Shear deformable doubly- and mono-symmetric composite I-beams

International Journal of Mechanical Sciences 53 (2011) 31–41

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

n Tel.:

E-m

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

Shear deformable doubly- and mono-symmetric composite I-beams

Nam-Il Kim n

Department of Civil and Environmental Engineering, Hanyang University, 17 Haengdang-dong, Seoungdong-gu, Seoul 133-791, South Korea

a r t i c l e i n f o

Article history:

Received 28 August 2009

Received in revised form

15 July 2010

Accepted 19 October 2010Available online 26 October 2010

Keywords:

Composite beam

Thin-walled

Stiffness matrix

Shear deformation

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijmecsci.2010.10.004

+82 31 290 7544; fax: +82 31 290 7548.

ail address: [email protected]

a b s t r a c t

A shear deformable beam element is developed for the coupled flexural and torsional analyses of thin-

walled composite I-beams with doubly- and mono-symmetric cross-sections. The present element

includes the transverse shear and the restrained warping induced shear deformation by using the first-

order shear deformation beam theory. Governing equations and force–displacement relations are derived

from the principle of minimum total potential energy. Then the explicit expressions for displacement

parameters are derived by applying the power series expansions of displacement components to

simultaneous ordinary differential equations. Finally, the element stiffness matrix is determined using

the force–displacement relations. In order to verify the accuracy and the superiority of the beam element

developed herein, the numerical solutions are presented and compared with the results obtained from the

isoparametric beam elements based on the Lagrangian interpolation polynomial, the detailed three-

dimensional analysis results using the shell elements of ABAQUS, and the solutions by other researchers.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The thin-walled composite I-beams have been used extensivelyin civil, marine and mechanical engineering as well as in aerospaceengineering. These structural components made of advancedcomposite materials are ideal for structural applications becauseof the high strength-to-weight and stiffness-to-weight ratios andtheir ability to be tailored to meet the design requirements ofstiffness and strength. These beams might be subjected to bendingand twisting types of loadings when used in above applications.Therefore, it becomes important to evaluate exactly their responsesunder these types of loadings. The first work with the contour-based cross-sectional analysis methods used in this study wasconducted by Vlasov [1]. Gjelsvik [2] presented an isotropic beamtheory identified with plate segments of the beam. The platedisplacements were related to the generalized beam displacementsthrough geometric consideration.

Up to the present, for the static analysis of composite beams, thefinite element method has been widely used because of itsversatility and accordingly a large amount of work [3–9] wasdevoted to the improvement of composite finite elements. Back andWill [3] developed the first-order shear deformable beam theoryand three different types of finite elements, namely, linear, quad-ratic and cubic elements to solve the governing equations. In theirstudy, they used reduced integration to alleviate shear locking. Lee[4] developed an analytical model to study the flexural behavior ofthin-walled composite beam with doubly symmetric I-section

ll rights reserved.

subjected to uniformly distributed vertical load. He presentedthe generalized displacements as a linear combination of the one-dimensional Lagrangian interpolation function. Also the two-noded C1 finite element of 8 DOF per node was developed bySubramanian [5] based on the higher order shear deformationtheory for flexural analysis of symmetric laminated compositebeams assuming a parabolic variation of transverse shear stressthrough the thickness of beams. Wu and Sun [6] derived the generalfinite element with 10 DOF per node for the thin-walled laminatedcomposite beams by applying the zero hoof strain assumption.Maddur and Chaturvedi [7] presented a modified Vlasov-type first-order shear deformation theory which can account for sheardeformations of open profile sections. And they simplified theirtheory for I-beams [8] as a special case and analyzed the deforma-tion response of I-sections made of orthotropic laminated compo-sites based on finite element procedure by using Lagrangeinterpolation function for the geometric coordinate variables andHermitian interpolation function for the unknown functions. Shiet al. [9] investigated the influence of the interpolation order ofbending strains on the solution accuracy of composite beamelement based on HSDT and presented a simple but accuratethird-order composite beam element using the assumed strainfinite element method.

The boundary element method (BEM), as an effective approachsolving the static problem of composite beams, has been used andsome works [10–12] was devoted to the improvement of beamelement in order to obtain the acceptable results. Mokos andSapountzakis [10] developed the BEM for the solution of the generaltransverse shear loading problem of composite beams of arbitrarycross-section. In their study, a stress function was introduced, whichfulfilled the equilibrium and compatibility equations and from which

www.elsevier.com/locate/ijmecsci

dx.doi.org/10.1016/j.ijmecsci.2010.10.004

mailto:[email protected]

dx.doi.org/10.1016/j.ijmecsci.2010.10.004

N.-I. Kim / International Journal of Mechanical Sciences 53 (2011) 31–4132

the transverse shear stresses at any interior point were obtained bydirect differentiation. Also the BEM for the nonuniform torsion ofcomposite bars of arbitrary constant cross-section subjected to aconcentrated or distributed twisting moment was developed bySapountzakis and Mokos [11]. Two boundary-value problems withrespect to variable along the beam angle of twist and to the warpingfunction with respect to the shear center were formulated. Sapount-zakis and Mokos [12] improved their previous study [11] byaccomplishing the evaluation of the secondary warping functionleading to the computation of the secondary shear stresses due towarping.

Considerable research efforts [13–17] have also been made bymany researchers to obtain analytical solutions for static analysis ofcomposite beams. Jung and Lee [13] presented a closed-formsolution for the static response of both symmetric and anti-symmetric lay-up I-beam with transverse shear coupling. Theanalysis included the effects of elastic coupling, shell wall thick-ness, transverse shear deformation, torsional warping, and con-strained warping. Pluzsik and Kollar [14] analyzed the effect ofshear deformation and restrained torsional warping for thin-walled beam with layups that are symmetrical or unsymmetrical,orthotropic or unorthotropic. Dufort et al. [15] proposed a simpleanalytical approach based on higher order theory that account forthe cross-section warping in beams under three-point bending.Song et al. [16] presented an analytical solution for the staticresponse of anisotropic composite I-beams loaded at their free endand highlighted the importance of a number of essential non-classical effects in composite I-beams. Kabir and Sherbourne [17]presented a theoretical study on the structural response of thin-walled open section beams made from mid-plane symmetric fiber-reinforced laminates based on the Vlasov-type linear hypothesis.

On the other hand, the exact element method based on thesolutions of ordinary differential equations was used as an effectivemethod solving the static problem of isotropic and compositebeams. The method of solution by power series that yields the exactstiffness matrix was first introduced by Eisenberger [18]. Murthyet al. [19] presented an accurate beam element with two node and 4DOF per node subjected to uniformly distributed load based on anexact solution of static governing equations. This beam theory wasbased on the higher order shear deformation theory.

The existing literatures reveal that even though a significantamount of research has been intensively conducted on the devel-opment of improved theories for the static analysis of compositebeam, to the best of author’s knowledge, there was no studyreported on the evaluation of stiffness matrix of thin-walledcomposite I-beams considering the fully coupled shear deforma-tion in the literature. It is well known that evaluating the stiffnessmatrix of the shear deformable thin-walled composite beam is verydifficult due to the complexities arising from coupling effects ofextensional, flexural, and torsional deformation.

Accordingly, the primary purpose of this study is to present theexplicit expression for the stiffness matrix of shear deformablethin-walled laminated composite I-beams with doubly- and mono-symmetric cross-sections. The important points of this study aresummarized as follows:

1.

ω ψ

A general shear deformable thin-walled composite beam theoryconsidering the effects of shear deformation and restrainedtorsional warping is developed for the flexural and torsionalanalyses.

2.
Governing equations and force–displacement relations arederived from the principle of minimum total potential energy.
3.

Fig. 1. Pictorial definitions of coordinates in thin-walled cross-section.

The element stiffness matrix of a shear deformable thin-walledcomposite beam is derived based on the power series expan-sions of displacement components.

4.
The isoparametric finite composite beam element is presentedbased on the Lagrangian interpolation polynomials.
5.
To demonstrate the accuracy and the validity of current study,the numerical solutions are presented and compared with thefinite element results using the isoparametric beam elementsand the shell elements of ABAQUS [20], and other researchers’available results. Additionally, the influences of shear deforma-tion, fiber orientation, and boundary conditions on the dis-placement and the twisting angle of beams are parametricallyinvestigated.
2. Derivation of governing equations and force–displacementrelations

2.1. Kinematics

The present theory uses two coordinate systems as shown inFig. 1: the first coordinate system is the orthogonal Cartesiancoordinate system (x,y,z), for which the y and z axes lie in the planeof the cross-section and the x-axis is parallel to the longitudinal axisof the beam; The second one is the contour coordinate S along theprofile of the section with its origin at any point O on the profilesection. The Z axis is normal to the middle surface of an elementand the s axis is tangent to the middle surface and is directed alongthe contour line of the cross-section. The point P denotes the poleaxis and r is theZ coordinate and q is the s coordinate of Q. Alsoo1 isthe rotation of cross-section about the axis of the beam andc is theangle tangent to the contour makes with the y-axis. In the presentbeam model, the following assumptions are made:

1.
The strains are assumed to be small. 2. The beam is linearly elastic and prismatic. 3. The contour of a cross-section does not deform in its plane. 4. The transverse shear strains and the warping shear are incor-
porated. It is assumed that they are uniform over the cross-sections.

5.
The normal stressss in the contour direction s is small comparedto the axial stress sx.
According to the assumption of in-plane rigid cross-section, themid-surface displacement components u and v at point Q in thecontour coordinate system can be expressed in terms of displace-ments Uy and Uz of the pole P in the y and z directions, respectively,and the rotation of cross-section o1 as follows:

uðs,xÞ ¼UyðxÞsincðsÞ�UzðxÞcoscðsÞ�o1ðxÞqðsÞ ð1aÞ

uðs,xÞ ¼UyðxÞcoscðsÞþUzðxÞsincðsÞþo1ðxÞrðsÞ ð1bÞ

The out-of-plane displacement w can be found from theassumption 4. The mid-surface shear strains in the contour can

N.-I. Kim / International Journal of Mechanical Sciences 53 (2011) 31–41 33

be expressed with respect to the transverse shear strains goyx, go

zx,and the warping shear strain go

f as follows:

gsxðs,xÞ ¼ goyx coscðsÞþgo

zx sincðsÞþgof rðsÞ ð2aÞ

gZxðs,xÞ ¼ goyx sincðsÞ�go

zx coscðsÞ�gofqðsÞ ð2bÞ

For each element of mid-surface, gsx can also be given from thedefinition of the shear strain as

gsxðs,xÞ ¼@u@xþ@w

@sð3Þ

By substituting Eq. (1b) into Eq. (3) and integrating with respectto s from the origin to an arbitrary point of the contour, the mid-surface axial displacement w can be obtained as

wðs,xÞ ¼UxðxÞ�o3ðxÞyðsÞþo2ðxÞzðsÞþ ff ð4Þ

where Ux represents the rigid body displacement along the x-axis;o2 ando3 denote the rotations about the y and z axes, respectively;f is the warping parameter and these parameters are given by

�o3 ¼ goyx�U0y ð5aÞ

o2 ¼ gozx�U0z ð5bÞ

f ¼ gof�o01 ð5cÞ

where the superscript ‘prime’ denotes differentiation with respectto x. The f in Eq. (4) is the sectional property called the sectorialarea or warping function, which can be determined by

fðsÞ ¼Z

rðsÞds ð6Þ

The displacement fields of any generic point on the profilesection can be expressed in terms of mid-surface displacements as

uðs,x,ZÞ ¼ uðs,xÞ ð7aÞ

uðs,x,ZÞ ¼ uðs,xÞ�ZLsðs,xÞ ð7bÞ

wðs,x,ZÞ ¼wðs,xÞ�ZLxðs,xÞ ð7cÞ

where Ls and Lx denote the rotations of a transverse normal aboutthe s and x, respectively. By using the assumption that the shearstrain gsZ should vanish at mid-surface, the function Ls can beobtained from the following relation:

gsZðs,xÞ ¼ gsZðs,x,ZÞ ¼ @u@Z þ

@u

@s¼�Lsþ

@u

@s¼ 0 ð8Þ

Similarly, the function Lx is determined by considering that themid-surface shear strain gZx is given by definition:

gZxðs,xÞ ¼ gZxðs,x,ZÞ ¼ @u

@xþ@w

@Z ¼ goyx sinc�go

zx cosc�gofq ð9Þ

By substituting Eq. (1a) into Eq. (9), the function Lx can bewritten as

Lx ¼o3 sincþo2 coscþ fq ð10Þ

Now the plate strain fields from the small displacement theoryof elasticity are given by diving the mid-surface strains and thecurvatures of the contour as follows:

esðs,x,ZÞ ¼ esðs,xÞ�Z ksðs,xÞ ð11aÞ

exðs,x,ZÞ ¼ exðs,xÞ�Z kxðs,xÞ ð11bÞ

gsxðs,x,ZÞ ¼ gsxðs,xÞ�Zksxðs,xÞ ð11cÞ

where

es ¼@u@s¼ 0, ks ¼

@Ls

@s¼ 0, ex ¼

@w

@x, kx ¼

@Lx

@x, ksx ¼

@Ls

@xþ@Lx

@s

ð12a2eÞ

In which es and ks are assumed to be zero, and ex, kx and ksx aremid-surface axial strain and biaxial curvatures, respectively.

The resulting strains can be obtained with respect to the beamstrain components from Eqs. (4), (10), and (11) as

ex ¼@w

@x�Zkx ¼U0x�o

03yþo02zþ f 0f�Zðo03 sincþo02 coscþ f 0qÞ

¼ eoxþðz�ZcoscÞkyþðyþZsincÞkzþðf�ZqÞkf ð13aÞ

gsx ¼ goyx coscþgo

zx sincþgof r�Zksx ð13bÞ

where eox ,ky,kz,kf, andksx are axial strain, biaxial curvatures in the

y and z directions, warping curvature, and twisting curvature in thebeam, respectively, defined as

eox ¼U0x, ky ¼�o03, kz ¼o02, kf ¼ f 0, ksx ¼ f�o01 ð14a2eÞ

2.2. Variational formulation

By applying the assumption that the shear stress in s–Zdirectionis small compared to the stresses in x and s–x directions, the strainenergy can be expressed as follows:

U ¼1

2

Z l

o

ZAðsxexþssxgsxÞdAdx ð15Þ

Substituting Eq. (13) into Eq. (15) and the variation of the strainenergy is expressed as

dU ¼

Z l

oF1deo

xþM2dkyþM3dkzþMfdkfþF2dgoyxþF3dgo

zx

�þT dgo

fþMt dksx

�dx ð16Þ

where F1 denotes the axial force; F2 and F3 are the shear forces in they and z directions, respectively; M2 and M3 are the bendingmoments about the y and z axes, respectively; Mf is the bimoment;T and Mt are the two contributions to the total twisting moment.These generalized forces and moments acting over the cross-section are related the stresses in the beam as

F1 ¼

ZAsx dsdZ ð17aÞ

M2 ¼

ZAsxðz�ZcoscÞdsdZ ð17bÞ

M3 ¼�

ZAsxðyþZsincÞdsdZ ð17cÞ

Mf ¼

ZAsxðf�ZqÞdsdZ ð17dÞ

F2 ¼

ZAssx coscdsdZ ð17eÞ

F3 ¼

ZAssx sincdsdZ ð17fÞ

T ¼

ZAssx r dsdZ ð17gÞ

Mt ¼�

ZAssxZdsdZ ð17hÞ

The variation of the potential energy due to external loads canbe written as follows:

dO¼�½p1dUxþp2dUyþp3dUzþm1do1þm2do2þm3do3þbdf �lo

ð18Þ


where p1, p2, and p3 are the concentrated loads and m1, m2, and m3

are the concentrated moments acting at the boundary; b is theconcentrated bimoment.

By using the principle of minimum total potential energy, thefollowing weak statement is obtained:

dP¼ dðUþOÞ

¼

Z l

ofF1dU0xþM2do02�M3do03þMfdf 0þF2dðU0y�o3Þ

þFzdðU0zþo2ÞþTdðf þo01ÞþMtdðf�o01Þgdx�½p1dUxþp2dUy

þp3dUzþm1do1þm2do2þm3do3þbdf �lo ¼ 0 ð19Þ

The relationship between the stress tensor and the plate strains canbe expressed in terms of the usual stiffness coefficients Qij for anyconstituent layer of the beam wall. The stress–strain relationship forthe kth lamina in an arbitrary s–x coordinate system is written as

sx

ss

ssx

8><>:

9>=>;

k

¼

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

264

375

k ex

es

gsx

8><>:

9>=>; ð20Þ

where Qij are the transformed reduced stiffness coefficients [21] andinclude the material properties of each layer. The stress–strain relation-ship can be simplified by adopting the zero hoof stress (ss¼0) or thezero hoof strain (es¼0) assumptions as

sx

ssx

( )k

¼Q�

11 Q�

16

Q�

16 Q�

66

" #kex

gsx

( )ð21Þ

where the condensed stiffness coefficients for the zero hoof stressassumption are given by

Q�

11 ¼Q11�Q

2

12

Q22

Q�

16 ¼ Q16�Q12Q26

Q22

, Q�

66 ¼Q66�Q

2

26

Q22

ð22a2cÞ

and for the zero hoof strain assumption, those are expressed as

Q�

11 ¼Q11, Q�

16 ¼ Q16, Q�

66 ¼Q66 ð23a2cÞ

Consequently, the constitutive equations for a laminate com-posite thin-walled member can be expressed by combiningEqs. (17), (21), and (13).

F1

M2

�M3

Mf

Mt

F2

F3

T

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼

E11 E12 E13 E14 E15 E16 E17 E18

E22 E23 E24 E25 E26 E27 E28

E33 E34 E35 E36 E37 E38

E44 E45 E46 E47 E48

E55 E56 E57 E58

E66 E67 E68

Symm: E77 E78

E88

2666666666666664

3777777777777775

U0xo02�o03

f 0

f�o01U0y�o3

U0zþo2

f þo01

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;ð24Þ

where Eij are laminate stiffnesses which depend on the cross-section of the composite.

Now, substituting Eq. (24) into the strain energy expressionterms in Eq. (19), the strain energy of the shear deformable thin-walled composite beam is as follows:

U ¼1

2

Z l

o½E11U02x þE22o022 þE33o023 þE44f 02þE55ðf�o01Þ

2

þE66ðU0y�o3Þ

2þE77ðU

0zþo2Þ

2þE88ðf þo01Þ

2þ2E12U0xo

02

�2E13U0xo03þ2E14U0xf 0þ2E15U0xðf�o

01Þþ2E16U0xðU

0y�o3Þ

þ2E17U0xðU0zþo2Þþ2E18U0xðf þo

01Þ�2E23o02o

03þ2E24o02f 0

þ2E25o02ðf�o01Þþ2E26o02ðU

0y�o3Þþ2E27o02ðU

0zþo2Þ

þ2E28o02ðf þo01Þ�2E34o03f 0�2E35o03ðf�o

01Þ�2E36o03ðU

0y�o3Þ

�2E37o03ðU0zþo2Þ�2E38o03ðf þo

01Þþ2E45f 0ðf�o01Þ

þ2E46f 0ðU0y�o3Þþ2E47f 0ðU0zþo2Þþ2E48f 0ðf þo01Þþ2E56ðU

0y�o3Þðf�o01Þþ2E57ðU

0zþo2Þðf�o01Þ

þ2E58ðf þo01Þðf�o01Þþ2E67ðU

0y�o3ÞðU

0zþo2Þ

þ2E68ðU0y�o3Þðf þo01Þþ2E78ðU

0zþo2Þðf þo01Þ�dx ð25Þ

The governing equations and the force–displacement relations canbe obtained by integrating the derivative of the varied quantities bypart and collecting the coefficients ofdUx,dUy,dUz,do1,do2,do3, anddf

in Eq. (19) as follows:

E11U00

xþE16U00

yþE17U00

zþðE18�E15Þo00

1þE12o00

2þE17o02�E13o00

3

�E16o03þE14f 00 þðE15þE18Þf 0 ¼ 0 ð26aÞ

E16U00

xþE66U00

yþE67U00

zþðE68�E56Þo00

1þE26o00

2þE67o02�E36o00

3

�E66o03þE46f 00 þðE56þE68Þf 0 ¼ 0 ð26bÞ

E17U00

xþE67U00

yþE77U00

zþðE78�E57Þo00

1þE27o00

2þE77o02�E37o00

3

�E67o03þE47f 00 þðE57þE78Þf 0 ¼ 0 ð26cÞ

ðE15�E18ÞU00

xþðE56�E68ÞU00

yþðE57�E78ÞU00

zþð2E58�E55�E88Þo00

1

þðE25�E28Þo00

2þðE57�E78Þo02þðE38�E35Þo00

3þðE68�E56Þo03

þðE45�E48Þf00 þðE55�E88Þf 0 ¼ 0 ð26dÞ

E12U00

x�E17U0xþE26U00

y�E67U0yþE27U00

z�E77U0zþðE28�E25Þo00

1

þðE57�E78Þo01þE22o00

2�E77o2�E23o00

3þðE37�E26Þo03

þE67o3þE24f 00 þðE25þE28�E47Þf 0�ðE57þE78Þf ¼ 0 ð26eÞ

E13U00

x�E16U0xþE36U00

y�E66U0yþE37U00

z�E67U0zþðE38�E35Þo00

1

þðE56�E68Þo01þE23o00

2þðE37�E26Þo02�E67o2�E33o00

3

þE66o3þE34f 00 þðE35þE38�E46Þf 0�ðE56þE68Þf ¼ 0 ð26fÞ

E14U00

x�ðE15þE18ÞU0xþE46U

00

y�ðE56þE68ÞU0yþE47U

00

z�ðE57þE78ÞU0z

þðE48�E45Þo00

1þðE55�E88Þo01þE24o00

2�ðE25þE28�E47Þo02

�ðE57þE78Þo2�E34o00

3þðE35þE38�E46Þo03

þðE56þE68Þo3þE44f 00�ðE55þ2E58þE88Þf ¼ 0 ð26gÞ

The above equations are most general form of the shear deformablethin-walled composite beam which are fully coupled implying that thebeam undergoes a coupled behavior involving bending, extension,twisting, transverse shearing, and restrained warping.

3. Stiffness matrix of shear deformable composite beam

3.1. Exact evaluation of displacement function

For the coupled flexural and torsional analyses of the thin-walled composite beams, the exact displacement functions areevaluated. For this, the displacement state vector consisting of 14displacement parameters is considered as follows:

d¼/Ux,U0x,Uy,U0y,Uz,U0z,o1,o01,o2,o02,o3,o03,f ,f 0STð27Þ

The solutions of seven displacement measures are taken as thefollowing infinite power series

Ux ¼X1n ¼ 0

anxn, Uy ¼X1n ¼ 0

bnxn, Uz ¼X1n ¼ 0

cnxn

o1 ¼X1n ¼ 0

dnxn, o2 ¼X1n ¼ 0

enxn, o3 ¼X1n ¼ 0

fnxn, f ¼X1n ¼ 0

gnxn

ð28a2gÞ


Substituting Eq. (28) into Eq. (26) and shifting the index ofpower of xn, the governing equations can be expressed in a matrixform as follows:

fanþ2,bnþ2,cnþ2,dnþ2,enþ2,fnþ2,gnþ2gT

¼ Znfan,anþ1,bn,bnþ1,cn,cnþ1,dn,dnþ1,en,enþ1,fn,fnþ1,gn,gnþ1gT

ð29Þ

where Zn is the 7�14 matrix function and then we put the initialintegration constant vector a as follows:

a¼ fao,a1,bo,b1,co,c1,do,d1,eo,e1,fo,f1,go,g1gT

ð30Þ

From Eq. (29), the following relation is obtained:

faiþ2,biþ2,ciþ2,diþ2,eiþ2,fiþ2,giþ2gT

¼ ZiNifai,aiþ1,bi,biþ1,ci,ciþ1,di,diþ1,ei,eiþ1,fi,fiþ1,gi,giþ1gT

ð31Þ

The terms for ai +2, bi +2, ci+ 2, di +2, ei + 2, fi + 2, and gi +2 converge tozero as i-N. The displacement state vector in Eq. (27) is expressedwith respect to the initial integration constant vector a by usingEqs. (28) and (31) as follows:

d¼Xna ð32Þ

where Xn denotes the 14�14 matrix function with the coefficientsof Ux, Uy, Uz,o1,o2,o3, and f.

Substituting the coordinates at two ends of member intoEq. (32), the nodal displacement vector Ue can be expressed as

Ue ¼Ha ð33Þ

The displacement state vector consisting of 14 displacementcomponents is evaluated from Eqs. (32) and (33) as follows:

d¼XnH�1Ue ð34Þ

It should be mentioned that Xn H�1 in Eq. (34) is the exactinterpolation function matrix not an approximate one. For evalua-tion of the displacement state vector, the calculation of thecoefficients by the recursive relations is continued, using atechnical computing program Mathematica [22], until the con-tribution of the next coefficient is less than an arbitrary smallnumber.

Fig. 2. Cross-section of the doubly symmetric I-beam.

3.2. Calculation of stiffness matrix

The element stiffness matrix of thin-walled composite beam isevaluated from the displacement state vector of the beam derivedin previous section. For this purpose, we consider the force–displacement relations which are derived from the strain energyin Eq. (25) and these relations are expressed in a matrix form asfollows:

f ¼ Sd ð35Þ

where S is the 7�14 matrix. By substituting Eq. (34) into Eq. (35),the nodal forces at two ends of element can be expressed withrespect to the nodal displacements as follows:

Fp¼�fð0Þ ¼�SXnð0ÞH

�1Ue ð36aÞ

Fq¼ fðlÞ ¼ SXnðlÞH

�1Ue ð36bÞ

Finally, the element stiffness matrix of thin-walled compositebeam is evaluated based on the linear relation between the nodaldisplacement parameters and the member forces as follows:

Fe ¼KUe ð37Þ

where

K¼�SXnð0ÞH

�1

SXnðlÞH�1

" #ð38Þ

It is noticeable that the element stiffness matrix in Eq. (38) isformed by the shape functions which are exact solutions of thegoverning equations. Therefore, the thin-walled composite beambased on the stiffness matrix developed by this study eliminatesdiscretization errors and is free from the shear and membranelocking.

4. Numerical examples

Numerical studies have been performed to illustrate the accu-racy and the validity of current shear deformable beam element forthe coupled flexural and torsional analyses of thin-walled compo-site I-beams with doubly- and mono-symmetric cross-sections.The solutions obtained from this study are compared with the finiteelement solutions using the two-noded isoparametric beam ele-ments [23] based on the reduced integration and the nine-nodelaminated shell elements (S9R5) by ABAQUS [20], and the solutionsby other researchers.

4.1. Doubly symmetric I-beam

Fig. 2 shows the doubly symmetric I-beam under consideration.The length of beam is 100 cm and all computations are carried outfor the glass-epoxy materials with the following elastic constants:E1¼53.78 GPa, E2¼E3¼17.93 GPa, G12¼G13¼8.96 GPa, G23¼3.45GPa, n12¼n13¼0.25, n23¼0.34. In which subscripts ‘1’ and ‘2’correspond to directions parallel and perpendicular to fibers,respectively. All constituent flanges and web are assumed to besymmetrically laminated with respect to its mid-plane. The totalthicknesses of top flange, bottom flange, and web are assumed to be0.208 cm and 16 layers with equal thickness are considered in twoflanges and web and the boundary conditions are: the clamped–free (C–F), simply–simply (S–S), clamped–simply (C–S), andclamped–clamped (C–C) boundary conditions.

The vertical displacements at free end of CF beam under the tipshear force �1 kN based on two different assumptions are pre-sented in Table 1 for the laminates considered. For comparison, theanalytical solutions by Park et al. [24] who neglected the sheardeformation effects and the finite element solutions by ABAQUSand 50 isoparametric composite beam elements are presented. ForABAQUS calculation, a total of 600 S9R5 shell elements (50 alongthe beam span and 12 through the cross-section) are used to obtain

Table 1Vertical displacement (cm) at free end of CF doubly symmetric I-beam (l/h¼20).

Stacking

sequence

Park et al. [24] ABAQUS FE analysis This study

ss¼0 es¼0

[15/�15]4S 4.521 4.618 4.610 4.611 4.422

[30/�30]4S 6.089 6.149 6.155 6.156 5.495

[45/�45]4S 8.795 8.833 8.854 8.855 7.492

[60/�60]4S 11.12 11.17 11.18 11.18 9.976

[75/�75]4S 12.07 12.16 12.15 12.16 11.65

[0/90]4S 6.093 6.213 6.201 6.201 6.106

Table 2Convergence of finite beam element solutions for vertical displacement (cm) at free

end of CF doubly symmetric I-beam (l/h¼20).

Stacking

sequence

No. of isoparametric beam elements

5 10 15 20 30 50

[15/�15]4S 4.5655 4.5994 4.6056 4.6078 4.6094 4.6102

[30/�30]4S 6.0951 6.1407 6.1492 6.1522 6.1543 6.1554

[45/�45]4S 8.7674 8.8334 8.8456 8.8498 8.8529 8.8545

[60/�60]4S 11.073 11.456 11.171 11.177 11.181 11.183

[75/�75]4S 12.035 12.125 12.142 12.148 12.152 12.154

[0/90]4S 6.1403 6.1860 6.1945 6.1975 6.1999 6.2007

0 10Fiber angle (degree)

0

2

4

6

8

10

1

3

5

7

9

Ver

tical

dis

plac

emen

t at m

id-s

pan

(×10

-3 c

m)

With shearWithout shear

3020 40 60 8050 70 90

Fig. 3. Variation of vertical displacement at mid-span for CC doubly symmetric

I-beam (l/h¼5).


0.4

0.8

1.2

1.6

2.0

2.4

0.6

1.0

1.4

1.8

2.2

Ver

tical

dis

plac

emen

t at m

id-s

pan

(×10

-1 c

m)


20 40 60 8030 50 70 90

Fig. 4. Variation of vertical displacement at mid-span for CC doubly symmetric

I-beam (l/h¼20).


the results. Also the convergence study for the isoparametric beamelement using the Lagrangian interpolation polynomials is per-formed and given in Table 2. It can be found from Tables 1 and 2 thatthe solutions developed by this study using only a single elementare in an excellent agreement with those using 50 isoparametricbeam elements. Also the correlation between results from thecurrent beam element with zero hoof stress assumption and thosefrom the ABAQUS analysis is seen to be excellent for the wholerange of fiber angle considered. It should be noted that both thisstudy and isoparametric beam formulation use the same elasticstrain energy. The difference of two methods is that this element isbased on the shape functions which satisfy the homogeneous formsof the equilibrium equations exactly. Therefore, it is possible toobtain the solutions though only a minimum number of beamelements are used. However, the solutions obtained from theisoparametric beam elements using the Lagrangian interpolationfunction which satisfies only displacement continuity at nodalpoint are approximate. Resultantly, as a number of isoparametricbeam element used in the beam structure increases, its solutionsconverge to those from the present beam element. It can also benoticed from Table 2 that the present approach exhibits betterresults than that of classical beam assumption which does notconsider shear deformation effects.

For CC beams with l/h¼5 and 20 under the vertical force �1 kNacting at mid-span, the variation of vertical displacement at mid-span is depicted in Figs. 3 and 4, respectively. From Figs. 3 and 4, it isinteresting to observe that the minimum displacements of beamsconsidering shear deformation are obtained at c¼331 for l/h¼5and at c¼01 for l/h¼20. On the other hand, for the case of beamsneglecting shear deformation, as the fiber angle increases, thecorresponding displacements increase for both l/h¼5 and 20. Asgraphs for the variation of displacement with respect to other threeboundary conditions are omitted due to space limitation, for beamsconsidering shear deformation with l/h¼5, the minimum displace-ments are obtained at c¼201 for both CF and SS beams. It isexhibited at c¼301 for CS beam. For longer span beams (l/h¼20),the displacements of beams with shear deformation increase withincrease of fiber angle change regardless of boundary conditions.

Also for beams without shear effect, the minimum displacementsare exhibited at c¼01 for all boundary conditions.

To investigate the shear deformation effect on the flexuralbehavior of beams with respect to the fiber angle change, therelative difference at mid-span between the displacement withshear deformation and without one are plotted in Figs. 5 and 6 forl/h¼5 and 20, respectively. It is observed from Figs. 5 and 6 that theshear effect is most significant at c¼01 and least significant atc¼571 for all boundary conditions. Also the shear effect for CFbeam is the same as that for SS beam.


20

40

60

80

100

30

50

70

90

(Uzw

s -U

zwos

)/Uzw

s × 1

00 (%

)

CFSSCSCC

20 40 60 8030 50 70 90

Fig. 5. Shear deformation effect on mid-span displacement for doubly symmetric

I-beams (l/h¼5).


0

10

20

30

5

15

25

(Uzw

s -U

zwos

)/Uzw

s ×

100

(%)

CFSSCSCC

20 40 60 8030 50 70 90

Fig. 6. Shear deformation effect on mid-span displacement for doubly symmetric

I-beams (l/h¼20).

Table 3Maximum twist angle (rad.) of SS doubly symmetric I-beam (l/h¼20).

Stacking

sequence

Lee and Lee [25] ABAQUS FE analysis This study

ss¼0 es¼0

[15/�15]4S 0.2073 0.2086 0.2075 0.2075 0.2074

[30/�30]4S 0.1563 0.1575 0.1565 0.1565 0.1560

[45/�45]4S 0.1396 0.1408 0.1397 0.1397 0.1389

[60/�60]4S 0.1571 0.1583 0.1572 0.1572 0.1560

[75/�75]4S 0.2080 0.2092 0.2082 0.2082 0.2074

[0/90]4S 0.2481 0.2498 0.2483 0.2483 0.2483


0.30

0.40

0.50

0.60

0.70

0.35

0.45

0.55

0.65

Twis

ting

angl

e (r

ad.)


20 40 60 8030 50 70 90

Fig. 7. Variation of twisting angle at right end for SS doubly symmetric I-beam.

under a twisting moment at right end (l/h¼5).


In Table 3, the maximum twist angles of SS beam under thetwisting moment 100 N cm acting at right end are presented forvarious laminated stacking sequences. In this case the DOFcorresponding to twisting angle at right end is released. FromTable 3, it can be found that the results obtained from the stiffnessmatrix by this study using a single element coincide with the finitebeam element solutions using the isoparametric beam elementsand are in good agreement with the solutions by ABAQUS and byLee and Lee [25] who did not consider shear effects. The variation oftwisting angle at right end is plotted in Fig. 7. It can be observedfrom Fig. 7 that the twisting angles decrease as the fiber angleincreases and the minimum twisting angle is exhibited at c¼451.

This is due to the fact that the stiffness components A66 and D66 inflanges and webs play an important role in torsional stiffness E55.Thus aligning the fiber angle around 451 leads to considerabledecrease of twisting angle. Also for CF, CS, and CC beams, theminimum twisting angles are obtained at c¼01. In Fig. 8, the sheardeformation effect on twisting angle of right end of beam under thetwisting moment acting at right end is presented. It is interesting tofind that the shear effect is negligible for SS beam and its effect forCF beam is the same as that for CS beam. It can also find that for CF,CS, and CC beams, the shear effects are minimum near 561 offiber angle.

Next the beams with l/h¼5 under the twisting moment 100 Ncm acting at mid-span are considered. The variation of twistingangle at mid-span for SS beam is depicted in Fig. 9. Unlike Fig. 7which shows the twisting angle at right end under the twistingmoment acting at right end, the twisting angle increases withincrease of fiber angle change for both beams with or without sheareffect. The shear effect with respect to fiber angle change ispresented in Fig. 10. It is observed from Fig. 10 that the sheareffect for SS beam is a little smaller than that for CF beam and itseffect for CS beam is larger than that for CF beam. Also theminimum shear effects occur near c¼571 for all boundaryconditions.


0.20

0.30

0.40

0.50

0.60

0.70

0.25

0.35

0.45

0.55

0.65

Twis

ting

angl

e (r

ad.)


20 40 60 8030 50 70 90

Fig. 9. Variation of twisting angle at mid-span for SS doubly symmetric I-beam.

under a twisting moment at mid-span (l/h¼5).


0

10

20

30

40

50

5

15

25

35

45

(ω1w

s -ω

1wos

)/ω1w

s × 1

00 (%

)

CFSSCSCC

20 40 60 8030 50 70 90

Fig. 10. Shear deformation effect on mid-span twisting angle for doubly symmetric

I-beams. under a twisting moment at mid-span (l/h¼5).

0 15Spanwise beam coordinate (cm)

0

0.002

0.004

0.006

0.008

0.01

0.001

0.003

0.005

0.007

0.009

Ben

ding

slo

pe (r

ad.)

This studyJung and Lee [13]NASTRAN (2D)Badir et al. [27]Experiment [26]

10 20 30 40 50 60 705 3525 45 55 65 75

Fig. 11. Bending slope along the beam length for doubly symmetric I-beam under a

tip shear force.


-4

0

4

8

12

16

20

-2

2

6

10

14

18

(ω1w

s -ω

1wos

)/ω1w

s × 1

00 (%

)

CFSSCSCC

20 40 60 8030 50 70 90

Fig. 8. Shear deformation effect on right end twisting angle for doubly symmetric

I-beams. under a twisting moment at right end (l/h¼5).


Now we have chosen the I-beam as a particular case from theexperimental study done by Chandra and Chopra [26]. The beamhas a length of 76.2 cm with the flange width of 2.54 cm and theheight of 1.27 cm and is clamped at its left end and warpingrestrained at both ends. The cross-section is made with graphite-epoxy (AS4/3501-6) material and its material constants are asfollows: E1¼141.9 GPa, E2¼E3¼9.79 GPa, G12¼G13¼6.0 GPa,G23¼6.0 GPa, n12¼n13¼0.42, n23¼0.5. The total thicknesses oftop and bottom flanges and web are 0.1016 cm and 8 layers withthe ply thickness of 0.0127 cm are considered in flanges and web.The I-beam is composed of top and bottom flanges with the un-

symmetric lay-up of [(0/90)2/(90/0)/15/15] and the web with thelay-up of [0/90]2S.

The bending slope along the beam length for the flexure–torsioncoupled I-beam under the tip shear force �1 N is plotted in Fig. 11.The present results are seen to be in excellent agreement withexperimental data obtained by Chandra and Chopra [26] as well asanalytical results of Jung and Lee [13], Badir et al. [27] and those ofNASTRAN 2D obtained by Jung and Lee [13]. The NASTRAN resultswere obtained by using 1200 (4-noded) CQUAD4 plate/shellelements. In addition, Fig. 12 shows the twisting angle along thebeam length of I-beam under the tip twisting moment 1 N cm. It is

0 5Spanwise beam coordinate (cm)

0.00

0.10

0.20

0.30

0.05

0.15

0.25

Twis

ting

angl

e (r

ad.)

This studyJung and Lee [13]NASTRAN (2D)Badir et al. [27]Experiment [26]

10 20 30 40 50 60 7015 25 35 45 55 65 75

Fig. 12. Twisting angle along the beam length for doubly symmetric I-beam under a

tip twisting moment.

Fig. 13. Cross-section of the mono-symmetric I-beam.

Table 4Stacking sequence of mono-symmetric I-beam.

Stacking

sequence

Top flange Bottom flange Web

QSISO [0/45/90/�45]2S [0/45/90/�45]3S [0/45/90/�45]S

ANG 0 [0]16 [0]24 [0]8

ANG15 [15/�15]4S [15/�15]6S [15/�15]2S

ANG30 [30/�30]4S [30/�30]6S [30/�30]2S

ANG45 [45/�45]4S [45/�45]6S [45/�45]2S

ANG60 [60/�60]4S [60/�60]6S [60/�60]2S

ANG75 [75/�75]4S [75/�75]6S [75/�75]2S

Table 5Vertical displacement (cm) at free end of CF mono-symmetric I-beam (l/h¼50).

Stacking

sequence

FE analysis ABAQUS This study

With shear Without shear ss¼0 es¼0

QSISO 14.254 14.216 14.25 14.255 13.401

ANG 0 8.2751 8.2223 8.278 8.2760 8.1046

ANG15 9.1446 9.1007 9.146 9.1455 8.7655

ANG30 12.289 12.257 12.28 12.290 10.959

ANG45 17.733 17.705 17.72 17.735 14.990

ANG60 22.408 22.376 22.40 22.410 19.980

ANG75 24.332 24.289 24.33 24.334 23.319

Table 6Vertical displacement (cm) at free end of CF mono-symmetric I-beam (l/h¼5).

Stacking

sequence

FE analysis ABAQUS This study

With shear Without shear ss¼0 es¼0

QSISO 1.8064 1.4216 1.850 1.8066 1.7212

ANG 0 1.3587 0.82223 1.426 1.3588 1.3417

ANG15 1.3583 0.91007 1.419 1.3583 1.3203

ANG30 1.5628 1.2256 1.608 1.5629 1.4298

ANG45 2.0705 1.7705 2.101 2.0706 1.7962

ANG60 2.5747 2.2376 2.606 2.5749 2.3319

ANG75 2.8770 2.4289 2.919 2.8772 2.7757


seen from Fig. 12 that the current predictions are correlated withanalytical solutions [13,27] and NASTRAN 2D analysis results. Theresults from this study are seen to be in good agreement withexperimental data near the left end, but the correlation is pooraround the beam tip. The reason for this discrepancy in theexperimental data may be due to the possible slip at the clampingend during the application of loading as well as other fabricationand/or testing error that might be occurred during the measure-ment process [13].

4.2. Mono-symmetric I-beam

In this example, the mono-symmetric I-beam as shown inFig. 13 is considered. The length of beam is 25 cm and the topand bottom flange widths are 3 and 4 cm, respectively. The heightof beam is 5 cm and the total thicknesses of top flange, bottomflange, and web are assumed to be 0.208, 0.312, and 0.104 cm,

respectively. The detailed stacking sequences of beam consideredare summarized in Table 4. The vertical tip displacement of CFbeam with l/h¼50 under the tip load �100 N is given in Table 5 andthat of CF beam with l/h¼5 under the tip load �10 kN is in Table 6.It is found from Tables 5 and 6 that the solutions from this studyusing only one element are in greatly agreement with those from 50isoparametric beam elements. And the results from the zero hoofstress assumption (ss¼0) are in close with ABAQUS solutions forboth beams with l/h¼50 and 5 for whole lamination schemesconsidered.

The shear effect on mid-span displacement of beams (l/h¼20)with various boundary conditions is presented in Fig. 14. In thiscase, the vertical load �1000 N is acted in mid-span of beam. FromFig. 14, the trend of variation of shear effect is similar to that in Fig. 6which depicts the shear effect for doubly symmetric I-beam. Theminimum shear effects are obtained at c¼571 for all boundaryconditions but the overall shear effects of mono-symmetricI-beams are larger than those of doubly symmetric I-beams asthe shear stiffness E77 of mono-symmetric I-beam is less than thatof doubly symmetric one.

Finally, the influence of zero hoof strain assumption (es¼0) onthe flexural behavior of CF mono-symmetric I-beam is investigatedwith respect to the fiber angle change, the top flange width, and the


0

4

8

12

16

20

24

2

6

10

14

18

22

(Cas

e1-C

ase2

)/Cas

e2×1

00 (%

)

20 40 60 8030 50 70 90

Fig. 15. Influence of zero hoof strain on vertical displacement at free end of CF

mono-symmetric I-beam. with respect to the fiber angle change (l/h¼50).

0.0 0.2b1 (cm)

0

10

20

5

15

25

(Cas

e1-C

ase2

)/Cas

e2×1

00 (%

)

0.4 0.8 1.2 1.6 2.00.6 1.0 1.4 1.8

Fig. 16. Influence of zero hoof strain on vertical displacement at free end of ANG45.

CF mono-symmetric I-beam with respect to b1 (l/h¼50).

0 10

10

12

14

16

18

20

9

11

13

15

17

19

(Cas

e1-C

ase2

)/Cas

e2×1

00 (%

)

ANG30ANG45ANG60

20 40 60 80 10030 50 70 90

Fig. 17. Influence of zero hoof strain on vertical displacement at free end of CF

mono-symmetric I-beam with respect to the slenderness ratio.


0

10

20

30

40

5

15

25

35

(Uzw

s -U

zwos

)/Uzw

s ×

100

(%)

CFSSCSCC

20 40 60 8030 50 70 90

Fig. 14. Shear deformation effect on mid-span displacement for mono-symmetric

I-beams (l/h¼20).


slenderness ratio. Fig. 15 shows the relative difference at mid-spandisplacement due to the assumption of es¼0 with respect to thefiber angle change. Also its influence for ANG45 beam is plotted inFig. 16 with respect to the top flange width. Here, Cases 1 and 2denote the displacements considering the zero hoof stress assump-tion and the zero hoof strain one, respectively. It can be found fromFigs. 15 and 16 that the influence of es¼0 has the maximum atc¼451 and its effect is regardless of the degree of mono-symme-tricity of cross-section. In Fig. 17, as the slenderness of beamincreases, the variation of influence of es¼0 is plotted for ANG30,ANG45, and ANG60 beams. It can be seen in Fig. 17 that the

influence of es¼0 keeps constant after l=h¼33 for beamsconsidered.

5. Conclusion

The stiffness matrix of shear deformable thin-walled compositebeams with doubly- and mono-symmetric cross-sections has beendeveloped based on power series expansion of displacementcomponents for the coupled flexural and torsional analyses. Thedisplacement fields are defined using the first-order shear


deformable beam theory considering the shear effects due to theshear forces and the restrained warping torsion. The governingequations and force–displacement relations are derived from theprinciple of minimum total potential energy. This stiffness matrixmethod uses the exact shape functions of the beam that arerepresented by converging infinite series. An advantage, which isoften overlooked but may be more important, is that the presentmethod can provide benchmark results when compared with otherresults obtained by the finite element or other approximatemethods.

Through numerical examples, the theory developed by thisstudy is validated comparing the flexural and torsional responseswith those by the finite element beam model using the Lagrangianinterpolation polynomials for all displacement parameters, thedetailed three-dimensional analysis results using the shell ele-ments of ABAQUS, and the available other researchers’ results.Good correlation is achieved for various beams with differentcross-sections, lamination schemes, and loading conditions con-sidered in this paper. The conclusions drawn from this study are asfollows:

(1)
Displacements and twisting angles obtained from the presentstiffness matrix method coincide with those from a largenumber of isoparametric beam elements and are in an excel-lent agreement with the ABAQUS results.
(2)
For shorter span beam with shear effect under the verticalforce, the minimum displacements are obtained with differentfiber angle with respect to the boundary conditions, but forlonger span beams, those are exhibited at c¼01 for allboundary conditions.
(3)
For shorter and longer span beams under the vertical force, theshear effect is most significant at c¼01 and least significant atc¼571 for both doubly- and mono-symmetric cross-sectionsand for all boundary conditions. Also the shear effect for CFbeam is the same as that for SS beam.
(4)
Even though the slenderness ratio of beam is the same, theoptimal fiber angle for the twisting angle varies with respect tothe location where the twisting moment is acted.
(5)
The influence of zero hoof strain has the maximum at c¼451and its effect is regardless of the degree of mono-symmetricityof cross-section. Also it remains constant after criticalslenderness ratio.
It is judged that the present numerical procedure provides arefined method for not only the evaluation of the stiffness matrix ofshear deformable thin-walled composite beam but also generalsolutions of simultaneous ordinary differential equations of thehigher order. Also this thin-walled composite beam elementeliminates discretization errors and is free from the shear lockingsince the displacement state vector satisfies the homogenous formof the simultaneous ordinary differential equations.

Acknowledgements

This research was supported by World Class University (WCU)program through the National Research Foundation of Korea

funded by the Ministry of Education, Science and Technology(R32-2008-000-20042-0).

References

[1] Vlasov VZ. Thin-walled elastic beams. 2nd ed.. Washington, DC: NationalScience Foundation; 1961.

[2] Gjelsvik A. The theory of thin-walled bars. New York: Wiley; 1981.[3] Back SY, Will KM. Shear-flexible thin-walled element for composite I-beams.

Engineering Structures 2008;30:1447–58.[4] Lee J. Flexural analysis of thin-walled composite beams using shear-deform-

able beam theory. Composite Structures 2005;70:212–22.[5] Subramanian P. Flexural analysis of symmetric laminated composite beams

using C1 finite element. Composite Structures 2001;54:121–6.[6] Wu X, Sun CT. Simplified theory for composite thin-walled beams. AIAA Journal

1992;30:2945–51.[7] Maddur SS, Chaturvedi SK. Laminated composite open profile sections: first

order shear deformation theory. Composite Structures 1999;45:105–14.[8] Maddur SS, Chaturvedi SK. Laminated composite open profile sections: non-

uniform torsion of I-sections. Composite Structures 2000;50:159–69.[9] Shi G, Lam KY, Tay TE. On efficient finite element modeling of composite beams

and plates using higher-order theories and an accurate composite beamelement. Composite Structures 1998;41:159–65.

[10] Mokos VG, Sapountzakis EJ. A BEM solution to transverse shear loading ofcomposite beams. International Journal of Solids and Structures 2005;42:3261–87.

[11] Sapountzakis EJ, Mokos VG. Nonuniform torsion of composite bars byboundary element method. Journal of Engineering Mechanics 2001;127:945–53.

[12] Sapountzakis EJ, Mokos VG. Warping shear stresses in nonuniform torsion ofcomposite bars by BEM. Computer Methods in Applied Mechanics andEngineering 2003;192:4337–53.

[13] Jung SN, Lee JY. Closed-form analysis of thin-walled composite I-beamsconsidering non-classical effects. Composite Structures 2003;60:9–17.

[14] Pluzsik A, Kollar LP. Effects of shear deformation and restrained warping on thedisplacements of composite beams. Journal of Reinforced Plastics and Com-posites 2002;21:1517–41.

[15] Dufort L, Drapier S, Grediac M. Closed-form for the cross-section warping inshort beams under three-point bending. Composite Structures 2001;52:223–46.

[16] Song O, Librescu L, Jeong NH. Static response of thin-walled composite I-beamsloaded at their free-end cross section: analytical solution. Composite Struc-tures 2001;52:55–65.

[17] Kabir MZ, Sherbourne AN. Shear strain effects on flexure and torsion of thin-walled pultruded composite beams. Canadian Journal of Civil Engineering1999;26:852–68.

[18] Eisenberger M. Exact static and dynamic stiffness matrices for general variablecross section members. AIAA Journal 1990;28:1105–9.

[19] Murthy MVVS, Mahapatra DR, Badarinarayana K, Gopalakrishnan S. A refinedhigher order finite element for asymmetric composite beams. CompositeStructures 2005;67:27–35.

[20] ABAQUS. Standard user’s manual, Version 6.1. Hibbit, Kalsson & Sorensen Inc.;2003.

[21] Jones RM. Mechanics of composite material. 2nd ed.. New York: Taylor &Francis; 1999.

[22] Wolfram S. Mathematica, a system for doing mathematics by computer.Redwood City, CA: Addison-Wesley; 1988.

[23] Bathe KJ. Finite element procedures. Englewood Cliffs, NJ: Prentice-Hall; 1996.[24] Park YS, Kwan HC, Shin DK. Bending analysis of symmetrically laminated

composite open section beam by Vlasov-type thin-walled beam theory. KoreaSociety of Civil Engineers Journal 2000;1:125–41.

[25] Lee J, Lee SH. Flexural–torsional behavior of thin-walled composite beams.Thin-Walled Structures 2004;42:1293–305.

[26] Chandra R, Chopra I. Experimental and theoretical analysis of compositeI-beams with elastic couplings. AIAA Journal 1991;29:2197–206.

[27] Badir AM, Berdichevsky VL, Armanios EA. Theory of composite thin-walledopened-cross-section beams. In: Proceedings of the 34th AIAA structures,Structural Dynamics, and Materials Conference, LaJolla, CA, 19-22 April 1993.p. 2761–70.

Shear deformable doubly- and mono-symmetric composite I-beams

Documents

Transcript of Shear deformable doubly- and mono-symmetric composite I-beams