nastran: user experience with four … 1196 april 1972 copy no. -? technical memorandum nastran:...

TG 1196

APRIL 1972

Copy No. -?

Technical Memorandum

NASTRAN: USER EXPERIENCEWITH FOUR EXAMPLE PROBLEMSby R. M. RIVELLO

WITH FOUR EXAMPLE PROBLEMS 7. Rielo/(Applied Physics Lab.) ,pr. 1972 139 p

CSCL 20K Unclas

G3/32 3678. 8

' 'eii'oduced by NATIONAt TECHNICAL:

<:_iNFORMATION SERVICE:U S Department of Commerce. Springfield VA 22151 -

THE JOHNS HOPKINS UNIVERSITY * APPLIED PHYSICS LABORATORY

Approved for public release; distribution unlimited.

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DOCUMENT CONTROL DATA - R & DrSeccIrity classification of title, body of abstract and indexing annotation nlut be entered when tile overall report is classified)

I. ORIGINA TING ACTIVITY (Corporate author) 20. REPORT SECURITY CLASSIFICATION

The Johns Hopkins University Applied Physics Lab. Unclassified8621 Georgia Avenue 2b. GROUP

Silver Spring. Md. 20910 n. a.3. REPORT TITLE

NASTRAN: User Experience with Four Example Problems

4 DESCRIPTIVE NOTES (Type of report and inclusive dates)

Technical Memorandum5. AUTHOR(S) (First name, middle initial, last name)

Robert M. Rivello

6- REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS

April 1972 129 178a. CON TRAC T OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S)

N00017-72-C-4401b. PROJECT NO. TG-1196

c. Sb. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)

d.

10. DISTRIBUTION STATEMENT .

Approved for public release; distribution unlimited

11. SUPPLEMENTARY NOTES i2. SPONSORING MILITARY ACTIVITY

n. a. Naval Ordnance Systems Command

1. AOSIL1. I

The NASTRAN computer program is used to solve four simple struc-tural problems. The problems are: (1) a simply-supported beam subjected tolateral loads, (2), a filamentary composite bar under the action of centrifugalforces, (3) a free beam subjected to lateral loads, and (4) the thermal bucklingof a simply-supported plate. Input and output data are given for each problem toassist the new NASTRAN user in preparing program input data. The results arecompared with solutions obtained by other methods. The example problems dis-closed errors in the plotting and thermal-buckling routines of the computer. Thesehave subsequently been corrected. The program was found to have broad capa-bilities and to offer many user conveniences.

I,

DD FNORV M 1473 UNCLASSIFIEDSecurity Classification

1 3. ABSTRACT

UNC LA SSIFIEDSecurity Classification

14.KEY WORDS

BucklingComputer plottingComputer programsFinite differences methodFinite element methodNASTRANStress analysisStress and deflectionStructural analysisSuperflywheelThermal stress analysis

UNC LA SSIFIEDSecurity Classification

[

TG 1196

APRIL 1972

Technical Memorandum

NASTRAN: USER EXPERIENCEWITH FOUR EXAMPLE PROBLEMSby R. M. RIVELLO

THE JOHNS HOPKINS UNIVERSITY *· APPLIED PHYSICS LABORATORY8621 Georgia Avenue o SilverSpring, Maryland o 20910Operating under Contract N00017-72-C-4401 with the Department of the Navy

Approved for public release; distribution unlimited.

_..,,,.

THE JOHNS HOPKINS UNIVERSITY PRECEDING PAGE BLANK NOT FILMEDAPPLIED PHYSICS LABORATORY

SILVER SPRING. MARYLAND

ABSTRACT

Four different structural problems are solved togain familiarity with the NASTRAN computer program.The problems are: (1) a simply-supported beam subjectedto lateral loads, (2) a rotating filamentary composite barunder the action of centrifugal forces, (3) a missile bodywith aerodynamic, gravitational, and inertial forces, and(4) a square simply-supported plate with in-plane tempera-ture changes capable of buckling the plate. Input and out-put data are given for each problem. These should be ofassistance to those who are using the program for thefirst time.

The results are compared with those obtained byother methods. However, except for the examples employingbeam elements in which the agreement is excellent, the ele-ment breakup chosen for convenience in obtaining programfamiliarity is too coarse to draw conclusions regarding theprogram accuracy. The example problems disclosed errorsin the plotting and thermal-buckling routines of the program.

The program was found to offer many user conveni-ences. Data preparation was straightforward once famil-iarity with the format appropriate to the problem was gained.The program documentation is complete for reference pur-poses, but is lacking in that it does not rapidly orient thenew user. The capabilities and conveniences of the programgreatly outweigh its inadequacies, and it is recommendedthat APL/JHU continue to update the program with eachnew release from NASA.

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TE OHNS HOPKINS UNv,,,,T PRECEDING PAGE BLANK NOT FILMAPPLIED PHYSICS LABORATORYSILVER SPRING. MARYLAND

CONTENTS

List of Illustrations vii

List of Tables xi

1 Introduction 1

2 Example Problems . 5Example Problem 1 . . 8Example Problem 2 .10Example Problem 3 . 15Example Problem 4 . . . 20

3. Discussion 29

Acknowledgment . . . . 31

References .33

Appendixes:

A. Example 1 - Simply-Supported Beam A-iB. Example 2 - Composite Flywheel B-1C. Example 3 - Missile Flight Loads C-1D. Example 4- Thermal Buckling D-1

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THE JOHNS HOPKINS UNIVERSITY PRECEDING PAGE BLANK NOT FITLMEDAPPLIED PHYSICS LABORATORY


ILLUSTRATIONS

1 Simply-Supported Beam, ExampleProblem 1. . 9

2 Finite-Element Idealizations, ExampleProblem 2 . 12

3 Comparison of Example Problem 2 Resultswith Results Given in Refs. 14 and 15 14

4 Example Problem 3 . 16

5 Finite-Element Breakup, ExampleProblem 3 . 16

6 Shear versus Body Station, ExampleProblem 3 . . . 17

7 Bending Moment versus Body Station,Example Problem 3 . . 17

8 Deflection versus Body Station, ExampleProblem 3 . . . 18

9 Example Problem 4 . 21

10 NASTRAN Idealization and TemperatureDistribution, Example Problem 4 21

11 Distribution of Prebuckling ThermalStresses axx' Example Problem 4 23

12 Distribution of Prebuckling ThermalStress a , Example Problem 4 . 24

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILV1ER SPRING. MARYLAND

13 Thermal-Buckling Mode Shape, ExampleProblem 4. 25

14 Finite Difference Mesh and TemperatureDistribution, Example Problem 4 26

A-1 Static Loading of Simply-Supported Beam;NASTRAN Example Problem 1: UndeformedShape . . . . A-19

A-2 Static Loading of Simply-Supported Beam,Concentrated Load at Center; NASTRANExample Problem 1: Static Deformation,Subcase 1, Load Set 1 . . . A-19

A-3 Static Loading of Simply-Supported Beam,Distributed Load; NASTRAN ExampleProblem 1: Static Deformation, Sub-case 2, Load Set 2 . . A-20

A-4 Static Loading of Simply-Supported Beam,Concentrated plus Distributed Load;NASTRAN Example Problem 1: StaticDeformation, Subcase 3, Load Set 3 A-20

B-1 Super Flywheel Stress Analysis, 1. 254cps; NASTRAN Example Problem 2:Undeformed Shape . . B-23

B-2 Super Flywheel Stress Analysis, 1. 254cps; NASTRAN Example Problem 2:Static Deformation, Subcase 1, LoadSet 1 . . . . B-24

B-3 Super Flywheel Stress Analysis, 1. 254cps; NASTRAN Example Problem 2:Static Deformation, Subcase 1, LoadSet 1 .. B-25

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APPLIED PHYSICS LABORATORYSILVER SPRING. MARYLAND

B-4 Super Flywheel Stress Analysis, 1. 254cps; NASTRAN Example Problem 2:Static Deformation, Subcase 1, LoadSet 1 . . . B-26

C-1 Flight Load Problem; NASTRAN ExampleProblem 3: Undeformed Shape C-17

C-2 Flight Load Problem; NASTRAN ExampleProblem 3: Static Deformation, Sub-case 1, Load Set 1 C-18

C-3 Flight Load Problem; NASTRAN ExampleProblem 3: Static Deformation, Sub-case 1, Load Set 1 . . C-19

D-1 Thermal Buckling of Square Plate;NASTRAN Example Problem 4: Unde-formed Shape . . . D-28

D-2 Thermal Buckling of Square Plate;NASTRAN Example Problem 4: StaticSolution, Static Deformation, Subcase1, Load Set 0 . . . . D-29

D-3 Thermal Buckling of Square Plate;NASTRAN Example Problem 4: BucklingSolution, Subcase 2, Mode 1, Eigen-value = 314.55923000 . . . D-30

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APPLIED PHYSICS LABORATORYSILVER SPRIWN. MARYLAND

PRECEDING PAGE BLANK NOT FILMED

TABLES

1 Program Options Used in ExampleProblems .

2 Example Problem 1: Comparison ofNASTRAN and Beam-Theory Results

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11

7



1. INTRODUCTION

The advent of high-speed digital computers has hada revolutionary effect upon the analysis of stress and de-flection of structures. Prior to the availability of thesecomputers, it was necessary to use simplifying approxi-mations to analyze structural components. Although ap-proximate theories such as those for beams, plates, andshells frequently give satisfactory results for simple struc-tural components, they are generally unsuitable for the ac-curate analysis of complex structures.

Matrix methods of structural analysis in which thecomplex structure is considered to be an assemblage ofsimple finite-sized elements were introduced in the late1950's (Refs. 1 and 2). Most of the matrix methods thathave been developed since then have used either forces ordisplacements as the unknowns. Of the methods that havebeen proposed, the so called "direct-stiffness" displace-ment method introduced in Ref. 2 has proven to be rela-tively simple to program and has given well-conditionedsolutions. As a result, it has been widely used and under-gone extensive development. Today many programs areavailable that use the direct-stiffness method and it doesnot appear economical for organizations contemplating theneed for such a program to develop their own unless theyrequire small special-purpose programs that will be suffi-ciently used to merit the costs of program development.However, in some cases it may be justifiable to write sim-ple, efficient programs for special types of problems in-stead of using a large, complex, and inefficient all-purposeprogram.

Of the general-purpose programs, the NASA STRuc-tural ANalysis (NASTRAN) program is capable of handlingthe largest number of different types of structural prob-lems. Several million dollars have been invested in the

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APPLIED PHYSICS LABORATORYSILVER SPRlNG. MARYLAND

development of this program and an annual expenditure ofapproximately one-half million dollars has been budgetedby NASA for its upkeep and improvement.

The diversity and complexity of structural problemsencountered by the Johns Hopkins University Applied Phys-ics Laboratory (APL) are such that the availability of ageneral-purpose program is a necessity. The capabilitiesof the nonproprietary NASTRAN program make it a naturalchoice. Furthermore, the Naval Ship Research and Devel-opment Center (NSRDC) had already installed Level 12. 0of the program on the IBM 360 computer at APL and, whenthe program was released by NASA to nongovernmentalagencies, the use of it was offered to APL.

Although the NASTRAN program had been installedon the APL computer, APL personnel had not been involvedand were, therefore, untrained in the use of the program.Self instruction in the use of the program is not a smallundertaking. The program has a total of twelve rigid for-mats to provide a general-purpose capability as follows:

1. Static Analysis

2. Static Analysis with Inertia Relief

3. Normal Modes Analysis

4. Differential Stiffness Analysis

5. Buckling Analysis

6. Piecewise Linear Analysis

7. Direct Complex Eigenvalue Analysis

8. Direct Frequency and Random Response

9. Direct Transient Response

10. Modal Complex Eigenvalue Analysis

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11. Modal Frequency and Random Response

12. Modal Transient Response

The program documentation that is available from theNASA COSMIC Office consists of a Theoretical Manual(Ref. 3), a'User's Manual (Ref. 4), a Programmer'sManual (Ref. 5), and a Demonstration Problem Manual(Ref. 6). The total number of pages in these documentsis roughly 3300. Unfortunately for the new user, themanuals are for reference rather than tutorial purposesand do not include indices.

The Demonstration Problem Manual gives a com-parison of NASTRAN results with solutions to the sameproblems obtained by other methods; however, this man-ual does not contain the input or output data for the NAS-TRAN solutions. As a result, it alone is of little valueas a learning aid. The NASTRAN tapes available fromCOSMIC contain a file of the bulk data card images for thedemonstration problems, but to obtain the full set of inputand output data it is necessary to prepare executive andcase control decks and run the problems. The NSRDCwas never successful in getting this portion of the tape torun on the APL computer. Furthermore, considerablecomputer expense would be entailed in running the 25demonstration problems.

After the four NASTRAN manuals were reviewed, itwas concluded that the only sensible way to become famil-iar with the program would be to prepare the data for sev-eral example problems and run them. The purpose of thisreport is to document the input and output data for the ex-ample problems, in the hope that the examples will behelpful to those who are learning to use the program.

Subsequent to running some of the example prob-lems, a set of the printouts of the demonstration problemsof Ref. 6 was obtained. These printouts (which were runwith NASTRAN Level 8. 1. 0) proved to be of considerablehelp, because there are several places where the User'sManual is unclear and some places where it is misleading.

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After the work documented in this report was com-pleted, a copy of a beginner's guide (Ref. 7) that is usedin the NASTRAN training courses taught by the MacNeal-Schwendler Corporation was obtained. The printouts forthe demonstration problems, the beginner's guide, andRefs. 8 through 12 should prove helpful to the new NASTRANuser. The BBE Project Office of APL has these refer-ences on file.

The job cards that are required to run NASTRANon the APL computer are described in Ref. 12. The newuser should also be aware of two newsletters that areperiodically published for NASTRAN users. One of these,the NASTRAN Newsletter, is published by NASA. Usersmay be placed on the distribution list for this newsletterby writing or phoning:

NASTRAN Systems Management OfficeMail Stop 188cNASA Langley Research CenterHampton, Va. 23365Telephone: (703) 827-2388

The other is the Navy Structures Computer Program NEWS-LETTER that is published by NSRDC and is available from:

NASTRAN Evaluation Project, Code 823Department of Applied Mathematics, NSRDCWashington, D. C. 20034

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2. EXAMPLE PROBLEMS

Example problems were chosen to exercise differ-ent rigid-format, structure-definition, loading, and out-put options of the program. The example problems are asfollows:

1. Static Deformations of a Simply-SupportedBeam

2. Static Deformation of a Composite Rectangular-Planform Flywheel

3. Static Deformation of a Missile Body Under theAction of Aerodynamic and Inertial Loads

4. Thermal Buckling of a Square Simply-SupportedPlate

The program features that are demonstrated by these prob-lems are:

1. Rigid Formats

a. Static Analysisb. Static Analysis, Inertia Reliefc. Buckling Analysis

2. Structure Definition Options

a. Single-Point Constraintsb. Free-Body Supportsc. Bar Elementsd. Quadrilateral Membrane Elementse. Quadrilateral Plate Elementsf. Thermally-Dependent Materialsg. Isotropic Materialsh. Anistropic Materials

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APPLIED PHYSICS LABORATORYSlLVER SPRING. MARYLAND

3. Static Loading

a. Concentrated Loadsb. Gravity Loadsc. Inertial Loadsd. Centrifugal Field Loade. Thermal Loadsf. Combined Loads

4. Output Options

a. Shape Plot of Undeformed Structureb. Shape Plot of Deformed Structurec. Vector Plot of Structural Displacementsd. Point Output Selectionse. Element Output Selectionsf. Subcase Level Request Changesg. Modal Plot

The program options that are used for each of the exampleproblems are shown in Table 1.

The primary purpose of the example problems isto gain familiarity with the NASTRAN data format ratherthan to obtain high accuracy in the computed results. Be-cause of this, crude structural modeling with few gridpoints and elements is used to reduce the data preparationand computing times. Improved accuracy could be obtainedby using smaller elements, especially in the regions oflarge stress gradients.

In initial runs of the example problems, it wasfound that the program would not write a tape for the CAL-COMP plotter. This difficulty was reported to NSRDC whomade the necessary programming corrections to obtainCALCOMP Model 565 plots. It is still not possible to ob-tain plots with the CALCOMP Model 765, the default modelfor the program. It was also found that there was an errorin the thermal-buckling portion of the program (this typeof problem was not included in the demonstration problemsof Ref. 6). The difficulty was reported to the NASTRANSystems Management Office and to NSRDC. Correctionswere made by NSRDC to the APL copy of the program.

-6-



Table 1

Program Options Used in Example Problems

Problem Proram OptionNo. lalb lc2a 2b2c2d2e2f2g2h 3a 3b3c3d3e3f 4aI4b 4c4d4e4f4g

1 X X X X XX XX X XXX

2 X X X X X XXXXX

3 XXX XX XX X X X XXX XX

4 X X X X X I X XI

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APPLIED PHYSICS LABORATORYSILVER SPRING. MAIRYLAND

Detailed descriptions of the example problems aregiven in the following subsections in which the computed resultsare compared with the results obtained by other methods.The computer and plotter outputs for each of the exampleproblems is given in the appendixes at the end of the re-port. The output contains an echo of the executive, case,and bulk data decks that constitute the input to the program.This input should serve as a useful guide to the new userin preparing program data.

EXAMPLE PROBLEM 1

The first example problem is for a uniform, sim-ply-supported beam. The computer and CALCOMP outputsfor the problem are given in Appendix A. The geometry,method of support, and material properties of the beamare shown in Fig. 1. The finite-element idealization con-sisting of four BAR (beam) elements and 5 grid points isalso shown in this figure. The origin of the basic coordi-nate system is taken at grid point 1.

Three static-load subcases are used: Subcase 1consists of a concentrated 100-lb load applied upward atgrid point 3, Subcase 2 consists of a 100 lb/in uniformlydistributed upward loading, and Subcase 3 is the sum ofthe loadings from subcases 1 and 2. The NASTRAN pro-gram does not have a distributed load bulk data card forBAR elements. In this example problem, the gravity-loading card GRAV is used to provide the distributed load-ing. A gravitional acceleration (gravity vector scale fac-tor) of 3.86 x 104 in/s 2 is used to produce a loading of 10lb/in with a cross sectional area of 1. 0 in2 and a materialspecific weight of 0. 1 lb/in 3 .

Vertical displacements and rotations (slopes) arecomputed at all grid points. Reactions are determined atall single-point constraints. Bending stresses are foundat both ends of each element at the four cross-sectionalpoints shown in Fig. 1. CALCOMP plot requests are madeboth for the undeformed structure and for the deformed

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y

(t1 t - (2)

20 IN TYPICAL

F--(3) + (4) .X

_1IA = 1.0 IN 2

I = 0.1 IN 4

E = 10 7 PSI

w = 0.1 LB/IN 3

( ) ELEMENT NUMBERS

O GRID POINT NUMBERS

STRESS CALCULATIONPOINT NUMBERS

FIG. 1 SIMPLY-SUPPORTED BEAM, EXAMPLE PROBLEM 1

-9-

IN Ap

I



structure for each of the three subcase loads. The result-ing CALCOMP plots are given at the end of Appendix A.

Bending deflection at the center of the beam andend slopes for subcases 1 and 2 were also computed.Stresses were calculated using bending-moment formulas fromRef. 13. The NASTRAN and beam-equation results arecompared in Table 2. It is seen that the NASTRAN resultsare in perfect agreement with those from beam theory forSubcase 1. This would be expected, since the BAR-ele-ment deflection function used in the program is a cubic, asis the actual deflection shape for a beam with concentratedloads. As a result, element size in this case does not af-fect accuracy. For Subcase 2 the stresses are in goodagreement, but the center deflection computed by NASTRANis 5% less than the exact solution given by beam theory.The deflection result is not surprising since, for a distrib-uted loading, the exact deflection curve for each elementis a quartic rather than a cubic as assumed in NASTRANand, in addition, the NASTRAN program uses lumped forcerather than consistent force matrices for distributedloads. Of course, accuracy could be improved by usingsmaller elements than those in the crude four-elementidealization used in the example.

EXAMPLE PROBLEM 2

In this second example problem the NASTRAN pro-gram is applied to a unidirectional graphite-epoxy com-posite bar rotating at a constant angular velocity about itscenter of gravity. Since the geometry, elastic properties,and loading are symmetric about both the longitudinal andlateral axes of the body, it is only necessary to model onequadrant of the bar. The finite-element idealization con-sists of 60 identical nonisotropic, quadrilateral, membrane(QDMEM) elements with 80 grid points as shown in Fig. 2a.The origin of the basic x-y coordinate system is taken atgrid point 1, the center of rotation. The angular velocityis 1. 264 rad/s.

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Table 2

Example Problem 1:Comparison of NASTRAN and Beam-Theory Results

Center Deflection End Slope Center Bending StressLoading (inches) (10-2 radians) (psi)

NASTRAN Beam Theory NASTRAN Beam Theory NASTRAN Beam Theory

Subcase 1 0.1333 0,1333 0.9999 1.0000 + 4 999 + 5 000

Subcase 2 0.3167 0.3333 2.5003 2.6667 ±10 001 ±10 000

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0.625 IN TYPICAL(

f-F_

(a) 80-GRID POINT - 60 ELEMENT NASTRAN IDEALIZATION

(b) 80-GRID POINT - 120 ELEMENT (80/120) IDEALIZATION USED IN REF. 15

L

~flit OX

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15 IN-

(c) 77-GRID POINT - 128 ELEMENT (77/128) IDEALIZATION USED IN REF. 15

FIG. 2 FINITE-ELEMENT IDEALIZATIONS, EXAMPLE PROBLEM 2

THE JOHNS HOPKINS UNIVERSITYAPPLIED PHYSICS LABORATORY

SILVIER SIING. MARYLAND

The symmetry conditions are imposed by applyingsingle-point constraints that prevent y-direction motion atgrid points on the x-axis and x-direction motion at gridpoints on the y-axis.

The computer printout and CALCOMP plots for theproblem are given in Appendix B. The centrifugal inertialforces are generated by using an RFORCE bulk-data card.Nondimensionalized values of the normal stresses axx andayy, and the shear stress Ux computed by the NASTRANprogram are given in Fig. 3 by the solid curves.

An exact solution to this problem does not exist.Weiss has obtained approximate solutions to the problem bythe collocation method (Ref. 14) and the finite-elementmethod (Ref. 15). These results are also shown in Fig. 3.Two different finite-element breakups were used in Ref. 15.In one of these (Fig. 2b) 80 grid points and 120 triangularmembrane elements (80/120) were used. In this case, thegrid points were chosen to coincide with those used in theNASTRAN solution. In the other idealization shown inFig. 2c, 77 grid points and 128 elements were used (77/128).

It is seen that the NASTRAN normal stresses are inexcellent agreement with the finite-element solutions ofRef. 15, but the agreement in the shearing stresses is notas good. There is considerable scatter in the shearingstresses computed by both the 80/120 and 77/128 idealiza-tions of Ref. 15. This scatter was not present in theNASTRAN results. Although they are not shown in Fig. 3,some of the values of the shear stress computed in Ref. 15were negative for x/a = 0. 6, and although the distributionshapes are similar, the collocation results do not agreewell with either the NASTRAN results or those of Ref. 15.This is especially true of the shear stress. The reasonsfor the lack of agreement are not apparent and the correctsolution is an open question. It appears, however, thatthe shear-stress calculations are very sensitive to theidealization.

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02 0.4 0.6 0.8

xla

FIG. 3 COMPARISON OF EXAMPLE PROBLEM 2REFS. 14 AND 15

RESULTS WITH RESULTS GIVEN IN

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1.8

1.6

1.4

1.2

1.0

0.8

N,

C'

c:

C',

n-

-Jcu

zC,

0z

O

z

z

0.6

0.4

0.2

00 1.0


APPLIED PHYSICS LABORATORYSlLVER SPRING. MARYLAND

EXAMPLE PROBLEM 3

The third example problem illustrates the applica-bility of NASTRAN to the calculation of shear-force, bend-ing-moment, bending-stress, and deflection distributionsof a missile body under the combined action of aerodynamic,gravitational, and inertial loads. To reduce data preparation,the simplified model (Fig. 4), which has constant cross-sec-tional properties and a uniform weight per unit length, wasused to represent the missile. The NASTRAN program,of course, can handle the cross-sectional and weight dis-tributions of an actual missile with no difficulty.

The finite-element idealization of the body by beam(BAR) elements is shown in Fig. 5. The computer print-out and CALCOMP plots for the problems are given inAppendix C. Since the missile is a free body, it is neces-sary to use a SUPORT card in the bulk-data deck. Dis-placements are computed for a free-body support at gridpoint 6. The modulus of elasticity is made temperaturedependent by the use of MAT 1, MATT 1, TABLEM 1, andTEMPD cards. The aerodynamic forces, gravitationalforces, and inertial forces are applied by using FORCE,GRAV, and SUPORT cards, respectively.

The NASTRAN results for shear, bending moment,and total deflection are given in Figs. 6 through 8. Tocheck these results the shear, bending moment, and bend-ing deformations were computed by numerical integrationof the beam equations

dVP dx' (1)

dMV = (2)dx

and

d2Wb Mb M (3)

dx 2 E1

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CROSS-SECTIONAL PROPERTIES: I = 100 IN4 , A = 4.4 IN 2

MATERIAL: STEEL, w= 0.3 LB/IN 3

NONSTRUCTURAL WEIGHT 8.68 LB/IN

FIG. 4 EXAMPLE PROBLEM 3

y

20 INTYPICAL

O GRID POINT NUMBERS( ) ELEMENT NUMBERS

FIG. 5 FINITE-ELEMENT BREAKUP, EXAMPLE PROBLEM 3

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THE JOHNS HOPKINS UNIVERSITYAPPLIED PHYSICS LABORATORY


:

U,-i-

FIG. 6 SHEAR vs. BODY STATION, EXAMPLE PROBLEM 3

-12

-oC

1-zUJw

02

-8

-4

040 80 120 160 200

BODY STATION (inches)

FIG. 7 BENDING MOMENT vs. BODY STATION, EXAMPLE PROBLEM 3

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40 80 120 160

BODY STATION (inches)

FIG. 8 DEFLECTION vs. BODY STATION, EXAMPLE PROBLEM 3

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-1.2

* -0.8UJ= -0,8C

z0

C-LUw

U-

-0.4

00 200

V = shear force (lb),

p = distributed load (lb/in),

M = bending moment (in-lb),

EI

Wb

bending rigidity (lb-in2),

bending deformation (in),

Shear deformations w were determined from a numericalintegration of the equation

dws

dxV

GKA '

where

G = shear modulus of elasticity (psi),

K = cross-sectional area factor for shear,(nondimensional),

and

A = cross-sectional area (in2 ).

(4)

The total deformation w was then computed from

W = Wb + w.bS (5)

Deformations were computed relative to a fictitious sup-port at grid point 6 as assumed in the NASTRAN computa-tion. The results of the laborious hand calculations usingEqs. 1 through 5 are also plotted in Figs. 6 through 8. Itis seen that the agreement in the two methods is very good.

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APPLIED PHYSICS LABORATORY

SILV-R SPRING. MARYLAND

where

and



EXAMPLE PROBLEM 4

Example problem 4 was chosen to illustrate the ap-plication of NASTRAN to thermal stress problems and,further, to demonstrate the method for computing thermalbuckling. The structure is a square, simply-supportedplate having uniform thickness. The geometry of the plateand the coordinate system are shown in Fig. 9. The plateis subjected to a temperature change T(x, y) that is sym-metric about both the x and y axes.

If the magnitude of the temperature change is greatenough, the thermal stresses induced in the plane of theplate will cause the plate to buckle. The mode shape ofthe buckle will be symmetric about the x and y axes.As a result of the above-noted symmetries, it is onlynecessary to model one quadrant of the plate if single-pointconstraints are used to prevent unsymmetrical deforma-tions along the x and y axes.

The idealization using quadrilateral plate (QUAD2)elements is shown in Fig. 10. This figure also shows thetemperature change at each of the grid points in terms ofT1, the temperature change at grid point 1. As shown,there is no temperature change at points on the simply-supported edges of the plate.

Difficulty was encountered in running the exampleproblem. The NASTRAN Demonstration Problem 5-1printout, run with NASTRAN Level 8. 1. 0, was used as aguide for preparation of the case control deck. In theLevel 8. 1. 0 printout the load card appears before Subcase1 (the statics solution). However, when the problem wasrun in this manner, there was a fatal-error message whichindicated that Subcase 2 (the buckling solution) containedboth a static load and a real eigenvalue method selection,and that one or the other must be removed. A check ofthe Level 12. 0 User's Manual (Ref. 4) indicated that theload (in this case TEMPERATURE(LOAD)) card must ap-pear in Subcase 1 rather than above the subcase level.

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E = 10 7 PSIv = 0.3a = 10 x 10-6 IN/IN OF

~I 4 4IN

I 1

~ -4 IN.- 4 IN -|1

'I _ i ~

0.1 IN

FIG. 9 EXAMPLE PROBLEM 4

OGRID POINTNUMBERS

() ELEMENTNUMBERS

FIG. 10 NASTRAN IDEALIZATION AND TEMPERATURE DISTRIBUTION, EXAMPLEPROBLEM 4

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Y

SIMPLY-SUPPORTEDEDGES



This change was made and the example problem wasrerun. The results contained only a static solution fol-lowed by "SYSTEM FATAL MESSAGE, 3001 ATTEMPT TOOPEN DATA SET **** IN SUBROUTINE DS1A, WHICH WASNOT DEFINED IN FIST." Since the cause appeared to be aNASTRAN program error, the difficulty was reported tothe NASTRAN System Management Office and NSRDC.Program corrections were made by NSRDC personnel whoreran the problem, using both the inverse and determinantmethods of eigenvalue extraction. The results of the twomethods did not agree. The NASA Goddard Space FlightCenter had experienced difficulty with the determinanteigenvalue method and had made corrections in their pro-gram for the IBM 360-95 computer. The problem was runat Goddard Space Flight Center and identical results wereobtained by both the inverse and determinant methods.NSRDC then made the necessary corrections to the programon the APL computer.

The NASTRAN results given in Figs. 11 to 13 andAppendix D were obtained after the corrections noted abovewere made. The problem was also solved by the finite-differences method. This method was first used to deter-mine the in-plane thermal stresses by obtaining an approxi-mate solution to the differential equation

V4 F = - EtV 2 T, (6)

subject to the boundary conditions F = aF/an = 0 alongthe boundary of the plate (Ref. 16), where F is the stressfunction defined by the equations

1 a2 F 1 2F 1 a2Fxx t 2 yy = and - . (7)xx t a2 2 xy t axay(ayxy t

The finite-difference mesh that was used in the solution isshown in Fig. 14. The normal stress axx and ayy deter-mined by the NASTRAN program and the finite-difference

- 22 -



La.

-

tX

xx

FIG. 11 DISTRIBUTION OF PREBUCKLING THERMAL STRESSES ax, EXAMPLEPROBLEM 4

- 23 -

y (inches)

FIG. 12 DISTRIBUTION OF PREBUCKLING THERMAL STRESS yy EXAMPLEPROBLEM 4

- 24 -



40

-

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1.0 1

0.8

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FIG. 13 THERMAL BUCKLING MODE SHAPE, EXAMPLE PROBLEM 4

- 25 -

SYM.-- _ _ -, X

FIG. 14 FINITE DIFFERENCE MESH AND TEMPERATURE DISTRIBUTION, EXAMPLEPROBLEM 4

- 26 -



fySYM

I

4-



method are compared-in Figs. 11 and 12. It is seen that,while the stress distributions computed by these twomethods are similar, the stresses computed by the finite-difference method are larger than those calculated byNASTRAN. It is not possible to say which method is moreaccurate as the grids that were used in both methodsare too coarse to expect accurate solutions. Whilethe grid that was used for the NASTRAN results istwice as fine as that used in the finite-difference solution,the NASTRAN program assumes that the temperaturewithin each quadrilateral, element is constant and equal tothe average of the temperatures at the grid points of theelement. As a result, a zero temperature change on theboundary is not attained. It is expected that this would re-duce the stresses computed by NASTRAN.

The finite-difference method was also used to solvefor the eigenvalue X and the mode shape w(x, y) of thedifferential equation

2F 2 a2F a 2 2F 2F w = 2 -2 + (8)

D 2 2 a x a y E _X-5-y 2 2V Da 22axa.y xay a ay

subject to the boundary conditions w = a2 w/an2 = 0 on theboundary (Ref. 16). In this equation X is a constant whichspecifies the magnitude of the temperature change accord-ing to the equation

T(x, y) = XT (x, y), (9)

where To(x, y) gives the spacial variation of the tempera-ture change normalized so that To(O, 0) = 1. The functionF o is related to F by

F(x, y) = XF (x, y). (10)0

- 27 -



Physically, X is the temperature at the point (0, 0) at whichthermal buckling of the plate occurs when it is subjected tothe temperature change T(x,y) = XTo(x,y).

The NASTRAN solution predicts that buckling willoccur when the temperature change at the point (0, 0)reaches 314. 7°F, while the finite-difference solution indi-cates that buckling will occur when the temperature changereaches 168. 8°F. As expected, the finite-difference methodpredicts a lower buckling temperature because it predictsgreater thermal stresses per degree of temperature change.The buckling mode shapes predicted by the two methods areshown in Fig. 13. It is seen that the agreement in the modeshapes is excellent.

- 28 -



3. DISCUSSION

The example problems demonstate only a few of themany capabilities of the NASTRAN program. Considerableeffort has gone into providing user convenience in the pro-gram. Once familiarity with a rigid format is gained, datapreparation is relatively simple and no programming skillsare required. The price of these user convenience featuresis, of course, increased computer run times. In largeproblems with many elements, bulk data preparation islaborious as there are no provisions for the automated gen-eration of grid point or element data.

Only the structural plot capabilities of the programwere used in the example problems. The case controlcards for these plots were found to be simple to prepare,since the program determines the appropriate scale, origin,and vantage point to be used for the plot. The x-y plot capa-bility extends the usefulness of the program; however, thereis no provision for stress-contour plots in two-dimensionalbodies such as those described in Ref. 17.

While the documentation is complete in the sensethat it provides a useful reference for those familiar with theprogram, it leaves something to be desired in aiding the newuser. A tutorial manual would be very helpful. The situa-tion for the new user is further aggravated by the lack of anindex for the NASTRAN manuals.

While the NASTRAN element library is extensive,there are programs with more and improved types of ele-ments. Fully compatible plate elements, consistent massand force matrices for all elements, and elements withvariable section properties and temperatures would allowadequate modeling with fewer elements.

- 29 -



The inadequacies of the program that have beenpointed out have been noted by others and are well known toNASA. While these inadequacies are more than compen-sated for by the capability and convenience of the program,many of them are expected to be removed in later levels ofthe program. All indications are that the program will be-come more general, convenient, and accurate, while re-quiring less computer time. Use of the program is expectedto become widespread both in government and industry.Under these circumstances it appears advisable for APL/JHU to constantly maintain its NASTRAN capability at thelatest current level of release.

- 30 -



ACKNOWLEDGMENT

The author is indebted to Messrs. James McKeeand Myles Hurwitz of NSRDC and to Mr. William Case andDr. James Mason of the NASA Goddard Space Flight Cen-ter for their assistance when troubles were encounteredin running the example problems described in this report.

- 31 -

PRECEDING PAGE BLANK NOT FTL,I TDTHE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER SPRING. MAMYLAND

REFERENCES

1. J. H. Argyris and S. Kelsey, Energy Theoremsand Structural Analysis, Butterworth and Co. Ltd.,London, 1960.

2. M. J. Turner, R. W. Clough, H. C. Martin, andL. J. Topp, "Stiffness and Deflection Analysis ofComplex Structures, " J. Aeron. Sci., Vol. 23,No. 9, September 1956, pp. 805-824.

3. R. H. MacNeal (Ed.), The NASTRAN TheoreticalManual, NASA SP-221, National Aeronautics andSpace Administration, Washington, D. C., Octo-ber 1969.

4. C. W. McCormick (Ed.) The NASTRAN User'sManual, NASA SP-222, National Aeronautics andSpace Administration, Washington, D. C., Octo-ber 1969.

5. F. J. Douglas (Ed.), The NASTRAN Programmer'sManual, NASA SP-223, National Aeronautics andSpace Administration, Washington, D. C., Octo-ber, 1969.

6. NASTRAN Demonstration Problem Manual, NASASP-224, National Aeronautics and Space Adminis-tration, Washington, D. C., October 1969.

7. C. W. McCormick, NASTRAN Beginner's Guide,MS 139-1, MacNeal-Schwendler Corp., LosAngeles, May 10, 1971.

8. T. C. Butler and D. Michel, NASTRAN, A Summaryof the Function and Capabilities of the NASA Struc-tural Analysis Computer System, NASA SP-260,National Aeronautics and Space Administration,Washington, D. C., 1971.

- 33 -

THE JOHNS HOP'KINS UNIVERSITY


9. E. Cuthill, et al., NASTRAN Evaluation Report,NSRDC Technical Note AML-49-70, Naval Ship Re-search and Development Center, Washington, D. C.,August 1970.

10. Proceedings of the Second Navv-NASTRAN Col-loquium, Naval Ship Research and DevelopmentCenter, Washington, D. C., December 1970.

11. NASTRAN: User's Experiences, NASA TM X-2378,National Aeronautics and Space Administration,Washington, D. C., September 13, 1971.

12. Navy Structures Computer Program NEWSLETTER,Vol. 1, No. 2, May 1970. (Pub. by Naval Ship Re-search and Development Center, Washington, D. C. ).

13. R. J. Roark, Formulas for Stress and Strain, 4thed., McGraw-Hill Book Co., New York, 1962.

14. R. O. Weiss, Elastic Analysis of a Rotating Aniso-tropic Bar Using the Method of Collocation, APL/JHU BFR-71-3, March 2, 1971.

15. R. O. Weiss, Analysis of Anistropic Flywheels byFinite Elements, APL/JHU BFR-71-09, September1, 1971.

16. R. M. Rivello, Theory and Analysis of FlightStructures, McGraw-Hill Book Co., New York,1969.

17. G. Dailey, A Computer Program to Plot ContourMaps of Stresses in Two-Dimensional StructuresAnalyzed by the Method of Finite Elements, APL/JHU TG 1128, July 1970.

- 34 -



APPENDIX A

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A -20



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nastran: user experience with four … 1196 april 1972 copy no. -? technical memorandum nastran:...

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