32) Free Vibration of Axially-loaded Thin-walled Composite Box Beams

Composite Structures 90 (2009) 233–241

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Composite Structures

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Free vibration of axially loaded thin-walled composite box beams

Thuc Phuong Vo, Jaehong Lee *

Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Republic of Korea

a r t i c l e i n f o

Article history:Available online 25 March 2009

Keywords:Thin-walled composite beamClassical lamination theoryFlexural–torsional coupled vibrationAxial force

0263-8223/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.03.010

* Corresponding author. Tel.: +82 2 3408 3287; faxE-mail address: [email protected] (J. Lee).

a b s t r a c t

A general analytical model applicable to flexural–torsional coupled vibration of thin-walled compositebox beams with arbitrary lay-ups under a constant axial force has been presented. This model is basedon the classical lamination theory and accounts for all the structural coupling coming from the materialanisotropy. Equations of motion are derived from the Hamilton’s principle. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtainedfor thin-walled composite box beams to investigate the effects of axial force, fiber orientation and mod-ulus ratio on the natural frequencies, load–frequency interaction curves and corresponding vibrationmode shapes.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Fiber-reinforced composite materials have been used over thepast few decades in a variety of structures. Composites have manydesirable characteristics, such as high ratio of stiffness andstrength to weight, corrosion resistance and magnetic transpar-ency. Thin-walled structural shapes made up of composite materi-als, which are usually produced by pultrusion, are beingincreasingly used in many engineering fields. However, the struc-tural behavior is very complex due to coupling effects as well aswarping-torsion and therefore, the accurate prediction of stabilitylimit state and dynamic characteristics is of the fundamentalimportance in the design of composite structures.

The theory of thin-walled members made of isotropic materialswas first developed by Vlasov [1] and Gjelsvik [2]. Up to the pres-ent, investigation into the stability and vibrational behavior ofthese members has received widespread attention and has beencarried out extensively. Closed-form solution for flexural and tor-sional natural frequencies, critical buckling loads of isotropicthin-walled bars are found in the literature (Timoshenko [3,4]and Trahair [5]). For some practical applications, earlier studieshave shown that the effect of axial force on the natural frequenciesand mode shapes is more pronounced than those of the sheardeformation and rotary inertia. Although a large number of studieshas been performed on the dynamic characteristics of axiallyloaded isotropic thin-walled beams, it should be noted that onlya few deal with thin-walled composite structures with arbitrarylay-ups. A literature survey on the subject shows that there ap-pears some works reported on the free vibration of axially loadedclosed-section thin-walled composite beams. Many numerical

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techniques have been used to solve the dynamic analysis of thesemembers. One of the most effective approach is to derive the exactstiffness matrices based on the solution of the differential equationof beam. Most of those studies adopted an analytical method thatrequired explicit expressions of exact displacement functions forgoverning equations. Banerjee [6,7] applied the exact dynamicstiffness matrix to perform the free vibration analysis of axiallyloaded composite Timoshenko beams. The works of Li et al.[8–11] deserved special attention because they developed the ana-lytical solution to determine the flexure–torsion coupled dynamicresponses of axially loaded thin-walled composite beam underconcentrated, distributed time-dependent loads and external sto-chastic excitations. The influences of axial force, Poisson effect, ax-ial deformation, shear deformation and rotary inertia werediscussed in their research. Kaya and Ozgumus [12] introducedthe differential transform method (DTM) to analyse the free vibra-tion response of an axially loaded, closed-section composite Timo-shenko beam which featured material coupling between flapwisebending and torsional vibrations. The effects of the bending–tor-sion coupling, the axial force and the slenderness ratio on the nat-ural frequencies were inspected. In the research of Banerjee and Liet al. and Kaya and Ozgumus [6–12], it was very effective in savingthe computing time due to the closed-form solution which can beeasily derived by the help of symbolic computation. However, theanalytical operations were often too complex to yield exact dis-placement functions in the case of solving a system of simulta-neous ordinary differential equations with many variables.Additionally, they considered only a cantilever glass-epoxy com-posite beam with rectangular cross-section in the numerical exam-ples. By using finite element method, Bank and Kao [13] analysedfree and forced vibration of thin-walled fibre reinforced compositematerial beams by using the Timoshenko beam theory. Song et al.[14] carried out the vibration and stability of pretwisted spinning

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234 T.P. Vo, J. Lee / Composite Structures 90 (2009) 233–241

thin-walled composite beams featuring bending–bending elasticcoupling. Recently, Cortinez et al. [15,16] presented a theoreticalmodel for the dynamic analysis of thin-walled composite beamswith initial stresses. Machado et al. [17] determined the regionsof dynamic instability of simply supported thin-walled compositebeam subjected to an axial excitation. The analysis was based ona small strain and moderate rotation theory, which was formulatedthrough the adoption of a second-order displacement field. In theirresearch [15–17], thin-walled composite beams for both open andclosed cross-sections and the shear flexibility (bending, non-uni-form warping) were incorporated. However, it was strictly validfor symmetric balanced laminates and especially orthotropic lam-inates. By using a boundary element method, Sapountzakis andTsiatas [18] solved the general flexural–torsional buckling andvibration problems of composite Euler–Bernoulli beams of arbi-trarily shaped cross-section. This method overcame the shortcom-ing of possible thin tube theory solution, which its utilization hadbeen proven to be prohibitive even in thin-walled homogeneoussections.

In this paper, which is an extension of the authors’ previousworks [19–21], flexural–torsional coupled vibration of thin-walledcomposite box beams with arbitrary lay-ups under a constant axialforce is presented. This model is based on the classical laminationtheory, and accounts for all the structural coupling coming fromthe material anisotropy. Equations of motion are derived fromthe Hamilton’s principle. A displacement-based one-dimensionalfinite element model is developed to solve the problem. Numericalresults are obtained for thin-walled composite box beams to inves-tigate the effects of axial force, fiber orientation and modulus ratioon the natural frequencies, load–frequency interaction curves andcorresponding vibration mode shapes.

2. Kinematics

The theoretical developments presented in this paper requiretwo sets of coordinate systems which are mutually interrelated.The first coordinate system is the orthogonal Cartesian coordinatesystem ðx; y; zÞ, for which the x- and y-axes lie in the plane of thecross-section and the z-axis parallel to the longitudinal axis ofthe beam. The second coordinate system is the local plate coordi-nate ðn; s; zÞ as shown in Fig. 1, wherein the n axis is normal tothe middle surface of a plate element, the s axis is tangent to themiddle surface and is directed along the contour line of thecross-section. The ðn; s; zÞ and ðx; y; zÞ coordinate systems are re-lated through an angle of orientation h as defined in Fig. 1. PointP is called the pole axis, through which the axis parallel to the zaxis is called the pole axis.

Fig. 1. Definition of coordinates in thin-walled closed sections.

To derive the analytical model for a thin-walled compositebeam, the following assumptions are made:

1. The contour of the thin wall does not deform in its own plane.2. The linear shear strain �csz of the middle surface is to have the

same distribution in the contour direction as it does in the St.Venant torsion in each element.

3. The Kirchhoff–Love assumption in classical plate theoryremains valid for laminated composite thin-walled beams.

4. Each laminate is thin and perfectly bonded.5. Local buckling is not considered.

According to assumption 1, the midsurface displacement com-ponents �u; �v at a point A in the contour coordinate system can beexpressed in terms of a displacements U;V of the pole P in thex; y directions, respectively, and the rotation angle U about the poleaxis,

�uðs; zÞ ¼ UðzÞ sin hðsÞ � VðzÞ cos hðsÞ �UðzÞqðsÞ ð1aÞ�vðs; zÞ ¼ UðzÞ cos hðsÞ þ VðzÞ sin hðsÞ þUðzÞrðsÞ ð1bÞ

These equations apply to the whole contour. The out-of-planeshell displacement �w can now be found from the assumption 2.For each element of middle surface, the shear strain become

�csz ¼@�v@zþ @

�w@s¼ U0ðzÞ FðsÞ

tðsÞ ð2Þ

where tðsÞ is thickness of contour box section, FðsÞ is the St. Venantcircuit shear flow.

After substituting for �v from Eq. (1b) and considering the fol-lowing geometric relations,

dx ¼ ds cos h ð3aÞdy ¼ ds sin h ð3bÞ

Eq. (2) can be integrated with respect to s from the origin to anarbitrary point on the contour,

�wðs; zÞ ¼WðzÞ � U0ðzÞxðsÞ � V 0ðzÞyðsÞ �U0ðzÞxðsÞ ð4Þ

where differentiation with respect to the axial coordinate z is de-noted by primes (0); W represents the average axial displacementof the beam in the z-direction; x and y are the coordinates of thecontour in the ðx; y; zÞ coordinate system; and x is the so-called sec-torial coordinate or warping function given by

xðsÞ ¼Z s

s�

rðsÞ � FðsÞtðsÞ

� �ds ð5aÞI

i

FðsÞtðsÞ ds ¼ 2Ai i ¼ 1; . . . ;n ð5bÞ

where rðsÞ is height of a triangle with the base ds; Ai is the area cir-cumscribed by the contour of the i circuit. The explicit forms of xðsÞand FðsÞ for box section are given in Ref. [19].

The displacement components u; v;w representing thedeformation of any generic point on the profile section are givenwith respect to the midsurface displacements �u; �v ; �w by theassumption 3.

uðs; z; nÞ ¼ �uðs; zÞ ð6aÞ

vðs; z;nÞ ¼ �vðs; zÞ � n@�uðs; zÞ@s

ð6bÞ

wðs; z;nÞ ¼ �wðs; zÞ � n@�uðs; zÞ@z

ð6cÞ

The strains associated with the small-displacement theory ofelasticity are given by

T.P. Vo, J. Lee / Composite Structures 90 (2009) 233–241 235

�s ¼ ��s þ n�js ð7aÞ�z ¼ ��z þ n�jz ð7bÞcsz ¼ �csz þ n�jsz ð7cÞ

where

��s ¼@�v@s

; ��z ¼@ �w@z

ð8aÞ

�js ¼ �@2�u@z2 ; �jz ¼ �

@2�u@z2 ; �jsz ¼ �2

@2�u@s@z

ð8bÞ

All the other strains are identically zero. In Eq. (8b), ��s and �js areassumed to be zero. ��z, �jz and �jsz are midsurface axial strain andbiaxial curvature of the shell, respectively. The above shell strainscan be converted to beam strain components by substituting Eqs.(1b), (4) and (6c) into Eq. (8b) as

��z ¼ ��z þ xjy þ yjx þxjx ð9aÞ�jz ¼ jy sin h� jx cos h� jxq ð9bÞ�jsz ¼ 2�vsz ¼ jsz ð9cÞ

where ��z ;jx;jy;jx and jsz are axial strain, biaxial curvaturesin the x- and y-direction, warping curvature with respect to theshear center, and twisting curvature in the beam, respectivelydefined as

��z ¼W 0 ð10aÞjx ¼ �V 00 ð10bÞjy ¼ �U00 ð10cÞjx ¼ �U00 ð10dÞjsz ¼ 2U0 ð10eÞ

The resulting strains can be obtained from Eqs. (7c) and (9c) as

�z ¼ ��z þ ðxþ n sin hÞjy þ ðy� n cos hÞjx þ ðx� nqÞjx ð11aÞ

csz ¼ nþ F2t

� �jsz ð11bÞ

3. Variational formulation

The total potential energy of the system can be stated, in itsbuckled shape, as

P ¼ UþV ð12Þ

where U is the strain energy

U ¼ 12

Zvðrz�z þ rszcszÞdv ð13Þ

After substituting Eq. (11b) into Eq. (13)

U ¼ 12

Zv

�rz ��z þ ðxþ n sin hÞjy þ ðy� n cos hÞjx þ ðx� nqÞjx� �

þrsz nþ F2t

� �jsz

dv ð14Þ

The variation of strain energy can be stated as

dU ¼Z l

0ðNzd�z þMydjy þMxdjx þMxdjx þMtdjszÞdz ð15Þ

where Nz;Mx;My;Mx;Mt are axial force, bending moments in the x-and y-direction, warping moment (bimoment), and torsional mo-ment with respect to the centroid, respectively, defined by integrat-ing over the cross-sectional area A as

Nz ¼Z

Arzdsdn ð16aÞ

My ¼Z

Arzðxþ n sin hÞdsdn ð16bÞ

Mx ¼Z

Arzðy� n cos hÞdsdn ð16cÞ

Mx ¼Z

Arzðx� nqÞdsdn ð16dÞ

Mt ¼Z

Arsz nþ F

2t

� �dsdn ð16eÞ

The potential of in-plane loads V due to transverse deflection

V ¼ 12

Zv

�r0z ½ðu0Þ

2 þ ðv 0Þ2�dv ð17Þ

where �r0z is the averaged constant in-plane edge axial stress, de-

fined by �r0z ¼ P0=A. The variation of the potential of in-plane loads

at the centroid is expressed by substituting the assumed displace-ment field into Eq. (17) as

dV ¼Z

v

P0

AU0dU0 þ V 0dV 0 þ ðq2 þ r2 þ 2rnþ n2ÞU0dU0 þ ðU0dU0�

þ U0dU0Þ½n cos h� ðy� ypÞ� þ ðU0dV 0 þ V 0dU0Þ½n cos h

þ x� xpÞ� �

dv ð18Þ

The kinetic energy of the system is given by

T ¼ 12

Zvqð _u2 þ _v2 þ _w2Þdv ð19Þ

where q is a density.The variation of the kinetic energy is expressed by substituting

the assumed displacement field into Eq. (19) as

dT ¼Z

vq _Ud _U þ _Vd _V þ _Wd _W þ ðq2 þ r2 þ 2rnþ n2Þ _Ud _Un

þ ð _Ud _U þ _Ud _UÞ½n cos h� ðy� ypÞ�

þ ð _Ud _V þ _Vd _UÞ½n cos hþ ðx� xpÞ�o

dv ð20Þ

In order to derive the equations of motion, Hamilton’s principleis used

dZ t2

t1

ðT�PÞdt ¼ 0 ð21Þ

Substituting Eqs. (15), (18) and (20) into Eq. (21), the followingweak statement is obtained

0 ¼Z t2

t1

Z l

0

�m0

_Wd _W þ ½m0_Uþ ðmc �my þm0ypÞ _U�d _U

þ ½m0_V þ ðms þmx �m0xpÞ _U�d _V þ ½ðmc �my þm0ypÞ _U

þ ðms þmx �m0xpÞ _V þ ðmp þm2 þ 2mxÞ _U�d _U

� P0 dU0ðU0 þU0ypÞ þ dV 0ðV 0 �U0xpÞ þ dU0ðU0 Ip

AþU0yp � V 0xpÞ

� ��

�NzdW 0 þMydU00 þMxdV 00 þMxdU00 � 2MtdU�

dzdt ð22Þ

The explicit expressions of inertia coefficients for composite boxsection are given in Ref. [21].

4. Constitutive equations

The constitutive equations of a kth orthotropic lamina in thelaminate co-ordinate system of section are given by

rz

rsz

� k

¼�Q �11

�Q �16�Q �16

�Q �66

" #k�z

csz

� ð23Þ


where Q �ij are transformed reduced stiffnesses. The transformed re-duced stiffnesses can be calculated from the transformed stiffnessesbased on the plane stress assumption and plane strain assumption.More detailed explanation can be found in Ref. [22].

The constitutive equations for bar forces and bar strains are ob-tained by using Eqs. (11b), (16e) and (23)

Nz

My

Mx

Mx

Mt

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

9>>>>>>=>>>>>>;¼

E11 E12 E13 E14 E15

E22 E23 E24 E25

E33 E34 E35

E44 E45

sym: E55

26666664

37777775

��zjy

jx

jx

jsz

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

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

ð24Þ

where Eij are stiffnesses of thin-walled composite beams and givenin Ref. [19].

5. Governing equations of motion

The governing equations of motion of the present study can bederived by integrating the derivatives of the varied quantities byparts and collecting the coefficients of dW; dU; dV and dU

N0z ¼ m0€W ð25aÞ

M00y þ P0ðU00 þU00ypÞ ¼ m0

€U þ ðmc �my þm0ypÞ€U ð25bÞM00

x þ P0ðV 00 �U00xpÞ ¼ m0€V þ ðms þmx �m0xpÞ€U ð25cÞ

M00x þ 2M0

t þ P0ðU00Ip

Aþ U00yp � V 00xpÞ

¼ ðmc �my þm0ypÞ€U þ ðms þmx �m0xpÞ€Vþ ðmp þm2 þ 2mxÞ€U ð25dÞ

The natural boundary conditions are of the form

dW : Nz ¼ P0 ð26aÞdU : My ¼ M0

y ð26bÞdU0 : M0

y ¼ M00y ð26cÞ

dV : Mx ¼ M0x ð26dÞ

dV 0 : M0x ¼ M00

x ð26eÞdU : M0

x þ 2Mt ¼ M00x ð26fÞ

dU0 : Mx ¼ M0x ð26gÞ

where P0;M00y ;M

0y ;M

00x ;M

0x ;M

00x and M0

x are prescribed values.Eq. (25a)–(25d) is most general form for flexural–torsional

vibration of thin-walled composite beams under a constant axialforce, and the dependent variables, W;U;V and U are fully coupled.By substituting Eqs. (10e) and (24) into Eq. (25a)–(25d), the expli-cit form of governing equations of motion can be obtained. If all thecoupling effects are neglected and the cross section is symmetricalwith respect to both x- and the y-axes, Eq. (25a)–(25d) can be sim-plified to the uncoupled differential equations as

ðEAÞcomW 00 ¼ qA €W ð27aÞ� ðEIyÞcomUiv þ P0U00 ¼ qA€U ð27bÞ� ðEIxÞcomViv þ P0V 00 ¼ qA€V ð27cÞ

� ðEIxÞcomUiv þ ðGJÞcom þ P0Ip

A

� �U00 ¼ qIp

€U ð27dÞ

From above equations, ðEAÞcom represents axial rigidity, ðEIxÞcom

and ðEIyÞcom represent flexural rigidities with respect to x- and y-axes, ðEIxÞcom represents warping rigidity, and ðGJÞcom, representstorsional rigidity of thin-walled composite beams, respectively,written as

ðEAÞcom ¼ E11 ð28aÞðEIyÞcom ¼ E22 ð28bÞðEIxÞcom ¼ E33 ð28cÞðEIxÞcom ¼ E44 ð28dÞðGJÞcom ¼ 4E55 ð28eÞ

It is well known that the three distinct load–frequency interac-tion curves corresponding to flexural buckling and natural fre-quencies in the x- and y-direction, and torsional buckling andnatural frequency, respectively. They are given by the orthotropysolution for simply supported boundary conditions [23]

xxxn ¼ xxn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� P0

Px

sð29aÞ

xyyn¼ xyn


Py

sð29bÞ

xhhn ¼ xhn


Ph

sð29cÞ

where xxn ;xynand xhn are corresponding flexural natural frequen-

cies in the x- and y-direction and torsional natural frequency [4].

xxn ¼n2p2

l2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEIyÞcom

qA

sð30aÞ

xyn¼ n2p2

l2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEIxÞcom

qA

sð30bÞ

xhn ¼npl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

qIp

n2p2

l2 ðEIxÞcom þ ðGJÞcom

� �sð30cÞ

and Px; Py and Ph are also corresponding flexural buckling loads inthe x- and y-direction and torsional buckling load [5], respectively.

Px ¼p2ðEIyÞcom

l2ð31aÞ

Py ¼p2ðEIxÞcom

l2 ð31bÞ

Ph ¼AIp

p2ðEIxÞcom

l2 þ ðGJÞcom

� �ð31cÞ

6. Finite element formulation

The present theory for thin-walled composite beams describedin the previous section was implemented via a displacement-basedfinite element method. The generalized displacements are ex-pressed over each element as a linear combination of the one-dimensional Lagrange interpolation function Wj and Hermite-cubicinterpolation function wj associated with node j and the nodalvalues

W ¼Xn

j¼1

wjWj ð32aÞ

U ¼Xn

j¼1

ujwj ð32bÞ

V ¼Xn

j¼1

v jwj ð32cÞ

U ¼Xn

j¼1

/jwj ð32dÞ


Substituting these expressions into the weak statement in Eq.(18), the finite element model of a typical element can be ex-pressed as the standard eigenvalue problem

ð½K� � P0½G� �x2½M�ÞfDg ¼ f0g ð33Þ

where ½K�; ½G� and ½M� are the element stiffness matrix, the elementgeometric stiffness matrix and the element mass matrix, respec-tively. The explicit forms of ½K�; ½G� and ½M� are given in Refs. [19–21].

In Eq. (33), fDg is the eigenvector of nodal displacements corre-sponding to an eigenvalue

fDg ¼ fW U VUgT ð34Þ

7. Numerical examples

A thin-walled composite box beam with length l = 8 m is con-sidered to investigate the effects of axial force, fiber orientationand modulus ratio on the natural frequencies, load–frequencyinteraction curves and the corresponding mode shapes. The geom-etry and stacking sequences of the box section are shown in Fig. 2,and the following engineering constants are used

E1=E2 ¼ 25;G12=E2 ¼ 0:6; m12 ¼ 0:25 ð35Þ

For convenience, the following nondimensional axial force and nat-ural frequency are used

P ¼ Pl2

b31tE2

ð36aÞ

�x ¼ xl2

b1

ffiffiffiffiffiqE2

rð36bÞ

The left and right webs are angle-ply laminates ½h=� h� and½�h=h� and the flanges laminates are assumed to be unidirectional(Fig. 2a). All the coupling stiffnesses are zero, but E25 does not van-ish due to unsymmetric stacking sequence of the webs. The lowestthree natural frequencies with and without the effect of axial forceare given in Table 1. The critical buckling loads and the natural fre-quencies without axial force agree completely with those of previ-ous papers [20,21], as expected. It can be shown from Table 1 thatthe change in the natural frequencies due to axial force is signifi-cant for all fiber angles. It is noticed that the natural frequencies in-crease as the axial force changes from compression ðP ¼ 0:5� PcrÞto tension ðP ¼ �0:5� PcrÞ which reveals that the compressiveforce has a softening effect on the natural frequencies while thetension force has a stiffening effect. The typical normal mode

Fig. 2. Geometry and stacking sequences

shapes corresponding to the lowest three natural frequencies withfiber angle h ¼ 30� for the case of a compressive axial forceðP ¼ 0:5� PcrÞ are illustrated in Figs. 3–5. The mode shapes forother cases of axial force (P ¼ 0 and P ¼ �0:5� Pcr) are similar tothe corresponding ones for the case of axial force ðP ¼ 0:5� PcrÞand are not plotted, although there is a little difference betweenthem. The lowest three interaction diagrams with the fiber angleh ¼ 0� and 30� obtained by finite element analysis and the ortho-tropy solution, which neglects the coupling effects of E25 fromEqs. 29a–c are plotted in Figs. 6 and 7. For unidirectional fiberdirection (Fig. 6), the smallest curve exactly corresponds to the firstflexural in x-direction and the larger ones correspond to the firstflexural in y-direction and the second flexural in x-direction ofthe orthotropy solution, respectively. However, as the fiber angleand axial compressive force increase, this order is changing. Itcan be explained partly by the interaction diagram between flex-ural buckling and natural frequency with the fiber angle h ¼ 30�

in Fig. 7. When the beam is subjected to small axial compressiveforce, the vibration modes 1 and 2 are the first flexural x- and y-direction (Figs. 3 and 4). Thus, the othotropy solution and the finiteelement analysis are identical. It is from Fig. 5 that the vibrationmode 3 exhibits double coupling (the second flexural mode in x-direction and torsional mode). Due to the small coupling stiffness-es E25, this mode becomes predominantly the second flexural x-direction mode, with a little contribution from torsion. Therefore,the results by the finite element analysis ðw3 � P3Þ and orthotropysolution ðwx2 � Px2 Þ are nearly identical in Fig. 7. It is indicated thatthe simple orthotropy solution is sufficiently accurate for thisstacking sequence. Characteristic of load–frequency interactioncurves is that the value of the axial force for which the natural fre-quency vanishes constitutes the critical buckling load. Thus, forh ¼ 30�, the first flexural buckling in minor axis occurs atP ¼ 13:88. Therefore, the lowest branch vanishes when P is slightlyover this value. As axial force increases, two interaction curveswy1� Py1

and wx2 � Px2 intersect at P ¼ 48:10, thus, after this value,vibration modes 2 and 3 change each other. Finally, the second andthird branch will also disappear when P is slightly over 54.53 and73.16, respectively. Figs. 6 and 7 explain the duality between flex-ural buckling and natural frequency. A comprehensive three-dimensional interaction diagram of natural frequency, axial com-pression and fiber angle is plotted in Fig. 8. Three groups of curvesare observed. The smallest group is for the first flexural mode in x-direction and the larger ones are for the first flexural mode in y-direction and flexural–torsional coupled mode, respectively.

The next example is the same as before except that in this case,the top flange and the left web laminates are ½h2�, while the bottom

of thin-walled composite box beam.

Table 1Effect of axial force on the first three natural frequencies with respect to the fiber angle change in the webs of a simply supported composite beam.

Fiber angle Buckling loads ðPcrÞ P ¼ 0:5� Pcr (compression) P = 0 (no axial force) P ¼ �0:5� Pcr (tension)

w1 w2 w3 w1 w2 w3 w1 w2 w3

0 36.009 7.696 16.704 40.725 10.884 18.392 43.536 13.330 19.937 46.17715 29.245 6.936 16.142 36.668 9.809 17.569 39.204 12.013 18.889 41.58630 13.549 4.721 14.750 24.965 6.677 15.487 26.691 8.177 16.191 28.31245 7.858 3.595 14.211 19.021 5.084 14.659 20.334 6.227 15.094 21.56860 6.670 3.312 14.097 17.527 4.685 14.481 18.738 5.737 14.855 19.87475 6.419 3.249 14.072 17.194 4.595 14.442 18.381 5.628 14.803 19.49690 6.375 3.238 14.068 17.136 4.580 14.436 18.319 5.609 14.795 19.430

Fig. 3. Mode shapes of the flexural and torsional components for the first modex1 ¼ 4:721 with the fiber angle 30� in the webs of a simply supported compositebeam under a compressive axial force P ¼ 0:5Pcr .

Fig. 4. Mode shapes of the flexural and torsional components for the second modex2 ¼ 14:750 with the fiber angle 30� in the webs of a simply supported compositebeam under a compressive axial force P ¼ 0:5Pcr .

Fig. 5. Mode shapes of the flexural and torsional components for the third modex3 ¼ 24:965 with the fiber angle 30� in the webs of a simply supported compositebeam under a compressive axial force P ¼ 0:5Pcr .

Fig. 6. Effect of axial force on the first three natural frequencies with the fiber angle0� in the webs of a simply supported composite beam.


flange and right web laminates are unidirectional (Fig. 2b). For thislay-up, the coupling stiffnesses E14; E15; E23; E25 and E35 become nomore negligibly small. Major effects of compressive axial force on

the natural frequencies are again seen in Table 2. Three-dimen-sional interaction diagram between flexural–torsional bucklingand natural frequency with respect to the fiber angle change is

Fig. 7. Effect of axial force on the first three natural frequencies with the fiber angle30� in the webs of a simply supported composite beam.

Fig. 8. Three-dimensional interaction diagram between axial force and the firstthree natural frequencies with respect to the fiber angle change in the webs of asimply supported composite beam.

Fig. 9. Three-dimensional interaction diagram between axial force and the firstthree natural frequencies with respect to the fiber angle change in the left web andtop flange of a simply supported composite beam.

Fig. 10. Effect of axial force on the first three natural frequencies with the fiberangle 30� in the left web and top flange of a simply supported composite beam.


shown in Fig. 9. Similar phenomena as the previous example can beobserved except that in this case all three groups are flexural–tor-sional coupled mode. The interaction diagram between flexural–torsional buckling and natural frequency by the finite elementanalysis and orthotropy solution with the fiber angle h ¼ 30� and60� are displayed in Figs. 10 and 11. It can be remarked again thatthe natural frequencies decrease with the increase of compressiveaxial forces, and the decrease becomes more quickly when axialforces are close to flexural–torsional buckling loads. For h ¼ 60�,at about P ¼ 7:92, 31.28 and 47.11, respectively, the natural

Table 2Effect of axial force on the first three natural frequencies with respect to the fiber angle c

Fiber angle Buckling loads ðPcrÞ P ¼ 0:5� Pcr (compression)

w1 w2 w3

0 36.009 7.696 16.704 40.72515 30.211 7.054 15.678 32.71730 17.016 5.295 13.099 24.08845 9.899 4.036 12.093 20.32460 7.918 3.609 11.892 18.95575 7.454 3.502 11.846 18.51790 7.370 3.482 11.837 18.424

frequencies become zero which implies that at these loads, flex-ural–torsional bucklings occur as a degenerate case of naturalvibration at zero frequency. As the fiber angle and compressive ax-ial force increases, the orthotropy solution and the finite elementanalysis solution show significantly discrepancy (Figs. 10 and 11).The typical normal mode shapes corresponding to the lowest threenatural frequencies with fiber angle h ¼ 60� for the case of com-pressive axial force ðP ¼ 0:5� PcrÞ are illustrated in Figs. 12–14. Rel-ative measures of flexural displacements and torsional rotation

hange in the left web and top flange of a simply supported composite beam.

P = 0 (no axial force) P ¼ �0:5� Pcr (tension)

w1 w2 w3 w1 w2 w3

10.884 18.392 43.536 13.330 19.937 46.1779.976 17.191 35.542 12.218 18.582 38.1547.488 14.129 26.285 9.170 15.089 28.3115.707 12.749 21.864 6.990 13.373 23.3025.104 12.427 20.282 6.251 12.941 21.5284.952 12.353 19.797 6.065 12.839 20.9994.924 12.338 19.696 6.031 12.820 20.891

Fig. 11. Effect of axial force on the first three natural frequencies with the fiberangle 60� in the left web and top flange of a simply supported composite beam.

Fig. 12. Mode shapes of the flexural and torsional components for the first modex1 ¼ 3:609 with the fiber angle 60� in the top flange and the left web of a simplysupported composite beam under a compressive axial force P ¼ 0:5Pcr .

Fig. 13. Mode shapes of the flexural and torsional components for the second modex2 ¼ 11:892 with the fiber angle 60� in the top flange and the left web of a simplysupported composite beam under a compressive axial force P ¼ 0:5Pcr .

Fig. 14. Mode shapes of the flexural and torsional components for the third modex3 ¼ 18:955 with the fiber angle 60� in the top flange and the left web of a simplysupported composite beam under a compressive axial force P ¼ 0:5Pcr .


show that all the modes are triply coupled mode (flexural mode inthe x- and y-directions and torsional mode). That is, the orthotropysolution is no longer valid for unsymmetrically laminated beams,and triply coupled flexural–torsional vibration should be consid-ered even for a doubly symmetric cross-section.

Finally, the effects of modulus ratio ðE1=E2Þ on the first five nat-ural frequencies of a cantilever thin-walled composite beam undera compressive axial force ðP ¼ 0:5� PcrÞ are investigated. Thestacking sequence of the flanges and webs are ½0=90�s (Fig. 2c).For this lay-up, all the coupling stiffnesses vanish and thus, thethree distinct vibration mode, flexural vibration in the x- and y-direction and torsional vibration are identified. It is observed fromFig. 15 that the natural frequencies xxx1 ;xyy1 ;xxx2 and xyy2 in-crease with increasing orthotropy ðE1=E2Þ. However, torsional nat-ural frequency is almost invariant and well above the other threetypes of natural frequencies, i.e. xxx1 ;xyy1

and xxx2 . It can be ex-

plained from Eqs. (29c) and (30c) that torsional frequency is dom-inated by torsional rigidity rather than warping rigidity. Moreover,effects of warping is negligibly small for box section. As ratio ofðE1=E2Þ increases, the order of the second flexural mode in the y-direction, the torsional mode change each other.

8. Concluding remarks

An analytical model is developed to study the flexural–torsionalcoupled vibration of thin-walled composite beams with arbitrarylay-ups under a constant axial force. This model is capable of pre-dicting accurately the natural frequencies and load–frequencyinteraction curves as well as corresponding vibration mode shapesfor various. To formulate the problem, a one-dimensional displace-

Fig. 15. Variation of the first five natural frequencies with respect to modulus ratiochange of a cantilever composite beam under a compressive axial force P ¼ 0:5Pcr .


ment-based finite element method is employed. All of the possiblevibration modes including the flexural mode in the x- and y-direction and the torsional mode, and fully coupled flexural–torsional mode are included in the analysis. The present model isfound to be appropriate and efficient in analyzing free vibrationproblem of thin-walled composite beams under a constant axialforce.

Acknowledgements

The support of the research reported here by a grant (code #06R&D B03) from Cutting-edge Urban Development Program fundedby the Ministry of Land, Transport and Maritime Affairs of Koreagovernment is gratefully acknowledged. The authors also wouldlike to thank the anonymous reviewers for their suggestions inimproving the standard of the manuscript.

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32) Free Vibration of Axially-loaded Thin-walled Composite Box Beams