-36)show that the very localized stress in that corner has been reduced only slightly.

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Figure 8-35 Displaced shape of the modified mesh.

Figure 8-36 Contours of maximum principal stress in the modified mesh.

Mesh design for large-distortion problems is more difficult than it is for small-displacement problems.A mesh must be produced where the shape of the elements is reasonable throughout the analysis, notjust at the start. You must estimate how the model will deform using experience, hand calculations, orthe results from a coarse finite element model.

8.8 Techniques for reducing volumetric lockingTwo techniques are employed to eliminate volumetric locking in this problem. The first involvesrefining the mesh in both corners at the bottom of the rubber model to reduce mesh distortion in theseareas. The second introduces a small amount of compressibility into the rubber material model.Provided the amount of compressibility is small, the results obtained with a nearly incompressiblematerial will be very similar to those obtained with an incompressible material; the presence ofcompressibility alleviates the volumetric locking.

Compressibility is introduced by setting the material constant D1 to a nonzero value. The value ischosen so that the initial Poisson's ratio, º0 , is close to 0.5. This can be verified using the equations


Figure 8-36 Contours of maximum principal stress in the modified mesh.

Mesh design for large-distortion problems is more difficult than it is for small-displacement problems.A mesh must be produced where the shape of the elements is reasonable throughout the analysis, notjust at the start. You must estimate how the model will deform using experience, hand calculations, orthe results from a coarse finite element model.

8.8 Techniques for reducing volumetric lockingTwo techniques are employed to eliminate volumetric locking in this problem. The first involvesrefining the mesh in both corners at the bottom of the rubber model to reduce mesh distortion in theseareas. The second introduces a small amount of compressibility into the rubber material model.Provided the amount of compressibility is small, the results obtained with a nearly incompressiblematerial will be very similar to those obtained with an incompressible material; the presence ofcompressibility alleviates the volumetric locking.

Compressibility is introduced by setting the material constant D1 to a nonzero value. The value ischosen so that the initial Poisson's ratio, º0 , is close to 0.5. This can be verified using the equations

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given in ``Hyperelastic behavior,'' Section 10.5.1 of the ABAQUS/Standard User's Manual, for ¹0 andK0 (the initial shear and bulk moduli, respectively) for the polynomial form of the strain energypotential. For example, using the hyperelastic material coefficients obtained earlier from the test data(see ``Results,'' Section 8.6.8), C10 = 176051 and C01 = 4332.63; setting D1 = 5.E-7 yields º0 =

0.46.

A model that incorporates the above features is shown in Figure 8-37 (the input file for this model isnot included in the appendix; however, it can be generated easily using ABAQUS/CAE).

Figure 8-37 Modified mesh with refinement at both corners.

The displaced shape associated with this model is shown in Figure 8-38. It is clear from this figure thatthe mesh distortion has been reduced significantly in the critical regions of the rubber model and thatthe volumetric locking has been eliminated.


8.9 Related ABAQUS examples� ``Pressurized rubber disc,'' Section 1.1.7 of the ABAQUS Benchmarks Manual

given in ``Hyperelastic behavior,'' Section 10.5.1 of the ABAQUS/Standard User's Manual, for ¹0 andK0 (the initial shear and bulk moduli, respectively) for the polynomial form of the strain energypotential. For example, using the hyperelastic material coefficients obtained earlier from the test data(see ``Results,'' Section 8.6.8), C10 = 176051 and C01 = 4332.63; setting D1 = 5.E-7 yields º0 =

0.46.

A model that incorporates the above features is shown in Figure 8-37 (the input file for this model isnot included in the appendix; however, it can be generated easily using ABAQUS/CAE).

Figure 8-37 Modified mesh with refinement at both corners.

The displaced shape associated with this model is shown in Figure 8-38. It is clear from this figure thatthe mesh distortion has been reduced significantly in the critical regions of the rubber model and thatthe volumetric locking has been eliminated.


8.9 Related ABAQUS examples� ``Pressurized rubber disc,'' Section 1.1.7 of the ABAQUS Benchmarks Manual

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� ``Necking of a round tensile bar,'' Section 1.1.9 of the ABAQUS Benchmarks Manual

� ``Fitting of rubber test data,'' Section 3.1.4 of the ABAQUS Benchmarks Manual

� ``Uniformly loaded, elastic-plastic plate,'' Section 3.2.1 of the ABAQUS Benchmarks Manual

8.10 Suggested readingThe following provides the interested user with additional references on material modeling.

General texts on materials

� M. F. Ashby and D. R. H. Jones, Engineering Materials, Pergamon Press, 1980.

� W. D. Callister, Materials Science & Engineering--An Introduction , John Wiley, 1994.

� K. J. Pascoe, An Introduction to the Properties of Engineering Materials , Van Nostrand,1978.

Plasticity

� Hibbitt, Karlsson & Sorensen, Metal Inelasticity in ABAQUS.

� J. Lubliner, Plasticity Theory, Macmillan Publishing Co., 1990.

� C. R. Calladine, Engineering Plasticity, Pergamon Press, 1969.

Rubber elasticity

� Hibbitt, Karlsson & Sorensen, Modeling Rubber with ABAQUS.

� A. Gent, Engineering with Rubber (How to Design Rubber Components) , HanserPublishers, 1992.

8.11 Summary� ABAQUS contains an extensive library to model the behavior of various engineering materials. It

includes models for metal plasticity and rubber elasticity.

� The stress-strain data for the metal plasticity model must be defined in terms of true stress and trueplastic strain.

� The metal plasticity model assumes incompressible plastic behavior once the material has yielded.This places some limitations on the element types that can be used in elastic-plastic simulations.

� Polynomial, Ogden, Arruda-Boyce, van der Waals, Mooney-Rivlin, neo-Hookean, reducedpolynomial, and Yeoh strain energy functions are available for rubber elasticity (hyperelasticity).All models allow the material coefficients to be determined directly from experimental test data.

� ``Necking of a round tensile bar,'' Section 1.1.9 of the ABAQUS Benchmarks Manual

� ``Fitting of rubber test data,'' Section 3.1.4 of the ABAQUS Benchmarks Manual

� ``Uniformly loaded, elastic-plastic plate,'' Section 3.2.1 of the ABAQUS Benchmarks Manual

8.10 Suggested readingThe following provides the interested user with additional references on material modeling.

General texts on materials

� M. F. Ashby and D. R. H. Jones, Engineering Materials, Pergamon Press, 1980.

� W. D. Callister, Materials Science & Engineering--An Introduction , John Wiley, 1994.

� K. J. Pascoe, An Introduction to the Properties of Engineering Materials , Van Nostrand,1978.

Plasticity

� Hibbitt, Karlsson & Sorensen, Metal Inelasticity in ABAQUS.

� J. Lubliner, Plasticity Theory, Macmillan Publishing Co., 1990.

� C. R. Calladine, Engineering Plasticity, Pergamon Press, 1969.

Rubber elasticity

� Hibbitt, Karlsson & Sorensen, Modeling Rubber with ABAQUS.

� A. Gent, Engineering with Rubber (How to Design Rubber Components) , HanserPublishers, 1992.

8.11 Summary� ABAQUS contains an extensive library to model the behavior of various engineering materials. It

includes models for metal plasticity and rubber elasticity.

� The stress-strain data for the metal plasticity model must be defined in terms of true stress and trueplastic strain.

� The metal plasticity model assumes incompressible plastic behavior once the material has yielded.This places some limitations on the element types that can be used in elastic-plastic simulations.

� Polynomial, Ogden, Arruda-Boyce, van der Waals, Mooney-Rivlin, neo-Hookean, reducedpolynomial, and Yeoh strain energy functions are available for rubber elasticity (hyperelasticity).All models allow the material coefficients to be determined directly from experimental test data.

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The test data must be specified as nominal stress and nominal strain values.

� Stability warnings may indicate that a hyperelastic material model is unsuitable for the strainranges you wish to analyze.

� The presence of symmetry can be used to reduce the size of a simulation since only part of thecomponent needs to be modeled. The effect of the rest of the component is represented by applyingappropriate boundary conditions.

� Mesh design for large-distortion problems is more difficult than for small-displacement problems.The elements in the mesh must not become too distorted at any stage of the analysis.

� Volumetric locking can be alleviated by permitting a small amount of compressibility. Care mustbe taken to ensure that the amount of compressibility introduced into the problem does not grosslyaffect the overall results.

� The X-Y plotting capabilities in ABAQUS/Viewer allow data in curves to be manipulated to createnew curves. Two curves or a curve and a constant can be added, subtracted, multiplied, or divided.Curves can also be differentiated, integrated, and combined.

The test data must be specified as nominal stress and nominal strain values.

� Stability warnings may indicate that a hyperelastic material model is unsuitable for the strainranges you wish to analyze.

� The presence of symmetry can be used to reduce the size of a simulation since only part of thecomponent needs to be modeled. The effect of the rest of the component is represented by applyingappropriate boundary conditions.

� Mesh design for large-distortion problems is more difficult than for small-displacement problems.The elements in the mesh must not become too distorted at any stage of the analysis.

� Volumetric locking can be alleviated by permitting a small amount of compressibility. Care mustbe taken to ensure that the amount of compressibility introduced into the problem does not grosslyaffect the overall results.

� The X-Y plotting capabilities in ABAQUS/Viewer allow data in curves to be manipulated to createnew curves. Two curves or a curve and a constant can be added, subtracted, multiplied, or divided.Curves can also be differentiated, integrated, and combined.

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9. DynamicsA static analysis is sufficient if you are interested in the long-term response of a structure to appliedloads. However, if the duration of the applied load is short, such as in an earthquake, or if the loadingis dynamic in nature, such as that from rotating machinery, you must perform a dynamic analysis.

9.1 IntroductionA dynamic simulation is one in which inertia forces are included in the dynamic equation ofequilibrium:

M Äu + I ¡ P= 0;

where

M

is the mass of the structure,

Äu

is the acceleration of the structure,

I

are the internal forces in the structure, and

P

are the applied external forces.

The expression in the equation shown above is nothing more than Newton's second law of motion(F = ma).

The inclusion of the inertial forces (M Äu) in the equation of equilibrium is the major differencebetween static and dynamic analyses. Another difference between the two types of simulations is in thedefinition of the internal forces, I . In a static analysis the internal forces arise only from thedeformation of the structure; in a dynamic analysis the internal forces contain contributions created byboth the motion (i.e., damping) and the deformation of the structure.

9.1.1 Natural frequencies and mode shapesThe simplest dynamic problem is that of a mass oscillating on a spring, as shown in Figure 9-1.

Figure 9-1 Mass-spring system.

9. DynamicsA static analysis is sufficient if you are interested in the long-term response of a structure to appliedloads. However, if the duration of the applied load is short, such as in an earthquake, or if the loadingis dynamic in nature, such as that from rotating machinery, you must perform a dynamic analysis.

9.1 IntroductionA dynamic simulation is one in which inertia forces are included in the dynamic equation ofequilibrium:

M Äu + I ¡ P= 0;

where

M

is the mass of the structure,

Äu

is the acceleration of the structure,

I

are the internal forces in the structure, and

P

are the applied external forces.

The expression in the equation shown above is nothing more than Newton's second law of motion(F = ma).

The inclusion of the inertial forces (M Äu) in the equation of equilibrium is the major differencebetween static and dynamic analyses. Another difference between the two types of simulations is in thedefinition of the internal forces, I . In a static analysis the internal forces arise only from thedeformation of the structure; in a dynamic analysis the internal forces contain contributions created byboth the motion (i.e., damping) and the deformation of the structure.

9.1.1 Natural frequencies and mode shapesThe simplest dynamic problem is that of a mass oscillating on a spring, as shown in Figure 9-1.

Figure 9-1 Mass-spring system.

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The internal force in the spring is given by ku so that its dynamic equation of motion is

mÄu + ku ¡ p= 0:

This mass-spring system has a natural frequency (in radians/time) given by

! =

rk

m:

If the mass is moved and then released, it will oscillate at this frequency. If the force is applied at thisfrequency, the amplitude of the displacement will increase dramatically--a phenomenon known asresonance.

Real structures have a large number of natural frequencies. Therefore, it is important to designstructures in such a way that the frequencies at which they may be loaded are not close to the naturalfrequencies. The natural frequencies can be determined by considering the dynamic response of theunloaded structure (P = 0 in the dynamic equilibrium equation). The equation of motion is then

M Äu+ I= 0:

For an undamped system I = Ku, so

M Äu +Ku= 0:

Solutions to this equation have the form

u = Áei!t :

Substituting this into the equation of motion yields the eigenvalue problem

KÁ = ¸MÁ;

where ¸ = !2 .

This system has n eigenvalues, where n is the number of degrees of freedom in the finite elementmodel. Let ¸j be the jth eigenvalue. Its square root, !j , is the natural frequency of the jth mode of thestructure, and Áj is the corresponding jth eigenvector. The eigenvector is also known as the modeshape because it is the deformed shape of the structure as it vibrates in the jth mode.

The internal force in the spring is given by ku so that its dynamic equation of motion is

mÄu + ku ¡ p= 0:

This mass-spring system has a natural frequency (in radians/time) given by

! =

rk

m:

If the mass is moved and then released, it will oscillate at this frequency. If the force is applied at thisfrequency, the amplitude of the displacement will increase dramatically--a phenomenon known asresonance.

Real structures have a large number of natural frequencies. Therefore, it is important to designstructures in such a way that the frequencies at which they may be loaded are not close to the naturalfrequencies. The natural frequencies can be determined by considering the dynamic response of theunloaded structure (P = 0 in the dynamic equilibrium equation). The equation of motion is then

M Äu+ I= 0:

For an undamped system I = Ku, so

M Äu +Ku= 0:

Solutions to this equation have the form

u = Áei!t :

Substituting this into the equation of motion yields the eigenvalue problem

KÁ = ¸MÁ;

where ¸ = !2 .

This system has n eigenvalues, where n is the number of degrees of freedom in the finite elementmodel. Let ¸j be the jth eigenvalue. Its square root, !j , is the natural frequency of the jth mode of thestructure, and Áj is the corresponding jth eigenvector. The eigenvector is also known as the modeshape because it is the deformed shape of the structure as it vibrates in the jth mode.

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In ABAQUS the *FREQUENCY procedure is used to extract the modes and frequencies of thestructure. This procedure is easy to use in that you need only specify the number of modes required orthe maximum frequency of interest.

9.1.2 Modal superpositionThe natural frequencies and mode shapes of a structure can be used to characterize its dynamicresponse to loads in the linear regime. The deformation of the structure can be calculated from acombination of the mode shapes of the structure using the modal superposition technique. Each modeshape is multiplied by a scale factor. The vector of displacements in the model, u, is defined as

u =

1Xi=1

®iÁi;

where ®i is the scale factor for mode Ái. This technique is valid only for simulations with smalldisplacements, linear elastic materials, and no contact conditions--in other words, linear problems.

In structural dynamic problems the response of a structure usually is dominated by a relatively smallnumber of modes, making modal superposition a particularly efficient method for calculating theresponse of such systems. Consider a model containing 10,000 degrees of freedom. Direct integrationof the dynamic equations of motion would require the solution of 10,000 simultaneous equations ateach point in time. If the structural response is characterized by 100 modes, only 100 equations need tobe solved every time step. Moreover, the modal equations are uncoupled, whereas the originalequations of motion are coupled. There is an initial cost in calculating the modes and frequencies, butthe savings obtained in the calculation of the response greatly outweighs the cost.

If nonlinearities are present in the simulation, the natural frequencies may change significantly duringthe analysis, and modal superposition cannot be employed. In this case direct integration of thedynamic equation of equilibrium is required, which is much more expensive than modal analysis.

A problem should have the following characteristics for it to be suitable for linear transient dynamicanalysis:

� The system should be linear: linear material behavior, no contact conditions, and no nonlineargeometric effects.

� The response should be dominated by relatively few frequencies. As the frequency content of theresponse increases, such as is the case in shock and impact problems, the modal superpositiontechnique becomes less effective.

� The dominant loading frequencies should be in the range of the extracted frequencies to ensurethat the loads can be described accurately.

� The initial accelerations generated by any suddenly applied loads should be described accuratelyby the eigenmodes.

� The system should not be damped heavily.

In ABAQUS the *FREQUENCY procedure is used to extract the modes and frequencies of thestructure. This procedure is easy to use in that you need only specify the number of modes required orthe maximum frequency of interest.

9.1.2 Modal superpositionThe natural frequencies and mode shapes of a structure can be used to characterize its dynamicresponse to loads in the linear regime. The deformation of the structure can be calculated from acombination of the mode shapes of the structure using the modal superposition technique. Each modeshape is multiplied by a scale factor. The vector of displacements in the model, u, is defined as

u =

1Xi=1

®iÁi;

where ®i is the scale factor for mode Ái. This technique is valid only for simulations with smalldisplacements, linear elastic materials, and no contact conditions--in other words, linear problems.

In structural dynamic problems the response of a structure usually is dominated by a relatively smallnumber of modes, making modal superposition a particularly efficient method for calculating theresponse of such systems. Consider a model containing 10,000 degrees of freedom. Direct integrationof the dynamic equations of motion would require the solution of 10,000 simultaneous equations ateach point in time. If the structural response is characterized by 100 modes, only 100 equations need tobe solved every time step. Moreover, the modal equations are uncoupled, whereas the originalequations of motion are coupled. There is an initial cost in calculating the modes and frequencies, butthe savings obtained in the calculation of the response greatly outweighs the cost.

If nonlinearities are present in the simulation, the natural frequencies may change significantly duringthe analysis, and modal superposition cannot be employed. In this case direct integration of thedynamic equation of equilibrium is required, which is much more expensive than modal analysis.

A problem should have the following characteristics for it to be suitable for linear transient dynamicanalysis:

� The system should be linear: linear material behavior, no contact conditions, and no nonlineargeometric effects.

� The response should be dominated by relatively few frequencies. As the frequency content of theresponse increases, such as is the case in shock and impact problems, the modal superpositiontechnique becomes less effective.

� The dominant loading frequencies should be in the range of the extracted frequencies to ensurethat the loads can be described accurately.

� The initial accelerations generated by any suddenly applied loads should be described accuratelyby the eigenmodes.

� The system should not be damped heavily.

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9.2 DampingIf an undamped structure is allowed to vibrate freely, the magnitude of the oscillation is constant. Inreality, however, energy is dissipated by the structure's motion, and the magnitude of the oscillationdecreases until the oscillation stops. This energy dissipation is known as damping. Damping is usuallyassumed to be viscous or proportional to velocity. The dynamic equilibrium equation can be rewrittento include damping as

M Äu + I ¡ P= 0;

I = Ku + C _u;

where

C

is the damping matrix for the structure and

_u

is the velocity of the structure.

The dissipation of energy is caused by a number of effects, including friction at the joints of thestructure and localized material hysteresis. Damping is a convenient way of including the importantabsorption of energy without modeling the effects in detail.

In ABAQUS the eigenmodes are calculated for the undamped system, yet most engineering problemsinvolve some kind of damping, however small. The relationship between the damped natural frequencyand the undamped natural frequency for each mode is

!d = !p

1¡ »2 ;

where!d

is the damped eigenvalue,

» = cco

is the fraction of critical damping,c

is the damping of that mode shape, andco

is the critical damping.

The eigenfrequencies of the damped system are very close to the corresponding quantities for theundamped system for small values of » (» < 0.1). As » increases, the undamped eigenfrequencies

9.2 DampingIf an undamped structure is allowed to vibrate freely, the magnitude of the oscillation is constant. Inreality, however, energy is dissipated by the structure's motion, and the magnitude of the oscillationdecreases until the oscillation stops. This energy dissipation is known as damping. Damping is usuallyassumed to be viscous or proportional to velocity. The dynamic equilibrium equation can be rewrittento include damping as

M Äu + I ¡ P= 0;

I = Ku + C _u;

where

C

is the damping matrix for the structure and

_u

is the velocity of the structure.

The dissipation of energy is caused by a number of effects, including friction at the joints of thestructure and localized material hysteresis. Damping is a convenient way of including the importantabsorption of energy without modeling the effects in detail.

In ABAQUS the eigenmodes are calculated for the undamped system, yet most engineering problemsinvolve some kind of damping, however small. The relationship between the damped natural frequencyand the undamped natural frequency for each mode is

!d = !p

1¡ »2 ;

where!d

is the damped eigenvalue,

» = cco

is the fraction of critical damping,c

is the damping of that mode shape, andco

is the critical damping.

The eigenfrequencies of the damped system are very close to the corresponding quantities for theundamped system for small values of » (» < 0.1). As » increases, the undamped eigenfrequencies

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become less accurate; and as » approaches 1, the use of undamped eigenfrequencies becomes invalid.

If a structure is critically damped (» = 1), after any disturbance it will return to its initial staticconfiguration as quickly as possible without overshooting ( Figure 9-2).

Figure 9-2 Damping.

9.2.1 Definition of damping in ABAQUSIn ABAQUS a number of different types of damping can be defined for a transient modal analysis:direct modal damping, Rayleigh damping, and composite modal damping.

Damping is defined for modal dynamic procedures by using the *MODAL DAMPING option. Thisoption is part of the step definition and allows different amounts of damping to be defined for eachmode. Direct, Rayleigh, and composite damping can all be defined this way.

Direct modal damping

The fraction of critical damping, », associated with each mode can be defined using directmodal damping. Typically, values in the range of 1% to 10% of critical damping are used.Direct modal damping allows you to define precisely the damping of each mode of the system.

The MODAL=DIRECT parameter on the *MODAL DAMPING option indicates that directmodal damping is being specified. For example, to define 4% of critical modal damping forthe first 10 modes and 5% for modes 11-20, include the following in the step definition:

*MODAL DAMPING, MODAL=DIRECT

1, 10, 0.04

11, 20, 0.05

Rayleigh damping

become less accurate; and as » approaches 1, the use of undamped eigenfrequencies becomes invalid.

If a structure is critically damped (» = 1), after any disturbance it will return to its initial staticconfiguration as quickly as possible without overshooting ( Figure 9-2).

Figure 9-2 Damping.

9.2.1 Definition of damping in ABAQUSIn ABAQUS a number of different types of damping can be defined for a transient modal analysis:direct modal damping, Rayleigh damping, and composite modal damping.

Damping is defined for modal dynamic procedures by using the *MODAL DAMPING option. Thisoption is part of the step definition and allows different amounts of damping to be defined for eachmode. Direct, Rayleigh, and composite damping can all be defined this way.

Direct modal damping

The fraction of critical damping, », associated with each mode can be defined using directmodal damping. Typically, values in the range of 1% to 10% of critical damping are used.Direct modal damping allows you to define precisely the damping of each mode of the system.

The MODAL=DIRECT parameter on the *MODAL DAMPING option indicates that directmodal damping is being specified. For example, to define 4% of critical modal damping forthe first 10 modes and 5% for modes 11-20, include the following in the step definition:


1, 10, 0.04

11, 20, 0.05

Rayleigh damping

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In Rayleigh damping the assumption is made that the damping matrix is a linear combinationof the mass and stiffness matrices,

C = ®M + ¯K;

where ® and ¯ are user-defined constants. Although the assumption that the damping isproportional to the mass and stiffness matrices has no rigorous physical basis, in practice thedamping distribution rarely is known in sufficient detail to warrant any other morecomplicated model. In general, this model ceases to be reliable for heavily damped systems;that is, above approximately 10% of critical damping.

The RAYLEIGH parameter on the *MODAL DAMPING option indicates that Rayleighdamping is to be used. For example, to define ® = 0.2525 and ¯ = 2.9 � 10-3 for modes 1-10and ® = 0.2727 and ¯ = 3.03 � 10-3 for modes 11-20, the following lines would be included inthe step definition:

*MODAL DAMPING, RAYLEIGH

1, 10, 0.2525, 2.9E-3

11, 20, 0.2727, 3.03E-3

Composite damping

In composite damping a fraction of critical damping is defined for each material, and acomposite damping value is found for the whole structure. This option is useful when manydifferent materials are present in the structure. Composite damping is not discussed further inthis guide.

9.2.2 Choosing damping valuesIn most linear dynamic problems the proper specification of damping is important to obtain accurateresults. However, damping is approximate in the sense that it models the energy absorbingcharacteristics of the structure without attempting to model the physical mechanisms that cause them.Therefore, it is difficult to determine the damping data required for a simulation. Occasionally, youmay have data available from dynamic tests, but often you will have to work with data gleaned fromreferences or experience. In such cases you should be very cautious in interpreting the results, and youshould use parametric studies to assess the sensitivity of the simulation to damping values.

9.3 Element selectionVirtually all of the elements in ABAQUS can be used in dynamic analyses. In general the rules forselecting the elements are the same as those for static simulations. However, for simulations of impactand blast loading, first-order elements should be used. They have a lumped mass formulation, which isbetter able to model the effect of stress waves than the consistent mass formulation used in thesecond-order elements.

9.4 Mesh design for dynamics

In Rayleigh damping the assumption is made that the damping matrix is a linear combinationof the mass and stiffness matrices,

C = ®M + ¯K;

where ® and ¯ are user-defined constants. Although the assumption that the damping isproportional to the mass and stiffness matrices has no rigorous physical basis, in practice thedamping distribution rarely is known in sufficient detail to warrant any other morecomplicated model. In general, this model ceases to be reliable for heavily damped systems;that is, above approximately 10% of critical damping.

The RAYLEIGH parameter on the *MODAL DAMPING option indicates that Rayleighdamping is to be used. For example, to define ® = 0.2525 and ¯ = 2.9 � 10-3 for modes 1-10and ® = 0.2727 and ¯ = 3.03 � 10-3 for modes 11-20, the following lines would be included inthe step definition:

*MODAL DAMPING, RAYLEIGH

1, 10, 0.2525, 2.9E-3

11, 20, 0.2727, 3.03E-3

Composite damping

In composite damping a fraction of critical damping is defined for each material, and acomposite damping value is found for the whole structure. This option is useful when manydifferent materials are present in the structure. Composite damping is not discussed further inthis guide.

9.2.2 Choosing damping valuesIn most linear dynamic problems the proper specification of damping is important to obtain accurateresults. However, damping is approximate in the sense that it models the energy absorbingcharacteristics of the structure without attempting to model the physical mechanisms that cause them.Therefore, it is difficult to determine the damping data required for a simulation. Occasionally, youmay have data available from dynamic tests, but often you will have to work with data gleaned fromreferences or experience. In such cases you should be very cautious in interpreting the results, and youshould use parametric studies to assess the sensitivity of the simulation to damping values.

9.3 Element selectionVirtually all of the elements in ABAQUS can be used in dynamic analyses. In general the rules forselecting the elements are the same as those for static simulations. However, for simulations of impactand blast loading, first-order elements should be used. They have a lumped mass formulation, which isbetter able to model the effect of stress waves than the consistent mass formulation used in thesecond-order elements.

9.4 Mesh design for dynamics

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When you are designing meshes for dynamic simulations, you need to consider the mode shapes thatwill be excited in the response and use a mesh that is able to represent those mode shapes adequately.This means that a mesh that is adequate for a static simulation may be unsuitable for calculating thedynamic response to loading that excites high frequency modes.

Consider, for example, the plate shown in Figure 9-3. The mesh of first-order shell elements isadequate for a static analysis of the plate under a uniform load and is also suitable for the prediction ofthe first mode shape. However, the mesh is clearly too coarse to be able to model the sixth modeaccurately.

Figure 9-3 Coarse mesh of plate.

Figure 9-4 shows the same plate modeled with a refined mesh of first-order elements. The displacedshape for the sixth mode now looks much better, and the frequency predicted for this mode is moreaccurate. If the dynamic loading on the plate is such that there is significant excitation of this mode,the refined mesh must be used; the results from the coarse mesh will not be accurate.

Figure 9-4 Fine mesh of plate.

9.5 Example: cargo crane--dynamic loadingThis example uses the same cargo crane that you analyzed in ``Example: cargo crane,'' Section 6.4, butyou have now been asked to investigate what happens when a load of 10 kN is dropped onto the liftinghook for 0.2 seconds. The connections at points A, B, C, and D (see Figure 9-11) can only withstand amaximum pull-out force of 100 kN. You have to decide whether or not any of these connections willbreak.

Figure 9-5 Cargo crane.

When you are designing meshes for dynamic simulations, you need to consider the mode shapes thatwill be excited in the response and use a mesh that is able to represent those mode shapes adequately.This means that a mesh that is adequate for a static simulation may be unsuitable for calculating thedynamic response to loading that excites high frequency modes.

Consider, for example, the plate shown in Figure 9-3. The mesh of first-order shell elements isadequate for a static analysis of the plate under a uniform load and is also suitable for the prediction ofthe first mode shape. However, the mesh is clearly too coarse to be able to model the sixth modeaccurately.

Figure 9-3 Coarse mesh of plate.

Figure 9-4 shows the same plate modeled with a refined mesh of first-order elements. The displacedshape for the sixth mode now looks much better, and the frequency predicted for this mode is moreaccurate. If the dynamic loading on the plate is such that there is significant excitation of this mode,the refined mesh must be used; the results from the coarse mesh will not be accurate.

Figure 9-4 Fine mesh of plate.

9.5 Example: cargo crane--dynamic loadingThis example uses the same cargo crane that you analyzed in ``Example: cargo crane,'' Section 6.4, butyou have now been asked to investigate what happens when a load of 10 kN is dropped onto the liftinghook for 0.2 seconds. The connections at points A, B, C, and D (see Figure 9-11) can only withstand amaximum pull-out force of 100 kN. You have to decide whether or not any of these connections willbreak.

Figure 9-5 Cargo crane.

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The short duration of the loading means that inertia effects are likely to be important, making dynamicanalysis essential. You are not given any information regarding the damping of the structure. Sincethere are bolted connections between the trusses and the cross bracing, the energy absorption causedby frictional effects is likely to be significant. Based on experience, you therefore choose 5% of criticaldamping in each mode.

The magnitude of the applied load versus time is shown in Figure 9-6.

Figure 9-6 Load-time characteristic.

9.5.1 Modifications to the input file--the model dataCopy the file crane.inp to dynamics.inp. The model data are the same as for the static analysiswith the modifications described below. These modifications are most easily made using an editor,although you may change the model in a preprocessor if you prefer.

The short duration of the loading means that inertia effects are likely to be important, making dynamicanalysis essential. You are not given any information regarding the damping of the structure. Sincethere are bolted connections between the trusses and the cross bracing, the energy absorption causedby frictional effects is likely to be significant. Based on experience, you therefore choose 5% of criticaldamping in each mode.

The magnitude of the applied load versus time is shown in Figure 9-6.

Figure 9-6 Load-time characteristic.

9.5.1 Modifications to the input file--the model dataCopy the file crane.inp to dynamics.inp. The model data are the same as for the static analysiswith the modifications described below. These modifications are most easily made using an editor,although you may change the model in a preprocessor if you prefer.

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Material

In dynamic simulations the density of every material must be specified so that the mass matrixcan be formed. The steel in the crane has a density of 7800 kg/m3. The beam elementproperties were defined using the *BEAM GENERAL SECTION option, so there are nomaterial property definitions in this input file. You need to specify the density using theDENSITY parameter on the *BEAM GENERAL SECTION option. For example,

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=OUTA, DENSITY=7800.

0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9

Add the DENSITY parameter to all the element property options.

If material data are defined using the *MATERIAL option, the density is included by using the*DENSITY option and giving the mass density on the data line. For example,


*ELASTIC

<Young's modulus>,<Poisson's ratio>

*DENSITY

<density>,

Initial conditions

In this example the structure has no initial velocities or accelerations, which is the default.However, if you wanted to define initial velocities, you could do so using the followingoption:

*INITIAL CONDITIONS, TYPE=VELOCITY

The nodes (or node sets), the direction, and the magnitude of the velocity are specified on thedata line, as follows:

<node or node set>,<dof>, <velocity>

For example:


NALL, 1, 10.0

would set the velocity in the 1-direction of all the nodes in node set NALL to 10 m/s.

Time variation of load

The magnitude of the concentrated load applied to the tip of the crane is time dependent asillustrated in Figure 9-6. The time dependence of a load is defined using the *AMPLITUDEoption. The *AMPLITUDE option must appear as part of the model data, even though the*CLOAD option referring to it is part of the history data.

Four pairs of time and magnitude data are given on each data line for the *AMPLITUDE

Material

In dynamic simulations the density of every material must be specified so that the mass matrixcan be formed. The steel in the crane has a density of 7800 kg/m3. The beam elementproperties were defined using the *BEAM GENERAL SECTION option, so there are nomaterial property definitions in this input file. You need to specify the density using theDENSITY parameter on the *BEAM GENERAL SECTION option. For example,


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9

Add the DENSITY parameter to all the element property options.

If material data are defined using the *MATERIAL option, the density is included by using the*DENSITY option and giving the mass density on the data line. For example,


*ELASTIC

<Young's modulus>,<Poisson's ratio>

*DENSITY

<density>,

Initial conditions

In this example the structure has no initial velocities or accelerations, which is the default.However, if you wanted to define initial velocities, you could do so using the followingoption:


The nodes (or node sets), the direction, and the magnitude of the velocity are specified on thedata line, as follows:

<node or node set>,<dof>, <velocity>

For example:


NALL, 1, 10.0

would set the velocity in the 1-direction of all the nodes in node set NALL to 10 m/s.

Time variation of load

The magnitude of the concentrated load applied to the tip of the crane is time dependent asillustrated in Figure 9-6. The time dependence of a load is defined using the *AMPLITUDEoption. The *AMPLITUDE option must appear as part of the model data, even though the*CLOAD option referring to it is part of the history data.

Four pairs of time and magnitude data are given on each data line for the *AMPLITUDE

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option, and a name is assigned to the amplitude curve using the NAME parameter. For yoursimulation the option block defining the amplitude curve should look similar to the following:

*AMPLITUDE, NAME=BOUNCE, VALUE=RELATIVE

0.0,0.0, 0.01,1.0, 0.2,1.0, 0.21,0.0

The name of the curve, BOUNCE, will be used to refer the loading option (*CLOAD) to thisamplitude curve. The actual load applied will be the product of the magnitude on the loadingoption and the amplitude on the BOUNCE curve. The parameter VALUE=RELATIVE is usedto indicate this approach. You can choose to define the absolute magnitude of the loading onthe *AMPLITUDE option by using VALUE=ABSOLUTE.

9.5.2 Modifications to the input file--the history dataThe history definition is substantially different from that in the static analysis. Therefore, delete theentire static step, and add a new history section as discussed below.

Two steps are required for this analysis. The first step calculates the natural frequencies and modeshapes of the structure. The second step then uses these data to calculate the transient dynamicresponse of the hoist. If you want to model any nonlinearities in this simulation, you must use the*DYNAMIC procedure. In this analysis we will assume that everything is linear.

Step 1 - Modes and frequencies

The *FREQUENCY procedure is used to calculate natural frequencies and mode shapes.ABAQUS offers the Lanczos and the subspace iteration eigenvalue extraction methods. TheLanczos method is generally faster when a large number of eigenmodes is required for asystem with many degrees of freedom. The subspace iteration method may be faster when onlya few (less than 20) eigenmodes are needed.

We use the Lanczos eigensolver in this analysis. The number of modes required is specified onthe data line of the *FREQUENCY option. Alternatively, it is possible to specify the minimumand maximum frequencies of interest, so that the step will complete once ABAQUS has foundall of the eigenvalues inside the specified range. A shift point may also be specified so thateigenvalues nearest the shift point will be extracted. By default, no minimum or maximumfrequency or shift is used. If the structure is not constrained completely, the shift value shouldbe set to a small negative value to remove numerical problems associated with rigid bodymotion.

The form of the *FREQUENCY option block when the Lanczos eigensolver is used is

*FREQUENCY, EIGENSOLVER=LANCZOS

<number of eigenvalues>,< min. frequency>,< max. frequency>,<shift

The step and procedure option blocks for this simulation are

*STEP, PERTURBATION

Frequency extraction of the first 30 modes.


option, and a name is assigned to the amplitude curve using the NAME parameter. For yoursimulation the option block defining the amplitude curve should look similar to the following:


0.0,0.0, 0.01,1.0, 0.2,1.0, 0.21,0.0

The name of the curve, BOUNCE, will be used to refer the loading option (*CLOAD) to thisamplitude curve. The actual load applied will be the product of the magnitude on the loadingoption and the amplitude on the BOUNCE curve. The parameter VALUE=RELATIVE is usedto indicate this approach. You can choose to define the absolute magnitude of the loading onthe *AMPLITUDE option by using VALUE=ABSOLUTE.

9.5.2 Modifications to the input file--the history dataThe history definition is substantially different from that in the static analysis. Therefore, delete theentire static step, and add a new history section as discussed below.

Two steps are required for this analysis. The first step calculates the natural frequencies and modeshapes of the structure. The second step then uses these data to calculate the transient dynamicresponse of the hoist. If you want to model any nonlinearities in this simulation, you must use the*DYNAMIC procedure. In this analysis we will assume that everything is linear.

Step 1 - Modes and frequencies

The *FREQUENCY procedure is used to calculate natural frequencies and mode shapes.ABAQUS offers the Lanczos and the subspace iteration eigenvalue extraction methods. TheLanczos method is generally faster when a large number of eigenmodes is required for asystem with many degrees of freedom. The subspace iteration method may be faster when onlya few (less than 20) eigenmodes are needed.

We use the Lanczos eigensolver in this analysis. The number of modes required is specified onthe data line of the *FREQUENCY option. Alternatively, it is possible to specify the minimumand maximum frequencies of interest, so that the step will complete once ABAQUS has foundall of the eigenvalues inside the specified range. A shift point may also be specified so thateigenvalues nearest the shift point will be extracted. By default, no minimum or maximumfrequency or shift is used. If the structure is not constrained completely, the shift value shouldbe set to a small negative value to remove numerical problems associated with rigid bodymotion.

The form of the *FREQUENCY option block when the Lanczos eigensolver is used is


<number of eigenvalues>,< min. frequency>,< max. frequency>,<shift

The step and procedure option blocks for this simulation are

*STEP, PERTURBATION

Frequency extraction of the first 30 modes.


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30,

In structural dynamic analysis the response is usually associated with the lower modes.However, enough modes should be extracted to provide a good representation of the dynamicresponse of the structure. One way of checking that a sufficient number of eigenvalues hasbeen extracted is to look at the total effective mass in each degree of freedom, which indicateshow much of the mass is active in each direction of the extracted modes. The effective massesare tabulated in the data file under the eigenvalue output. Ideally, the sum of the modaleffective masses for each mode in each direction should be at least 90% of the total mass. Thisis discussed further in ``Effect of the number of modes,'' Section 9.6.

Boundary conditions

The boundary conditions are the same as in the static analysis.

*BOUNDARY

ATTACH, ENCASTRE

Output

By default, ABAQUS writes the mode shapes to the output database (.odb) file so that theycan be plotted using ABAQUS/Viewer. The nodal displacements for each mode shape arenormalized so that the maximum displacement is unity. Therefore, these results, and thecorresponding stresses and strains, are not physically meaningful: they should be used only forrelative comparisons.

In this analysis we do not require any nodal or elemental output to be written to the data file.Therefore, set FREQUENCY=0 as follows:



Terminate the step with

*END STEP

Step 2 - Transient dynamics

The *MODAL DYNAMIC procedure is used for transient modal dynamic analysis. The fixedtime increment and the total step time are given on the data line for this option. The total timeof the simulation is 0.5 seconds with a constant increment of 0.005 seconds. The format of thisdata line is basically the same as that for *STATIC. However, in this case we must be carefulto ensure that we give real values of time; in dynamic analysis time is a real, physicalquantity.

The form of the *STEP and *MODAL DYNAMIC option blocks for this simulation should be

*STEP, PERTURBATION

Simulation of Load Dropped on Crane

*MODAL DYNAMIC

30,

In structural dynamic analysis the response is usually associated with the lower modes.However, enough modes should be extracted to provide a good representation of the dynamicresponse of the structure. One way of checking that a sufficient number of eigenvalues hasbeen extracted is to look at the total effective mass in each degree of freedom, which indicateshow much of the mass is active in each direction of the extracted modes. The effective massesare tabulated in the data file under the eigenvalue output. Ideally, the sum of the modaleffective masses for each mode in each direction should be at least 90% of the total mass. Thisis discussed further in ``Effect of the number of modes,'' Section 9.6.

Boundary conditions

The boundary conditions are the same as in the static analysis.

*BOUNDARY

ATTACH, ENCASTRE

Output

By default, ABAQUS writes the mode shapes to the output database (.odb) file so that theycan be plotted using ABAQUS/Viewer. The nodal displacements for each mode shape arenormalized so that the maximum displacement is unity. Therefore, these results, and thecorresponding stresses and strains, are not physically meaningful: they should be used only forrelative comparisons.

In this analysis we do not require any nodal or elemental output to be written to the data file.Therefore, set FREQUENCY=0 as follows:



Terminate the step with

*END STEP

Step 2 - Transient dynamics

The *MODAL DYNAMIC procedure is used for transient modal dynamic analysis. The fixedtime increment and the total step time are given on the data line for this option. The total timeof the simulation is 0.5 seconds with a constant increment of 0.005 seconds. The format of thisdata line is basically the same as that for *STATIC. However, in this case we must be carefulto ensure that we give real values of time; in dynamic analysis time is a real, physicalquantity.

The form of the *STEP and *MODAL DYNAMIC option blocks for this simulation should be

*STEP, PERTURBATION

Simulation of Load Dropped on Crane

*MODAL DYNAMIC

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0.005, 0.5

Damping

5% of critical damping should be used in all 30 modes extracted in the first step. Specify thisinput using the *MODAL DAMPING option block, as follows:


1, 30, 0.05

Loading

Apply the concentrated point load to the tip of the crane at node 104 in the negative global2-direction. The *EQUATION constraint between nodes 104 and 204 in degree of freedom 2means that the load will be carried equally by both nodes and, thus, by both halves of thecrane. The concentrated load is defined using the *CLOAD option. Use the parameterAMPLITUDE=BOUNCE to indicate that the amplitude curve named BOUNCE (previouslydefined as part of the model data) should be used to define the time varying magnitude of theload during the step. Your input should look similar to the following:

*CLOAD, AMPLITUDE=BOUNCE

104, 2, -1.0E4

The actual magnitude of the load applied at any point in time is obtained by multiplying themagnitude given on the *CLOAD option (-10,000 N) and the value of the BOUNCE amplitudecurve at that time.

Boundary conditions

The same boundary conditions that were applied in Step 1 are still in effect for this step. Sincethe boundary conditions cannot be changed between a *FREQUENCY step and anysubsequent modal dynamic steps, no boundary conditions should be specified.

Output

Dynamic analyses usually require many more increments than static analyses to complete. As aconsequence, the volume of output from dynamic analyses can be very large, and you shouldcontrol the output requests to keep the results files to a reasonable size.

You can estimate the size of the restart file using the approximate sizes given near the bottomof the data file during a datacheck analysis.

In this example request output of the deformed shape to the output database file at the end ofevery fifth increment. There will be 100 increments (0.5/0.005); therefore, there will be 20frames of output if you use the following:


The displacements of the loaded nodes, which are grouped into a node set named TIP, and thereaction forces at the fixed nodes, which are grouped into a node set named ATTACH, arewritten as history data to the output database file every increment so that a higher resolution of

0.005, 0.5

Damping

5% of critical damping should be used in all 30 modes extracted in the first step. Specify thisinput using the *MODAL DAMPING option block, as follows:


1, 30, 0.05

Loading

Apply the concentrated point load to the tip of the crane at node 104 in the negative global2-direction. The *EQUATION constraint between nodes 104 and 204 in degree of freedom 2means that the load will be carried equally by both nodes and, thus, by both halves of thecrane. The concentrated load is defined using the *CLOAD option. Use the parameterAMPLITUDE=BOUNCE to indicate that the amplitude curve named BOUNCE (previouslydefined as part of the model data) should be used to define the time varying magnitude of theload during the step. Your input should look similar to the following:


104, 2, -1.0E4

The actual magnitude of the load applied at any point in time is obtained by multiplying themagnitude given on the *CLOAD option (-10,000 N) and the value of the BOUNCE amplitudecurve at that time.

Boundary conditions

The same boundary conditions that were applied in Step 1 are still in effect for this step. Sincethe boundary conditions cannot be changed between a *FREQUENCY step and anysubsequent modal dynamic steps, no boundary conditions should be specified.

Output

Dynamic analyses usually require many more increments than static analyses to complete. As aconsequence, the volume of output from dynamic analyses can be very large, and you shouldcontrol the output requests to keep the results files to a reasonable size.

You can estimate the size of the restart file using the approximate sizes given near the bottomof the data file during a datacheck analysis.

In this example request output of the deformed shape to the output database file at the end ofevery fifth increment. There will be 100 increments (0.5/0.005); therefore, there will be 20frames of output if you use the following:


The displacements of the loaded nodes, which are grouped into a node set named TIP, and thereaction forces at the fixed nodes, which are grouped into a node set named ATTACH, arewritten as history data to the output database file every increment so that a higher resolution of

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these data will be available. In dynamic analyses we are also concerned about the energydistribution in the model and what form the energy takes. Kinetic energy is present in themodel as a result of the motion of the mass; strain energy is present as a result of thedisplacement of the structure; energy is also dissipated through damping. We can output thekinetic energy (ALLKE), strain energy (ALLSE), energy dissipated through damping(ALLVD), external work on the entire model (ALLWK), and the total energy balance in themodel (ETOTAL). The history portion of the output request is written as follows:

*NSET, NSET=TIP

104, 204


*NODE OUTPUT, NSET=TIP

U,

*NODE OUTPUT, NSET=ATTACH

RF,

*ENERGY OUTPUT

ALLKE, ALLSE, ALLVD, ALLWK, ETOTAL

No results should be printed in the data file.



Complete the input of the step.

*END STEP

9.5.3 Running the analysisThe input file is called dynamics.inp (an example is listed in ``Cargo crane - dynamic loading,''Section A.9). Use the following command to run the analysis in the background:

abaqus job=dynamics

9.5.4 ResultsExamine the status (.sta) file and printed output data (.dat) file to evaluate the analysis results.

Status file

Looking at the contents of the status file, dynamics.sta, we can see that the time incrementassociated with the single increment in Step 1 is very small. A *FREQUENCY step uses notime, because time is not relevant in a frequency extraction step. The contents of the status fileare shown below.




ITERS FREQ

these data will be available. In dynamic analyses we are also concerned about the energydistribution in the model and what form the energy takes. Kinetic energy is present in themodel as a result of the motion of the mass; strain energy is present as a result of thedisplacement of the structure; energy is also dissipated through damping. We can output thekinetic energy (ALLKE), strain energy (ALLSE), energy dissipated through damping(ALLVD), external work on the entire model (ALLWK), and the total energy balance in themodel (ETOTAL). The history portion of the output request is written as follows:

*NSET, NSET=TIP

104, 204



U,


RF,

*ENERGY OUTPUT


No results should be printed in the data file.



Complete the input of the step.

*END STEP

9.5.3 Running the analysisThe input file is called dynamics.inp (an example is listed in ``Cargo crane - dynamic loading,''Section A.9). Use the following command to run the analysis in the background:

abaqus job=dynamics

9.5.4 ResultsExamine the status (.sta) file and printed output data (.dat) file to evaluate the analysis results.

Status file

Looking at the contents of the status file, dynamics.sta, we can see that the time incrementassociated with the single increment in Step 1 is very small. A *FREQUENCY step uses notime, because time is not relevant in a frequency extraction step. The contents of the status fileare shown below.




ITERS FREQ

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1 1 1 0 1 0 0.00 1.00e-36 1.000e-36

2 1 1 0 1 0 0.00 0.00500 0.005000

2 2 1 0 1 0 0.00 0.0100 0.005000

2 3 1 0 1 0 0.00 0.0150 0.005000

2 4 1 0 1 0 0.00 0.0200 0.005000

2 5 1 0 1 0 0.00 0.0250 0.005000

2 6 1 0 1 0 0.00 0.0300 0.005000

....

2 94 1 0 1 0 0.00 0.470 0.005000

2 95 1 0 1 0 0.00 0.475 0.005000

2 96 1 0 1 0 0.00 0.480 0.005000

2 97 1 0 1 0 0.00 0.485 0.005000

2 98 1 0 1 0 0.00 0.490 0.005000

2 99 1 0 1 0 0.00 0.495 0.005000

2 100 1 0 1 0 0.00 0.500 0.005000

Ten iterations are required to obtain 30 eigenvalues in Step 1. Generally, extracting a largenumber of eigenvalues requires many iterations and, therefore, increases the computation time.

The output in the status file for Step 2 shows that the time increment size is constantthroughout the step and that each increment requires only one iteration. Since modal dynamicanalysis involves the linear superposition of the mode shapes, no iterating is required. For thesame reason, the message file contains no information about equilibrium or residuals.

Data file

The primary results for Step 1 are the extracted eigenvalues, participation factors, andeffective mass, as shown below.

The highest frequency extracted is 89 Hz. The period associated with this frequency is 0.0112seconds, which is comparable to the fixed time increment of 0.005 seconds. There is no pointin extracting modes whose period is substantially smaller than the time increment used.Conversely, the time increment must be capable of resolving the highest frequencies ofinterest.

1 1 1 0 1 0 0.00 1.00e-36 1.000e-36

2 1 1 0 1 0 0.00 0.00500 0.005000

2 2 1 0 1 0 0.00 0.0100 0.005000

2 3 1 0 1 0 0.00 0.0150 0.005000

2 4 1 0 1 0 0.00 0.0200 0.005000

2 5 1 0 1 0 0.00 0.0250 0.005000

2 6 1 0 1 0 0.00 0.0300 0.005000

....

2 94 1 0 1 0 0.00 0.470 0.005000

2 95 1 0 1 0 0.00 0.475 0.005000

2 96 1 0 1 0 0.00 0.480 0.005000

2 97 1 0 1 0 0.00 0.485 0.005000

2 98 1 0 1 0 0.00 0.490 0.005000

2 99 1 0 1 0 0.00 0.495 0.005000

2 100 1 0 1 0 0.00 0.500 0.005000

Ten iterations are required to obtain 30 eigenvalues in Step 1. Generally, extracting a largenumber of eigenvalues requires many iterations and, therefore, increases the computation time.

The output in the status file for Step 2 shows that the time increment size is constantthroughout the step and that each increment requires only one iteration. Since modal dynamicanalysis involves the linear superposition of the mode shapes, no iterating is required. For thesame reason, the message file contains no information about equilibrium or residuals.

Data file

The primary results for Step 1 are the extracted eigenvalues, participation factors, andeffective mass, as shown below.

The highest frequency extracted is 89 Hz. The period associated with this frequency is 0.0112seconds, which is comparable to the fixed time increment of 0.005 seconds. There is no pointin extracting modes whose period is substantially smaller than the time increment used.Conversely, the time increment must be capable of resolving the highest frequencies ofinterest.

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The column for generalized mass lists the mass of a single degree of freedom systemassociated with that mode.

The table of participation factors (see Figure 9-7) indicates the predominant degrees offreedom in which the modes act.

Figure 9-7 Table of participation factors.

The table of effective mass (see Figure 9-8) indicates the amount of mass active in each degreeof freedom for any one mode.

Figure 9-8 Table of effective mass.

The column for generalized mass lists the mass of a single degree of freedom systemassociated with that mode.

The table of participation factors (see Figure 9-7) indicates the predominant degrees offreedom in which the modes act.

Figure 9-7 Table of participation factors.

The table of effective mass (see Figure 9-8) indicates the amount of mass active in each degreeof freedom for any one mode.

Figure 9-8 Table of effective mass.

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The total mass of the model is given earlier in the data file and is 425.5 kg.

TOTAL MASS OF MODEL ASSOCIATED WITH EACH DIRECTION

(MASS MAY BE DIFFERENT IN DIFFERENT DIRECTIONS IF POINT MASSES HA

DIRECTION 1 DIRECTION 2 DIRECTI

425.5202 425.5202 425.52

To ensure that enough modes have been used, the total effective mass in each direction shouldbe a large proportion of the mass of the model (say 90%). However, some of the mass of themodel is associated with nodes that are constrained. This constrained mass is approximatelyone-quarter of the mass of all the elements attached to the constrained nodes, which, in thiscase, is approximately 29 kg. Therefore, the mass of the model that is able to move is 396.5kg. The effective mass in the x-, y-, and z-directions is 6%, 98%, and 97%, respectively, of themass that can move. The total effective mass in the 2- and 3-directions is well above the 90%recommended earlier; the total effective mass in the 1-direction is much lower. However, sincethe loading is applied in the 2-direction, the response in the 1-direction is not significant.

The data file does not contain any results for the modal dynamics step, because all of the datafile output requests were turned off.

9.5.5 PostprocessingWhen you are in the directory containing the output database file dynamics.odb, type the command

abaqus viewer odb=dynamics


The total mass of the model is given earlier in the data file and is 425.5 kg.

TOTAL MASS OF MODEL ASSOCIATED WITH EACH DIRECTION

(MASS MAY BE DIFFERENT IN DIFFERENT DIRECTIONS IF POINT MASSES HA

DIRECTION 1 DIRECTION 2 DIRECTI

425.5202 425.5202 425.52

To ensure that enough modes have been used, the total effective mass in each direction shouldbe a large proportion of the mass of the model (say 90%). However, some of the mass of themodel is associated with nodes that are constrained. This constrained mass is approximatelyone-quarter of the mass of all the elements attached to the constrained nodes, which, in thiscase, is approximately 29 kg. Therefore, the mass of the model that is able to move is 396.5kg. The effective mass in the x-, y-, and z-directions is 6%, 98%, and 97%, respectively, of themass that can move. The total effective mass in the 2- and 3-directions is well above the 90%recommended earlier; the total effective mass in the 1-direction is much lower. However, sincethe loading is applied in the 2-direction, the response in the 1-direction is not significant.

The data file does not contain any results for the modal dynamics step, because all of the datafile output requests were turned off.

9.5.5 PostprocessingWhen you are in the directory containing the output database file dynamics.odb, type the command

abaqus viewer odb=dynamics


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Plotting mode shapes

You can visualize the deformation mode associated with a given natural frequency by plottingthe mode shape associated with that frequency.

To select a mode and plot the corresponding mode shape:



2. Select the first step (Frequency extraction of the first 30 modes) from theStep field.

Select Mode 1 in the Frame field.


ABAQUS/Viewer displays the deformed model shape associated with the first vibration mode,as shown in Figure 9-9.

Figure 9-9 Mode 1.

4. Select the third mode from the Frame Selector dialog box.

5. Click OK.

ABAQUS/Viewer displays the third mode shape shown in Figure 9-10, and the FrameSelector dialog box disappears.

Plotting mode shapes

You can visualize the deformation mode associated with a given natural frequency by plottingthe mode shape associated with that frequency.

To select a mode and plot the corresponding mode shape:



2. Select the first step (Frequency extraction of the first 30 modes) from theStep field.

Select Mode 1 in the Frame field.


ABAQUS/Viewer displays the deformed model shape associated with the first vibration mode,as shown in Figure 9-9.

Figure 9-9 Mode 1.

4. Select the third mode from the Frame Selector dialog box.

5. Click OK.

ABAQUS/Viewer displays the third mode shape shown in Figure 9-10, and the FrameSelector dialog box disappears.

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Figure 9-10 Mode 3.

Animation of results

You will animate the analysis results. First create a scale factor animation of the thirdeigenmode. Then create a time-history animation of the transient results.

To create a scale factor animation of an eigenmode:

1. From the main menu bar, select Animate->Scale Factor; or use the tool in thetoolbox.

ABAQUS/Viewer displays the third mode shape and steps through different deformation scalefactors ranging from 0 to 1.

ABAQUS/Viewer also displays the movie player controls on the left side of the prompt area.

You can use these controls to start, stop, and step through the animation. From left to right,these controls perform the following functions: play, stop, first image, previous image,next image, and last image.

Figure 9-10 Mode 3.

Animation of results

You will animate the analysis results. First create a scale factor animation of the thirdeigenmode. Then create a time-history animation of the transient results.

To create a scale factor animation of an eigenmode:

1. From the main menu bar, select Animate->Scale Factor; or use the tool in thetoolbox.

ABAQUS/Viewer displays the third mode shape and steps through different deformation scalefactors ranging from 0 to 1.

ABAQUS/Viewer also displays the movie player controls on the left side of the prompt area.

You can use these controls to start, stop, and step through the animation. From left to right,these controls perform the following functions: play, stop, first image, previous image,next image, and last image.

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2. In the prompt area, click to stop the animation.

To create a time-history animation of the transient results:

1. From the main menu bar, select Options->Animation to view the animation options.

ABAQUS/Viewer displays the Animation Options dialog box.

2. Click the Time History tab.

The Select steps for animation field appears.

3. Select the second step (Simulation of Drop Load on Crane).

4. Click OK to accept the selection and to close the dialog box.

5. From the main menu bar, select Animate->Time History; or use the tool from thetoolbox.

ABAQUS/Viewer displays the movie player controls on the left side of the prompt area.

6. In the prompt area, click to start the animation.

ABAQUS/Viewer steps through each available frame of the second step. The state blockindicates the current step and increment throughout the animation. After the last increment ofthis step is reached, the animation process repeats itself.

7. You can customize the deformed shape plot while the animation is running.

a. Display the Deformed Plot Options dialog box.

b. Choose Uniform from the Deformation Scale Factor field.

c. Enter 15.0 as the deformation scale factor value.

d. Click Apply to apply your change.

ABAQUS/Viewer now steps through the frames in the second step with a deformationscale factor of 15.0.

e. Choose Auto-compute from the Deformation Scale Factor field.

f. Click OK to apply your change and to close the Deformed Options dialog box.

ABAQUS/Viewer now steps through the frames in the second step with a defaultdeformation scale factor of 0.8.

Determining the peak pull-out force

2. In the prompt area, click to stop the animation.

To create a time-history animation of the transient results:

1. From the main menu bar, select Options->Animation to view the animation options.

ABAQUS/Viewer displays the Animation Options dialog box.

2. Click the Time History tab.

The Select steps for animation field appears.

3. Select the second step (Simulation of Drop Load on Crane).

4. Click OK to accept the selection and to close the dialog box.

5. From the main menu bar, select Animate->Time History; or use the tool from thetoolbox.

ABAQUS/Viewer displays the movie player controls on the left side of the prompt area.

6. In the prompt area, click to start the animation.

ABAQUS/Viewer steps through each available frame of the second step. The state blockindicates the current step and increment throughout the animation. After the last increment ofthis step is reached, the animation process repeats itself.

7. You can customize the deformed shape plot while the animation is running.

a. Display the Deformed Plot Options dialog box.

b. Choose Uniform from the Deformation Scale Factor field.

c. Enter 15.0 as the deformation scale factor value.

d. Click Apply to apply your change.

ABAQUS/Viewer now steps through the frames in the second step with a deformationscale factor of 15.0.

e. Choose Auto-compute from the Deformation Scale Factor field.

f. Click OK to apply your change and to close the Deformed Options dialog box.

ABAQUS/Viewer now steps through the frames in the second step with a defaultdeformation scale factor of 0.8.

Determining the peak pull-out force

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To find the peak pull-out force at the attachment points, create an X-Y plot of the reactionforce in the 1-direction (variable RF1) at the attached nodes. This involves plotting multiplecurves at the same time.

To plot multiple curves:


ABAQUS/Viewer displays the ODB History Output dialog box.

2. Select the following curves (using [Ctrl]+Click) from the Output Variables field:

Reaction Force: RF1 at Node 100 in NSET ATTACH




3. Click Plot.

ABAQUS/Viewer displays the selected curves.


ABAQUS/Viewer displays the Viewport Annotations dialog box.


6. Click OK to apply your change and to close the dialog box.

ABAQUS/Viewer displays the minimum and maximum values.

The resulting plot (which has been customized) is shown in Figure 9-11. The curves for thetwo nodes at the top of each truss (nodes 107 and 207) are almost a reflection of those for thenodes on the bottom of each truss (nodes 100 and 200).

Figure 9-11 History of the reaction forces at the attached nodes.

To find the peak pull-out force at the attachment points, create an X-Y plot of the reactionforce in the 1-direction (variable RF1) at the attached nodes. This involves plotting multiplecurves at the same time.

To plot multiple curves:


ABAQUS/Viewer displays the ODB History Output dialog box.

2. Select the following curves (using [Ctrl]+Click) from the Output Variables field:





3. Click Plot.

ABAQUS/Viewer displays the selected curves.


ABAQUS/Viewer displays the Viewport Annotations dialog box.


6. Click OK to apply your change and to close the dialog box.

ABAQUS/Viewer displays the minimum and maximum values.

The resulting plot (which has been customized) is shown in Figure 9-11. The curves for thetwo nodes at the top of each truss (nodes 107 and 207) are almost a reflection of those for thenodes on the bottom of each truss (nodes 100 and 200).

Figure 9-11 History of the reaction forces at the attached nodes.

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At the attachment points at the top of each truss structure, the peak tensile force is around 80kN, which is below the 100 kN capacity of the connection. Keep in mind that a negativereaction force in the 1-direction means that the member is being pulled away from the wall.The lower attachments are in compression (positive reaction force) while the load is applied,but oscillate between tension and compression after the load has been removed. The peaktensile force is about 50 kN, well below the allowable value. To find this value, probe the X-Yplot.

To query the X-Y plot:

1. Select Tools->Query.


2. Select Probe values in the Visualization Queries field if it is not already selected.

3. Click OK.


4. Select the point indicated in Figure 9-11.

The Y -coordinate of this point is -51.1 kN. This corresponds to the value of the reaction forcein the 1-direction.

9.6 Effect of the number of modesFor this simulation 30 modes were used to represent the dynamic behavior of the structure. The totalmodal effective mass for all of these modes was well over 90% of the mass of the structure that canmove in the y- and z-directions, indicating that the dynamic representation is adequate.

At the attachment points at the top of each truss structure, the peak tensile force is around 80kN, which is below the 100 kN capacity of the connection. Keep in mind that a negativereaction force in the 1-direction means that the member is being pulled away from the wall.The lower attachments are in compression (positive reaction force) while the load is applied,but oscillate between tension and compression after the load has been removed. The peaktensile force is about 50 kN, well below the allowable value. To find this value, probe the X-Yplot.

To query the X-Y plot:

1. Select Tools->Query.


2. Select Probe values in the Visualization Queries field if it is not already selected.

3. Click OK.


4. Select the point indicated in Figure 9-11.

The Y -coordinate of this point is -51.1 kN. This corresponds to the value of the reaction forcein the 1-direction.

9.6 Effect of the number of modesFor this simulation 30 modes were used to represent the dynamic behavior of the structure. The totalmodal effective mass for all of these modes was well over 90% of the mass of the structure that canmove in the y- and z-directions, indicating that the dynamic representation is adequate.

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Figure 9-12 shows the displacement in degree of freedom 2 at node 104 versus time and illustrates theeffect of using fewer modes on the quality of the results. If you look at the table of effective mass, youwill see that the first significant mode in the 2-direction is mode 3, which accounts for the lack ofresponse when only two modes are used. The displacement of node 104 in degree of freedom 2 for theanalyses using five modes and 30 modes is similar after 0.2 seconds; however, the early responsediffers, suggesting that there are significant modes in the range of 5-30 relating to the early response.When five modes are used, the total modal effective mass in the 2-direction is only 57% of themoveable mass.

Figure 9-12 Effect of different numbers of modes on the results.

9.7 Effect of dampingIn this simulation we used 5% of critical damping in all modes. This value was chosen fromexperience, based on the fact that the bolted connections between the trusses and the cross bracingmight absorb significant energy as a result of local frictional effects. In cases such as this, whereaccurate data are not available, it is important to investigate the effect of the choices that youmake.

Figure 9-13 compares the history of the reaction force at one of the top connections when 1%, 5%, and10% of critical damping are used.

Figure 9-13 Effect of damping on the results.

Figure 9-12 shows the displacement in degree of freedom 2 at node 104 versus time and illustrates theeffect of using fewer modes on the quality of the results. If you look at the table of effective mass, youwill see that the first significant mode in the 2-direction is mode 3, which accounts for the lack ofresponse when only two modes are used. The displacement of node 104 in degree of freedom 2 for theanalyses using five modes and 30 modes is similar after 0.2 seconds; however, the early responsediffers, suggesting that there are significant modes in the range of 5-30 relating to the early response.When five modes are used, the total modal effective mass in the 2-direction is only 57% of themoveable mass.

Figure 9-12 Effect of different numbers of modes on the results.

9.7 Effect of dampingIn this simulation we used 5% of critical damping in all modes. This value was chosen fromexperience, based on the fact that the bolted connections between the trusses and the cross bracingmight absorb significant energy as a result of local frictional effects. In cases such as this, whereaccurate data are not available, it is important to investigate the effect of the choices that youmake.

Figure 9-13 compares the history of the reaction force at one of the top connections when 1%, 5%, and10% of critical damping are used.

Figure 9-13 Effect of damping on the results.

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As expected, the oscillations at lower damping levels do not diminish as quickly as those at higherdamping levels, and the peak force is higher in the models with lower damping. With damping even aslow as 1%, the peak pull-out force is 85 kN, which is still less than the strength of the connection (100kN). Therefore, the cargo crane should retain its integrity under this drop load.

9.8 Other dynamic proceduresWe now briefly review the other dynamic procedures available in ABAQUS/Standard--namely linearmodal dynamics and nonlinear dynamics.

9.8.1 Linear modal dynamicsThere are several other linear, dynamic procedures in ABAQUS/Standard that employ the modalsuperposition technique. Unlike *MODAL DYNAMIC, which calculates the response in the timedomain, these procedures provide results in the frequency domain, which can give additional insightinto the behavior of the structure.

A complete description of these procedures is given in the ABAQUS/Standard User's Manual.

Steady-state dynamics

The *STEADY STATE DYNAMICS option calculates the amplitude and phase of thestructure's response caused by harmonic excitation over a user-specified range of frequencies.Typical examples include the following:

� The response of car engine mounts over a range of engine operating speeds.

� Rotating machinery in buildings.

� Components on aircraft engines.

Response spectrum

The *RESPONSE SPECTRUM option provides an estimate of the peak response

As expected, the oscillations at lower damping levels do not diminish as quickly as those at higherdamping levels, and the peak force is higher in the models with lower damping. With damping even aslow as 1%, the peak pull-out force is 85 kN, which is still less than the strength of the connection (100kN). Therefore, the cargo crane should retain its integrity under this drop load.

9.8 Other dynamic proceduresWe now briefly review the other dynamic procedures available in ABAQUS/Standard--namely linearmodal dynamics and nonlinear dynamics.

9.8.1 Linear modal dynamicsThere are several other linear, dynamic procedures in ABAQUS/Standard that employ the modalsuperposition technique. Unlike *MODAL DYNAMIC, which calculates the response in the timedomain, these procedures provide results in the frequency domain, which can give additional insightinto the behavior of the structure.

A complete description of these procedures is given in the ABAQUS/Standard User's Manual.

Steady-state dynamics

The *STEADY STATE DYNAMICS option calculates the amplitude and phase of thestructure's response caused by harmonic excitation over a user-specified range of frequencies.Typical examples include the following:

� The response of car engine mounts over a range of engine operating speeds.

� Rotating machinery in buildings.

� Components on aircraft engines.

Response spectrum

The *RESPONSE SPECTRUM option provides an estimate of the peak response

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(displacement, stress, etc.) when a structure is subjected to dynamic motion of its fixed points.Such motion is known as "base motion"; an example is a seismic event causing groundmotion. Typically the method is used when an estimate of the peak response is required fordesign purposes.

Random response

The *RANDOM RESPONSE option predicts the response of a system subjected to randomcontinuous excitation. The excitation is expressed in a statistical sense using a power spectraldensity function. Examples of random response analysis include the following:

� The response of an airplane to turbulence.

� The response of a structure to noise, such as that emitted by a jet engine.

9.8.2 Nonlinear dynamicsAs mentioned earlier, the *MODAL DYNAMIC procedure is suitable only for linear problems. Whennonlinear dynamic response is of interest, the equation of motion must be integrated directly. Thedirect integration dynamic procedure in ABAQUS is *DYNAMIC. When this procedure is used, themass, damping, and stiffness matrices are assembled and the equation of dynamic equilibrium is solvedat each point in time. Since these operations are computationally intensive, direct integration dynamicsis more expensive than the modal methods.

The nonlinear dynamic procedure in ABAQUS/Standard uses implicit time integration. It is suitablefor nonlinear structural dynamics problems, for example, in which a sudden event initiates the dynamicresponse, such as an impact, and the structural response involves large amounts of energy beingdissipated by plasticity or viscous damping. In such studies the high frequency response, which isimportant initially, is damped out rapidly by the dissipative mechanisms in the model.

An alternative for nonlinear dynamic analyses is the explicit dynamics procedure available inABAQUS/Explicit. The explicit algorithm propagates the solution as a stress wave through the model,one element at a time. Thus, it is most suitable for applications in which stress wave effects areimportant and in which the event time being simulated is short (typically less than one second).

Another advantage associated with the explicit algorithm is that it can model discontinuousnonlinearities such as contact and failure more easily than ABAQUS/Standard. Large, highlydiscontinuous problems are often more easily modeled with ABAQUS/Explicit, even if the response isquasi-static.

9.9 Related ABAQUS examples� ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2 of the ABAQUS

Example Problems Manual

� `Èxplosively loaded cylindrical panel,'' Section 1.3.3 of the ABAQUS Benchmarks Manual

� `Èigenvalue analysis of a cantilever plate,'' Section 1.4.6 of the ABAQUS Benchmarks Manual

(displacement, stress, etc.) when a structure is subjected to dynamic motion of its fixed points.Such motion is known as "base motion"; an example is a seismic event causing groundmotion. Typically the method is used when an estimate of the peak response is required fordesign purposes.

Random response

The *RANDOM RESPONSE option predicts the response of a system subjected to randomcontinuous excitation. The excitation is expressed in a statistical sense using a power spectraldensity function. Examples of random response analysis include the following:

� The response of an airplane to turbulence.

� The response of a structure to noise, such as that emitted by a jet engine.

9.8.2 Nonlinear dynamicsAs mentioned earlier, the *MODAL DYNAMIC procedure is suitable only for linear problems. Whennonlinear dynamic response is of interest, the equation of motion must be integrated directly. Thedirect integration dynamic procedure in ABAQUS is *DYNAMIC. When this procedure is used, themass, damping, and stiffness matrices are assembled and the equation of dynamic equilibrium is solvedat each point in time. Since these operations are computationally intensive, direct integration dynamicsis more expensive than the modal methods.

The nonlinear dynamic procedure in ABAQUS/Standard uses implicit time integration. It is suitablefor nonlinear structural dynamics problems, for example, in which a sudden event initiates the dynamicresponse, such as an impact, and the structural response involves large amounts of energy beingdissipated by plasticity or viscous damping. In such studies the high frequency response, which isimportant initially, is damped out rapidly by the dissipative mechanisms in the model.

An alternative for nonlinear dynamic analyses is the explicit dynamics procedure available inABAQUS/Explicit. The explicit algorithm propagates the solution as a stress wave through the model,one element at a time. Thus, it is most suitable for applications in which stress wave effects areimportant and in which the event time being simulated is short (typically less than one second).

Another advantage associated with the explicit algorithm is that it can model discontinuousnonlinearities such as contact and failure more easily than ABAQUS/Standard. Large, highlydiscontinuous problems are often more easily modeled with ABAQUS/Explicit, even if the response isquasi-static.

9.9 Related ABAQUS examples� ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2 of the ABAQUS

Example Problems Manual

� `Èxplosively loaded cylindrical panel,'' Section 1.3.3 of the ABAQUS Benchmarks Manual

� `Èigenvalue analysis of a cantilever plate,'' Section 1.4.6 of the ABAQUS Benchmarks Manual

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9.10 Suggested reading

� Clough, R. W. and J. Penzien, Dynamics of Structures, McGraw-Hill, 1975.

� NAFEMS Ltd., A Finite Element Dynamics Primer, 1993.

� Spence, P. W. and C. J. Kenchington, The Role of Damping in Finite Element Analysis , ReportR0021, NAFEMS Ltd., 1993.

9.11 Summary� Dynamic analyses include the effect of the structure's inertia.

� The *FREQUENCY procedure extracts the natural frequencies and mode shapes of the structure.

� The mode shapes can then be used to determine the dynamic response of linear systems by modalsuperposition. This technique is efficient, but it cannot be used for nonlinear problems.

� Linear dynamic procedures are available to calculate the transient response to transient loading,the steady-state response to harmonic loading, the peak response to base motion, and the responseto random loading.

� You should extract enough modes to obtain an accurate representation of the dynamic behavior ofthe structure. The total modal effective mass should be at least 90% of the mass that can move toproduce accurate results.

� You can define direct modal damping, Rayleigh damping, and composite modal damping.However, since the natural frequencies and mode shapes are based on the undamped structure, thestructure being analyzed should be only lightly damped.

� Modal techniques are not suitable for nonlinear dynamic simulations. Direct time integration(*DYNAMIC) methods must be used in these situations.

� The *AMPLITUDE option allows any time variation of loads or prescribed boundary conditions tobe defined.

� Mode shapes and transient results can be animated in ABAQUS/Viewer. This provides a usefulway of understanding the response of dynamic and nonlinear static analyses.

9.10 Suggested reading

� Clough, R. W. and J. Penzien, Dynamics of Structures, McGraw-Hill, 1975.

� NAFEMS Ltd., A Finite Element Dynamics Primer, 1993.

� Spence, P. W. and C. J. Kenchington, The Role of Damping in Finite Element Analysis , ReportR0021, NAFEMS Ltd., 1993.

9.11 Summary� Dynamic analyses include the effect of the structure's inertia.

� The *FREQUENCY procedure extracts the natural frequencies and mode shapes of the structure.

� The mode shapes can then be used to determine the dynamic response of linear systems by modalsuperposition. This technique is efficient, but it cannot be used for nonlinear problems.

� Linear dynamic procedures are available to calculate the transient response to transient loading,the steady-state response to harmonic loading, the peak response to base motion, and the responseto random loading.

� You should extract enough modes to obtain an accurate representation of the dynamic behavior ofthe structure. The total modal effective mass should be at least 90% of the mass that can move toproduce accurate results.

� You can define direct modal damping, Rayleigh damping, and composite modal damping.However, since the natural frequencies and mode shapes are based on the undamped structure, thestructure being analyzed should be only lightly damped.

� Modal techniques are not suitable for nonlinear dynamic simulations. Direct time integration(*DYNAMIC) methods must be used in these situations.

� The *AMPLITUDE option allows any time variation of loads or prescribed boundary conditions tobe defined.

� Mode shapes and transient results can be animated in ABAQUS/Viewer. This provides a usefulway of understanding the response of dynamic and nonlinear static analyses.

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10. Multiple Step AnalysisThe general goal of an ABAQUS simulation is to determine the response of the model to the appliedloads. In ABAQUS the term load refers to nonzero boundary conditions or applied displacements, aswell as point loads or pressure loads. In some cases these loads are relatively simple, such as a singleset of point loads on a structure. In other problems the loads applied to a structure can be verycomplex. For instance, different loads may be applied to different portions of the model in a particularsequence over some period of time, or the magnitude of the loads may vary as a function of time. Theterm load history is used to refer to such complex loading of a model.

In ABAQUS the user divides the complete load history of the simulation into a number of steps. Eachstep is a period of "time," specified by the user, for which ABAQUS calculates the response of themodel to a particular set of loads and boundary conditions. The user must specify the type of response,known as the analysis procedure, during each step and may change analysis procedures from step tostep. For example, static dead loads, perhaps gravitational loads, could be applied to a structure in onestep, and the dynamic response of the loaded structure to earthquake accelerations could be calculatedin the next step.

ABAQUS divides all of its analysis procedures into two main groups: linear perturbation and general.ABAQUS makes a very clear distinction between perturbation and general analysis proceduresbecause loading conditions and "time" are defined differently for the two cases. Furthermore, theresults from each type of procedure should be interpreted differently.

The response of the model during a general analysis procedure, known as a general step, may be eithernonlinear or linear. In a step that uses a perturbation procedure, which is called a perturbation step, theresponse can only be linear. ABAQUS treats such steps as a linear perturbation about the preloaded,predeformed state created by any previous general steps; therefore, its capability for doing linearsimulations is rather more general than that of a purely linear analysis program.

10.1 General (nonlinear) analysis proceduresThe starting point for each general step is the deformed state at the end of the last general step.Therefore, the state of the model evolves in a sequence of general steps as it responds to the loadsdefined in each step. The initial conditions specified using the *INITIAL CONDITIONS option definethe starting point for the first general step in the simulation.

All general analysis procedures share the same concepts for applying loads and defining "time."

10.1.1 Time in general analysis stepsABAQUS has two measures of time in a simulation. The total time increases throughout all generalsteps and is the accumulation of the total step time from each general step. Each step also has its owntime scale (known as the step time), which begins at zero for each step. Time varying loads andboundary conditions can be specified in terms of either time scale. The time scales for an analysiswhose history is divided into three steps, each 100 seconds long, are shown in Figure 10-1.

10. Multiple Step AnalysisThe general goal of an ABAQUS simulation is to determine the response of the model to the appliedloads. In ABAQUS the term load refers to nonzero boundary conditions or applied displacements, aswell as point loads or pressure loads. In some cases these loads are relatively simple, such as a singleset of point loads on a structure. In other problems the loads applied to a structure can be verycomplex. For instance, different loads may be applied to different portions of the model in a particularsequence over some period of time, or the magnitude of the loads may vary as a function of time. Theterm load history is used to refer to such complex loading of a model.

In ABAQUS the user divides the complete load history of the simulation into a number of steps. Eachstep is a period of "time," specified by the user, for which ABAQUS calculates the response of themodel to a particular set of loads and boundary conditions. The user must specify the type of response,known as the analysis procedure, during each step and may change analysis procedures from step tostep. For example, static dead loads, perhaps gravitational loads, could be applied to a structure in onestep, and the dynamic response of the loaded structure to earthquake accelerations could be calculatedin the next step.

ABAQUS divides all of its analysis procedures into two main groups: linear perturbation and general.ABAQUS makes a very clear distinction between perturbation and general analysis proceduresbecause loading conditions and "time" are defined differently for the two cases. Furthermore, theresults from each type of procedure should be interpreted differently.

The response of the model during a general analysis procedure, known as a general step, may be eithernonlinear or linear. In a step that uses a perturbation procedure, which is called a perturbation step, theresponse can only be linear. ABAQUS treats such steps as a linear perturbation about the preloaded,predeformed state created by any previous general steps; therefore, its capability for doing linearsimulations is rather more general than that of a purely linear analysis program.

10.1 General (nonlinear) analysis proceduresThe starting point for each general step is the deformed state at the end of the last general step.Therefore, the state of the model evolves in a sequence of general steps as it responds to the loadsdefined in each step. The initial conditions specified using the *INITIAL CONDITIONS option definethe starting point for the first general step in the simulation.

All general analysis procedures share the same concepts for applying loads and defining "time."

10.1.1 Time in general analysis stepsABAQUS has two measures of time in a simulation. The total time increases throughout all generalsteps and is the accumulation of the total step time from each general step. Each step also has its owntime scale (known as the step time), which begins at zero for each step. Time varying loads andboundary conditions can be specified in terms of either time scale. The time scales for an analysiswhose history is divided into three steps, each 100 seconds long, are shown in Figure 10-1.

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Figure 10-1 Step and total time for a simulation.

10.1.2 Specifying loads in general stepsIn general steps the loads must be specified as total values, not incremental values. For example, if aconcentrated load has a value of 1000 N in the first step and it is increased to 3000 N in the secondgeneral step, the magnitude given on the *CLOAD option in the two steps should be 1000 N and 3000N, not 1000 N and 2000 N.

Modifying loads from step to step

Applying a load in ABAQUS requires more than just providing its magnitude and direction.You must also specify how these new loads interact with the existing loads and boundaryconditions of the same type that were defined in previous general steps. All the loading andboundary condition options--such as *BOUNDARY, *CLOAD, and *DLOAD--use the OPparameter to indicate how the loads they define interact with the existing loads of that type.The parameter can be set to OP=MOD or OP=NEW. ABAQUS assumes OP=MOD if no valueis provided for the OP parameter.

Using OP=MOD causes the loads defined in the current general step to modify the same typesof loads already applied to the model in previous general steps. ABAQUS maintains any loadsthat are not specifically modified in the current step at the magnitudes they had at the end ofthe last general step. For example, consider a cantilever beam modeled with two B22 elements(see Figure 10-2) with concentrated loads of 1000 N applied to nodes 3 and 5 in the firstgeneral step.

Figure 10-2 Loads applied to a beam in the first general step (Step 1).

In the next general step (Step 2) loads of 2000 N are specified on nodes 4 and 5 using theOP=MOD parameter. Thus, these loads modify those applied in Step 1. The loading applied tothe model at the end of Step 2 is shown in Figure 10-3.

Figure 10-3 Loads applied in Step 2 with OP=MOD.

Figure 10-1 Step and total time for a simulation.

10.1.2 Specifying loads in general stepsIn general steps the loads must be specified as total values, not incremental values. For example, if aconcentrated load has a value of 1000 N in the first step and it is increased to 3000 N in the secondgeneral step, the magnitude given on the *CLOAD option in the two steps should be 1000 N and 3000N, not 1000 N and 2000 N.

Modifying loads from step to step

Applying a load in ABAQUS requires more than just providing its magnitude and direction.You must also specify how these new loads interact with the existing loads and boundaryconditions of the same type that were defined in previous general steps. All the loading andboundary condition options--such as *BOUNDARY, *CLOAD, and *DLOAD--use the OPparameter to indicate how the loads they define interact with the existing loads of that type.The parameter can be set to OP=MOD or OP=NEW. ABAQUS assumes OP=MOD if no valueis provided for the OP parameter.

Using OP=MOD causes the loads defined in the current general step to modify the same typesof loads already applied to the model in previous general steps. ABAQUS maintains any loadsthat are not specifically modified in the current step at the magnitudes they had at the end ofthe last general step. For example, consider a cantilever beam modeled with two B22 elements(see Figure 10-2) with concentrated loads of 1000 N applied to nodes 3 and 5 in the firstgeneral step.

Figure 10-2 Loads applied to a beam in the first general step (Step 1).

In the next general step (Step 2) loads of 2000 N are specified on nodes 4 and 5 using theOP=MOD parameter. Thus, these loads modify those applied in Step 1. The loading applied tothe model at the end of Step 2 is shown in Figure 10-3.

Figure 10-3 Loads applied in Step 2 with OP=MOD.


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Using OP=NEW causes ABAQUS to remove all existing loads of that type and only apply theloads specified in this current step to the model. If OP=NEW is specified on the *CLOADoption in Step 2, the loading on our example beam is shown in Figure 10-4.

Figure 10-4 Loads applied in Step 2 with OP=NEW.

Be very careful when you use the OP=NEW parameter on the *BOUNDARY option toremove a boundary constraint from your model. All boundary constraints are removed fromthe model, not just the one you want removed; therefore, you must respecify all the boundaryconditions that should remain active in the model. Remember that the boundary conditionsthat you specify must provide enough constraints to prevent rigid body motions in allcomponents of your model. Failure to do so will cause ABAQUS to issue numericalsingularity warnings and leads to excessive displacements.

10.2 Linear perturbation analysisThe starting point for a linear perturbation step is called the base state of the model. If the first step in asimulation is a linear perturbation step, the base state is the state of the model specified using the*INITIAL CONDITIONS option. Otherwise, the base state is the state of the simulation at the end ofthe last general step prior to the linear perturbation step. Although the response of the structure duringthe perturbation step is by definition linear, the model may have a nonlinear response in previousgeneral steps. For models with a nonlinear response in the prior general steps, ABAQUS uses thecurrent elastic modulus (tangent modulus) as the linear stiffness for perturbation procedures (seeFigure 10-5). The loads in the perturbation step should be sufficiently small that the model's responsewould not deviate much from that predicted with the tangent modulus. If the simulation includescontact, the contact state between two surfaces does not change during a perturbation step: points thatwere closed in the base state remain closed, and points that were open remain open.

Figure 10-5 The tangent modulus used as the stiffness in linear perturbation steps that occur after

Using OP=NEW causes ABAQUS to remove all existing loads of that type and only apply theloads specified in this current step to the model. If OP=NEW is specified on the *CLOADoption in Step 2, the loading on our example beam is shown in Figure 10-4.

Figure 10-4 Loads applied in Step 2 with OP=NEW.

Be very careful when you use the OP=NEW parameter on the *BOUNDARY option toremove a boundary constraint from your model. All boundary constraints are removed fromthe model, not just the one you want removed; therefore, you must respecify all the boundaryconditions that should remain active in the model. Remember that the boundary conditionsthat you specify must provide enough constraints to prevent rigid body motions in allcomponents of your model. Failure to do so will cause ABAQUS to issue numericalsingularity warnings and leads to excessive displacements.

10.2 Linear perturbation analysisThe starting point for a linear perturbation step is called the base state of the model. If the first step in asimulation is a linear perturbation step, the base state is the state of the model specified using the*INITIAL CONDITIONS option. Otherwise, the base state is the state of the simulation at the end ofthe last general step prior to the linear perturbation step. Although the response of the structure duringthe perturbation step is by definition linear, the model may have a nonlinear response in previousgeneral steps. For models with a nonlinear response in the prior general steps, ABAQUS uses thecurrent elastic modulus (tangent modulus) as the linear stiffness for perturbation procedures (seeFigure 10-5). The loads in the perturbation step should be sufficiently small that the model's responsewould not deviate much from that predicted with the tangent modulus. If the simulation includescontact, the contact state between two surfaces does not change during a perturbation step: points thatwere closed in the base state remain closed, and points that were open remain open.

Figure 10-5 The tangent modulus used as the stiffness in linear perturbation steps that occur after


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general, nonlinear steps.

10.2.1 Time in linear perturbation stepsIf another general step follows a perturbation step, it uses the state of the model at the end of the lastgeneral step as its starting point, not the state of the model at the end of the perturbation step. Thus, theresponse from a linear perturbation step has no permanent effect on the simulation. Therefore,ABAQUS does not include the step time of linear perturbation steps into the total time for the analysis.In fact, what ABAQUS actually does is to define the step time of a perturbation step to be very small(10-36) so that it has no effect when it is added to the total accumulated time. The exception to this ruleis the *MODAL DYNAMIC procedure.

10.2.2 Specifying loads in linear perturbation stepsLoads and prescribed boundary conditions given in linear perturbation steps are always local to thatstep. The load magnitudes (including the magnitudes of prescribed boundary conditions) given in alinear perturbation step are always the increment (perturbation) of the load, not the total magnitude.All loads and boundary conditions that were in effect at the end of the last general step act on themodel in the perturbation step, but they will not be reported in any of the output variables. Therefore,the total load applied to the model in a linear perturbation step is the sum of the "dead" loads from thelast general step and the perturbation loads applied in the current step. Likewise, the output from sucha step is the perturbation value only. The total value of the variable is the sum of this perturbationvalue and the value from the base state.

As an example of a complex load history that includes a mixture of general and perturbation steps,consider the manufacture and use of a stainless steel sink, shown schematically in Figure 10-6.

Figure 10-6 Steps in the manufacture and use of a sink.

general, nonlinear steps.

10.2.1 Time in linear perturbation stepsIf another general step follows a perturbation step, it uses the state of the model at the end of the lastgeneral step as its starting point, not the state of the model at the end of the perturbation step. Thus, theresponse from a linear perturbation step has no permanent effect on the simulation. Therefore,ABAQUS does not include the step time of linear perturbation steps into the total time for the analysis.In fact, what ABAQUS actually does is to define the step time of a perturbation step to be very small(10-36) so that it has no effect when it is added to the total accumulated time. The exception to this ruleis the *MODAL DYNAMIC procedure.

10.2.2 Specifying loads in linear perturbation stepsLoads and prescribed boundary conditions given in linear perturbation steps are always local to thatstep. The load magnitudes (including the magnitudes of prescribed boundary conditions) given in alinear perturbation step are always the increment (perturbation) of the load, not the total magnitude.All loads and boundary conditions that were in effect at the end of the last general step act on themodel in the perturbation step, but they will not be reported in any of the output variables. Therefore,the total load applied to the model in a linear perturbation step is the sum of the "dead" loads from thelast general step and the perturbation loads applied in the current step. Likewise, the output from sucha step is the perturbation value only. The total value of the variable is the sum of this perturbationvalue and the value from the base state.

As an example of a complex load history that includes a mixture of general and perturbation steps,consider the manufacture and use of a stainless steel sink, shown schematically in Figure 10-6.

Figure 10-6 Steps in the manufacture and use of a sink.


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The sink is formed from sheet steel using a punch, a die, and a blank holder. This forming simulationwill consist of a number of general steps. Typically, the first step may involve the application of blankholder pressure, and the punching operation will be simulated in the second step. The third step willinvolve the removal of the tooling, allowing the sink to spring back into its final shape. Each of thesesteps is a general step since together they model a sequential load history, where the starting conditionfor each step is the ending condition from the previous step. These steps obviously include manynonlinear effects (plasticity, contact, large deformations). At the end of the third step, the sink willcontain residual stresses and inelastic strains caused by the forming process. Its thickness will alsovary as a direct result of the manufacturing process.

The sink is then installed: boundary conditions would be applied around the edge of the sink where itis attached to the worktop. The response of the sink to a number of different loading conditions may be

The sink is formed from sheet steel using a punch, a die, and a blank holder. This forming simulationwill consist of a number of general steps. Typically, the first step may involve the application of blankholder pressure, and the punching operation will be simulated in the second step. The third step willinvolve the removal of the tooling, allowing the sink to spring back into its final shape. Each of thesesteps is a general step since together they model a sequential load history, where the starting conditionfor each step is the ending condition from the previous step. These steps obviously include manynonlinear effects (plasticity, contact, large deformations). At the end of the third step, the sink willcontain residual stresses and inelastic strains caused by the forming process. Its thickness will alsovary as a direct result of the manufacturing process.

The sink is then installed: boundary conditions would be applied around the edge of the sink where itis attached to the worktop. The response of the sink to a number of different loading conditions may be


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of interest and has to be simulated. For example, a simulation may need to be performed to ensure thatthe sink does not break if someone stands on it. Step 4 would, therefore, be a linear perturbation stepanalyzing the static response of the sink to a local pressure load. Remember that the results from Step 4will be perturbations from the state of the sink after the forming process; do not be surprised if thedisplacement of the center of the sink in this step is only 2 mm, but you know that the sink deformedmuch more than that since the start of the forming simulation. This 2 mm deflection is just theadditional deformation from the sink's final configuration after the forming process (i.e., the end ofStep 3) caused by the weight of the person. The total deflection, measured from the undeformed sheet'sconfiguration, is the sum of this 2 mm and the deflection at the end of Step 3.

The sink may also be fitted with a waste disposal unit, so its steady-state dynamic response to aharmonic load at certain frequencies must be simulated. Step 5 would, therefore, be a second linearperturbation step using the *STEADY STATE DYNAMICS, DIRECT procedure with a load applied atthe point of attachment of the disposal unit. The base state for this step is the state at the end of theprevious general step--that is, at the end of the forming process (Step 3). The response in the previousperturbation step (Step 4) is ignored. The two perturbation steps are, therefore, separate, independentsimulations of the sink's response to loads applied to the base state of the model.

If another general step is included in the analysis, the condition of the structure at the start of the stepis that at the end of the previous general step (Step 3). Step 6 could, therefore, be a general step withloads modeling the sink being filled with water. The response in this step is probably linear but couldbe nonlinear if required. Following this general step, Step 7 could be a simulation repeating theanalysis performed in Step 4. However, in this case the base state (the state of the structure at the endof the previous general step) is the state of the model at the end of Step 6. Therefore, the response willbe that of a full sink, rather than an empty one. Performing another steady-state dynamics simulationwould produce inaccurate results because the mass of the water, which would change the responseconsiderably, would not be considered in the analysis.

The following procedures in ABAQUS/Standard are always linear perturbation steps:

� *BUCKLE

� *FREQUENCY

� *MODAL DYNAMIC

� *RANDOM RESPONSE

� *RESPONSE SPECTRUM

� *STEADY STATE DYNAMICS

The *STATIC procedure can be either a general or linear perturbation procedure. Include thePERTURBATION parameter on the *STEP option to make a static step a linear perturbationprocedure.

10.3 Example: vibration of a piping systemIn this example you will study the vibrational frequencies of a 5 m long section of a piping system. The

of interest and has to be simulated. For example, a simulation may need to be performed to ensure thatthe sink does not break if someone stands on it. Step 4 would, therefore, be a linear perturbation stepanalyzing the static response of the sink to a local pressure load. Remember that the results from Step 4will be perturbations from the state of the sink after the forming process; do not be surprised if thedisplacement of the center of the sink in this step is only 2 mm, but you know that the sink deformedmuch more than that since the start of the forming simulation. This 2 mm deflection is just theadditional deformation from the sink's final configuration after the forming process (i.e., the end ofStep 3) caused by the weight of the person. The total deflection, measured from the undeformed sheet'sconfiguration, is the sum of this 2 mm and the deflection at the end of Step 3.

The sink may also be fitted with a waste disposal unit, so its steady-state dynamic response to aharmonic load at certain frequencies must be simulated. Step 5 would, therefore, be a second linearperturbation step using the *STEADY STATE DYNAMICS, DIRECT procedure with a load applied atthe point of attachment of the disposal unit. The base state for this step is the state at the end of theprevious general step--that is, at the end of the forming process (Step 3). The response in the previousperturbation step (Step 4) is ignored. The two perturbation steps are, therefore, separate, independentsimulations of the sink's response to loads applied to the base state of the model.

If another general step is included in the analysis, the condition of the structure at the start of the stepis that at the end of the previous general step (Step 3). Step 6 could, therefore, be a general step withloads modeling the sink being filled with water. The response in this step is probably linear but couldbe nonlinear if required. Following this general step, Step 7 could be a simulation repeating theanalysis performed in Step 4. However, in this case the base state (the state of the structure at the endof the previous general step) is the state of the model at the end of Step 6. Therefore, the response willbe that of a full sink, rather than an empty one. Performing another steady-state dynamics simulationwould produce inaccurate results because the mass of the water, which would change the responseconsiderably, would not be considered in the analysis.

The following procedures in ABAQUS/Standard are always linear perturbation steps:

� *BUCKLE

� *FREQUENCY

� *MODAL DYNAMIC

� *RANDOM RESPONSE

� *RESPONSE SPECTRUM

� *STEADY STATE DYNAMICS

The *STATIC procedure can be either a general or linear perturbation procedure. Include thePERTURBATION parameter on the *STEP option to make a static step a linear perturbationprocedure.

10.3 Example: vibration of a piping systemIn this example you will study the vibrational frequencies of a 5 m long section of a piping system. The


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pipe is made of steel and has an outer diameter of 18 cm and a 2 cm wall thickness (see Figure 10-7).

Figure 10-7 Portion of piping system being analyzed.

It is clamped firmly at one end and can move only axially at the other end. This 5 m portion of thepiping system may be subjected to harmonic loading at frequencies up to 50 Hz. The lowest vibrationalmode of the unloaded structure is 40.1 Hz, but this value does not consider how the loading applied tothe piping structure may affect its response. To ensure that the section does not resonate, you havebeen asked to determine the magnitude of the in-service load that is required so that its lowestvibrational mode is higher than 50 Hz. You are told that the section of pipe will be subjected to axialtension when in service. Start by considering a load magnitude of 4 MN.

The lowest vibrational mode of the pipe will be a sine wave deformation in any direction transverse tothe pipe axis because of the symmetry of the structure's cross-section. You use three-dimensional beamelements to model the pipe section.

10.3.1 Coordinate systemThe default, global coordinate system is used. Place the origin at the left end of the pipe section, andmake the axis of the pipe and the global 1-axis coincident, as shown in Figure 10-7.

10.3.2 Mesh designModel the pipe section with a uniformly spaced mesh of 30 second-order, pipe elements ( PIPE32). Thenode and element numbers of the model used in this discussion are shown in Figure 10-8.

Figure 10-8 Node and element numbers (both increase by 1 from left to right).

10.3.3 Preprocessing--creating the modelYou can create the mesh for this example using your preprocessor, or if you prefer, you can use theABAQUS mesh generation options shown in ``Vibration of a piping system,'' Section A.10.

pipe is made of steel and has an outer diameter of 18 cm and a 2 cm wall thickness (see Figure 10-7).

Figure 10-7 Portion of piping system being analyzed.

It is clamped firmly at one end and can move only axially at the other end. This 5 m portion of thepiping system may be subjected to harmonic loading at frequencies up to 50 Hz. The lowest vibrationalmode of the unloaded structure is 40.1 Hz, but this value does not consider how the loading applied tothe piping structure may affect its response. To ensure that the section does not resonate, you havebeen asked to determine the magnitude of the in-service load that is required so that its lowestvibrational mode is higher than 50 Hz. You are told that the section of pipe will be subjected to axialtension when in service. Start by considering a load magnitude of 4 MN.

The lowest vibrational mode of the pipe will be a sine wave deformation in any direction transverse tothe pipe axis because of the symmetry of the structure's cross-section. You use three-dimensional beamelements to model the pipe section.

10.3.1 Coordinate systemThe default, global coordinate system is used. Place the origin at the left end of the pipe section, andmake the axis of the pipe and the global 1-axis coincident, as shown in Figure 10-7.

10.3.2 Mesh designModel the pipe section with a uniformly spaced mesh of 30 second-order, pipe elements ( PIPE32). Thenode and element numbers of the model used in this discussion are shown in Figure 10-8.

Figure 10-8 Node and element numbers (both increase by 1 from left to right).

10.3.3 Preprocessing--creating the modelYou can create the mesh for this example using your preprocessor, or if you prefer, you can use theABAQUS mesh generation options shown in ``Vibration of a piping system,'' Section A.10.


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10.3.4 Reviewing the input file--the model dataThe model definition--including the model description, node and element definitions, sectionproperties, and material properties--is discussed next.

Model description

Check that the *HEADING option includes a suitable title in the data lines. Yours should looklike the following:

*HEADING

Analysis of a 5 meter long pipe under tensile load

Pipe has OD of 180 mm and ID of 140 mm

S.I. Units


Check that the correct element type (PIPE32) has been used and that the element set names aresuitably descriptive.

*ELEMENT, TYPE=PIPE32, ELSET=PIPE

Create node sets containing the nodes at either end of the pipe section. The following optionblocks create the node sets for the model shown in Figure 10-8:

*NSET, NSET=LEFT

1

*NSET, NSET=RIGHT

61

Beam properties

The *BEAM SECTION, SECTION=PIPE option will be used with the PIPE32 elements. Theouter radius (90 mm) and the wall thickness (20 mm) are needed to define this beam sectiontype geometrically. It is easier to define the orientation of the beam section geometry for thismodel than it was for the cargo crane model in the earlier chapters because the pipe section issymmetric. Define the approximate n1-direction as the vector (0., 0., -1.0). In this model theactual n1-vector will coincide with this approximate vector. The following option blockshould be in your model:

*BEAM SECTION, ELSET=PIPE, MATERIAL=STEEL, SECTION=PIPE

0.09, 0.02

0.0, 0.0, -1.0

Material data

The option blocks defining the material behavior of the steel pipe in your model should looksimilar to the following:


10.3.4 Reviewing the input file--the model dataThe model definition--including the model description, node and element definitions, sectionproperties, and material properties--is discussed next.

Model description

Check that the *HEADING option includes a suitable title in the data lines. Yours should looklike the following:

*HEADING



S.I. Units


Check that the correct element type (PIPE32) has been used and that the element set names aresuitably descriptive.

*ELEMENT, TYPE=PIPE32, ELSET=PIPE

Create node sets containing the nodes at either end of the pipe section. The following optionblocks create the node sets for the model shown in Figure 10-8:

*NSET, NSET=LEFT

1

*NSET, NSET=RIGHT

61

Beam properties

The *BEAM SECTION, SECTION=PIPE option will be used with the PIPE32 elements. Theouter radius (90 mm) and the wall thickness (20 mm) are needed to define this beam sectiontype geometrically. It is easier to define the orientation of the beam section geometry for thismodel than it was for the cargo crane model in the earlier chapters because the pipe section issymmetric. Define the approximate n1-direction as the vector (0., 0., -1.0). In this model theactual n1-vector will coincide with this approximate vector. The following option blockshould be in your model:


0.09, 0.02

0.0, 0.0, -1.0

Material data

The option blocks defining the material behavior of the steel pipe in your model should looksimilar to the following:



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*ELASTIC

200.E9, 0.3

You must define the density of the steel material (7800 kg/m3) because eigenmodes andeigenfrequencies are being extracted in this simulation and a mass matrix is needed for thisprocedure. Therefore, the following option block must follow the *ELASTIC option block:

*DENSITY

7800.0

10.3.5 Reviewing the input file--the history dataIn this simulation you need to investigate the eigenmodes and eigenfrequencies of the steel pipesection when a 4 MN tensile load is applied. Therefore, the load history data will be split into twosteps:Step 1. General step: Apply a 4 MN tensile force.Step 2. Linear perturbation step: Calculate modes and frequencies.

The actual magnitude of time in these steps will have no effect on the results; unless the modelincludes damping or rate-dependent material properties, "time" has no physical meaning in a staticanalysis procedure. Therefore, use a step time of 1.0 in the general analysis steps.

Step 1 - Apply a 4 MN tensile force

The options necessary to define the first analysis step--including the procedure definition,boundary conditions, loading, and output requests--are reviewed.

Step and analysis procedure definition

The first step is a general static step that includes the effect of geometric nonlinearity. Specifyan initial increment size that is 1/10 the total step time, causing ABAQUS to apply 10% of theload in the first increment. Use the following option blocks to define the analysis procedure,and remember to provide a meaningful description of the step because this makes reviewingthe load history much easier:


Apply axial tensile load of 4.0MN

*STATIC

0.1, 1.0

Boundary conditions

The pipe section is clamped at its left end (node 1 in the model shown in Figure 10-8). It isalso clamped at the other end; however, the axial force must be applied at this end, so onlydegrees of freedom 2 to 6 are constrained.

*BOUNDARY

LEFT, 1, 6

RIGHT, 2, 6

*ELASTIC

200.E9, 0.3

You must define the density of the steel material (7800 kg/m3) because eigenmodes andeigenfrequencies are being extracted in this simulation and a mass matrix is needed for thisprocedure. Therefore, the following option block must follow the *ELASTIC option block:

*DENSITY

7800.0

10.3.5 Reviewing the input file--the history dataIn this simulation you need to investigate the eigenmodes and eigenfrequencies of the steel pipesection when a 4 MN tensile load is applied. Therefore, the load history data will be split into twosteps:Step 1. General step: Apply a 4 MN tensile force.Step 2. Linear perturbation step: Calculate modes and frequencies.

The actual magnitude of time in these steps will have no effect on the results; unless the modelincludes damping or rate-dependent material properties, "time" has no physical meaning in a staticanalysis procedure. Therefore, use a step time of 1.0 in the general analysis steps.

Step 1 - Apply a 4 MN tensile force

The options necessary to define the first analysis step--including the procedure definition,boundary conditions, loading, and output requests--are reviewed.


The first step is a general static step that includes the effect of geometric nonlinearity. Specifyan initial increment size that is 1/10 the total step time, causing ABAQUS to apply 10% of theload in the first increment. Use the following option blocks to define the analysis procedure,and remember to provide a meaningful description of the step because this makes reviewingthe load history much easier:



*STATIC

0.1, 1.0

Boundary conditions

The pipe section is clamped at its left end (node 1 in the model shown in Figure 10-8). It isalso clamped at the other end; however, the axial force must be applied at this end, so onlydegrees of freedom 2 to 6 are constrained.

*BOUNDARY

LEFT, 1, 6

RIGHT, 2, 6


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Tensile loading

Apply a 4 MN tensile force to the right end of the pipe section such that it deforms in thepositive axial (global 1) direction. Forces are applied, by default, in the global coordinatesystem. Therefore, the *CLOAD option block looks like

*CLOAD

RIGHT, 1, 4.0E6

In this case the load is applied directly to the node set defined earlier. Use the node set namefrom your model or the node number in the *CLOAD option in your input file.

Output requests

Write data to the restart file every 10 increments. In addition, write the preselected field dataevery 10 increments as well as the stress components and stress invariants for element 25 ashistory data to the output database file. Use the FREQUENCY parameter to suppress the nodeand element printout to the data file. The following option blocks define these outputrequests:

*ELSET, ELSET=ELEMENT25

25

*RESTART, WRITE, FREQUENCY=10


*OUTPUT, HISTORY

*ELEMENT OUTPUT, ELSET=ELEMENT25

S, SINV



End the step with the *END STEP option.

Step 2 - Extract modes and frequencies

The second step extracts the natural frequencies of the extended pipe. The required options arediscussed below.


In the second step you need to calculate the eigenmodes and eigenfrequencies of the pipe in itsloaded state. The eigenfrequency extraction procedure (*FREQUENCY option) used in thisstep is a linear perturbation procedure. Although only the first (lowest) eigenmode is ofinterest to you, extract the first eight eigenmodes for the model. Specify this number on thedata line of the *FREQUENCY option block. Since a small number of eigenvalues isrequested, use the subspace iteration eigensolver. The option blocks to define the analysisprocedure should look similar to the following:

*STEP, PERTURBATION, NLGEOM

Extract modes and frequencies

*FREQUENCY, EIGENSOLVER=SUBSPACE

Tensile loading

Apply a 4 MN tensile force to the right end of the pipe section such that it deforms in thepositive axial (global 1) direction. Forces are applied, by default, in the global coordinatesystem. Therefore, the *CLOAD option block looks like

*CLOAD

RIGHT, 1, 4.0E6

In this case the load is applied directly to the node set defined earlier. Use the node set namefrom your model or the node number in the *CLOAD option in your input file.

Output requests

Write data to the restart file every 10 increments. In addition, write the preselected field dataevery 10 increments as well as the stress components and stress invariants for element 25 ashistory data to the output database file. Use the FREQUENCY parameter to suppress the nodeand element printout to the data file. The following option blocks define these outputrequests:


25



*OUTPUT, HISTORY


S, SINV



End the step with the *END STEP option.

Step 2 - Extract modes and frequencies

The second step extracts the natural frequencies of the extended pipe. The required options arediscussed below.


In the second step you need to calculate the eigenmodes and eigenfrequencies of the pipe in itsloaded state. The eigenfrequency extraction procedure (*FREQUENCY option) used in thisstep is a linear perturbation procedure. Although only the first (lowest) eigenmode is ofinterest to you, extract the first eight eigenmodes for the model. Specify this number on thedata line of the *FREQUENCY option block. Since a small number of eigenvalues isrequested, use the subspace iteration eigensolver. The option blocks to define the analysisprocedure should look similar to the following:





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8

Loads

You require the natural frequencies of the extended pipe section. This does not involve theapplication of any perturbation loads, and the fixed boundary conditions are carried over fromthe previous general step. Therefore, you do not need to specify any loads or boundaryconditions in this step.

Output requests

Any output requests required in a linear perturbation step must be redefined since the requestsfrom the previous general step do not carry over. You want data to be written to the restart andoutput database files but do not want node or element output to be written to the data file. Thefollowing option blocks define these requests:

*RESTART, WRITE

*OUTPUT, FIELD, VARIABLE=PRESELECT



Again, mark the termination of the step definition with the *END STEP option.

10.3.6 Running the analysisStore the input option blocks in a file called pipe.inp. The input file that creates the model shownin Figure 10-8 is listed in ``Vibration of a piping system,'' Section A.10. Run the analysis in thebackground using the command

abaqus job=pipe

10.3.7 Status fileCheck the status file as the job is running. When the analysis completes, the contents of the status filewill look similar to


STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF DO

DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF MONIT

ITERS FREQ

1 1 1 0 1 1 0.100 0.100 0.1000

1 2 1 0 1 1 0.200 0.200 0.1000

1 3 1 0 1 1 0.350 0.350 0.1500

1 4 1 0 1 1 0.575 0.575 0.2250

1 5 1 0 1 1 0.913 0.913 0.3375

1 6 1 0 1 1 1.00 1.00 0.08750

2 1 1 0 4 0 1.00 1.00e-36 1.000e-36

Both steps are shown, and the time associated with the linear perturbation step (Step 2) is very small:

8

Loads

You require the natural frequencies of the extended pipe section. This does not involve theapplication of any perturbation loads, and the fixed boundary conditions are carried over fromthe previous general step. Therefore, you do not need to specify any loads or boundaryconditions in this step.

Output requests

Any output requests required in a linear perturbation step must be redefined since the requestsfrom the previous general step do not carry over. You want data to be written to the restart andoutput database files but do not want node or element output to be written to the data file. Thefollowing option blocks define these requests:

*RESTART, WRITE




Again, mark the termination of the step definition with the *END STEP option.

10.3.6 Running the analysisStore the input option blocks in a file called pipe.inp. The input file that creates the model shownin Figure 10-8 is listed in ``Vibration of a piping system,'' Section A.10. Run the analysis in thebackground using the command

abaqus job=pipe

10.3.7 Status fileCheck the status file as the job is running. When the analysis completes, the contents of the status filewill look similar to




ITERS FREQ

1 1 1 0 1 1 0.100 0.100 0.1000

1 2 1 0 1 1 0.200 0.200 0.1000

1 3 1 0 1 1 0.350 0.350 0.1500

1 4 1 0 1 1 0.575 0.575 0.2250

1 5 1 0 1 1 0.913 0.913 0.3375

1 6 1 0 1 1 1.00 1.00 0.08750

2 1 1 0 4 0 1.00 1.00e-36 1.000e-36

Both steps are shown, and the time associated with the linear perturbation step (Step 2) is very small:


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the *FREQUENCY procedure, or any linear perturbation procedure, does not contribute to the generalloading history of the model.

10.3.8 PostprocessingRun ABAQUS/Viewer using the command

abaqus viewer odb=pipe

Deformed shapes from the linear perturbation steps

When ABAQUS/Viewer starts, it automatically uses the last available frame on the outputdatabase file. The results from the second step of this simulation are the natural mode shapesof the pipe and the corresponding natural frequencies. Plot the first mode shape.

To plot the first mode shape:



2. Select step Extract modes and frequencies and frame Mode 1.

3. Click OK.

4. From the main menu bar, select Plot->Deformed Shape.

5. To superimpose the undeformed model shape on the deformed model shape, click DeformedOptions in the prompt area.



7. Click OK.

The undeformed and deformed model shapes are displayed.

8. To include node symbols for the undeformed shape plot, select Options->Undeformed.

The Undeformed Plot Options dialog box appears; by default, the Basic tab is selected.Click the Labels tab, and toggle on Show node symbols.

It is now possible to select the color, shape, and size of the symbol. Change the color to greenand the symbol shape to a solid circle.

9. Click OK.

10. Include node symbols for the deformed shape plot using the same procedure described for theundeformed shape plot, selecting Options->Deformed from the main menu.

The default view is isometric. Try rotating the model to find a better view of the first

the *FREQUENCY procedure, or any linear perturbation procedure, does not contribute to the generalloading history of the model.

10.3.8 PostprocessingRun ABAQUS/Viewer using the command

abaqus viewer odb=pipe

Deformed shapes from the linear perturbation steps

When ABAQUS/Viewer starts, it automatically uses the last available frame on the outputdatabase file. The results from the second step of this simulation are the natural mode shapesof the pipe and the corresponding natural frequencies. Plot the first mode shape.

To plot the first mode shape:



2. Select step Extract modes and frequencies and frame Mode 1.

3. Click OK.

4. From the main menu bar, select Plot->Deformed Shape.

5. To superimpose the undeformed model shape on the deformed model shape, click DeformedOptions in the prompt area.



7. Click OK.

The undeformed and deformed model shapes are displayed.

8. To include node symbols for the undeformed shape plot, select Options->Undeformed.

The Undeformed Plot Options dialog box appears; by default, the Basic tab is selected.Click the Labels tab, and toggle on Show node symbols.

It is now possible to select the color, shape, and size of the symbol. Change the color to greenand the symbol shape to a solid circle.

9. Click OK.

10. Include node symbols for the deformed shape plot using the same procedure described for theundeformed shape plot, selecting Options->Deformed from the main menu.

The default view is isometric. Try rotating the model to find a better view of the first


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eigenmode. You should be able to rotate the model to obtain a plot similar to the one shown inFigure 10-9.

Figure 10-9 First and second eigenmode shapes of the pipe section under the tensile load.

Since this is a linear perturbation step, the undeformed shape is the shape of the structure inthe base state. This makes it easy to see the motion relative to the base state. Use the ODBFrame options in the prompt area to plot the other mode shapes. You will discover that thismodel has many repeated eigenmodes. This is a result of the symmetric nature of the pipe'scross-section. Some of the higher vibrational mode shapes are shown in Figure 10-10.

Figure 10-10 Shapes of eigenmodes 3 through 6.

The natural frequency associated with each eigenmode will be reported in the plot title. Thelowest natural frequency of the pipe section when the 4 MN tensile load is applied is 47.1 Hz.The tensile loading has increased the stiffness of the pipe and, thus, increased the vibrationalfrequencies of the pipe section. This lowest natural frequency is within the frequency range ofthe harmonic loads; therefore, resonance of the pipe may be a problem when it is used withthis loading.

You, therefore, need to continue the simulation and apply additional tensile load to the pipesection until you find the magnitude that raises the natural frequency of the pipe section to anacceptable level. Rather than repeating the analysis and increasing the applied axial load, youcan use the restart capability in ABAQUS to continue the load history of a prior simulation ina new analysis.

10.4 Restarting the analysisMulti-step simulations need not be defined in a single input file. Indeed, it is usually desirable to run a

eigenmode. You should be able to rotate the model to obtain a plot similar to the one shown inFigure 10-9.

Figure 10-9 First and second eigenmode shapes of the pipe section under the tensile load.

Since this is a linear perturbation step, the undeformed shape is the shape of the structure inthe base state. This makes it easy to see the motion relative to the base state. Use the ODBFrame options in the prompt area to plot the other mode shapes. You will discover that thismodel has many repeated eigenmodes. This is a result of the symmetric nature of the pipe'scross-section. Some of the higher vibrational mode shapes are shown in Figure 10-10.

Figure 10-10 Shapes of eigenmodes 3 through 6.

The natural frequency associated with each eigenmode will be reported in the plot title. Thelowest natural frequency of the pipe section when the 4 MN tensile load is applied is 47.1 Hz.The tensile loading has increased the stiffness of the pipe and, thus, increased the vibrationalfrequencies of the pipe section. This lowest natural frequency is within the frequency range ofthe harmonic loads; therefore, resonance of the pipe may be a problem when it is used withthis loading.

You, therefore, need to continue the simulation and apply additional tensile load to the pipesection until you find the magnitude that raises the natural frequency of the pipe section to anacceptable level. Rather than repeating the analysis and increasing the applied axial load, youcan use the restart capability in ABAQUS to continue the load history of a prior simulation ina new analysis.

10.4 Restarting the analysisMulti-step simulations need not be defined in a single input file. Indeed, it is usually desirable to run a


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complex simulation in stages. This allows you to examine the results and confirm that the analysis isperforming as expected before continuing with the next stage. The restart file allows an ABAQUSsimulation to be restarted and the model's response to additional load history to be calculated.

10.4.1 Writing a restart fileThe *RESTART option controls the writing of the restart file. While the option can appear anywherein the input file, it normally appears as part of a step definition. As with other output request options,the values defined in a *RESTART option apply during the current step and any subsequent generalsteps, until the option is modified in a later step. The option

*RESTART, WRITE, FREQUENCY=<n>

writes data to the restart file every nth increment. Restart data are also written at the end of each step,regardless of whether the last increment is divisible by n. If the FREQUENCY parameter is omitted,data are written every increment.

Restart files can become very large for large models, so it is often useful to include the OVERLAYparameter to control the size of the restart file. This parameter allows data to be overwritten on therestart file during a step. This means that at the end of the analysis there is only one set of restart datafor each step, corresponding to the state of the model at the end of each step. However, if the analysisis interrupted for some reason, such as a computer malfunction, the analysis can be continued from thepoint where restart data were last written.

10.4.2 Reading a restart fileWhen restarting a simulation from the end of a previous analysis, use the READ parameter on the*RESTART option. You can also use the STEP and INC parameters to specify the particular point inthe simulation's load history from which to restart the analysis. When performing a restart simulation,the *RESTART option should appear immediately after the *HEADING option. No model data canappear in this restart input file since the model data for the analysis will be read from the restart file.Only node set definitions, element set definitions, amplitude definitions, and additional history datacan be given in the restart input file.

Continuing an interrupted run

The new analysis continues directly from the specified step and increment of the previousanalysis. If the given step and increment do not correspond to the end of the previous analysis,ABAQUS will simulate all of the remaining previously defined load history data before tryingto simulate any new load history data provided in the input file. Therefore, if an analysis wasinterrupted by a computer malfunction, the following input file would complete the analysis asoriginally defined:

*HEADING

Restart of interrupted run

*RESTART, READ, STEP=<step>, INC=<increment>

Continuing with additional steps

complex simulation in stages. This allows you to examine the results and confirm that the analysis isperforming as expected before continuing with the next stage. The restart file allows an ABAQUSsimulation to be restarted and the model's response to additional load history to be calculated.

10.4.1 Writing a restart fileThe *RESTART option controls the writing of the restart file. While the option can appear anywherein the input file, it normally appears as part of a step definition. As with other output request options,the values defined in a *RESTART option apply during the current step and any subsequent generalsteps, until the option is modified in a later step. The option

*RESTART, WRITE, FREQUENCY=<n>

writes data to the restart file every nth increment. Restart data are also written at the end of each step,regardless of whether the last increment is divisible by n. If the FREQUENCY parameter is omitted,data are written every increment.

Restart files can become very large for large models, so it is often useful to include the OVERLAYparameter to control the size of the restart file. This parameter allows data to be overwritten on therestart file during a step. This means that at the end of the analysis there is only one set of restart datafor each step, corresponding to the state of the model at the end of each step. However, if the analysisis interrupted for some reason, such as a computer malfunction, the analysis can be continued from thepoint where restart data were last written.

10.4.2 Reading a restart fileWhen restarting a simulation from the end of a previous analysis, use the READ parameter on the*RESTART option. You can also use the STEP and INC parameters to specify the particular point inthe simulation's load history from which to restart the analysis. When performing a restart simulation,the *RESTART option should appear immediately after the *HEADING option. No model data canappear in this restart input file since the model data for the analysis will be read from the restart file.Only node set definitions, element set definitions, amplitude definitions, and additional history datacan be given in the restart input file.

Continuing an interrupted run

The new analysis continues directly from the specified step and increment of the previousanalysis. If the given step and increment do not correspond to the end of the previous analysis,ABAQUS will simulate all of the remaining previously defined load history data before tryingto simulate any new load history data provided in the input file. Therefore, if an analysis wasinterrupted by a computer malfunction, the following input file would complete the analysis asoriginally defined:

*HEADING

Restart of interrupted run

*RESTART, READ, STEP=<step>, INC=<increment>

Continuing with additional steps


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If the previous analysis completed successfully and, having viewed the results, you want toadd additional steps to the load history, the specified step and increment should be the last stepand last increment of the previous analysis. Alternatively, they can be omitted and by defaultABAQUS will read the last available data in the restart file. The *RESTART option isfollowed by any new step definitions.

*HEADING

Add new step data

*RESTART, READ, STEP=<last step>, INC=<last increment>

*STEP

... new step definition...

*END STEP

Changing an analysis

Sometimes, having viewed the results of the previous analysis, you may want to restart theanalysis from an intermediate point and change the remaining load history data in somemanner--for example, to add more output requests, to change the loading, or to adjust theanalysis controls. Often this is necessary when a step has exceeded its maximum number ofincrements. If the analysis is restarted as described above, ABAQUS thinks that the analysis ispartway through a step, tries to complete the step, and promptly exceeds the maximum numberof increments again.

In such situations the END STEP parameter should be included on the *RESTART option toindicate that the current step should be terminated at the step and increment specified on the*RESTART option and all previously defined history data should be ignored. The simulationmay then continue with the new steps defined after the *RESTART option. For example, if astep allowed only a maximum of 20 increments, which was exceeded, the following restartinput file would allow ABAQUS to restart the simulation and finish the applied load:

*HEADING

Continue an analysis that exceeded the maximum number of

increments

*RESTART, READ, STEP=<step>, INC=20, END STEP

*STEP, INC=100

... repeat step definition...

END STEP

In this situation the entire step definition, including applied loads and boundary conditions,should be identical to that specified in the original run with the following exceptions:

� The number of increments should be increased.

� The total time of the new step should be the total time of the original step less the timecompleted in the first run. For example, if the time of the step as originally specified was100 seconds and the analysis ran out of increments at a step time of 20 seconds, theduration of the step in the restart analysis should be 80 seconds.

If the previous analysis completed successfully and, having viewed the results, you want toadd additional steps to the load history, the specified step and increment should be the last stepand last increment of the previous analysis. Alternatively, they can be omitted and by defaultABAQUS will read the last available data in the restart file. The *RESTART option isfollowed by any new step definitions.

*HEADING

Add new step data

*RESTART, READ, STEP=<last step>, INC=<last increment>

*STEP

... new step definition...

*END STEP

Changing an analysis

Sometimes, having viewed the results of the previous analysis, you may want to restart theanalysis from an intermediate point and change the remaining load history data in somemanner--for example, to add more output requests, to change the loading, or to adjust theanalysis controls. Often this is necessary when a step has exceeded its maximum number ofincrements. If the analysis is restarted as described above, ABAQUS thinks that the analysis ispartway through a step, tries to complete the step, and promptly exceeds the maximum numberof increments again.

In such situations the END STEP parameter should be included on the *RESTART option toindicate that the current step should be terminated at the step and increment specified on the*RESTART option and all previously defined history data should be ignored. The simulationmay then continue with the new steps defined after the *RESTART option. For example, if astep allowed only a maximum of 20 increments, which was exceeded, the following restartinput file would allow ABAQUS to restart the simulation and finish the applied load:

*HEADING

Continue an analysis that exceeded the maximum number of

increments

*RESTART, READ, STEP=<step>, INC=20, END STEP

*STEP, INC=100

... repeat step definition...

END STEP

In this situation the entire step definition, including applied loads and boundary conditions,should be identical to that specified in the original run with the following exceptions:

� The number of increments should be increased.

� The total time of the new step should be the total time of the original step less the timecompleted in the first run. For example, if the time of the step as originally specified was100 seconds and the analysis ran out of increments at a step time of 20 seconds, theduration of the step in the restart analysis should be 80 seconds.


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� Any amplitude definitions specified in terms of step time need to be respecified to reflectthe new time scale of the step. Amplitude definitions specified in terms of total time donot need to be changed, provided the modifications given above are used.

The magnitudes of any loads or prescribed boundary conditions remain unchanged since theyare always total values in general analysis steps.

10.5 Example: restarting the pipe vibration analysisTo demonstrate how to restart an analysis, take the pipe section example in ``Example: vibration of apiping system,'' Section 10.3, and restart the simulation, adding two additional steps of load history.The first simulation predicted that the piping section would be vulnerable to resonance when extendedaxially; you must now determine how much additional axial load will increase the pipe's lowestvibrational frequency to an acceptable level.

Step 3 will be a general step that increases the axial load on the pipe to 8 MN, and Step 4 will calculatethe eigenmodes and eigenfrequencies again.

Create a new input file, called pipe-2.inp, and add the option blocks discussed below.

10.5.1 Reviewing the input file--the model dataThe only model data required are the *HEADING option and a *RESTART option to read the restartdata from the end of the previous analysis. ABAQUS reads all other model data, such as node andelement definitions, directly from the restart file. Add the following option blocks to your new inputfile:

*HEADING

Increase tensile load on the piping system

and determine lowest frequency.

*RESTART, READ

Neither the INCREMENT nor the STEP parameter is included on the *RESTART, READ option sinceby default ABAQUS will read the data for the last increment written to the restart file. Since you arecontinuing the simulation from the end of the previous analysis, no parameters are needed.

10.5.2 Reviewing the input file--the history dataThe history data consist of two steps. Apply a tensile load (8 MN) to the pipe section in Step 3. Thefollowing option block must be placed in Step 3:

*CLOAD

RIGHT, 1, 8.0E6

Set the initial time increment in Step 3 to 1/10 the total step time, which should be 1.0. Step 4 is anexact copy of Step 2 from the previous analysis. All of the load history option blocks necessary todefine this restart analysis are shown below.


� Any amplitude definitions specified in terms of step time need to be respecified to reflectthe new time scale of the step. Amplitude definitions specified in terms of total time donot need to be changed, provided the modifications given above are used.

The magnitudes of any loads or prescribed boundary conditions remain unchanged since theyare always total values in general analysis steps.

10.5 Example: restarting the pipe vibration analysisTo demonstrate how to restart an analysis, take the pipe section example in ``Example: vibration of apiping system,'' Section 10.3, and restart the simulation, adding two additional steps of load history.The first simulation predicted that the piping section would be vulnerable to resonance when extendedaxially; you must now determine how much additional axial load will increase the pipe's lowestvibrational frequency to an acceptable level.

Step 3 will be a general step that increases the axial load on the pipe to 8 MN, and Step 4 will calculatethe eigenmodes and eigenfrequencies again.

Create a new input file, called pipe-2.inp, and add the option blocks discussed below.

10.5.1 Reviewing the input file--the model dataThe only model data required are the *HEADING option and a *RESTART option to read the restartdata from the end of the previous analysis. ABAQUS reads all other model data, such as node andelement definitions, directly from the restart file. Add the following option blocks to your new inputfile:

*HEADING



*RESTART, READ

Neither the INCREMENT nor the STEP parameter is included on the *RESTART, READ option sinceby default ABAQUS will read the data for the last increment written to the restart file. Since you arecontinuing the simulation from the end of the previous analysis, no parameters are needed.

10.5.2 Reviewing the input file--the history dataThe history data consist of two steps. Apply a tensile load (8 MN) to the pipe section in Step 3. Thefollowing option block must be placed in Step 3:

*CLOAD

RIGHT, 1, 8.0E6

Set the initial time increment in Step 3 to 1/10 the total step time, which should be 1.0. Step 4 is anexact copy of Step 2 from the previous analysis. All of the load history option blocks necessary todefine this restart analysis are shown below.



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Apply 8 MN axial tensile load

*STATIC

0.1,1.

*CLOAD

RIGHT, 1, 8.0E6



*OUTPUT, HISTORY


S, SINV



*END STEP




8,

*RESTART, WRITE




*END STEP

The complete input file for this restart analysis is listed in ``Restarting the pipe vibration analysis, ''Section A.11.

10.5.3 Running the analysisWhen running a simulation that will need to read data from a restart file, you must specify the rootname of the restart file, without the .res extension, with the oldjob parameter on the ABAQUScommand line. Thus, use the following command to run this restart analysis:

abaqus job=pipe-2 oldjob=pipe

10.5.4 Status fileAgain, check the status file as the job is running. When the analysis completes, the contents of thestatus file will look like




ITERS FREQ

3 1 1 0 1 1 1.10 0.100 0.1000

3 2 1 0 1 1 1.20 0.200 0.1000

3 3 1 0 1 1 1.35 0.350 0.1500

3 4 1 0 1 1 1.58 0.575 0.2250

Apply 8 MN axial tensile load

*STATIC

0.1,1.

*CLOAD

RIGHT, 1, 8.0E6



*OUTPUT, HISTORY


S, SINV



*END STEP




8,

*RESTART, WRITE




*END STEP

The complete input file for this restart analysis is listed in ``Restarting the pipe vibration analysis, ''Section A.11.

10.5.3 Running the analysisWhen running a simulation that will need to read data from a restart file, you must specify the rootname of the restart file, without the .res extension, with the oldjob parameter on the ABAQUScommand line. Thus, use the following command to run this restart analysis:

abaqus job=pipe-2 oldjob=pipe

10.5.4 Status fileAgain, check the status file as the job is running. When the analysis completes, the contents of thestatus file will look like




ITERS FREQ

3 1 1 0 1 1 1.10 0.100 0.1000

3 2 1 0 1 1 1.20 0.200 0.1000

3 3 1 0 1 1 1.35 0.350 0.1500

3 4 1 0 1 1 1.58 0.575 0.2250


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3 5 1 0 1 1 1.91 0.913 0.3375

3 6 1 0 1 1 2.00 1.00 0.08750

4 1 1 0 6 0 2.00 1.00e-36 1.000e-36

This analysis starts at Step 3 since Steps 1 and 2 were completed in the previous analysis. There arenow two output database (.odb) files associated with this simulation. Data for Steps 1 and 2 are in thefile pipe.odb; data for Steps 3 and 4 are in the file pipe-2.odb. When plotting results inABAQUS/Viewer, you need to remember which results are stored in each file, and you need to ensurethat ABAQUS/Viewer is using the correct output database file.

10.5.5 Postprocessing the restart analysis resultsStart ABAQUS/Viewer and specify that the output database file from the restart analysis should beused by giving the following command:

abaqus viewer odb=pipe-2

Plotting the eigenmodes of the pressurized pipe

Plot the same six eigenmode shapes of the pipe section for this simulation as were plotted inthe previous analysis. The eigenmode shapes can be plotted using the procedures described forthe original analysis. These eigenmodes and their natural frequencies are shown in Figure10-11.

Figure 10-11 Shapes and frequencies of eigenmodes 1 through 6 with 8 MN tensile load.

Under 8 MN of axial load, the lowest mode is now at 53.1 Hz, which is greater than therequired minimum of 50 Hz. If you want to find the exact load at which the lowest mode isjust above 50 Hz, you can repeat this restart analysis and change the value of the appliedload.

Plotting X-Y graphs for selected steps

3 5 1 0 1 1 1.91 0.913 0.3375

3 6 1 0 1 1 2.00 1.00 0.08750

4 1 1 0 6 0 2.00 1.00e-36 1.000e-36

This analysis starts at Step 3 since Steps 1 and 2 were completed in the previous analysis. There arenow two output database (.odb) files associated with this simulation. Data for Steps 1 and 2 are in thefile pipe.odb; data for Steps 3 and 4 are in the file pipe-2.odb. When plotting results inABAQUS/Viewer, you need to remember which results are stored in each file, and you need to ensurethat ABAQUS/Viewer is using the correct output database file.

10.5.5 Postprocessing the restart analysis resultsStart ABAQUS/Viewer and specify that the output database file from the restart analysis should beused by giving the following command:

abaqus viewer odb=pipe-2

Plotting the eigenmodes of the pressurized pipe

Plot the same six eigenmode shapes of the pipe section for this simulation as were plotted inthe previous analysis. The eigenmode shapes can be plotted using the procedures described forthe original analysis. These eigenmodes and their natural frequencies are shown in Figure10-11.

Figure 10-11 Shapes and frequencies of eigenmodes 1 through 6 with 8 MN tensile load.

Under 8 MN of axial load, the lowest mode is now at 53.1 Hz, which is greater than therequired minimum of 50 Hz. If you want to find the exact load at which the lowest mode isjust above 50 Hz, you can repeat this restart analysis and change the value of the appliedload.

Plotting X-Y graphs for selected steps


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Use the results stored in the output database files, pipe.odb and pipe-2.odb, to plot thehistory of von Mises stress in the pipe for the whole simulation.

To generate a history plot of the von Mises stress in element 25 for the restart analysis:



2. Plot the von Mises stress history of element 25 at integration point 1 at an angle of 0° withrespect to the local 1-axis of the element.

The plot traces the von Mises stress history of the element in the restart analysis. Since therestart is a continuation of an earlier job, it is often useful to view the results from the entire(original and restarted) analysis.

To generate a history plot of the von Mises stress in element 25 for the entire analysis:

1. Save the current plot by selecting Result->History Output from the main menu. Ensure thatthe correct quantity has been selected, and click Save As. Name the plot RESTART.

2. From the main menu bar, select File->Open; or use the tool in the toolbar to open the filepipe.odb in ABAQUS/Viewer.

3. Following the procedure outlined above, save the plot of the von Mises stress history ofelement 25. Name this plot ORIGINAL.

4. From the main menu bar, select Tools->XY Data->Manager.

The ORIGINAL and RESTART plots are listed in the XY Data Manager dialog box.

To create the plot over the correct time scale, you need to shift the RESTART curve along theX-axis by an amount equal to the total time (X-coordinate) of the ORIGINAL curve.

5. Click Create in the XY Data Manager dialog box. Select Operate on XY Data, and clickContinue.


6. Enter the following in the text field that appears at the top of the dialog box:

swap(swap("RESTART")+1.0)

In ABAQUS/Viewer, a numerical shift is possible only for the Y-axis. Thus, we swap the axesof curve RESTART and add a value of 1.0 (the maximum X-value from the ORIGINAL curve)to the swapped Y-axis. The axes are then swapped again. As you gain more experience inmanipulating X-Y data, you may find it useful to use the tools in the Operators dialog box.These tools reduce the amount of keyboard input required to create a logical expression in thetext field.

Use the results stored in the output database files, pipe.odb and pipe-2.odb, to plot thehistory of von Mises stress in the pipe for the whole simulation.

To generate a history plot of the von Mises stress in element 25 for the restart analysis:



2. Plot the von Mises stress history of element 25 at integration point 1 at an angle of 0° withrespect to the local 1-axis of the element.

The plot traces the von Mises stress history of the element in the restart analysis. Since therestart is a continuation of an earlier job, it is often useful to view the results from the entire(original and restarted) analysis.

To generate a history plot of the von Mises stress in element 25 for the entire analysis:

1. Save the current plot by selecting Result->History Output from the main menu. Ensure thatthe correct quantity has been selected, and click Save As. Name the plot RESTART.

2. From the main menu bar, select File->Open; or use the tool in the toolbar to open the filepipe.odb in ABAQUS/Viewer.

3. Following the procedure outlined above, save the plot of the von Mises stress history ofelement 25. Name this plot ORIGINAL.

4. From the main menu bar, select Tools->XY Data->Manager.

The ORIGINAL and RESTART plots are listed in the XY Data Manager dialog box.

To create the plot over the correct time scale, you need to shift the RESTART curve along theX-axis by an amount equal to the total time (X-coordinate) of the ORIGINAL curve.

5. Click Create in the XY Data Manager dialog box. Select Operate on XY Data, and clickContinue.


6. Enter the following in the text field that appears at the top of the dialog box:

swap(swap("RESTART")+1.0)

In ABAQUS/Viewer, a numerical shift is possible only for the Y-axis. Thus, we swap the axesof curve RESTART and add a value of 1.0 (the maximum X-value from the ORIGINAL curve)to the swapped Y-axis. The axes are then swapped again. As you gain more experience inmanipulating X-Y data, you may find it useful to use the tools in the Operators dialog box.These tools reduce the amount of keyboard input required to create a logical expression in thetext field.


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7. Click Plot Expression to see the shifted plot. Save this plot as RESTART-SHIFTED.

8. In the XY Data Manager dialog box, select the ORIGINAL and RESTART-SHIFTED plotswith [Ctrl]+Click, and click Plot to create a plot of von Mises stress history of element 25 forthe entire simulation.

9. To change the style of the line, click XY Curve Options in the prompt area.


10. For the RESTART-SHIFTED curve, select a dotted line style.

11. Click Dismiss.

12. To change the plot titles, click XY Plot Options in the prompt area.

The XY Plot Options dialog box appears; by default, the Scale tab is selected. Click theTitles tab.

13. Click Title Source for the X-axis, and select User-specified. Change the title to TOTALTIME. Similarly, change the Y-axis title to STRESS INVARIANT - MISES.

14. Click OK.

The plot created by these commands is shown in Figure 10-12.

Figure 10-12 History of von Mises stress in element 25.

The von Mises stress history of the same element during Step 3 can be plotted by itself byselecting only the RESTART-SHIFTED curve (see Figure 10-13).

7. Click Plot Expression to see the shifted plot. Save this plot as RESTART-SHIFTED.

8. In the XY Data Manager dialog box, select the ORIGINAL and RESTART-SHIFTED plotswith [Ctrl]+Click, and click Plot to create a plot of von Mises stress history of element 25 forthe entire simulation.

9. To change the style of the line, click XY Curve Options in the prompt area.


10. For the RESTART-SHIFTED curve, select a dotted line style.

11. Click Dismiss.

12. To change the plot titles, click XY Plot Options in the prompt area.

The XY Plot Options dialog box appears; by default, the Scale tab is selected. Click theTitles tab.

13. Click Title Source for the X-axis, and select User-specified. Change the title to TOTALTIME. Similarly, change the Y-axis title to STRESS INVARIANT - MISES.

14. Click OK.

The plot created by these commands is shown in Figure 10-12.

Figure 10-12 History of von Mises stress in element 25.

The von Mises stress history of the same element during Step 3 can be plotted by itself byselecting only the RESTART-SHIFTED curve (see Figure 10-13).


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Figure 10-13 History of von Mises stress in element 25 during Step 3.

10.6 Related ABAQUS examples

� ``Deep drawing of a cylindrical cup,'' Section 1.3.5 of the ABAQUS Example Problems Manual

� ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2 of the ABAQUSExample Problems Manual


� ``Random response to jet noise excitation, '' Section 1.4.10 of the ABAQUS Benchmarks Manual

10.7 Summary� An ABAQUS simulation can include any number of steps.

� An analysis step is a period of "time" during which the response of the model to a specified set ofloads and boundary conditions is calculated. The character of this response is determined by theparticular analysis procedure used during the step.

� The response of a structure in a general analysis step may be either linear or nonlinear.

� The starting condition for each general step is the ending condition of the previous general step.Thus, the model's response evolves during a sequence of general steps in a simulation.

� Linear perturbation steps calculate the linear response of the structure to a perturbation load. Theresponse is reported relative to the base state defined by the condition of the model at the end ofthe last general step.

� In general steps the OP parameter on any loading options--such as *BOUNDARY, *CLOAD, and*DLOAD--controls how the values specified with these options interact with those values defined

Figure 10-13 History of von Mises stress in element 25 during Step 3.

10.6 Related ABAQUS examples


� ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2 of the ABAQUSExample Problems Manual


� ``Random response to jet noise excitation, '' Section 1.4.10 of the ABAQUS Benchmarks Manual

10.7 Summary� An ABAQUS simulation can include any number of steps.

� An analysis step is a period of "time" during which the response of the model to a specified set ofloads and boundary conditions is calculated. The character of this response is determined by theparticular analysis procedure used during the step.

� The response of a structure in a general analysis step may be either linear or nonlinear.

� The starting condition for each general step is the ending condition of the previous general step.Thus, the model's response evolves during a sequence of general steps in a simulation.

� Linear perturbation steps calculate the linear response of the structure to a perturbation load. Theresponse is reported relative to the base state defined by the condition of the model at the end ofthe last general step.

� In general steps the OP parameter on any loading options--such as *BOUNDARY, *CLOAD, and*DLOAD--controls how the values specified with these options interact with those values defined


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in previous steps.

� Analyses can be restarted as long as a restart file is saved. Restart files can be used to continue aninterrupted analysis or to add additional load history to the simulation.

in previous steps.

� Analyses can be restarted as long as a restart file is saved. Restart files can be used to continue aninterrupted analysis or to add additional load history to the simulation.


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11. ContactMany engineering problems involve contact between two or more components. In these problems aforce normal to the contacting surfaces acts on the two bodies when they touch each other. If there isfriction between the surfaces, shear forces may be created that resist the tangential motion (sliding) ofthe bodies. The general aim of contact simulations is to identify the areas on the surfaces that are incontact and to calculate the contact pressures generated.

In a finite element analysis contact conditions are a special class of discontinuous constraint, allowingforces to be transmitted from one part of the model to another. The constraint is discontinuous becauseit is applied only when the two surfaces are in contact. When the two surfaces separate, no constraint isapplied. The analysis has to be able to detect when two surfaces are in contact and apply the contactconstraints accordingly. Similarly, the analysis must be able to detect when two surfaces separate andremove the contact constraints.

In an ABAQUS contact simulation surfaces must be created on the various components in the model.The pairs of surfaces that may contact each other, known as contact pairs, must be identified. Finally,the constitutive models governing the interactions between the various surfaces must be defined. Thesesurface interaction definitions include behavior such as friction.

11.1 Interaction between surfacesThe interaction between contacting surfaces consists of two components: one normal to the surfacesand one tangential to the surfaces. The tangential component consists of the relative motion (sliding) ofthe surfaces and, possibly, frictional shear stresses.

11.1.1 Interaction normal to the surfacesThe distance separating two surfaces is called the clearance. The contact constraint is applied when theclearance between two surfaces becomes zero. There is no limit in the contact formulation on themagnitude of contact pressure that can be transmitted between the surfaces. The surfaces separatewhen the contact pressure between them becomes zero or negative, and the constraint is removed. Thissurface interaction behavior, referred to as "hard" contact, is summarized in the contactpressure-clearance relationship shown in Figure 11-1.

Figure 11-1 Contact pressure-clearance relationship for "hard" contact.

11. ContactMany engineering problems involve contact between two or more components. In these problems aforce normal to the contacting surfaces acts on the two bodies when they touch each other. If there isfriction between the surfaces, shear forces may be created that resist the tangential motion (sliding) ofthe bodies. The general aim of contact simulations is to identify the areas on the surfaces that are incontact and to calculate the contact pressures generated.

In a finite element analysis contact conditions are a special class of discontinuous constraint, allowingforces to be transmitted from one part of the model to another. The constraint is discontinuous becauseit is applied only when the two surfaces are in contact. When the two surfaces separate, no constraint isapplied. The analysis has to be able to detect when two surfaces are in contact and apply the contactconstraints accordingly. Similarly, the analysis must be able to detect when two surfaces separate andremove the contact constraints.

In an ABAQUS contact simulation surfaces must be created on the various components in the model.The pairs of surfaces that may contact each other, known as contact pairs, must be identified. Finally,the constitutive models governing the interactions between the various surfaces must be defined. Thesesurface interaction definitions include behavior such as friction.

11.1 Interaction between surfacesThe interaction between contacting surfaces consists of two components: one normal to the surfacesand one tangential to the surfaces. The tangential component consists of the relative motion (sliding) ofthe surfaces and, possibly, frictional shear stresses.

11.1.1 Interaction normal to the surfacesThe distance separating two surfaces is called the clearance. The contact constraint is applied when theclearance between two surfaces becomes zero. There is no limit in the contact formulation on themagnitude of contact pressure that can be transmitted between the surfaces. The surfaces separatewhen the contact pressure between them becomes zero or negative, and the constraint is removed. Thissurface interaction behavior, referred to as "hard" contact, is summarized in the contactpressure-clearance relationship shown in Figure 11-1.

Figure 11-1 Contact pressure-clearance relationship for "hard" contact.

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The dramatic change in contact pressure that occurs when a contact condition changes from "open" (apositive clearance) to "closed" (clearance equal to zero) sometimes makes it difficult to completecontact simulations. Some techniques to overcome difficulties with contact simulations are discussedlater in this chapter. Other sources of information include ``Common difficulties associated withcontact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual, and the Contact inABAQUS/Standard lecture notes.

11.1.2 Sliding of the surfacesIn addition to determining whether contact has occurred at a particular point, the analysis also mustcalculate the relative sliding of the two surfaces. This can be a very complex calculation; therefore,ABAQUS makes a distinction between analyses where the magnitude of sliding is small and thosewhere the magnitude of sliding may be finite. It is much less expensive computationally to modelproblems where the sliding between the surfaces is small. What constitutes "small sliding" is oftendifficult to define, but a general guideline to follow is that problems where a point contacting a surfacedoes not slide more than a small fraction of a typical element dimension can use the "small-sliding"approximation.

11.1.3 FrictionIf the two interacting surfaces are rough, the analysis may need to take frictional forces, which resistthe relative sliding of the surfaces, into account. Coulomb friction is a common friction model used todescribe the interaction of contacting surfaces. The model characterizes the frictional behavior betweenthe surfaces using a coefficient of friction, ¹. The product ¹p, where p is the contact pressure betweenthe two surfaces, gives the limiting frictional shear stress for the contacting surfaces. The contactingsurfaces will not slip (slide relative to each other) until the shear stress across their interface equals thelimiting frictional shear stress, ¹p. For most surfaces ¹ is normally less than unity. The solid line inFigure 11-2 summarizes the behavior of the Coulomb friction model: there is zero relative motion(slip) of the surfaces when they are sticking (the shear stresses are below ¹p).

Figure 11-2 Frictional behavior.

The dramatic change in contact pressure that occurs when a contact condition changes from "open" (apositive clearance) to "closed" (clearance equal to zero) sometimes makes it difficult to completecontact simulations. Some techniques to overcome difficulties with contact simulations are discussedlater in this chapter. Other sources of information include ``Common difficulties associated withcontact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual, and the Contact inABAQUS/Standard lecture notes.

11.1.2 Sliding of the surfacesIn addition to determining whether contact has occurred at a particular point, the analysis also mustcalculate the relative sliding of the two surfaces. This can be a very complex calculation; therefore,ABAQUS makes a distinction between analyses where the magnitude of sliding is small and thosewhere the magnitude of sliding may be finite. It is much less expensive computationally to modelproblems where the sliding between the surfaces is small. What constitutes "small sliding" is oftendifficult to define, but a general guideline to follow is that problems where a point contacting a surfacedoes not slide more than a small fraction of a typical element dimension can use the "small-sliding"approximation.

11.1.3 FrictionIf the two interacting surfaces are rough, the analysis may need to take frictional forces, which resistthe relative sliding of the surfaces, into account. Coulomb friction is a common friction model used todescribe the interaction of contacting surfaces. The model characterizes the frictional behavior betweenthe surfaces using a coefficient of friction, ¹. The product ¹p, where p is the contact pressure betweenthe two surfaces, gives the limiting frictional shear stress for the contacting surfaces. The contactingsurfaces will not slip (slide relative to each other) until the shear stress across their interface equals thelimiting frictional shear stress, ¹p. For most surfaces ¹ is normally less than unity. The solid line inFigure 11-2 summarizes the behavior of the Coulomb friction model: there is zero relative motion(slip) of the surfaces when they are sticking (the shear stresses are below ¹p).

Figure 11-2 Frictional behavior.

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The discontinuity between the two states--sticking or slipping--can result in convergence problemsduring the simulation. You should include friction in your simulations only when it has a significantinfluence on the response of the model. If your contact simulation with friction encountersconvergence problems, one of the first modifications you should try in diagnosing the difficulty is torerun the analysis without friction.

Modeling the ideal friction behavior can be very difficult; therefore, by default, ABAQUS uses apenalty friction formulation with an allowable "elastic slip," shown by the dotted line in Figure 11-2.The "elastic slip" is the small amount of relative motion between the surfaces that occurs when thesurfaces should be sticking. ABAQUS automatically chooses the penalty stiffness (the slope of thedotted line) so that this allowable "elastic slip" is a very small fraction of the characteristic elementlength. The penalty friction formulation works well for most problems, including most metal formingapplications. In those problems where the ideal stick-slip frictional behavior must be included, the"Lagrange" friction model can be used. The "Lagrange" friction model is more expensive in terms ofthe computer resources used because ABAQUS uses additional variables for each surface node withfrictional contact. In addition, the solution converges more slowly so that additional iterations areusually required. This friction model is not discussed in this guide.

Often the friction coefficient at the initiation of slipping from a sticking condition is different from thefriction coefficient during established sliding. The former is typically referred to as the static frictioncoefficient, and the latter is referred to as the kinetic friction coefficient. In ABAQUS/Standard anexponential decay law is available to model the transition between static and kinetic friction. Thisfriction model is not discussed in this guide.

The inclusion of friction in a model adds unsymmetric terms to the system of equations being solved. If¹ is less than about 0.2, the magnitude and influence of these terms are quite small and the regular,symmetric solver works well. For higher coefficients of friction, the unsymmetric solver is invokedautomatically because it will improve the convergence rate. The unsymmetric solver can also beselected by including the UNSYMM=YES parameter on the *STEP option. The unsymmetric solverrequires twice as much computer memory and scratch disk space as the symmetric solver.

11.2 Contact definition in ABAQUS

The discontinuity between the two states--sticking or slipping--can result in convergence problemsduring the simulation. You should include friction in your simulations only when it has a significantinfluence on the response of the model. If your contact simulation with friction encountersconvergence problems, one of the first modifications you should try in diagnosing the difficulty is torerun the analysis without friction.

Modeling the ideal friction behavior can be very difficult; therefore, by default, ABAQUS uses apenalty friction formulation with an allowable "elastic slip," shown by the dotted line in Figure 11-2.The "elastic slip" is the small amount of relative motion between the surfaces that occurs when thesurfaces should be sticking. ABAQUS automatically chooses the penalty stiffness (the slope of thedotted line) so that this allowable "elastic slip" is a very small fraction of the characteristic elementlength. The penalty friction formulation works well for most problems, including most metal formingapplications. In those problems where the ideal stick-slip frictional behavior must be included, the"Lagrange" friction model can be used. The "Lagrange" friction model is more expensive in terms ofthe computer resources used because ABAQUS uses additional variables for each surface node withfrictional contact. In addition, the solution converges more slowly so that additional iterations areusually required. This friction model is not discussed in this guide.

Often the friction coefficient at the initiation of slipping from a sticking condition is different from thefriction coefficient during established sliding. The former is typically referred to as the static frictioncoefficient, and the latter is referred to as the kinetic friction coefficient. In ABAQUS/Standard anexponential decay law is available to model the transition between static and kinetic friction. Thisfriction model is not discussed in this guide.

The inclusion of friction in a model adds unsymmetric terms to the system of equations being solved. If¹ is less than about 0.2, the magnitude and influence of these terms are quite small and the regular,symmetric solver works well. For higher coefficients of friction, the unsymmetric solver is invokedautomatically because it will improve the convergence rate. The unsymmetric solver can also beselected by including the UNSYMM=YES parameter on the *STEP option. The unsymmetric solverrequires twice as much computer memory and scratch disk space as the symmetric solver.

11.2 Contact definition in ABAQUS

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The first step in defining contact between two structures in ABAQUS is to create surfaces using the*SURFACE option. Next, pairs of surfaces that may contact each other are created using the*CONTACT PAIR option. Each contact pair refers to a surface interaction definition, which is createdwith the *SURFACE INTERACTION option. A contact pressure-clearance relationship and frictionproperties can be assigned to a surface interaction definition.

11.2.1 Defining surfacesSurfaces are created with the *SURFACE option by identifying all of the element faces that form thesurface. This is done in much the same way as defining distributed pressure loads.

Surfaces on continuum elements

A two-dimensional, first-order continuum element, such as CPE4, has four faces consisting ofthe segments defined by nodes 1-2, 2-3, 3-4, and 4-1, respectively, as shown in Figure 11-3.The face identifiers consist of the letter "S" followed by the face number. For example, use thefollowing option block to include face 2 of the element shown in Figure 11-3 in a surfacecalled FLANGE1:

*SURFACE, NAME=FLANGE1

5, S2

Figure 11-3 Face numbers on a two-dimensional, first-order (4-node) element.

As is the case for many options in ABAQUS, both element numbers and element sets can beused; the use of element sets can make the definition of large surfaces much easier. It is validto specify both element sets and individual elements in the same *SURFACE option block, asshown in Figure 11-4 and the example below. The surface TOPSURF consists of the elementfaces shown in Figure 11-4 and is created as follows:

*ELSET, ELSET=TOP, GENERATE

5, 8

*SURFACE, NAME=TOPSURF

TOP, S3

5, S4

8, S2

The first step in defining contact between two structures in ABAQUS is to create surfaces using the*SURFACE option. Next, pairs of surfaces that may contact each other are created using the*CONTACT PAIR option. Each contact pair refers to a surface interaction definition, which is createdwith the *SURFACE INTERACTION option. A contact pressure-clearance relationship and frictionproperties can be assigned to a surface interaction definition.

11.2.1 Defining surfacesSurfaces are created with the *SURFACE option by identifying all of the element faces that form thesurface. This is done in much the same way as defining distributed pressure loads.

Surfaces on continuum elements

A two-dimensional, first-order continuum element, such as CPE4, has four faces consisting ofthe segments defined by nodes 1-2, 2-3, 3-4, and 4-1, respectively, as shown in Figure 11-3.The face identifiers consist of the letter "S" followed by the face number. For example, use thefollowing option block to include face 2 of the element shown in Figure 11-3 in a surfacecalled FLANGE1:

*SURFACE, NAME=FLANGE1

5, S2

Figure 11-3 Face numbers on a two-dimensional, first-order (4-node) element.

As is the case for many options in ABAQUS, both element numbers and element sets can beused; the use of element sets can make the definition of large surfaces much easier. It is validto specify both element sets and individual elements in the same *SURFACE option block, asshown in Figure 11-4 and the example below. The surface TOPSURF consists of the elementfaces shown in Figure 11-4 and is created as follows:

*ELSET, ELSET=TOP, GENERATE

5, 8


TOP, S3

5, S4

8, S2

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Figure 11-4 Face numbers and elements that form the surface TOPSURF.

ABAQUS can determine the free faces of two- and three-dimensional continuum elementsautomatically and use them to create a surface. To use this capability, simply include all theelements whose free faces make up the surface on the data lines of the *SURFACE option.Either element sets or individual element numbers can be used. If elements in the interior ofthe body are included, ABAQUS will ignore them. For example, the surface shown in Figure11-4 could also be defined using


TOP,

Surfaces on shell and rigid elements

For shell elements and rigid elements you must specify which side of the element forms thecontact surface. The side in the direction of the positive element normal is called SPOS, whilethe side in the direction of the negative element normal is called SNEG, as shown in Figure11-5. As discussed in Chapter 5, "Using Shell Elements," the connectivity of an elementdefines the positive element normal. The positive element normals can be viewed inABAQUS/Viewer.

Figure 11-5 Surfaces on a two-dimensional shell or rigid element.

The following option block defines surface SURF1 as the surface composed of all the SPOSfaces of the elements in element set SHELLS:

*SURFACE, NAME=SURF1

SHELLS, SPOS

Restrictions on the types of surfaces that can be created in ABAQUS are discussed in theABAQUS/Standard User's Manual; please read them before beginning a contact simulation.

11.2.2 Contact pairsDefine possible contact between two surfaces in an ABAQUS simulation by giving the surface names

Figure 11-4 Face numbers and elements that form the surface TOPSURF.

ABAQUS can determine the free faces of two- and three-dimensional continuum elementsautomatically and use them to create a surface. To use this capability, simply include all theelements whose free faces make up the surface on the data lines of the *SURFACE option.Either element sets or individual element numbers can be used. If elements in the interior ofthe body are included, ABAQUS will ignore them. For example, the surface shown in Figure11-4 could also be defined using


TOP,

Surfaces on shell and rigid elements

For shell elements and rigid elements you must specify which side of the element forms thecontact surface. The side in the direction of the positive element normal is called SPOS, whilethe side in the direction of the negative element normal is called SNEG, as shown in Figure11-5. As discussed in Chapter 5, "Using Shell Elements," the connectivity of an elementdefines the positive element normal. The positive element normals can be viewed inABAQUS/Viewer.

Figure 11-5 Surfaces on a two-dimensional shell or rigid element.

The following option block defines surface SURF1 as the surface composed of all the SPOSfaces of the elements in element set SHELLS:

*SURFACE, NAME=SURF1

SHELLS, SPOS

Restrictions on the types of surfaces that can be created in ABAQUS are discussed in theABAQUS/Standard User's Manual; please read them before beginning a contact simulation.

11.2.2 Contact pairsDefine possible contact between two surfaces in an ABAQUS simulation by giving the surface names

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on the *CONTACT PAIR option. When you define a contact pair, you must decide whether themagnitude of the relative sliding will be small or finite. The default is the more general finite-slidingformulation. The small-sliding formulation is appropriate if the relative motion of the two surfaces isless than a small proportion of the characteristic length of an element face. The small-slidingformulation is selected by including the SMALL SLIDING parameter on the *CONTACT PAIRoption. Using the small-sliding formulation will result in a more efficient analysis.

Each contact pair must refer to a surface interaction definition, in much the same way that eachelement must refer to an element property definition. Use the INTERACTION parameter on the*CONTACT PAIR option to refer to a *SURFACE INTERACTION option where the different surfaceinteraction models, such as *FRICTION, can be defined.

The following example:

*CONTACT PAIR, INTERACTION=FRIC, SMALL SLIDING

FLANGE1, FLANGE2

*SURFACE INTERACTION, NAME=FRIC

*FRICTION

0.1,

specifies that surfaces FLANGE1 and FLANGE2 might interact with each other and that the amount ofrelative sliding that occurs will be considered to be small. The interaction between the surfacesincludes friction, with a coefficient of friction of 0.1.

11.2.3 Slave and master surfacesABAQUS uses a pure master-slave contact algorithm: nodes on one surface (the slave) cannotpenetrate the segments that make up the other surface (the master), as shown in Figure 11-6.

Figure 11-6 The master surface can penetrate the slave surface.

The algorithm places no restrictions on the master surface; it can penetrate the slave surface betweenslave nodes, as shown in Figure 11-6. The order of the two surfaces given on the *CONTACT PAIRoption is important because it determines which surface is the master surface and which is the slavesurface. The first surface is taken to be the slave surface, and the second is the master surface.

A consequence of this strict master-slave formulation is that you must be careful to select the slave andmaster surfaces correctly to achieve the best possible contact simulation. Some simple rules to followare:

on the *CONTACT PAIR option. When you define a contact pair, you must decide whether themagnitude of the relative sliding will be small or finite. The default is the more general finite-slidingformulation. The small-sliding formulation is appropriate if the relative motion of the two surfaces isless than a small proportion of the characteristic length of an element face. The small-slidingformulation is selected by including the SMALL SLIDING parameter on the *CONTACT PAIRoption. Using the small-sliding formulation will result in a more efficient analysis.

Each contact pair must refer to a surface interaction definition, in much the same way that eachelement must refer to an element property definition. Use the INTERACTION parameter on the*CONTACT PAIR option to refer to a *SURFACE INTERACTION option where the different surfaceinteraction models, such as *FRICTION, can be defined.

The following example:


FLANGE1, FLANGE2


*FRICTION

0.1,

specifies that surfaces FLANGE1 and FLANGE2 might interact with each other and that the amount ofrelative sliding that occurs will be considered to be small. The interaction between the surfacesincludes friction, with a coefficient of friction of 0.1.

11.2.3 Slave and master surfacesABAQUS uses a pure master-slave contact algorithm: nodes on one surface (the slave) cannotpenetrate the segments that make up the other surface (the master), as shown in Figure 11-6.

Figure 11-6 The master surface can penetrate the slave surface.

The algorithm places no restrictions on the master surface; it can penetrate the slave surface betweenslave nodes, as shown in Figure 11-6. The order of the two surfaces given on the *CONTACT PAIRoption is important because it determines which surface is the master surface and which is the slavesurface. The first surface is taken to be the slave surface, and the second is the master surface.

A consequence of this strict master-slave formulation is that you must be careful to select the slave andmaster surfaces correctly to achieve the best possible contact simulation. Some simple rules to followare:

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� the slave surface should be the more finely meshed surface; and

� if the mesh densities are similar, the slave surface should be the surface with the softer underlyingmaterial.

11.2.4 Small and finite slidingWhen using the small-sliding formulation, ABAQUS establishes the relationship between the slavenodes and the master surface at the beginning of the simulation. ABAQUS determines which segmenton the master surface will interact with each node on the slave surface. It maintains these relationshipsthroughout the analysis, never changing which master surface segments interact with which slavenodes. If geometric nonlinearity is included in the model by using the NLGEOM parameter on the*STEP option, the small-sliding algorithm accounts for any rotation and deformation of the mastersurface and updates the load path through which the contact forces are transmitted. If geometricnonlinearity is not included in the model, any rotation or deformation of the master surface is ignoredand the load path remains fixed.

The finite-sliding contact formulation requires that ABAQUS constantly determine which part of themaster surface is in contact with each slave node. This is a very complex calculation, especially if boththe contacting bodies are deformable. The structures in such simulations can be either two- orthree-dimensional. ABAQUS can also model the finite-sliding self-contact of a single deformablebody. Such a situation occurs when a structure folds over onto itself. This capability is available onlyfor two-dimensional (plane strain, plane stress, and axisymmetric) problems.

The finite-sliding formulation for contact between a deformable body and a rigid surface is not ascomplex as the finite-sliding formulation for two deformable bodies. Finite-sliding simulations wherethe master surface is rigid can be performed for both two- and three-dimensional models.

11.2.5 Element selectionWhen selecting elements for contact analyses, it is better, in general, to use first-order elements forthose parts of a model that will form a slave surface. Second-order elements can cause problems incontact simulations because of the way these elements calculate consistent nodal loads for a constantpressure. The consistent nodal loads for a constant pressure, P , on a second-order, two-dimensionalelement with area A are shown in Figure 11-7.

Figure 11-7 Equivalent nodal loads for a constant pressure on a two-dimensional, second-order

element.

� the slave surface should be the more finely meshed surface; and

� if the mesh densities are similar, the slave surface should be the surface with the softer underlyingmaterial.

11.2.4 Small and finite slidingWhen using the small-sliding formulation, ABAQUS establishes the relationship between the slavenodes and the master surface at the beginning of the simulation. ABAQUS determines which segmenton the master surface will interact with each node on the slave surface. It maintains these relationshipsthroughout the analysis, never changing which master surface segments interact with which slavenodes. If geometric nonlinearity is included in the model by using the NLGEOM parameter on the*STEP option, the small-sliding algorithm accounts for any rotation and deformation of the mastersurface and updates the load path through which the contact forces are transmitted. If geometricnonlinearity is not included in the model, any rotation or deformation of the master surface is ignoredand the load path remains fixed.

The finite-sliding contact formulation requires that ABAQUS constantly determine which part of themaster surface is in contact with each slave node. This is a very complex calculation, especially if boththe contacting bodies are deformable. The structures in such simulations can be either two- orthree-dimensional. ABAQUS can also model the finite-sliding self-contact of a single deformablebody. Such a situation occurs when a structure folds over onto itself. This capability is available onlyfor two-dimensional (plane strain, plane stress, and axisymmetric) problems.

The finite-sliding formulation for contact between a deformable body and a rigid surface is not ascomplex as the finite-sliding formulation for two deformable bodies. Finite-sliding simulations wherethe master surface is rigid can be performed for both two- and three-dimensional models.

11.2.5 Element selectionWhen selecting elements for contact analyses, it is better, in general, to use first-order elements forthose parts of a model that will form a slave surface. Second-order elements can cause problems incontact simulations because of the way these elements calculate consistent nodal loads for a constantpressure. The consistent nodal loads for a constant pressure, P , on a second-order, two-dimensionalelement with area A are shown in Figure 11-7.

Figure 11-7 Equivalent nodal loads for a constant pressure on a two-dimensional, second-order

element.

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The contact algorithms base important decisions on the forces acting on the slave nodes. It is difficultfor the algorithms to tell if the force distribution shown in Figure 11-7 represents a constant contactpressure or an actual variation across the element. The equivalent nodal forces for a three-dimensional,second-order brick element are even more confusing because they do not even have the same sign for aconstant pressure, making it impossible for the contact algorithm to work correctly. Therefore, to avoidsuch problems, ABAQUS automatically adds a midface node to any face of a second-order,three-dimensional brick or wedge element that defines a slave surface. The equivalent nodal forces fora second-order element face with a midface node have the same sign for a constant pressure, althoughthey still differ considerably in magnitude.

The equivalent nodal forces for applied pressures on first-order elements always have a consistent signand magnitude; therefore, there is no ambiguity about the contact state that a given distribution ofnodal forces represents.

11.3 Contact algorithmUnderstanding the algorithm ABAQUS uses to solve contact problems will help you understand thediagnostic output in the message file and carry out contact simulations successfully.

The contact algorithm in ABAQUS/Standard, which is shown in Figure 11-8, is built around theNewton-Raphson technique discussed in Chapter 7, "Nonlinearity."

Figure 11-8 Contact logic.

The contact algorithms base important decisions on the forces acting on the slave nodes. It is difficultfor the algorithms to tell if the force distribution shown in Figure 11-7 represents a constant contactpressure or an actual variation across the element. The equivalent nodal forces for a three-dimensional,second-order brick element are even more confusing because they do not even have the same sign for aconstant pressure, making it impossible for the contact algorithm to work correctly. Therefore, to avoidsuch problems, ABAQUS automatically adds a midface node to any face of a second-order,three-dimensional brick or wedge element that defines a slave surface. The equivalent nodal forces fora second-order element face with a midface node have the same sign for a constant pressure, althoughthey still differ considerably in magnitude.

The equivalent nodal forces for applied pressures on first-order elements always have a consistent signand magnitude; therefore, there is no ambiguity about the contact state that a given distribution ofnodal forces represents.

11.3 Contact algorithmUnderstanding the algorithm ABAQUS uses to solve contact problems will help you understand thediagnostic output in the message file and carry out contact simulations successfully.

The contact algorithm in ABAQUS/Standard, which is shown in Figure 11-8, is built around theNewton-Raphson technique discussed in Chapter 7, "Nonlinearity."

Figure 11-8 Contact logic.

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ABAQUS examines the state of all contact pairs at the start of each increment to establish whetherslave nodes are open or closed. In Figure 11-8, p denotes the contact pressure at a slave node and hdenotes the penetration of a slave node into the master surface. If a node is closed, ABAQUSdetermines whether it is sliding or sticking. ABAQUS applies a constraint for each closed node andremoves constraints from any node where the contact state changes from closed to open. ABAQUSthen carries out an iteration and updates the configuration of the model using the calculatedcorrections.

Before checking for equilibrium of forces or moments, ABAQUS first checks for changes in thecontact conditions at the slave nodes. Any node where the clearance after the iteration becomesnegative or zero has changed status from open to closed. Any node where the contact pressurebecomes negative has changed status from closed to open. If any contact changes are detected in thecurrent iteration, ABAQUS labels it a severe discontinuity iteration and no equilibrium checks arecarried out.

ABAQUS modifies the contact constraints to reflect the change in contact status after the first iterationand tries a second iteration. ABAQUS repeats the procedure until an iteration is completed with nochanges in contact status. This iteration becomes the first equilibrium iteration, and ABAQUSperforms the normal equilibrium convergence checks. If the convergence checks fail, ABAQUSperforms another iteration. Every time a severe discontinuity iteration occurs, ABAQUS resets theinternal count of equilibrium iterations to zero. This iteration count is used to determine if anincrement should be abandoned due to a slow convergence rate. ABAQUS repeats the entire processuntil convergence is achieved, as summarized in Figure 11-8.

The summary for each completed increment in the message and status files shows how many iterationswere severe discontinuity iterations and how many were equilibrium iterations. The total number of

ABAQUS examines the state of all contact pairs at the start of each increment to establish whetherslave nodes are open or closed. In Figure 11-8, p denotes the contact pressure at a slave node and hdenotes the penetration of a slave node into the master surface. If a node is closed, ABAQUSdetermines whether it is sliding or sticking. ABAQUS applies a constraint for each closed node andremoves constraints from any node where the contact state changes from closed to open. ABAQUSthen carries out an iteration and updates the configuration of the model using the calculatedcorrections.

Before checking for equilibrium of forces or moments, ABAQUS first checks for changes in thecontact conditions at the slave nodes. Any node where the clearance after the iteration becomesnegative or zero has changed status from open to closed. Any node where the contact pressurebecomes negative has changed status from closed to open. If any contact changes are detected in thecurrent iteration, ABAQUS labels it a severe discontinuity iteration and no equilibrium checks arecarried out.

ABAQUS modifies the contact constraints to reflect the change in contact status after the first iterationand tries a second iteration. ABAQUS repeats the procedure until an iteration is completed with nochanges in contact status. This iteration becomes the first equilibrium iteration, and ABAQUSperforms the normal equilibrium convergence checks. If the convergence checks fail, ABAQUSperforms another iteration. Every time a severe discontinuity iteration occurs, ABAQUS resets theinternal count of equilibrium iterations to zero. This iteration count is used to determine if anincrement should be abandoned due to a slow convergence rate. ABAQUS repeats the entire processuntil convergence is achieved, as summarized in Figure 11-8.

The summary for each completed increment in the message and status files shows how many iterationswere severe discontinuity iterations and how many were equilibrium iterations. The total number of

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iterations for an increment is the sum of these two.

By separating the two types of iterations, you can see how well ABAQUS is coping with the contactcalculations and how well it is achieving equilibrium. If the number of severe discontinuity iterationsis high but there are few equilibrium iterations, ABAQUS is having difficulty determining the propercontact conditions. By default, ABAQUS abandons any increment where it needs more than twelvesevere discontinuity iterations and tries the increment again with a smaller increment size. If there areno severe discontinuity iterations, the contact state is not changing from increment to increment.

11.4 Example: flange connectionIn this example you will investigate the behavior of a preliminary design for the flanged joint shown inFigure 11-9. The top flange is made from steel and the bottom flange from aluminum. The joint carriesan axial load of 200 kN. An O-ring will be used to help seal the joint. In the preliminary design theO-ring will be placed 0.01 m from the inside of the flange.

Figure 11-9 Sketch of axisymmetric flanged joint under axial load.

You have been asked to determine the distance that the flanges separate at the seal position so that anappropriately sized seal can be selected. You should also use the computational model to confirm thatthe overall sizing of the flange is adequate.

11.4.1 Constraining rigid body motionThere appears to be no need for boundary conditions in this simulation. The forces applied to eachflange are supposed to be equal in magnitude and opposite in direction; thus, the loading should be

iterations for an increment is the sum of these two.

By separating the two types of iterations, you can see how well ABAQUS is coping with the contactcalculations and how well it is achieving equilibrium. If the number of severe discontinuity iterationsis high but there are few equilibrium iterations, ABAQUS is having difficulty determining the propercontact conditions. By default, ABAQUS abandons any increment where it needs more than twelvesevere discontinuity iterations and tries the increment again with a smaller increment size. If there areno severe discontinuity iterations, the contact state is not changing from increment to increment.

11.4 Example: flange connectionIn this example you will investigate the behavior of a preliminary design for the flanged joint shown inFigure 11-9. The top flange is made from steel and the bottom flange from aluminum. The joint carriesan axial load of 200 kN. An O-ring will be used to help seal the joint. In the preliminary design theO-ring will be placed 0.01 m from the inside of the flange.

Figure 11-9 Sketch of axisymmetric flanged joint under axial load.

You have been asked to determine the distance that the flanges separate at the seal position so that anappropriately sized seal can be selected. You should also use the computational model to confirm thatthe overall sizing of the flange is adequate.

11.4.1 Constraining rigid body motionThere appears to be no need for boundary conditions in this simulation. The forces applied to eachflange are supposed to be equal in magnitude and opposite in direction; thus, the loading should be

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self-equilibrating. However, with no boundary conditions, there will be some small out-of-balance netforce acting on the model due to the finite precision of computers. If the model is used in a staticsimulation with no boundary conditions (only applied forces), this small net force would causeunlimited rigid body motion of the model.

Such rigid body motion is known mathematically as a numerical singularity . When ABAQUS detectsnumerical singularities in a simulation, it prints a solver problem message in the message file. Suchmessages have the following format:

***WARNING: SOLVER PROBLEM.

NUMERICAL SINGULARITY WHEN PROCESSING NODE 302

D.O.F. 2 RATIO = 1.14541E+14

The node whose node number is printed in the message lies in that portion of the mesh that isunconstrained. The degree of freedom indicates the direction in which rigid body motion can occur.

Generally the results of a simulation containing numerical singularities should not be accepted asvalid. To prevent numerical singularities in a static analysis, you must provide enough boundaryconditions to prevent all rigid body motions of each component in the model. Rigid body motions canconsist of both translations and rotations of the components. The potential rigid body motions dependon the dimensionality of the model.Dimensionality Possible Rigid Body MotionThree-dimensional Translation in the 1-, 2-, and 3-directions.

Rotation about the 1-, 2-, and 3-axes.Axisymmetric Translation in the 2-direction.Plane stress Translation in the 1- and 2-directions.Plane strain Rotation about the 3-axis.

The flanges need to be constrained axially (the global 2-direction) to prevent rigid body motion duringthe simulation. Because the flanges are bolted together, only a single node in the model needs aboundary condition in the axial direction. The applied loading in this simulation will cause the reactionforce at the constrained node to be approximately zero; therefore, the boundary condition can beapplied to almost any node in the model.

11.4.2 Coordinate systemUse the default axisymmetric coordinate system for this analysis. The global 1-axis is the radialdirection, and the global 2-axis is the axis of symmetry in the model. The location of the origin for themodel should be at the juncture of the two flanges as shown in Figure 11-9.

11.4.3 Mesh designYou should consider the type of element you will use before you design your mesh. When choosing anelement type, you must consider several aspects of your model such as the model's geometry, the typeof deformation that will be seen, the loads being applied, etc. The following are the important points toconsider in this simulation:

� The contact between the flanges. Whenever possible, first-order elements should be used forcontact simulations.

self-equilibrating. However, with no boundary conditions, there will be some small out-of-balance netforce acting on the model due to the finite precision of computers. If the model is used in a staticsimulation with no boundary conditions (only applied forces), this small net force would causeunlimited rigid body motion of the model.

Such rigid body motion is known mathematically as a numerical singularity . When ABAQUS detectsnumerical singularities in a simulation, it prints a solver problem message in the message file. Suchmessages have the following format:

***WARNING: SOLVER PROBLEM.

NUMERICAL SINGULARITY WHEN PROCESSING NODE 302

D.O.F. 2 RATIO = 1.14541E+14

The node whose node number is printed in the message lies in that portion of the mesh that isunconstrained. The degree of freedom indicates the direction in which rigid body motion can occur.

Generally the results of a simulation containing numerical singularities should not be accepted asvalid. To prevent numerical singularities in a static analysis, you must provide enough boundaryconditions to prevent all rigid body motions of each component in the model. Rigid body motions canconsist of both translations and rotations of the components. The potential rigid body motions dependon the dimensionality of the model.Dimensionality Possible Rigid Body MotionThree-dimensional Translation in the 1-, 2-, and 3-directions.

Rotation about the 1-, 2-, and 3-axes.Axisymmetric Translation in the 2-direction.Plane stress Translation in the 1- and 2-directions.Plane strain Rotation about the 3-axis.

The flanges need to be constrained axially (the global 2-direction) to prevent rigid body motion duringthe simulation. Because the flanges are bolted together, only a single node in the model needs aboundary condition in the axial direction. The applied loading in this simulation will cause the reactionforce at the constrained node to be approximately zero; therefore, the boundary condition can beapplied to almost any node in the model.

11.4.2 Coordinate systemUse the default axisymmetric coordinate system for this analysis. The global 1-axis is the radialdirection, and the global 2-axis is the axis of symmetry in the model. The location of the origin for themodel should be at the juncture of the two flanges as shown in Figure 11-9.

11.4.3 Mesh designYou should consider the type of element you will use before you design your mesh. When choosing anelement type, you must consider several aspects of your model such as the model's geometry, the typeof deformation that will be seen, the loads being applied, etc. The following are the important points toconsider in this simulation:

� The contact between the flanges. Whenever possible, first-order elements should be used forcontact simulations.

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� Significant bending of the flanges is expected under the applied loading. Fully integratedfirst-order elements exhibit shear locking when subjected to bending deformation. Therefore,either reduced-integration or incompatible mode elements should be used.

� The mesh can use regularly shaped elements so the sensitivity of incompatible mode elements todistortion will not be a problem.

Based on these considerations, CAX4I elements should be used in this model.

The mesh in your model must be exactly the same as the mesh shown in Figure 11-10 because nodesare required at specific locations to represent the bolts and O-ring in the physical structure.

The node and element numbers referred to in this section are shown in Figure 11-11.

Figure 11-10 Suggested mesh for the flanged joint model.

� Significant bending of the flanges is expected under the applied loading. Fully integratedfirst-order elements exhibit shear locking when subjected to bending deformation. Therefore,either reduced-integration or incompatible mode elements should be used.

� The mesh can use regularly shaped elements so the sensitivity of incompatible mode elements todistortion will not be a problem.

Based on these considerations, CAX4I elements should be used in this model.

The mesh in your model must be exactly the same as the mesh shown in Figure 11-10 because nodesare required at specific locations to represent the bolts and O-ring in the physical structure.

The node and element numbers referred to in this section are shown in Figure 11-11.

Figure 11-10 Suggested mesh for the flanged joint model.

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Figure 11-11 Node and element numbers in the flanges.

These numbers correspond to those generated by the input file in ``Flange connection,'' Section A.12.If you generate your model using a preprocessor, your node and element numbers may be different.

11.4.4 Creating the model--preprocessingCreate the mesh for the two flanges. Suitable ABAQUS mesh generation options are given in ``Flangeconnection,'' Section A.12, if you do not want to use a preprocessor. If you use a preprocessor, do notequivalence the nodes along the contacting surfaces of the two flanges. Doing so will make itimpossible to model the contact between the two flanges. If the preprocessor does not let you specifyincompatible mode elements, use regular first-order, axisymmetric elements; you can change theelement type in the input file later. It is important that you assign different materials to the top andbottom flanges because this should cause the preprocessor to create two element sets that can be usedto assign different material properties to the two parts of the model. The material constants you shoulduse are given in the next section.

11.4.5 Reviewing the input file--the model dataThe model data for the flange connection problem--including the node and element definitions, sectionand material properties, and constraints--are reviewed.

Model description

Provide a suitable description of the flange model, such as the one shown below, in the*HEADING option.

*HEADING

Axisymmetric flange connection

SI units (N, m, kg, s)


There should be two *ELEMENT option blocks in the model since the two flanges are madeof different materials. Give the element sets created with each *ELEMENT option meaningfulnames. If there is only one *ELEMENT option block, make sure that there are element sets for

Figure 11-11 Node and element numbers in the flanges.

These numbers correspond to those generated by the input file in ``Flange connection,'' Section A.12.If you generate your model using a preprocessor, your node and element numbers may be different.

11.4.4 Creating the model--preprocessingCreate the mesh for the two flanges. Suitable ABAQUS mesh generation options are given in ``Flangeconnection,'' Section A.12, if you do not want to use a preprocessor. If you use a preprocessor, do notequivalence the nodes along the contacting surfaces of the two flanges. Doing so will make itimpossible to model the contact between the two flanges. If the preprocessor does not let you specifyincompatible mode elements, use regular first-order, axisymmetric elements; you can change theelement type in the input file later. It is important that you assign different materials to the top andbottom flanges because this should cause the preprocessor to create two element sets that can be usedto assign different material properties to the two parts of the model. The material constants you shoulduse are given in the next section.

11.4.5 Reviewing the input file--the model dataThe model data for the flange connection problem--including the node and element definitions, sectionand material properties, and constraints--are reviewed.

Model description

Provide a suitable description of the flange model, such as the one shown below, in the*HEADING option.

*HEADING

Axisymmetric flange connection



There should be two *ELEMENT option blocks in the model since the two flanges are madeof different materials. Give the element sets created with each *ELEMENT option meaningfulnames. If there is only one *ELEMENT option block, make sure that there are element sets for

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each flange. Check that the preprocessor used the correct element type on the *ELEMENToption. Change the value of the TYPE parameter to CAX4I if the preprocessor did not use thisvalue. The two *ELEMENT options in your model should look like those shown below.

*ELEMENT, TYPE=CAX4I, ELSET=TOPFLANG

*ELEMENT, TYPE=CAX4I, ELSET=BOTFLANG

Section properties

Since the two flanges are made from different materials, two element property definitions arerequired, each referring to a different material property. Make sure that the appropriatematerial definition is assigned to the appropriate element set in the *SOLID SECTIONoptions.

*SOLID SECTION, ELSET=TOPFLANG, MATERIAL=STEEL

*SOLID SECTION, ELSET=BOTFLANG, MATERIAL=ALU

Material properties

Two linear elastic material definitions are needed. The top flange is made from steel ( E=200GPa, º=0.3), and the bottom flange is made from aluminum (E=70 GPa, º=0.2). Make surethat the names used match those referred to by the MATERIAL parameters on the *SOLIDSECTION options. The material definitions should look similar to those shown below.


*ELASTIC

200.0E9, 0.3

*MATERIAL, NAME=ALU

*ELASTIC

70.0E9, 0.2


We also need to model the effect of the bolts that hold the flanges together. A very simplisticapproach is used in this simulation. A node from the top flange is joined to a node on thebottom flange with a TIE-type MPC. In the mesh from ``Flange connection,'' Section A.12,nodes 18 and 218 lie on the centerline of the bolt. Therefore, the following *MPC optionblock simulates the bolt:

*MPC

TIE, 18, 218

Nodes 7 and 207, which lie on the surfaces of the top and bottom flanges, respectively, are notinvolved in the constraint because they will be included in the contact definition. A constraintcannot be applied to a slave node; therefore, the next pair of nodes on the bolt centerline isused in the constraint definition.

A more realistic model of this flange would use a three-dimensional mesh. In such a modelcyclic symmetry could be utilized so that only a portion of the flange would have to be

each flange. Check that the preprocessor used the correct element type on the *ELEMENToption. Change the value of the TYPE parameter to CAX4I if the preprocessor did not use thisvalue. The two *ELEMENT options in your model should look like those shown below.


*ELEMENT, TYPE=CAX4I, ELSET=BOTFLANG

Section properties

Since the two flanges are made from different materials, two element property definitions arerequired, each referring to a different material property. Make sure that the appropriatematerial definition is assigned to the appropriate element set in the *SOLID SECTIONoptions.



Material properties

Two linear elastic material definitions are needed. The top flange is made from steel ( E=200GPa, º=0.3), and the bottom flange is made from aluminum (E=70 GPa, º=0.2). Make surethat the names used match those referred to by the MATERIAL parameters on the *SOLIDSECTION options. The material definitions should look similar to those shown below.


*ELASTIC

200.0E9, 0.3

*MATERIAL, NAME=ALU

*ELASTIC

70.0E9, 0.2


We also need to model the effect of the bolts that hold the flanges together. A very simplisticapproach is used in this simulation. A node from the top flange is joined to a node on thebottom flange with a TIE-type MPC. In the mesh from ``Flange connection,'' Section A.12,nodes 18 and 218 lie on the centerline of the bolt. Therefore, the following *MPC optionblock simulates the bolt:

*MPC

TIE, 18, 218

Nodes 7 and 207, which lie on the surfaces of the top and bottom flanges, respectively, are notinvolved in the constraint because they will be included in the contact definition. A constraintcannot be applied to a slave node; therefore, the next pair of nodes on the bolt centerline isused in the constraint definition.

A more realistic model of this flange would use a three-dimensional mesh. In such a modelcyclic symmetry could be utilized so that only a portion of the flange would have to be

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modeled. The bolt loading option could also be used.


Assume that the nodes at the ends of each flange deform as a rigid body in the axial direction.Use linear constraints to model this condition by tying the axial displacements of the threenodes at the end of the flange together. Two linear constraint equations are needed for eachflange; make sure that the same node is used as the second node in each equation. Add anoption block similar to the following to your model--remember to change the node numbers toreflect those used in your mesh:

*EQUATION

2,

112,2,1.0, 111,2,-1.0

2,

113,2,1.0, 111,2,-1.0

2,

312,2,1.0, 311,2,-1.0

2,

313,2,1.0, 311,2,-1.0

These constraint equations tie the axial (degree of freedom 2) displacements of nodes 112 and113 to that of node 111 and tie the axial displacements of nodes 312 and 313 to that of node311.

11.4.6 Contact definitionThe options to define the contact interaction are now discussed.

Surface definitions

Create an element set containing the elements adjacent to the desired surface on the bottomflange. The following input creates the appropriate element set for the model shown in Figure11-11:

*ELSET, ELSET=FLBOT, GENERATE

101,110,1

You will probably have different element numbers in your model.

Define the surface on the bottom flange, SURFBOT, by using this element set in the*SURFACE option. Do not use a face identifier: ABAQUS will create the surface using thefree faces of the elements in the element set. The input in your model should be similar to thefollowing:

*SURFACE, NAME=SURFBOT, TRIM=YES

FLBOT,

The TRIM parameter is used to remove the corner extensions that would have been included

modeled. The bolt loading option could also be used.


Assume that the nodes at the ends of each flange deform as a rigid body in the axial direction.Use linear constraints to model this condition by tying the axial displacements of the threenodes at the end of the flange together. Two linear constraint equations are needed for eachflange; make sure that the same node is used as the second node in each equation. Add anoption block similar to the following to your model--remember to change the node numbers toreflect those used in your mesh:

*EQUATION

2,

112,2,1.0, 111,2,-1.0

2,

113,2,1.0, 111,2,-1.0

2,

312,2,1.0, 311,2,-1.0

2,

313,2,1.0, 311,2,-1.0

These constraint equations tie the axial (degree of freedom 2) displacements of nodes 112 and113 to that of node 111 and tie the axial displacements of nodes 312 and 313 to that of node311.

11.4.6 Contact definitionThe options to define the contact interaction are now discussed.

Surface definitions

Create an element set containing the elements adjacent to the desired surface on the bottomflange. The following input creates the appropriate element set for the model shown in Figure11-11:


101,110,1

You will probably have different element numbers in your model.

Define the surface on the bottom flange, SURFBOT, by using this element set in the*SURFACE option. Do not use a face identifier: ABAQUS will create the surface using thefree faces of the elements in the element set. The input in your model should be similar to thefollowing:


FLBOT,

The TRIM parameter is used to remove the corner extensions that would have been included

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in the surface definition. Such extensions are undesirable on slave surfaces, and the surface onthe bottom flange will be the slave surface of the contact pair. More details on trimmingsurfaces can be found in the ABAQUS/Standard User's Manual.

Create the surface for the top flange using the same approach, except do not trim this surface.

*ELSET, ELSET=FLTOP, GENERATE

1,10,1

*SURFACE, NAME=SURFTOP, TRIM=NO

FLTOP,

Creating the contact pair

Having defined both contact surfaces, use the *CONTACT PAIR option to indicate that theymay interact with each other during the analysis. There should be little relative sliding of thetwo flanges so include the SMALL SLIDING parameter on the *CONTACT PAIR option. Inthis simulation assume that the coefficient of friction between the surfaces is 0.1. Use the*FRICTION option to define the friction between the surfaces. This option must be used inconjunction with the *SURFACE INTERACTION option. Make the surface of the softeraluminum flange the slave surface. The input defining the contact pair in your model should besimilar to the following:


SURFBOT, SURFTOP


*FRICTION

0.1,

11.4.7 Reviewing the input file--the history dataYou do not need to include the effects of geometric nonlinearity in your model because thedisplacements and strains of the flanges should be minimal. Use a total step time of 1.0 on the*STATIC option block. In most contact simulations the size of the first increment should be limited toabout 10% of the total step time. However, in this analysis ABAQUS is only going to need severaliterations to determine the correct contact state. It can do this as easily with 100% of the load appliedas it can with 10% of the load applied, so you can reduce the computational cost of the analysis byusing an initial increment size of 1.0. The step and procedure option blocks in your model should looklike

*STEP, INC=20

*STATIC

1.0, 1.0

Loading

A 200 kN axial load must be applied to the end of each flange. For the model shown in Figure11-11, the correct input is

in the surface definition. Such extensions are undesirable on slave surfaces, and the surface onthe bottom flange will be the slave surface of the contact pair. More details on trimmingsurfaces can be found in the ABAQUS/Standard User's Manual.

Create the surface for the top flange using the same approach, except do not trim this surface.


1,10,1


FLTOP,

Creating the contact pair

Having defined both contact surfaces, use the *CONTACT PAIR option to indicate that theymay interact with each other during the analysis. There should be little relative sliding of thetwo flanges so include the SMALL SLIDING parameter on the *CONTACT PAIR option. Inthis simulation assume that the coefficient of friction between the surfaces is 0.1. Use the*FRICTION option to define the friction between the surfaces. This option must be used inconjunction with the *SURFACE INTERACTION option. Make the surface of the softeraluminum flange the slave surface. The input defining the contact pair in your model should besimilar to the following:


SURFBOT, SURFTOP


*FRICTION

0.1,

11.4.7 Reviewing the input file--the history dataYou do not need to include the effects of geometric nonlinearity in your model because thedisplacements and strains of the flanges should be minimal. Use a total step time of 1.0 on the*STATIC option block. In most contact simulations the size of the first increment should be limited toabout 10% of the total step time. However, in this analysis ABAQUS is only going to need severaliterations to determine the correct contact state. It can do this as easily with 100% of the load appliedas it can with 10% of the load applied, so you can reduce the computational cost of the analysis byusing an initial increment size of 1.0. The step and procedure option blocks in your model should looklike

*STEP, INC=20

*STATIC

1.0, 1.0

Loading

A 200 kN axial load must be applied to the end of each flange. For the model shown in Figure11-11, the correct input is

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*CLOAD

111, 2, 200.E3

311, 2, -200.E3

The nodes in your model will probably have different numbers.

Boundary conditions

This is a static simulation, so boundary conditions are needed to prevent rigid body motions.Since the model is axisymmetric, no constraint is needed to prevent rigid body motion in theradial direction. There are no constraints in the physical problem in the axial direction becausethe loads applied to the flanges are supposed to be in equilibrium. However, as discussedearlier, the numerical model cannot rely on the applied loads to balance; therefore, a singleboundary condition is needed to prevent rigid body motion in the axial direction.

The location of this constraint is not very important since the reaction force at the node shouldbe very small; however, the following guidelines should be followed:

� Boundary conditions should not be applied to nodes on contact surfaces because they canconflict with the contact constraints.

� Boundary conditions should not be applied to nodal degrees of freedom that have beeneliminated with linear constraint equations, such as the axial degrees of freedom of thenodes at the ends of the two flanges. If you apply boundary conditions to an eliminateddegree of freedom, ABAQUS will issue an error message in the data file.

� So that we can check the reaction force at the constraint easily, boundary conditionsshould not be applied to nodes where loads are applied.

Therefore, node 100 in the model shown in Figure 11-11 is constrained in the 2-direction.

*BOUNDARY

100, 2

Output requests

Print the reaction forces in the data file to check that the reaction force at the constrained nodeis zero. This will confirm that the approach used to avoid numerical problems caused by rigidbody motions does not violate the physics of the real structure. No other results should beprinted in the data file; use the FREQUENCY parameter and the *EL PRINT option tosuppress printed element output. The input options for these requests are


RF


Detailed information about the contact state of each contact pair can be obtained in both thedata (.dat) file and the results (.fil) file. Printed contact output is available using the*CONTACT PRINT option. Output of contact variables to the results file is available usingthe *CONTACT FILE option. Output for specific contact surfaces can be requested with either

*CLOAD

111, 2, 200.E3

311, 2, -200.E3

The nodes in your model will probably have different numbers.

Boundary conditions

This is a static simulation, so boundary conditions are needed to prevent rigid body motions.Since the model is axisymmetric, no constraint is needed to prevent rigid body motion in theradial direction. There are no constraints in the physical problem in the axial direction becausethe loads applied to the flanges are supposed to be in equilibrium. However, as discussedearlier, the numerical model cannot rely on the applied loads to balance; therefore, a singleboundary condition is needed to prevent rigid body motion in the axial direction.

The location of this constraint is not very important since the reaction force at the node shouldbe very small; however, the following guidelines should be followed:

� Boundary conditions should not be applied to nodes on contact surfaces because they canconflict with the contact constraints.

� Boundary conditions should not be applied to nodal degrees of freedom that have beeneliminated with linear constraint equations, such as the axial degrees of freedom of thenodes at the ends of the two flanges. If you apply boundary conditions to an eliminateddegree of freedom, ABAQUS will issue an error message in the data file.

� So that we can check the reaction force at the constraint easily, boundary conditionsshould not be applied to nodes where loads are applied.

Therefore, node 100 in the model shown in Figure 11-11 is constrained in the 2-direction.

*BOUNDARY

100, 2

Output requests

Print the reaction forces in the data file to check that the reaction force at the constrained nodeis zero. This will confirm that the approach used to avoid numerical problems caused by rigidbody motions does not violate the physics of the real structure. No other results should beprinted in the data file; use the FREQUENCY parameter and the *EL PRINT option tosuppress printed element output. The input options for these requests are


RF


Detailed information about the contact state of each contact pair can be obtained in both thedata (.dat) file and the results (.fil) file. Printed contact output is available using the*CONTACT PRINT option. Output of contact variables to the results file is available usingthe *CONTACT FILE option. Output for specific contact surfaces can be requested with either

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option by using the MASTER and SLAVE parameters. If neither of these parameters is used,results are printed for all of the contact surfaces in the model. As with other output options,use the FREQUENCY parameter to limit the amount of output. The default contact variablesgiven as output areCPRESS Contact pressure.CSHEAR1 Frictional stress in the local 1-direction.CSHEAR2 Frictional stress in the local 2-direction (three-dimensional models

only).CSLIP1 Accumulated relative tangential motion in the local 1-direction.CSLIP2 Accumulated relative tangential motion in the local 2-direction

(three-dimensional models only).COPEN Contact openings.

The local 1- and 2-directions on the surface are calculated using the same method fordetermining the local material directions in shell elements.

The *PRINT, CONTACT=YES option prints the contact status of all the slave surface nodesin the message file at the start of the analysis and prints a message in every severediscontinuity iteration stating which nodes changed status. This is particularly useful to checkwhether surfaces have been defined correctly at the start of the simulation and to helpdetermine where problems occur during the analysis.

The contact output option blocks needed in your model are

*CONTACT PRINT

*PRINT, CONTACT=YES

Write the preselected field data to the output database (.odb) file so that you can useABAQUS/Viewer to produce contour plots of stress and contact pressure. Include thefollowing output requests in your input file:

*OUTPUT, FIELD, OP=NEW, VARIABLE=PRESELECT, FREQUENCY=1

Finally, complete the step with

*END STEP

11.4.8 Running the analysisSave the model input in a file called flange.inp. Run the analysis in the background.

abaqus job=flange

If you have not created the model from within a preprocessor, the input data described here areincluded in ``Flange connection,'' Section A.12.

Status file

The analysis completes in five iterations as shown in the status file.



option by using the MASTER and SLAVE parameters. If neither of these parameters is used,results are printed for all of the contact surfaces in the model. As with other output options,use the FREQUENCY parameter to limit the amount of output. The default contact variablesgiven as output areCPRESS Contact pressure.CSHEAR1 Frictional stress in the local 1-direction.CSHEAR2 Frictional stress in the local 2-direction (three-dimensional models

only).CSLIP1 Accumulated relative tangential motion in the local 1-direction.CSLIP2 Accumulated relative tangential motion in the local 2-direction

(three-dimensional models only).COPEN Contact openings.

The local 1- and 2-directions on the surface are calculated using the same method fordetermining the local material directions in shell elements.

The *PRINT, CONTACT=YES option prints the contact status of all the slave surface nodesin the message file at the start of the analysis and prints a message in every severediscontinuity iteration stating which nodes changed status. This is particularly useful to checkwhether surfaces have been defined correctly at the start of the simulation and to helpdetermine where problems occur during the analysis.

The contact output option blocks needed in your model are

*CONTACT PRINT

*PRINT, CONTACT=YES

Write the preselected field data to the output database (.odb) file so that you can useABAQUS/Viewer to produce contour plots of stress and contact pressure. Include thefollowing output requests in your input file:

*OUTPUT, FIELD, OP=NEW, VARIABLE=PRESELECT, FREQUENCY=1

Finally, complete the step with

*END STEP

11.4.8 Running the analysisSave the model input in a file called flange.inp. Run the analysis in the background.

abaqus job=flange

If you have not created the model from within a preprocessor, the input data described here areincluded in ``Flange connection,'' Section A.12.

Status file

The analysis completes in five iterations as shown in the status file.



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ITERS FREQ

1 1 1 3 2 5 1.00 1.00 1.000

ABAQUS required three iterations to establish the correct contact conditions in the model;i.e., whether or not nodes on the bottom flange were contacting the top flange. The fourthiteration did not produce any changes in the model's contact condition, so ABAQUS checkedthe force equilibrium and found that the solution in this iteration did not satisfy theequilibrium convergence conditions. A converged solution was obtained in the fifth iteration.ABAQUS was able to apply 100% of the load successfully in a single increment because themajor source of nonlinearity in the model was the contact between the flanges. Thus, onceABAQUS determined the correct contact state, it easily found the solution for the load appliedto the flanges.

Message file

The *PRINT, CONTACT=YES option should be used in contact simulations because itprovides detailed information in the message file about the changes in the model's contactconditions. The initial contact condition for all of the contact pairs is listed at the top of themessage file. This information is also available in the data (.dat) file after a datacheckanalysis. In this simulation the surfaces coincide initially. Usually surfaces in contact arereported as overclosed by a very small amount.

DETAILED OUTPUT OF CONTACT CHANGES DURING ITERATION REQUESTED

PRINT OF INCREMENT NUMBER, TIME, ETC., TO THE MESSAGE FILE EVE

SLAVE SURFACE SURFBOT MASTER SURFACE SURFTOP NODE NUMBER 210 INITIA











ABAQUS prints the node number of every slave node whose contact status changes in asevere discontinuity iteration, as well as the contact pair to which it belongs, to the messagefile. In the first iteration of this simulation, six slave nodes had a negative contact pressure,indicating that their contact status should be changed from closed to open.


SLAVE SURFACE SURFBOT MASTER SURFACE SURFTOP NODE NUMBER 210 IS NO

SLAVE SURFACE SURFBOT MASTER SURFACE SURFTOP NODE NUMBER 211

OPENS - CONTACT PRESSURE/FORCE IS -4.32341E+05


ITERS FREQ

1 1 1 3 2 5 1.00 1.00 1.000

ABAQUS required three iterations to establish the correct contact conditions in the model;i.e., whether or not nodes on the bottom flange were contacting the top flange. The fourthiteration did not produce any changes in the model's contact condition, so ABAQUS checkedthe force equilibrium and found that the solution in this iteration did not satisfy theequilibrium convergence conditions. A converged solution was obtained in the fifth iteration.ABAQUS was able to apply 100% of the load successfully in a single increment because themajor source of nonlinearity in the model was the contact between the flanges. Thus, onceABAQUS determined the correct contact state, it easily found the solution for the load appliedto the flanges.

Message file

The *PRINT, CONTACT=YES option should be used in contact simulations because itprovides detailed information in the message file about the changes in the model's contactconditions. The initial contact condition for all of the contact pairs is listed at the top of themessage file. This information is also available in the data (.dat) file after a datacheckanalysis. In this simulation the surfaces coincide initially. Usually surfaces in contact arereported as overclosed by a very small amount.

DETAILED OUTPUT OF CONTACT CHANGES DURING ITERATION REQUESTED

PRINT OF INCREMENT NUMBER, TIME, ETC., TO THE MESSAGE FILE EVE












ABAQUS prints the node number of every slave node whose contact status changes in asevere discontinuity iteration, as well as the contact pair to which it belongs, to the messagefile. In the first iteration of this simulation, six slave nodes had a negative contact pressure,indicating that their contact status should be changed from closed to open.





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SEVERE DISCONTINUITY ITERATION 1 ENDS

CONTACT CHANGE SUMMARY: 0 CLOSURES 6 OPENINGS.

ABAQUS removes the contact constraints from these six nodes and performs anotheriteration, during which two more slave nodes lose contact with the top flange.







After making the necessary modification, ABAQUS tries a third iteration; this time only onenode has a change in contact condition. Node 211, which had opened in the first iteration,closes again after the third iteration. ABAQUS detects no changes in contact in the next(fourth) iteration and, therefore, carries out the normal equilibrium convergence checks. Thesolution satisfies the force residual tolerance check, but the displacement correction was toolarge compared to the largest displacement increment. ABAQUS refers to this iteration asequilibrium iteration 1.


AVERAGE FORCE 2.357E+04 TIME AVG. FORCE

LARGEST RESIDUAL FORCE 13.1 AT NODE 11



DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREM

















ABAQUS removes the contact constraints from these six nodes and performs anotheriteration, during which two more slave nodes lose contact with the top flange.







After making the necessary modification, ABAQUS tries a third iteration; this time only onenode has a change in contact condition. Node 211, which had opened in the first iteration,closes again after the third iteration. ABAQUS detects no changes in contact in the next(fourth) iteration and, therefore, carries out the normal equilibrium convergence checks. Thesolution satisfies the force residual tolerance check, but the displacement correction was toolarge compared to the largest displacement increment. ABAQUS refers to this iteration asequilibrium iteration 1.



LARGEST RESIDUAL FORCE 13.1 AT NODE 11



DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREM

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The second equilibrium iteration produces a converged solution for the first increment.





LARGEST CORRECTION TO DISP. 6.692E-10 AT NODE 11






11.4.9 ResultsReview the results by examining the printed output data (.dat) file.

Data file

The reaction force at the node that was constrained to prevent rigid body motion is printed atthe end of the data (.dat) file.

N O D E O U T P U T

THE FOLLOWING TABLE IS PRINTED FOR ALL NODES

NODE FOOT- RF1 RF2

NOTE

100 0.0000E+00 5.0932E-11

MAXIMUM 0.0000E+00 5.0932E-11

AT NODE 1 100

MINIMUM 0.0000E+00 0.0000E+00

AT NODE 1 1

The reaction force is 5.0932 � 10-11 N, which is 16 orders of magnitude smaller than theapplied load of 200 kN and, thus, is effectively zero. This result confirms that the appliedboundary condition, which is not present in the physical problem being simulated, has littleeffect on the solution.

The data file also contains the output from the *CONTACT PRINT option, as shown belowfor the last increment of the simulation. The parts of the contact surface under the bolted area

The second equilibrium iteration produces a converged solution for the first increment.





LARGEST CORRECTION TO DISP. 6.692E-10 AT NODE 11






11.4.9 ResultsReview the results by examining the printed output data (.dat) file.

Data file

The reaction force at the node that was constrained to prevent rigid body motion is printed atthe end of the data (.dat) file.

N O D E O U T P U T

THE FOLLOWING TABLE IS PRINTED FOR ALL NODES

NODE FOOT- RF1 RF2

NOTE

100 0.0000E+00 5.0932E-11

MAXIMUM 0.0000E+00 5.0932E-11

AT NODE 1 100

MINIMUM 0.0000E+00 0.0000E+00

AT NODE 1 1

The reaction force is 5.0932 � 10-11 N, which is 16 orders of magnitude smaller than theapplied load of 200 kN and, thus, is effectively zero. This result confirms that the appliedboundary condition, which is not present in the physical problem being simulated, has littleeffect on the solution.

The data file also contains the output from the *CONTACT PRINT option, as shown belowfor the last increment of the simulation. The parts of the contact surface under the bolted area

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are closed and sticking. Some shear stress, which developed due to the friction between thesurfaces, is transmitted from one surface to the other. The maximum separation of the flangesis 0.28 mm and occurs at the location where the O-ring will be placed.

11.4.10 PostprocessingExamine the results graphically using ABAQUS/Viewer. To begin ABAQUS/Viewer, use thecommand:

abaqus viewer odb=flange



You will plot the deformed model shape after assigning different colors to each of the flangecomponents. This will allow you to easily distinguish between the two components.

To change the color of the components:

1. From the main menu bar, select Tools->Color Code.

The Color Code dialog box appears.

2. Select TOPFLANG from the list of element sets.

3. Select RED for the display color.

4. Similarly, specify CYAN as the display color for element set BOTFLANG.



are closed and sticking. Some shear stress, which developed due to the friction between thesurfaces, is transmitted from one surface to the other. The maximum separation of the flangesis 0.28 mm and occurs at the location where the O-ring will be placed.

11.4.10 PostprocessingExamine the results graphically using ABAQUS/Viewer. To begin ABAQUS/Viewer, use thecommand:

abaqus viewer odb=flange



You will plot the deformed model shape after assigning different colors to each of the flangecomponents. This will allow you to easily distinguish between the two components.

To change the color of the components:

1. From the main menu bar, select Tools->Color Code.

The Color Code dialog box appears.

2. Select TOPFLANG from the list of element sets.

3. Select RED for the display color.

4. Similarly, specify CYAN as the display color for element set BOTFLANG.



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1. From the main menu bar, select Plot->Deformed Shape; or click the icon in thetoolbox to plot the deformed shape.

2. Click Deformed Options in the prompt area.


3. Click Uniform, and enter a uniform deformation scale factor of 10.0.

4. Choose Exterior visible edges, and set the render style to Shaded.


ABAQUS/Viewer displays the deformed model shape at the end of the simulation, as shownin Figure 11-12. The flange colors reflect the changes made earlier.

Figure 11-12 Deformed model shape at the end of the analysis.

From Figure 11-12 we can see the basic results of the analysis. The flanges have been pulledapart at their inner radius, but the bolts have held the outer part of the flanges together. Theplot clearly shows the need for an O-ring to help seal this joint.

Reporting values of axial displacement

The separation of the flanges at the seal position can also be determined using

1. From the main menu bar, select Plot->Deformed Shape; or click the icon in thetoolbox to plot the deformed shape.



3. Click Uniform, and enter a uniform deformation scale factor of 10.0.



ABAQUS/Viewer displays the deformed model shape at the end of the simulation, as shownin Figure 11-12. The flange colors reflect the changes made earlier.

Figure 11-12 Deformed model shape at the end of the analysis.

From Figure 11-12 we can see the basic results of the analysis. The flanges have been pulledapart at their inner radius, but the bolts have held the outer part of the flanges together. Theplot clearly shows the need for an O-ring to help seal this joint.

Reporting values of axial displacement

The separation of the flanges at the seal position can also be determined using

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ABAQUS/Viewer.

To report the values of displacement:



2. From the Visualization Queries list, select Probe Values; click OK.


3. Select Nodes, and select Deformed Coordinates in the Current Probe Values field.

4. Select the nodes at the seal position: place the cursor over node 2 and click mouse button 1 tosave the deformed coordinates. Similarly, select node 202 and save its deformed coordinates.

The deformed coordinates are displayed in the Selected Probe Values field.

The difference between the 2-direction coordinate values is the separation of the flanges at theseal position.

5. When you have selected all of the desired nodes, exit from probe mode by clicking mousebutton 2 anywhere in the viewport or by clicking the tool in the prompt area.

Contouring the von Mises stress

You will contour the von Mises stress in the flanges to assess whether or not the material hasyielded.


1. From the main menu bar, select Plot->Contours; or click in the toolbox to displaycontours of von Mises stress.



3. Drag the Contour Intervals slider bar to change the number of contour intervals to 7.


5. Click the Shape tab.

The Shape options become available.

6. Click Uniform, and specify a uniform displacement magnification factor of 10.0.


ABAQUS/Viewer.

To report the values of displacement:



2. From the Visualization Queries list, select Probe Values; click OK.


3. Select Nodes, and select Deformed Coordinates in the Current Probe Values field.

4. Select the nodes at the seal position: place the cursor over node 2 and click mouse button 1 tosave the deformed coordinates. Similarly, select node 202 and save its deformed coordinates.

The deformed coordinates are displayed in the Selected Probe Values field.

The difference between the 2-direction coordinate values is the separation of the flanges at theseal position.

5. When you have selected all of the desired nodes, exit from probe mode by clicking mousebutton 2 anywhere in the viewport or by clicking the tool in the prompt area.

Contouring the von Mises stress

You will contour the von Mises stress in the flanges to assess whether or not the material hasyielded.


1. From the main menu bar, select Plot->Contours; or click in the toolbox to displaycontours of von Mises stress.



3. Drag the Contour Intervals slider bar to change the number of contour intervals to 7.


5. Click the Shape tab.

The Shape options become available.

6. Click Uniform, and specify a uniform displacement magnification factor of 10.0.


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To move the contour legend closer to the model:


2. Click the Legend tab.


3. In the Upper left corner field, set % Viewport X to 15 and % Viewport Y to 90.


The legend is moved closer to the model, as shown in Figure 11-13.

Figure 11-13 Contours of von Mises stress in the flanges.

The stresses in the two flanges are similar since both are carrying the same load. The peakstress is around 150 MPa. This stress can be compared against the yield stress of the materialsto check that the flanges will not yield. It can also be used to determine the fatigue life of thecomponents. However, the mesh used for this analysis is very coarse; it omits geometricdetails, such as fillet radii, and only includes the effect of the bolts in an approximate manner.Therefore, stress values obtained from this model must be used with caution. A more accurateanalysis should be carried out if this preliminary simulation raises any questions about theperformance of the flanges.

11.5 Rigid surfaces

To move the contour legend closer to the model:


2. Click the Legend tab.


3. In the Upper left corner field, set % Viewport X to 15 and % Viewport Y to 90.


The legend is moved closer to the model, as shown in Figure 11-13.

Figure 11-13 Contours of von Mises stress in the flanges.

The stresses in the two flanges are similar since both are carrying the same load. The peakstress is around 150 MPa. This stress can be compared against the yield stress of the materialsto check that the flanges will not yield. It can also be used to determine the fatigue life of thecomponents. However, the mesh used for this analysis is very coarse; it omits geometricdetails, such as fillet radii, and only includes the effect of the bolts in an approximate manner.Therefore, stress values obtained from this model must be used with caution. A more accurateanalysis should be carried out if this preliminary simulation raises any questions about theperformance of the flanges.

11.5 Rigid surfaces

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Rigid surfaces are two- or three-dimensional geometric structures that cannot deform but that cantranslate or rotate as rigid bodies. They are useful for modeling components that are much stiffer thanany other component in the simulation. They are also very cost-effective since the only variablesassociated with a rigid surface are the translations and rotations on a single node, known as the rigidbody reference node. In addition, ABAQUS does not need to calculate the stiffnesses or stresses withinthe rigid body. A rigid surface is always the master surface in a contact pair.

There are two methods for defining rigid surfaces in ABAQUS: using analytical rigid surfaces or usingrigid elements.

11.5.1 Analytical rigid surfacesAnalytical rigid surfaces are defined using the *SURFACE option and have three basic forms. Thespecific form of analytical rigid surface is specified with the value of the TYPE parameter. SetTYPE=SEGMENTS to define a two-dimensional, segmented rigid surface. The cross-section of thesurface is defined in the two-dimensional plane of the model using straight lines, circular arcs, andparabolic arcs. The cross-section of a three-dimensional rigid surface is defined in a user-specifiedplane in the same manner used for two-dimensional surfaces. Then, this cross-section is swept aroundan axis (TYPE=REVOLUTION) to form a surface or extruded along a vector to form a longthree-dimensional surface (TYPE=CYLINDER) as shown in Figure 11-14.

Figure 11-14 Analytical rigid surfaces.

The benefit of these analytical rigid surfaces is that they are defined by only a small number ofgeometric points and are computationally efficient. However, in three dimensions the range of shapesthat can be created with them is limited.

11.5.2 Rigid elementsA more general method for creating rigid surfaces is to use rigid elements to define the surfacegeometry. Although ABAQUS/Standard allows other elements-not just rigid elements-to be part of arigid body, in the discussion that follows, we will focus on the rigid elements available in ABAQUS,which are shown in Figure 11-15.

Figure 11-15 Rigid elements in two and three dimensions.

Rigid surfaces are two- or three-dimensional geometric structures that cannot deform but that cantranslate or rotate as rigid bodies. They are useful for modeling components that are much stiffer thanany other component in the simulation. They are also very cost-effective since the only variablesassociated with a rigid surface are the translations and rotations on a single node, known as the rigidbody reference node. In addition, ABAQUS does not need to calculate the stiffnesses or stresses withinthe rigid body. A rigid surface is always the master surface in a contact pair.

There are two methods for defining rigid surfaces in ABAQUS: using analytical rigid surfaces or usingrigid elements.

11.5.1 Analytical rigid surfacesAnalytical rigid surfaces are defined using the *SURFACE option and have three basic forms. Thespecific form of analytical rigid surface is specified with the value of the TYPE parameter. SetTYPE=SEGMENTS to define a two-dimensional, segmented rigid surface. The cross-section of thesurface is defined in the two-dimensional plane of the model using straight lines, circular arcs, andparabolic arcs. The cross-section of a three-dimensional rigid surface is defined in a user-specifiedplane in the same manner used for two-dimensional surfaces. Then, this cross-section is swept aroundan axis (TYPE=REVOLUTION) to form a surface or extruded along a vector to form a longthree-dimensional surface (TYPE=CYLINDER) as shown in Figure 11-14.

Figure 11-14 Analytical rigid surfaces.

The benefit of these analytical rigid surfaces is that they are defined by only a small number ofgeometric points and are computationally efficient. However, in three dimensions the range of shapesthat can be created with them is limited.

11.5.2 Rigid elementsA more general method for creating rigid surfaces is to use rigid elements to define the surfacegeometry. Although ABAQUS/Standard allows other elements-not just rigid elements-to be part of arigid body, in the discussion that follows, we will focus on the rigid elements available in ABAQUS,which are shown in Figure 11-15.

Figure 11-15 Rigid elements in two and three dimensions.

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Use the R2D2 elements for two-dimensional plane stress and plane strain problems and the RAX2elements for axisymmetric simulations. Use the R3D3 and R3D4 elements for modelingthree-dimensional rigid surfaces. These elements are all created in a similar manner to regular two- andthree-dimensional shell elements but have rigid body property definitions associated with them. In fact,if your preprocessor does not support rigid elements, use shell elements to mesh the desired rigidsurfaces and then change the element type to the appropriate rigid element and element propertyoptions in an editor.

Once a surface has been meshed with rigid elements, the *SURFACE option is used to create thesurface. A rigid surface can be defined on either the positive ( SPOS) or negative (SNEG) face of therigid element. The connectivity of the rigid element dictates which element face is the positive face andwhich is the negative face.

11.5.3 Rigid body reference nodeAll rigid surfaces, however they are created, must be bound to a rigid body and assigned a rigid bodyreference node. Analytical surfaces are bound to a rigid body using the *RIGID BODY option. Use theANALYTICAL SURFACE parameter to bind an analytical rigid surface to a rigid body, and use theELSET parameter to associate a group of rigid elements with a rigid body. Use the REF NODEparameter on the *RIGID BODY option to assign the rigid body reference node.

As mentioned previously, the rigid body reference node controls the motion of the entire rigid surface.This node has both translational and rotational degrees of freedom. Thus, if a rigid surface should beconstrained completely, translational and rotational boundary conditions must be applied to the rigidbody reference node. Forces and moments can also be applied to the rigid body reference node and,thus, to the rigid surface.

The position of the rigid body reference node is not important unless rotations are applied to thesurface or reaction moments about a certain axis through the surface are desired. In either of thesesituations the node should be placed such that it lies on the desired axis through the surface.

11.5.4 Modeling issues for rigid surfacesThere are a number of issues that you should consider when modeling contact problems that have rigidsurfaces. These issues are discussed in detail in the ABAQUS/Standard User's Manual, but summariesof some of the more important issues are described here.

The rigid surface should be large enough to ensure that slave nodes do not slide off and "fall behind"the surface. If this happens, the solution usually will fail to converge. Extending the rigid surface orincluding corners along the perimeter (see Figure 11-16) will prevent slave nodes from falling behindthe master surface.

Use the R2D2 elements for two-dimensional plane stress and plane strain problems and the RAX2elements for axisymmetric simulations. Use the R3D3 and R3D4 elements for modelingthree-dimensional rigid surfaces. These elements are all created in a similar manner to regular two- andthree-dimensional shell elements but have rigid body property definitions associated with them. In fact,if your preprocessor does not support rigid elements, use shell elements to mesh the desired rigidsurfaces and then change the element type to the appropriate rigid element and element propertyoptions in an editor.

Once a surface has been meshed with rigid elements, the *SURFACE option is used to create thesurface. A rigid surface can be defined on either the positive ( SPOS) or negative (SNEG) face of therigid element. The connectivity of the rigid element dictates which element face is the positive face andwhich is the negative face.

11.5.3 Rigid body reference nodeAll rigid surfaces, however they are created, must be bound to a rigid body and assigned a rigid bodyreference node. Analytical surfaces are bound to a rigid body using the *RIGID BODY option. Use theANALYTICAL SURFACE parameter to bind an analytical rigid surface to a rigid body, and use theELSET parameter to associate a group of rigid elements with a rigid body. Use the REF NODEparameter on the *RIGID BODY option to assign the rigid body reference node.

As mentioned previously, the rigid body reference node controls the motion of the entire rigid surface.This node has both translational and rotational degrees of freedom. Thus, if a rigid surface should beconstrained completely, translational and rotational boundary conditions must be applied to the rigidbody reference node. Forces and moments can also be applied to the rigid body reference node and,thus, to the rigid surface.

The position of the rigid body reference node is not important unless rotations are applied to thesurface or reaction moments about a certain axis through the surface are desired. In either of thesesituations the node should be placed such that it lies on the desired axis through the surface.

11.5.4 Modeling issues for rigid surfacesThere are a number of issues that you should consider when modeling contact problems that have rigidsurfaces. These issues are discussed in detail in the ABAQUS/Standard User's Manual, but summariesof some of the more important issues are described here.

The rigid surface should be large enough to ensure that slave nodes do not slide off and "fall behind"the surface. If this happens, the solution usually will fail to converge. Extending the rigid surface orincluding corners along the perimeter (see Figure 11-16) will prevent slave nodes from falling behindthe master surface.

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Figure 11-16 Extending rigid surfaces to prevent convergence problems.

The deformable mesh must be refined enough to interact with any feature on the rigid surface. There isno point in having a 10 mm wide feature on the rigid surface if the deformable elements that willcontact it are 20 mm across: the rigid feature will just penetrate into the deformable surface as shownin Figure 11-17.

Figure 11-17 Modeling small features on the rigid surface.

With a sufficiently refined mesh on the deformable surface, ABAQUS will prevent the rigid surfacefrom penetrating the slave surface.

The contact algorithm in ABAQUS/Standard requires the master surface of a contact pair to besmooth. Rigid surfaces are always the master surface and so should always be smoothed. ABAQUSdoes not smooth discrete rigid surfaces. The level of refinement controls the smoothness of a discreterigid surface. Analytical rigid surfaces can be smoothed using the FILLET RADIUS parameter on the*SURFACE option to define a fillet radius that is used to smooth any sharp corners in the rigid surfacedefinition (see Figure 11-18.)

Figure 11-18 Smoothing an analytical rigid surface.

The rigid surface normal must always point toward the deformable surface with which it will interact.If it does not, ABAQUS will detect severe overclosures at all of the nodes on the deformable surfaceand probably will terminate the simulation.

Figure 11-16 Extending rigid surfaces to prevent convergence problems.

The deformable mesh must be refined enough to interact with any feature on the rigid surface. There isno point in having a 10 mm wide feature on the rigid surface if the deformable elements that willcontact it are 20 mm across: the rigid feature will just penetrate into the deformable surface as shownin Figure 11-17.

Figure 11-17 Modeling small features on the rigid surface.

With a sufficiently refined mesh on the deformable surface, ABAQUS will prevent the rigid surfacefrom penetrating the slave surface.

The contact algorithm in ABAQUS/Standard requires the master surface of a contact pair to besmooth. Rigid surfaces are always the master surface and so should always be smoothed. ABAQUSdoes not smooth discrete rigid surfaces. The level of refinement controls the smoothness of a discreterigid surface. Analytical rigid surfaces can be smoothed using the FILLET RADIUS parameter on the*SURFACE option to define a fillet radius that is used to smooth any sharp corners in the rigid surfacedefinition (see Figure 11-18.)

Figure 11-18 Smoothing an analytical rigid surface.

The rigid surface normal must always point toward the deformable surface with which it will interact.If it does not, ABAQUS will detect severe overclosures at all of the nodes on the deformable surfaceand probably will terminate the simulation.

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The normals for an analytical rigid surface are defined as the directions obtained by the 90°counterclockwise rotation of the vectors from the beginning to the end of each line and circularsegment forming the surface (see Figure 11-19).

Figure 11-19 Normals for an analytical rigid surface.

The normals for a rigid surface created from rigid elements are defined by the faces specified on the*SURFACE option creating the surface.

11.6 Example: forming a channelThis simulation of the forming of a channel in a long metal sheet illustrates the use of rigid surfacesand some of the more complex techniques often required for a successful contact analysis.

The problem consists of a strip of deformable material, called the blank, and the tools--the punch, die,and blank holder--that contact the blank. The tools can be modeled as rigid surfaces because they aremuch stiffer than the blank. Figure 11-20 shows the basic arrangement of the components.

Figure 11-20 Forming analysis.

The blank is 1 mm thick and is squeezed between the blank holder and the die. The blank holder forceis 440 kN. This force, in conjunction with the friction between the blank and blank holder and theblank and die, controls how the blank material is drawn into the die during the forming process. Youhave been asked to determine the forces acting on the punch during the forming process. You alsomust assess how well the channel is formed with these particular settings for the blank holder forceand the coefficient of friction between the tools and blank.

The normals for an analytical rigid surface are defined as the directions obtained by the 90°counterclockwise rotation of the vectors from the beginning to the end of each line and circularsegment forming the surface (see Figure 11-19).

Figure 11-19 Normals for an analytical rigid surface.

The normals for a rigid surface created from rigid elements are defined by the faces specified on the*SURFACE option creating the surface.

11.6 Example: forming a channelThis simulation of the forming of a channel in a long metal sheet illustrates the use of rigid surfacesand some of the more complex techniques often required for a successful contact analysis.

The problem consists of a strip of deformable material, called the blank, and the tools--the punch, die,and blank holder--that contact the blank. The tools can be modeled as rigid surfaces because they aremuch stiffer than the blank. Figure 11-20 shows the basic arrangement of the components.

Figure 11-20 Forming analysis.

The blank is 1 mm thick and is squeezed between the blank holder and the die. The blank holder forceis 440 kN. This force, in conjunction with the friction between the blank and blank holder and theblank and die, controls how the blank material is drawn into the die during the forming process. Youhave been asked to determine the forces acting on the punch during the forming process. You alsomust assess how well the channel is formed with these particular settings for the blank holder forceand the coefficient of friction between the tools and blank.

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A two-dimensional, plane strain model will be used. The assumption that there is no strain in theout-of-plane direction of the model is valid if the structure is long in this direction. Only half of thechannel needs to be modeled because the forming process is symmetric about a plane along the centerof the channel.

The dimensions of the various components are shown in Figure 11-21.

Figure 11-21 Dimensions, in m, of the components in the forming simulation.

11.6.1 Coordinate systemTwo-dimensional, plane strain models are defined, by default, in the global 1-2 plane as shown inFigure 11-21. For the forming simulation, place the origin of this plane at the bottom left-hand cornerof the blank (Figure 11-21). The 1-direction will be normal to the symmetry plane, which is located atx = 0.

11.6.2 Mesh designThe mesh for this simulation can be divided into the deformable blank and the rigid tools.

Blank

Once again, the element type should be selected before the mesh is designed. Using theselection process discussed in ``Mesh design,'' Section 11.4.3, incompatible mode elements areselected. The mesh used for the blank should consist of two rows of 100 CPE4I elements (seeFigure 11-22). Two rows of elements are used so that better resolution of the deformationthrough the thickness of the blank will be obtained.

A two-dimensional, plane strain model will be used. The assumption that there is no strain in theout-of-plane direction of the model is valid if the structure is long in this direction. Only half of thechannel needs to be modeled because the forming process is symmetric about a plane along the centerof the channel.

The dimensions of the various components are shown in Figure 11-21.

Figure 11-21 Dimensions, in m, of the components in the forming simulation.

11.6.1 Coordinate systemTwo-dimensional, plane strain models are defined, by default, in the global 1-2 plane as shown inFigure 11-21. For the forming simulation, place the origin of this plane at the bottom left-hand cornerof the blank (Figure 11-21). The 1-direction will be normal to the symmetry plane, which is located atx = 0.

11.6.2 Mesh designThe mesh for this simulation can be divided into the deformable blank and the rigid tools.

Blank

Once again, the element type should be selected before the mesh is designed. Using theselection process discussed in ``Mesh design,'' Section 11.4.3, incompatible mode elements areselected. The mesh used for the blank should consist of two rows of 100 CPE4I elements (seeFigure 11-22). Two rows of elements are used so that better resolution of the deformationthrough the thickness of the blank will be obtained.

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Figure 11-22 Mesh.

The node and element numbers for the mesh shown in Figure 11-23 are from the model of thisproblem given in ``Forming a channel,'' Section A.13. These node and element numbers areused in the discussion that follows.

Figure 11-23 Node and element numbers for the blank.

Tools

The tools are modeled with rigid surfaces. Analytical rigid surfaces are used to model theblank holder and the punch. R2D2 rigid elements are used to model the die. The mesh for thedie is shown in Figure 11-24. The curved corner is modeled with 10 elements: this is done tocreate a sufficiently smooth surface that will capture the corner geometry accurately. Alwaysuse a sufficient number of elements to model such curves. Create three rigid body referencenodes, one for each tool. Place them roughly in the center of each tool.

Figure 11-24 Node and element numbers for the die in the vicinity of the corner.

Figure 11-22 Mesh.

The node and element numbers for the mesh shown in Figure 11-23 are from the model of thisproblem given in ``Forming a channel,'' Section A.13. These node and element numbers areused in the discussion that follows.

Figure 11-23 Node and element numbers for the blank.

Tools

The tools are modeled with rigid surfaces. Analytical rigid surfaces are used to model theblank holder and the punch. R2D2 rigid elements are used to model the die. The mesh for thedie is shown in Figure 11-24. The curved corner is modeled with 10 elements: this is done tocreate a sufficiently smooth surface that will capture the corner geometry accurately. Alwaysuse a sufficient number of elements to model such curves. Create three rigid body referencenodes, one for each tool. Place them roughly in the center of each tool.

Figure 11-24 Node and element numbers for the die in the vicinity of the corner.

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11.6.3 Preprocessing--creating the modelUse a preprocessor to generate the mesh for the blank and the rigid die. The analytical rigid surfacesfor the punch and blank holder can also be defined using a preprocessor such as ABAQUS/CAE.

If your preprocessor does not directly support rigid elements, mesh the die with 2-node beam elementssuch as B21. Once the ABAQUS input file has been created, change the TYPE parameter on the*ELEMENT option from B21 to R2D2, and delete any beam section options.

If you do not want to use a preprocessor, the ABAQUS input options for the model shown in Figure11-23 and Figure 11-24 are given in ``Forming a channel,'' Section A.13.

11.6.4 Reviewing the input file--the model dataWe first review the model definition, including the node and element definitions, section and materialproperties, and rigid body definition.

Model description

Provide a relevant description of the simulation and model on the *HEADING option.

*HEADING

Analysis of the forming of a channel

SI units (N, kg, m, s)


Check that the preprocessor used the correct element type for the blank and for the rigidelements modeling the die. Provide meaningful element set names, such as BLANK and DIE,for the two groups of elements. The *ELEMENT options in your model should look like

*ELEMENT, TYPE=CPE4I, ELSET=BLANK

*ELEMENT, TYPE=R2D2, ELSET=DIE

Create node sets so that parts of the model can be constrained and moved easily. Put the nodesthat are located on the centerline of the blank and have symmetric constraints into a node setcalled CENTER.

11.6.3 Preprocessing--creating the modelUse a preprocessor to generate the mesh for the blank and the rigid die. The analytical rigid surfacesfor the punch and blank holder can also be defined using a preprocessor such as ABAQUS/CAE.

If your preprocessor does not directly support rigid elements, mesh the die with 2-node beam elementssuch as B21. Once the ABAQUS input file has been created, change the TYPE parameter on the*ELEMENT option from B21 to R2D2, and delete any beam section options.

If you do not want to use a preprocessor, the ABAQUS input options for the model shown in Figure11-23 and Figure 11-24 are given in ``Forming a channel,'' Section A.13.

11.6.4 Reviewing the input file--the model dataWe first review the model definition, including the node and element definitions, section and materialproperties, and rigid body definition.

Model description

Provide a relevant description of the simulation and model on the *HEADING option.

*HEADING




Check that the preprocessor used the correct element type for the blank and for the rigidelements modeling the die. Provide meaningful element set names, such as BLANK and DIE,for the two groups of elements. The *ELEMENT options in your model should look like

*ELEMENT, TYPE=CPE4I, ELSET=BLANK


Create node sets so that parts of the model can be constrained and moved easily. Put the nodesthat are located on the centerline of the blank and have symmetric constraints into a node setcalled CENTER.

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*NSET, NSET=CENTER

1,102,203

Place the nodes along the middle of the blank toward the right-hand side of the model, wherethe blank lies between the blank holder and die, into node set MIDRIGHT.

*NSET, NSET=MIDRIGHT, GENERATE

155,202,1

Place the nodes along the middle of the sheet toward the left-hand side of the model,underneath the punch, into node set MIDLEFT.

*NSET, NSET=MIDLEFT, GENERATE

102,147,1

Again, the node numbers in these option blocks are for the model in Figure 11-23; your nodenumbers may be different.

These last two node sets will be used to apply temporary constraints to the blank as thesimulation establishes the contact between the blank and the tools.

Define two element sets, BLANK_B and BLANK_T, containing the lower and upper rows ofelements in the blank. These will be used to define the contact surfaces on the blank.

*ELSET, ELSET=BLANK_B, GENERATE

1,100,1

*ELSET, ELSET=BLANK_T, GENERATE

101,200,1

Section and material properties for the blank

The blank is made from a high-strength steel (elastic modulus of 210.0 GPa, º = 0.3). Itsinelastic stress-strain behavior is shown in Figure 11-25. The material undergoes considerablework hardening as it deforms plastically. It is likely that plastic strains will be large in thisanalysis; therefore, hardening data are provided up to 50% plastic strain.

Figure 11-25 Yield stress vs. plastic strain.

*NSET, NSET=CENTER

1,102,203

Place the nodes along the middle of the blank toward the right-hand side of the model, wherethe blank lies between the blank holder and die, into node set MIDRIGHT.


155,202,1

Place the nodes along the middle of the sheet toward the left-hand side of the model,underneath the punch, into node set MIDLEFT.


102,147,1

Again, the node numbers in these option blocks are for the model in Figure 11-23; your nodenumbers may be different.

These last two node sets will be used to apply temporary constraints to the blank as thesimulation establishes the contact between the blank and the tools.

Define two element sets, BLANK_B and BLANK_T, containing the lower and upper rows ofelements in the blank. These will be used to define the contact surfaces on the blank.


1,100,1


101,200,1

Section and material properties for the blank

The blank is made from a high-strength steel (elastic modulus of 210.0 GPa, º = 0.3). Itsinelastic stress-strain behavior is shown in Figure 11-25. The material undergoes considerablework hardening as it deforms plastically. It is likely that plastic strains will be large in thisanalysis; therefore, hardening data are provided up to 50% plastic strain.

Figure 11-25 Yield stress vs. plastic strain.

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11-315

The blank is going to undergo significant rotation as it deforms. Reporting the values of stressand strain in a coordinate system that rotates with the blank's motion will make it much easierto interpret the results. Therefore, an *ORIENTATION option should be used to create acoordinate system that is aligned initially with the global coordinate system but moves withthe elements as they deform. The following input options are needed to define the blank'selement and material properties:

*SOLID SECTION,MATERIAL=STEEL,ORIENTATION=LOCAL,

ELSET=BLANK

*ORIENTATION, NAME=LOCAL

1.,0.,0., 0.,1.,0.

3,0


*ELASTIC

2.1E11,0.3

*PLASTIC

400.E6, 0.0E-2

420.E6, 2.0E-2

500.E6,20.0E-2

600.E6,50.0E-2

Rigid body

The R2D2 elements forming the surface of the die must be tied together to form a rigid body.To do so, specify the element set containing all of the elements on the *RIGID BODY option.Use the REF NODE parameter to assign a rigid body reference node to the elements. Add thefollowing input to your model:

*RIGID BODY, ELSET=DIE, REF NODE=9000

The values for the ELSET and REF NODE parameters reflect the model shown in ``Forming achannel,'' Section A.13, and may need to be changed to reflect your data. If you forgot to createa rigid body reference node for the die when building the model in the preprocessor, add thefollowing line to your input file:

The blank is going to undergo significant rotation as it deforms. Reporting the values of stressand strain in a coordinate system that rotates with the blank's motion will make it much easierto interpret the results. Therefore, an *ORIENTATION option should be used to create acoordinate system that is aligned initially with the global coordinate system but moves withthe elements as they deform. The following input options are needed to define the blank'selement and material properties:

*SOLID SECTION,MATERIAL=STEEL,ORIENTATION=LOCAL,

ELSET=BLANK


1.,0.,0., 0.,1.,0.

3,0


*ELASTIC

2.1E11,0.3

*PLASTIC

400.E6, 0.0E-2

420.E6, 2.0E-2

500.E6,20.0E-2

600.E6,50.0E-2

Rigid body

The R2D2 elements forming the surface of the die must be tied together to form a rigid body.To do so, specify the element set containing all of the elements on the *RIGID BODY option.Use the REF NODE parameter to assign a rigid body reference node to the elements. Add thefollowing input to your model:


The values for the ELSET and REF NODE parameters reflect the model shown in ``Forming achannel,'' Section A.13, and may need to be changed to reflect your data. If you forgot to createa rigid body reference node for the die when building the model in the preprocessor, add thefollowing line to your input file:

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*NODE, NSET=REFDIE

9000,0.1,-0.06

11.6.5 Contact definitionsThe contact definitions for each part of the model are discussed here.

Rigid surfaces

The surface of the die is created from the mesh of R2D2 elements. These elements are in anelement set called DIE. Specify the side of the elements that form the contact surface. Thesurface is on the positive side of the elements created in the model in ``Forming a channel,''Section A.13, so the following input defines the surface DIE on the mesh of rigid elements:

*SURFACE, NAME=DIE

DIE, SPOS

You may have defined the rigid elements differently; check your element definitions todetermine whether the surface lies on the positive or negative side of the elements.

When rigid elements are used to define a rigid surface, the contact direction is determined bythe nodal normal, which is the average of the normals of the elements attached to the node.Precise normals can be specified by using the *NORMAL option. This option allows you tospecify the exact nodal normal that is used at a given node on a given contact surface. In thissimulation use it to specify the nodal normals at nodes 9006 and 9016 to better define thecontact direction. The effect of specifying the normals to the surface is illustrated in Figure11-26. For the sake of clarity, only two elements are used to represent the corner of the die inthis figure, instead of the 10 that are actually used.

Figure 11-26 Effect of specifying the normals to the surface.

Add the following option block to your input file (if necessary, change the node numbers tovalues appropriate for your model):

*NODE, NSET=REFDIE

9000,0.1,-0.06

11.6.5 Contact definitionsThe contact definitions for each part of the model are discussed here.

Rigid surfaces

The surface of the die is created from the mesh of R2D2 elements. These elements are in anelement set called DIE. Specify the side of the elements that form the contact surface. Thesurface is on the positive side of the elements created in the model in ``Forming a channel,''Section A.13, so the following input defines the surface DIE on the mesh of rigid elements:

*SURFACE, NAME=DIE

DIE, SPOS

You may have defined the rigid elements differently; check your element definitions todetermine whether the surface lies on the positive or negative side of the elements.

When rigid elements are used to define a rigid surface, the contact direction is determined bythe nodal normal, which is the average of the normals of the elements attached to the node.Precise normals can be specified by using the *NORMAL option. This option allows you tospecify the exact nodal normal that is used at a given node on a given contact surface. In thissimulation use it to specify the nodal normals at nodes 9006 and 9016 to better define thecontact direction. The effect of specifying the normals to the surface is illustrated in Figure11-26. For the sake of clarity, only two elements are used to represent the corner of the die inthis figure, instead of the 10 that are actually used.

Figure 11-26 Effect of specifying the normals to the surface.

Add the following option block to your input file (if necessary, change the node numbers tovalues appropriate for your model):

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11-317

The blank holder and the punch are modeled with analytical rigid surfaces. A rigid bodyreference node will be assigned to each of these surfaces when they are created. If you did notcreate these rigid body reference nodes, add the following option block to your model:

*NODE, NSET=REFPUNCH

7000, 0.000, 0.061

*NODE, NSET=REFHOLD

8000, 0.1,0.06

Each node is placed in a node set to make the input file easy to read. While someoneunfamiliar with the specific mesh used in the simulation may not know why boundaryconditions are applied to node 7000, they might be able to guess why boundary conditionswere applied to the REFPUNCH node.

To ensure that an analytical rigid surface's normals point toward the deformable surfaces thatthe rigid surface will contact, the segments composing the rigid surface must be defined in aparticular order. For example, to create the correct normals for the surface PUNCH, define thesurface from the top right corner to the bottom left corner of the punch. Add the followinginput to your model to create the surface PUNCH:

*SURFACE, TYPE=SEGMENTS, NAME=PUNCH, FILLET RADIUS=0.001

START,0.050, 0.061

LINE, 0.050, 0.007

CIRCL,0.045, 0.002, 0.045,0.007

LINE,-0.010, 0.002

*RIGID BODY, ANALYTICAL SURFACE=PUNCH, REF NODE=7000

The parameter TYPE=SEGMENTS specifies that a two-dimensional rigid surface is beingdefined. The NAME parameter specifies the name of the surface, PUNCH. The data linesdefine the geometry of the surface. The first data line always has the word "START" followedby the 1- and 2-coordinates of the starting point for the surface. The subsequent lines defineline, circular, and parabolic segments. For this surface the second data line defines a straightline from the start position (0.05, 0.061) to (0.050, 0.007). The third data line defines acircular arc from the end of the straight line (0.05, 0.007) to (0.045, 0.002) with the center ofthe circle located at (0.045, 0.007). The last data line defines a straight line from the end of thearc to (-0.010, 0.002).

This definition should produce a smooth rigid surface but, to be safe, the FILLET RADIUSparameter specifies that a 1 mm fillet radius should be used to smooth any discontinuities inthe surface definition. It is always good practice to add the FILLET RADIUS parameter to the

The blank holder and the punch are modeled with analytical rigid surfaces. A rigid bodyreference node will be assigned to each of these surfaces when they are created. If you did notcreate these rigid body reference nodes, add the following option block to your model:


7000, 0.000, 0.061

*NODE, NSET=REFHOLD

8000, 0.1,0.06

Each node is placed in a node set to make the input file easy to read. While someoneunfamiliar with the specific mesh used in the simulation may not know why boundaryconditions are applied to node 7000, they might be able to guess why boundary conditionswere applied to the REFPUNCH node.

To ensure that an analytical rigid surface's normals point toward the deformable surfaces thatthe rigid surface will contact, the segments composing the rigid surface must be defined in aparticular order. For example, to create the correct normals for the surface PUNCH, define thesurface from the top right corner to the bottom left corner of the punch. Add the followinginput to your model to create the surface PUNCH:


START,0.050, 0.061

LINE, 0.050, 0.007

CIRCL,0.045, 0.002, 0.045,0.007

LINE,-0.010, 0.002


The parameter TYPE=SEGMENTS specifies that a two-dimensional rigid surface is beingdefined. The NAME parameter specifies the name of the surface, PUNCH. The data linesdefine the geometry of the surface. The first data line always has the word "START" followedby the 1- and 2-coordinates of the starting point for the surface. The subsequent lines defineline, circular, and parabolic segments. For this surface the second data line defines a straightline from the start position (0.05, 0.061) to (0.050, 0.007). The third data line defines acircular arc from the end of the straight line (0.05, 0.007) to (0.045, 0.002) with the center ofthe circle located at (0.045, 0.007). The last data line defines a straight line from the end of thearc to (-0.010, 0.002).

This definition should produce a smooth rigid surface but, to be safe, the FILLET RADIUSparameter specifies that a 1 mm fillet radius should be used to smooth any discontinuities inthe surface definition. It is always good practice to add the FILLET RADIUS parameter to the

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definition of any analytical rigid surface.

The *RIGID BODY option is used to bind the analytical surface to a rigid body with its rigidbody reference node specified by the REF NODE parameter and the surface referred to by itsname, using the ANALYTICAL SURFACE parameter.

The rigid surface for the blank holder is defined in a similar way. Add the following optionblock to your model to define the surface on the blank holder:

*SURFACE, TYPE=SEGMENTS, NAME=HOLDER, FILLET RADIUS=0.001

START,0.110, 0.001

LINE, 0.056,0.001

CIRCL,0.051,0.006, 0.056,0.006

LINE, 0.051,0.060

*RIGID BODY, ANALYTICAL SURFACE=HOLDER, REF NODE=8000

All of the rigid surfaces in this simulation extend beyond the deformable blank to ensure thatthere is no possibility that slave nodes will slide behind any of them. The initial configurationof these surfaces and the locations of their reference nodes are shown in Figure 11-27.

Figure 11-27 Rigid body reference nodes.

Deformable surfaces

Using the two element sets defined on the blank, create a contact surface on the top of theblank, called BLANK_T, and one on the bottom, called BLANK_B. If you use the automaticfree surface generation capability, the option blocks will look like

*SURFACE, NAME=BLANK_B, TRIM=YES

BLANK_B,

*SURFACE, NAME=BLANK_T, TRIM=YES

BLANK_T,

definition of any analytical rigid surface.

The *RIGID BODY option is used to bind the analytical surface to a rigid body with its rigidbody reference node specified by the REF NODE parameter and the surface referred to by itsname, using the ANALYTICAL SURFACE parameter.

The rigid surface for the blank holder is defined in a similar way. Add the following optionblock to your model to define the surface on the blank holder:


START,0.110, 0.001

LINE, 0.056,0.001

CIRCL,0.051,0.006, 0.056,0.006

LINE, 0.051,0.060


All of the rigid surfaces in this simulation extend beyond the deformable blank to ensure thatthere is no possibility that slave nodes will slide behind any of them. The initial configurationof these surfaces and the locations of their reference nodes are shown in Figure 11-27.

Figure 11-27 Rigid body reference nodes.

Deformable surfaces

Using the two element sets defined on the blank, create a contact surface on the top of theblank, called BLANK_T, and one on the bottom, called BLANK_B. If you use the automaticfree surface generation capability, the option blocks will look like

*SURFACE, NAME=BLANK_B, TRIM=YES

BLANK_B,

*SURFACE, NAME=BLANK_T, TRIM=YES

BLANK_T,

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11-319

Contact pairs

Contact must be defined between the top of the blank and the punch, the top of the blank andthe blank holder, and the bottom of the blank and the die. The rigid surface must be the mastersurface in each of these contact pairs. Each contact pair must refer to a *SURFACEINTERACTION option that defines a surface interaction model governing how the surfaces ofthe contact pair interact with each other. Multiple contact pairs can refer to the same*SURFACE INTERACTION option.

In this example we assume that the friction coefficient is zero between the blank and thepunch. The friction coefficient between the blank and the other two tools is assumed to be 0.1.Therefore, two *SURFACE INTERACTION options must be used in the model: one withfriction and one without. Frictionless contact is the default in ABAQUS, so no *FRICTIONoption is needed in the surface interaction definition for the BLANK_T-PUNCH contact pair.

The option blocks to define the contact pairs and surface interactions in your model will looklike

*CONTACT PAIR, INTERACTION=FRIC

BLANK_B, DIE


BLANK_T, HOLDER

*CONTACT PAIR, INTERACTION=NOFRIC

BLANK_T, PUNCH


*FRICTION

0.1,

*SURFACE INTERACTION, NAME=NOFRIC

11.6.6 Reviewing the input file--the history dataThere are two major sources of difficulty in contact analyses: rigid body motion of the componentsbefore contact conditions constrain them and sudden changes in contact conditions, which lead tosevere discontinuity iterations as ABAQUS tries to establish the exact condition of all contact surfaces.Therefore, wherever possible, take precautions to avoid these situations.

Removing rigid body motion is not particularly difficult. Simply ensure that there are enoughconstraints to prevent all rigid body motions of all the components in the model. This may mean usingboundary conditions initially to get the components into contact, instead of applying loads directly.Using this approach may require more steps than originally anticipated, but the solution of the problemshould proceed more smoothly.

Unless a dynamic impact event is being simulated, always try to establish contact between componentsin a reasonably gentle manner, avoiding large overclosures and rapid changes in contact pressure.Again, this usually means adding additional steps to the analysis to move components into contactbefore fully loading them. This approach, although requiring more steps, should minimize convergence

Contact pairs

Contact must be defined between the top of the blank and the punch, the top of the blank andthe blank holder, and the bottom of the blank and the die. The rigid surface must be the mastersurface in each of these contact pairs. Each contact pair must refer to a *SURFACEINTERACTION option that defines a surface interaction model governing how the surfaces ofthe contact pair interact with each other. Multiple contact pairs can refer to the same*SURFACE INTERACTION option.

In this example we assume that the friction coefficient is zero between the blank and thepunch. The friction coefficient between the blank and the other two tools is assumed to be 0.1.Therefore, two *SURFACE INTERACTION options must be used in the model: one withfriction and one without. Frictionless contact is the default in ABAQUS, so no *FRICTIONoption is needed in the surface interaction definition for the BLANK_T-PUNCH contact pair.

The option blocks to define the contact pairs and surface interactions in your model will looklike


BLANK_B, DIE


BLANK_T, HOLDER


BLANK_T, PUNCH


*FRICTION

0.1,


11.6.6 Reviewing the input file--the history dataThere are two major sources of difficulty in contact analyses: rigid body motion of the componentsbefore contact conditions constrain them and sudden changes in contact conditions, which lead tosevere discontinuity iterations as ABAQUS tries to establish the exact condition of all contact surfaces.Therefore, wherever possible, take precautions to avoid these situations.

Removing rigid body motion is not particularly difficult. Simply ensure that there are enoughconstraints to prevent all rigid body motions of all the components in the model. This may mean usingboundary conditions initially to get the components into contact, instead of applying loads directly.Using this approach may require more steps than originally anticipated, but the solution of the problemshould proceed more smoothly.

Unless a dynamic impact event is being simulated, always try to establish contact between componentsin a reasonably gentle manner, avoiding large overclosures and rapid changes in contact pressure.Again, this usually means adding additional steps to the analysis to move components into contactbefore fully loading them. This approach, although requiring more steps, should minimize convergence

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11-320

difficulties and, therefore, make the solution far more efficient. With these points in mind, we can nowdesign the history data for this example.

Step 1

The first stage of the process is to hold the blank between the blank holder and the die. At thestart of the first increment of the analysis, contact may not be fully established between thevarious components, even though their surfaces are coincident initially. Problems can occurwhen contact is not fully established: components may undergo rigid body motion; or thecontact status may oscillate between open and closed, which is known as chattering.Chattering is especially common in contact analyses with multiple components.

To avoid rigid body motions and chattering in this simulation, fix the midplane of the blank inthe vertical direction to prevent the blank from moving initially. The midplane of the blank isused because constraints should not be applied to nodes that are also part of a contact surface.If a contact node is constrained by a boundary condition in the direction that contact occurs,there will be two constraints applied to a single degree of freedom. This can cause numericalproblems, and ABAQUS may issue a zero pivot warning message in the message file.

One method of establishing contact is to apply a force to the blank holder. However, the blankholder may undergo rigid body motion in the vertical direction because contact between theblank holder and the blank may not be fully established. Therefore, it is better to move theblank holder by means of an applied displacement with a *BOUNDARY option. This ensuresfirm contact between these two surfaces. In addition, move the die slightly upward to establishfirm contact between it and the blank. The motion of the tools in the first step, including themagnitudes, is summarized in Figure 11-28.

Figure 11-28 Loads and constraints in Step 1.

Constrain the blank holder and die in degrees of freedom 1 and 6, where degree of freedom 6is the rotation in the plane of the model. All of the boundary conditions for the rigid surfacesare applied to their respective rigid body reference nodes. Constrain the punch completely, andapply symmetric boundary constraints on the nodes of the blank lying on the symmetry plane(node set CENTER).

difficulties and, therefore, make the solution far more efficient. With these points in mind, we can nowdesign the history data for this example.

Step 1

The first stage of the process is to hold the blank between the blank holder and the die. At thestart of the first increment of the analysis, contact may not be fully established between thevarious components, even though their surfaces are coincident initially. Problems can occurwhen contact is not fully established: components may undergo rigid body motion; or thecontact status may oscillate between open and closed, which is known as chattering.Chattering is especially common in contact analyses with multiple components.

To avoid rigid body motions and chattering in this simulation, fix the midplane of the blank inthe vertical direction to prevent the blank from moving initially. The midplane of the blank isused because constraints should not be applied to nodes that are also part of a contact surface.If a contact node is constrained by a boundary condition in the direction that contact occurs,there will be two constraints applied to a single degree of freedom. This can cause numericalproblems, and ABAQUS may issue a zero pivot warning message in the message file.

One method of establishing contact is to apply a force to the blank holder. However, the blankholder may undergo rigid body motion in the vertical direction because contact between theblank holder and the blank may not be fully established. Therefore, it is better to move theblank holder by means of an applied displacement with a *BOUNDARY option. This ensuresfirm contact between these two surfaces. In addition, move the die slightly upward to establishfirm contact between it and the blank. The motion of the tools in the first step, including themagnitudes, is summarized in Figure 11-28.


Constrain the blank holder and die in degrees of freedom 1 and 6, where degree of freedom 6is the rotation in the plane of the model. All of the boundary conditions for the rigid surfacesare applied to their respective rigid body reference nodes. Constrain the punch completely, andapply symmetric boundary constraints on the nodes of the blank lying on the symmetry plane(node set CENTER).

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11-321

The effects of geometric nonlinearity must be considered in this simulation, so include theNLGEOM parameter on the *STEP option. The step should complete in one increment, so onthe *STATIC option set the initial time increment and the total time to be equal. To limit theamount of output, write the preselected field output only at the end of the step, and do notprint any results to the data (.dat) file. Additionally, we do not need any history output in thepresent step. Output for variables that we need to track using history data can be requested,when needed, at a later stage in the analysis. Use the *PRINT, CONTACT=YES option towrite contact diagnostics to the message file.

The complete step definition in your model should be similar to the following:

*STEP, INC=10, NLGEOM

Push the blank holder and die together

*STATIC

1.0,1.0

*BOUNDARY

MIDRIGHT,2

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,2,,-1.E-8

REFHOLD ,6

*PRINT, CONTACT=YES



*OUTPUT, FIELD, OP=NEW, FREQUENCY=99, VARIABLE=PRESELECT

*END STEP

Step 2

Step 1 established contact between the blank and the blank holder and die, which fullyconstrains the blank in the 2-direction. Therefore, the boundary conditions on the right half ofthe midplane of the blank (node set MIDRIGHT) can now be removed, as shown in Figure11-29. Remove the boundary constraints by using the OP=NEW parameter on the*BOUNDARY option; however, you must respecify all of the constraints that remain ineffect.


The effects of geometric nonlinearity must be considered in this simulation, so include theNLGEOM parameter on the *STEP option. The step should complete in one increment, so onthe *STATIC option set the initial time increment and the total time to be equal. To limit theamount of output, write the preselected field output only at the end of the step, and do notprint any results to the data (.dat) file. Additionally, we do not need any history output in thepresent step. Output for variables that we need to track using history data can be requested,when needed, at a later stage in the analysis. Use the *PRINT, CONTACT=YES option towrite contact diagnostics to the message file.

The complete step definition in your model should be similar to the following:


Push the blank holder and die together

*STATIC

1.0,1.0

*BOUNDARY

MIDRIGHT,2

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,2,,-1.E-8

REFHOLD ,6

*PRINT, CONTACT=YES




*END STEP

Step 2

Step 1 established contact between the blank and the blank holder and die, which fullyconstrains the blank in the 2-direction. Therefore, the boundary conditions on the right half ofthe midplane of the blank (node set MIDRIGHT) can now be removed, as shown in Figure11-29. Remove the boundary constraints by using the OP=NEW parameter on the*BOUNDARY option; however, you must respecify all of the constraints that remain ineffect.


Contact

11-322

This step should also complete in a single increment because the only change being made tothe model is the removal of the vertical constraints on the blank; therefore, set the initial timeincrement and the total time to be equal. The output requests do not need to be changed. Theinput data defining the second step are, therefore,


Remove the middle constraint

*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,2,,-1.E-8

REFHOLD ,6

*END STEP

Step 3

The magnitude of the blank holder force is a controlling factor in many forming processes;therefore, it needs to be introduced as a variable load in the analysis. In Step 3 remove theboundary condition used to move the blank holder down and replace it with a force specifiedusing a *CLOAD option, as shown in Figure 11-30.


This step should also complete in a single increment because the only change being made tothe model is the removal of the vertical constraints on the blank; therefore, set the initial timeincrement and the total time to be equal. The output requests do not need to be changed. Theinput data defining the second step are, therefore,



*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,2,,-1.E-8

REFHOLD ,6

*END STEP

Step 3

The magnitude of the blank holder force is a controlling factor in many forming processes;therefore, it needs to be introduced as a variable load in the analysis. In Step 3 remove theboundary condition used to move the blank holder down and replace it with a force specifiedusing a *CLOAD option, as shown in Figure 11-30.


Contact

11-323

There should be no problems replacing the applied displacement on the blank holder with aforce because there is firm contact between the blank holder and the blank. In this simulationthe required blank holder force is 440 kN. Generally, care should be taken to ensure that thenewly applied force is of the same order as the reaction force generated from the boundarycondition so that the contact condition does not change significantly.

Again, use the OP=NEW parameter on the *BOUNDARY option to remove the constraint onthe blank holder in degree of freedom 2, and respecify the remaining boundary conditions.This step will also complete in a single increment, so set the initial time increment to be equalto the total time. The output requests do not need to be changed.

The input for Step 3 is


Apply prescribed force on blank holder

*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,6

*CLOAD

REFHOLD,2,-4.4E5

*END STEP

Step 4

At the beginning of the analysis the punch and the blank are separated to avoid anyinterference while contact is established between the blank and the die and blank holder. Inthis step the punch is moved down in the 2-direction just enough to achieve contact with theblank, as shown in Figure 11-31.

There should be no problems replacing the applied displacement on the blank holder with aforce because there is firm contact between the blank holder and the blank. In this simulationthe required blank holder force is 440 kN. Generally, care should be taken to ensure that thenewly applied force is of the same order as the reaction force generated from the boundarycondition so that the contact condition does not change significantly.

Again, use the OP=NEW parameter on the *BOUNDARY option to remove the constraint onthe blank holder in degree of freedom 2, and respecify the remaining boundary conditions.This step will also complete in a single increment, so set the initial time increment to be equalto the total time. The output requests do not need to be changed.

The input for Step 3 is


Apply prescribed force on blank holder

*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,ENCASTRE

REFHOLD ,1

REFHOLD ,6

*CLOAD

REFHOLD,2,-4.4E5

*END STEP

Step 4

At the beginning of the analysis the punch and the blank are separated to avoid anyinterference while contact is established between the blank and the die and blank holder. Inthis step the punch is moved down in the 2-direction just enough to achieve contact with theblank, as shown in Figure 11-31.

Contact

11-324


The vertical constraint on node set MIDLEFT is removed, and a small pressure is applied tothe surface of the elements in element set BLANK_T to pull them onto the surface of thepunch. It can be difficult to choose the proper magnitude of pressure to apply. In this analysiswe will pick a pressure magnitude (1000 Pa) that produces a force on the blank that is threeorders of magnitude lower than the blank holder force. A positive pressure acts into thesurface; however, in this simulation we want the pressure to act out of the surface, so use anegative pressure. This technique is used to prevent chattering of the BLANK_T and PUNCHsurfaces. Leave the field output requests and the load on the blank holder unchanged in thisstep, and add a request for history output of the reaction force on the reference node of thepunch.

Set the initial time increment to 10% of the total time since the contact condition may beharder to establish in this step. The following input options should be used for Step 4:


Move the punch down a little while applying a small

pressure to BLANK_T

*STATIC

0.1,1.0

*BOUNDARY, OP=NEW

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,1

REFPUNCH,2,,-1.000001E-3

REFPUNCH,6

REFHOLD ,1

REFHOLD ,6

*DLOAD

BLANK_T, P3, -1000.0E0

*OUTPUT, HISTORY, OP=ADD, FREQUENCY=1

*NODE OUTPUT, NSET=REFPUNCH


The vertical constraint on node set MIDLEFT is removed, and a small pressure is applied tothe surface of the elements in element set BLANK_T to pull them onto the surface of thepunch. It can be difficult to choose the proper magnitude of pressure to apply. In this analysiswe will pick a pressure magnitude (1000 Pa) that produces a force on the blank that is threeorders of magnitude lower than the blank holder force. A positive pressure acts into thesurface; however, in this simulation we want the pressure to act out of the surface, so use anegative pressure. This technique is used to prevent chattering of the BLANK_T and PUNCHsurfaces. Leave the field output requests and the load on the blank holder unchanged in thisstep, and add a request for history output of the reaction force on the reference node of thepunch.

Set the initial time increment to 10% of the total time since the contact condition may beharder to establish in this step. The following input options should be used for Step 4:


Move the punch down a little while applying a small

pressure to BLANK_T

*STATIC

0.1,1.0

*BOUNDARY, OP=NEW

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFDIE ,6

REFPUNCH,1

REFPUNCH,2,,-1.000001E-3

REFPUNCH,6

REFHOLD ,1

REFHOLD ,6

*DLOAD

BLANK_T, P3, -1000.0E0



Contact

11-325

RF,

*END STEP

Step 5

Remove the pressure load applied to element set BLANK_T and move the punch down tocomplete the forming operation, as shown in Figure 11-32. When a pressure load is removedin a static analysis, the magnitude is ramped down to zero over the duration of the step.Therefore, the pressure will continue to pull on the surface BLANK_T and hold it tightlyagainst the punch, maintaining contact between the surfaces, particularly in the early part ofthe step. This helps prevent chattering between the punch and the blank at the beginning of theforming process. As long as the force created by the pressure is small compared to the forcerequired to move the punch, it will have little effect on the solution.


Between the frictional sliding, the changing contact conditions, and the inelastic materialbehavior, there is significant nonlinearity in this step; therefore, set the maximum number ofincrements to a large value. Set the initial time increment to be 0.0001, the total step time to be1.0, and the minimum time increment to be 10-6. Therefore, ABAQUS can take smaller timeincrements during the highly nonlinear parts of the response without terminating the analysis.

Reset the FREQUENCY parameter on the *OUTPUT, FIELD option so that field data arewritten to the output database file every 20 increments. This allows the response of the blankto be evaluated at several points during the forming operation. The input for Step 5 is


Move the punch down to full extent

*STATIC

.0001,1.0,1.E-6

*BOUNDARY

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFPUNCH,1

RF,

*END STEP

Step 5

Remove the pressure load applied to element set BLANK_T and move the punch down tocomplete the forming operation, as shown in Figure 11-32. When a pressure load is removedin a static analysis, the magnitude is ramped down to zero over the duration of the step.Therefore, the pressure will continue to pull on the surface BLANK_T and hold it tightlyagainst the punch, maintaining contact between the surfaces, particularly in the early part ofthe step. This helps prevent chattering between the punch and the blank at the beginning of theforming process. As long as the force created by the pressure is small compared to the forcerequired to move the punch, it will have little effect on the solution.


Between the frictional sliding, the changing contact conditions, and the inelastic materialbehavior, there is significant nonlinearity in this step; therefore, set the maximum number ofincrements to a large value. Set the initial time increment to be 0.0001, the total step time to be1.0, and the minimum time increment to be 10-6. Therefore, ABAQUS can take smaller timeincrements during the highly nonlinear parts of the response without terminating the analysis.

Reset the FREQUENCY parameter on the *OUTPUT, FIELD option so that field data arewritten to the output database file every 20 increments. This allows the response of the blankto be evaluated at several points during the forming operation. The input for Step 5 is



*STATIC

.0001,1.0,1.E-6

*BOUNDARY

CENTER ,XSYMM

REFDIE ,1

REFDIE ,2,, 1.E-8

REFPUNCH,1

Contact

11-326

REFPUNCH,2,,-0.03

REFPUNCH,6

REFHOLD ,1

REFHOLD ,6

*DLOAD, OP=NEW

*OUTPUT, FIELD, OP=ADD, FREQUENCY=20, VARIABLE=PRESELECT

*END STEP

11.6.7 Troubleshooting contact analysesContact analyses are generally more difficult to complete than just about any other type of simulationin ABAQUS. Therefore, it is important to understand all of the options available to help you withcontact analyses.

If a contact analysis runs into difficulty, the first thing to check is whether the contact surfaces aredefined correctly. The easiest way to do this is to run a datacheck analysis and plot the surfacenormals in ABAQUS/Viewer. You can plot all of the normals, for both surfaces and structuralelements, on either the deformed or the undeformed plots. Use the Normals options in theUndeformed Options and Deformed Options dialog boxes to do this, and confirm that the surfacenormals are in the correct directions.

ABAQUS may still have some problems with contact simulations, even when the contact surfaces areall defined correctly. One reason for these problems may be the default convergence tolerances andlimits on the number of iterations: they are quite rigorous. In contact analyses it is sometimes better toallow ABAQUS to iterate a few more times rather than abandon the increment and try again. This iswhy ABAQUS makes the distinction between severe discontinuity iterations and equilibrium iterationsduring the simulation.

The *PRINT, CONTACT=YES option is essential for almost every contact analysis. The informationthis option provides in the message file can be vital for spotting mistakes or problems. For example,chattering can be spotted because the same slave node will be seen to be involved in all of the severediscontinuity iterations. If you see this, you will have to modify the mesh in the region around thatnode or add constraints to the model. Contact data in the message file can also identify regions whereonly a single slave node is interacting with a surface. This is a very unstable situation and can causeconvergence problems. Again, you should modify the model to increase the number of elements insuch regions.

11.6.8 Running the analysisSave the input in the file channel.inp (see ``Forming a channel,'' Section A.13). Run the analysisin the background since it will take several minutes to an hour to complete.

abaqus job=channel

Check the status and message files while the job is running to see how it is progressing.

Status file

REFPUNCH,2,,-0.03

REFPUNCH,6

REFHOLD ,1

REFHOLD ,6

*DLOAD, OP=NEW


*END STEP

11.6.7 Troubleshooting contact analysesContact analyses are generally more difficult to complete than just about any other type of simulationin ABAQUS. Therefore, it is important to understand all of the options available to help you withcontact analyses.

If a contact analysis runs into difficulty, the first thing to check is whether the contact surfaces aredefined correctly. The easiest way to do this is to run a datacheck analysis and plot the surfacenormals in ABAQUS/Viewer. You can plot all of the normals, for both surfaces and structuralelements, on either the deformed or the undeformed plots. Use the Normals options in theUndeformed Options and Deformed Options dialog boxes to do this, and confirm that the surfacenormals are in the correct directions.

ABAQUS may still have some problems with contact simulations, even when the contact surfaces areall defined correctly. One reason for these problems may be the default convergence tolerances andlimits on the number of iterations: they are quite rigorous. In contact analyses it is sometimes better toallow ABAQUS to iterate a few more times rather than abandon the increment and try again. This iswhy ABAQUS makes the distinction between severe discontinuity iterations and equilibrium iterationsduring the simulation.

The *PRINT, CONTACT=YES option is essential for almost every contact analysis. The informationthis option provides in the message file can be vital for spotting mistakes or problems. For example,chattering can be spotted because the same slave node will be seen to be involved in all of the severediscontinuity iterations. If you see this, you will have to modify the mesh in the region around thatnode or add constraints to the model. Contact data in the message file can also identify regions whereonly a single slave node is interacting with a surface. This is a very unstable situation and can causeconvergence problems. Again, you should modify the model to increase the number of elements insuch regions.

11.6.8 Running the analysisSave the input in the file channel.inp (see ``Forming a channel,'' Section A.13). Run the analysisin the background since it will take several minutes to an hour to complete.

abaqus job=channel

Check the status and message files while the job is running to see how it is progressing.

Status file

Contact

11-327

This analysis should take over 200 increments to complete. The top of the status file is shownbelow.




ITERS FREQ

1 1 1 0 2 2 1.00 1.00 1.000

2 1 1 0 1 1 2.00 1.00 1.000

3 1 1 0 2 2 3.00 1.00 1.000

4 1 1 0 2 2 3.10 0.100 0.1000

4 2 1 0 1 1 3.20 0.200 0.1000

4 3 1 0 1 1 3.35 0.350 0.1500

4 4 1 0 1 1 3.58 0.575 0.2250

4 5 1 0 1 1 3.91 0.913 0.3375

4 6 1 3 1 4 4.00 1.00 0.08750

5 1 4 8 1 9 4.00 1.56e-06 1.563e-06

5 2 1 3 1 4 4.00 3.13e-06 1.563e-06

ABAQUS has a very difficult time determining the contact state in the first increment of Step5. It needs four attempts before it finds the proper configuration of the PUNCH and BLANK_Tsurfaces. Once it finds the correct configuration, ABAQUS only needs a single iteration toachieve equilibrium. After this difficult start, ABAQUS quickly increases the increment size toa more reasonable value. The end of the status file is shown below.

5 456 1 0 2 2 4.99 0.990 0.0003959

5 457 1 0 1 1 4.99 0.991 0.0005938

5 458 1 0 1 1 4.99 0.991 0.0008907

5 459 1 0 2 2 4.99 0.993 0.001336

5 460 2 1 3 4 4.99 0.993 0.0005010

5 461 1 0 1 1 4.99 0.994 0.0007515

5 462 1 0 3 3 5.00 0.995 0.001127

5 463 1 0 2 2 5.00 0.997 0.001691

5 464 1 1 4 5 5.00 0.999 0.002536

5 465 1 1 3 4 5.00 1.00 0.0006165

This simulation, unlike the previous one, contains many severe discontinuity iterations. Themessage file will be quite large because of the number of iterations in the analysis. Although itmight be tempting to limit the information written to this file, generally this should not bedone because this information is the main source of diagnostic data that ABAQUS providesduring the simulation.

11.6.9 PostprocessingBecause of the length of this simulation, no printed output was requested. All results will be studiedusing an interactive postprocessor. Use the following command to run ABAQUS/Viewer:

This analysis should take over 200 increments to complete. The top of the status file is shownbelow.




ITERS FREQ

1 1 1 0 2 2 1.00 1.00 1.000

2 1 1 0 1 1 2.00 1.00 1.000

3 1 1 0 2 2 3.00 1.00 1.000

4 1 1 0 2 2 3.10 0.100 0.1000

4 2 1 0 1 1 3.20 0.200 0.1000

4 3 1 0 1 1 3.35 0.350 0.1500

4 4 1 0 1 1 3.58 0.575 0.2250

4 5 1 0 1 1 3.91 0.913 0.3375

4 6 1 3 1 4 4.00 1.00 0.08750

5 1 4 8 1 9 4.00 1.56e-06 1.563e-06

5 2 1 3 1 4 4.00 3.13e-06 1.563e-06

ABAQUS has a very difficult time determining the contact state in the first increment of Step5. It needs four attempts before it finds the proper configuration of the PUNCH and BLANK_Tsurfaces. Once it finds the correct configuration, ABAQUS only needs a single iteration toachieve equilibrium. After this difficult start, ABAQUS quickly increases the increment size toa more reasonable value. The end of the status file is shown below.

5 456 1 0 2 2 4.99 0.990 0.0003959

5 457 1 0 1 1 4.99 0.991 0.0005938

5 458 1 0 1 1 4.99 0.991 0.0008907

5 459 1 0 2 2 4.99 0.993 0.001336

5 460 2 1 3 4 4.99 0.993 0.0005010

5 461 1 0 1 1 4.99 0.994 0.0007515

5 462 1 0 3 3 5.00 0.995 0.001127

5 463 1 0 2 2 5.00 0.997 0.001691

5 464 1 1 4 5 5.00 0.999 0.002536

5 465 1 1 3 4 5.00 1.00 0.0006165

This simulation, unlike the previous one, contains many severe discontinuity iterations. Themessage file will be quite large because of the number of iterations in the analysis. Although itmight be tempting to limit the information written to this file, generally this should not bedone because this information is the main source of diagnostic data that ABAQUS providesduring the simulation.

11.6.9 PostprocessingBecause of the length of this simulation, no printed output was requested. All results will be studiedusing an interactive postprocessor. Use the following command to run ABAQUS/Viewer:

Contact

11-328

abaqus viewer odb=channel


Deformed model shape and contour plots

The basic result of this simulation is the deformation of the blank and the plastic strain causedby the forming process. We can plot the deformed model shape and the plastic strain usingABAQUS/Viewer, as described below.


1. From the main menu bar, select Plot->Deformed Shape; or click the tool in thetoolbox.

The deformed model shape at the end of Step 5 is plotted. You can remove the die and thepunch from the display and visualize just the blank.



3. Select the element set BLANK from the list of available element sets, and click to

replace the current display group with the specified element set. Click , if necessary, to fitthe model in the viewport.




6. Click OK to apply the settings and to close the dialog box. The resulting plot is shown inFigure 11-33.

Figure 11-33 Deformed shape of blank at the end of Step 5.

To plot the contours of equivalent plastic strain:

abaqus viewer odb=channel


Deformed model shape and contour plots

The basic result of this simulation is the deformation of the blank and the plastic strain causedby the forming process. We can plot the deformed model shape and the plastic strain usingABAQUS/Viewer, as described below.


1. From the main menu bar, select Plot->Deformed Shape; or click the tool in thetoolbox.

The deformed model shape at the end of Step 5 is plotted. You can remove the die and thepunch from the display and visualize just the blank.



3. Select the element set BLANK from the list of available element sets, and click to

replace the current display group with the specified element set. Click , if necessary, to fitthe model in the viewport.




6. Click OK to apply the settings and to close the dialog box. The resulting plot is shown inFigure 11-33.

Figure 11-33 Deformed shape of blank at the end of Step 5.

To plot the contours of equivalent plastic strain:

Contact

11-329

1. From the main menu bar, select Plot->Contours; or click the tool from the toolbox todisplay contours of von Mises stress.




4. Click OK to apply these settings.



6. Select PEEQ from the Output Variable list.

PEEQ is an integrated measure of plastic strain. A non-integrated measure of plastic strain isPEMAG. PEEQ and PEMAG are equal for proportional loading.

7. Click OK to apply these settings. Use the tool to zoom into any region of interest in theblank, as shown in Figure 11-34.

Figure 11-34 Contours of the scalar plastic strain variable PEEQ in one corner of the blank.

The maximum plastic strain is 29.3%. Compare this with the failure strain of the material todetermine if the material will tear during the forming process.

1. From the main menu bar, select Plot->Contours; or click the tool from the toolbox todisplay contours of von Mises stress.







6. Select PEEQ from the Output Variable list.

PEEQ is an integrated measure of plastic strain. A non-integrated measure of plastic strain isPEMAG. PEEQ and PEMAG are equal for proportional loading.

7. Click OK to apply these settings. Use the tool to zoom into any region of interest in theblank, as shown in Figure 11-34.

Figure 11-34 Contours of the scalar plastic strain variable PEEQ in one corner of the blank.

The maximum plastic strain is 29.3%. Compare this with the failure strain of the material todetermine if the material will tear during the forming process.

Contact

11-330

History plots of the reaction forces on the blank and punch

It is important to check that the force required to push the punch into the blank is much largerthan the force created by the pressure applied to the surface of the blank. The force on theblank from the applied pressure is approximately 100 N (1000 Pa � 0.1 m � 1.0 m) at the startof Step 5. The solid line in Figure 11-35 shows the variation of the reaction force RF2 at thepunch's rigid body reference node.

Figure 11-35 Force on punch.

To create a history plot of the reaction force:

1. From the main menu bar, select Plot->History Output.

A history plot of the reaction force in the 1-direction appears.

2. From the main menu bar, select Results->History Output.


3. From the list of available variables, select Reaction force: RF2 at Node 7000 inNSET REFPUNCH.

4. Click Plot to create a history plot of RF2.


6. To label the axes, click XY Plot Options in the prompt area.


History plots of the reaction forces on the blank and punch

It is important to check that the force required to push the punch into the blank is much largerthan the force created by the pressure applied to the surface of the blank. The force on theblank from the applied pressure is approximately 100 N (1000 Pa � 0.1 m � 1.0 m) at the startof Step 5. The solid line in Figure 11-35 shows the variation of the reaction force RF2 at thepunch's rigid body reference node.

Figure 11-35 Force on punch.

To create a history plot of the reaction force:

1. From the main menu bar, select Plot->History Output.

A history plot of the reaction force in the 1-direction appears.

2. From the main menu bar, select Results->History Output.


3. From the list of available variables, select Reaction force: RF2 at Node 7000 inNSET REFPUNCH.

4. Click Plot to create a history plot of RF2.


6. To label the axes, click XY Plot Options in the prompt area.


Contact

11-331

7. Click the Titles tab, and select User-specified from the Title source pull-down lists for theX-Axis and Y-Axis labels.

8. Specify Reaction Force - RF2 as the Y-Axis label, and specify Total Time as theX-Axis label.


The punch force, shown in Figure 11-35, rapidly increases to about 150 kN during Step 5,which runs from a total time of 4.0 to 5.0. The punch force clearly is much larger than theforce (100 N) created by the pressure load.

Plotting contours on surfaces

ABAQUS/Viewer includes a number of features designed specifically for postprocessingcontact analyses. Within ABAQUS/Viewer, the Display Group feature can be used to collectsurfaces into display groups, similar to element and node sets.

To display contact surface normal vectors:

1. From the main menu bar, select Plot->Undeformed Shape; or click in the toolbox toplot the undeformed model shape.

2. From the main menu bar, select Tools->Display Group->Create; or click in thetoolbox.


3. Select the surfaces BLANK_T and PUNCH from the list of available surfaces, and click

to replace the current display group with the specified surfaces.



5. Click the Normals tab, and toggle on Show normals to turn on the display of the normalvectors.

6. In the Style section, set Length to Short.


Use the tool, if necessary, to zoom into any region of interest, as shown in Figure 11-36.

Figure 11-36 Surface normals.

7. Click the Titles tab, and select User-specified from the Title source pull-down lists for theX-Axis and Y-Axis labels.

8. Specify Reaction Force - RF2 as the Y-Axis label, and specify Total Time as theX-Axis label.


The punch force, shown in Figure 11-35, rapidly increases to about 150 kN during Step 5,which runs from a total time of 4.0 to 5.0. The punch force clearly is much larger than theforce (100 N) created by the pressure load.

Plotting contours on surfaces

ABAQUS/Viewer includes a number of features designed specifically for postprocessingcontact analyses. Within ABAQUS/Viewer, the Display Group feature can be used to collectsurfaces into display groups, similar to element and node sets.

To display contact surface normal vectors:

1. From the main menu bar, select Plot->Undeformed Shape; or click in the toolbox toplot the undeformed model shape.

2. From the main menu bar, select Tools->Display Group->Create; or click in thetoolbox.


3. Select the surfaces BLANK_T and PUNCH from the list of available surfaces, and click

to replace the current display group with the specified surfaces.



5. Click the Normals tab, and toggle on Show normals to turn on the display of the normalvectors.

6. In the Style section, set Length to Short.


Use the tool, if necessary, to zoom into any region of interest, as shown in Figure 11-36.

Figure 11-36 Surface normals.

Contact

11-332

To contour the contact pressure:

1. From the main menu bar, select Plot->Contours; or click in the toolbox to plot thecontours of plastic strain again.



3. Select the surface PUNCH from the list of available surfaces, and click to remove thesurface from the current display group.

Now the surface BLANK_T is the only member of the display group.



5. Select CPRESS BLANK_T/PUNCH from the Output Variable list.


To better visualize contours of surface-based variables in two-dimensional models, extrude theplane strain elements to construct the equivalent three-dimensional view. You can sweepaxisymmetric elements in a similar fashion.


The ODB Display Options dialog box appears.

8. Select the Sweep & Extrude tab to access the Sweep & Extrude options.

9. In the Model Extrude region, toggle on Extrude 2D elements; and set the Depth to 0.05 toextrude the model for the purpose of displaying contours.


To contour the contact pressure:

1. From the main menu bar, select Plot->Contours; or click in the toolbox to plot thecontours of plastic strain again.



3. Select the surface PUNCH from the list of available surfaces, and click to remove thesurface from the current display group.

Now the surface BLANK_T is the only member of the display group.



5. Select CPRESS BLANK_T/PUNCH from the Output Variable list.


To better visualize contours of surface-based variables in two-dimensional models, extrude theplane strain elements to construct the equivalent three-dimensional view. You can sweepaxisymmetric elements in a similar fashion.


The ODB Display Options dialog box appears.

8. Select the Sweep & Extrude tab to access the Sweep & Extrude options.

9. In the Model Extrude region, toggle on Extrude 2D elements; and set the Depth to 0.05 toextrude the model for the purpose of displaying contours.


Contact

11-333

Rotate the model using the tool to display the model from a suitable view, such as theone shown in Figure 11-37.

Figure 11-37 Contact pressure.

11.7 Related ABAQUS examples� ``Indentation of a crushable foam specimen with a hemispherical punch, '' Section 1.1.7 of the


� ``Pressure penetration analysis of an air duct kiss seal, '' Section 1.1.17 of the ABAQUS ExampleProblems Manual


11.8 Summary� Contact analyses require a careful, logical approach. Divide the analysis into several steps if

necessary, and apply the loading slowly making sure that the contact conditions are wellestablished.

� In general, it is best to use a separate step for each part of the analysis even if it is just to changeboundary conditions to loads. You will almost certainly end up with more steps than anticipated,but the model should converge much more easily. Contact analyses are much more difficult tocomplete if you try to apply all the loads in one step.

� Achieve stable contact conditions between all components before applying the working loads tothe structure. If necessary, apply temporary boundary conditions, which may be removed at a laterstage. The final results should be unaffected, provided that the constraints produce no permanent

Rotate the model using the tool to display the model from a suitable view, such as theone shown in Figure 11-37.

Figure 11-37 Contact pressure.

11.7 Related ABAQUS examples� ``Indentation of a crushable foam specimen with a hemispherical punch, '' Section 1.1.7 of the


� ``Pressure penetration analysis of an air duct kiss seal, '' Section 1.1.17 of the ABAQUS ExampleProblems Manual


11.8 Summary� Contact analyses require a careful, logical approach. Divide the analysis into several steps if

necessary, and apply the loading slowly making sure that the contact conditions are wellestablished.

� In general, it is best to use a separate step for each part of the analysis even if it is just to changeboundary conditions to loads. You will almost certainly end up with more steps than anticipated,but the model should converge much more easily. Contact analyses are much more difficult tocomplete if you try to apply all the loads in one step.

� Achieve stable contact conditions between all components before applying the working loads tothe structure. If necessary, apply temporary boundary conditions, which may be removed at a laterstage. The final results should be unaffected, provided that the constraints produce no permanent

Contact

11-334

deformation.

� Do not apply boundary conditions to nodes on contact surfaces that constrain the node in thedirection of contact. If there is friction, do not constrain these nodes in any degree of freedom: zeropivot messages may result.

� Always try to use first-order elements for contact simulations.

deformation.

� Do not apply boundary conditions to nodes on contact surfaces that constrain the node in thedirection of contact. If there is friction, do not constrain these nodes in any degree of freedom: zeropivot messages may result.

� Always try to use first-order elements for contact simulations.

Contact

11-335

Appendix A: Example Input FilesThis appendix lists complete input files for the examples contained in this guide. You can get a copy ofany of these input files with the command


where <file name> does not include the extension .inp.

A.1 Overhead hoist frame

frame.inp*HEADING


SI Units



**

** Model definition

**

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.


11, 101,102

12, 102,103

13, 101,104

14, 102,104

15, 102,105

16, 103,105

17, 104,105

*SOLID SECTION, ELSET=FRAME, MATERIAL=STEEL

1.963E-5,


*ELASTIC

200.E9, 0.3

**

** History data

**

*STEP, PERTURBATION

Appendix A: Example Input FilesThis appendix lists complete input files for the examples contained in this guide. You can get a copy ofany of these input files with the command


where <file name> does not include the extension .inp.

A.1 Overhead hoist frame

frame.inp*HEADING


SI Units



**

** Model definition

**

*NODE

101, 0., 0., 0.

102, 1., 0., 0.

103, 2., 0., 0.

104, 0.5, 0.866, 0.

105, 1.5, 0.866, 0.


11, 101,102

12, 102,103

13, 101,104

14, 102,104

15, 102,105

16, 103,105

17, 104,105

*SOLID SECTION, ELSET=FRAME, MATERIAL=STEEL

1.963E-5,


*ELASTIC

200.E9, 0.3

**

** History data

**

*STEP, PERTURBATION

0-336

10kN central load

*STATIC

*BOUNDARY

101, ENCASTRE

103, 2

*CLOAD

102, 2, -10.E3

*NODE PRINT

U,

RF,

*EL PRINT

S,

**********************************

** OUTPUT FOR HKS QA PURPOSES

**********************************

*EL FILE

S,

*NODE FILE

U, RF

*END STEP

A.2 Connecting lug

lug.inp*HEADING



**

** Nodal coordinates

**

*NODE

841, -0.025, 0.025, 0.

857, -0.025, -0.025, 0.

3241, -0.100, 0.025, 0.

3257, -0.100, -0.025, 0.

*NGEN, NSET=LEFT

3241,3257,1

*NGEN, NSET=RIGHT

841, 857,1

*NFILL, NSET=ALLN

RIGHT, LEFT, 24, 100

**

10kN central load

*STATIC

*BOUNDARY

101, ENCASTRE

103, 2

*CLOAD

102, 2, -10.E3

*NODE PRINT

U,

RF,

*EL PRINT

S,

**********************************


**********************************

*EL FILE

S,

*NODE FILE

U, RF

*END STEP

A.2 Connecting lug

lug.inp*HEADING



**


**

*NODE

841, -0.025, 0.025, 0.

857, -0.025, -0.025, 0.

3241, -0.100, 0.025, 0.

3257, -0.100, -0.025, 0.

*NGEN, NSET=LEFT

3241,3257,1

*NGEN, NSET=RIGHT

841, 857,1

*NFILL, NSET=ALLN


**

0-337

*NODE, SYSTEM=C

1, 0.015, -90., 0.

64, 0.015, -95.625, 0.

*NGEN, LINE=C, NSET=HOLE

1, 64, 1, , 0., 0., 0., 0., 0., 1.

*NODE, SYSTEM=C

801, 0.025, -90., 0.

833, 0.025, 90., 0.

*NGEN, LINE=C

801, 833, 1, , 0., 0., 0., 0., 0., 1.

*NODE

865,0.,-2.5E-02,0.

**

*NGEN

833, 841,1

857, 865,1

**

*NSET, NSET=GEN1, GENERATE

801, 864, 1

*NFILL

HOLE, GEN1, 8, 100

*NSET, NSET=ALLN, GENERATE

1, 3257

**

*NCOPY, SHIFT, CHANGE NUMBER=5000, OLD SET=ALLN

0., 0., 0.005

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.010

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.015

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.020

0., 0., 0., 0., 0., 1., 0.

**

** Additional node sets

**


3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

*NODE, SYSTEM=C

1, 0.015, -90., 0.

64, 0.015, -95.625, 0.


1, 64, 1, , 0., 0., 0., 0., 0., 1.

*NODE, SYSTEM=C

801, 0.025, -90., 0.

833, 0.025, 90., 0.

*NGEN, LINE=C

801, 833, 1, , 0., 0., 0., 0., 0., 1.

*NODE

865,0.,-2.5E-02,0.

**

*NGEN

833, 841,1

857, 865,1

**


801, 864, 1

*NFILL

HOLE, GEN1, 8, 100


1, 3257

**


0., 0., 0.005

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.010

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.015

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.020

0., 0., 0., 0., 0., 1., 0.

**


**


3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

0-338

18241, 18257, 4

23241, 23257, 2

*NSET, NSET=HOLEBOT, GENERATE

1, 20001, 5000

**

** Element connectivity

**

*ELEMENT, TYPE=C3D20R, ELSET=HOLEIN

1, 1,401,405,5, 10001,10401,10405,10005, 201,403,205,3,

10201,10403,10205,10003, 5001,5401,5405,5005

16, 61,461,401,1, 10061,10461,10401,10001, 261,463,201,63,

10261,10463,10201,10063, 5061,5461,5401,5001

*ELGEN, ELSET=HOLE

1, 15,4,1, 2,400,100

*ELGEN, ELSET=HOLE

16, 2,400,100

*ELEMENT, TYPE=C3D20R

201, 841,1241,1245,845, 10841,11241,11245,10845, 1041,1243,1045,

843,11041,11243,11045, 10843, 5841,6241,6245,5845

*ELGEN, ELSET=BLOCK

201, 6,400,1, 4,4,10

*ELSET, ELSET=LUG1

BLOCK,HOLE

*ELCOPY, OLDSET=LUG1, ELEMENT SHIFT=1000, SHIFTNODES=10000,

NEWSET=LUG2

**

** Additional element sets

**

*ELSET, ELSET=LUG

LUG1,LUG2

*ELSET, ELSET=BUILTIN, GENERATE

206, 236,10

1206,1236,10

*ELSET, ELSET=PRESS, GENERATE

1, 4

13, 16

1001,1004

1013,1016

************ END OF MESH GENERATION COMMANDS ***********

**

** Physical and material properties

**


18241, 18257, 4

23241, 23257, 2


1, 20001, 5000

**


**


1, 1,401,405,5, 10001,10401,10405,10005, 201,403,205,3,

10201,10403,10205,10003, 5001,5401,5405,5005

16, 61,461,401,1, 10061,10461,10401,10001, 261,463,201,63,

10261,10463,10201,10063, 5061,5461,5401,5001

*ELGEN, ELSET=HOLE

1, 15,4,1, 2,400,100

*ELGEN, ELSET=HOLE

16, 2,400,100


201, 841,1241,1245,845, 10841,11241,11245,10845, 1041,1243,1045,

843,11041,11243,11045, 10843, 5841,6241,6245,5845

*ELGEN, ELSET=BLOCK

201, 6,400,1, 4,4,10

*ELSET, ELSET=LUG1

BLOCK,HOLE


NEWSET=LUG2

**


**

*ELSET, ELSET=LUG

LUG1,LUG2


206, 236,10

1206,1236,10


1, 4

13, 16

1001,1004

1013,1016


**


**


0-339


*ELASTIC

200.E9, 0.3

**

** History definition

**

*STEP, PERTURBATION

*STATIC

**

** Boundary conditions

**

*BOUNDARY

LHEND, ENCASTRE

**

** Loads

**

*DLOAD

PRESS, P6, 5.E7

**

** Output requests

**


U2,


RF,


S, MISES

**********************************


**********************************

*END STEP

A.3 Skew plate

skew.inp*HEADING



**


**

*NODE, NSET=N1


*ELASTIC

200.E9, 0.3

**


**

*STEP, PERTURBATION

*STATIC

**


**

*BOUNDARY

LHEND, ENCASTRE

**

** Loads

**

*DLOAD

PRESS, P6, 5.E7

**

** Output requests

**


U2,


RF,


S, MISES

**********************************


**********************************

*END STEP

A.3 Skew plate

skew.inp*HEADING



**


**

*NODE, NSET=N1

0-340

1, 0.0, 0.0, 0.0

*NODE, NSET=N13

7, 0.0, 40.0E-2, 0.0

*NGEN, NSET=ENDA

1,7,1

*NCOPY, CHANGENUMBER=700, OLDSET=ENDA, NEWSET=ENDB, SHIFT

100.0E-2, 57.7E-2, 0.0

0.0,0.0,0.0,1.0,0.0,0.0,0.0

*NFILL, NSET=ALLNODES

ENDA, ENDB, 14, 50

*NSET, NSET=MIDSPAN, GENERATE

351,357,2

**


**


1, 1,101,103,3,51,102,53,2

*ELGEN, ELSET=PLATE

1, 3,2,1, 7,100,10

**


**


ORIENTATION=SKEW

0.8E-2,


10.0E-2, 5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

3, 0.0


*ELASTIC

30.0E9,0.3

**

** Local coordinate system

**


10.0E-2, 5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

**

** History data--END OF MODEL DATA--

**

*STEP, PERTURBATION


*STATIC

**

1, 0.0, 0.0, 0.0

*NODE, NSET=N13

7, 0.0, 40.0E-2, 0.0

*NGEN, NSET=ENDA

1,7,1


100.0E-2, 57.7E-2, 0.0

0.0,0.0,0.0,1.0,0.0,0.0,0.0


ENDA, ENDB, 14, 50


351,357,2

**


**


1, 1,101,103,3,51,102,53,2

*ELGEN, ELSET=PLATE

1, 3,2,1, 7,100,10

**


**


ORIENTATION=SKEW

0.8E-2,


10.0E-2, 5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

3, 0.0


*ELASTIC

30.0E9,0.3

**


**


10.0E-2, 5.77E-2,0.0, -5.77E-2,10.0E-2,0.0

**

** History data--END OF MODEL DATA--

**

*STEP, PERTURBATION


*STATIC

**

0-341


**

*BOUNDARY

ENDA,ENCASTRE

ENDB,2,6

**

** Pressure loads

**

*DLOAD

PLATE,P,-20000.0

**

** Output requests

**

*OUTPUT,FIELD,OP=NEW

*NODE OUTPUT

U,

*ELEMENT OUTPUT

S,

*OUTPUT,HISTORY,OP=NEW

*NODE OUTPUT,NSET=MIDSPAN

U,

*EL PRINT

S,


RF,

*NODE PRINT, SUMMARY=NO, TOTALS=YES, GLOBAL=YES

RF,


U,

*END STEP

A.4 Cargo crane

crane.inp*HEADING



**


**

*PREPRINT,MODEL=YES

*NODE


**

*BOUNDARY

ENDA,ENCASTRE

ENDB,2,6

**

** Pressure loads

**

*DLOAD

PLATE,P,-20000.0

**

** Output requests

**

*OUTPUT,FIELD,OP=NEW

*NODE OUTPUT

U,

*ELEMENT OUTPUT

S,

*OUTPUT,HISTORY,OP=NEW

*NODE OUTPUT,NSET=MIDSPAN

U,

*EL PRINT

S,


RF,

*NODE PRINT, SUMMARY=NO, TOTALS=YES, GLOBAL=YES

RF,


U,

*END STEP

A.4 Cargo crane

crane.inp*HEADING



**


**

*PREPRINT,MODEL=YES

*NODE

0-342

100 , 0, 0, 1

104, 8,1.5,.1

107,0,1,1

*NGEN, NSET=TRUSSA

100,104

104,107

*NCOPY, CHANGE NUMBER=100, OLDSET=TRUSSA, REFLECT=MIRROR,

NEW SET=TRUSSB

0.,0.,0.,1.,0.,0.

0.,1.,0.

*NCOPY, CHANGE NUMBER=200, OLDSET=TRUSSA, SHIFT

0.,0.,0.

0.,0.,0.,0.,0.,1.,0.

*NCOPY, CHANGE NUMBER=200, OLDSET=TRUSSB, SHIFT

0.,0.,0.

0.,0.,0.,0.,0.,1.,0.

*ELEMENT, TYPE=B33, ELSET=OUTA

100, 100,101

101, 101,102

102, 102,103

103, 103,104

104, 104,105

105, 105,106

106, 106,107

*ELEMENT, TYPE=B33, ELSET=BRACEA

110, 101,107

111, 101,106

112, 102,106

113, 102,105

114, 103,105

*ELEMENT, TYPE=B33, ELSET=OUTB

200, 200,201

201, 201,202

202, 202,203

203, 203,204

204, 204,205

205, 205,206

206, 206,207

*ELEMENT, TYPE=B33, ELSET=BRACEB

210, 201,207

211, 201,206

212, 202,206

213, 202,205

100 , 0, 0, 1

104, 8,1.5,.1

107,0,1,1

*NGEN, NSET=TRUSSA

100,104

104,107


NEW SET=TRUSSB

0.,0.,0.,1.,0.,0.

0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.


100, 100,101

101, 101,102

102, 102,103

103, 103,104

104, 104,105

105, 105,106

106, 106,107


110, 101,107

111, 101,106

112, 102,106

113, 102,105

114, 103,105


200, 200,201

201, 201,202

202, 202,203

203, 203,204

204, 204,205

205, 205,206

206, 206,207


210, 201,207

211, 201,206

212, 202,206

213, 202,205

0-343

214, 203,205

*ELEMENT, TYPE=B33, ELSET=CROSSEL

300, 306,406

301, 306,401

302, 301,401

303, 302,402

304, 402,305

305, 305,405

306, 405,303

307, 303,403

**

****** END OF MESH GENERATION COMMANDS ******

**

**

** Define element normals that would be altered by averaging

**

*NORMAL, TYPE=ELEMENT

113, 102, -0.3986, 0.9114, 0.1025

113, 105, -0.3986, 0.9114, 0.1025

213, 202, -0.3986, 0.9114, -0.1025

213, 205, -0.3986, 0.9114, -0.1025

103, 104, -0.1820, 0.9829, 0.0205

203, 204, -0.1820, 0.9829, -0.0205

*ELSET, ELSET=TRUSSA

OUTA,BRACEA

*ELSET, ELSET=TRUSSB

OUTB,BRACEB

**

*NSET, NSET=ATTACH

100,107,200,207

**


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9

214, 203,205


300, 306,406

301, 306,401

302, 301,401

303, 302,402

304, 402,305

305, 305,405

306, 405,303

307, 303,403

**


**

**


**


113, 102, -0.3986, 0.9114, 0.1025

113, 105, -0.3986, 0.9114, 0.1025

213, 202, -0.3986, 0.9114, -0.1025

213, 205, -0.3986, 0.9114, -0.1025

103, 104, -0.1820, 0.9829, 0.0205

203, 204, -0.1820, 0.9829, -0.0205


OUTA,BRACEA


OUTB,BRACEB

**

*NSET, NSET=ATTACH

100,107,200,207

**


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9

0-344


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

**


0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

**

** Multi-point constraints

**

*MPC

PIN,101,301

PIN,102,302

PIN,103,303

PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

**

** Constraint equations

**

*EQUATION

2,

104,1,1.0, 204,1,-1.0

2,

104,2,1.0, 204,2,-1.0

**

** History data

**

*STEP, PERTURBATION

Static tip load

*STATIC

**


**

*BOUNDARY

ATTACH, ENCASTRE

**


0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

**


0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

**


**

*MPC

PIN,101,301

PIN,102,302

PIN,103,303

PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

**


**

*EQUATION

2,

104,1,1.0, 204,1,-1.0

2,

104,2,1.0, 204,2,-1.0

**

** History data

**

*STEP, PERTURBATION

Static tip load

*STATIC

**


**

*BOUNDARY

ATTACH, ENCASTRE

**

0-345

** Loading

**

*CLOAD

104,2,-1.0E4

**

** Output requests

**

*OUTPUT,FIELD

*NODE OUTPUT

U, RF

*ELEMENT OUTPUT

SF,

*NODE PRINT

U,

RF,

*EL PRINT

SF,

**********************************


**********************************

*NODE FILE

U, RF

*EL FILE

SF,

*END STEP

A.5 Nonlinear skew plate

skew_nl.inp*HEADING

Nonlinear Elastic Skew Plate. 20 kPa Load.


**


**

*NODE, NSET=N1

1, 0.0, 0.0, 0.0

*NODE, NSET=N13

7, 0.0, 40.0E-2, 0.0

*NGEN, NSET=ENDA

1,7,1


** Loading

**

*CLOAD

104,2,-1.0E4

**

** Output requests

**

*OUTPUT,FIELD

*NODE OUTPUT

U, RF

*ELEMENT OUTPUT

SF,

*NODE PRINT

U,

RF,

*EL PRINT

SF,

**********************************


**********************************

*NODE FILE

U, RF

*EL FILE

SF,

*END STEP

A.5 Nonlinear skew plate

skew_nl.inp*HEADING

Nonlinear Elastic Skew Plate. 20 kPa Load.


**


**

*NODE, NSET=N1

1, 0.0, 0.0, 0.0

*NODE, NSET=N13

7, 0.0, 40.0E-2, 0.0

*NGEN, NSET=ENDA

1,7,1


0-346

100.0E-2, 57.7E-2, 0.0

0.0,0.0,0.0,1.0,0.0,0.0,0.0


ENDA, ENDB, 14, 50


351,357,2

**


**


1, 1,101,103,3,51,102,53,2

*ELGEN, ELSET=PLATE

1, 3,2,1, 7,100,10

**

************** END OF MESH GENERATION COMMANDS ***********

**


**


ORIENTATION=SKEW

0.8E-2,


100.0E-2,57.7E-2,0.0, -57.7E-2,100.0E-2,0.0

3, 0.0


*ELASTIC

30.0E9,0.3

**


**


100.0E-2,57.7E-2,0.0, -57.7E-2,100.0E-2,0.0

**

** History data

**



*STATIC

0.1, 1.0

**


**

*BOUNDARY

100.0E-2, 57.7E-2, 0.0

0.0,0.0,0.0,1.0,0.0,0.0,0.0


ENDA, ENDB, 14, 50


351,357,2

**


**


1, 1,101,103,3,51,102,53,2

*ELGEN, ELSET=PLATE

1, 3,2,1, 7,100,10

**

************** END OF MESH GENERATION COMMANDS ***********

**


**


ORIENTATION=SKEW

0.8E-2,


100.0E-2,57.7E-2,0.0, -57.7E-2,100.0E-2,0.0

3, 0.0


*ELASTIC

30.0E9,0.3

**


**


100.0E-2,57.7E-2,0.0, -57.7E-2,100.0E-2,0.0

**

** History data

**



*STATIC

0.1, 1.0

**


**

*BOUNDARY

0-347

ENDA,ENCASTRE

ENDB,2,6

**

** Pressure loads

**

*DLOAD

PLATE,P,-20000E+0

**

** Output requests

**




U,


U,


RF,


**********************************


**********************************

*END STEP

A.6 Connecting lug with plasticity

lug_plas.inp*HEADING

Steel Connecting Lug including elastic-plastic material

definition


**


**

*NODE

841, -0.025, 0.025, 0.

857, -0.025, -0.025, 0.

3241, -0.100, 0.025, 0.

3257, -0.100, -0.025, 0.

*NGEN, NSET=LEFT

3241,3257,1

*NGEN, NSET=RIGHT

ENDA,ENCASTRE

ENDB,2,6

**

** Pressure loads

**

*DLOAD

PLATE,P,-20000E+0

**

** Output requests

**




U,


U,


RF,


**********************************


**********************************

*END STEP

A.6 Connecting lug with plasticity

lug_plas.inp*HEADING

Steel Connecting Lug including elastic-plastic material

definition


**


**

*NODE

841, -0.025, 0.025, 0.

857, -0.025, -0.025, 0.

3241, -0.100, 0.025, 0.

3257, -0.100, -0.025, 0.

*NGEN, NSET=LEFT

3241,3257,1

*NGEN, NSET=RIGHT

0-348

841, 857,1

*NFILL, NSET=ALLN


**

*NODE, SYSTEM=C

1, 0.015, -90., 0.

64, 0.015, -95.625, 0.


1, 64, 1, , 0., 0., 0., 0., 0., 1.

*NODE, SYSTEM=C

801, 0.025, -90., 0.

833, 0.025, 90., 0.

*NGEN, LINE=C

801, 833, 1, , 0., 0., 0., 0., 0., 1.

*NODE

865,0.,-2.5E-02,0.

**

*NGEN

833, 841,1

857, 865,1

**


801, 864, 1

*NFILL

HOLE, GEN1, 8, 100


1, 3257

**


0., 0., 0.005

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.010

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.015

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.020

0., 0., 0., 0., 0., 1., 0.

**


**

841, 857,1

*NFILL, NSET=ALLN


**

*NODE, SYSTEM=C

1, 0.015, -90., 0.

64, 0.015, -95.625, 0.


1, 64, 1, , 0., 0., 0., 0., 0., 1.

*NODE, SYSTEM=C

801, 0.025, -90., 0.

833, 0.025, 90., 0.

*NGEN, LINE=C

801, 833, 1, , 0., 0., 0., 0., 0., 1.

*NODE

865,0.,-2.5E-02,0.

**

*NGEN

833, 841,1

857, 865,1

**


801, 864, 1

*NFILL

HOLE, GEN1, 8, 100


1, 3257

**


0., 0., 0.005

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.010

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.015

0., 0., 0., 0., 0., 1., 0.


0., 0., 0.020

0., 0., 0., 0., 0., 1., 0.

**


**

0-349


3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

18241, 18257, 4

23241, 23257, 2


1, 20001, 5000

**


**


1, 1,401,405,5, 10001,10401,10405,10005, 201,403,205,3,

10201,10403,10205,10003, 5001,5401,5405,5005

16, 61,461,401,1, 10061,10461,10401,10001, 261,463,201,63,

10261,10463,10201,10063, 5061,5461,5401,5001

*ELGEN, ELSET=HOLE

1, 15,4,1, 2,400,100

*ELGEN, ELSET=HOLE

16, 2,400,100


201, 841,1241,1245,845, 10841,11241,11245,10845, 1041,1243,1045,

843,11041,11243,11045, 10843, 5841,6241,6245,5845

*ELGEN, ELSET=BLOCK

201, 6,400,1, 4,4,10

*ELSET, ELSET=LUG1

BLOCK,HOLE


NEWSET=LUG2

**


**

*ELSET, ELSET=LUG

LUG1,LUG2


206, 236,10

1206,1236,10


**


**




3241, 3257, 2

8241, 8257, 4

13241, 13257, 2

18241, 18257, 4

23241, 23257, 2


1, 20001, 5000

**


**


1, 1,401,405,5, 10001,10401,10405,10005, 201,403,205,3,

10201,10403,10205,10003, 5001,5401,5405,5005

16, 61,461,401,1, 10061,10461,10401,10001, 261,463,201,63,

10261,10463,10201,10063, 5061,5461,5401,5001

*ELGEN, ELSET=HOLE

1, 15,4,1, 2,400,100

*ELGEN, ELSET=HOLE

16, 2,400,100


201, 841,1241,1245,845, 10841,11241,11245,10845, 1041,1243,1045,

843,11041,11243,11045, 10843, 5841,6241,6245,5845

*ELGEN, ELSET=BLOCK

201, 6,400,1, 4,4,10

*ELSET, ELSET=LUG1

BLOCK,HOLE


NEWSET=LUG2

**


**

*ELSET, ELSET=LUG

LUG1,LUG2


206, 236,10

1206,1236,10


**


**



0-350

*ELASTIC

200.E9, 0.3

*PLASTIC

380.E6, 0.0

**

*ELSET, ELSET=EL206

206,


1, 4

13, 16

1001,1004

1013,1016

**


**

*STEP, INC=25

*STATIC

0.2, 1.0

**


**

*BOUNDARY

LHEND, ENCASTRE

**

** Loads

**

*DLOAD

PRESS, P6, 1.E+08

**

** Output requests

**






U,


S, SINV, PE, E

**********************************


**********************************

*END STEP

*ELASTIC

200.E9, 0.3

*PLASTIC

380.E6, 0.0

**

*ELSET, ELSET=EL206

206,


1, 4

13, 16

1001,1004

1013,1016

**


**

*STEP, INC=25

*STATIC

0.2, 1.0

**


**

*BOUNDARY

LHEND, ENCASTRE

**

** Loads

**

*DLOAD

PRESS, P6, 1.E+08

**

** Output requests

**






U,


S, SINV, PE, E

**********************************


**********************************

*END STEP

0-351

A.7 Axisymmetric mount

mount.inp*HEADING



*PREPRINT, ECHO=YES, HISTORY=YES, MODEL=YES

**


**

*NODE

1, 0.01, 0.0

31, 0.06, 0.0

101, 0.01, 0.005

131, 0.06, 0.005

1501, 0.01, 0.03

1531, 0.0528301,0.03

*NGEN

101,1501,100

*NGEN, LINE=C

131,1531,100,,0.1,0.03

*NSET, NSET=LEFT, GENERATE

1,1501,100

*NSET, NSET=RIGHT, GENERATE

31,1531,100

*NFILL, NSET=RUBBER

LEFT,RIGHT,30,1

**


**

*NSET, NSET=BOTTOM, GENERATE

1,31,1

*NSET, NSET=MIDDLE, GENERATE

1501,1531,1

**


**

*ELEMENT, TYPE=CAX4I

1,1,2,102,101

*ELGEN, ELSET=PLATE

1, 30,1,1

*ELEMENT, TYPE=CAX4H

A.7 Axisymmetric mount

mount.inp*HEADING



*PREPRINT, ECHO=YES, HISTORY=YES, MODEL=YES

**


**

*NODE

1, 0.01, 0.0

31, 0.06, 0.0

101, 0.01, 0.005

131, 0.06, 0.005

1501, 0.01, 0.03

1531, 0.0528301,0.03

*NGEN

101,1501,100

*NGEN, LINE=C

131,1531,100,,0.1,0.03

*NSET, NSET=LEFT, GENERATE

1,1501,100

*NSET, NSET=RIGHT, GENERATE

31,1531,100

*NFILL, NSET=RUBBER

LEFT,RIGHT,30,1

**


**

*NSET, NSET=BOTTOM, GENERATE

1,31,1

*NSET, NSET=MIDDLE, GENERATE

1501,1531,1

**


**

*ELEMENT, TYPE=CAX4I

1,1,2,102,101

*ELGEN, ELSET=PLATE

1, 30,1,1

*ELEMENT, TYPE=CAX4H

0-352

101, 101,102,202,201

*ELGEN, ELSET=RUBBER

101, 30,1,1, 14,100,100

**


**



*MATERIAL,NAME=RUBBER


*UNIAXIAL TEST DATA

0.054E6, 0.0380

0.152E6, 0.1338

0.254E6, 0.2210

0.362E6, 0.3450

0.459E6, 0.4600

0.583E6, 0.6242

0.656E6, 0.8510

0.73E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621


*ELASTIC

2.0E11,0.3

**

** History data

**


*STATIC

.01,1.0

**


101, 101,102,202,201

*ELGEN, ELSET=RUBBER

101, 30,1,1, 14,100,100

**


**



*MATERIAL,NAME=RUBBER


*UNIAXIAL TEST DATA

0.054E6, 0.0380

0.152E6, 0.1338

0.254E6, 0.2210

0.362E6, 0.3450

0.459E6, 0.4600

0.583E6, 0.6242

0.656E6, 0.8510

0.73E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621


*ELASTIC

2.0E11,0.3

**

** History data

**


*STATIC

.01,1.0

**


0-353

**

*BOUNDARY

MIDDLE, 2,2, 0.0

**

** Pressure load

**

*DLOAD

PLATE,P1, 0.5E6

**

** Output requests

**



*NSET, NSET=NODEOUT

1,

*OUTPUT, HISTORY


U,


**********************************


**********************************

*END STEP

A.8 Test fit of hyperelastic material data

single_elem2.inp*HEADING

Single element model to test fit of hyperelastic test data

*NODE,NSET=ALL

1,

2,1.

3,1.,1.,

4,0.,1.,

5,0.,0.,1.

6,1.,0.,1.

7,1.,1.,1.

8,0.,1.,1.

*NSET,NSET=FACE1

1,2,3,4

*NSET,NSET=FACE2

5,6,7,8

**

*BOUNDARY

MIDDLE, 2,2, 0.0

**

** Pressure load

**

*DLOAD

PLATE,P1, 0.5E6

**

** Output requests

**



*NSET, NSET=NODEOUT

1,

*OUTPUT, HISTORY


U,


**********************************


**********************************

*END STEP

A.8 Test fit of hyperelastic material data

single_elem2.inp*HEADING

Single element model to test fit of hyperelastic test data

*NODE,NSET=ALL

1,

2,1.

3,1.,1.,

4,0.,1.,

5,0.,0.,1.

6,1.,0.,1.

7,1.,1.,1.

8,0.,1.,1.

*NSET,NSET=FACE1

1,2,3,4

*NSET,NSET=FACE2

5,6,7,8

0-354

*NSET,NSET=FACE3

1,2,5,6

*NSET,NSET=FACE4

2,

*NSET,NSET=FACE42

3,6,7

*NSET,NSET=FACE5

3,4,7,8

*NSET,NSET=FACE6

4,1,8,5

*EQUATION

**

** Since the S11 output is Cauchy or true stress, we need to

** determine the nominal stress for post-processing.

** Nodes 3,6,7 are tied to node 2 in dof 1 so that:

** Nominal stress (dof 1) = RF1 (@ node 2) / Original area

** (w/c is 1 x 1 = 1)

2,

FACE42,1,1, 2,1,-1

*ELEMENT,TYPE=C3D8RH,ELSET=ONE

1,1,2,3,4,5,6,7,8

*SOLID SECTION,ELSET=ONE,MATERIAL=TREL

*MATERIAL,NAME=TREL

**

** To use the OGDEN model, simply replace the POLYNOMIAL

** parameter with the OGDEN parameter.

** To change the strain energy function order, change the N

** parameter.

**

*HYPERELASTIC,N=1,POLYNOMIAL,TEST DATA INPUT

*UNIAXIAL TEST DATA

0.054E6, 0.038

0.152E6, 0.1338

0.254E6, 0.221

0.362E6, 0.345

0.459E6, 0.46

0.583E6, 0.6242

0.656E6, 0.851

0.73E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

*NSET,NSET=FACE3

1,2,5,6

*NSET,NSET=FACE4

2,

*NSET,NSET=FACE42

3,6,7

*NSET,NSET=FACE5

3,4,7,8

*NSET,NSET=FACE6

4,1,8,5

*EQUATION

**

** Since the S11 output is Cauchy or true stress, we need to

** determine the nominal stress for post-processing.

** Nodes 3,6,7 are tied to node 2 in dof 1 so that:

** Nominal stress (dof 1) = RF1 (@ node 2) / Original area

** (w/c is 1 x 1 = 1)

2,

FACE42,1,1, 2,1,-1

*ELEMENT,TYPE=C3D8RH,ELSET=ONE

1,1,2,3,4,5,6,7,8

*SOLID SECTION,ELSET=ONE,MATERIAL=TREL

*MATERIAL,NAME=TREL

**

** To use the OGDEN model, simply replace the POLYNOMIAL

** parameter with the OGDEN parameter.

** To change the strain energy function order, change the N

** parameter.

**

*HYPERELASTIC,N=1,POLYNOMIAL,TEST DATA INPUT

*UNIAXIAL TEST DATA

0.054E6, 0.038

0.152E6, 0.1338

0.254E6, 0.221

0.362E6, 0.345

0.459E6, 0.46

0.583E6, 0.6242

0.656E6, 0.851

0.73E6, 1.4268

*BIAXIAL TEST DATA

0.089E6, 0.0200

0.255E6, 0.1400

0.503E6, 0.4200

0-355

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621

**

** In every step the step time and magnitude of displacement are

** equal so that history plots of RF at node 2 actually

** correspond to nominal strain vs nominal stress.

**

*STEP,NLGEOM,INC=50000

Step 1: Uniaxial Tension

*STATIC,DIRECT

.02, 1.8

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE6,1

FACE4,1,1, 1.8

*ENERGY PRINT

*EL PRINT,FREQUENCY=500

S,

E,

*NODE PRINT,FREQUENCY=500

U,RF

*OUTPUT,FIELD, OP=ADD,FREQUENCY=1

*NODE OUTPUT

U,RF

*OUTPUT,HISTORY,OP=ADD,FREQUENCY=1

*NODE OUTPUT, NSET=ALL

U,RF

*END STEP

*STEP,NLGEOM,INC=20

Step 2: Unload

*STATIC,DIRECT

.1, 1.8

*BOUNDARY,OP=MOD

FACE4,1

*OUTPUT,FIELD, OP=NEW,FREQUENCY=0

0.958E6, 1.4900

1.703E6, 2.7500

2.413E6, 3.4500

*PLANAR TEST DATA

0.055E6, 0.0690

0.324E6, 0.2828

0.758E6, 1.3862

1.269E6, 3.0345

1.779E6, 4.0621

**

** In every step the step time and magnitude of displacement are

** equal so that history plots of RF at node 2 actually

** correspond to nominal strain vs nominal stress.

**


Step 1: Uniaxial Tension

*STATIC,DIRECT

.02, 1.8

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE6,1

FACE4,1,1, 1.8

*ENERGY PRINT


S,

E,


U,RF


*NODE OUTPUT

U,RF



U,RF

*END STEP

*STEP,NLGEOM,INC=20

Step 2: Unload

*STATIC,DIRECT

.1, 1.8

*BOUNDARY,OP=MOD

FACE4,1


0-356

*OUTPUT,HISTORY,OP=NEW,FREQUENCY=0

*END STEP


Step 3: Biaxial Tension

*STATIC,DIRECT

.02, 1.7

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE6,1

FACE4,1,1, 1.7

FACE5,2,2, 1.7

*ENERGY PRINT


S,

E,


U,RF


*NODE OUTPUT

U,RF



U,RF

*END STEP

*STEP,NLGEOM,INC=20

Step 4: Unload

*STATIC,DIRECT

.1, 1.7

*BOUNDARY,OP=MOD

FACE4,1,1,

FACE5,2,2,



*END STEP


Step 5: Planar Tension (Pure Shear)

*STATIC,DIRECT

.02, 1.8

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE5,2


*END STEP


Step 3: Biaxial Tension

*STATIC,DIRECT

.02, 1.7

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE6,1

FACE4,1,1, 1.7

FACE5,2,2, 1.7

*ENERGY PRINT


S,

E,


U,RF


*NODE OUTPUT

U,RF



U,RF

*END STEP

*STEP,NLGEOM,INC=20

Step 4: Unload

*STATIC,DIRECT

.1, 1.7

*BOUNDARY,OP=MOD

FACE4,1,1,

FACE5,2,2,



*END STEP


Step 5: Planar Tension (Pure Shear)

*STATIC,DIRECT

.02, 1.8

*BOUNDARY,OP=NEW

FACE1,3

FACE3,2

FACE5,2

0-357

FACE6,1

FACE4,1,1, 1.8

*ENERGY PRINT


S,

E,


U,RF


*NODE OUTPUT

U,RF



U,RF

*END STEP

A.9 Cargo crane - dynamic loading

dynamics.inp*HEADING

3-D model of light-service cargo crane--Dynamic drop load


**


**

*NODE

100 , 0, 0, 1

104, 8,1.5,.1

107,0,1,1

*NGEN, NSET=TRUSSA

100,104

104,107


NEW SET=TRUSSB

0.,0.,0.,1.,0.,0.

0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.

FACE6,1

FACE4,1,1, 1.8

*ENERGY PRINT


S,

E,


U,RF


*NODE OUTPUT

U,RF



U,RF

*END STEP

A.9 Cargo crane - dynamic loading

dynamics.inp*HEADING

3-D model of light-service cargo crane--Dynamic drop load


**


**

*NODE

100 , 0, 0, 1

104, 8,1.5,.1

107,0,1,1

*NGEN, NSET=TRUSSA

100,104

104,107


NEW SET=TRUSSB

0.,0.,0.,1.,0.,0.

0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.


0.,0.,0.

0.,0.,0.,0.,0.,1.,0.

0-358


100, 100,101

101, 101,102

102, 102,103

103, 103,104

104, 104,105

105, 105,106

106, 106,107


110, 101,107

111, 101,106

112, 102,106

113, 102,105

114, 103,105


200, 200,201

201, 201,202

202, 202,203

203, 203,204

204, 204,205

205, 205,206

206, 206,207


210, 201,207

211, 201,206

212, 202,206

213, 202,205

214, 203,205


300, 306,406

301, 306,401

302, 301,401

303, 302,402

304, 402,305

305, 305,405

306, 405,303

307, 303,403

**


**

**


**


100, 100,101

101, 101,102

102, 102,103

103, 103,104

104, 104,105

105, 105,106

106, 106,107


110, 101,107

111, 101,106

112, 102,106

113, 102,105

114, 103,105


200, 200,201

201, 201,202

202, 202,203

203, 203,204

204, 204,205

205, 205,206

206, 206,207


210, 201,207

211, 201,206

212, 202,206

213, 202,205

214, 203,205


300, 306,406

301, 306,401

302, 301,401

303, 302,402

304, 402,305

305, 305,405

306, 405,303

307, 303,403

**


**

**


**

0-359


113, 102, -0.3986, 0.9114, 0.1025

113, 105, -0.3986, 0.9114, 0.1025

213, 202, -0.3986, 0.9114, -0.1025

213, 205, -0.3986, 0.9114, -0.1025

103, 104, -0.1820, 0.9829, 0.0205

203, 204, -0.1820, 0.9829, -0.0205


OUTA,BRACEA


OUTB,BRACEB

**

*NSET, NSET=ATTACH

100,107,200,207

**


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=OUTB, DENSITY=7800.

0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEA, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEB, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

**

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=CROSSEL, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

**


**

*MPC

PIN,101,301

PIN,102,302

PIN,103,303


113, 102, -0.3986, 0.9114, 0.1025

113, 105, -0.3986, 0.9114, 0.1025

213, 202, -0.3986, 0.9114, -0.1025

213, 205, -0.3986, 0.9114, -0.1025

103, 104, -0.1820, 0.9829, 0.0205

203, 204, -0.1820, 0.9829, -0.0205


OUTA,BRACEA


OUTB,BRACEB

**

*NSET, NSET=ATTACH

100,107,200,207

**


0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=OUTB, DENSITY=7800.

0.10,0.05,0.005,0.005,0.005,0.005

-0.1118, 0.0, 0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEA, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, -0.9936

200.E9,80.E9

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=BRACEB, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

-0.1118, 0.0, 0.9936

200.E9,80.E9

**

*BEAM GENERAL SECTION, SECTION=BOX, ELSET=CROSSEL, DENSITY=7800.

0.03,0.03,0.003,0.003,0.003,0.003

0.0,1.0,0.0

200.E9,80.E9

**


**

*MPC

PIN,101,301

PIN,102,302

PIN,103,303

0-360

PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

**


**

*EQUATION

2,

104,1,1.0, 204,1,-1.0

2,

104,2,1.0, 204,2,-1.0

**

** Time variation of loading

**


0., 0., 0.01, 1.0, 0.2, 1.0, 0.21, 0.

**

** History data

**

**

** Step 1: Extract nodes and frequencies

**

*STEP, PERTURBATION

Frequency extraction of the first 30 modes


**

** Change 30 to 5 (or 2) for 5 (or 2) modes

** Note that one needs to change the "30"

** after the *MODAL DAMPING card

**

30,

**


**

*BOUNDARY

ATTACH, ENCASTRE

**

** Output requests

**

PIN,105,305

PIN,106,306

PIN,201,401

PIN,202,402

PIN,203,403

PIN,205,405

PIN,206,406

**


**

*EQUATION

2,

104,1,1.0, 204,1,-1.0

2,

104,2,1.0, 204,2,-1.0

**

** Time variation of loading

**


0., 0., 0.01, 1.0, 0.2, 1.0, 0.21, 0.

**

** History data

**

**

** Step 1: Extract nodes and frequencies

**

*STEP, PERTURBATION

Frequency extraction of the first 30 modes


**


** Note that one needs to change the "30"

** after the *MODAL DAMPING card

**

30,

**


**

*BOUNDARY

ATTACH, ENCASTRE

**

** Output requests

**

0-361



*END STEP

**

** Step 2: Transient dynamic analysis

**

*STEP, PERTURBATION

Simulation of Drop Load on Crane

*MODAL DYNAMIC

0.005, 0.5

**

** Damping

**


**


** Change 0.05 to 0.1 (or 0.01) for 10% (or 1%) damping

**

1, 30, 0.05

**

** Loads

**


104, 2, -1.0E4

**

** Output requests

**


*NSET, NSET=TIP

104, 204



U,


RF,

*ENERGY OUTPUT




**

**********************************


**********************************



*END STEP

**

** Step 2: Transient dynamic analysis

**

*STEP, PERTURBATION

Simulation of Drop Load on Crane

*MODAL DYNAMIC

0.005, 0.5

**

** Damping

**


**


** Change 0.05 to 0.1 (or 0.01) for 10% (or 1%) damping

**

1, 30, 0.05

**

** Loads

**


104, 2, -1.0E4

**

** Output requests

**


*NSET, NSET=TIP

104, 204



U,


RF,

*ENERGY OUTPUT




**

**********************************


**********************************

0-362

*NODE FILE, FREQUENCY=20

U, V, A

*END STEP

A.10 Vibration of a piping system

pipe.inp*HEADING



S.I. Units

**


**

*NODE

1,0.

61, 5.0

*NGEN

1,61

**


**

*ELEMENT, TYPE=PIPE32

1, 1, 2, 3

*ELGEN, ELSET=PIPE

1, 30,2,1

**

*NSET, NSET=LEFT

1,

*NSET, NSET=RIGHT

61,

**


**


0.09, 0.02

0.0E0, 0.0E0, -1.0E0


*ELASTIC

200.0E9, 0.3

*DENSITY

7800.,

*NODE FILE, FREQUENCY=20

U, V, A

*END STEP

A.10 Vibration of a piping system

pipe.inp*HEADING



S.I. Units

**


**

*NODE

1,0.

61, 5.0

*NGEN

1,61

**


**

*ELEMENT, TYPE=PIPE32

1, 1, 2, 3

*ELGEN, ELSET=PIPE

1, 30,2,1

**

*NSET, NSET=LEFT

1,

*NSET, NSET=RIGHT

61,

**


**


0.09, 0.02

0.0E0, 0.0E0, -1.0E0


*ELASTIC

200.0E9, 0.3

*DENSITY

7800.,

0-363

**

** Step 1 - General step

**



*STATIC

0.1, 1.0

*BOUNDARY

LEFT, 1, 6

RIGHT, 2, 6

*CLOAD

RIGHT, 1, 4.0E6


25,



*OUTPUT, HISTORY


S, SINV



**********************************


**********************************

*NODE FILE, FREQ=10

U,

*EL FILE, FREQ=10

SINV, S

*END STEP

**

** Step 2 - Linear perturbation step

**

*STEP, PERTURBATION



8,

*RESTART, WRITE




*END STEP

A.11 Restarting the pipe vibration analysis

**


**



*STATIC

0.1, 1.0

*BOUNDARY

LEFT, 1, 6

RIGHT, 2, 6

*CLOAD

RIGHT, 1, 4.0E6


25,



*OUTPUT, HISTORY


S, SINV



**********************************


**********************************

*NODE FILE, FREQ=10

U,

*EL FILE, FREQ=10

SINV, S

*END STEP

**


**

*STEP, PERTURBATION



8,

*RESTART, WRITE




*END STEP

A.11 Restarting the pipe vibration analysis

0-364

pipe-2.inp*HEADING



*RESTART, READ

**


**


Apply 8MN axial tensile load

*STATIC

0.1, 1.

*CLOAD

RIGHT, 1, 8.0E6



*OUTPUT, HISTORY


S, SINV



**********************************


**********************************

*END STEP

**


**




8,

*RESTART, WRITE




**********************************


**********************************

*END STEP

A.12 Flange connection

pipe-2.inp*HEADING



*RESTART, READ

**


**


Apply 8MN axial tensile load

*STATIC

0.1, 1.

*CLOAD

RIGHT, 1, 8.0E6



*OUTPUT, HISTORY


S, SINV



**********************************


**********************************

*END STEP

**


**




8,

*RESTART, WRITE




**********************************


**********************************

*END STEP

A.12 Flange connection

0-365

flange.inp*HEADING

Axisymmetric Flange Connection


**


**

*NODE

1, 0.03, 0.

11, 0.13, 0.

111, 0.03, 0.1

121, 0.13, 0.1

*NGEN, NSET=BOT

1,11,1

*NGEN, NSET=TOP

111,121,1

*NFILL, NSET=TOPFLANG

BOT,TOP,10,11

*NCOPY, REFLECT=LINE, OLDSET=TOPFLANG, NEWSET=FLANGE2,

CHANGE=200

0.,0.,0., 1.,0.,0.

**


**


1, 1, 2,13,12

21, 23,24,35,34

*ELGEN, ELSET=TOPFLANG

1, 10,1,1, 2,11,10


21, 2,1,1, 8,11,10

*ELCOPY, ELEMENT SHIFT=100, OLDSET=TOPFLANG, NEWSET=BOTFLANG,

SHIFT NODES=200, REFLECT

**

******************** END OF MESH GENERATION COMMANDS ********

**

**


**



flange.inp*HEADING

Axisymmetric Flange Connection


**


**

*NODE

1, 0.03, 0.

11, 0.13, 0.

111, 0.03, 0.1

121, 0.13, 0.1

*NGEN, NSET=BOT

1,11,1

*NGEN, NSET=TOP

111,121,1

*NFILL, NSET=TOPFLANG

BOT,TOP,10,11

*NCOPY, REFLECT=LINE, OLDSET=TOPFLANG, NEWSET=FLANGE2,

CHANGE=200

0.,0.,0., 1.,0.,0.

**


**


1, 1, 2,13,12

21, 23,24,35,34


1, 10,1,1, 2,11,10


21, 2,1,1, 8,11,10

*ELCOPY, ELEMENT SHIFT=100, OLDSET=TOPFLANG, NEWSET=BOTFLANG,

SHIFT NODES=200, REFLECT

**

******************** END OF MESH GENERATION COMMANDS ********

**

**


**



0-366


*ELASTIC

200.E+9, 0.3

*MATERIAL, NAME=ALU

*ELASTIC

70.E+9, 0.2

**

** Multi-point constraints and constraint equations

**

*MPC

TIE, 18, 218

*EQUATION

2,

112,2,1.0, 111,2,-1.0

2,

113,2,1.0, 111,2,-1.0

2,

312,2,1.0, 311,2,-1.0

2,

313,2,1.0, 311,2,-1.0

**

** Contact Definition

**


1,10,1


FLTOP,


101,110,1


FLBOT,


SURFBOT, SURFTOP


*FRICTION

0.1,

**

** History Definition--End of Model data--

**

*STEP, INC=20

*STATIC

1.0, 1.0

**


*ELASTIC

200.E+9, 0.3

*MATERIAL, NAME=ALU

*ELASTIC

70.E+9, 0.2

**

** Multi-point constraints and constraint equations

**

*MPC

TIE, 18, 218

*EQUATION

2,

112,2,1.0, 111,2,-1.0

2,

113,2,1.0, 111,2,-1.0

2,

312,2,1.0, 311,2,-1.0

2,

313,2,1.0, 311,2,-1.0

**

** Contact Definition

**


1,10,1


FLTOP,


101,110,1


FLBOT,


SURFBOT, SURFTOP


*FRICTION

0.1,

**

** History Definition--End of Model data--

**

*STEP, INC=20

*STATIC

1.0, 1.0

**

0-367

** Loading

**

*CLOAD

111, 2, 200.E3

311, 2, -200.E3

**


**

*BOUNDARY

100, 2

**

** Output requests

**


RF,


*CONTACT PRINT

*PRINT, CONTACT=YES


**********************************


**********************************

*END STEP

A.13 Forming a channel

channel.inp*HEADING



**

** Nodal coordinates for blank

**

*NODE

1, 0., 0.

101, 0.1, 0.

203, 0., 0.001

303, 0.1, 0.001

*NGEN, NSET=BOT

1,101,1

*NGEN, NSET=TOP

203,303,1

** Loading

**

*CLOAD

111, 2, 200.E3

311, 2, -200.E3

**


**

*BOUNDARY

100, 2

**

** Output requests

**


RF,


*CONTACT PRINT

*PRINT, CONTACT=YES


**********************************


**********************************

*END STEP

A.13 Forming a channel

channel.inp*HEADING



**

** Nodal coordinates for blank

**

*NODE

1, 0., 0.

101, 0.1, 0.

203, 0., 0.001

303, 0.1, 0.001

*NGEN, NSET=BOT

1,101,1

*NGEN, NSET=TOP

203,303,1

0-368

*NFILL

BOT,TOP,2,101

**

** Element connectivity for blank

**

*ELEMENT, TYPE=CPE4I

1, 1,2,103,102

*ELGEN, ELSET=BLANK

1, 100,1,1, 2,101,100

**

** Node and element sets

**

*NSET, NSET=CENTER

1,102,203


155,202,1


102,147,1


1,100,1


101,200,1

**

** Nodal coordinates and rigid body reference node for die

**

*NODE

9001, 0.051, -0.06

9006, 0.051, -0.005

9016, 0.056, 0.

9021, 0.11, 0.

*NGEN

9001,9006,1

9016,9021,1

*NGEN, LINE=C

9006,9016,1,,0.056, -0.005, 0.

*NODE, NSET=REFDIE

9000,0.1,-0.06

**

** Rigid elements for die

**


9001, 9001,9002

*ELGEN, ELSET=DIE

*NFILL

BOT,TOP,2,101

**

** Element connectivity for blank

**

*ELEMENT, TYPE=CPE4I

1, 1,2,103,102

*ELGEN, ELSET=BLANK

1, 100,1,1, 2,101,100

**

** Node and element sets

**

*NSET, NSET=CENTER

1,102,203


155,202,1


102,147,1


1,100,1


101,200,1

**

** Nodal coordinates and rigid body reference node for die

**

*NODE

9001, 0.051, -0.06

9006, 0.051, -0.005

9016, 0.056, 0.

9021, 0.11, 0.

*NGEN

9001,9006,1

9016,9021,1

*NGEN, LINE=C

9006,9016,1,,0.056, -0.005, 0.

*NODE, NSET=REFDIE

9000,0.1,-0.06

**

** Rigid elements for die

**


9001, 9001,9002

*ELGEN, ELSET=DIE

0-369

9001, 20,1,1


*NORMAL, TYPE=CONTACT SURFACE

DIE, 9006,-1,0,0

DIE, 9016, 0,1,0

**

** Rigid blankholder

**

*NODE, NSET=REFHOLD

8000, 0.1,0.06


START,0.110, 0.001

LINE, 0.056,0.001

CIRCL,0.051,0.006, 0.056,0.006

LINE, 0.051,0.060


**

** Rigid punch

**


7000, 0.000, 0.061


START,0.050, 0.061

LINE, 0.050, 0.007

CIRCL,0.045, 0.002, 0.045,0.007

LINE,-0.010, 0.002


**


**

*SOLID SECTION, MATERIAL=STEEL, ORIENTATION=LOCAL, ELSET=BLANK


1.,0.,0., 0.,1.,0.

3,0


*ELASTIC

2.1E11,0.3

*PLASTIC

400.E6, 0.0E-2

420.E6, 2.0E-2

500.E6,20.0E-2

600.E6,50.0E-2

**

9001, 20,1,1


*NORMAL, TYPE=CONTACT SURFACE

DIE, 9006,-1,0,0

DIE, 9016, 0,1,0

**

** Rigid blankholder

**

*NODE, NSET=REFHOLD

8000, 0.1,0.06


START,0.110, 0.001

LINE, 0.056,0.001

CIRCL,0.051,0.006, 0.056,0.006

LINE, 0.051,0.060


**

** Rigid punch

**


7000, 0.000, 0.061


START,0.050, 0.061

LINE, 0.050, 0.007

CIRCL,0.045, 0.002, 0.045,0.007

LINE,-0.010, 0.002


**


**

*SOLID SECTION, MATERIAL=STEEL, ORIENTATION=LOCAL, ELSET=BLANK


1.,0.,0., 0.,1.,0.

3,0


*ELASTIC

2.1E11,0.3

*PLASTIC

400.E6, 0.0E-2

420.E6, 2.0E-2

500.E6,20.0E-2

600.E6,50.0E-2

**

0-370

** Contact definition

**

*SURFACE, NAME=BLANK_B

BLANK_B, S1

*SURFACE, NAME=BLANK_T

BLANK_T, S3

*SURFACE, NAME=DIE

DIE, SPOS


BLANK_B, DIE


BLANK_T, HOLDER


BLANK_T, PUNCH


*FRICTION

0.1,


**

** Step 1

**

*STEP, INC=10, NLGEOM, UNSYMM=YES

Push the blankholder and die together

*STATIC

1.0,1.0

*BOUNDARY

MIDRIGHT ,2,2

MIDLEFT, 2, 2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,2,2,-1.E-8

REFHOLD ,6,6

*PRINT, CONTACT=YES




*END STEP

**

** Step 2

** Contact definition

**

*SURFACE, NAME=BLANK_B

BLANK_B, S1

*SURFACE, NAME=BLANK_T

BLANK_T, S3

*SURFACE, NAME=DIE

DIE, SPOS


BLANK_B, DIE


BLANK_T, HOLDER


BLANK_T, PUNCH


*FRICTION

0.1,


**

** Step 1

**


Push the blankholder and die together

*STATIC

1.0,1.0

*BOUNDARY

MIDRIGHT ,2,2

MIDLEFT, 2, 2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,2,2,-1.E-8

REFHOLD ,6,6

*PRINT, CONTACT=YES




*END STEP

**

** Step 2

0-371

**



*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2,2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,2,2,-1.E-8

REFHOLD ,6,6

*END STEP

**

** Step 3

**


Apply prescribed force on blankholder

*STATIC

1.0,1.0

*BOUNDARY,OP=NEW

MIDLEFT ,2,2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,6,6

*CLOAD

REFHOLD,2,-4.4E5

*END STEP

**

** Step 4

**


Move punch down a little whilst applying small pressure to

BLANK_T

*STATIC

0.1,1.0

**



*STATIC

1.0,1.0

*BOUNDARY, OP=NEW

MIDLEFT ,2,2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,2,2,-1.E-8

REFHOLD ,6,6

*END STEP

**

** Step 3

**


Apply prescribed force on blankholder

*STATIC

1.0,1.0

*BOUNDARY,OP=NEW

MIDLEFT ,2,2

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,ENCASTRE

REFHOLD ,1,1

REFHOLD ,6,6

*CLOAD

REFHOLD,2,-4.4E5

*END STEP

**

** Step 4

**


Move punch down a little whilst applying small pressure to

BLANK_T

*STATIC

0.1,1.0

0-372

*BOUNDARY, OP=NEW

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,1,1

REFPUNCH,2,2,-1.00000E-3

REFPUNCH,6,6

REFHOLD ,1,1

REFHOLD ,6,6

*DLOAD

BLANK_T, P3, -1000.0E0



RF,

*END STEP

**

** Step 5

**



*STATIC

.0001,1.0,1.E-6

*BOUNDARY

CENTER ,XSYMM

REFDIE ,1,1

REFDIE ,2,2, 1.E-8

REFDIE ,6,6

REFPUNCH,1,1

REFPUNCH,2,2,-0.03

REFPUNCH,6,6

REFHOLD ,1,1

REFHOLD ,6,6

*DLOAD, OP=NEW


*ELSET, ELSET=QA, GENERATE

50, 75, 1

150, 175, 1

*END STEP

0-373

getting started with abaqus standard

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