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The 1D CUF Models, a brief Overview


The MUL2Team

July 8, 2011
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Theories of Structures at a Glance

Classical Model Pioneers (Beam)

Leonardo Da Vinci

Euler

De Saint

Venant

Timoshenko
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Classical Beam Theory Features

Low computational cost.

Reliable results for

slender, compact, and

homogenous structures in

bending.

Poor accuracy in

analyzing short,thin-walled, and

non-homogenous

structures.

The analysis of different loading conditions (e.g. torsion) requires

the use ofad hocmodels which areproblem dependent(Is

shear present? Which is the cross-section shape? Etc.)
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Washizus Statement: Arbitrary Rich 1D Models

An arbitrary rich

displacement field couldlead to more and more

accurate results

independently of the

problem characteristics.

Challenging goal: how to find a way to make the choice of the

number of terms arbitrary?
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Our Proposal: the CUF Models for 1D (Beam) and 2D

(Plate/Shell) Theories

The Carrera Unified Formulation, CUF, allows us to use any

order expansions of the unknown variables.

Main Features

Hierarchical structure, that is, the order of the theory can be

chosen as input.Different expansion types can be adopted (e.g. Taylor,

Lagrange).

Arbitrary geometries and boundary conditions can be used.

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Outline

1 CUF Theoretical Introduction

2 Taylor-Based 1D Models

3 Lagrange-Based 1D Models, LE

4 Guidelines and Recommendations

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CUF Theoretical Introduction

The Carrera Unified Formulation, CUF

Main feature of CUF

Problem equations and matrices are obtained by means of a few

fundamental nucleiwhich are formally independent of the order,

N, of the model.

Main capability

Hierarchical The order of the model is a free-parameter.

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CUF Description

Unified Formulation of the Displacement Field

u= Fu, =1, 2, ...., MF= f(x, z), wherexand zare the cross-section coordinates (1D -Beam)

F= f(z),zis the thickness coordinate (2D - Plate/Shell)

N-order model for Beams, (x and z are the cross-section coordinates)

ux=ux1+x ux2+z ux3+x2 ux4+xz ux5+z

2 ux6+...uy=uy1+x uy2+z uy3+x

2 uy4+xz uy5+z2 uy6+...

uz=uz1

+x uz2

+z uz3

+x2 uz4

+xz uz5

+z2 uz6

+...

N-order model for Plates, (z is the thickness coordinate)

ux=ux1+z ux2+z2 ux3+z

3 ux4+...uy=uy1+z uy2+z

2 uy3+z3 uy4+...

uz=uz1+z uz2+z2

uz3+z3

uz4+...

The 1D CUF Models a brief Overview

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A Decade of CUF Plate/Shell Models

E. Carrera,

Developments, ideas and evaluations basedupon the Reissners mixed variational theorem inthe modeling of multilayered plates and shells,Applied Mechanics Reviews, Vol. 54, 2001, pp.301-329.

E. Carrera,

Theories and finite elements for multilayeredplates and shells: a unified compact formulationwith numerical assessment and benchmarking,Archives of Computational Methods inEngineering, Vol. 10, No. 3, 2003, pp. 216-296.


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Fundamental Nuclei

Analytical Solution (ANLT)

The Principle of Virtual Displacements

Li= l

uT Ks us dy+ uT s usy=Ly=0

L k

pxx, L k

pzx

=

l

uxpkxxE

kz

, pkzxE

kx

dy

,s: Expansion function

indexes.

Navier-type solution.

Finite Element Formulation (FEM)Lint= qTiK

ijsqsj

Line= qTiMijsqsj

Lext= PuT

i,j: Shape function

indexes.

,s: Expansion function

indexes.

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Assembly for a giveni,jpair (FEM)

Kzx Kzy

KyyKyxKzz

Kyz

Kxx Kxy Kxz

{3 x 3 Nucleus

{ 3 x M x M locks-B

= 1

= 2, M-1

= M

}

}}

s = 1 s = 2, M-1 s = M

} }}

A Component of the Stiffness Matrix Fundamental Nucleus

Kijsxx =

C22

F,xFs,xd

l

NiNjdy+C66

F,zFs,zd

l

NiNjdy+

C44 FFsdl Ni,yNj,ydy

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Taylor-Based 1D Models

The Taylor CUF 1D Models, TE

ux= ux1 +x ux2+z ux3 +x2 ux4+xz ux5 +z

2 ux6+...uy= uy1 +x uy2+z uy3 +x

2 uy4+xz uy5 +z2 uy6+...

uz= uz1

N=0+ x uz2+z uz3

N=1+ x2 uz4+xz uz5+z

2 uz6

N=2+...

Important feature

Classical models, such asTimoshenko, are obtainable as

particular cases of the linear model.

Main Tags

Isotropic Thin-Walled FGM

Static Free Vibrations Aeroelasticity

Arlequin

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e CU ode s, a b e O e e


Torsion of a Compact Section Structure (FEM)

E. Carrera, G. Giunta, P.

Nali, and M. Petrolo,Refined beam elementswith arbitrarycross-section geometries,

Computers andStructures, Volume 88,Issues 5-6, 2010, Pages283-293.

EBBM and TBM: null displacements!

(a) N= 1 (b) N= 2

(c) N= 3 (d) N= 4

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,


Torsion of a Wing Model (FEM)

E. Carrera, G. Giunta, P.

Nali, and M. Petrolo,Refined beam elementswith arbitrarycross-section geometries,

Computers andStructures, Volume 88,

Issues 5-6, 2010, Pages283-293.

Torsion analysis of a wing

via a fourth-order model

DeformedUndeformed

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Point Load on a Thin-Walled Cylinder (FEM)

z

xO

y

d

E. Carrera, G. Giunta, M.

Petrolo,A Modern and CompactWay to FormulateClassical and AdvancedBeam Theories,In: Developments inComputational StructuresTechnology, Ch. 4. DOI:10.4203/csets.25.4. 2010.

F

Load N= 11

UndeformedShell

N = 11N = 4

TBM

Computational Cost

CUF 1D, 7000 DOFs

Shell 50000 DOFs

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Lobe Modes of a Thin-Walled Cylinder (FEM)

At least a third-order model is necessary to detect these modal

shapes!

Two Lobes Three Lobes

Two Lobes, 2D Three Lobes, 2D

First two-lobe frequency

Theory DOFs f[Hz]

EBBM 155

TBM 155

N = 1 279 N = 2 558

N = 3 930 38.755

N = 4 1395 25.156

N = 5 1953 20.501

N = 6 2604 20.450

N = 7 3348 17.363

Shell 49500 17.406

Solid 174000 18.932

E. Carrera, G. Giunta, M.

Petrolo,A Modern and CompactWay to FormulateClassical and AdvancedBeam Theories,In: Developments in

Computational StructuresTechnology,Ch. 4.DOI:10.4203/csets.25.4. 2010.

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Functionally Graded Material Beam (ANLT)

G. Giunta, S. Belouettar,

E. Carrera,Analysis of FGM beamsby means of a unifiedformulation,IOP Conference Series:Material Science andEngineering, Accepted.

FEM 3D

N=1

N=2

N=4

Mid-Span

Bending Stress

-40

-30

-20

-10

0

10

20

FEM 3D

-40

-30

-20

-10

0

10

20

N= 1

Shear Stress

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

FEM 3D

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

N= 5

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Arlequin Method in the Framework of CUF 1D Theories

Main idea: higher-order elements are employed only in the

portion of the structure in which lower-order theories would

yield inaccurate results.

E.g.: inA1a first-order model, in A2a fourth-order model. Sis

the overlapping volume.

The global mechanical problem is solved by merging together

the two sub-domains via the Arlequin method implemented bymeans of CUF.

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Arlequin Method Numerical Results (FEM)

Arlequina

Arlequinb

101 ux uy 102 uz 10

1 xx 10

1 xy 10

1 yy DOF

N=4 3.741 2.544 2.118 1.425 8.461 5.214 2745

N=1 3.749 2.522 0.053 1.446 5.000 4.870 549

Arlequina 3.729 2.537 2.116 1.424 5.000 5.217 1197

Arlequinb 3.716 2.547 0.056 1.444 8.352 4.807 1197

a Refined elements near the loading application zone.b Refined elements near the simply support. DOF: degrees of freedom.

Shear Stress

3D

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

N = 4

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Arlequinb

F. Biscani, G. Giunta, S. Belouettar, E. Carrera, H. Hu,

Variable kinematic beam elements coupled via Arlequin method,

Composite Structures, Accepted.

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Aeroelastic Coupling (@ SDSU with Dr. Luciano Demasi)

CUF 1D models are particularly eligible for Aeroelastic

Applications since:

They are computationally cheap;

They provide accurate shell-like displacement fields.

Aerodynamic Models

VLMSteady

+ DLMUnsteady

Flutter Solution

Quartic Approximation of the Aerodynamic Kernel + G-Method

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Static Aeroelastic Analysis of a Swept Tapered Wing

A. Varello, M. Petrolo, E.Carrera,A Refined 1D FE Modelfor the Application toAeroelasticity ofComposite Wings,In: IV InternationalConference onComputational Methodsfor Coupled Problems in

Science and Engineering,2011.

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Effect of the Sweep Angle on Flutter

0

20

40

60

80

100

-30 -20 -10 0 10 20 30

Velocity,m/s

Sweep Angle, Deg

CUF

Koo

0

20

40

60

80

100

-30 -20 -10 0 10 20 30

Frequency,Hz

Sweep Angle, Deg

CUF

Koo

20

25

30

35

40

45

50 55 60 65 70

f[Hz]

Mode 3Mode 4

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

50 55 60 65 70

Damping

V [m/s]

Mode 3Mode 4

Koo, K.N.

Aeroelastic Characteristics of Double-Swept Isotropic and Composite Wings,Journal of Aircraft, 38(2), 343-348, 2001

Petrolo M., Carrera E. and Demasi L.

An Advanced Unified Aeroelastic Formulation based on 1D Higher-Order Finite Elements,In: 15th International Forum on Aeroelasticity and Structural Dynamics 2011 - IFASD2011, 26-30 June 2011,Paris,2011


L B d 1D M d l LE
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Lagrange-Based 1D Models, LE

Lagrange-Based 1D Models: Displacement Unknowns

only

ux=L uxuy=L uyuz=L uz

L9 polynomials - Isoparametric

L =1

4(r2+r r)(s

2+ ...

Main TagsOpen

Cross-sections Thin-Walled

Local Model

Refinements

Shell-Like BCs Solid-Like BCs Composite


L B d 1D M d l LE
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A Typical Cross-Section Mesh

z

xy

b

h

t


Lagrange Based 1D Models LE
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Open Cross-Sections

z

xy

b

h

t

E. Carrera, M. Petrolo,

Refined Beam Elementswith only DisplacementVariables and Plate/ShellCapabilities,Submitted.

SolidBeam

ONLY

Lagrange1D models

can detect

such a

result!

Computational Cost

CUF 1D, 6500 DOFsSolid 130000 DOFs


Lagrange-Based 1D Models LE
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Shell-Like BCs

BCs on the lateral

edges

SolidBeam

Computational Cost

CUF 1D, 7000 DOFsSolid 43000 DOFs


Lagrange-Based 1D Models LE
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Lagrange Based 1D Models, LE

Solid-Like BCs

Clamped Point

Computational Cost

CUF 1D, 7500 DOFs

Solid 85000 DOFs

SolidBeam


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Lagrange Based 1D Models, LE

Arbitrary BCs: Panel Flutter by means of CUF 1D

Carrera E. , Zappino E. , Augello G. , Ferrarese A. and Montabone M.

Aeroelastic Analysis of Versatile Thermal Insulation Panels for Launchers Applications,In: ESA - 7th European Aerothermodynamics Symposium on Space Veicles2011,Bruges,Belgium.


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g g ,

Composite Structures: ESL and LW Descriptions

Equivalent Single Layer:

Taylor and Lagrange

s

Layer 1

s

Layer 2

s

Layer 3

s

Multilayer

MultilayeredStructure

3 X 3 FundamentalNucleus Array

Assembled ESLMatrix

Layer-Wise: Lagrange

s

Layer 1Layer 2

Layer 3

s

Multilayer

Multilayered

Structure

3 X 3 FundamentalNucleus Array

AssembledLW Matrix

s

s


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g g

Composite Longeron

Computational Cost

CUF, 12 L9 8000 DOFs

Solid 250000 DOFs

Timoshenko

-2.245e-006

-2.24e-006

-2.235e-006

-2.23e-006

-2.225e-006

-2.22e-006

-2.215e-006

-2.21e-006

-2.205e-006

CUF

SOLID

-2.85e-006

-2.8e-006

-2.75e-006

-2.7e-006

-2.65e-006

-2.6e-006

-2.55e-006

-2.5e-006

-2.45e-006

-2.8e-006

-2.75e-006

-2.7e-006

-2.65e-006

-2.6e-006

-2.55e-006

-2.5e-006

-2.45e-006

-2.4e-006

-2.35e-006

-2.3e-006

E. Carrera, M. Petrolo,

Refined One-Dimensional Formulations for LaminatedStructure Analysis,

AIAA Journal, In Press.


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The Very Last Extension: Component-Wise Approach, an

Outlook

CUF Lagrange 1D models can describe each Layer but they

can also model Fibers and Matrix.
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