efficient laminated composite beam element subjected to variable axial force for coupled stability...

Acta MechDOI 10.1007/s00707-013-1081-1

Nam-Il Kim · Jaehong Lee

Efficient laminated composite beam element subjectedto variable axial force for coupled stability analysis

Received: 31 July 2013 / Revised: 1 November 2013© Springer-Verlag Wien 2014

Abstract A numerically efficient laminated composite beam element subjected to a variable axial force ispresented for a coupled stability analysis. The analytical technique is used to present the thin-walled laminatedcomposite beam theory considering the transverse shear and the restrained warping-induced shear deformationbased on an orthogonal Cartesian coordinate system. The elastic strain energy and the potential energy dueto the variable axial force are introduced. The equilibrium equations are derived from the energy principle,and explicit expressions for the displacement parameters are presented using the power series expansions ofdisplacement components. Finally, the member stiffness matrix is determined using the force–displacementrelations. In order to verify accuracy and efficiency of the beam element developed in this study, numericalresults are presented and compared with results from other researchers and the finite beam element results,and the detailed finite shell element analysis results using ABAQUS; especially, the influence of variable axialforces, the fiber orientation, and boundary conditions on the buckling behavior of the laminated compositebeams is parametrically investigated.

1 Introduction

Fiber-reinforced plastics have been increasingly used in the fields of civil, architectural, mechanical, andaeronautical engineering due to the evidence that composite materials have many advantages with respect tothe isotropic counterparts. The most well-known advantages of composite materials are their high strengthand stiffness to weight, corrosion resistance, magnetic transparency, and enhanced fatigue life. The design ofcomposite thin-walled members is often governed by stability considerations because of their high strength-to-stiffness ratio and material properties. Thus, the accurate prediction of stability limit state and bucklingcharacteristics is of fundamental importance in the design of thin-walled composite structures.

Up to the present, investigations on the stability behavior of thin-walled beams have been performedextensively, and the theory of thin-walled beams made of isotropic materials was first developed by Vlosov[1] and Gjelsvik [2]. The closed-form solutions for the flexural and torsional buckling of isotropic thin-walledbeams are found in the literature [3,4]. Bauld and Tzeng [5] carried out the first consistent study dealingwith a static analysis of thin-walled orthotropic beams. They used a Vlasov-type linear hypothesis in order toderive the beam theory neglecting shear effects. Based on Vlasov hypothesis, Pandey et al. [6] investigated

N.-I. Kim · J. Lee (B)Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku,Seoul 143-747, South KoreaE-mail: [email protected].: +82-2-34083287

N.-I. KimE-mail: [email protected].: +82-10-66687656

N.-I. Kim, J. Lee

the flexural–torsional stability of thin-walled composite I-section beams under various support conditions. Inorder to study the stability problem of thin-walled composite beams, Lin et al. [7] developed a finite elementhaving 7 DOFs at each node and performed a parametric study of optimum fiber direction for improving thelateral buckling of pultruded I-beams. They considered the influence of the in-plane shear strain on the stabilityof beams. A displacement-based one-dimensional finite element model for the lateral buckling of compositeI- and channel beams was developed by Lee et al. [8] and Lee and Kim [9]. They expressed displacementsover each element as a linear combination of the one-dimensional Lagrange interpolation function for the axialdisplacement and Hermite cubic interpolation function for the flexural and torsional displacements.

On the other hand, several researchers performed a stability analysis of composite beams consideringthe shear deformation effect. Librescu and Song [10] and Oin and Librescu [11] investigated the cross-ply,circumferentially uniform stiffness, and circumferentially asymmetric stiffness for a thin-walled beam basedon the extended Galerkin’s method. They showed that the non-classical effects such as transverse shear andnon-uniformity of membrane shear stiffness could significantly influence the accuracy of the predictions.Bhaskar and Librescu [12] presented a flexural buckling analysis of the extension-twist coupled beam withsingle cell. Matsunaga [13] applied some truncated approximate theories to solve the eigenvalue problemsfor the simply supported laminated composite beam. Back and Will [14] developed the two-, three-, andfour-node isoparametric beam elements using the reduced integration for the buckling analysis of thin-walledbisymmetric and monosymmetric I-beams.

The existing literatures reveal that although a large number of studies dealing with buckling problems oflaminated composite beams have been performed, there was no study reported on the exact solutions for thecoupled buckling analysis of thin-walled laminated composite beams in the literature. It is well known that theexact evaluation (i.e., with arbitrary precision) of the coupled buckling loads of the thin-walled composite beamssubject to variable axial force is very difficult due to the complexities arising from the variable coefficientswithin 7 differential equations.

In this study, in order to calculate the exact (or with arbitrary precision) buckling loads of the thin-walledlaminated composite beams under variable axial force, a numerically efficient beam element is developed basedon the power series methodology. The present beam allows for the shear flexibility due to the bending as wellas the non-uniform torsion warping. Three types of cross sections such as the bisymmetric I-, monosymmetricI-, and channel sections are considered. In order to show accuracy and usefulness of the model and presentseries approach, some comparisons with former finite beam element approaches as well as shell elements ofthe commercial software ABAQUS [15] are made. Particularly, the effect of the variable axial force gradienton the critical buckling loads is investigated.

2 Laminated composite beam theory

2.1 Kinematics

For the stability analysis of thin-walled laminated composite beams, the following assumptions are adopted.

1. The thin-walled composite beam is linearly elastic and prismatic.2. The contour of a cross section does not deform in its plane.3. Each laminate is thin and perfectly bonded.4. Local and prebuckling buckling is not considered.

Figure 1 shows the two sets of coordinate systems, which are mutually interrelated. The first coordinatesystem is the orthogonal Cartesian coordinate system (x, y, z) with the x axis parallel to the length of thebeam. The second one is the local plate coordinate (x, n, s), wherein the n axis is normal to the middle surfaceof the plate element, and the s axis is tangent to the middle surface and is directed along the contour line of thecross section. The (x, n, s) and (x, y, z) coordinate systems are related by an angle of orientation ψ as definedin Fig. 1.

As a consequence of thin-walled beam assumptions, the longitudinal and transverse displacements [16] ofthe arbitrary mid-surface point can be written as

u(x, y, z) = u(x)− ω3(x)y + ω2(x)z + f (x)φ, (1.1)

v(x, y, z) = v(x)− (z − ez)ω1(x), (1.2)

w(x, y, z) = w(x)+ (y − ey)ω1(x) (1.3)

Efficient laminated composite beam element

Fig. 1 Coordinate system in thin-walled cross section

where u, v, w are the rigid body translations, and ω1, ω2, ω3 are the rigid body rotations; f and φ are theparameter defining warping of the cross section and the warping function, respectively; ey and ez are thecoordinates of the pole P in the y and z axes, respectively.

The displacement fields of any generic point in the flanges and web can be expressed with respect to themid-surface displacements as

U f (x, y, z) = u f (x, y)+ n∂w f

∂x(x, y), (2.1)

V f (x, y, z) = v f (x, y) + n∂w f

∂y(x, y), (2.2)

W f (x, y, z) = w f (x, y), (2.3)

and

Uw(x, y, z) = uw(x, z)− n∂vw

∂x(x, z), (3.1)

Vw(x, y, z) = vw(x, z), (3.2)

Ww(x, y, z) = ww(x, z)− n∂vw

∂z(x, z) (3.3)

where the superscripts “ f ” and “w” denote the flange and the web, respectively. The beam strain fieldsassociated with the displacement fields of Eqs. (2) and (3) can be expressed as

εx = ∂U

∂x= εo

x + (z − n cosψ) κy + (y + n sinψ) κz + (φ + nq) κφ, (4.1)

γxy = ∂U

∂y+ ∂V

∂x= γ o

xy − γ oφ (z − ez)+ κxsn cosψ, (4.2)

γxz = ∂U

∂z+ ∂W

∂x= γ o

xz + γ oφ (y − ey)+ κxsn sinψ (4.3)

where εox is the axial strain; γ o

xy and γ oxz are the transverse shear strains in the x−y and x−z planes, respectively;

γ oφ is the warping shear strain; κy and κz are the biaxial curvatures in the y and z directions, respectively; κφ

and κxs are the warping curvature with respect to the shear center and the twist curvature, respectively; q isthe s coordinate of Q. This strain and the curvatures are defined by

εox = u′, κy = ω′

2, κz = −ω′3, κφ = f ′, κxs = 2ω′

1 (5)

where the superscript “prime” indicates differentiation with respect to x .

N.-I. Kim, J. Lee

2.2 Variational formulation

The total potential energy of the composite beam is the sum of the strain energy S and the potential energyG due to the variable axial force as follows:

= S +G . (6)

From the basic assumptions for thin-walled components, the strain energy of the deformed plate is as follows:

S = 1

2

∫

V

(σxεx + τxyγxy + τxzγxz

)dV (7)

where σx , τxy , and τxz are the normal and shear stresses, respectively. Substituting Eq. (4) into Eq. (7), thevariation form of the strain energy in Eq. (7) can be written as

δS =l∫

o

(F1 δε

ox + F2 δγ

oxy + F3 δγ

oxz + M2 δκy + M3 δκz + Mφ δκφ + T δγ o

φ + Mt δκsx

)dx (8)

where δ and l are the variation symbol and the length of the beam, respectively; F1 is the axial force; F2 andF3 are the shear forces in the y and z directions, respectively; M2 and M3 are the bending moments about they and z axes, respectively; Mφ is the bimoment; T and Mt are two contributions to the total twist moment. Theexpressions of the generalized forces and moments acting over the cross section are expressed as follows:

F1 =∫

A

σx dy dz, (9.1)

F2 =∫

A

τxydy dz, (9.2)

F3 =∫

A

τxzdy dz, (9.3)

M2 =∫

A

σx (z − n cosψ) dy dz, (9.4)

M3 = −∫

A

σx (y + n sinψ) dy dz, (9.5)

Mφ =∫

A

σx (φ + n q) dy dz, (9.6)

T =∫

A

[τxz(y − ey)− τxy(z − ez)

]dy dz, (9.7)

Mt =∫

A

(τxyn cosψ + τxzn sinψ

)dy dz. (9.8)

For a laminated composite material, the plate stress–strain relationship [17] for the kth lamina of twoflanges can be expressed in the orthogonal Cartesian coordinate system as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σf

x

σf

y

τf

yz

τf

xz

τf

xy

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

k

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Q f11 Q f

12 0 0 Q f16

Q f12 Q f

22 0 0 Q f26

0 0 Q f44 Q f

45 0

0 0 Q f45 Q f

55 0

Q f16 Q f

26 0 0 Q f66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εfx

εfy

γf

yz

γf

xz

γf

xy

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10)


where Q fi j are the transformed reduced stiffness coefficients of the flanges. Adopting the free stress (σ f

y = 0)

and free strain (ε fy = 0) assumptions in the contour direction, the constitutive equations of the flanges can be

expressed as:⎧⎨⎩σ

fx

τf

xy

⎫⎬⎭

k

=⎡⎣ Q∗ f

11 Q∗ f16

Q∗ f16 Q∗ f

66

⎤⎦

k ⎧⎨⎩ε

fx

γf

xy

⎫⎬⎭ (11)

where the condensed stiffness coefficients for the free stress (σ fy = 0) in the contour direction are given by

Q∗ f11 = Q f

11 − Q f 2

12

Q f22

, (12.1)

Q∗ f16 = Q f

16 − Q f12 Q f

26

Q f22

, (12.2)

Q∗ f66 = Q f

66 − Q f 2

26

Q f22

. (12.3)

For the free stain (ε fy = 0) in the contour direction, they are expressed as

Q∗ f11 = Q f

11, (13.1)

Q∗ f16 = Q f

16, (13.2)

Q∗ f66 = Q f

66. (13.3)

Similarly, the plate stress–strain relationship for the web is expressed as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

σwx

σwz

τwyz

τwxy

τwxz

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

k

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Qw11 Qw

12 0 0 Qw16

Qw12 Qw

22 0 0 Qw26

0 0 Qw44 Qw

45 0

0 0 Qw45 Qw

55 0

Qw16 Qw

26 0 0 Qw66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

k ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

εwx

εwz

γ wyz

γ wxy

γ wxz

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (14)

Based on the free stress (σwz = 0) and free strain (εwz = 0) assumptions in the contour direction, the constitutiveequations for the web are written as:⎧⎨

⎩σwx

τwxz

⎫⎬⎭

k

=⎡⎣ Q∗w

11 Q∗w16

Q∗w16 Q∗w

66

⎤⎦

k ⎧⎨⎩εwx

γ wxz

⎫⎬⎭ . (15)

Substituting Eqs. (11) and (15) into Eq. (9), the constitutive equations for the laminated composite beam canbe expressed as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

F1

M2

−M3

Mφ

Mt

F2

F3

T

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

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

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εox

κy

κz

κφ

κxs

γ oxy

γ oxz

γ oφ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(16)

N.-I. Kim, J. Lee

where Ei j are the laminate stiffnesses that depend on the cross section of the composite. Substitution of Eq. (16)into Eq. (8) leads to the following variation of the strain energy for the shear deformable laminated compositebeam:

δS =l∫

o

[E11u′δu′ + E22 ω

′2δω

′2 + E33 ω

′3δω

′3 + E44 f ′δ f ′ + 4E55ω

′1δω

′1 + E66

(v′ − ω3

)δ(v′ − ω3

)

+ E77(w′ + ω2

)δ(w′ + ω2

)+ E88(ω′

1 + f)δ(ω′

1 + f)+ E12

(ω′

2δu′ + u′δω′

2

)− E13

(ω′


3

)+ E14(

f ′δu′ + u′δ f ′)+ 2E15(ω′


1

)+ E16

{(v′ − ω3

)δu′ + u′δ

(v′ − ω3

)}+ E17{(w′ + ω2

)δu′ + u′δ

(w′ + ω2

)}+ E18

{(ω′

1 + f)δu′ + u′δ

(ω′

1 + f)}− E23

(ω′

3δω′2 + ω′

2δω′3

)+ E24(

f ′δω′2 + ω′

2δ f ′)+ 2E25

(ω′

2δω′1 + ω′

1δω′2

)+ E26{(v′ − ω3

)δω′

2 + ω′2δ(v′ − ω3

)}+ E27

{(w′ + ω2

)δω′

2 + ω′2δ(w′ + ω2

)}+ E28{(ω′

1 + f)δω′

2 + ω′2δ(ω′

1 + f)}

− E34(

f ′δω′3 + ω′

3δ f ′)− 2E35(ω′

3δω′1 + ω′

1δω′3

)− E36{(v′ − ω3

)δω′

3

+ ω′3δ(v′ − ω3

)}−E37{(w′ + ω2

)δω′

3 + ω′3δ(w′ + ω2

)}− E38{(ω′

1 + f)δω′

3 + ω′3δ(ω′

1 + f)}

+ 2E45(

f ′δω′1 + ω′

1δ f ′)+ E46{(v′ − ω3

)δ f ′ + f ′δ

(v′ − ω3

)}+ E47{(w′

+ ω2) δ f ′ + f ′δ(w′ + ω2

)}+ E48{(ω′

1 + f)δ f ′ + f ′δ

(ω′

1 + f)}+ 2E56

{(v′ − ω3

)δω′

1

+ ω′1δ(v′ − ω3

)}+ 2E57{(w′ + ω2

)δω′

1 + ω′1δ(w′ + ω2

)}+ 2E58{(ω′

1 + f)δω′

1

+ ω′1δ(ω′

1 + f)}+ E67

{(w′ + ω2

)δ(v′ − ω3

)+ (v′ − ω3

)δ(w′ + ω2

)}+ E68

{(ω′

1 + f)δ(v′ − ω3

)+ (v′ − ω3

)δ(ω′

1 + f)}+ E78

{(ω′

1 + f)δ(w′ + ω2

)+ (

w′ + ω2)δ(ω′

1 + f)}]

dx . (17)

Next, the variational form of the geometric potential energy due to the variable axial force o F can be expressedfrom the study by Kim et al. [16] as follows:

δ∏G

=l∫

o

[o F{v′δv′ + w′δw′ − ey

(ω′

1δw′ + w′δω′

1

)+ ez(ω′

1δv′ + v′δω′

1

)} +o Mpω′1δω

′1

]dx (18)

where o Mp denotes the second-order coupling torque resulting from the geometric nonlinearity. This propertyis caused by the horizontal component of axial stress due to the inclination of the cross section as a result ofdifferent warping. The twisting moment due to this one will weaken the torsional rigidity if the axial stress iscompressive. If the value of an axial force o F is assumed to be linearly variable along the longitudinal axis ofthe beam, o F and o Mp can be expressed as

o F = ξ x + ζ, (19.1)

o Mp =(

Iy + Iz

A+ e2

y + e2z

)(ξ x + ζ )+

(Iφyy + Iφzz

Iφ

)o Mφ (19.2)

where ξ and ζ are gradient and constant, respectively, of the axial force; Iy and Iz are the moments of inertiaabout y and z axes, respectively; Iφ is the warping moment of inertia; Iφyy and Iφzz are the moments of inertiadue to the warping of the cross section defined by

Iy =∫

A

z2dA, Iz =∫

A

y2dA, Iφ =∫

A

φ2 dA, Iφyy =∫

A

φz2 dA, Iφzz =∫

A

φy2 dA. (20)

In order to derive stability equations and force–displacement relationships of the laminated composite beam,the extended Hamilton’s principle is used by

t2∫

t1

δ

(∏S

+∏G

−∏ext

)dt = 0 (21)


where∏

ext is the work done by external forces. Substituting Eq. (19) into Eq. (18), the following weakstatement is obtained:

t2∫

t1

l∫

o

[F1δu

′ + F2δ(v′ − ω3

)+ F3δ(w′ + ω2

)+ M2δω′2 − M3δω

′3 + Mφδ f ′ + T δ

(ω′

1 + f)+ 2Mtδω

′1

+ (ξ x + ζ ){v′δv′ + w′δw′ − ey

(ω′

1δw′ + w′δω′

1

)+ ez(ω′

1δv′ + v′δω′

1

)}

+(

Iy + Iz

A+ e2

y + e2z

)(ξ x + ζ ) ω′

1δω′1 + Iφyy + Iφzz

Iφo Mφω1δω

′1 − FeδUT

e

]dxdt = 0 (22)

where Ue and Fe signify the element displacement and force vectors, respectively.Finally, stability equations can be derived by integrating the derivative of the varied quantities by part and

by collecting the coefficients of δu, δv, δw, δω1, δω2, δω3, and δ f in Eq. (22) as follows:

F ′1 = 0, (23.1)

F ′2 + (ξ x + ζ )

(v′′ + ezω

′′1

)+ ξ(v′ + ezω

′1

) = 0, (23.2)

F ′3 + (ξ x + ζ )

(w′′ − eyω

′′1

)+ ξ(w′ − eyω

′1

) = 0, (23.3)

2M ′t + T ′ + (ξ x + ζ )

(ezv

′′ − eyw′′)+ ξ

(ezv

′ − eyw′)+

{(Iy + Iz

A+ e2

y + e2z

)(ξ x + ζ )

+(

Iφyy + Iφzz

Iφ

)o Mφ

}ω′′

1 + ξ

(Iy + Iz

A+ e2

y + e2z

)ω′

1 = 0, (23.4)

M ′2 − F3 = 0, (23.5)

M ′3 + F2 = 0, (23.6)

M ′φ + T = 0. (23.7)

The force–displacement relations can also be obtained as the natural boundary conditions as follows:

δu : F1, (24.1)

δv : F2 + (ξ x + ζ )(v′ + ezω

′1

), (24.2)

δw : F3 + (ξ x + ζ )(w′ − eyω

′1

), (24.3)

δω1 : 2Mt + T + (ξ x + ζ )(ezv

′ − eyw′)+

{(Iy + Iz

A+ e2

y + e2z

)(ξ x + ζ ) +

(Iφyy + Iφzz

Iφ

)o Mφ

}ω′

1,

(24.4)

δω2 : M2, (24.5)

δω3 : M3, (24.6)

δ f : Mφ. (24.7)

If the axial force is constant and the material is isotropic, then the cross section is bisymmetric with respectto y and z axes, in which all the coupling effects are neglected. In this case, the stability equations in Eq. (24)can be simplified to the following uncoupled differential equations:

E Au′ = 0, (25.1)

G Ay(v′′ − ω′

3

)+ ζv′′ = 0, (25.2)

G Az(w′′ + ω′

2

)+ ζw′′ = 0, (25.3)

G Jω′′1 + G Ar

(ω′′

1 + f ′)+ ζ(Iy + Iz

)A

ω′1 = 0, (25.4)

E Iyω′2 − G Az

(w′ + ω2

) = 0, (25.5)

E Izω′′3 + G Ay

(v′′ − ω3

) = 0, (25.6)

E Iφ f ′′ − G Ar(ω′

1 + f) = 0 (25.7)

N.-I. Kim, J. Lee

where E and G are the Young’s modulus and the shear modulus, respectively; J is the St. Venant torsionalconstant; Ay, Az , and Ar are the effective shear areas given by

Ay = I 2z∫

sB2

zth

ds, Az = I 2

y∫s

B2y

thds, Ar = I 2

φ∫s

B2r

thds

(26)

where th is the thickness of flanges or web, and

By =∫

s

zthds, Bz =∫

s

ythds, Br =∫

s

φthds. (27)

It is known that the orthotropic beam with bisymmetric cross section may undergo flexural buckling ineither of the two planes of symmetry and torsional buckling. Three distinct approximated buckling loads forthe orthotropic beam with clamped boundary conditions at both ends are given by Kollá and Springer [18],

ζy =(

l2

4π2 E Iz+ 1

G Ay

)−1

, (28.1)

ζz =(

l2

4π2 E Iy+ 1

G Az

)−1

, (28.2)

ζω1 = A

Iy + Iz

{(l2

4π2 E Iφ+ 1

G Ar

)−1

+ G J

} 12

(28.3)

where ζy, ζz , and ζω1 are flexural buckling loads in y and z directions and torsional one, respectively.

3 Power series methodology

In view of the fact that in the present problem the axial force varies continuously along the domain leadingto the second-order differential system with variable coefficients, one has to employ power series producttogether with the power series form of higher derivatives. Thus, the 7 displacement parameters are taken asthe following infinite power series:

{u, v, w, ω1, ω2, ω3, f } =∞∑

i=0

{ai , bi , ci , di , ei , fi , gi } xi (29)

where ai , bi , ci , di , ei , fi , and gi are unknown coefficients. Substituting Eq. (29) into (23) and shifting theindex of power of xi , the stability equations can be expressed in the form of series and are presented in“Appendix A.” These equations are compactly written as follows:

∞∑i=0

{ai+2, bi+2, ci+2, di+2, ei+2, fi+2, gi+2}T

=∞∑

i=0

� {ai , ai+1, bi , bi+1, ci , ci+1, di , di+1, ei , ei+1, fi , fi+1, gi , gi+1}T . (30)

Based on Eq. (30), in case of i = 0, 1 and i = j ( j ≥ 2), we can obtain the following recurrence relations:For i = 0

{a2, b2, c2, d2, e2, f2, g2}T = �0 {a0, a1, b0, b1, c0, c1, d0, d1, e0, e1, f0, f1, g0, g1}T = M2a; (31)

For i = 1

{a3, b3, c3, d3, e3, f3, g3}T = �1 {a1, a2, b1, b2, c1, c2, d1, d2, e1, e2, f1, f2, g1, g2}T = M3a; (32)


For i = j ( j ≥ 2)

{ai+2, bi+2, ci+2, di+2, ei+2, fi+2, gi+2}T

= �i {ai , ai+1, bi , bi+1, ci , ci+1, di , di+1, ei , ei+1, fi , fi+1, gi , gi+1}T = Mi+2a (33)

where a denotes the initial integration constant vector given by

a = {a0, a1, b0, b1, c0, c1, d0, d1, e0, e1, f0, f1, g0, g1}T, (34)

and the detailed 14 × 14 matrix Mi+2 is given in “Appendix B.” It is noted that the terms for ai+2, bi+2, ci+2,di+2, ei+2, fi+2, and gi+2 converge to zero as i → ∞. Theoretically, i → ∞, however, for practical purposesi may be an arbitrary large integer. In this study, the technical computer software Mathematica [19] is used,and the calculation of the coefficients by the recursive relations Eq. (33) is continued until the contribution ofthe next coefficient is less than an arbitrary small number, which has been chosen as 10−8.

Now, the displacement state vector consisting of 14 displacement parameters is defined by

d = ⟨u, u′, v, v′, w,w′, ω1, ω

′1, ω2, ω

′2, ω3, ω

′3, f, f ′⟩T . (35)

The displacement state vector d can be expressed with respect to a by using Eqs. (29) and (33) as follows:

d = Xa (36)

where X denotes the 14×14 matrix function with coefficients of u, v, w, ω1, ω2, ω3, and f . Let q be the nodaldisplacement with 14 DOFs at two ends of the beam as follows:

q = {u(0), v(0), w(0), ω1(0), ω2(0), ω3(0), f (0), u(l), v(l), w(l), ω1(l), ω2(l), ω3(l), f (l)}T . (37)

Substituting coordinates of the ends of components x = 0, l into Eq. (36), the nodal displacement q can beobtained by

q = Ha. (38)

Elimination of a from Eq. (36) using Eq. (38) yields the generalized displacement consisting of 14 displacementcomponents,

d = XH−1q (39)

where XH−1 denotes the interpolation function matrix. Hence, the “exact” interpolation functions for the 14displacement parameters are obtained up to the preset approach.

Next, the member stiffness matrix of the laminated composite beam is evaluated based on the generalizeddisplacement of the beam. Substituting Eq. (39) into the force–displacement relations in Eq. (24), the nodalforces at two ends x = 0, l can be expressed with respect to q as follows:

F = Sd = SXH−1q = Kq (40)

where K is the 14×14 member stiffness matrix. It is noted that the stiffness matrix for the complete structure isassembled as usual from the individual member stiffness. The boundary conditions of the structure are appliedto the member stiffness matrix, resulting in the stiffness matrix K for the structure’s free DOFs. This matrix isdependent on the variable forces o F and o Mp. When the structure buckles, the stiffness matrix of the structurebecomes singular. The determinant of K is determined for a set of increasing values of o F . The buckling loadis bounded between the successive values of o F for which the determinant changes sign. In this study, theRegula-Falsi search method [20] is employed to find buckling loads up to the desired accuracy.

N.-I. Kim, J. Lee

4 Finite element method

For comparison, the isoparametric beam element with 7 DOFs per node is used. The present finite beam elementutilizes the same shape functions for all translational, rotational, and warping displacements. Substituting shapefunctions and cross-sectional properties into Eq. (21) and integrating along the element length, the total potentialenergy of the finite beam element is obtained as

∏F

= 1

2UT

e(Ke + Kg

)Ue − UT

e Fe (41)

where Ke and Kg are the element elastic stiffness and geometric stiffness matrices, respectively, in localcoordinate. The explicit forms of any block matrices Ke and Kg are presented in “Appendix C.” To alleviateshear locking, in the present work, the reduced integration scheme in which the quadrature rule is taken tobe one order lower than the normal one is employed. Then, using the conventional transformation matrix anddirect stiffness method, the global matrix equation is expressed for the stability analysis,

(KE + λ KG)U = 0, (42)

where KE and KG are the global elastic stiffness and geometric stiffness matrices, respectively; λ is theminimum eigenvalue. The critical buckling load can be obtained as the product of the eigenvalue and theapplied load.

5 Results and discussion

A numerical investigation has been carried out to illustrate the accuracy and superiority of the current beamelement for the coupled stability analysis of composite beams with the bisymmetric, monosymmetric, andchannel sections subjected to the variable axial force at the pole of the cross sections. The results obtainedfrom this study are compared with those from other researchers and the finite element solutions using theisoparametric beam elements and the shell elements by ABAQUS.

5.1 Bisymmetric I-beams

The simply supported (SS) bisymmetric I-section with a span of 6.0 m subjected to the constant axial forceis considered. The width of flanges and the height of web are 0.6 and 0.6 m, respectively. The flanges and theweb are made of 4 layers with each ply 7.5 mm in thickness and are assumed to be symmetrically laminatedwith respect to their middle plane. The material is graphite–epoxy (AS4/3501) whose material properties arepresented in Table 1. The subscripts “1” and “2” correspond to directions parallel and perpendicular to thefibers, respectively. In Table 2, the critical buckling loads obtained from the present analysis using a singleelement are compared with the finite element results of Vo and Lee [21] adopting the free stress assumption inthe contour direction based on the local plate coordinate system. It can be found from Table 2 that the presentsolutions are in good agreement with the results in Vo and Lee [21].

Next, the clamped-free (CF) I-beams made with the glass–epoxy material are considered. The total 16layers with equal thickness are used in two flanges and web, and the configuration of the cross section and its

Table 1 Material properties in the numerical examples

Properties Graphite–epoxy Glass-epoxy

Young’s modulus (GPa)E1 144 53.78E2 9.65 17.93

Shear modulus (GPa)G12 = G13 4.14 8.96G23 3.45 3.45

Poisson’s ratioν12 = ν13 0.3 0.25ν23 0.5 0.34


Table 2 Buckling loads of the SS bisymmetric I-beams made with graphite–epoxy (×106 N), (l/h = 10)

Layups This study Vo and Lee [21]

With shear Without shear With shear

σs = 0 εs = 0 σs = 0 σs = 0

[0]4 33.18 33.34 42.69 30.38[0/90]S 19.85 19.87 22.90 20.63[45/− 45]S 4.44 12.99 4.45 4.41

(a)

(b) (c)

Fig. 2 Configurations of cross sections under consideration. a Bisymmetric I-section, b monosymmetric I-section, c channelsection

Table 3 Buckling loads of the CF bisymmetric I-beams made with glass–epoxy (N), (l/h = 15)

Layups This study ABAQUS

σs = 0 εs = 0

[0]16 1,032.6 1,054.5 1,032.2[15/− 15]4S 933.0 973.7 933.3[30/− 30]4S 692.9 777.3 694.1[45/− 45]4S 479.7 567.7 481.1[60/− 60]4S 379.6 425.8 380.7[75/− 75]4S 349.7 364.9 350.6[90/− 90]4S 344.4 351.7 345.2

material constants are given in Fig. 2a and Table 1, respectively. For beams with l/h = 15 under constant axialforce, the buckling loads from this study are presented in Table 3 and compared with results from ABAQUS.For ABAQUS calculation, a total of 180 nine-noded shell elements (S9R5) (6 through the cross section) are

N.-I. Kim, J. Lee

Table 4 Convergence of the finite beam element solutions for the buckling loads of the CF bisymmetric I-beam with [45/−45]4Slayup (N), (l/h = 15, R = 2)

Boundary conditions No. of isoparametric beam elements This study

5 10 20 30 40 50 60

CF 286.63 282.59 281.60 281.41 281.35 281.32 281.30 281.26SS 1,360.1 1,290.7 1,274.2 1,271.2 1,270.1 1,269.6 1,269.4 1,268.8

0 20 40 60 8010 30 50 70 90

Fiber angle(degree)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.1

0.3

0.5

0.7

0.9

1.1

Buc

klin

glo

ad(×

103

N)

R=1R=2R=3R=4R=5

(a)

1 2 3 4 5

Force ratio, R

0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.3

0.5

0.7

0.9

1.1

Red

uctio

n ra

tio

CFSSCSSCCC

(b)

Fig. 3 The buckling loads of CF beams and the reduction ratio of buckling loads with [0]16 layup for bisymmetric I-beams(l/h = 15). a The buckling load (Y). b Reduction ratio of buckling loads with [0]16 layup

used to obtain results. It can be found from Table 3 that the correlation of the present results considering sheareffects based on the free stress (σs = 0) assumption and results from ABAQUS is seen to be excellent for alllayups considered. The maximum deviation with the free strain (εs = 0) assumption is about 18.3 % at 45◦ offiber angle.


Table 5 Buckling loads of the CF monosymmetric I-beams made with glass–epoxy (N), (l/h = 15)


σs = 0 εs = 0

[0]16 1,796.2 1,834.4 1,792.5[15/− 15]4S 1,623.7 1,694.4 1,621.6[30/− 30]4S 1,206.5 1,353.4 1,206.9[45/− 45]4S 835.4 988.6 836.7[60/− 60]4S 661.1 741.6 661.9[75/− 75]4S 608.9 635.5 609.5[90/− 90]4S 599.6 612.4 600.1

Table 6 Convergence of the finite beam element solutions for the buckling loads of the CF monosymmetric I-beam with[45/−45]4S layup (N), (l/h = 15, R = 2)


5 10 20 30 40 50 60

CF 499.11 492.09 490.36 490.04 489.93 489.88 489.85 489.79SS 2,365.3 2,244.7 2,216.1 2,210.8 2,209.0 2,208.1 2,207.7 2,206.6

To show the superiority of the beam element developed by this study, the convergence quality of the finiteelement presented in the previous section is and presented in Table 4, in which the axial force ratio R denotes theaxial force at the right end divided by that at the left end. From Table 4, one can see examined that the bucklingloads calculated with the finite element approach converge monotonically to the exact solutions furnished withthe power series methodology. Even the results with 60 elements do not reach the exact values given by thepower series methodology, although the finite element approach is good for practical purpose. Figure 3 showsthe variation of the buckling loads for CF beams with increase in R and the reduction ratio of buckling loadsfor beams with [0]16 layup with respect to the various boundary conditions. The reduction ratio means theratio of buckling load under variable axial force to that under constant axial force (R = 1). The boundaryconditions under consideration are as follows: the clamped-free (CF), simply–simply (SS), clamped–simply(CS), simply–clamped (SC), and clampan–clamped (CC) boundary conditions. The capital letter Y indicatesthe flexural mode in y direction. It can be observed from Fig. 3a that the buckling loads decrease with anincrease in the fiber angle change ψ since the flexural stiffness E33 decreases with an increase in ψ . Thebuckling load decreases with an increase in R, as expected. It can also be found from Fig. 3b that the effectof the axial force ratio on buckling loads is the largest for CF beam and the lowest for SC beam. Its effect onthe SS beam is the same as that on the CC beam. The reduction ratio is irrespective of ψ and the beam length,although the results are not presented here.

5.2 Monosymmetric I-beams

Next, the monosymmetric I-beams, as seen in Fig. 2b, made with glass–epoxy material are considered. The16 layers with each ply 0.13 mm in thickness are used in flanges and web. For CF beams with l/h of 15 underR = 1, the buckling loads evaluated by this study are presented in Table 5 and are compared with results fromABAQUS. It can be found from Table 5 that the present results based on the σs = 0 assumption are in goodagreement with the results from ABAQUS with the maximum error 0.2 % at 0◦ of fiber angle. Similar to thebisymmetric I-beams, the maximum deviation due to the εs = 0 assumption is 18.3 % at 45◦. In Table 6, theconvergence quality of the isoparametric beam elements is presented for CF and SS beams with R = 2. Itis seen that the isoparametric beam elements more than 60 are required to obtain satisfactory results for bothboundary conditions. Figure 4a, b shows the buckling loads for CF beams and the reduction ratio of beamswith [0]16 layup, respectively. It can be observed from Fig. 4a that the flexural–torsional buckling loads of themonosymmetric I-beam are larger than those of bisymmetric I-beams. It can also be observed from Fig. 4bthat the effect of the axial force ratio is the largest and the lowest for CF and SC beams, respectively, similarto that of bisymmetric I-beams. However, the axial force ratio effect for a CC beam is higher than that for anSS beam due to the coupling between the flexural and torsional modes.

N.-I. Kim, J. Lee

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.0

0.4

0.8

1.2

1.6

2.0

0.2

0.6

1.0

1.4

1.8

Buc

klin

glo

ad( ×

103

N)

R=1R=2R=3R=4R=5

(a)

1 2 3 4 5

Force ratio, R

0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.3

0.5

0.7

0.9

1.1

Red

uctio

n ra

tio

CFSSCSSCCC

(b)

Fig. 4 The buckling loads of CF beams and the reduction ratio of buckling loads with [0]16 layup for monosymmetric I-beams(l/h = 15). a The buckling load (YT). b Reduction ratio of buckling loads with [0]16 layup

Table 7 Buckling loads of the CF channel beams made with glass–epoxy (N), (l/h = 15)


σs = 0 εs = 0

[0]16 2,562.3 2,616.7 2,546.7[15/− 15]4S 2,315.6 2,416.4 2,306.0[30/− 30]4S 1,720.3 1,929.7 1,719.0[45/− 45]4S 1,191.2 1,409.6 1,192.4[60/− 60]4S 942.6 1,057.4 943.0[75/− 75]4S 868.2 906.1 867.8[90/− 90]4S 855.0 873.2 854.2


Table 8 Convergence of the finite beam element solutions for the buckling loads of the CF channel beams with [45/ − 45]4Slayup (N), (l/h = 15, R = 2)


5 10 20 30 40 50 60

CF 711.68 701.66 699.19 698.74 698.58 698.51 698.47 698.38SS 3,372.9 3,200.9 3,160.1 3,152.6 3,150.0 3,148.8 3,148.1 3,146.7

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.0

0.5

1.0

1.5

2.0

2.5

3.0B

uckl

ing

load

(×10

3N

)

R=1R=2R=3R=4R=5

(a)

1 2 3 4 5

Force ratio, R

0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.3

0.5

0.7

0.9

1.1

Red

uctio

n ra

tio

CFSSCSSCCC

(b)

Fig. 5 The buckling loads of CF beams and the reduction ratio of buckling loads with [0]16 layup for channel beams (l/h = 15)a The buckling load (Y). b Reduction ratio of buckling loads with [0]16 layup

5.3 Channel beams

Figure 2c shows the configuration of the channel section under consideration, and the material properties arethe same as in the previous example. For beams with l/h = 15 and CF boundary condition under R = 1,

N.-I. Kim, J. Lee

the calculated buckling loads are presented in Table 7. It is seen that the flexural buckling loads in y directionconsidering σs = 0 assumption are in good agreement with those from ABAQUS. From the convergence testof the finite beam elements shown in Table 8, the considerable numbers of the finite beam elements are neededto reach results obtained from this study using a single element for two CF and SS beams. Furthermore, thebuckling loads of CF beams and the reduction ratio of beams are presented in Fig. 5a,b, respectively. It is seenthat the buckling loads of channel beams are larger than those of the bisymmetric and monosymmetric I-beamsas shown in Fig. 5a. The reduction ratio due to the axial force gradient for an SS beam is the same as that fora CC beam since the coupling effect is negligible. It is also seen that the effect of the axial force gradient onbuckling loads for a CF beam is significantly larger than that of the bisymmetric and monosymmetric I-beams.

6 Conclusions

For the coupled stability analysis of the thin-walled laminated composite beams with bisymmetric, monosym-metric, and channel beams subjected to the variable axial force, the numerical algorithm to calculate the exactbuckling loads is developed. Based on the orthogonal Cartesian coordinate system, the general theoretical beammodel is presented considering the transverse shear and the restrained warping. The calculation process forbuckling loads has been carried out appealing to the power series methodology, which gives exact eigenvalues.The current power series method can offer advantages with respect to conventional finite element methodsusing the approximate shape functions where a lot of elements have to be used to get accurate buckling loads.By numerical examples, the effect of the axial force gradient on the decoupled and coupled buckling loads isnewly investigated with respect to various boundary conditions. The axial force gradient on buckling loads isthe largest for CF boundary conditions and the lowest for an SC one, and especially, it is significant for an CFbeam with channel section.

Acknowledgments The support of the research reported here by the Basic Science Research Program by the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0019373 & 2012R1A2A1A01007405) is gratefully acknowledged.

Appendix A

Stability equations in the form of series:∞∑

i=1

[E11(i + 2)(i + 1)ai+2 + E16(i + 2)(i + 1)bi+2 + E17(i + 2)(i + 1)ci+2 + (2E15 + E18)(i + 2)(i + 1)di+2

+ E12(i + 2)(i + 1)ei+2 + E17(i + 1)ei+1 − E13(i + 2)(i + 1) fi+2 − E16(i + 1) fi+1

+ E14(i + 2)(i + 1)gi+2 +E18(i + 1)gi+1]

xi = 0, (A-1)∞∑

i=1


+ E26(i + 2)(i + 1)ei+2 + E67(i + 1)ei+1 − E36(i + 2)(i + 1) fi+2 − E66(i + 1) fi+1 + E46(i + 2)(i + 1)gi+2

+ 2E68(i + 1)gi+1 + ξ(i + 1)ibi+1 + ζ(i + 2)(i + 1)bi+2 + ξez(i + 1)idi+1 + ζez(i + 2)(i + 1)di+2

+ ξ(i + 1)bi+1 + ξez(i + 1)di+1]

xi = 0, (A-2)∞∑

i=1


+ E27(i + 2)(i + 1)ei+2 + E77(i + 1)ei+1 − E37(i + 2)(i + 1) fi+2 − E67(i + 1) fi+1 + E47(i + 2)(i + 1)gi+2

+ E78(i + 1)gi+1 + ξ(i + 1)ici+1 + ζ(i + 2)(i + 1)ci+2 − ξey(i + 1)idi+1 − ζey(i + 2)(i + 1)di+2

+ ξ(i + 1)ci+1 − ξey(i + 1)di+1]

xi = 0, (A-3)∞∑

i=1

[(2E15 + E18)(i + 2)(i + 1)ai+2 + (2E56 + E68)(i + 2)(i + 1)bi+2 + (2E57 + E78)(i + 2)(i + 1)ci+2

+ (4E55 + 4E58 + E88)(i + 2)(i + 1)di+2 + (2E25 + E28)(i + 2)(i + 1)ei+2 + (2E57 + E78)(i + 1)ei+1

− (2E35 + E38)(i + 2)(i + 1) fi+2 − (2E56 + E68)(i + 1) fi+1 + (2E45 + E48)(i + 2)(i + 1)gi+2


+ (2E58 + E88)(i + 1)gi+1 − ξey(i + 1)ici+1 − ζey(i + 2)(i + 1)ci+2 + ξez(i + 1)ibi+1 + ζez(i + 2)(i + 1)bi+2

− ξey(i + 1)ci+1 + ξez(i + 1)bi+1 + ξ{(

Iy + Iz)/A + e2

y + e2z

}(i + 1)2di+1

+ ζ{(

Iy + Iz)/A + e2

y + e2z

}(i + 2)(i + 1)di+2 + (

Iφyy + Iφzz)/Iφ

o Mφ(i + 2)(i + 1)di+2]

xi = 0, (A-4)

∞∑i=1

[E12(i + 2)(i + 1)ai+2 − E17(i + 1)ai+1 + E26(i + 2)(i + 1)bi+2 − E67(i + 1)bi+1 + E27(i + 2)(i + 1)ci+2

− E77(i + 1)ci+1 + (2E25 + E28)(i + 2)(i + 1)di+2 − (2E57 + E78)(i + 1)di+1 + E22(i + 2)(i + 1)ei+2

− E77ei − E23(i + 2)(i + 1) fi+2 + (E37 − E26)(i + 1) fi+1 + E67 fi + E24(i + 2)(i + 1)gi+2

+ (E28 − E47)(i + 1)gi+1 − E78gi]

xi = 0, (A-5)∞∑

i=1



+ (E37 − E26)(i + 1)ei+1 − E67ei − E33(i + 2)(i + 1) fi+2 + E66 fi + E34(i + 2)(i + 1)gi+2

+ (E38 − E46)(i + 1)gi+1 − E68gi]

xi = 0, (A-6)∞∑

i=1



− (E28 − E47)(i + 1)ei+1 − E78ei − E34(i + 2)(i + 1) fi+2 + (E38 − E46)(i + 1) fi+1 + E68 fi

+ E44(i + 2)(i + 1)gi+2 −E88gi ] xi = 0. (A-7)

Appendix B

Expressions of matrix Mi+2:

Mi+2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Mi (1, 1) Mi (1, 2) Mi (1, 3) Mi (1, 12) Mi (1, 13) Mi (1, 14)

Mi+1(1, 1) Mi+1(1, 2) Mi+1(1, 3) Mi+1(1, 12) Mi+1(1, 13) Mi+1(1, 14)

Mi (2, 1) Mi (2, 2) Mi (2, 3) Mi (2, 12) Mi (2, 13) Mi (2, 14)

Mi+1(2, 1) Mi+1(2, 2) Mi+1(2, 3) Mi+1(2, 12) Mi+1(2, 13) Mi+1(2, 14)

Mi (3, 1) Mi (3, 2) Mi (3, 3) Mi (3, 12) Mi (3, 13) Mi (3, 14)

Mi+1(3, 1) Mi+1(3, 2) Mi+1(3, 3) Mi+1(3, 12) Mi+1(3, 13) Mi+1(3, 14)

Mi (4, 1) Mi (4, 2) Mi (4, 3) · · · · · · · · · · · · Mi (4, 12) Mi (4, 13) Mi (4, 14)

Mi+1(4, 1) Mi+1(4, 2) Mi+1(4, 3) Mi+1(4, 12) Mi+1(4, 13) Mi+1(4, 14)

Mi (5, 1) Mi (5, 2) Mi (5, 3) Mi (5, 12) Mi (5, 13) Mi (5, 14)

Mi+1(5, 1) Mi+1(5, 2) Mi+1(5, 3) Mi+1(5, 12) Mi+1(5, 13) Mi+1(5, 14)

Mi (6, 1) Mi (6, 2) Mi (6, 3) Mi (6, 12) Mi (6, 13) Mi (6, 14)

Mi+1(6, 1) Mi+1(6, 2) Mi+1(6, 3) Mi+1(6, 12) Mi+1(6, 13) Mi+1(6, 14)

Mi (7, 1) Mi (7, 2) Mi (7, 3) Mi (7, 12) Mi (7, 13) Mi (7, 14)

Mi+1(7, 1) Mi+1(7, 2) Mi+1(7, 3) Mi+1(7, 12) Mi+1(7, 13) Mi+1(7, 14)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-8)

N.-I. Kim, J. Lee

where Mi ( j, k) denotes the component of Mi at j th row and kth column, and the matrix M1 is expressed asfollows:

M1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

· 1 · · · · · · · · · · · ·· · · · · · · · · · · · · ·· · · 1 · · · · · · · · · ·· · · · · · · · · · · · · ·· · · · · 1 · · · · · · · ·· · · · · · · · · · · · · ·· · · · · · · 1 · · · · · ·· · · · · · · · · · · · · ·· · · · · · · · · 1 · · · ·· · · · · · · · · · · · · ·· · · · · · · · · · · 1 · ·· · · · · · · · · · · · · ·· · · · · · · · · · · · · 1· · · · · · · · · · · · · ·

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (A-9)

Appendix C

Terms in the “upper-triangle” part of the symmetric linear and geometric stiffness matrices:

ke11 =

L∫

o

E11 N ′αN ′

β dx,

ke12 =

L∫

o

E16 N ′αN ′

β dx,

ke13 =

L∫

o

E17 N ′αN ′

β dx,

ke14 =

L∫

o

(2E15 + E18) N ′αN ′

β dx,

ke15 =

L∫

o

(E12 N ′

αN ′β + E17 N ′

αNβ)

dx,

ke16 = −

L∫

o

(E13 N ′

αN ′β + E16 N ′

αNβ)

dx,

ke17 =

L∫

o

(E14 N ′

αN ′β + E18 N ′

αNβ)

dx,

ke22 =

L∫

o

E66 N ′αN ′

β dx,


ke23 =

L∫

o

E67 N ′αN ′

β dx,

ke24 =

L∫

o

(2E56 + E68) N ′αN ′

β dx,

ke25 =

L∫

o

(E26 N ′

αN ′β + E67 N ′

αNβ)

dx,

ke26 = −

L∫

o

(E36 N ′

αN ′β + E66 N ′

αNβ)

dx,

ke27 =

L∫

o

(E46 N ′

αN ′β + E68 N ′

αNβ)

dx,

ke33 =

L∫

o

E77 N ′αN ′

β dx,

ke34 =

L∫

o

(2E57 + E78) N ′αN ′

β dx,

ke35 =

L∫

o

(E27 N ′

αN ′β + E77 N ′

αNβ)

dx . (A-10)

ke36 = −

L∫

o

(E37 N ′

αN ′β + E67 N ′

αNβ)

dx,

ke37 =

L∫

o

(E47 N ′

αN ′β + E78 N ′

αNβ)

dx,

ke44 =

L∫

o

(4E55 + 4E58 + E88) N ′αN ′

β dx,

ke45 =

L∫

o

[(2E25 + E28) N ′

αN ′β + (2E57 + E78) N ′

αNβ]

dx,

ke46 = −

L∫

o

[(2E35 + E38) N ′

αN ′β + (2E56 + E68) N ′

αNβ]

dx,

ke47 =

L∫

o

[(2E45 + E48) N ′

αN ′β + (2E58 + E88) N ′

αNβ]

dx,

ke55 =

L∫

o

[E22 N ′

αN ′β + E27

(N ′αNβ + NαN ′

β

)+ E77 NαNβ

]dxm,

N.-I. Kim, J. Lee

ke56 = −

L∫

o

[E23 N ′

αN ′β + E26 N ′

αNβ + E37 NαN ′β + E67 NαNβ

]dx,

ke57 =

L∫

o

[E24 N ′

αN ′β + E28 N ′


]dx,

ke66 =

L∫

o

[E33 N ′

αN ′β + E36


β

)+ E66 NαNβ

]dx,

ke67 = −

L∫

o

[E34 N ′

αN ′β + E38 N ′


]dx,

ke77 =

L∫

o

[E44 N ′

αN ′β + E48


β

)+ E88 NαNβ

]dx,

kg22 =

L∫

o

N ′αN ′

β dx,

kg24 =

L∫

o

ez N ′αN ′

β dx,

kg33 =

L∫

o

N ′αN ′

β dx,

kg34 = −

L∫

o

ey N ′αN ′

β dx,

kg44 =

L∫

o

(Iy + Iz

A+ e2

y + e2z

)N ′αN ′

β dx .

References

1. Vlasov, V.Z.: Thin Walled Elastic Beams. 2nd edn. Israel Program for Scientific Transactions, Jerusalem (1961)2. Gjelsvik, A.: The Theory of Thin-Walled Bars. Wiley, New York (1981)3. Timoshenko, S.V., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961)4. Trahair, N.S.: Flexural–Torsional Buckling of Structures. CRC Press, London (1993)5. Bauld, N.R., Tzeng, L.: A Vlasov theory for fiber-reinforced beams with thin-walled open cross sections. Int. J. Solids

Struct. 20, 277–297 (1984)6. Pandey, M.D., Kabir, M.Z., Sherbourne, A.N.: Flexural–torsional stability of thin-walled composite I-section beams. Com-

pos. Eng. 5, 321–342 (1995)7. Lin, Z.M., Polyzois, D., Shah, A.: Stability of thin-walled pultruded structural members by finite element method. Thin-

Walled Struct. 24, 1–18 (1996)8. Lee, J., Kim, S.E., Hong, K.: Lateral buckling of I-section composite beams. Eng. Struct. 24, 955–964 (2002)9. Lee, J., Kim, S.E.: Lateral buckling analysis of thin-walled laminated channel-section beams. Compos. Struct. 56, 391–

399 (2002)10. Librescu, L., Song, O.: Thin-Walled Composite Beams. Springer, Netherlands (2006)11. Qin, Z., Librescu, L.: On a shear deformable theory of anisotropic thin-walled beams: Further contribution and valida-

tions. Compos. Struct. 56, 345–358 (2002)12. Bhaskar, K., Librescu, L.: Buckling under axial compression of thin-walled composite beams exhibiting extension-twist

coupling. Compos. Struct. 31, 203–212 (1995)


13. Matsunaga, H.: Vibration and buckling of multilayered composite beams according to higher order deformation theories. J.Sound Vib. 246, 47–62 (2001)

14. Back, S.Y., Will, K.M.: Shear–flexible thin-walled element for composite I-beams. Eng. Struct. 30, 1447–1458 (2008)15. ABAQUS: Standard user’s manual, Ver. 6.1, Hibbit, Kalsson & Sorensen Inc. (2003)16. Kim, M.Y., Chang, S.P., Kim, S.B.: Spatial stability and free vibration of shear flexible thin-walled elastic beams. I: Analytical

approach. Int. J. Numer. Methods Eng. 37, 4097–4115 (1994)17. Wang, S.: Free vibration analysis of skew fibre-reinforced composite laminates based on first-order shear deformation plate

theory. Comput. Struct. 63, 525–538 (1997)18. Kollá, L.P., Springer, G.S.: Mechanics of Composite Structures. Cambridge University Press, Cambridge (2003)19. Wolfram Mathematica 7. Wolfram Research Inc., Illinois (2009)20. Wendroff, B.: Theoretical Numerical Analysis. Academic Press, New York (1966)21. Vo, T.P., Lee, J.: On sixfold coupled buckling of thin-walled composite beams. Compos. Struct. 90, 295–303 (2009)

efficient laminated composite beam element subjected to variable axial force for coupled stability...

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