(Qu YG)a Variational Formulation for Dynamic Analysis of Composite Laminated Beams Based on a General Higher-Order Shear Deformation Theory

7/27/2019 (Qu YG)a Variational Formulation for Dynamic Analysis of Composite Laminated Beams Based on a General Higher

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A variational formulation for dynamic analysis of composite laminated

beams based on a general higher-order shear deformation theory

Yegao Qu , Xinhua Long, Hongguang Li, Guang Meng

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e i n f o

Article history:Available online 15 March 2013

Keywords:

Composite laminated beam

Variational formulation

Higher-order shear deformation theory

Free vibration

Transient vibration

a b s t r a c t

This paper presents a general formulation for free and transient vibration analyses of composite lami-nated beams with arbitrary lay-ups and any boundary conditions. A modified variational principle com-

bined with a multi-segment partitioning technique is employed to derive the formulation based on ageneral higher-order shear deformation theory. The material couplings of bending-stretching, bending-

twist, and stretching-twist as well as the Poissons effect are taken into account. A considerable numberof free and transient vibration solutions are presented for cross- and angle-ply laminated beams with var-ious geometric and material parameters. Different combinations of free, simply-supported, pinned,

clamped and elastic-supported boundary conditions are examined. The validity of the formulation is con-

firmed by comparing the present solutions with analytical and experimental results available in the lit-erature and the ones obtained from finite element analyses. The accuracy of several higher-order sheardeformable beam theories for predicting the vibrations of laminated beams has been ascertained. Results

of parametric studies for composite beams with different orthotropic ratios, fiber orientations, layer

numbers and boundary conditions are also discussed. The present formulation is versatile in the sensethat it is capable of accommodating a variety of beam theories available in the literature, and allowsthe use of different polynomials as admissible functions for composite beams, such as the Chebyshev

and Legendre orthogonal polynomials, and the ordinary power polynomials. Moreover, it permits to deal

with the linear vibration problems for thin and thick beams subjected to dynamic loads and boundaryconditions of arbitrary type.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

Composite laminated beams are extensively used in aircraftstructures, space vehicles, turbo-machines and other industrial

applications due to their high strength-to-weight and stiffness-to-weight ratios. It is well known that laminated beams in theseapplications often operate in complex environmental conditionsand are commonly exposed to a variety of dynamic excitations

which may result in excessive vibration and fatigue damage. Athorough understanding of the vibration behaviors of laminatedbeams is therefore of particular importance. Despite the many con-tributions to the analysis of laminated beams, the establishment of

reliable and efficient modeling techniques for simulating the dy-namic behaviors of generally layered composite beams remains achallenging task and is the focus of the present study.

The development of accurate beam theories has been the sub-

ject of significant research interest for many years, and a largeamount of beam models have been proposed based on different

assumptions and approximations. Excellent overviews of thesetypes of models may be found in Kapania and Raciti [1], Ghugaland Shimpi[2],and Vinson and Sierakowski[3]. Variational princi-ples, including the classical principles (either displacements or

stresses as unknowns) and mixed principles (e.g. displacementsand stresses simultaneously as unknowns), are usually employedto derive the consistent governing equations and boundary condi-tions for the theoretical models; see [414]for details and proofs.

Since a general displacement-based theory will be used in the pres-ent study, a brief review related to the displacement-based theo-ries is given below. Physically, laminated beams with generallayer-ups are three-dimensional (3D) structures for which the

methods of linear elasticity theory may be applied[15]. However,it is well recognized that the solutions of the 3D elasticity equa-tions for composite beams are difficult to obtain and in most casesare even unattainable. Typically, researchers make suitable

assumptions concerning the kinematics of deformation or the stateof stress through the thickness of the beams, and reduce the 3Dbeam problems to various 1D representations with reasonableaccuracy, such as the equivalent single layer (ESL) model and the

layer-wise (LW) model. Following the ordinary classification of

0263-8223/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.02.032

Corresponding author. Tel.: +86 021 34206332; fax: +86 021 34206814.

E-mail addresses:[email protected],[email protected](Y. Qu).

Composite Structures 102 (2013) 175192

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the ESL models, there are mainly three major categories, i.e., theclassical beam theory (CBT), the first-order beam theory (FBT),

and the higher-order beam theory (HBT). The CBT known asEulerBernoulli beam theory is the simplest one and is applicableto slender composite beams. For thick beams, the CBT underesti-mates deflection and overestimates natural frequency due to

ignoring the transverse shear deformation effect [13]. In order

to take into account the effects of transverse shear deformationfor the analysis of moderately thick beams, the FBT by Timoshenkohas been developed. In this theory, transverse shear strain distribu-

tion is assumed to be constant through the beam thickness and,thus, requires a shear correction factor to appropriately representthe strain energy of deformation [13,16,17]. It has been shownthat the accuracy of the FBT solutions will be strongly dependent

on the shear correction factor, and the value of this factor is nota constant but changes with material properties, layer sequences,loading cases, boundary conditions, etc. The limitations of classical

beam theory and first-order beam theory stimulated the develop-ment of higher-order shear deformation theories to avoid the useof shear correction factors, to include correct cross sectionalwarping and to get the realistic variation of the transverse shear

strains and stresses through the thickness of beam. A number ofhigh-order theories with different shear strain shape functions(including polynomial functions[1822], trigonometric functions[2325], exponential functions[26,27], etc.) have been proposed.

Although some shape functions mentioned above were initiallydeveloped for elastic plates or shells, application of these functionsto composite beams is immediate. In the LW models, each layer ina laminated beam is considered to be a separate beam, and com-

patibility conditions are applied between adjacent layers. Thesemodels provide realistic descriptions of kinematics at the ply leveland yield accurate stress results for composite laminated beams,but suffer from an excessive number of displacement variables in

proportion to the number of layers and hence are not suitable forpractical applications, especially when optimization studies areconcerned. There also exist some special layer-wise theories, often

called zigzag theories, containing a constant number of unknownvariables irrespective of the number of layers in laminated beams.In these theories, the additional unknowns are eliminated byenforcing the continuity of the transverse shear stress componentsat the interfaces between adjacent layers and by satisfying the zeroshear traction conditions on the top and bottom surfaces of the

beams. Examples of LW theories are those found in the articles[2831].

The dynamic analysis of composite laminated beams based onvarious beam theories has been the subject of significant research

activities during the past few years. Early studies have been com-piled in the excellent review paper by Kapania and Raciti [32]. Inorder to properly focus on the features and emphasis of the presentpaper, a brief review is herein given to the works which are mainly

devoted to the free and transient vibration analyses of laminatedbeams. The dynamic analysis of laminated beams is mostly re-stricted to free vibrations. A number of analytical and computa-tional methods have been developed and proposed to handle the

free vibration problems of laminated beams. They include, butare not limited to, the closed-form solution[3,16,21,22,33,34], var-iational method[3539], dynamic stiffness method[4044], trans-fer matrix method [4547], differential quadrature method

[48,49], meshless method [50], finite difference method [5153],and finite element method[5456]. It should be mentioned thatthe Poissons effect, which is often neglected in the one-dimensional laminated beam analysis, has very significant influ-

ence on the vibration analysis of composite beams with generallayer-ups. The incorporation of this effect involves either

correcting the relations of the generalized force/moment resultantsand generalized strains of laminated beams or correcting the

constitutive equations of a 3D anisotropic body [57]. The transientvibration analysis of composite laminated beams has received less

attention compared to the free vibrations. Marur and Kant [58]developed a finite element model with seven degrees of freedomper node for predicting the transient dynamic responses of com-posite beams. The governing equations of motion were solved

using the central difference predictor technique to obtain the re-

sponse history at different time steps. Sokolinsky and Nutt [59]presented a discretized formulation based on an implicit finite dif-ference method for the time-domain response analysis of sandwich

beams. Khdeir[60] investigated the transient vibrations of cross-ply laminated beams by using a generalized modal approach inconjunction with a general higher-order beam theory. Arvin et al.[61] performed a finite element analysis to obtain the structural re-

sponses of a composite sandwich beam with viscoelastic core.Kapuria and Alam[62]developed a 1D beam finite element withelectric degrees of freedom for the dynamic analysis of hybrid pie-

zoelectric beams by using the layer-wise theory. The Newmark di-rect time integration method was employed to obtain the transientresponses of the composite beams. Tagarielli et al. [63] reportedthe finite element solutions for the dynamic shock responses of

fully clamped monolithic and sandwich beams. Kiral [64]used athree-dimensional finite element model together with the New-mark integration method to obtain the dynamic response of com-posite beams subjected to moving loads. Mohebpour et al. [65]

investigated the dynamic responses of composite laminated beamssubjected to a moving oscillator by using the first-order beam the-ory and finite element method. Based on the mode superpositionmethod, Jafari-Talookolaei et al. [66] studied the dynamic re-

sponses of a delaminated beam due to a moving oscillatory mass.alim[67] performed a forced vibration analysis of non-uniformcomposite beams subjected to impulsive loads. The solutions ob-tained in the Laplace domain were transformed to the time domain

by using the Durbins inverse Laplace transform method. Accordingto the comprehensive survey of the literature, it is found that mostof the previous efforts were restricted to laminated beams with

limited sets of classical boundary conditions (e.g., the free, sim-ply-supported and clamped edges). Actually, the boundary condi-tions of a composite beam may not always be classical inengineering applications. This may become one of the mainsources of discrepancy when the comparison between theory andexperiment is made. Moreover, the existence of various higher-or-

der shear deformable beam theories gives rise to a problem thatone may be easily inundated by the abundance of the availablemodels or choices. Although the free vibration results for lami-nated beams based on some shear deformation theories have been

presented and compared by Aydogdu [36,37], to the best of ourknowledge, the discrepancies of various higher-order theories inpredicting the transient responses of laminated beams have notbeen investigated. It should be further remarked here that many

of the commonly used methods, such as the closed-form method[3,16,21,22,33,34] and the dynamic stiffness method[4044], aremainly restricted to free vibration analysis; they soon becomecumbersome when one wants to deal with the dynamic response

problems of composite beams under arbitrary loading cases.The primary objective of the present investigation is to devel-

op a unified formulation for free and transient vibration analysesof generally layered laminated beams with arbitrary combina-

tions of classical and non-classical boundary conditions. A modi-fied variational principle in conjunction with a multi-segmentpartitioning technique is employed to derive the formulationbased on a general higher-order shear beam theory. The elastic

couplings of the bending-stretching, bending-twist and stretch-ing-twist with the Poissons effect are taken into account. The for-

mulation is particularly attractive since one can choose differentpolynomials as admissible displacement and rotation functions

176 Y. Qu et al. / Composite Structures 102 (2013) 175192


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for a practical beam analysis. This fact is confirmed through theapplication of the following four types of polynomials, i.e. the

Chebyshev orthogonal polynomials of first and second kind, theLegendre orthogonal polynomials of first kind, and the ordinarypower polynomials. To validate the convergence, efficiency andaccuracy the proposed formulation, as well as to explore the lim-

its of its applicability, a considerable number of numerical exam-

ples are given for the free vibrations of cross- and angle-plylaminated beams with different combinations of free, simply-sup-ported, pinned, clamped and elastic-supported boundary condi-

tions. The present solutions are validated through comparisonswith previously published analytical and experimental results inthe literature and those solutions obtained from the finite ele-ment analyses. As to the transient vibration analyses, laminated

beams subjected to distributed rectangular and exponential pulseloads are examined. The accuracy of a variety of higher-ordershear deformable beam theories for predicting the vibration

behaviors of laminated beams has been investigated. The effectsof the fiber orientation, layer number and boundary conditionon the transient vibration responses of laminated beams are alsodiscussed. As it will become evident in what follows, the present

formulation is capable of dealing with the linear vibration prob-lems of composite laminated beams subjected to dynamic loadsand boundary conditions of arbitrary type.

2. Mathematical formulations

A laminated beam made of an arbitrary number of perfectly

bonded orthotropic layers is shown inFig. 1. A Cartesian co-ordi-nate system (ox,y,z) is defined on the central axis of the beam,where thex-axis is taken along the central-line with the y-axis in

the width direction and the z-axis in the thickness direction. Thecomposite beam has a length ofL with a rectangular cross-sectionofb h.

It is assumed that the deformations of the laminated beam are

characterized by its center-line and take place in the xzplane. Thedisplacement field for the beam based on a general shear deform-able theory takes the form:

~ux;z; t ux; t fz@w

@x gz#x; t

~wx;z; t wx; t1

whereu and w represent center-line displacement components ofthe beam along the x andzdirections, respectively;# is a unknown

function that represent the effect of transverse shear strain on thecenter-line of the beam. f(z) and g(z) denote the shape functionsthat determine the distribution of axial and transverse strain andstress through the beam thickness. trepresents the time variable.

The linear strains associated with the proposed displacementfield as given in Eq.(1) under the assumptions of small deforma-

tion and small rotation are given by:

ex e0x fe

1x ge

2x ; cxz

fc0xz gc1xz 2:a-b

where

e0x @u

@x; e1x

@2w

@x2; e2x

@#

@x; c0xz

@w

@x; c1xz # 3:a

f1 @f

@z; g

@g

@z 3:b

All further equationssuch as constitutive equations and equations ofmotion are derived using the general displacement field in Eq.(1)soas to ensurethe applicability of thepresent formulation and solution

methodology to various beam theories. It should be noted that thedisplacement field in Eq. (1) contains the displacement fields ofCBT, FBT and some HBTs as special cases. This is achieved by choos-ing the shape functionsf(z) andg(z) as follows: for CBT,f(z) =zand

g(z) = 0; for FBT,f(z) = 0 and g(z) =z. For higher-order beam theories,these shape functions are generally evaluated by implementing thetransverse shear stress boundary conditions so that the transverseshear stresses on the top and bottom surfaces of the beam vanish,

i.e.,cxzjz= h/2= 0. The choice of these shape functions is not unique.In practice, it is based on the satisfaction of certain mechanical con-straints of the problem considered and, in general, characterizes the

degree of sophistication and accuracy of the resulting beam theory.Table 1shows a number of HBTs in terms of different shape func-

tionsg(z), whenf(z) is taken as z. In order to express these theoriesin a concise manner, acronyms are used herein as shown inTable 1.It should be remarked here that the shape functions employed inHBT[V], HBT[VE1] andHBT[VE2] do notcomply exactly with the tangen-

tial stress-free boundary conditions on the top and bottom surfacesof beams. Regarding the tangential trigonometric function proposedby Mantari et al. [75,76], HBT[M2]and HBT[M3]respectively representthe higher-order beam theories for m= 1/(5h) and m= p/(2h).Similarly, for the shape functions obtained by Viola et al. [74]andEl Meiche et al.[78], HBT[VE1]and HBT[VE2]denote the beam modelsfor n= 1 and n= 1/[cosh (p /2) 1], respectively, whereas HBT[M5]and HBT[M6] indicate the beam theories when the parameter m is

respectively taken asm =6 andm =7 in the hybrid shape func-tions developed by Mantari and Guedes Soares[80]. If the exponen-tial shape functions of Karama et al. [26] and Aydogdu[27] arecompared, it is found that they are mathematically equivalent

descriptions [77]; therefore, a symbol HBT[KA] is just used herein.It is noted that the displacement fields of these HBTs are formulatedbased on shear rotation as variable sincef(z) is taken as z. Actually,a number of HBTs can also be formulated by taking the rotation of

normal as unknown variable[82]. For instance, the Reddys third-order shear deformation theory [20], which is indicated as HBT[R]in this paper, can be easily recovered from Eq. (1) by choosing

f(z) =4z3/(3h2) andg(z) =z4z3/(3h2). Although new shape func-

tions are attainable, only the ones presented in Table 1 are examinedin this paper. The development of new beam theories derived from

the general displacement field has been left as a subject for futureinvestigation.

Fig. 1. Geometry and co-ordinate system of a laminated beam.

Y. Qu et al. / Composite Structures 102 (2013) 175192 177


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While reducing the beam problem to a one-dimensional prob-lem, it is found that the Poissons effect has very significant influ-

ence on the vibration analysis of composite laminated beams,especially for those angle-ply beams with large ply angles. By

ignoring the force and moment resultants in the width directionand correcting the generalized force-strain relations, one obtainsthe constitutive equations for one-dimensional laminated beamas (for more details, seeAppendix A)

Nx

Mx

Px

264

375 A11 B11 E11B11 D11 F11

E11 F11 H11

264

375

e0xe1xe2x

264

375; Qxz

Pxz

A55 D55

D55 F55

c0xzc1xz

" #

4:a-b

whereNxandMxare the stretching force resultant and bending mo-ment resultant, respectively; Px represents the high-order bendingmoment resultant. Qxz and Pxzare shear force resultants. A11; B11,

etc., are the coefficients of the matrix A BC1BT and the matricesA, B and Care given by

in which

Aij; Bij; Dij; Eij; Fij; Hij bXNk1

Z zk1zk

1;f;f2;g;fg;g2Qkijdz; i;j 1; 2;6 6:a

A55; D55; F55 bXNk1

Z zk1zk

f2;fg;g2Qk55dz 6:b

where Qkij denotes the transformed reduced stiffness. For the kth

orthotropic lamina, Qkij can be written as[83]:

Qk11 Qk11 cos

4 hk 2 Qk12 2Qk66

sin

2hk cos2 hk Qk22 sin

4hk

Qk12 Q

k11Q

k22 4Q

k66

sin

2

hk

cos2

hk

Qk12sin

4

hk

cos4

hk

Qk22 Qk11 sin

4h

k 2 Qk12 2Qk66

sin

2h

k cos2 hk Qk22 cos4 h

k

Qk16 Qk11 Q

k122Q

k66

sinhk cos3 hk Qk12 Q

k222Q

k66

sin3 hk coshk

Qk26 Qk11 Q

k122Q

k66

sin

3hk coshk Qk12 Q

k222Q

k66

sinhk cos3 hk

Qk66 Qk11 Q

k222Q

k12 2Q

k66

sin

2hk cos2 hk Q

k66sin

4hk cos4 hk

Qk55 Qk55 cos

2 hk Qk44 sin

2hk 7:a-g

wherehk is the fiber orientation angle ofkth lamina with respect to

the x-axis of the beam. The elastic constants Qkij in the principalmaterial coordinate system are expressed as follows:

Qk11

Ek11 lk12l

k21

; Qk12

lk12Ek2

1 lk12lk21

; Qk22

Ek21 lk12l

k21

;

Qk44 Gk23; Q

k55 G

k13; Q

k66 G

k12 8:a-f

where Ek1,Ek2,G

k12,G

k23,G

k13, and l

k12are the engineering parameters of

thekth lamina[83].

2.1. Variational formulation for laminated beams

As mentioned previously, variational statements can be usedto establish the governing equations for composite beams. Theseequations could either be given in a strong form or a weak form.

The strong form for the vibration problem of a composite beam

consists of partial differential equations of motion (hold in each

Table 1

Several shape functions proposed in the literature.

Model Shape functions Acronyms

Kaczkowski[68],Panc[69]and Reissner[70] 5z4 1

43h2

z2

HBT[KPR]

Levinson[18], Murty[19]and Reddy[20] z 1 43h2

z2

HBT[LMR]

Levy[71], Stein[72], Touratier[23,24] hp sin

phz

HBT[LST]

Mantari et al.[73]sin

p

hz

e

12

cos phz

p

2hz

HBT[M1]

Viola et al.[74] 2hptan

p2hz

HBT[V]

Mantari et al.[75,76] tanmz zm sec2 mh2

; m 15h ;p2h

HBT[M2] HBT[M3]

Karama et al.[26], Aydogdu[27] ze2z=h2

(za2z=h2= lna ,"a> 0) HBT[KA]

Mantari et al.[77] z2:852z=h2

0:028z HBT[M4]

Viola et al.[74],El Meiche et al. [78] n hp sinh p

hz

z

; n 1; 1coshp=21

n o HBT[VE1]HBT [VE2]

Soldatos[79] h sinh zh

zcosh 12

HBT[S]

Mantari et al.[80] sinh zh

em cosh

zh z

h cosh 12

msinh2 12

h iem cosh

12 ; m f6; 7g HBT[M5] HBT[M6]

Akavci and Tanrikulu[81] 3p2 h tanh

zh

3p2 zsec h

2 12

HBT[AT1]

Akavci and Tanrikulu[81] zsec h p z2

h2

zsec h p4

1 p2 tanh

p4

HBT[AT2]

A

A11 B11 E11

B11 D11 F11

E11 F11 H11

264

375; B

A12 A16 B12 B16 E12 E16

B12 B16 D12 D16 F12 F16

E12 E16 F12 F16 H12 H16

264

375; C

A22 A26 B22 B26 E22 E26

A26 A66 B26 B66 E26 E66

B22 B26 D22 D26 B22 F26

B26 B66 D26 D66 F26 F66

E22 E26 F22 F26 H22 H26

E26 E66 F26 F66 H26 H66

2666666664

3777777775

5:a-c



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point of the beam) and associated boundary and initial condi-tions. The solutions to the strong form satisfy the differential

equations, boundary and initial conditions exactly and must beas smooth as required by the differential equations and bound-ary terms. Closed form solutions are limited to few simple load-ing cases and boundary conditions. In the weak form, additional

approximations are generally introduced for the analysis and the

resulting equations may be written as a system of algebraicequations, which lead to a solution that is easier to obtain thanby solving the partial differential equations (i.e., the strong form)

directly. In this work, an efficient and accurate variational meth-od proposed by the authors [8487] for predicting the dynamicbehaviors of isotropic and composite shells is employed for thevibration analysis of composite beams. To this end, a laminated

beam is equally divided into N0 beam segments in the xdirection; then the vibration problem of the beam is character-ized using a modified variational principle, which involves seek-

ing the minimum of a modified variational functional as[86,87]

P Z t1

t0X

N0

i1

TiUiWidt Z t1

t0Xi;i1

Pkj dt 9

whereTiand Uiare the maximum kinetic energy and strain energyof a beam segment, respectively. Wi is the work done by external

loads. The subscript i indicates the beam segment number. Pkjde-notes the interface potential on the common boundaries of adjacentbeam segments (i) and (i+ 1). t0 and t1 are two specified times.Based on the general displacement field in Eq.(1),the maximum ki-

netic energy of the ith beam segment is given as

T1

2

Zli

q0_u2 q0 _w

2 q2@_w

@x

2 q5 _#

2

"

2 q1_u@_w

@x

q3_u_# q4@_w

@x

_# dx 10where the dot above a variable represents differentiation with re-

spect to time. li is the length of the ith beam segment. q0, q1, q2,q3,q4 and q5 are the inertia terms, defined by:

q0;q1;q2;q3;q4;q5 bXNk1

Z zk1zk

qk1;f;f2;g;fg;g2dz 11

whereqk is the mass density per unit volume of the kth lamina.The strain energy of the ith beam segment associated with the

general shear deformable theory is expressed as

Ui 1

2

Zli

Nxe0x Mxe

1x Pxe

2x Qxzc

0xzPxzc

1xz

dx 12

where the strains, curvatures, force and moment resultants are de-fined by Eqs.(3-4).All external loads are assumed to act on the center-line of each

beam segment. Then, the work done by external forces for the ithbeam segment is

Wi

Zli

fu;iuifw;iwidx 13

wherefu,i and fw,i are the distributed forces along the beam lengthapplied in thex and zdirections, respectively.

In constructing the interface potential Pkjin Eq. (9), we followthe similar way as was done in the authors earlier works[8487]

for elastic shells, where a modified variational principle (MVP) inconjunction with the least-squares weighted residual method(LSWRM) was proposed to impose the essential interface continu-ity constraints. By doing so, we have

Pkj fuNxHu fwQxzHwfrMxHr f#PxH#jxxi

1

2 fujuH

2ufwjwH

2v

frjrH2wf#j#H

2#

xxi

14

in which

Qxz Qxz@Mx@x

15

The first expression in the right-hand side ofPkj is derived by

means of the MVP to relax the enforcement of the interface conti-nuity constraints (for more details, the reader is referred to refer-ences [84,88]). The second expression is obtained from theLSWRM, and the existence of this term is twofold: to ensure a

numerically stable operation for the multi-segment decompositionof composite beams, and to ensure a unified formulation that candeal with non-classical boundary constraints. Hu, Hw, Hrand H#are the essential continuity equations on the common interfaces,given by: Hu=uiui+1, Hw=wiwi+1, Hr=@wi/@x@ wi+1/@xand H#=#i #i+1. jt(t=u, w, r,#) are pre-assigned weightedparameters. In practice, the weighted parameters taken to be

102107Ewill lead to reasonably converged solutions, withEbeingthe maximum elastic modulus in the principal coordinate direction

[86,87]. Hence,j= 103Ehas been adopted to present all the resultsin the following analysis. ft are the parameters defining various

boundary conditions. For the case of two adjacent beam segments,ft= 1; while for the case of geometric boundaries, values offt aredefined inTable 2. An arbitrary set of classical boundary conditionsat the two ends of a composite beam can be obtained by an appro-

priate choice of the values offt. The essential boundary terms forthese classical boundary conditions at the ends of a compositebeam are also given along with the table.

As noted in the Introduction, the mechanical interaction of the

composite beam and the elastic foundation is an important issuein practical engineering applications. For such elastic supportedbeams, vibration results based on alternative mathematical meth-ods are very rare in the literature. A careful examination of the sec-

ond expression of the right-hand side of Eq. (14)reveals that theweighted parameters jt may be viewed as physical stiffnesses ofthe elastic foundation along translational and/or rotational direc-tions. This offers an important advantage of the present formula-

tion since non-classical boundary conditions (elastic restraints) atthe two ends of a composite beam can be easily incorporated inthe present model by removing the modified variational term fromEq.(14).

2.2. Equations of motion and solution methodology

The main advantage of the modified variational functional P in

Eq.(9)is that the choice of the displacement and rotation functionsfor each beam segment is greatly simplified, and any linearly inde-

pendent, complete basis functions may be employed. The reasonlies in the fact that both the interface continuity and geometricboundary conditions in a laminated beam are relaxed and enforcedthrough the MVP and the LSWRM, and there is no need to explicitlysatisfy the natural conditions on these interfaces and boundaries

for the displacement and rotation functions. The functional Ppermits the use of the same functions for each beam segment.

Table 2

Values offt(t= u,w, r,#) for classical boundary conditions.

Boundary set Essential conditions fu fw fr f#

Free No constraints 0 0 0 0

Simply-supported u=w= 0 1 1 0 0

Pinned u=w=@w/@x= 0 1 1 1 0

Clamped u=w=@w/@x=#= 0 1 1 1 1



6/18

The displacement and rotation components of each beam segmentcan be written in the forms:

ux; t XPp0

Tpx~upt Uxut; wx; t XPp0

Tpx ~wpt Wxwt

#x; t XP

p0

Tpx ~#pt Hx#t

16

Note the subscripti is omitted here for the sake of brevity. Tp(x) is

theporder complete polynomials, andPis the highest degree takenin the polynomials. ~up, ~wp and ~#p are the generalized coordinatevariables.U(x), W(x) and H(x) are the function vectors; u , w and# are the generalized coordinate vectors. Four sets of polynomials

are independently used to expand the displacement and rotationcomponents of each beam segment in the x direction. They are:

(a) Chebyshev orthogonal polynomials of first kind (COPFK) [89]

T0x 1; T1x x; Tp1x 2xTpx Tp1x; for pP2

17

(b) Chebyshev orthogonal polynomials of second kind (COPSK)[89]

T0x 1; T1x 2x; Tp1x 2xTpx Tp1x; forpP2

18

(c) Legendre orthogonal polynomials of first kind (LOPFK) [89]

T0x 1; T1x x; p 1Tp1x

2p 1xTpx pTp1x; forpP2 19

(d) Ordinary power polynomials (OPP)

Tpx xp; for p 0;1;2;. . . 20

It should be remarked here that the COPFK, COPSK and LOPFKare complete and orthogonal series defined on the interval ofx2[1,1]. A coordinate transformation from x (for the ith beamsegment, x2[xi,xi+1]) to xx2 1; 1 needs to be introduced to

implement the present analysis, i.e., x gax gb,ga= (xi+1xi)/2and gb= (xi+1+xi)/2. Similarly, a mapping between x2[xi ,xi+1]and x2 0; 1 is used for the OPP since it is a stable and completerepresentation only on the [0,1] interval. In doing so, we obtain

the relation: x xi1xixxi.Upon inserting Eqs.(10), (12),(13) and (14) into the Eq.(9)and

setting the variation of the preceding functional P to zero (i.e.,dP = 0) with respect to the generalized coordinate vectors u, wand #, one finds the equations of motion for a laminated beam as

Mq K KkKjq F 21

where q is the global generalized coordinate vector, given as:

q uT1;wT1;#

T1;u

T2;w

T2;#

T2; . . . ;u

TN0

;wTN0 ;#TN0

h iT. Mand K are, respec-

tively, the disjoint generalized mass and stiffness matrices. Kk and

Kj are the generalized interface stiffness matrices introduced bythe MVP and the LSWRM, respectively. The elements in the abovematrices are listed inAppendix B. The task related to the free vibra-tion problems of laminated beams can be easily solved by assuming

harmonic motion and removing the external loads from Eq. (21).Regarding the transient vibration problems, the Newmark methodwith integration parameters c0= 0.5 and c1= 0.25 is employed toobtain the time domain solutions for composite beams. To avoid

repetition readers may consult Refs. [8487]for more details aboutthe solution methods outlined above.

3. Results and discussion

A considerable number of numerical examples are now pre-sented to confirm the reliability, convergence, efficiency, and accu-racy of the proposed formulation. In order to simplify thepresentation, F, S, P, C, and E represent free, simply-supported, pin-

ned, clamped and elastic supports, respectively. Three types of

elastic end support conditions, indicated by the symbols E

I

, E

II

andEIII, are considered herein for laminated beams.EI type is con-sidered to be axial elastic (i.e., u 0, w=@ w/@x=#= 0) and is

characterized by a stiffness constant ku per unit length in the xdirection. On the contrary, support type EII only allows elasticallyrestrained displacement in the zdirection (i.e., w 0, u=@w/@

x=#= 0) and a transverse restrained stiffnesskw is prescribed forthis boundary condition. When both axial and transverse displace-ments of the beam end are elastically restrained (i.e.,u 0,w 0,@w/@x=#= 0), the end support is denoted by the symbol E III. In

what follows, a simple letter string will be used to describe thetypes of the boundary conditions imposed on the two ends of abeam, e.g., the symbol FC denotes the beam having free andclamped end conditions atx= 0 and x=L, respectively. The numer-

ical results to be discussed are appropriately grouped into two ma-jor categories: free vibration and transient vibration responses. Tovalidate the present approach, the results are compared with avail-able published results in the literature and those solutions ob-

tained from the finite element analyses.

3.1. Free vibration analysis of laminated beams

The performance characteristics of the proposed formulation forthe free vibration analysis of composite beams are evaluated in thissection. The first example is slender and thick orthotropic (00)beams with simply-supported boundary conditions. The followingmaterial properties and geometric dimensions are used:E1= 144.80 GPa, E2= 9.65 GPa, G23= 3.45 GPa,G12=G13= 4.14 GPa,

l12= 0.3 and q= 1389.23 kg/m3; for beam I (slender beam),L= 0.762 m, L/b=L/h= 120; for beam II (thick beam), L = 0.381 m,L/b=L/h= 15. Convergence studies of the natural frequencies forthe two beams are carried out to determine the optimal numberof beam segments required for satisfactory solutions, as shown in

Table 3. For these calculations, eight terms (i.e., P= 7) of the firstkind Chebyshev orthogonal polynomials are used for each beamsegment. The present results obtained from HBT[LMR]are comparedwith those exact solutions employing CBT[3]and FBT[16], and fi-

nite element results based on HBT[R] [54]. It is observed that thepresent method furnishes stable and rapid convergence character-istics as the number of beam segments is increased. Review of theconvergence rate of the natural frequencies indicates that good

accuracy of the vibration frequencies can be achieved with little

segments divided in the laminated beams. In fact, if a decomposi-tion of two segments (i.e., N0= 2) in the orthotropic beams is used,the frequencies of mode order up to 6 are still converged to two

decimal places. When 20 segments are used for the beams, theaccuracy of high-order vibration modes can be greatly improved.To limit the length of the paper, these converged high-order fre-quencies are not here presented. Further, there is excellent agree-

ment between the HBT[LMR]results of the present formulation andthose HBT[R]solutions[54]obtained from finite element modeling,confirming the high accuracy of the present approach. For the slen-der beam, the difference between the CBT and present solutions is

negligible. Thus the CBT is certainly accurate in this case. However,for thick beams, one should not expect the CBT to produce accuratepredictions of the natural frequencies; for example, the discrep-

ancy between the CBT and present HBT[LMR] results for the fifthmode of the thick beam reaches more than 117.3%.



7/18

In order to further verify the convergence and accuracy of thepresent formulation, the first five natural frequencies of cross-ply(00/900/00/900) and angle-ply (300/500/300/500) laminated com-posite beams with respect to different number of beam segments

are respectively listed in Tables 4 and 5. The material propertiesand geometric dimensions of the beams follow the same configura-tion as defined in the aforementioned Beam II, whereas CCboundary conditions are considered herein for the beams. Each

layer is assumed to be of the same thickness. Similar to the previ-ous examples, HBT[LMR] is employed for the theoretical computa-tions, and the convergence study for the beams starts withsegment number N0= 2, which is increased progressively to

N0= 100. From Tables 4 and 5, the convergence trends of the

numerical results for the beams are obvious. For the cross-plybeam, the present results are compared with those FBT solutionsobtained by the modified varational method [38]and the dynamicstiffness method[42], whereas for the angle-ply beam, the present

solutions are compared with the corresponding values obtained byHBT[LST][43]and HBT[R] [54]. In both cases, it is observed that theresults obtained using two beam segments (i.e., N0= 2) are close tothe reference results. The small discrepancies in the results are

probably due to the fact that different beam theories and solutionapproaches were used in the literature. It should be noted that thefourth mode frequency inTable 5given by Ref. [54]may be mis-printed by the authors. Table 6shows the comparison of present

results with those HBT[LST]solutions reported by Jun and Hongxing

Table 3

Convergence of natural frequencies (kHz) for orthotropic beams with different number of beam segments N0 (beam I: L = 0.762 m, L/b=L/h= 120; beam II, L = 0.381 m, L/b=L/

h= 15; boundary conditions: SS).

Model Mode no. Number of beam segments Ref.[3] CBT Ref.[16]FBT Ref.[54]HBT [R]

N0= 2 N0= 4 N0= 6 N0= 8 N0= 12 N0= 20 N0= 50

Beam I 1 0.0506 0.0506 0.0506 0.0506 0.0506 0.0506 0.0506 0.051 0.051 0.051

2 0.2015 0.2015 0.2015 0.2015 0.2015 0.2015 0.2015 0.203 0.203 0.202

3 0.4507 0.4507 0.4507 0.4507 0.4507 0.4507 0.4507 0.457 0.454 0.453

4 0.7946 0.7946 0.7946 0.7946 0.7946 0.7946 0.7946 0.812 0.804 0.799

5 1.2301 1.2286 1.2286 1.2286 1.2286 1.2286 1.2286 1.269 1.262 1.238

Beam II 1 0.7533 0.7533 0.7533 0.7533 0.7533 0.7533 0.7533 0.813 0.755 0.756

2 2.5470 2.5470 2.5470 2.5470 2.5470 2.5470 2.5470 3.250 2.548 2.554

3 4.7328 4.7328 4.7328 4.7328 4.7328 4.7328 4.7328 7.314 4.716 4.742

4 7.0251 7.0251 7.0251 7.0251 7.0251 7.0251 7.0251 13.002 6.960 7.032

5 9.3534 9.3501 9.3501 9.3500 9.3500 9.3500 9.3500 20.316 9.194 9.355

Table 4

Convergence of natural frequencies (Hz) for cross-ply (00/900/00/900) laminated beams with different number of beam segments N0 (L= 0.381 m, L/b=L/h= 15; boundary

conditions: CC).

Mode no. Number of beam segments Ref.[38]FBT Ref.[42]FBT

N0= 2 N0= 4 N0= 6 N0= 8 N0= 12 N0= 20 N0= 50 N0= 100

1 1056.1 1054.4 1054.1 1054.1 1054.0 1054.0 1054.0 1054.0 1062.2 1054.4

2 2526.7 2518.0 2516.6 2516.3 2516.2 2516.3 2516.3 2516.3 2612.0 2509.2

3 4336.7 4314.1 4310.8 4310.1 4309.9 4309.9 4309.9 4309.9 4293.7 4281.5

4 6323.5 6285.9 6280.1 6279.0 6278.6 6278.7 6278.7 6278.7 6309.0 6215.8

5 8400.0 8359.7 8351.4 8349.7 8349.3 8349.4 8349.3 8349.3 8245.7 8239.9

Table 5

Convergence of natural frequencies (Hz) for angle-ply (300/500/300/500) laminated beams with different number of beam segments N0 (L= 0.381 m, b= h= L/15; boundary

conditions: CC).

Mode no. Number of beam segments Ref.[43]HBT[LST] Ref.[54]HBT [R]

N0= 2 N0= 4 N0= 6 N0= 8 N0= 12 N0= 20 N0= 50 N0= 100

1 637.9 637.4 637.3 637.2 637.2 637.2 637.2 637.2 638.5 640.5

2 1657.7 1654.5 1653.7 1653.4 1653.2 1653.2 1653.2 1653.2 1657.3 1666.8

3 3040.0 3029.2 3026.5 3025.6 3025.1 3024.9 3025.0 3025.0 3034.0 3059.54 4677.5 4654.1 4648.3 4646.2 4645.1 4644.8 4644.8 4644.8 4661.2 3397.8

5 4960.9 4960.9 4960.9 4960.9 4960.8 4960.8 4960.8 4960.8 4784.6 4712.5

Table 6

Natural frequencies (Hz) for angle-ply (300/500/300/500) laminated beams with different boundary conditions (L= 0.381 m, b = h= L/15).

Mode no. FF FC SS CS

Ref.[43]HBT [LST] Present HBT[LMR] Ref.[43]HBT[LST] Present HBT[LMR] Ref.[43]HBT[LST] Present HBT[LMR] Ref.[43]HBT[LST] Present HBT[LMR]

1 659.3 659.2 105.3 105.1 294.8 294.9 450.5 449.8

2 1738.6 1735.2 637.6 636.4 1132.4 1130.2 1389.9 1387.0

3 3213.4 3207.2 1698.0 1694.2 2414.4 2409.6 2724.1 2717.5

4 4784.8 4948.0 2392.3 2480.6 4012.3 4001.1 4340.4 4327.8

5 4961.0 4961.6 3121.0 3113.2 4784.2 4952.9 4784.3 4956.1



8/18

[43] for the angle-ply (300/500/300/500) laminated composite beamwith four sets of boundary conditions, namely, the FF, FC, SSand CS. The present results are obtained by using N0= 4. Again,

one can see that the present HBT[LMR]solutions are in good corre-lation with the reference results.

Investigations carried out above demonstrate that the freevibration behaviors of composite laminated beams are well charac-terized by the HBT[LMR]. Comparisons of numerical results based ondifferent beam theories for angle-ply (h) laminated beams withfree end conditions are shown in Table 7. The material propertiesand geometric dimensions of the beams are given as follows:E1= 37.41 GPa, E2= 13.67 GPa, G12= 5.478 GPa, G13= 6.03 GPa,G23= 6. 666 GPa, l12= 0.3; for the (0

0) beam: L= 0.21008 m,b= 16.96103m, h= 3.33103m, q= 1975.2 kg/m3; for the(450) beam: L= 0.11179 m, b= 12.7103m, h= 3.38103 m,q= 1968.9 kg/m3; for the (900) beam: L= 0.11180 m,b= 12.68103 m, h= 3.37103 m, q= 1965.9 kg/m3. Experi-mental results reported by Ritchie et al. [90] for the laminatedbeams are also included inTable 7for additional comparison pur-poses. It is observed that the present HBT solutions are in closeagreement with the experimental results. The max discrepancy be-

tween the HBT results and the experimental data does not exceedthan 2.8% for the worst case and in most case is less than 1%. Forlower-order vibration modes, there seem to be almost no differ-ences in the solutions of all HBTs; however, the vibration results

of HBT[V], HBT[VE1] and HBT[VE2] deviate significantly from other

HBT solutions when higher-order vibration modes are concerned.The discrepancies in the results may be attributed to the violationof the tangential stress-free boundary conditions for the above

three types of HBTs. Another interesting observation is that theHBT[VE1] and HBT[VE2] produce identical results for all numericalcases. This is anticipated since they use the similar shape functions

just with different multiplying coefficients. Moreover, the HBT[VE1]and HBT[VE2]predict frequencies that are higher than the frequen-cies based on other HBTs. A detailed comparison between the pres-ent HBT results and those HBT[LST]solutions of Jun and Hongxing[43]for the (450) beam with other four sets of boundary conditions(i.e., FC, SS, CS and CC) are listed inTable 8. From this table the

accuracy of the results of the proposed formulation is remarkable.There is no significant difference between the present HBT resultsand the HBT[LST]solutions reported by Jun and Hongxing [43], ex-cept for the results of those HBTs (i.e., HBT [V], HBT[VE1], HBT[VE2],

HBT[M5], and HBT[M6]) which do not satisfy zero shear traction con-ditions on the top and bottom surfaces of the beam.

Having shown the validity of the present approach, we now turnour attention to the parametric studies on the free vibrations oflaminated beams. In view of the large number of parameters that

could be studied, it is decidedto limit the scope of the current workto an investigation of just two variables, namely the effects of ply-orientation and material anisotropy on the natural frequencies oflaminated beams. Unless otherwise stated, the following material

parameters are used: E1/E2= 15, E2= 10 GPa, G12/E2= 0.5,

Table 7

Comparison of natural frequencies (Hz) obtained with different theories for angle-ply ( h) laminated beams (boundary conditions: F-F).

Ply-angle Mode HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]

00 1 337.02 337.04 337.04 337.05 337.05 337.15 337.05 337.04 337.04

2 926.06 926.07 926.07 926.08 926.19 927.22 926.07 926.09 926.09

3 1806.65 1806.69 1806.69 1806.71 1807.16 1811.48 1806.68 1806.76 1806.77

4 2967.26 2967.27 2967.26 2967.33 2968.58 2980.75 2967.27 2967.47 2967.50

5 4397.15 4397.17 4397.17 4397.31 4400.13 4427.45 4397.17 4397.61 4397.71

6 6083.53 6083.57 6083.57 6083.84 6089.34 6142.21 6083.56 6084.40 6084.62

HBT[M4] HBT[VE1] HBT[VE2] HBT[S] HBT[M5] HBT[M6] HBT[AT1] HBT[AT2] Ref.[90]

1 337.05 337.32 337.32 337.05 337.04 337.05 337.04 337.03 337.7

2 926.09 928.88 928.88 926.07 926.11 926.13 926.07 926.10 926.7

3 1806.78 1818.40 1818.40 1806.66 1806.86 1806.95 1806.70 1806.81 1812.3

4 2967.52 3000.37 3000.37 2967.24 2967.75 2967.99 2967.30 2967.60 2979.1

5 4397.76 4471.94 4471.94 4397.05 4398.25 4398.80 4397.23 4397.92 4411.1

6 6084.72 6229.32 6229.32 6083.41 6085.69 6086.70 6083.70 6084.99 6104.8

450 HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]1 764.88 765.31 765.31 765.29 765.34 765.68 765.31 765.33 765.31

2 2096.99 2097.37 2097.36 2097.38 2097.72 2100.96 2097.37 2097.44 2097.44

3 4077.13 4077.41 4077.41 4077.48 4078.84 4092.08 4077.40 4077.64 4077.67

4 6667.09 6667.26 6667.27 6667.46 6671.23 6707.87 6667.25 6667.87 6667.98

5 9829.59 9829.69 9829.68 9830.11 9838.49 9919.14 9829.68 9830.97 9831.27


1 765.32 766.18 766.18 765.26 765.33 765.26 765.32 765.32 767.9

2 2097.44 2106.12 2106.12 2097.37 2097.51 2097.50 2097.38 2097.47 2091.9

3 4077.71 4113.36 4113.36 4077.44 4077.93 4078.17 4077.44 4077.80 4155.9

4 6668.07 6767.32 6767.32 6667.04 6668.71 6669.46 6667.36 6668.26 6762.7

5 9831.48 10051.61 10051.61 9829.20 9832.92 9834.53 9829.89 9831.85 10006.0

900 HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]1 728.82 728.61 728.63 728.62 728.65 728.91 728.61 728.62 728.63

2 1997.99 1997.84 1997.83 1997.84 1998.13 2000.78 1997.84 1997.87 1997.89

3 3886.91 3886.87 3886.85 3886.93 3888.05 3898.92 3886.87 3887.03 3887.07

4 6362.23 6362.01 6362.00 6362.15 6365.27 6395.50 6362.02 6362.50 6362.61

5 9390.71 9390.85 9390.82 9391.20 9398.10 9464.88 9390.82 9391.89 9392.19


1 728.62 729.31 729.31 728.81 728.66 728.61 728.61 728.63 731.2

2 1997.89 2005.01 2005.01 1997.90 1997.95 1997.96 1997.82 1997.91 1994.8

3 3887.09 3916.40 3916.40 3886.89 3887.28 3887.47 3886.87 3887.15 3901.5

4 6362.64 6444.41 6444.41 6362.10 6363.18 6363.75 6362.05 6362.81 6315.7

5 9392.29 9574.03 9574.04 9391.01 9393.50 9394.78 9391.00 9392.61 9317.8



9/18

G13 /E2=G23/E2= 0.3846, l12 = 0.25 and q= 1600 kg/m3. The

HBT[LMR] and a decomposition of four segments in the laminatedbeams are employed for all numerical computations. The first four

mode frequencies of four-layer symmetric angle-ply (h, h, h,h)beams are given for different ply orientations and boundary condi-tions, as shown inTable 9. The natural frequencies are presentedusing a dimensionless form:x xL2=h

ffiffiffiffiffiffiffiffiffiffiffiq=E1

p . It is observed that

the frequency parameters decrease with increasing the ply orienta-tion angle for all boundary conditions. Moreover, the elastic bound-ary conditionhas a profound influence on the natural frequencies of

the laminated beams, and the CC boundary condition provides thelargest frequencies for all cases.The fundamental frequency parameters x xL2=h

ffiffiffiffiffiffiffiffiffiffiffiq=E2

p for

antisymmetric cross-ply laminated beams with different layer

numbers and orthotropic ratios E1/E2 are shown inTable 10. It isclearly shown from this table that, for a certain layer-up, the fre-quency values increase with an increase of the E1/E2 ratio for allclassical boundary conditions. This is true because an increase in

elastic modulus strengthens the beam stiffness. Further, for eachset of classical boundary conditions, as the number of layers in-creases, the fundamental frequency parameter increases. The rea-son for this is that as the number of layers increases, the

magnitude of the coupling stiffnesses B11 and E11 in Eq. (4) de-crease. It is noted that for the antisymmetric cross-ply laminates,the coupling stiffnesses reach their maximum values with two lay-

ers and become zero with an infinite number of layers. As a conse-

quence, the less is the extension-bending coupling, then the largerfundamental frequency parameters are obtained. However, the dif-ferences are insignificant for beams that are constructed of a larger

number of layers. For theEIIIEIII boundary conditions, it is of par-ticular interest to note that the frequency parameters of the beamsare not particularly affected by the change of the layer numbersand E1/E2 ratios. This is expected since the fundamental vibration

frequencies of the beams are dominant by the elastic supportconditions.

As far as the computational aspects of the study are concerned,

the current formulation is presented in a very general form thatleaves open the opportunity of using it in conjunction with differ-ent polynomials. It has been proved in the aforementioned studiesthat the free vibration behaviors of laminated beams are well char-

acterized by the Chebyshev orthogonal polynomials of first kind(COPFK). In order to evaluate the versatility of the formulationfor the application of different basis functions, the relative fre-quency discrepancies between the COPFK solutions and the results

of COPSK, LOPFK and OPP for the antisymmetric cross-ply (00/900)2laminated beam with different boundary conditions are illustratedinFig. 2. The frequency results of the first fifty modes are exam-ined. In all the numerical cases, the HBT[LMR] is considered and

the order of the polynomials is taken as: P= 8. The discrepancy isdefined as: Relative discrepancy xd xCOPFK=xCOPFK, wherethe subscript d denotes the COPSK, LOPFK and OPP. The number

of segments, N0, decomposed in the laminated beam is chosen

Table 8

Comparison of natural frequencies (Hz) obtained with different theories for the angle-ply (450/450/450/450) laminated beam subjected to different boundary conditions

(L= 0.11179 m, b = 12.7 103m, h = 3.38 103m).

Boundary Mode HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]

FC 1 120.58 120.57 120.57 120.57 120.58 120.60 120.58 120.58 120.57

2 752.61 752.61 752.60 752.61 752.70 753.52 752.61 752.63 752.62

3 2093.93 2093.93 2093.93 2093.96 2094.52 2099.85 2093.93 2094.02 2094.04

4 4065.98 4066.00 4066.00 4066.09 4068.08 4086.84 4066.00 4066.29 4066.38

HBT[M4] HBT[VE1] HBT[VE2] HBT[S] HBT[M5] HBT[M6] HBT[AT1] HBT[AT2] Ref.[43]1 120.57 120.62 120.62 120.57 120.58 120.58 120.58 120.57 120.5

2 752.62 754.81 754.81 752.60 752.64 752.61 752.61 752.62 752.5

3 2094.05 2108.44 2108.44 2093.93 2094.15 2093.92 2093.94 2094.07 2093.8

4 4066.42 4117.33 4117.33 4065.99 4066.78 4066.02 4066.05 4066.49 4065.7

HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]SS 1 338.20 338.20 338.20 338.20 338.21 338.33 338.20 338.20 338.20

2 1347.02 1347.00 1347.01 1347.01 1347.20 1349.00 1347.01 1347.04 1347.04

3 3009.51 3009.52 3009.52 3009.56 3010.48 3019.38 3009.52 3009.67 3009.69

4 5299.07 5299.05 5299.05 5299.18 5301.99 5329.25 5299.05 5299.49 5299.57


1 338.20 338.50 338.50 338.20 338.21 338.20 338.20 338.20 338.1

2 1347.04 1351.86 1351.86 1347.01 1347.08 1347.01 1347.01 1347.05 1347.0

3 3009.71 3033.65 3033.65 3009.53 3009.88 3009.51 3009.55 3009.77 3009.5

4 5299.63 5373.40 5373.39 5299.05 5300.12 5299.15 5299.12 5299.79 5299.1

HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]CS 1 527.32 527.32 527.32 527.33 527.35 527.86 527.32 527.33 527.33

2 1698.97 1698.98 1698.98 1699.00 1699.32 1703.43 1698.98 1699.04 1699.06

3 3514.93 3514.94 3514.93 3515.03 3516.45 3532.07 3514.94 3515.16 3515.25

4 5945.46 5945.45 5945.44 5945.67 5949.70 5991.27 5945.43 5946.04 5946.29


1 527.34 528.63 528.63 527.32 527.35 527.32 527.36 527.34 527.3

2 1699.07 1709.88 1709.88 1698.98 1699.14 1698.99 1699.22 1699.08 1698.9

3 3515.30 3557.13 3557.12 3514.95 3515.57 3514.92 3515.88 3515.35 3514.7

4 5946.38 6059.00 6059.00 5945.45 5947.15 5945.47 5947.97 5946.53 5944.8

HBT[R] HBT[KPR] HBT[LMR] HBT[LST] HBT[M1] HBT[V] HBT[M2] HBT[M3] HBT[KA]CC 1 763.39 763.38 763.38 763.39 763.48 764.78 763.39 763.40 763.41

2 2088.73 2088.74 2088.73 2088.78 2089.45 2097.01 2088.74 2088.84 2088.89

3 4054.70 4054.77 4054.77 4054.91 4057.27 4081.73 4054.77 4055.11 4055.28

4 6621.53 6621.63 6621.62 6621.96 6627.94 6686.94 6621.62 6622.44 6622.89


1 763.42 766.80 766.80 763.38 763.44 763.39 763.47 763.42 763.3

2 2088.91 2109.06 2109.06 2088.73 2089.05 2088.73 2089.20 2088.93 2088.5

3 4055.35 4121.44 4121.44 4054.77 4055.80 4054.77 4056.28 4055.40 4054.14 6623.03 6784.44 6784.43 6621.61 6624.13 6621.62 6625.32 6623.12 6619.9



10/18

as:N0= 50. It can be observed fromFig. 2that the four sets of poly-nomials lead to almost identical results, and the absolute maxi-mum discrepancy does not exceed 1106 for the worst case.

This reveals, regardless of the boundary condition being consid-

ered, that the accuracy of the present formulation is not particu-larly affected by the employed basis functions.

Concerning the present formulation used in the free vibration

analysis, one may assert that it offers a simple yet powerful

Table 9

Frequencies parametersx xL2=hffiffiffiffiffiffiffiffiffiffiffiq=E1

p of four-layer angle-ply (h, h, h,h) laminated beams with different boundary conditions (L/h= 15, b= h= 2.5 102m; elastic

support stiffness: ku= 1.5 105 N/m, kw= 2.5 10

5 N/m).

h Mode no. Boundary conditions

SS PP CC FC S-C PC FP S-P EIEI EIIEII EIIIEIII

00 1 2.6297 3.0928 4.7851 0.9764 3.6824 3.9264 0.4204 2.8597 0.5196 0.6659 0.5196

2 8.7727 9.3764 10.7268 5.1117 9.7960 10.0892 4.2905 9.0725 4.7851 2.7942 0.6659

3 16.1350 16.9016 17.7455 11.8961 16.9551 17.3349 11.2284 16.5143 10.7268 8.8238 2.7942

4 23.8003 24.7482 25.2337 19.3963 24.5222 24.9960 19.0247 24.2687 17.7455 16.1629 8.8238

150 1 1.8773 2.3122 3.7170 0.6844 2.7496 2.9832 0.3419 2.0938 0.5196 0.6629 0.5196

2 6.7199 7.2621 8.8099 3.8516 7.7941 8.0604 3.1286 6.9903 3.7170 2.1011 0.6629

3 13.1319 13.7852 15.0281 9.4723 14.1043 14.4265 8.6752 13.4565 8.8099 6.7864 2.1011

4 20.1919 20.9678 21.8363 16.1212 21.0284 21.4132 15.5248 20.5767 15.0281 13.1662 6.7864

300 1 1.2251 1.6057 2.6062 0.4409 1.8589 2.0677 0.2556 1.4147 0.5195 0.6553 0.5195

2 4.6455 5.1094 6.5985 2.6261 5.6167 5.8494 2.0976 4.8780 2.6062 1.5447 0.6553

3 9.6804 10.2105 11.8104 6.8506 10.7593 11.0224 6.0651 9.9450 6.5985 4.7416 1.5447

4 15.7425 16.3389 17.7947 12.3123 16.7873 17.0825 11.4991 16.0397 11.8104 9.7262 4.7416

450 1 0.8745 1.2163 1.9202 0.3132 1.3466 1.5367 0.2022 1.0444 0.5194 0.6429 0.5194

2 3.3982 3.8195 5.0431 1.9097 4.1982 4.4127 1.5362 3.6099 1.9202 1.2816 0.6429

3 7.3167 7.7888 9.3411 5.1343 8.3232 8.5601 4.4869 7.5530 5.0431 3.5300 1.2816

4 12.3174 12.8324 14.5082 8.1271 13.4189 13.6756 8.1271 12.5748 9.3411 7.3771 3.5300

600 1 0.7583 1.0817 1.6787 0.2713 1.1720 1.3527 0.1821 0.9188 0.5193 0.6348 0.5193

2 2.9650 3.3674 4.4547 1.6638 3.6846 3.8907 1.3470 3.1673 1.6787 1.2032 0.6348

3 6.4408 6.8908 8.3408 4.5099 7.3789 7.6057 3.9349 6.6663 4.4547 3.1163 1.20324 10.9539 11.4409 12.8083 6.4044 12.0221 12.2657 6.4044 11.1975 8.3408 6.5095 3.1163

750 1 0.7325 1.0523 1.6241 0.2620 1.1330 1.3118 0.1777 0.8910 0.5192 0.6325 0.5192

2 2.8675 3.2667 4.3189 1.6086 3.5677 3.7723 1.3052 3.0682 1.6241 1.1865 0.6325

3 6.2400 6.6865 8.1043 4.3675 7.1591 7.3844 3.8112 6.4638 4.3189 3.0241 1.1865

4 10.6340 11.1165 12.2105 6.1054 11.6891 11.9306 6.1054 10.8754 8.1043 6.3110 3.0241

900 1 0.7302 1.0496 1.6191 0.2611 1.1294 1.3081 0.1773 0.8885 0.5192 0.6322 0.5192

2 2.8586 3.2576 4.3064 1.6036 3.5569 3.7615 1.3014 3.0592 1.6191 1.1850 0.6322

3 6.2215 6.6678 8.0823 4.3544 7.1388 7.3640 3.7999 6.4452 4.3064 3.0157 1.1850

4 10.6044 11.0867 12.1670 6.0837 11.6581 11.8996 6.0837 10.8457 8.0823 6.2927 3.0157

Table 10

Frequencies parametersx xL2=hffiffiffiffiffiffiffiffiffiffiffiq=E2

p for antisymmetric cross-ply laminated beams with different layer numbers and E1/E2 ratios (L/h= 15, b = h= 2.5 10

2m; elastic

support stiffness: ku= 1.5 105 N/m, kw= 2.5 10

5 N/m).

Lay-up E1/E2 Boundary conditions

SS PP CC FS F-P FC SP SC PC EIIIEIII

(00/900) 5 4.6399 5.8843 8.6667 6.1638 0.9211 1.4280 5.2389 6.2935 7.0727 2.0118

10 5.9576 7.2026 9.9944 7.1700 1.0737 1.6682 6.5460 7.4858 8.3192 2.0120

15 6.9269 8.2074 11.0100 7.9627 1.1866 1.8577 7.5284 8.4025 9.2775 2.0121

20 8.4007 9.7901 12.6369 9.2809 1.3600 2.1743 9.0526 9.8638 10.8115 2.0122

(00/900)2 5 4.8152 6.1047 9.9582 7.2218 0.9519 1.6733 5.4522 7.1230 7.8584 2.0119

10 6.3992 7.6945 12.5325 9.4130 1.1284 2.1993 7.0362 9.1690 9.9197 2.0122

15 7.5990 8.9048 14.3085 11.0828 1.2536 2.6095 8.2395 10.6551 11.4174 2.0122

20 9.4267 10.7603 16.7204 13.6405 1.4372 3.2581 10.0784 12.7976 13.5809 2.0123

(00/900)4 5 4.8475 6.2312 10.2589 7.4651 0.9769 1.7292 5.5359 7.3155 8.0815 2.0120

10 6.4647 7.9329 13.0785 9.8951 1.1760 2.3127 7.1942 9.5331 10.3405 2.0122

15 7.6883 9.2156 14.9869 11.7276 1.3162 2.7643 8.4462 11.1225 11.9579 2.0123

20 9.5485 11.1659 17.5255 14.4997 1.5195 3.4725 10.3490 13.3787 14.2560 2.0123

(00/900)8 5 4.8550 6.2733 10.3334 7.5249 0.9861 1.7430 5.5619 7.3631 8.1424 2.0120

10 6.4793 8.0100 13.2128 10.0124 1.1937 2.3402 7.2420 9.6223 10.4535 2.0122

15 7.7078 9.3147 15.1540 11.8838 1.3394 2.8017 8.5076 11.2367 12.1020 2.0123

20 9.5748 11.2934 17.7256 14.7071 1.5501 3.5241 10.4282 13.5212 14.4358 2.0123

(00/900)20 5 4.8570 6.2859 10.3542 7.5416 0.9890 1.7468 5.5696 7.3763 8.1599 2.0120

10 6.4833 8.0329 13.2503 10.0450 1.1991 2.3478 7.2560 9.6471 10.4857 2.0122

15 7.7131 9.3439 15.2007 11.9272 1.3464 2.8120 8.5255 11.2686 12.1430 2.0123

20 9.5819 11.3309 17.7816 14.7647 1.5593 3.5384 10.4512 13.5609 14.4870 2.0123

(00/900)40 5 4.8573 6.2878 10.3572 7.5440 0.9894 1.7473 5.5707 7.3782 8.1624 2.0120

10 6.4838 8.0362 13.2556 10.0497 1.1998 2.3489 7.2580 9.6506 10.4903 2.0122

15 7.7139 9.3482 15.2073 11.9334 1.3474 2.8135 8.5281 11.2731 12.1489 2.0123

20 9.5829 11.3363 17.7896 14.7729 1.5606 3.5404 10.4545 13.5666 14.4943 2.0123



11/18

alternative to other complex analytical techniques for providing

reasonably accurate vibration frequencies of laminated beams.The use of the generalized displacement field ensures the applica-

bility of the present formulation and solution methodology to avariety of beam theories. Not only the lower-order frequenciesbut the high-order frequencies can be obtained by using a small

number of beam segments. Moreover, unlike most existing tech-niques, the current method offers a unified solution for a varietyof boundary conditions, including both the classical and non-classical cases. For different boundary conditions, there is no need

to reformulate the beam matrix in the present method, and onlyvery simple algebraic operations at the geometrical boundariesneed to be re-evaluated. This leads to reduced computational ef-forts, and is simple and efficient to implement. It is of interest to

note that, irrespective of the boundary condition being considered,the accuracy of the present formulation is not particularly affectedby the employed polynomials. This makes the choice of admissible

functions very flexible, and can be considered as one distinguished

feature of the present formulation.

3.2. Forced vibration analysis of laminated beams: transient responses

This subsection is devoted to the transient response analysis of

laminated beams subjected to transverse dynamic loads. The fol-lowing material properties are used throughout the investigation:E1/E2= 15,E2= 9.5 GPa,G12/E2=G13/E2= 0.5, G23/E2= 0.35, l12= 0.3and q= 1450 kg/m3. All layers are assumed to have the same thick-ness. Two commonly time-dependent dynamic loads, i.e., the rect-angular pulse and exponential pulse (see Fig. 3), are considered.For the rectangular pulse, the loading function is described as:q=q0[H(t)H(tT0)], where q0 and T0 are respectively the mag-

nitude and duration of the applied load, and H(t) is the Heavisidestep function. The exponential pulse is given as: q =q

0est, which

may be used to simulate the laminated beam subjected to a blast

0 10 20 30 40 50-6

-3

0

3

6

9

COPSK

LOPFK

OPP

COPSK

LOPFK

OPP

(c)(b)

Relatived

iscrepancy(x10-7)

Relativedis

crepancy(x10-7)

Relatived

iscrepancy(x10-7)

Relatived

iscrepancy(x10-7)

Mode number

0 10 20 30 40 50

Mode number

0 10 20 30 40 50

Mode number

0 10 20 30 40 50

Mode number

0 10 20 30 40 50

Mode number

0 10 20 30 40 50

Mode number

COPSK

LOPFK

OPP

(a)

-6

-3

0

3

6

-3

0

3

-9

-6

-3

0

3

6

9

12

Relativedis

crepancy(x10-7)

Relativedis

crepancy(x10-7)

-9

-6

-3

0

3

6

9

12COPSK

LOPFK

OPP

COPSK

LOPFK

OPP

(f)(e)COPSK

LOPFK

OPP

(d)

-12

-9

-6

-3

0

3

6

9

12

15

Fig. 2. Frequency discrepancies of orthogonal polynomials for the antisymmetric cross-ply (0 0/900)2laminated beam with different boundary conditions: (a) SS; (b) PP; (c)

CC; (d) F-C; (e) S-C; (f) P-C.

Fig. 3. Applied load types for a laminated beam.



12/18

0.00 0.03 0.06 0.09 0.12 0.15-3

-2

-1

0

1

2

3

4

Deflection(x10-4m)

Deflection(x10-4m)

Time (s)

0.00 0.03 0.06 0.09 0.12 0.15

Time (s)

Present: N0=2

Present: N0=4

Present: N0=8

ANSYS

(a)

-3

-2

-1

0

1

2

3

4

5Present: N0=2

Present: N0=4

Present: N0=8

ANSYS

(b)

Fig. 4. Transient deflection responses of the orthotropic beam subjected to the rectangular pulse: (a) CC; (b) EII EII.

0.0 0.1 0.2 0.3 0.4 0.5 0.6-1.5

-1.0

-0.5

0.0

0.5

1.0

Deflection(x10-2m)

Time (s)

HBT[R]

HBT[KPR]

HBT[LMR]

HBT[LST]

HBT[M1]

HBT[V]

HBT[M2]

HBT[M3]

HBT[KA]

HBT[M4]

HBT[VE1]

HBT[VE2]

HBT[S]

HBT[M5]

HBT[M6]

HBT[AT1]

HBT[AT2]

Fig. 5. Transient deflection responses of angle-ply (450) beam subjected to uniform rectangular pulse and SS boundary conditions: L/h= 150,b=h= 0.01 m;T0= 0.3 s,x1= 0,

x2= L.

0.000 0.005 0.010 0.015 0.020

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Deflection(x10-5m)

Time (s)

HBT[R]

HBT[KPR]

HBT[LMR]

HBT[LST]

HBT[M1]

HBT[V]

HBT[M2]

HBT[M3]

HBT[KA]

HBT[M4]

HBT[VE1]

HBT[VE2]

HBT[S]

HBT[M5]

HBT[M6]

HBT[AT1]

HBT[AT2]

Fig. 6. Transient deflection responses of angle-ply (450) beam subjected to uniform rectangular pulse and SS boundary conditions:: L/h= 30,b=h= 0.01 m;T0= 0.01 s,x1= 0,x2= L.



13/18

loading if the characteristic parameter sis adjusted to approximatethe loading curve from a real blast test. These two loadings may

have arbitrary distributions in the spatial domain. However, forthe sake of brevity, only uniform and sinusoidal distributions ofthe loadings along the length of the beam are examined herein. Un-

less otherwise stated, the COPFK (P= 7) and HBT[LMR]are employedfor the following theoretical calculations, and the transient re-

sponses are measured at the center of the beams in the transversedirection. All the initial conditions, i.e., displacements and veloci-

ties, are taken as zero.The first testing example concerns an orthotropic beam sub-

jected to the rectangular pulsefw(x,t) =q0[H(t)H(tT0)], which

is assumed to be uniformly distributed along the length of the beam(i.e., x1= 0 and x2=L) with intensity q0=1 N/m and duration

0.0000 0.0006 0.0012 0.0018 0.0024 0.0030-2.8

-2.1

-1.4

-0.7

0.0

0.7

Deflection(x10-7m)

Time (s)

HBT[R]

HBT[KPR]

HBT[LMR]

HBT[LST]

HBT[M1]

HBT[V]

HBT[M2]

HBT[M3]

HBT[KA]

HBT[M4]

HBT[VE1]

HBT[VE2]

HBT[S]

HBT[M5]

HBT[M6]

HBT[AT1]

HBT[AT2]0.0020 0.0022

-0.4

-0.3

-0.2

-0.1

Fig. 7. Transient deflection responses of angle-ply (45 0) beam subjected to uniform rectangular pulse and SS boundary conditions:: L/h= 10, b =h= 0.01 m; T0= 0.0015 s,

x1= 0, x2= L.

0.000 0.002 0.004 0.006 0.008 0.010 0.012-6

-4

-2

0

2

4

6

8

(a)

Deflection(x10-6m)

Deflection(x10-6m)

Time (s)

=00

=150

=300

=450

=600

=750

=900

0.000 0.002 0.004 0.006 0.008

-2

-1

0

1

2

3

(b)

Time (s)

=00

=150

=300

=450

=600

=750

=900

0.000 0.002 0.004 0.006 0.008-2

-1

0

1

2

(c)

D

eflection(x10-6m)

D

eflection(x10-5m)

Time (s)

=00

=150

=300

=450

=600

=750

=900

0.00 0.01 0.02 0.03 0.04 0.05-2

-1

0

1

2

3

(d)

Time (s)

=00

=150

=300

=450

=600

=750

=900

Fig. 8. Effects of ply orientation on the centre deflection of angle-ply laminated beams with different boundary conditions: (a) SS; (b) PP; (c) CC; (d) EIIIEIII.



14/18

T0= 0.015 s.In this example, geometrical properties of the beamaregiven as:L/h= 150, b=h= 0.01 m. Two sets of boundary conditionsare examined, namely the CC and EIIEII. In the EII-EII boundaryconditions, the elastic support stiffness at the ends of the beam is

taken as: kv= 3.5 106 N/m. Comparisons are made with finite ele-

ment analysis ANSYS to check the validity of the present theoreticalformulation. A mesh of 150 BEAM188 elements is used for theorthotropic beam for reasonably converged results. A full calcula-

tion procedure (direct solver) is employed in ANSYS. The deflectionresponses for the beam with CC and EIIEII boundary conditionsare illustrated in Fig. 4a and b, respectively. The time incrementDtin the figures is taken to be 0.15 ms. It is observed that the pres-

ent formulation gives fairly accurate predictions of the transientvibration responses, and a decomposition of two beam segmentsis adequate for converged results for the two boundary conditions.

To examine the accuracy of different theories for predicting

transient responses of laminated beams, Figs. 57 depict thetime-history deflections of angle-ply (450) laminated beams withdifferent length-to-thickness ratios (L/h= 150,30,10). In all cases,SS boundary conditions are considered, and the uniformly distrib-uted rectangular pulse is employed with intensity q0=1 N/m.

The numerical results show that, in the case of slender and moder-ately thick beams (i.e.,L/h= 150,30), the discrepancies of the trans-verse deflection responses obtained by different higher-order sheardeformable beam theories are negligible. This implies that all the

shear functions defined in Table 1 may be used in the modelingof the displacement field for slender and moderately thick lami-nated beams. However, for thick beams (e.g., L/h= 10), the vibra-tion responses based on HBT[V], HBT[VE1] and HBT[VE2] tend to

deviate from those calculated by the rest of HBTs. The differences

in the results are probably due to the fact that the three sets ofHBTs violate the free surface boundary conditions. It is interestingto note that for laminated beams with pinned boundary conditions,the deflection responses determined by HBT[VE1]and HBT[VE2]devi-

ate significantly from other HBTs even for moderately thick beams.However, those results for pinned supported beams are omittedherein for the brevity of the presentation.

Havingalready established the accuracy of the present solutions,

parametric studies are conducted to investigate the transient vibra-tion behaviorsof laminatedbeamssubjected to an exponential load-ing. The influence of ply orientation and layer number on thedynamic responses of laminated beams will be discussed in detail

based on the HBT[LMR]. The loading has a sinusoidal distribution inthe spatial domain, defined as: fw(x, t) =q0cos (px/L)e

st(x1= 0,x2=L,see Fig.3), andthe magnitude q0 and characteristic parameter

s of the applied load are respectively given as: q0=1 N/m ands= 800. The geometric details of the beam are: L= 0.25 m,b=h= 0.01 m. Four sets of boundary conditions are examined,namely the SS, PP, CC and the EIII EIII. In theEIIIEIII boundaryconditions, the elastic support stiffnesses are taken as: ku= 1.5105 N/m andkv= 2.510

5 N/m.

Fig.8 shows theeffect of theply orientationhonthedeflectionre-sponses of angle-ply (h) laminated beams with different boundaryconditions. The depicted results reveal that the increase of the fiberorientation plays a beneficial role toward increasing the amplitude

and the oscillation period of transient deflection responses for thebeams with three combinations of classical boundary conditions,i.e., SS, PP and CC. This is expected since an increase in the fiberorientation will lead to a decrease in the bending stiffness of the

beam. In the case of the EIIIEIII boundary conditions, the fundamen-

0.000 0.002 0.004 0.006 0.008

-1.0

-0.5

0.0

0.5

1.0

1.5

Deflection(x10-6m)

Deflection(x10-6m)

Deflection(x10-6m)

Time (s)

(00/90

0)

(00/90

0)8

(00/90

0)16

(00/90

0)64

(a)

0.000 0.001 0.002 0.003 0.004 0.005 0.006-1.0

-0.5

0.0

0.5

1.0

Time (s)

(00/90

0)

(00/90

0)8

(00/90

0)16

(00/90

0)64

(b)

0.000 0.001 0.002 0.003 0.004 0.005-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Time (s)

(00/90

0)

(00/90

0)8

(00/90

0)16

(00/90

0)64

(c)

0.00 0.01 0.02 0.03 0.04 0.05 0.06-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Deflection(x10-5m)

Time (s)

(00/90

0)

(00/90

0)8

(00/90

0)16

(00/90

0)64

(d)

Fig. 9. Effects of layer number on the centre deflection of antisymmetric cross-ply laminated beams with different boundary conditions: (a) SS; (b) PP; (c) CC; (d) EIIIEIII.



15/18

tal bending modes are dominant by the elastic boundary, and theamplitude and oscillation period of the displacement responses

therefore are not affected by the change of the fiber orientation.The effects of layer number on the transient deflection re-

sponses of antisymmetric cross-ply laminated beams are illus-trated in Fig. 9. It can be seen from the figures that the

laminated beam of (00/900) tends to yield the largest deflection

amplitudes for the SS, PP and CC boundary conditions. This isbecause the extension-bending coupling terms for antisymmetriccross-ply beams attain their maximum value with two layers.

However, in the case of the EIIIEIII boundary conditions, the dis-crepancies of the deflection response curves for the laminatedbeams with different layer-ups are not quite obvious. Anotherobservation worth noting is that the amplitudes and oscillation

periods of the time response curves seem very insignificantly af-fected by an increase in the layer number.

4. Conclusions

This paper introduces a general computational algorithm forvibration analysis of composite laminated beams with arbitraryboundary conditions. A modified variational principle in conjunc-

tion with a multi-segment partitioning technique is employed toderive the formulation based on a general higher-order beam the-ory. The displacement and rotation fields of each beam segmentare expanded by means of polynomials. The Poissons effect and

the material couplings of the bending-stretching, bending-twist,and stretching-twist are considered. The applicability and versatil-ity of the present formulation are illustrated for free and transientvibrations of laminated beams with various geometric and mate-

rial parameters under different combinations of classical andnon-classical boundary conditions. It is found that the presentmethod furnishes stable and rapid convergence characteristics as

the number of beam segments is increased, and the numerical re-sults based on the present formulation are in excellent agreementwith corresponding results published in the literature and those

solutions obtained from the finite element analyses. The accuracyof several higher-order shear deformable beam theories for pre-dicting the vibrations of laminated beams has been confirmed.

What is of particular importance is that, regardless of the boundarycondition being considered, the accuracy of the present formula-tion is not particularly affected by the employed polynomials. This

makes the choice of admissible functions very flexible, and can beconsidered as one distinguished feature of the present formulation.

In views of the accuracy, efficiency and versatility of the presentformulation, it offers a potential alternative to the conventional

analytical and numerical approaches for initial and boundary valueproblems of composite laminated beams.

Appendix A. Constitutive equations for one-dimensional

laminated beam

The one-dimensional laminated beam equations that account

for Poissons effect are derived based on a general higher-orderplate theory with the following displacement field:

~ux;y;z; t ux;y; t fz@w

@xgz#x;y; t

~vx;y;z; t vx;y; t fz@w

@y gzux;y; t

~wx;y;z; t wx;y; t

A:1

where u, v,w, #andu are the five unknown functions of middle sur-face of the plate, while f(z) and g(z) represents shape functionsdetermining the distribution of the transverse shear strains andstresses along the thickness.

The linear strain expressions derived from the displacementmodel of Eq.(A.1)are as follows:

ex e0x fe1x ge2x ; ey e0y fe1y ge2y ; exy e0xyfe1xyge2xy;

cxzfj0xz gj

1xz; cyz

fj0yz gj1yz A:2

where

e0x @u

@x; e1x

@2w

@x2; e2x

@#

@x; e0y

@v

@y; e1y

@2w

@y2;

e2y @u@y

; e0xy@v

@x

@u

@y; e1xy 2

@2w

@x@y; e2xy

@u@x

@#

@y A:3

j0xz@w

@x; j1xz #; j

0yz

@w

@y; j1yz u A:4

f1 @f

@z

; g@g

@z

A:5

The laminated plate constitutive equations based on the general

higher-order shear deformation theory can be expressed as

For a one-dimensional composite laminated beam, it is as-sumed that the y-direction is free of stresses, i.e., Ny=Nxy=My= -Mxy=Py=Pxy= 0, while the mid-plane strains, and bending and

twisting curvatures corresponding to y-direction are assumed tobe nonzero. The strains and curvatures e0y ; e

1y ; e

2y ; e

0xy; e

1xy; e

2xy

can

be solved in terms of e0

x ; e1

x ; e2

x

, resulting in the followingrelation:

Nx

Ny

Nxy

Mx

My

Mxy

Px

Py

Pxy

266666666666666664

377777777777777775

A11 A12 A16 B11 B12 B16 E11 E12 E16

A12 A22 A26 B12 B22 B26 E12 E22 E26

A16 A26 A66 B16 B26 B66 E16 E26 E66

B11 B12 B16 D11 D12 D16 F11 F12 F16

B12 B22 B26 D12 D22 D26 F12 F22 F26

B16 B26 B66 D16 D26 D66 F16 F26 F66

E11 E12 E16 F11 F12 F16 H11 H12 H16

E12 E22 E26 F12 F22 F26 H12 H22 H26

E16 E26 E66 F16 F26 F66 H16 H26 H66

266666666666666664

377777777777777775

e0xe0ye0xye1xe1y

e1xye2xe2ye2xy

26666666666666666664

37777777777777777775

;Qxz

Pxz

A55 D55

D55 F55

j0xz

j1

xz" # A:6a-b



16/18

Nx

Mx

Px

264

375 A11 B11 E11B11 D11 F11

E11 F11 H11

264

375

e0xe1xe2x

264

375 A:7

Appendix B. Generalized mass and stiffness matrices of

composite laminated beam

The disjoint generalized mass and stiffness matrices of a lami-

nated beam are respectively given by

M diagM1;M2;. . .;Mi;. . .;MN0 ;

K diagK1;K2;. . .;Ki;. . .;KN0 B:1

where the sub-matricesMiand Kiare the mass and stiffness matri-ces of the ith beam segment.

Mi

Zli

Miuu Miuw M

iu#

Mi;Tuw Miww M

iw#

Mi;Tu# Mi;Tw# M

i##

2664

3775dl; Ki

Zli

Kiuu Kiuw K

iu#

Ki;Tuw Kiww K

iw#

Ki;Tu# Ki;Tw# K

i##

2664

3775dlB:2

The elements of the mass matrices are:

Miuu q0UTU; Miuw q1U

T@W

@x ;

Miu# q3UTH; Miww q0W

TW q2@WT

@x

@W

@x ;

Miw# q4@WT

@x H; Mi## q5H

TH

B:3

According to the FSDT, the elements of the stiffness matrices are:

Kiuu A11@UT

@x

@U

@x; Kiuw B11

@UT

@x

@2W

@x2 ; Kiu# E11

@UT

@x

@H

@x;

Kiww A55@WT

@x

@W

@x D11

@2WT

@x2@2W

@x2 ;

Kiw# D55@WT@x

H F11 @2WT@x2

@H@x

; Ki## F55HTH H11 @

HT

@x@H@x

B:4

The generalized interface stiffness matrix Kkand Kjintroducedby the MVP and LSWRM are obtained through the assembly of all

interface matrices. The interface matrixKik and Kij at the interface

location ofx =xi is given below

Kik

Kuiui Kuiwi Kui#i Kuiui1 Kui wi1 Kui#i1

KTui wi Kwi wi Kwi#i Kwi ui1 Kwi wi1 Kwi#i1

KTui#i KTwi#i

K#i#i K#iui1 K#iwi1 K#i#i1

KTuiui1 KTwiui1

KT#i ui1 0 0 0

K

T

ui wi1 K

T

wi wi1 K

T

#i wi10 0 0

KTui#i1 KTwi#i1

KT#i#i1 0 0 0

2666666666664

3777777777775

xxi

;

Kij

Kuiui 0 0 Kui ui1 0 0

0 Kwi wi 0 0 Kwiwi1 0

0 0 K#i#i 0 0 K#i#i1

KTui ui1 0 0 Kui1ui1 0 0

0 KTwi wi1 0 0 Kwi1wi1 0

0 0 KT#i#i1 0 0 K#i1#i1

2666666666664

3777777777775

xxi

B:5

where the sub-matrices are expanded as

Kui ui fuA11@UTi@x Ui U

Ti

@Ui@x

!;

Kui wi B11 fw@2UTi@x2

Wifr@UTi@x

@Wi@x

fuUTi

@2Wi@x2

! B:6a-b

Kui#i

E11

f#

@UTi

@x H

if

uUT

i

@Hi

@x !;

Kui ui1 fuA11@UTi@x Ui1;

Kui wi1 fwB11@2UTi@x2

Wi1frB11@UTi@x

@Wi1@x

B:6c-e

Kui#i1 f#E11@UTi@x

Hi1 B:6f

Kwi wi fw A55@WTi@x WiW

Ti

@Wi@x

! D11

@3WTi@x3

WiWTi

@3Wi@x3

!" #

frD11@2WTi@x2

@Wi@x

@WTi@x

@2Wi@x2

! B:6g

Kwi#i F11 f#@2WTi@x2

Hi fr@WTi@x

@Hi

@x

! fwW

Ti D55HiF11

@2Hi

@x2

!;

Kwi ui1 fuB11@2WTi@x2

Ui1 B:6h-i

Kwi wi1 fw A55@WTi@x

D11@3WTi@x3

!Wi1frD11

@2WTi@x2

@Wi1@x

;

Kwi#i1 f#F11@2WTi@x2

Documents

(Qu YG)a Variational Formulation for Dynamic Analysis of Composite Laminated Beams Based on a General Higher-Order Shear Deformation Theory