1

Optimal Design of Thin-Walled Laminated

Beams with Geometrically Nonlinear

Behaviour

A. J. Valido, J. Barradas Cardoso, P. P. Moita

EngOpt 2014 - 4th International Conference on Engineering Optimization

Lisbon, 8-11 September 2014

2

• Finite element model for design optimization of composite laminated

thin-walled beam structures with geometrically nonlinear behaviour

accounting warping and post-critical behaviour;

• Objective

• Thin flat panels corresponding to a symmetriclaminates;

• Eight node quadratic isoparametric finite elementsin order to determine its bending-torsion properties;

• The structural response is obtained with basis on thegeneral thin-walled beam theory;

• For sensitivity analysis the cross-section is modeledthroughout design elements that geometricallycoincide with the laminates;

• Adjoint variable method

• First forward finite difference method

• Design optimization performed throughout nonlinear programming techniques;

• Lamina orientation and cross-section global geometry are considered as design variables.

1

2

ti

xi

yi

zi

qi

Laminate i

y

z

s

n

E

SzS

yS

Y

Z

yE

zE

Laminate i

Master point K

K

3

• Bending-torsion cross-section properties

• Eight node quadratic isoparametric finite elements in order to

determine its bending-torsion properties;

GA G dA= ò yGS G z dA= ò = òzGS G y dA

2yyGI G z dA= ò 2

zzGI G y dA= ò EJ E dAf = fòyEJ E y dAf = fò zEJ E z dAf = fò 2EJ E dAff = fò

, ,y yGJ G dAf = fò , ,

z zGJ G dAf = fò , ,zy zGJ G y dAf = fò

, ,yz yGJ G z dAf = fò ( )2

, , ,y y yGJ G dAf f = fò ( )2

, , ,z z zGJ G dAf f = fò

EA E dA= ò yES E z dA= ò zES E y dA= ò2

yyEI E z dA= ò yzEI E yz dA= ò 2zzEI E y dA= ò

dA dy dz=

y y yGk GA G dA·= Ñf Ñfò z z zGk GA G dA·= Ñf Ñfò pGk GI G dAw w w·= Ñf Ñfò

y

z

• Are given as integrals based on the cross-section geometry, on

the warping functions and on the individual stiffness of the

panels.

4

Saint-Venant torsional stiffness

Location of elastic centerLocation of shear center

( )2 2 , ,E E E

z yGJ G y z y z dA G dA= + + f - f = - fò ò

z

E

ESy

EA= y

E

ESz

EA=

For bending properties11

1m

xE Eta

® º =

For torsion properties

3

66

12b

xyopen part G Gt d

ì ® º =ïïïíï

66

1m

xyclosed part G Gta

íïï ® º =ïî

are the equivalent elastic constants

of the laminate, where the compliance coefficients

aij and dij are obtained by inverting the classical

laminate constitutive equations. The superscripts

“m” and “b” indicate the membrane and bending

modes, respectively.

b

xyopen part G Gt d

® º =

minat

2

f f-= -

-

zz yzz y

S

yy zz yz

EI EJ EI EJy

EI EI EI

2

f f-=

-

yy yzy z

S

yy zz yz

EI EJ EI EJz

EI EI EI

f = f+ -ES Sy z z y

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• The contour of the cross section does not deform in its own plane;

• The shearing deformation of the middle surface is zero in each panel;

• Each panel behaves as thin plate. This implies that the Kirchoff hypothesis is valid for each plate element;

• Strains are small but large displacements and rotations are allowed.

The beams are made from an assembly of thin flat-layered panels, each panel

corresponding to a symmetric laminate

Displacement at a generic point:

• Assumptions and kinematics

1

2

ti

xi

yi

zi

qi

Laminate i

y

z

s

n

E

SzS

yS

Y

Z

yE

zE

Laminate i

0

( )

( )

¢= - q + q -fq

= - - q

= + - q

z y x

S S x

S S x

u u y z

v v z z

w w y y

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Updated Lagrangean formulation

Incremental decompositions

Linearized incremental equilibrium equation

Linearization

• Nonlinear structural analysis

0

+D +D +D +D

+D +D

× ×

×

d = d

+ d G

ò òò

et t t t t t t t t t

t t t

t t t t t

t T

d V d V

d

S f u

T u

P

P

P

u

Configuration at

time t=0

Configuration

at time t

Configuration at

time t+Dt

Z

Y

X

tu

t+Dt u

tx

0x

t+Dt x

Linearized incremental equilibrium equation

Linearization

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• Two-node Hermitean finite beam element

• Seven degrees of freedom per node

• Linear displacement field is adopted for u and a

cubic displacement field is adopted for the other

displacements

Incremental vectors of nodal displacements and nodal forces

discretization

• Beam element model

B

Fx

u

x

y

z

Mx

Mz

My

Fy

Fz

qy

qz

qxq 'x

v

w

{ }1 1 1 x1 y1 z1 x1 2 2 2 x2 y2 z2 x2u v w θ θ θ θ u v w θ θ θ θ¢ ¢=T

U

{ }x1 y1 z1 x1 y1 z1 1 x2 y2 z2 x2 y2 z2 2= F F F M M M B F F F M M M BT

F

( )+ =t t

L NL tK K U P

( ) 0× × × ×d + d = d + d Gò ò òe s et t t t

t t L t N t t Td V d V dS f u T u

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• Generalized displacement control method (Yang and Shieh, 1990)

Generalized stiffness parameter:

1/ 2i 1 1 1D l =D l GSP

1 1T 1 1

1 1T 1

ˆ ˆ

ˆ ˆ-

D D=D Di i

GSPU U

U U

Lo

ad

Displacement 0

‘+’ Di $U1

Di-1 1$U

Di-1 1$U

‘+’

‘+’

‘-’

‘+’

‘-’

Di-1 1$U

Di-1 1$U

Di-1 1$U

Di-1 1$U

Di $U1

Di $U1

Di $U1

Di $U1 Di $U1

0

1

0 Displacement

GS

P

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• Design sensitivity analysis

• For sensitivity analysis the cross-section is modeledthroughout design elements that geometricallycoincide with the laminates;

Total design variation

An objective or a constraint may be represented by

a functional ,

Y Y Yd = d + dYddA dr ds

Y

Adjoint variable method, may be derived by stating:

• To calculate the sensitivities w.r.t to lamina orientation.

Wa is the state equation where the virtual fields are replaced by adjoint fieldsY W= - ®aL

First forward finite difference method

• To calculate the sensitivities w.r.t. to cross section geometry.

1

2

ti

xi

yi

zi

qi

Laminate i

y

z

s

n

E

SzS

yS

Y

Z

yE

zE

Laminate i

Master point K

K

10

1, 2,…, 14 – to define the cross-section

I, J, K, P, Q – Master points

The position of the master points is treated as dv

12 3

45 6

7 8 9 10

11 12 13 14

I

J

K

P

Q

• Changing cross-sectional geometry

K

J

I

P

Q

A general design arbitrary configuration defined by:

- Master points

- Medium lines orientation unit vectors

( , )= - -IJ J I J I IJy y z z Le

2 2( ) ( )= - + -IJ J I J IL y y z z

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2000

x

y

A B

z

4.8 kN

• Numerical Examples

Example 1 – Beam with asymmetric channel section

• The cross-section is discretized by 40 eight node quadratic isoparametric finite elements.

• The beam is discretized by 8 finite elements of equal length.

• Two symmetric laminates with 8 layers of equal thickness h=0. 625 mm,

(total thickness t=5 mm) E1 = 140 GPa; E2 = 10 GPa; n12 = 0.3; G12 = 5 GPa

[ ]S

0 / 45 / 45 / 0-

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Problem:

min

. . 500

. . , , ,

B

P P J J

A

s t w mm

d v Y Z Y Z

£

-

13

14

2000

x

y

A B

z

Fy=0.001F

F

• The cross-section is discretized by 64 isoparametric quadratic 8

node finite elements.

• The beam is discretized by 8 finite elements of equal length.

• Three symmetric laminates with ten layers of equal thickness

h=0.4 mm.

Example 2 – Column with asymmetric thin-walled open section

[ ]S

0 / 45 / 45 / 45 / 45- -

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Problem 1:

• The nonlinear critical load constraint is

replaced by a displacement constraint;

5300crF N=

min

. . 7000 12

. . ,

cr B

P P

A

s t F N v mm

d v Y Z

= ® £

-

16

7000crF N=

5300crF N=

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Problem 2:

• The nonlinear critical load

constraint is replaced by a

displacement constraint;

1 1 2 2 3 3

2,3 4,5 2,3 4,5 2,3 4,5

max

. . 7000 12

. . , , , , ,

cr B

GJ

s t F N v mm

d v

= ® £

- q q q q q q

7000crF N=

5300crF N=

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• A finite element model for optimal design of composite laminated thin-walled beam structures, with

geometrically nonlinear behavior, including post critical behavior and accounting for Warping

deformation;

• Based on an Updated Lagrangean formulation;

• To define the load-deflection path, a generalized displacement control method has been

implemented;

• The thin-walled cross-sections are modeled as assemblies of flat symmetric laminated panels and

their bending-torsion properties are defined in terms of the cross-section geometry, warping

function and properties of the corresponding laminate at each point;

• An eight-node quadratic two-dimensional isoparametric finite element modeling is used in order to

determine the bending-torsion properties;

• The structural discretization is formulated throughout three-dimensional two-node Hermitean

finite beam elements, with 7 d.o.f. per node;

• Final Comments

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• Design elements are used for design sensitivity analysis and optimization. Geometrically these

elements coincide with the laminates;

• Lamina orientations and cross-section global geometry are considered as design variables;

• The nonlinear critical load constraint was replaced by a displacement constraint;

• The last example shows that the critical load of laminate composite beam is strongly dependent on

the lamina orientation, hence this orientation is a fundamental parameter to these structures.