ARTICLE IN PRESS

Thin-Walled Structures 47 (2009) 1363–1372

Contents lists available at ScienceDirect

Thin-Walled Structures

0263-82

doi:10.1

� Corr

E-m

avalido@

journal homepage: www.elsevier.com/locate/tws

Finite element analysis of thin-walled composite laminated beams withgeometrically nonlinear behavior including warping deformation

J.E. Barradas Cardoso a,�, Nuno M.B. Benedito b, Anıbal J.J. Valido b

a Instituto Superior Tecnico, Departamento de Engenharia Mecanica, Av. Rovisco Pais, 1049-001 Lisboa, Portugalb Escola Superior de Tecnologia, Instituto Politecnico de Setubal, Campus do IPS, Estefanilha, 2914-508 Setubal, Portugal

a r t i c l e i n f o

Article history:

Received 7 December 2007

Received in revised form

14 July 2008

Accepted 11 March 2009Available online 27 May 2009

Keywords:

Thin-walled beams

Composites

Nonlinear structures

Finite elements

31/$ - see front matter & 2009 Elsevier Ltd. A

016/j.tws.2009.03.002

esponding author. Tel.: +351 218417725.

ail addresses: barradas@ist.utl.pt (J.E.B. Cardo

est.ips.pt (A.J.J. Valido).

a b s t r a c t

A finite element model for structural analysis of composite laminated thin-walled beam structures, with

geometrically nonlinear behavior, including torsion warping deformation, is presented. A general

continuum formulation considering the updated Lagrangean procedure and a generalized displacement

control method, are used to describe the deformation of the structure. The beam cross-section geometry

is discretized by quadratic isoparametric finite elements to determine its bending-torsion properties.

The structural discretization is performed throughout three-dimensional two-node Hermitean finite

beam elements, with seven degrees-of-freedom per node. Several applications are presented,

addressing the influence of lamina orientation on the structural behavior.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The demanding for more economic structures is generallytranslated in the production of lighter structures and it hasbrought up the needs of the utilization of composite materials andthe consideration of large structural deformations. Today, the useof composite materials is generalized to a large variety ofstructures due to their good strength/weight and stiffness/weightratios when compared to conventional materials.

Thin-walled beams are also applied to various structures, dueto their large axial stiffness and large bending stiffness associatedto a small weight. These features may be reinforced if compositematerials are used in the beam construction. Since thin-walledbeams are slender, stability problems should be considered intheir design. On the other hand, in the case of thin-walled opencross-sections, the torsion stiffness is lower and the warpingeffects may be relevant.

In a standard finite element analysis of stiffened plate and shellstructures it is necessary of a large number of degrees-of-freedom indiscretizing them. Alternatively, some of these structures may beanalyzed by a two-phase strategy: (i) finite element analysis ofcross-section geometry to determine the bending and torsionproperties of the structure as a beam; and (ii) finite element analysisof the structure based on a general thin-walled beam theory.

ll rights reserved.

so),

Thin-walled beams made of isotropic materials have beenstudied by many authors [1–16]. Some works have been devotedto the analysis of the flexural-torsion properties of thin-walledcomposite beams [17–19]. In [20,21], closed-form solutions forsolid and thin-walled composite beams were presented.

Vlasov’s theory was extended for the bending and twisting ofthin-walled composite beams with open cross-section made fromsymmetric fiber-reinforced laminates [22,23] and for compositebeams with arbitrary geometric and material sectional properties[24,25]. A geometrically nonlinear theory for composite thin-walled beams accounting for finite flexural displacements andarbitrarily large twist angle has been presented in [26]. In [27], ananalytical model to the analysis of flexural, torsion and flexural-torsion buckling of a thin-walled I-section composite beamsubject to axial load is developed. The bending-torsion behaviorof thin-walled composite beams has also been studied in [28–31].In [25,32,33], thin-walled composite beams based on the varia-tional asymptotical beam sectional (VABS) approach had beenstudied. This approach is used to rigorously split the geometricallynonlinear three-dimensional (3D) elasticity problem into a linear2D cross-section analysis and a nonlinear 1D beam analysis,instead of invoking ad hoc kinematic assumptions. In [34], animproved stability theory for the spatially coupled stabilityanalysis of thin-walled composite beams with arbitrary lamina-tion is derived, and a numerical method evaluates the exactelement stiffness matrix. In [35,36,38,39], the influence oftransverse and torsion shear deformation on buckling loads ofcomposite beams is studied. In [37], the study of this influence isextended for post-buckling analysis.

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The present work presents a finite element model for structuralanalysis of composite laminated thin-walled beam structures, withgeometrically nonlinear behavior, including post-critical behaviorand warping deformation. To achieve this goal, an updatedLagrangean description for the nonlinear structural deformation isused, and the structural analysis is performed by a displacement-controlled continuation method [40]. This method has been usedsuccessfully by the authors for nonlinear analysis of beam structuresas well as truss structures, including post-critical behavior [41–43].In this paper, the thin-walled beams are composed by composite flatpanels with symmetric and balanced stacking sequences. The cross-section bending-torsion properties are integrals based on the cross-section geometry, on the warping function and on the individualstiffness of the panels that constitute the cross-section, whichdepend by their side on the laminate configuration and laminamaterial distribution. Though the laminates present symmetric andbalanced stacking sequences, these sequences may change indifferent laminates, i.e., the cross-section thin walls can havedifferent lay-up distributions provided they keep symmetry andbalance. The cross-section geometry is discretized by quadraticisoparametric finite elements to determine its bending-torsionproperties [18]. The structural discretization is formulated through-out three-dimensional two-node Hermitean finite beam elements.Numerical examples are performed on open cross-section beams,where the structural behavior and particularly the critical loadmagnitude related to the lamina orientation are investigated.

2. Model assumptions

The present work addresses to straight thin-walled compositebeams of arbitrary cross-section with geometrically nonlinearbehavior.

The beams are made from an assembly of thin flat-layeredpanels, each panel corresponding to a laminate (Fig. 1). A beamreferential system (x, y, z) for the member geometry and cross-section properties for which the y and z axes lie in the plane of thecross-section and the x-axis is parallel to the longitudinal axis ofthe beam, a local panel coordinate system (x, n, s), wherein the n-axis is normal to the middle surface of a panel and the s-axis isdirected along the cross-section contour middle line, and amaterial axes system (1, 2) for each ply, are used. The point (ys,zs) define the location of the shear center with respect to theelastic center.

The following assumptions are adopted:

(1)

x

z

the contour does not deform in its own plane;

(2) the warping function is dependent on the material arrange-

ment of the laminates;

y

xi

1

2

Laminate i

ni

s

t

si

x, u0

y, v0

z, w0

n

s

E

S ys

zs

�x

Fig. 1. Thin-walled composite beam coordinate systems.

(3)

the shearing deformation of the middle surface is zero in eachpanel;

(4)

each panel behaves as a thin plate. This implies that theKirchoff hypothesis is valid for each plate element;

(5)

each laminated panel is symmetric and balanced; (6) the strains are small but large displacements and rotations are

allowed; and

(7) the effect of the shear deformation due torsion and bending is

negligible.

3. Nonlinear structural analysis

Using the updated Lagrangean formulation to describe themotion of the continuum [44], and matrix and tensor notations,we get the linearized incremental virtual work equation at theload level t asZðtS � dt�L þ

tr � dt�NÞ dtV ¼

Ztf � dudtV þ

ZtT

0� dudtGT (1)

and the incremental constitutive law

tS ¼ Ete (2)

where tS, f, u and tT0 are, respectively, the stress, body force,

displacement and prescribed surface traction increments, tr is theCauchy stress measure, E is the tangent constitutive tensor, andthe Green-Lagrange strain tensor increment is given as

te ¼ teL þ teN ; teL ¼ tbðuT Þ; teN ¼

12tZðuT ;uT Þ (3)

with tb and tg operators defined as

tbðÞ ¼1

2tr ðÞ� �

þ tr ðÞ� �T

þ tr ðÞ� �

tr tuT� �� �T

þ tr tuT� �� �

tr ðÞ� �Tn o

tZða; bÞ ¼1

2tr að Þ� �

tr bð Þ� �T

þ tr bð Þ� �

tr að Þ� �Tn o

(4)

In the foregoing equations tu is the displacement field at theload level t,r ¼ (@/@y, @/@z), is the space gradient operator, d refersto arbitrary variation of the state fields and ‘ � ’ refers to thestandard tensor product. The left superscript and the leftsubscript, stand respectively for the configurations where thequantities are measured and referred to. If the configurationswhere the quantities are measured and referred to are the same,only the superscript is used.

To define the load–deflection path, a generalized displacementcontrol method [40] has been implemented.

4. Cross-section analysis

By using a thin-walled beam theory, the points of the cross-section have three properties: their coordinates (y, z) on the planeof the section and a warping function that depends on thosecoordinates o(y, z). In the case of thin-walled beams made of thinflat laminated panels, the properties of the corresponding laminateat each point have also to be considered. These distributed pointproperties are integrated over the cross-section area, giving thefollowing cross-section bending-torsion-independent properties

EA ¼

ZEn dA; ESy ¼

ZEnz dA; ESz ¼

ZEny dA;

EIyy ¼

ZEnz2 dA; EIyz ¼

ZEnyz dA; EIzz ¼

ZEny2 dA;

EJo ¼

ZEnodA; EJyo ¼

ZEnyodA; EJzo ¼

ZEnzodA;

EJoo ¼

ZEno2 dA (5)

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J.E.B. Cardoso et al. / Thin-Walled Structures 47 (2009) 1363–1372 1365

where commas stand for partial derivatives. If the coordinates y

and z are central and the warping function is referred to the shearcenter and normalized, then

ESy ¼ ESz ¼ EJo ¼ EJyo ¼ EJzo ¼ 0 (6)

From the properties presented in Eq. (5), let us distinguish thewarping property EJoo, as an independent property, the combina-tion of others to define the St. Venant torsion stiffness as

GJ ¼

ZGn y2

þ z2þ yo;z � zo;y

� �dA ¼

ZGn r �roð Þ � rdA

¼

ZGn y2 þ z2 þ yo;z � zo;y� �

dA ¼

ZGn r �roð Þ � rdA (7)

where o and o are the warping functions referred, respectively, tothe shear and elastic centers and

r ¼ r � rS ¼ ðy;�zÞ; y ¼ y� yS; z ¼ z� zS (8)

where r ¼ (z, �y) is a vector of coordinates with respect to theelastic center.

The location of the elastic center is given as

yE ¼ESz

EA; zE ¼

ESy

EA(9)

and the location of the shear center is given as

yS ¼ �EIzzEJzo � EIyzEJyo

EIyyEIzz � EI2

yz

; zS ¼EIyyEJyo � EIyzEJzo

EIyyEIzz � EI2

yz

(10)

We may note the warping properties EJyo and EJzo in Eq. (10) todetermine the position of the shear center, are dependent on thewarping function referred to the elastic center.

In the determination of the bending cross-section properties,the equivalent membrane longitudinal modulus Ex

m is used, sincefor thin-walled beams the normal stresses due to bendingcorrespond essentially to a membrane effect in the laminate.According to the classical torsion theory of isotropic thin-walledopen cross-section beams, the shear stresses within the elementthickness varies linearly from a positive to a negative value at thesurfaces and zero at the mid-plane. In this case, the appliedtorsion moment is resisted by an equilibrating system ofdistributed moments in the panels, representing a bending modeof response. Hence, to calculate the torsion properties of the openparts of the cross-section, the corresponding shear modulus valueshould be the bending equivalent value Gxs

b. For closed cross-sections, the stress distribution is assumed constant throughoutthe element thickness, corresponding to a membrane mode ofresponse. So, to calculate the torsion properties of the closed partsof the cross-section, we must use the membrane equivalent shearmodulus Gxs

m. For a symmetric laminate, these equivalentmodules are given as

En� Em

x ¼1

ta11; Gn� Gb

xs ¼12

t3d66

or Gn� Gm

xs ¼1

ta66(11)

where t is the laminate thickness, and the compliance coefficientsaij and dij are obtained by inverting the laminate constitutiveequations. The superscripts m and b indicate the membrane andbending modes, respectively [45].

To determine the warping function o(y, z), one uses the stressvariational equilibrium equationZðsxy;y þ sxz;zÞdodA ¼ 0 (12)

that after integration and substitution of the stress displacementrelations of elasticity for torsion leads to

dP �Z

Gnro � rddA�

ZGnr � rdodA ¼ 0 (13)

For numerical implementation of the Eq. (13), the cross-sectiongeometry is discretized with quadratic isoparametric finiteelements. The authors’ code is used to solve in a generalizedform, the warping function and the bending-torsion properties ofeither solid, either thin-walled open or closed composite cross-sections. It extends the code implemented in [6,7] for isotropicmaterials.

5. Beam element model

The structural discretization is formulated throughout three-dimensional two-node Hermitean finite beam elements withseven degrees-of-freedom per node. To uncouple the governingequations for torsion and flexure, two reference lines are used: thecentroidal axis (line of elastic centers E) for stretching andbending components, and the line of shear centers S for shear,twisting and warping components (Fig. 2). The incrementalvectors of nodal displacements and nodal forces are, respectively,

U ¼ fu1 v1 w1 yx1 yy1 yz1 y0x1 u2 v2 w2 yx2 yy2 yz2 y0x2 gT

(14)

and

F ¼ f Fx1 Fy1 Fz1 Mx1 My1 Mz1 B1 Fx2 Fy2 Fz2 Mx2 My2 Mz2 B2 gT

(15)

In Eq. (14), uk, vk and wk are the element k-node incrementaldisplacement components of the elastic center, respectively, in thex, y and z directions; yxk, yyk and yzk, are the element k-nodeincremental rotations about x, y and z axis, respectively; and theright upper coma stands for the derivative with respect to x. In Eq.(15), Fxk, Fyk and Fzk are the element k-node incremental forcecomponents, respectively, in the x, y and z directions; Mxk, Myk andMzk are the element k-node incremental moments around x-, y-and z-axis, respectively; and Bk is the incremental bimoment atthe node k of the element.

With reference to Fig. 1, the incremental displacement of anypoint of the cross-section is given by

uxðx; y; zÞ ¼ uðxÞ � yyzðxÞ þ zyyðxÞ þoðy; zÞwðxÞuyðx; y; zÞ ¼ vðxÞ � ðz� zSÞyxðxÞ

uzðx; y; zÞ ¼ wðxÞ þ ðy� ySÞyxðxÞ (16)

where w represents the intensity of warping. Assuming the effectof the shear deformation due torsion and bending is negligible,thus

w ¼ �yx;x ¼ �dyx=dx and yy ¼ �wS;x ¼ �dwS=dx; yz ¼ �vS;x

¼ �dvS=dx (17)

The beam Green strain measure at load level t can be given bythe vector

t �

Þ

¼t�xx 2t�xy 2t�xz

n oT(18)

and its increment is given by the Eqs. (3) and (4) as

t e¼t eLþt eN �

t�xx

2t�xy

2t�xz

8><>:

9>=>; ¼

texx

texy

texz

8><>:

9>=>;þ

tZxx

tZxy

tZxz

8><>:

9>=>; (19)

ARTICLE IN PRESS

z

y

100

20

40

E43

.8

6.2

2000

x

y

A B

z

Fz = 0.001F

F

t

Fig. 3. Cantilever beam, cross-section geometry and finite element mesh.

x

v1

v2

u1

�'

w1

w2

u2

E1

E2

z

y

S2

S1

x

E1

E2

z

y

S2

S1

tB1

tMy

tFx1

tFy1

tFz1

tMy2 tFy2

tFz2

tFx2

tMz2

tMx2

tB2

tMx1

tMz1

x1

�'x2

�y1

�y2

�z2

�x2

�x1

�z1

1

Fig. 2. Nodal displacements and nodal forces for the beam element.

J.E.B. Cardoso et al. / Thin-Walled Structures 47 (2009) 1363–13721366

where

t�xx ¼@ux

@xþ@ux

@x

@tux

@xþ@uy

@x

@tuy

@xþ@uz

@x

@tuz

@xþ

1

2

@ux

@x

� 2"

þ@uy

@x

� 2

þ@uz

@x

� 2#

2t�xy ¼@ux

@yþ@uy

@x

þ@ux

@x

@tux

@yþ@tux

@x

@ux

@yþ@uy

@x

@tuy

@yþ@tuy

@x

@uy

@yþ@uz

@x

@tuz

@y

þ@tuz

@x

@uz

@yþ

@ux

@x

@ux

@yþ@uy

@x

@uy

@yþ@uz

@x

@uz

@y

�

2t�xz ¼@ux

@zþ@w

@x

þ@ux

@x

@tux

@zþ@tux

@x

@ux

@zþ@uy

@x

@tuy

@zþ@tuy

@x

@uy

@zþ@uz

@x

@tuz

@z

þ@tuz

@x

@uz

@zþ

@ux

@x

@ux

@zþ@uy

@x

@uy

@zþ@uz

@x

@uz

@z

� (20)

The increments of the stress components are obtained,considering the linear part of the strain incremental vector inEq. (2), as

tSxx

tSxy

tSxz

8><>:

9>=>; ¼

E� 0 0

0 G� 0

0 0 G�

264

375

texx

texy

texz

8><>:

9>=>; (21)

where E* ¼ Exm and G* ¼ Gxs

m (if the point belongs to the closedpart of the cross-section) or G* ¼ Gxs

b (if the point belongs to thecross-section open part), are the equivalent laminate modulusdefined in Section 3.

Using the finite element modeling, it is adopted a lineardisplacement field for u(x) and a cubic displacement field for theother generalized displacements. Then, the Eq. (1) of incrementalvirtual work becomes

ðtKL þtKNLÞU ¼ tP (22)

where tKL and tKNL are, respectively, the linear and the nonlinear(or geometric) parts of the; tangent stiffness matrix at load level t,

tP is the incremental vector of external forces and U is theincremental nodal displacement vector.

6. Numerical examples

6.1. Thin-walled asymmetric cross-section cantilever beam

A thin-walled asymmetrical channel-section cantilever beamshown in Fig. 3 is considered. To calculate the bending-torsioncross-section properties, the cross-section is discretized by 36eight node quadratic isoparametric finite elements. The beam isdiscretized by eight finite elements of equal length.

In a first case, an isotropic section of thickness t ¼ 5 mm isconsidered. The material is given by E ¼ 300 MPa and G ¼ 115

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MPa. The location of the shear center was calculated asyS ¼ �26.3 mm and zS ¼ �12.8 mm. As the cross-section isasymmetric, the column instability will occur in a flexural-torsionmode. The critical load was evaluated and compared with thetheoretical one given by [8]

Fy ¼p2EIyy

K2y L2

; Fz ¼p2EIzz

K2z L2

; Fy ¼A

IpsGJ þ

p2EIoo

K2yL2

!(23)

The Fig. 4 shows the load–deflection curves, where thetheoretical value Fcr ¼ 14.02 N has been considered. One may

Fig. 4. Load vs. displacement curves for asymmetrical channel-section isotr

-0.5

2000

x

y

A B

z

Fz = 0.001F

F

Fig. 5. Negative perturbation force and load vs. twisting curve for a

3

6

9

12

15

18

F [

N]

2000

x

y

A B

z

Fz = 0.001F

F

Fig. 6. Positive perturbation force and load vs. twisting curve for a

observe the agreement between this value and the calculatedvalue. In [46], the curve F=Fcr � yxB

was calculated for the samebeam. We may observe the agreement between its trajectory in[46] and the trajectory shown in Fig. 4.

In a second case the same structure was considered, but withthe cross-section formed by three equal laminates with eightlayers [y/�y/y/�y]S and total thickness t ¼ 1 mm. The materialproperties are E1 ¼140 GPa, E2 ¼ 10 GPa, n12 ¼ 0.3 and G12 ¼ 5GPa.

Considering the laminate configuration [45/�45/45/�45]S anda negative perturbation force as indicated in Fig. 5, the critical

opic cantilever beam: (a) F=Fcr � yxB, (b) F/Fcr�wB/L and (c) F/Fcr�vB/L.

0

30

60

90

120

150

180

θx_B [rad]

F [

N]

-0.4 -0.3 -0.2 -0.1 0.0

symmetrical channel cross-section composite cantilever beam.

0

0

0

0

0

0

0

0.00 0.05 0.10 0.15 0.20 0.25

θx_B [rad]

symmetrical channel cross-section composite cantilever beam.

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J.E.B. Cardoso et al. / Thin-Walled Structures 47 (2009) 1363–13721368

load obtained is Fcr ¼ 147.79 N. The load vs. twisting curve, F � yxB,

is also presented in Fig. 5.Considering a positive perturbation force, the critical load

obtained is 149.87 N. This case is represented in Fig. 6. Comparingthe load vs. twisting curves of Figs. 5 and 6, we observe the post-critical behavior of the beam is quite different.

The linear critical buckling load was also obtained byeigenvalue analysis using ANSYS code, as Fcr ¼ 147.24 N, wherethe same beam was meshed by 800 laminated shell elements(SHELL99). The Fig. 7 shows the correspondent buckling mode. Wemay note that the beam tip twisting rotation is positive, so thiscritical load value should be compared with the one obtainedwhen a positive perturbation force was considered.

DISPLACEMENTSTEP = 1SUB = 1FREQ = 147.242DMX = 1.405

JUL 7 200615:24:41

C assimetrico composito (eigenbuckling analysis)

1

Z

Fig. 7. Buckling mode obtained by eigenvalue analysis using ANSYS for

asymmetrical channel cross-section composite cantilever beam.

0100200300400500600

0Lamina orientation [°]

Crit

ical

Loa

d [N

]

15 30 45 60 75 90

Fig. 8. Critical load vs. lamina orientation and laminate equivalent membrane long

composite cantilever beam.

50

2000

x

y

A

B

z

Fz

Fig. 9. Channel cross-sect

The influence of the lamina orientation on the critical load wasalso investigated. The Fig. 8 shows the variation of the critical loadand the variation of the laminate equivalent membranelongitudinal modulus with the lamina orientation. The criticalload variation follows nearly the same trajectory as the variationof the laminate equivalent membrane modulus. This may indicatethe mode corresponding to the critical load is essentially a flexuralmode.

6.2. Channel cross-section cantilever beam

This problem is concerned with the bending and twisting ofthe channel-section cantilever beam shown is Fig. 9. The beam issubject to a transverse force Fz applied at the elastic center of thefree end. To calculate the bending-torsion cross-sectionproperties, the cross-section is discretized by 44 eight nodequadratic isoparametric finite elements. The beam is discretizedby eight finite elements of equal length.

Two laminates identified in the Fig. 9 as 1 and 2, with fourlayers [y/�y]S and total thickness t ¼ 3 mm, forms the cross-section. The material properties E1 ¼ 48.3 GPa, E2 ¼ 19.8 GPa,n12 ¼ 0.27, G12 ¼ 8.96 GPa, corresponding to S2-glass/epoxy areused. Two material architectures are considered: a unidirectional01 lay-up and an angle-ply lay-up [45/�45]s. Four cases corre-sponding to combinations of these two material architectures areconsidered, as indicated in Table 1. The elastic center location d,the shear center location ys, the torsion stiffness value GJ and thewarping property value EJoo are also presented in the Table 1. Dueto application of the force Fz, the beam undergoes bending in theplane zx, coupled bending–twisting. Fig. 10 shows the free endcurves load vs. twisting for each case.

We may observe that the unidirectional lay-up in web andflanges (case 1) leads to the smaller torsion stiffness and to thelargest warping property; otherwise, the angle-ply lay-up in bothweb and flanges (case 2) has the opposite effect. Comparing to the

020406080

100120140160

Ex_

m [G

Pa]

0Lamina orientation [°]

15 30 45 60 75 90

itudinal modulus vs. lamina orientation for asymmetrical channel cross-section

z

yE

50

d

ys

S

t = 3

1

1

2

ion cantilever beam.

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unidirectional lay-up (case 1), the cross-section torsion stiffnessand warping property are more affected by the lamina orientationof laminate 1 (case 4) than by the lamina orientation of thelaminate 2 (case 3). The twisting rotation reaches the greatervalue in the case 1 and the smaller value in the case 4. However,we may note that in the case 4 the torsion moment is smaller thanin the other cases, since the elastic center and the shear center arecloser.

6.3. Thin-walled cruciform cross-section beam

In this example, the critical loads corresponding to flexural andtorsion instability modes of the cruciform cross-section beam are

x

yz

2000

A B

Fy = 0.001F

Fx

t =

Fig. 11. Cruciform cross-section cantilever beam and loading to e

Fig. 10. Curves load vs. twisting of the free end for channel cross-section cantilever

beam.

Table 1Results for the channel cross-section cantilever beam.

Case Laminates lay-up d (mm) ys (mm) GJ (N mm2) EJoo (N mm4)

1 1-[01/01]s 32.18 37.01 0.11598E+8 0.21899E+13

2-[01/01]s

2 1-[451/�451]s 32.18 37.01 0.17216E+8 0.11275E+13

2-[451/�451]s

3 1-[01/01]s 29.34 41.05 0.13342E+8 0.19628E+13

2-[451/�451]s

4 1-[451/�451]s 35.83 31.37 0.15432E+8 0.13214E+13

2-[01/01]s

determined. Two laminates identified in the figures as 1 and 2,with eight layers [y/�y/y/�y]S and total thickness t ¼ 2 mm, formthe cross-section. The material properties are E1 ¼140 GPa,E2 ¼ 10 GPa, n12 ¼ 0.3, G12 ¼ 5 GPa.

To calculate the bending-torsion cross-section properties, thecross-section is discretized by 44 eight node quadratic isopara-metric finite elements. The beam is discretized by eight hermiteantwo-node beam finite elements of equal length.

In a first case, for a cantilever beam, to determine the criticalload corresponding the bending mode, a lateral perturbation forceFy ¼ 0.001F is considered (Fig. 11). Considering the configurationof both laminates as [45/�45/45/�45]S, the theoretical value ofthe critical load obtained using the Eq. (23), where Kz is theequivalency factor related to the Euler column, is Fcr ¼ 394.67 N.

y100

60

z

2

2

1

valuate the critical load corresponding to the flexural mode.

0.0

0.5

1.0

1.5

2.0

-1.0VB/L

F/Fc

r

-0.8 -0.6 -0.4 -0.2 0.0

Fig. 12. Load vs. displacement curve for the cruciform cross-section cantilever

beam corresponding to the bending mode.

0

500

1000

1500

2000

2500

3000

3500

0Lamina orientation [°]

Fcr [

N]

Laminate 1 (Laminate 2 at 0°)Laminate 2 (Laminate 1 at 0°)

15 30 45 60 75 90

Fig. 13. Critical load vs. lamina orientation of laminates 1 and 2 for the cruciform

cross-section cantilever beam corresponding to the bending mode.

ARTICLE IN PRESS

x

yz

A B Fx

2000

t = 2

100

60

Mx = 0.001Fmm 2

1

y

z

Fig. 14. Cruciform cross-section cantilever beam and loading to evaluate the critical load corresponding to the torsional mode.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00θx [rad]

F/Fc

r

0.03 0.06 0.09 0.12 0.15

Fig. 15. Load vs. twisting curve for the cruciform cross-section cantilever beam

corresponding to the buckling torsion mode.

0

5

10

15

20

25

0Lamina orientation [°]

Crit

ical

Loa

d [K

N]

15 30 45 60 75 90

Fig. 16. Critical load vs. lamina orientation of both laminates simultaneously for

the cruciform cross-section cantilever beam.

0.0E+002.0E+064.0E+066.0E+068.0E+061.0E+071.2E+071.4E+071.6E+07

0Lamina orientation [°]

GJ

[N.m

m^2

]

15 30 45 60 75 90

Fig. 17. Torsion stiffness vs. lamina orientation of both laminates simultaneously

for the cruciform cross-section.

0.0E+001.0E+092.0E+093.0E+094.0E+095.0E+096.0E+097.0E+098.0E+099.0E+091.0E+10

0Lamina orientation [°]

EJω

ωω

ω [N

.mm

^4]

15 30 45 60 75 90

Fig. 18. Warping property vs. lamina orientation of both laminates simultaneously

for the cruciform cross-section.

J.E.B. Cardoso et al. / Thin-Walled Structures 47 (2009) 1363–13721370

The Fig. 12 shows the relation between the normalized appliedload and the normalized tip displacement in the perturbationforce direction. We may verify that the critical load value obtainedcorresponds to the theoretical one.

The influence of the lamina orientation of the laminates 1 and2 on the critical load is shown in Fig. 13. The dashed curverepresents the variation of the critical load vs. lamina orientationof laminate 1, keeping the lamina orientation of laminated 2 equalto 01. The continuous curve represents the variation of the criticalload vs. lamina orientation of laminate 2, keeping the laminaorientation of laminated 1 equal to 01.

From Fig. 13, we verify that when laminate 1 is unidirectional(01), the critical load varies between Fcr ¼ 3110 N, for an unidirec-tional (01) configuration of the laminate 2, and Fcr ¼ 228 N, whenconfiguration of laminate 2 is [90/90/90/90]S.

Keeping laminate 2 unidirectional (01), the influence of laminaorientation of the laminate 1 on the critical load is imperceptible(dashed line). The value of the critical load is approximatelyFcr ¼ 3110 N for any configuration of laminate 1. This was apredictable situation, since the laminate 2 has a great contributionto the bending stiffness of the cross-section, in contrast tolaminate 1, which contribution is insignificant.

In a second case, also for a cantilever beam, to determine thecritical load corresponding the torsion mode, a perturbationmoment Mx/mm ¼ 0.001F is considered, as indicated in Fig. 14.Considering the configuration [45/�45/45/�45]S for bothlaminates, the theoretical critical load value obtained using Eq.(23), where Ky is the equivalency factor related to the Eulercolumn to torsion and EIps is the polar bending stiffness related tothe shear center, is Fcr ¼ 21568 N.

The Fig. 15 presents the variation of the tip twisting rotationwith the applied load normalized by the theoretical value. We

ARTICLE IN PRESS

x

yz

1000

A B

Fy = 0.001F

Fx

C

1000

A B

Fy = 0.001F

Fx

C

Fig. 19. Cruciform cross-section beam and loading to evaluate the critical load corresponding to the bending mode; (a) clamped–clamped and (b) clamped–simple

supported.

Fig. 20. Load vs. displacement curves for the cruciform cross-section beam corresponding to the bending mode; (a) clamped–clamped and (b) clamped–simple supported.

J.E.B. Cardoso et al. / Thin-Walled Structures 47 (2009) 1363–1372 1371

may verify that the critical load value obtained corresponds to thetheoretical one.

The variation of the critical load vs. lamina orientation of bothlaminates simultaneous is presented in Fig. 16.

The Eq. (23) show the influence of the torsion stiffness andwarping property terms on the beam critical load. Taking theproperties of the unidirectional (01) laminate as reference in thiscase, where the beam length is L ¼ 2000 mm, we may concludethat the torsion stiffness GJ ¼ 2:14� 106 N mm2 has muchmore influence on the beam critical load than the termðp2EJooÞ=ðK

2yL2Þ ¼ 0:0058� 106 N mm2 corresponding to the

warping property.The Figs. 17 and 18 show, respectively, the variation of the

torsion stiffness and warping property with the laminaorientation of both laminates simultaneously. Comparing Figs.16 and 17, we conclude that the variation of the critical loadfollows the same trajectory as the variation of the torsion stiffness.

Now, the critical load and the post-buckling path aredetermined for a clamped–clamped (Fig. 19a) and aclamped–simple supported beam (Fig. 19b), considering theconfiguration [45/�45/45/�45]S for both laminates. Thenonlinear response is represented, respectively, in Fig. 20a andb, where the curves were normalized with respect to thetheoretical critical load values obtained by the Eq. (23) and tothe mid-span displacement. These values are Fcr ¼ 6314.7 N andFcr ¼ 3321.8 N for the clamped–clamped beam and for theclamped–simple supported beam, respectively. The figures showa good agreement between numerical and theoretical values.

7. Concluding remarks

A finite element model for structural analysis of compositelaminated thin-walled beam structures with geometrically non-

linear behavior, based on an updated Lagrangean formulation, hasbeen presented. Warping deformation is included. To define theload–deflection path, a generalized displacement control methodhas been implemented. The thin-walled cross-sections aremodeled as assemblies of flat symmetric laminated panels andtheir bending-torsion properties are defined in terms of the cross-section geometry, warping function and properties of thecorresponding laminate at each point. The cross-section geometryis discretized by quadratic isoparametric finite elements todetermine its bending-torsion properties. The structural beammodeling is formulated throughout three-dimensional two-nodehermitean finite beam elements.

The warping function is dependent not only on the cross-section geometry but as well as on the cross-section materialdistribution.

The critical load for an isotropic cross-section beam has beencalculated and is in agreement with its theoretical value. Theinfluence of the lamina orientation on the structural behavior aswell as on the critical load of composite laminated beams hasbeen studied. As one expected, the critical load of laminatecomposite beam is strongly dependent on the lamina orientation,hence this orientation is a fundamental parameter to thesestructures. Also, the bending mode critical load variation vs.lamina orientation follows nearly the same trajectory as thevariation of the laminate equivalent membrane longitudinalmodulus. On the other hand, the torsion mode critical loadvariation vs. lamina orientation follows nearly the same trajectoryas the variation of the St. Venant torsion stiffness.

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