Mechanics Research Communications 35 (2008) 497–516

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Mechanics Research Communications

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

Flexural-torsional buckling analysis of composite beams by BEMincluding shear deformation effect

E.J. Sapountzakis *, J.A. DourakopoulosInstitute of Structural Analysis and Seismic Research, School of Civil Engineering, National Technical University of Athens, Zografou Campus,GR-157 80 Athens, Greece

a r t i c l e i n f o

Article history:Received 30 November 2007Received in revised form 6 June 2008Available online 18 June 2008

Keywords:Flexural-torsional bucklingTimoshenko beamNonuniform torsionWarpingFlexuralBarComposite beamTwistBoundary element methodShear deformation

0093-6413/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.mechrescom.2008.06.007

* Corresponding author. Tel.: +30 2107721718; faE-mail addresses: cvsapoun@central.ntua.gr (E.J.

a b s t r a c t

In this paper, a boundary element method is developed for the general flexural-torsionallinear buckling analysis of Timoshenko beams of arbitrarily shaped composite cross-sec-tion. The composite beam consists of materials in contact, each of which can surround afinite number of inclusions. The materials have different elasticity and shear moduli withsame Poisson’s ratio and are firmly bonded together. The beam is subjected to a compres-sive centrally applied load together with arbitrarily axial, transverse and/or torsional dis-tributed loading, while its edges are restrained by the most general linear boundaryconditions. The resulting boundary value problem, described by three coupled ordinary dif-ferential equations, is solved employing a boundary integral equation approach. Besidesthe effectiveness and accuracy of the developed method, a significant advantage is thatthe method can treat composite beams of both thin and thick walled cross-sections takinginto account the warping along the thickness of the walls, while the displacements as wellas the stress resultants are computed at any cross-section of the beam using the respectiveintegral representations as mathematical formulae. All basic equations are formulated withrespect to the principal shear axes coordinate system, which does not coincide with theprincipal bending one in a nonsymmetric cross-section. To account for shear deformations,the concept of shear deformation coefficients is used. Six coupled boundary value problemsare formulated with respect to the transverse displacements, to the angle of twist, to theprimary warping function and to two stress functions and solved using the analog equationmethod, a BEM based method. Several beams are analysed to illustrate the method anddemonstrate its efficiency. The significant influence of the boundary conditions and theshear deformation effect on the buckling load are investigated through examples withgreat practical interest.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Elastic stability of beams is one of the most important criteria in the design of structures subjected to compressive loads.This beam buckling analysis becomes much more complicated in the case the cross-section’s centroid does not coincide withits shear center (asymmetric beams), leading to the formulation of the flexural-torsional buckling problem. Also, compositestructural elements consisting of a relatively weak matrix material reinforced by stronger inclusions or of materials in con-tact are of increasing technological importance. Steel beams or columns totally encased in concrete, fiber-reinforced mate-rials or concrete plates stiffened by steel beams are most common examples. Moreover, unless the beam is very ‘‘thin” the

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x: +30 2107721720.Sapountzakis), jdour@mail.ntua.gr (J.A. Dourakopoulos).

498 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

error incurred from the ignorance of the effect of shear deformation is substantial, and an accurate analysis requires its inclu-sion in it.

The first published work on flexural-torsional buckling appeared in 1899 by Michell (1899) and Prandtl (1899) for thin,rectangular, solid beams. Since then, the flexural-torsional buckling problem of thin-walled homogeneous beams (based onthe assumptions of the thin tube theory) (Vlasov, 1961; Timoshenko and Gere, 1961; Rao and Carnegie, 1970; Mei, 1970;Hodges and Peters, 1975; Reissner, 1979; Milisavljevic, 1995; Hodges, 2001) or of symmetrical cross-section beams ignoringwarping (Orloske et al., 2006) has been studied by many researchers noting that Reissner (1979) was the first who includedtransverse shear in his analysis. Moreover, the flexural-torsional buckling problem in the case of composite beams of thin-walled or laminated cross-sections has also been examined ignoring (Lee and Kim, 2001; Sapkás and Kollár, 2002) or takinginto account (Kollár, 2001; Machado and Cortínez, 2005; Yu et al., 2002; Cortínez and Piovan, 2006) shear deformation effectemploying again the assumptions of the thin tube theory and using either the ‘refined models’ or shell stress resultants. Tothe authors’ knowledge publications on the solution to the general flexural-torsional buckling analysis of Timoshenko beamsof arbitrarily shaped composite cross-section do not exist.

In this investigation, an integral equation technique is developed for the solution of the aforementioned problem. The com-posite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials havedifferent elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The beam is subjected to acompressive centrally applied load together with arbitrarily axial, transverse and torsional distributed loading, while its edgesare restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three cou-pled ordinary differential equations, is solved employing the concept of the analog equation (Katsikadelis, 2002). According tothis method, the three coupled fourth order hyperbolic partial differential equations are replaced by three uncoupled onessubjected to fictitious load distributions under the same boundary conditions. All basic equations are formulated with respectto the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetriccross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary valueproblems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping functionand to two stress functions and solved using the analog equation method (Katsikadelis, 2002), a BEM based method. Theessential features and novel aspects of the present formulation compared with previous ones are summarized as follows:

(i) The proposed method can be applied to beams having an arbitrary composite constant cross-section and not to a nec-essarily thin-walled one.

(ii) All basic equations are formulated with respect to the principal shear axes coordinate system, which does not neces-sarily coincide with the principal bending one.

(iii) Shear deformation effect is taken into account on the flexural-torsional buckling analysis of beams of nonsymmetricconstant composite cross-section avoiding the restrictions of the thin-walled theory.

(iv) Torsional warping arising from nonuniform torsion is taken into account.(v) The beam is supported by the most general linear boundary conditions including elastic support or restraint.

(vi) The proposed method overcomes the shortcoming of the induced error in the case of the utilization of a thin tube the-ory solution.

(vii) The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko and Goodier’s(1984) and Cowper’s (1966) definitions, for which several authors (Schramm et al., 1994; Schramm et al., 1997) havepointed out that one obtains unsatisfactory results or definitions given by other researchers (Stephen, 1980; Hutch-inson, 2001), for which these factors take negative values.

(viii) With the exception of the structural models presented by Machado and Cortínez (2005), Cortínez and Piovan (2006)which make use of shell stress resultants, previous formulations concerning composite beams of thin-walled cross-sections or laminated cross-sections are analyzing these beams using the ‘refined models’. However, these modelsdo not satisfy the continuity conditions of transverse shear stress at layer interfaces and assume that the transverseshear stress along the thickness coordinate remains constant, leading to the fact that kinematic or static assumptionscannot be always valid.

(ix) The proposed method employs a pure BEM approach (requiring only boundary discretization) resulting in line or par-abolic elements instead of area elements of the FEM solutions (requiring the whole cross-section to be discretized intotriangular or quadrilateral area elements), while a small number of line elements are required to achieve highaccuracy.

Several beams are analysed to illustrate the method and demonstrate its efficiency. The significant influence of theboundary conditions and the shear deformation effect on the buckling load are investigated through examples with greatpractical interest.

2. Statement of the problem

Let us consider a prismatic beam of length l (Fig. 1), of constant arbitrary cross-section of area A. The cross-section con-sists of materials in contact, each of which can surround a finite number of inclusions, with modulus of elasticity Ej and shear

Fig. 1. Prismatic element of an arbitrarily shaped composite cross section occupying region X (a) subjected in bending and torsional loading (b).

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 499

modulus Gj, occupying the regions Xj ðj ¼ 1;2; . . . ;KÞ of the y; z plane (Fig. 1). The materials of these regions are assumedhomogeneous, isotropic and linearly elastic. Let also the boundaries of the nonintersecting regions Xj be denoted byCjðj ¼ 1;2; . . . ;KÞ. These boundary curves are piecewise smooth, i.e. they may have a finite number of corners. In Fig. 1aCYZ is the coordinate system through the cross-section’s centroid C, while yC , zC are its coordinates with respect to Syz prin-cipal shear system of axes through the cross-section’s shear center S. The beam is subjected to a compressive load �P, to thecombined action of the arbitrarily distributed axial loading pX ¼ pXðXÞ, transverse loading pY ¼ pYðXÞ, pZ ¼ pZðXÞ acting in theY and Z directions, respectively, and to the arbitrarily distributed twisting moment mx ¼ mxðxÞ (Fig. 1b).

Under the aforementioned loading the displacement field of the beam with respect to the Syz system of axes is given as

�uðx; y; zÞ ¼ uðxÞ þ hYðxÞZ � hZðxÞY þdhxðxÞ

dx/P

Sðy; zÞ ð1aÞ

�vðx; y; zÞ ¼ vðxÞ � zhxðxÞ ð1bÞ�wðx; y; zÞ ¼ wðxÞ þ yhxðxÞ ð1cÞ

and therefore the displacement components of the cross-section’s centroid can be written as

uC ¼ uðxÞ þ dhxðxÞdx

/PSðy; zÞ ð2aÞ

vC ¼ vðxÞ � zChxðxÞ ð2bÞwC ¼ wðxÞ þ yChxðxÞ ð2cÞ

where uðxÞ, vðxÞ and wðxÞ are the beam axial and transverse displacements of the shear center S with respect to x, y and zaxes, respectively, hY , hZ are the angles of rotation of the cross-section due to bending, dhx=dx denotes the rate of changeof the angle of twist hx regarded as the torsional curvature and /P

S is the primary warping function with respect to the shearcenter S of the cross-section of the beam (Sapountzakis and Mokos, 2003).

500 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

Moreover, according to the linear theory of beams (small deflections), the angles of rotation of the deflection line withrespect to the shear center ðxÞ and to the centroid ðbÞ in the x–z and x–y planes of the beam subjected to the aforementionedloading and taking into account shear deformation effect satisfy the following relations

sinxy ’ xy ¼ �dwdx¼ hY � cxz; sinxz ’ xz ¼ �

dvdx¼ �hZ � cxy ð3a;bÞ

cos by � 1; cos bz � 1 ð3c;dÞ

sin by ’ �dwC

dx; sin bz ’ �

dvC

dxð3e; fÞ

while employing the stress–strain relations of the three-dimensional elasticity after ignoring the shear strain due to warping,the arising shear stress resultants Qz, Q y are given as

Qz ¼XK

j¼1

ZXj

sxzdXj ¼ G1Azdwdxþ hY

� �ð4aÞ

Qy ¼XK

j¼1

ZXj

sxydXj ¼ G1Aydvdx� hZ

� �ð4bÞ

where the first material is considered as reference material, cxz, cxy are the additional angles of rotation of the cross-sectiondue to shear deformation (Fig. 2a) and G1Az, G1Ay are the cross-section’s shear rigidities of the Timoshenko’s beam theory, with

Az ¼ jzAG ¼ 1az

AG ¼ 1az

XK

j¼1

GjAj

G1ð5aÞ

Ay ¼ jyAG ¼ 1ay

AG ¼ 1ay

XK

j¼1

GjAj

G1ð5bÞ

the shear areas with respect to z, y axes, respectively, jz, jy are the shear correction factors and az, ay the shear deformationcoefficients. It is worth here noting that the reduction of Eq. (4) using the shear modulus G1 of the first material, could beachieved using any other material, considering it as reference material.

Referring to Fig. 2, the stress resultants Rx, Ry, Rz acting in the x, y, z directions, respectively, are related to the axial N andthe shear Qy, Qz forces as

Rx ¼ N cos bþ Q z sin by þ Qy sin bz ð6aÞRy ¼ �N sin bz þ Q y cos bz ð6bÞRz ¼ �N sin by þ Q z cos by ð6cÞ

which by virtue of the small deflection theory and Eqs. (2) and (3) become

Rx ¼ N � QzdwC

dx� Q y

dvC

dxð7aÞ

Ry ¼ NdvC

dxþ Qy ¼ N

dvdx� zC

dhx

dx

� �þ Qy ð7bÞ

Rz ¼ NdwC

dxþ Q z ¼ N

dwdxþ yC

dhx

dx

� �þ Q z ð7cÞ

The second and third terms in the right hand side of Eq. (7a), express the influence of the shear forces Qy, Qz on the horizontalstress resultant Rx. However, since initial shear forces are not taken into account, these terms can be neglected since Q y, Qz

are much smaller than N (Rothert and Gensichen, 1987; Ramm and Hofmann, 1995) and thus Eq. (7a) can be written as

Rx ’ N ð8Þ

Employing Eq. (1a) to the strain–displacement equations of the three-dimensional elasticity and ignoring axial deformations,the normal strain component ex can be written as

ex ¼dhY

dxZ � dhZ

dxY þ d2hx

dx2 /PS ð9Þ

and the arising bending moments MY , MZ are given as

MY ¼XK

j¼1

ZXj

EiexZ dXj ¼ E1IYYdhY

dx� E1IYZ

dhZ

dxð10aÞ

MZ ¼ �XK

j¼1

ZXj

EiexY dXj ¼ E1IZZdhZ

dx� E1IYZ

dhY

dxð10bÞ

Fig. 2. Displacements (a) and equilibrium of an element in the xz (b) and xy (c) planes.

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 501

where

IYY ¼XK

j¼1

Ej

E1

ZXj

Z2dXj; IZZ ¼XK

j¼1

Ej

E1

ZXj

Y2dXj; IYZ ¼XK

j¼1

Ej

E1

ZXj

YZdXj ð11a;b; cÞ

are the moments and the product of inertia of the cross-section with respect to its centroid C. Substituting Eqs. (4a) and (4b)in Eqs. (10a) and (10b) the bending moments MY , MZ can be written as

MY ¼ E1IYY1

G1Az

dQ z

dx� d2w

dx2

!� E1IYZ

d2vdx2 �

1G1Ay

dQ y

dx

!ð12aÞ

MZ ¼ E1IZZd2vdx2 �

1G1Ay

dQ y

dx

!� E1IYZ

1G1Az

dQ z

dx� d2w

dx2

!ð12bÞ

The governing equations of the problem at hand will be derived by considering the equilibrium of the deformed element.Thus, referring to Fig. 2 we obtain

502 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

dRx

dxþ pX ¼ 0;

dRy

dxþ pY ¼ 0;

dRz

dxþ pZ ¼ 0 ð13a;b; cÞð13a—cÞ

dMY

dx� Qz ¼ 0;

dMZ

dxþ Q y ¼ 0 ð13d; eÞ

Substituting Eqs. (8), (7b), and (7c) into Eqs. (13a–c) we obtain

dNdx¼ �pX ð14aÞ

dQ y

dxþ dN

dxovox� zC

ohx

ox

� �þ N

o2vox2 � zC

o2hx

ox2

!þ pY ¼ 0 ð14bÞ

dQ z

dxþ dN

dxowoxþ yC

ohx

ox

� �þ N

o2wox2 þ yC

o2hx

ox2

!þ pZ ¼ 0 ð14cÞ

Substituting Eqs. (14b) and (14c) into Eqs. (12a) and (12b) we obtain the expressions of the bending moments MY , MZ as

MY ¼ �E1IYYd2wdx2 �

E1IYY

G1AzpZ � pX

dwdxþ yC

dhx

dx

� �þ N

d2wdx2 þ yC

d2hx

dx2

! !

� E1IYZd2vdx2 �

E1IYZ

G1AypY � pX

dvdx� zC

dhx

dx

� �þ N

d2vdx2 � zC

d2hx

dx2

! !ð15aÞ

MZ ¼ E1IZZd2vdx2 þ

E1IZZ

G1AypY � pX

dvdx� zC

dhx

dx

� �þ N

d2vdx2 � zC

d2hx

dx2

! !

þ E1IYZd2wdx2 þ

E1IYZ

G1AzpZ � pX

dwdxþ yC

dhx

dx

� �þ N

d2wdx2 þ yC

d2hx

dx2

! !ð15bÞ

subsequently the expressions of the shear forces Qy, Q z employing Eqs. (13d,e) as

Qy ¼ �E1IZZd3vdx3 � E1IYZ

d3wdx3

� E1IZZ

G1Ay

dpY

dx� dpX

dxdvdx� zC

dhx

dx

� �� 2pX

d2vdx2 � zC

d2hx

dx2

!þ N

d3vdx3 � zC

d3hx

dx3

! !

� E1IYZ

G1Az

dpZ

dx� dpX

dxdwdxþ yC

dhx

dx

� �� 2pX

d2wdx2 þ yC

d2hx

dx2

!þ N

d3wdx3 þ yC

d3hx

dx3

! !ð16aÞ

Qz ¼ �E1IYYd3wdx3 � E1IYZ

d3vdx3 �

� E1IYY

G1Az

dpZ

dx� dpX

dxdwdxþ yC

dhx

dx

� �� 2pX

d2wdx2 þ yC

d2hx

dx2

!þ N

d3wdx3 þ yC

d3hx

dx3

! !

� E1IYZ

G1Ay

dpY

dx� dpX

dxdvdx� zC

dhx

dx

� �� 2pX

d2vdx2 � zC

d2hx

dx2

!þ N

d3vdx3 � zC

d3hx

dx3

! !ð16bÞ

and eliminating these forces from Eqs. (14b) and (14c) we obtain the first two coupled partial differential equations of theproblem of the beam under consideration subjected to the combined action of axial, bending and torsional loading as

E1IZZd4vdx4 þ E1IYZ

d4wdx4 þ

E1IZZ

G1Ay

d2pY

dx2 �d2pX

dx2

dvdx� zC

dhx

dx

� �� 3

dpX

dxd2vdx2 � zC

d2hx

dx2

!� 3pX

d3vdx3 � zC

d3hx

dx3

!"

þNd4vdx4 � zC

d4hx

dx4

!#þ E1IYZ

G1Az

d2pZ

dx2 �d2pX

dx2

dwdxþ yC

dhx

dx

� �� 3

dpX

dxd2wdx2 þ yC

d2hx

dx2

!� 3pX

d3wdx3 þ yC

d3hx

dx3

!"

þNd4wdx4 þ yC

d4hx

dx4

!#� pY þ pX

dvdx� zC

dhx

dx

� �� N

d2vdx2 � zC

d2hx

dx2

!¼ 0 ð17aÞ

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 503

E1IYYd4wdx4 þ E1IYZ

d4vdx4 þ

E1IYY

G1Az

d2pZ

dx2 �d2pX

dx2

dwdxþ yC

dhx

dx

� �� 3

dpX

dxd2wdx2 þ yC

d2hx

dx2

!� 3pX

d3wdx3 þ yC

d3hx

dx3

!"

þ Nd4wdx4 þ yC

d4hx

dx4

!#þ E1IYZ

G1Ay

d2pY

dx2 �d2pX

dx2

dvdx� zC

dhx

dx

� �� 3

dpX

dxd2vdx2 � zC

d2hx

dx2

!� 3pX

d3vdx3 � zC

d3hx

dx3

!"

þ Nd4vdx4 � zC

d4hx

dx4

!#� pZ þ pX

dwdxþ yC

dhx

dx

� �� N

d2wdx2 þ yC

d2hx

dx2

!¼ 0 ð17bÞ

Finally, the angles of rotation of the cross-section due to bending hY , hZ are given from Eqs. (4a) and (4b) as

hY ¼ �dwdx� E1IYY

G1Az

d3wdx3 �

E1IYZ

G1Az

d3vdx3

� E1IYY

G21A2

z

dpZ

dx� dpX

dxdwdxþ yC

dhx

dx

� �� 2pX

d2wdx2 þ yC

d2hx

dx2

!þ N

d3wdx3 þ yC

d3hx

dx3

!" #

� E1IYZ

G21AyAz

dpY

dx� dpX

dxdvdx� zC

dhx

dx

� �� 2pX

d2vdx2 � zC

d2hx

dx2

!þ N

d3vdx3 � zC

d3hx

dx3

!" #ð18aÞ

hZ ¼dvdxþ E1IZZ

G1Ay

d3vdx3 þ

E1IYZ

G1Ay

d3wdx3

þ E1IZZ

G21A2

y

dpY

dx� dpX

dxdvdx� zC

dhx

dx

� �� 2pX

d2vdx2 � zC

d2hx

dx2

!þ N

d3vdx3 � zC

d3hx

dx3

!" #

þ E1IYZ

G21AyAz

dpZ

dx� dpX

dxdwdxþ yC

dhx

dx

� �� 2pX

d2wdx2 þ yC

d2hx

dx2

!þ N

d3wdx3 þ yC

d3hx

dx3

!" #ð18bÞ

Equilibrium of torsional moments along x axis of the beam element, after taking into account the additional shear stressesdue to the presence of the axial force N (Timoshenko and Gere, 1961), which employing Eq. (2) are written as

sjxz ¼ rj d �w

dx

� �¼ rj dw

dxþ y

dhx

dx

� �ð19aÞ

sjxy ¼ rj d�v

dx

� �¼ rj dv

dx� z

dhx

dx

� �ð19bÞ

and the corresponding arising additional twisting moment

Mt;add ¼Z

Xðsxzy� sxyzÞdX ¼ NyC

dwdx� NzC

dvdxþ N

IS

AE

dhx

dxð20Þ

where

IS ¼XK

j¼1

y2CAE

j þ z2CAE

j þ IjYY þ Ij

ZZ

� �ð21Þ

AE ¼Pk

j¼1EjAj

E1ð22Þ

leads to the third (coupled with the previous two) partial differential equation of the problem of the beam under consider-ation as

� dMt

dxþ dMt;add

dx

� �¼ mx þ pZyc � pY zc ð23Þ

which employing Eq. (20) and having in mind that the torsional moment Mt is given as (Sapountzakis and Mokos, 2003)

Mt ¼ �E1CSd3hx

dx3 þ G1Itdhx

dxð24Þ

can be written as

E1CSd4hx

dx4 � G1Itd2hx

dx2 � N ycd2wdx2 � zc

d2vdx2 þ

IS

AE

d2hx

dx2

!¼ mx þ pZyC � pY zC � pX yc

dwdx� zc

dvdx

� �� pX

IS

AE

dhx

dxð25Þ

504 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

where IS is the polar moment of inertia with respect to the shear center S, E1CS and G1It are the cross-section’s warping andtorsional rigidities, respectively, with CS, It being its warping and torsion constants, respectively, given as (Sapountzakis andMokos, 2003)

CS ¼XK

j¼1

Ej

E1

ZXj

ðuPSÞ

2dXj ð26aÞ

It ¼XK

j¼1

Gj

G1

ZXj

y2 þ z2 þ youP

S

oz� z

ouPS

oy

� �dXj ð26bÞ

It is worth here noting that the primary warping function uPS ðy; zÞ can be established by solving independently the Neumann

problem (Sapountzakis and Mokos, 2003)

r2uPS ¼ 0 in X ¼ [K

j¼1Xj ð27Þ

GiouP

Son

� �i� Gj

ouPS

on

� �j¼ 1

2 ðGi � GjÞoðq2

S Þos

� �j

on Cjðj ¼ 1;2; . . . ;KÞ ð28Þ

where r2 ¼ o2=ox2 þ o2=oy2 is the Laplace operator; Gi ¼ 0 at the free part of the boundary of Xj region or Gi is the shearmodulus of Xi region at the common part of the boundaries of Xi and Xj regions; qS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pis the distance of a point

on the boundary Cj from the shear center S; ðo=onÞj denotes the directional derivative normal to the boundary Cj andðo=osÞj denotes differentiation with respect to its arc length s.

As it is already mentioned, Eqs. (17a), (17b), and (25) constitute the governing equations of the beam subjected to thecombined action of axial, bending and torsional loading taking into account shear deformation effect. For the special caseof an axially compressive load N ¼ �P ðpX ¼ 0Þ. The aforementioned equations are also subjected to the pertinent boundaryconditions of the problem, which are given as

a1vðxÞ þ a2RyðxÞ ¼ a3; �a1hZðxÞ þ �a2MZðxÞ ¼ �a3 ð29a;bÞ

b1wðxÞ þ b2RzðxÞ ¼ b3;�b1hYðxÞ þ �b2MY ðxÞ ¼ �b3 ð30a;bÞ

c1hxðxÞ þ c2MtðxÞ ¼ c3; �c1dhxðxÞ

dxþ �c2MwðxÞ ¼ �c3 ð31a;bÞ

at the beam ends x ¼ 0; l, where Ry, Rz and MZ , MY are the reactions and bending moments with respect to y and z axes,respectively, obtained from Eqs. (7b), (7c), and (15a), (15b), (16a), (16b) as

Ry ¼ �E1IZZd3vdx3 �

E1IZZ

G1AyN

d3vdx3 � zC

d3hx

dx3

!� E1IYZ

d3wdx3

� E1IYZ

G1AzN

d3wdx3 þ yC

d3hx

dx3

!þ N

dvdx� zC

dhx

dx

� �ð32aÞ

Rz ¼ �E1IYYd3wdx3 �

E1IYY

G1AzN

d3wdx3 þ yC

d3hx

dx3

!� E1IYZ

d3vdx3

� E1IYZ

G1AyN

d3vdx3 � zC

d3hx

dx3

!þ N

dwdxþ yC

dhx

dx

� �ð32bÞ

MY ¼ �E1IYYd2wdx2 �

E1IYY

G1AzN

d2wdx2 þ yC

d2hx

dx2

!� E1IYZ

d2vdx2 �

E1IYZ

G1AyN

d2vdx2 � zC

d2hx

dx2

!ð33aÞ

MZ ¼ E1IZZd2vdx2 þ

E1IZZ

G1AyN

d2vdx2 � zC

d2hx

dx2

!þ E1IYZ

d2wdx2 þ

E1IYZ

G1AzN

d2wdx2 þ yC

d2hx

dx2

!ð33bÞ

the angles of rotation due to bending hY , hZ are evaluated from Eq. (18) as

hY ¼ �dwdx� E1IYY

G1Az

d3wdx3 �

EIYY

G21A2

z

Nd3wdx3 þ yC

d3hx

dx3

!� E1IYZ

G1Az

d3vdx3

� E1IYZ

G21AyAz

Nd3vdx3 � zC

d3hx

dx3

!ð34Þ

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 505

hZ ¼dvdxþ E1IZZ

G1Ay

d3vdx3 þ

E1IZZ

G21A2

y

Nd3vdx3 � zC

d3hx

dx3

!þ E1IYZ

G1Ay

d3wdx3

þ E1IYZ

G21AyAz

Nd3wdx3 þ yC

d3hx

dx3

!ð35Þ

while in Eq. (31b) Mw is the warping moment given as (Sapountzakis and Mokos, 2003)

Mw ¼ �E1CSd2hx

dx2 ð36Þ

Finally, ak; �ak; bk;�bk; ck; �ck ðk ¼ 1;2;3Þ are functions specified at the beam ends x ¼ 0; l. Eqs. (29)–(31) describe the most gen-

eral linear boundary conditions associated with the problem at hand and can include elastic support or restraint. It is appar-ent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derivedfrom these equations by specifying appropriately these functions (e.g. for a clamped edge it is a1 ¼ b1 ¼ c1 ¼ 1,�a1 ¼ �b1 ¼ �c1 ¼ 1, a2 ¼ a3 ¼ b2 ¼ b3 ¼ c2 ¼ c3 ¼ �a2 ¼ �a3 ¼ �b2 ¼ �b3 ¼ �c2 ¼ �c3 ¼ 0).

The solution of the boundary value problem given from Eqs. (17), (25) subjected to the boundary conditions (29)–(31)which represents the flexural-torsional buckling of beams, presumes the evaluation of the shear deformation coefficientsay, az, corresponding to the principal shear axes coordinate system Syz. These coefficients are established equating theapproximate formula of the shear strain energy per unit length (Schramm et al., 1997)

Uappr: ¼ayQ2

y

2AGG1

þ azQ 2z

2AGG1

ð37Þ

with the exact one given from

Uexact ¼XK

j¼1

E1

Ej

ZXj

ðsxzÞ2j þ ðsxyÞ2j2G1

dXj ð38Þ

and are obtained as (Mokos and Sapountzakis, 2005)

ay ¼1jy¼ AG

E1D2

XK

j¼1

ZXj

EjððrHÞj � eÞ � ððrHÞj � eÞdXj ð39aÞ

az ¼1jz¼ AG

E1D2

XK

j¼1

ZXj

EjððrUÞj � dÞ � ððrUÞj � dÞdXj ð39bÞ

where ðsxzÞj; ðsxyÞj are the transverse (direct) shear stress components, ðrÞ � iY ðo=oYÞ þ iZðo=oZÞ is a symbolic vector withiY ; iZ the unit vectors along Y and Z axes, respectively, D is given from

D ¼ 2ð1þ mÞðIYY IZZ � I2YZÞ ð40Þ

m is the Poisson ratio of the cross-section materials, e and d are vectors defined as

e ¼ m IYYY2 � Z2

2� IYZYZ

!" #iY þ m IYY YZ þ IYZ

Y2 � Z2

2

!" #iZ ð41aÞ

d ¼ m IZZYZ � IYZY2 � Z2

2

!" #iY þ �m IZZ

Y2 � Z2

2þ IYZYZ

!" #iZ ð41bÞ

and HðY ; ZÞ, UðY; ZÞ are stress functions, which are evaluated from the solution of the following Neumann type boundaryvalue problems (Mokos and Sapountzakis, 2005):

ðr2HÞj ¼ 2ðIYZZ � IYY YÞ in Xjðj ¼ 1;2; . . . ;KÞ ð42aÞ

EjoHon

� �j � Ei

oHon

� �i ¼ ðEj � EiÞn � e on Cjðj ¼ 1;2; . . . ;KÞ ð42bÞ

ðr2UÞj ¼ 2ðIYZY � IZZZÞ in Xjðj ¼ 1;2; . . . ;KÞ ð43aÞ

EjoUon

� �j � Ei

oUon

� �i ¼ ðEj � EiÞn � d on Cjðj ¼ 1;2; . . . ;KÞ ð43bÞ

where Ei is the modulus of elasticity of the Xi region at the common part of the boundaries of Xj and Xi regions, or Ei ¼ 0 atthe free part of the boundary of Xj region, while ðo=onÞj � nY ðo=oYÞj þ nZðo=oZÞj denotes the directional derivative normal tothe boundary Cj. The vector n normal to the boundary Cj is positive if it points to the exterior of the Xj region, while thenormal derivatives across the interior boundaries vary discontinuously. It is also worth here noting that the boundary con-

506 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

ditions (28), (42b), (43b) have been derived from the physical consideration that the traction vectors in the direction of thenormal vector n on the interfaces separating the j and i different materials are equal in magnitude and opposite in direction,while it vanishes on the free surface of the beam.

3. Integral representations – numerical solution

3.1. For the transverse v, w displacements and the angle of twist hx

According to the precedent analysis, the flexural-torsional buckling problem of composite beams reduces in establishingthe displacement components vðxÞ, wðxÞ and hxðxÞ having continuous derivatives up to the fourth order with respect to x,satisfying the coupled governing equations (17) and (25) inside the beam and the boundary conditions (29)–(31) at the beamends x ¼ 0; l.

Eqs. (17) and (25) are solved using the analog equation method (Katsikadelis, 2002) as it is developed for hyperbolic dif-ferential equations (Sapountzakis, 2005; Sapountzakis and Tsiatas, 2007). This method is applied for the problem at hand asfollows. Let vðxÞ, wðxÞ and hxðxÞ be the sought solution of the aforementioned boundary value problem. Setting asu1ðxÞ ¼ vðxÞ, u2ðxÞ ¼ wðxÞ, u3ðxÞ ¼ hxðxÞ and differentiating these functions four times with respect to x yields

d4ui

dx4 ¼ qiðxÞ ði ¼ 1;2;3Þ ð44Þ

Eq. (44) indicate that the solution of Eqs. (17) and (25) can be established by solving Eq. (44) under the same boundary con-ditions (29)–(31), provided that the fictitious load distributions qiðxÞ ði ¼ 1;2;3Þ are first established. These distributions canbe determined using BEM as follows:

The solution of Eq. (44) is given in integral form as

uiðxÞ ¼Z l

0qiu

�dx� u�d3ui

dx3 �du�

dxd2ui

dx2 þd2u�

dx2

dui

dx� d3u�

dx3 ui

" #l

0

ð45Þ

where u� is the fundamental solution given as

u� ¼ 112

l3 2þ rl

3 � 3rl

2� �ð46Þ

with r ¼ x� n, x, n points of the beam, which is a particular singular solution of the equation

d4u�

dx4 ¼ dðx� nÞ ð47Þ

Employing Eq. (46) the integral representation (45) can be written as

uiðxÞ ¼Z l

0qiK4ðrÞdn� K4ðrÞ

d3ui

dx3 þK3ðrÞd2ui

dx2 þK2ðrÞdui

dxþK1ðrÞui

" #l

0

ð48Þ

where the kernels KjðrÞ; ðj ¼ 1;2;3;4Þ are given as

K1ðrÞ ¼ �12

sgnrl

ð49aÞ

K2ðrÞ ¼ �12

l 1� rl

� �ð49bÞ

K3ðrÞ ¼ �14

l2 rl

rl

� 2� �

sgnrl

ð49cÞ

K4ðrÞ ¼1

12l3 2þ r

l

3 � 3rl

2� �ð49dÞ

Notice that in Eq. (48) for the line integral it is r ¼ x� n, x, n points inside the beam, whereas for the rest terms it is r ¼ x� f, xinside the beam, f at the beam ends 0, l.

Differentiating Eq. (48) with respect to x, results in the integral representations of the derivatives of ui as

duiðxÞdx

¼Z l

0qiK3ðrÞdn� K3ðrÞ

d3ui

dx3 þK2ðrÞd2ui

dx2 þK1ðrÞdui

dx

" #l

0

ð50aÞ

d2uiðxÞdx2 ¼

Z l

0qiK2ðrÞdn� K2ðrÞ

d3ui

dx3 þK1ðrÞd2ui

dx2

" #l

0

ð50bÞ

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 507

d3uiðxÞdx3 ¼

Z l

0qiK1ðrÞdn� K1ðrÞ

d3ui

dx3

" #l

0

ð50cÞ

d4uiðxÞdx4 ¼ qiðxÞ ð50dÞ

The integral representations (48) and (50), when applied for the beam ends ð0; lÞ, together with the boundary condi-tions (29)–(31) are employed to express the unknown boundary quantities uiðfÞ, ui;xðfÞ, ui;xxðfÞ and ui;xxxðfÞ ðf ¼ 0; lÞ interms of qi. This is accomplished numerically as follows. The interval ð0; lÞ is divided into L equal elements (Fig. 3), onwhich qiðxÞ is assumed to vary according to certain law (constant, linear, parabolic, etc.). The constant element assumptionis employed here as the numerical implementation becomes very simple and the obtained results are very good. Employ-ing the aforementioned procedure for the coupled boundary conditions (29) and (30) the following set of linear equationsis obtained:

D11 0 0 D14 0 0 0 D18

0 D22 D23 D24 0 0 D27 D28

E31 E32 E33 E34 0 0 0 00 E42 E43 E44 0 0 0 00 0 0 D54 D55 0 0 D58

0 0 D63 D64 0 D66 D67 D68

0 0 0 0 E31 E32 E33 E34

0 0 0 0 0 E42 E43 E44

266666666666664

377777777777775

u1

u1;x

u1;xx

u1;xxx

u2

u2;x

u2;xx

u2;xxx

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

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;¼

a3

�a3

00b3

�b3

00

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

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;þ

00F3

F4

0000

266666666666664

377777777777775q1 þ

000000F3

F4

266666666666664

377777777777775q2 ð51Þ

while for the boundary conditions (31) we have

E11 E12 0 E14

0 E22 E23 0E31 E32 E33 E34

0 E42 E43 E44

2666437775

u3

u3;x

u3;xx

u3;xxx

8>>><>>>:9>>>=>>>; ¼

c3

�c3

00

8>>><>>>:9>>>=>>>;þ

00F3

F4

2666437775q3 ð52Þ

where D11, D14, D18, D22, D23, D24, D27, D28, D54, D55, D58, D63, D64, D66, D67, D68, E22, E23, E1j, ðj ¼ 1;2;4Þ are 2� 2 known squarematrices including the values of the functions aj; �aj; bj;

�bj; cj; �cj ðj ¼ 1;2Þ of Eqs. (29)–(31); a3, �a3, b3, �b3, c3, �c3 are 2� 1 knowncolumn matrices including the boundary values of the functions a3; �a3; b3;

�b3; c3; �c3 of Eqs. (29)–(31); Ejk,ðj ¼ 3;4; k ¼ 1;2;3;4Þ are square 2� 2 known coefficient matrices resulting from the values of the kernels KjðrÞðj ¼ 1;2;3;4Þ at the beam ends and Fj ðj ¼ 3;4Þ are 2� L rectangular known matrices originating from the integration ofthe kernels on the axis of the beam. Moreover

ui ¼ fuið0Þ; uiðlÞgT ð53aÞ

ui;x ¼duið0Þ

dx;

duiðlÞdx

�T

ð53bÞ

ui;xx ¼d2uið0Þ

dx2 ;d2uiðlÞ

dx2

( )T

ð53cÞ

ui;xxx ¼d3uið0Þ

dx3 ;d3uiðlÞ

dx3

( )T

ð53dÞ

are vectors including the two unknown boundary values of the respective boundary quantities and qi ¼ fqi1qi

2 . . . qiLg

T

ði ¼ 1;2;3Þ is the vector including the L unknown nodal values of the fictitious load.

Nodal points

2

x

1 L

l

Fig. 3. Discretization of the beam interval and distribution of the nodal points.

508 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

Discretization of Eqs. (48) and (50) and application to the L collocation points yields

ui ¼ C4qi � ðH1ui þH2ui;x þH3ui;xx þH4ui;xxxÞ ð54aÞui;x ¼ C3qi � ðH1ui;x þH2ui;xx þH3ui;xxxÞ ð54bÞui;xx ¼ C2qi � ðH1ui;xx þH2ui;xxxÞ ð54cÞui;xxxx ¼ C1qi �H1ui;xxxx ð54dÞui;xxxx ¼ qi ð54eÞ

where Cj ðj ¼ 1;2;3;4Þ are L� L known matrices; Hj ðj ¼ 1;2;3;4Þ are L� 2 also known matrices and ui, ui;x, ui;xx, ui;xxxx, ui;xxxx

are vectors including the values of uiðxÞ and their derivatives at the L nodal points.The above equations, after eliminating the boundary quantities employing Eqs. (51) and (52), can be written as

ui ¼ Tiqi þ Tijqj þ ti; i; j ¼ 1;2; i 6¼ j ð55aÞu3 ¼ T3q3 þ t3 ð55bÞui;x ¼ Tixqi þ Tijxqj þ tix; i; j ¼ 1;2; i 6¼ j ð55cÞu3;x ¼ T3xq3 þ t3x ð55dÞui;xx ¼ Tixxqi þ Tijxxqj þ tixx; i; j ¼ 1;2; i 6¼ j ð55eÞu3;xx ¼ T3xxq3 þ t3xx ð55fÞui;xxx ¼ Tixxxqi þ Tijxxxqj þ tixxx; i; j ¼ 1;2; i 6¼ j ð55gÞu3;xxx ¼ T3xxxq3 þ t3xxx ð55hÞui;xxxx ¼ qi; i ¼ 1;2;3 ð55iÞ

where Ti, Tix, Tixx, Tixxx, Tij, Tijx, Tijxx, Tijxxx are known L� L matrices and ti, tix, tixx, tixxx are known L� 1 matrices. It is worth herenoting that for homogeneous boundary conditions ða3 ¼ �a3 ¼ b3 ¼ �b3 ¼ c3 ¼ �c3 ¼ 0Þ it is ti ¼ tix ¼ tixx ¼ tixxx ¼ 0.

In the conventional BEM, the load vectors qi are known and Eq. (55) are used to evaluate uiðxÞ and their derivatives at theL nodal points. This, however, can not be done here since qi are unknown. For this purpose, 3L additional equations are de-rived, which permit the establishment of qi. These equations result by applying Eqs. (17) and (25) to the L collocation points,leading to the formulation of the following set of 3L simultaneous equations

ðA� NBþ CÞq1

q2

q3

8><>:9>=>; ¼ f ð56Þ

where the 3L� 3L matrices A, B, C are given as

A ¼E1IZZ E1IYZ 0E1IYZ E1IYY 0

0 0 E1CS � G1ItT3xx

264375 ð57aÞ

B ¼

� 1G1Ay

E1IZZ þ T1xx � 1G1Az

E1IYZ þ T12xxzC

G1AyE1IZZ � yC

G1AzE1IYZ � zCT3xx

� 1G1Ay

E1IYZ þ T21xx � 1G1Az

E1IYY þ T2xxzC

G1AyE1IYZ � yC

G1AzE1IYY þ yCT3xx

yCT21xx � zCT1xx yCT2xx � zCT12xxIS

AE T3xx

26643775 ð57bÞ

C ¼C11 C12 C13

C21 C22 C23

C31 C32 C33

264375 ð57cÞ

the Cij L� L matrices are evaluated from the expressions

C11 ¼ pXT1x �E1IZZ

G1AyðpX;xxT1x þ 3pX;xT1xx þ 3pXT1xxxÞ

��E1IYZ

G1AzðpX;xxT21x þ 3pX;xT21xx þ 3pXT21xxxÞ

ð58aÞ

C12 ¼ pXT12x �E1IZZ

G1AyðpX;xxT12x þ 3pX;xT12xx þ 3pXT12xxxÞ

��E1IYZ

G1AzðpX;xxT2x þ 3pX;xT2xx þ 3pXT2xxxÞ

ð58bÞ

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 509

C13 ¼ �zCpXT3x þE1IZZ

G1AyðzCpX;xxT3x þ 3zCpX;xT3xx þ 3zCpXT3xxxÞ

�� E1IYZ

G1AzðyCpX;xxT3x þ 3yCpX;xT3xx þ 3yCpXT3xxxÞ

ð58cÞ

C21 ¼ pXT21x �E1IYY

G1AzðpX;xxT21x þ 3pX;xT21xx þ 3pXT21xxxÞ

�� E1IYZ

G1AyðpX;xxT1x þ 3pX;xT1xx þ 3pXT1xxxÞ

ð58dÞ

C22 ¼ pXT2x �E1IYY

G1AzðpX;xxT2x þ 3pX;xT2xx þ 3pXT2xxxÞ

�� E1IYZ

G1AyðpX;xxT12x þ 3pX;xT12xx þ 3pXT12xxxÞ

ð58eÞ

C23 ¼ yCpXT3x �E1IYY

G1AzðyCpX;xxT3x þ 3yCpX;xT3xx þ 3yCpXT3xxxÞ

�þ E1IYZ

G1AyðzCpX;xxT3x þ 3zCpX;xT3xx þ 3zCpXT3xxxÞ

ð58fÞ

C31 ¼ ½yCpXT21x � zCpXT1x� ð58gÞC32 ¼ ½yCpXT2x � zCpXT12x� ð58hÞ

C33 ¼IS

AE pXT3x

� ð58iÞ

and the 3L� 1 column matrix f is given as

f ¼f1

f2

f3

8><>:9>=>;þ

00

G1Itt3xx

8><>:9>=>;þ N

t1xx � zCt3xx

t2xx þ yCt3xx

yCt2xx � zCt1xx þ IS

AE t3xx

8><>:9>=>; ð59Þ

with

f1 ¼ pY � pXðt1x � zCt3xÞ �E1IZZ

G1AyðpY ;xx � pX;xxðt1x � zCt3xÞ

� 3pX;xðt1xx � zCt3xxÞ � 3pXðt1xxx � zCt3xxxÞÞ �E1IYZ

G1AzðpZ;xx ð60aÞ

� pX;xxðt2x þ yCt3xÞ � 3pX;xðt2xx þ yCt3xxÞ � 3pXðt2xxx þ yCt3xxxÞÞ

f2 ¼ pZ � pXðt2x þ yCt3xÞ �E1IYY

G1AzðpZ;xx � pX;xxðt2x þ yCt3xÞ

� 3pX;xðt2xx þ yCt3xxÞ � 3pXðt2xxx þ yCt3xxxÞÞ �E1IYZ

G1AyðpY ;xx ð60bÞ

� pX;xxðt1x � zCt3xÞ � 3pX;xðt1xx � zCt3xxÞ � 3pXðt1xxx � zCt3xxxÞÞ

f3 ¼mx þ pZyC � pY zC þ zCpXt1x � yCpXt2x �IS

AE pXt3x ð60cÞ

In the above set of equations the matrices E1IYY , E1IZZ , E1IYZ , E1CS, G1It are L� L diagonal matrices including the values ofthe corresponding quantities, respectively, at the L nodal points. Moreover, pX , pX;x, pX;xx are diagonal matrices and pY , pY;xx,pZ , pZ;xx and mx are vectors containing the values of the external loading and their derivatives at these points.

Solving the linear system of Eq. (56) for the fictitious load distributions q1, q2, q3 the displacements and their derivativesin the interior of the beam are computed using Eq. (55).

3.1.1. Buckling equationIn this case it is a3 ¼ �a3 ¼ b3 ¼ �b3 ¼ c3 ¼ �c3 ¼ 0 (homogeneous boundary conditions) and pX ¼ pX;x ¼ pX;xx ¼

pY ¼ pY;xx ¼ pZ ¼ pZ;xx ¼mx ¼ 0, N ¼ �P. Thus, Eq. (56) becomes

ðAþ PBÞq1

q2

q3

8><>:9>=>; ¼ 0 ð61Þ

The condition that Eq. (61) has a nontrivial solution yields the buckling equation

detðAþ PBÞ ¼ 0 ð62Þ

510 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

3.2. For the primary warping function /PS

The integral representations and the numerical solution for the evaluation of the angle of twist hx assume that the warp-ing CS and torsion It constants given from Eqs. (26a) and (26b) are already established. Eqs. (26a) and (26b) indicate that theevaluation of the aforementioned constants presumes that the primary warping function /P

S at any interior point of the do-main X of the cross-section of the beam is known. Once /P

S is established, CS and It constants are evaluated by converting thedomain integrals into line integrals along the boundary employing the following relations

Table 1Bucklin

E2=E1

b2 ¼ h2

0.512

b2 ¼ h2

0.512

b2 ¼ h2

0.512

CS ¼ �1E1

XK

j¼1

ZCj

ðBÞj Ejo/P

S

on

!j

� Eio/P

S

on

!i

24 35ds on Cj ðj ¼ 1;2; . . . ;KÞ ð63aÞ

It ¼1

G1

XK

j¼1

ZCj

ðGj � GiÞ ðyz2 � zðuPS ÞjÞ cos bþ ðzy2 þ yðuP

S ÞjÞ sin bh i

ds on Cj ðj ¼ 1;2; . . . ;KÞ ð63bÞ

and using constant boundary elements for the approximation of these line integrals. In Eqs. (63a) and (63b) Cj

ðj ¼ 1;2; . . . ;KÞ is an interface between regions Xj and Xi, while Ej ¼ Gj ¼ 0 in the case Cj is a free boundary. Moreover, inthese equations the normal n to the boundary Cj points to the exterior of the region Xj and Cj is traveled only once, whileðBðy; zÞÞj is a fictitious function defined as the solution of the following Neumann problem

ðr2BÞj ¼ ðuPS Þj in Xj ðj ¼ 1;2; . . . ;KÞ ð64Þ

EjoBon

� �j � Ei

oBon

� �i ¼ 0 on Cj ðj ¼ 1;2; . . . ;KÞ ð65Þ

C

b2

b1

h2

h1

S2E

z,Z

y

Y

2E

1E

Fig. 4. Cross-section of the composite beam of Example 1.

g load P (kN) of the composite beam of Example 1

Hinged–hinged Fixed–hinged Fixed–fixed

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

Presentstudy

FEM Presentstudy

FEM Presentstudy

FEM

¼ 0:02 m479 476 502 530 527 541 573 570 556783 780 815 834 831 841 882 879 849

1487 1483 1490 1547 1544 1512 1612 1609 1520

¼ 0:08 m7046 6845 6949 13,343 12,586 13,010 22,288 20,571 21,164

13,558 13,089 13,288 25,306 23,646 24,550 39,929 36,892 38,11727,076 25,828 26,299 50,698 46,425 48,521 75,155 69,625 72,321

¼ 0:20 m24,098 23,524 23,591 49,235 46,666 47,666 96,012 87,510 92,80039,442 38,082 38,301 80,601 74,580 76,766 157,223 137,653 147,28457,031 54,383 54,978 116,685 105,070 108,940 228,201 190,992 206,875

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 511

The evaluation of the primary warping function uPS and the fictitious function Bðy; zÞ is accomplished using BEM as this is

presented in Sapountzakis and Mokos (2001) and in Sapountzakis and Mokos (2003), respectively.

3.3. For the stress functions HðY ; ZÞ and UðY ; ZÞ

The evaluation of the stress functions HðY; ZÞ and UðY ; ZÞ is accomplished using BEM as this is presented in Mokos andSapountzakis (2005). Moreover, since the flexural-torsional buckling problem of composite beams is solved by the BEM, thedomain integrals for the evaluation of the area, the bending moments of inertia and the shear deformation coefficients (Eqs.(39a) and (39b)) have to be converted to boundary line integrals, in order to maintain the pure boundary character of themethod. This is achieved using integration by parts, the Gauss theorem and the Green identity (Mokos and Sapountzakis,2005).

Fig. 5. 3-D views of the buckling mode shapes of the FEM solution of the beams of Example 1 with E2/E1 = 0.5 (numbers in parentheses correspond to theFEM solution).

512 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

4. Numerical examples

On the basis of the analytical and numerical procedures presented in the previous sections, a computer program has beenwritten and representative examples have been studied to demonstrate the efficiency and the range of applications of thedeveloped method. In all the examples treated, each cross-section has been analysed employing N ¼ 300 constant boundaryelements along the boundary of the cross section, which are enough to ensure convergence at the calculation of the sectionalconstants, while the beam interval is divided into L ¼ 60 constant equal elements.

Example 1. A monosymmetric beam of length l ¼ 3:0 m, with a composite cross section consisting of three rectangular partsðb1 ¼ h1 ¼ 0:40 mÞ in contact (reference material 1: E1 ¼ 3:0� 107 kN=m2, m1 ¼ 0:20, materials 2, 3: E2 ¼ E3, m2 ¼ m3 ¼ 0:20Þ,as this is shown in Fig. 4, has been studied. Three different types starting from a thin-walled and ending with a thick-walledcross-section are considered, that is (i) b2 ¼ h2 ¼ 0:02 m, (ii) b2 ¼ h2 ¼ 0:08 m and (iii) b2 ¼ h2 ¼ 0:20 m. In Table 1 thecomputed values of the buckling load P for the cases of hinged–hinged, fixed–hinged and fixed–fixed boundary conditionsand for various values of the ratio E2=E1 are presented taking into account or ignoring shear deformation effect as comparedwith those obtained from a FEM solution (MSC/NASTRAN, 1999) employing 2600, 21,600 and 6000 solid brick elements forthe three cases, respectively (the buckling mode shapes of the latter are presented in Fig. 5). From the obtained results theinfluence of the inclusion of the aforementioned effect is remarkable leading to the conclusion that it has to be taken intoaccount. Moreover, it can be concluded that the influence of the boundary conditions on the buckling load is significant,while the buckling load is increasing monotonically with the ratio E2=E1.

Example 2. To demonstrate the range of applications of the proposed method a slab-and-beam structure of lengthl ¼ 40:0 m, with a composite cross section consisting of a rectangular concrete C20/25 plate (reference material 1:E1 ¼ 2:9� 107 kN=m2, m1 ¼ 0:20Þ stiffened by two concrete C35/45 I-section beams (material 2: E2 ¼ 3:35� 107 kN=m2,m2 ¼ 0:20Þ, as this is shown in Fig. 6, has been studied. In Table 2 the geometrical and inertia properties of the compositecross-section are presented together with its warping, torsion constants and its shear deformation coefficients referred to

25cm

15cm 7cm

177cm

20cm35cm

90cm

35cm

330cm

C35/45

C20/25 C

Z,z

Y

S

50cm

30cm 30cm

50cm

90cm320cm

190cm y

Fig. 6. Cross-section of the composite slab-and-beam structure of Example 2.

Table 2Geometric, inertia, torsion and warping constants and shear deformation coefficients of the composite cross-section of example 2

IYY ¼ 4:049 m4 ay ¼ 3:349IZZ ¼ 17:182 m4 az ¼ 3:618IYZ ¼ 0:00 m4 ayz ¼ 0:0CS ¼ 16:572 m6 yC ¼ 0:0 mIt ¼ 0:205 m4 zC ¼ �1:55 mIS ¼ 31:403 m4 AG ¼ 4:246 m2

Table 3Buckling load P (kN) of the composite beam of Example 2

Hinged–hinged Fixed–hinged Fixed–fixed

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

674,313 661,137 1,082,168 1,045,833 1,834,828 1,744,819

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 513

its principal shear system of axes (the directions of which coincide with the principal bending ones due to the monosym-metric property of the cross-section). In Table 3 the computed values of the buckling load P for the cases of hinged–hinged,fixed–hinged and fixed–fixed boundary conditions are presented taking into account or ignoring shear deformation effect.From the obtained results the influence of both the aforementioned effect and the boundary conditions is once moreremarkable.

Example 3. To demonstrate the range of applications of the proposed method, three different cases of a nonsymmetric beamof length l ¼ 1:0 m, with a rectangular composite cross-section consisting of a rectangular part (reference material 1:E1 ¼ 2:9� 107 kN=m2, m1 ¼ 0:30Þ stiffened by an L-section of unequal legs (material 2: E2 ¼ 2:1� 108 kN=m2, m2 ¼ 0:30Þ hav-ing three different thickness t values, namely t ¼ t1 ¼ 1 cm (composite cross-section properties AG ¼ 3:188� 10�2 m2,It ¼ 6:800� 10�5 m4, CS ¼ 1:057� 10�8 m6, IS ¼ 1:147� 10�4 m4Þ, t ¼ t2 ¼ 3 cm (composite cross-section propertiesAG ¼ 5:394� 10�2 m2, It ¼ 8:611� 10�5 m4, CS ¼ 2:420� 10�8 m6, IS ¼ 2:024� 10�4 m4Þ, t ¼ t3 ¼ 5 cm (composite cross-section properties AG ¼ 8:181� 10�2 m2, It ¼ 1:137� 10�4 m4, CS ¼ 4:009� 10�8 m6, IS ¼ 2:518� 10�4 m4Þ, as this is shownin Fig. 7, has been studied. Since the proposed method requires the coordinate system CYZ through the cross-section’s cen-troid C to have Y, Z axes parallel to the principal shear axes, in the first column of Tables 4–6 the geometric, the inertia con-stants and the shear deformation coefficients of the three aforementioned cases of the examined cross-section are given with

C

b=10.5cm

h=15

.5cm

t

t

S

Sθ

Principal shear axes (at S)

Y

Z~

~

y

z

Sθ

1

2

2

Fig. 7. Cross-section of the nonsymmetric composite beam of Example 3.

Table 4Geometric, inertia constants and shear deformation coefficients of the composite cross-section of Example 3 for t ¼ t1 ¼ 1 cm

Coordinate system CeY eZ Coordinate system CYZ

IeYeY ¼ 7:7371� 10�5 m4 IYY ¼ 7:8757� 10�5 m4

IeZeZ ¼ 3:5671� 10�5 m4 IZZ ¼ 3:4286� 10�5 m4

IeYeZ ¼ �7:9462� 10�6 m4 IYZ ¼ �1:8532� 10�6 m4

a~y ¼ 1:649 ay ¼ 1:654a~z ¼ 1:426 az ¼ 1:422a~y~z ¼ 3:233� 10�2 ayz ¼ 0:00~yC ¼ 5:92� 10�3 m yC ¼ 6:46� 10�3 m~zC ¼ 4:30� 10�3 m zC ¼ 3:43� 10�3 mhS ¼ 0:14 rad –

Table 5Geometric, inertia constants and shear deformation coefficients of the composite cross-section of Example 3 for t ¼ t2 ¼ 3 cm

Coordinate system CeY eZ Coordinate system CYZ

IeYeY ¼ 1:3362� 10�4 m4 IYY ¼ 1:3805� 10�4 m4

IeZeZ ¼ 5:4703� 10�5 m4 IZZ ¼ 5:0266� 10�5 m4

IeYeZ ¼ �2:6147� 10�5 m4 IYZ ¼ �1:7718� 10�5 m4

a~y ¼ 1:828 ay ¼ 1:831a~z ¼ 1:530 az ¼ 1:527a~y~z ¼ 3:045� 10�2 ayz ¼ 0:00~yC ¼ 1:06� 10�2 m yC ¼ 1:16� 10�2 m~zC ¼ 1:12� 10�2 m zC ¼ 1:01� 10�2 mhS ¼ 0:10 rad –

Table 6Geometric, inertia constants and shear deformation coefficients of the composite cross-section of Example 3 for t ¼ t3 ¼ 5 cm

Coordinate system CeY eZ Coordinate system CYZ

IeYeY ¼ 1:7038� 10�4 m4 IYY ¼ 1:4297� 10�4 m4

IeZeZ ¼ 6:6739� 10�5 m4 IZZ ¼ 9:4151� 10�5 m4

IeYeZ ¼ �3:2453� 10�5 m4 IYZ ¼ �5:6061� 10�5 m4

a~y ¼ 1:516 ay ¼ 1:527a~z ¼ 1:415 az ¼ 1:404a~y~z ¼ �3:457� 10�2 ayz ¼ 0:00~yC ¼ 8:41� 10�3 m yC ¼ 4:94� 10�3 m~zC ¼ 1:04� 10�2 m zC ¼ 1:25� 10�2 mhS ¼ �0:30 rad –

Table 7Buckling load P (kN) of the composite beam of Example 3

Hinged–hinged Fixed–hinged Fixed–fixed

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

Without sheardeformation

With sheardeformation

Presentstudy

FEM Presentstudy

FEM Presentstudy

FEM

t ¼ t1 ¼ 1 cm9791 9364 9430 20,029 18,154 18,824 39,158 33,126 36,128

t ¼ t2 ¼ 3 cm13,392 12,917 12,916 27,373 25,268 25,931 53,422 46,580 50,000

t ¼ t3 ¼ 5 cm16,423 15,994 15,948 33,572 31,654 32,371 65,539 59,202 63,341

Table 8Buckling load P(kN) of the composite beam of Example 3, for various beam lengths

Beam length(m)

Hinged–hinged Fixed–fixed

Without sheardeformation

With sheardeformation

Discrepancy(%)

Without sheardeformation

With sheardeformation

Discrepancy(%)

t ¼ t1 ¼ 1 cm1.00 9791 9364 4.56 39,158 33,126 18.210.90 12,087 11,444 5.62 48,334 39,464 22.480.80 15,297 14,281 7.11 61,155 47,616 28.43

t ¼ t2 ¼ 3 cm1.00 13,392 12,917 3.68 53,422 46,580 14.690.90 16,530 15,811 4.55 65,877 55,776 18.110.80 20,915 19,778 5.75 83,224 67,736 22.87

t ¼ t3 ¼ 5 cm1.00 16,423 15,994 2.68 65,539 59,202 10.700.90 20,271 19,621 3.31 80,835 71,408 13.200.80 25,649 24,618 4.19 102,157 87,554 16.68

514 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

Fig. 8. 3D views of the buckling mode shapes of the FEM solution of the beams of Example 3 (numbers in parentheses correspond to the FEM solution).

E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516 515

respect to an original coordinate system CeY eZ , followed by the evaluation of the angle of rotation hS (Mokos and Sapountza-kis, 2005) giving the final coordinate system CYZ and the new geometric, inertia constants and shear deformation coefficientsgiven in the second column of the aforementioned tables. In Tables 7 and 8 the computed values of the buckling load P forvarious boundary conditions are presented taking into account or ignoring shear deformation effect. More specifically, todemonstrate the accuracy of the proposed method, in Table 7 the obtained results are compared with those obtained froma FEM solution (MSC/NASTRAN, 1999) employing 16,500, 17,600 and 16,000 solid brick elements for the three cases, respec-tively (the corresponding buckling mode shapes of the latter method are presented in Fig. 8), while in Table 8 the discrep-ancy of the obtained results taking into account or ignoring shear deformation effect for various beam lengths demonstratesthe influence of this effect.

5. Concluding remarks

In this paper a boundary element method is developed for the general flexural-torsional linear buckling analysis of Tim-oshenko beams of arbitrarily shaped composite cross-section without initial effects. The main conclusions that can be drawnfrom this investigation are:

516 E.J. Sapountzakis, J.A. Dourakopoulos / Mechanics Research Communications 35 (2008) 497–516

(a) The numerical technique presented in this investigation is well suited for computer aided analysis for compositebeams of arbitrary cross-section, subjected to any linear boundary conditions and to an arbitrarily distributed or con-centrated loading.

(b) The proposed method can treat composite beams of both thin and thick walled cross-sections taking into account thewarping along the thickness of the walls, while the displacements as well as the stress resultants are computed at anycross-section of the beam using the respective integral representations as mathematical formulae.

(c) All basic equations are formulated with respect to the principal shear axes coordinate system, which does not neces-sarily coincide with the principal bending one.

(d) The discrepancy of the obtained results arising from the ignorance of shear deformation especially in thick-walledcross-sections is remarkable and necessitates its inclusion in these cases.

(e) The developed procedure retains the advantages of a BEM solution over a pure domain discretization method since itrequires only boundary discretization.

Acknowledgements

This work has been funded by the Project PENED 2003. The project is cofinanced 75% of public expenditure through EC –European Social Fund and 25% of public expenditure through Ministry of Development – General Secretariat of Research andTechnology and through private sector, under measure 8.3 of OPERATIONAL PROGRAM ‘‘COMPETITIVENESS” in the 3rd Com-munity Support Program.

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