analysis of laminated composite beams using layerwise displacement theories.pdf

8/14/2019 Analysis of Laminated Composite Beams Using Layerwise Displacement Theories.pdf

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Analysis of laminated composite beamsusing layerwise displacement theories

Masoud Tahani *

Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran

Available online 30 March 2006

Abstract

Within the displacement field of a layerwise theory, two laminated beam theories for beams with general lamination are developed. Inthe first theory, an existing layerwise laminated plate theory is adapted to laminated beams. The procedure used in the second theory issimple and straightforward and similar to the one used in the development of plate and shell theories. These theories can also be used indeveloping simpler theories such as classical, first, and higher-order shear deformation laminated beam theories. Equations of motionsare obtained by using Hamiltons principle. For the assessment of the accuracy of these theories, analytical solutions for static bendingand free vibration are developed and compared with those of an existing three-dimensional elasticity solution of cross-ply laminates incylindrical bending and with the three-dimensional finite element analysis for angle-ply laminates.2006 Elsevier Ltd. All rights reserved.

Keywords: Composite beam; Layerwise theory; Analytical solution

1. Introduction

The use of fiber-reinforced composite laminates has gen-erally increased in weight sensitive applications such asaerospace and automotive structures because of their highspecific strength and high specific stiffness. By comparisonwith the analysis of laminated plates and shells, the workdone so far in the area of fiber-reinforced composite beamsis limited. Beam like structural components made of com-posite materials are being increasingly used in engineeringapplications. Because of their complex behavior in theanalysis of such structures some technical aspects must be

taken into consideration. For example, ignoring the trans-verse shear deformation in the classical lamination theories(CLTs) results in an underestimate in deflection and anoverestimate in natural frequencies. The first and higher-order shear deformation theories are improvements toclassical theories. In these theories transverse shear defor-mation through the thickness of the structure is not

ignored. Another aspect in the analysis of composite struc-tures is the existence of couplings among stretching, shear-ing, bending, and twisting. These couplings can significantlychange the response of composite structures and, thus, haveto be considered.

A survey of developments in the vibration analysis oflaminated beams and plates has been presented by Kapaniaand Raciti [1,2]. Miller and Adams [3] have studied thevibration of clamped-free unidirectional beams withoutincluding shear deformation and rotary inertia. Teoh andHuang[4] have presented a theoretical analysis for the freevibrations of the same beams by including the transverse

shear and rotary inertia and compared their numericalresults with the experimental results presented by Abarcarand Cunniff[5].

As far as the development of a laminated beam theory isconcerned, two different approaches are adopted in the lit-erature. In the first approach the lateral (the y-direction)displacement of the beam is simply neglected. This way,the couplings between in-plane shearing and stretchingand between bending and twisting are ignored. Such a the-ory is often used for isotropic beams [6]and cross-ply (0and 90 layers) laminated beams, [714]. Abramovich

0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2006.02.019

* Tel.: +98 511 876 3304; fax: +98 511 882 9541.E-mail address:[email protected]

www.elsevier.com/locate/compstruct

Composite Structures 79 (2007) 535547
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[15], Chandrashekhara et al.[16], Bhimaraddi and Chandr-ashekhara [17], and Sheinman [18] also have used thisapproach for angle-ply (handhlayers) laminated beams.In fact, the theory has been developed in these papers is forthe cylindrical bending of laminated plates and not for thebending of laminated beams. In the second approach a

laminated beam theory is developed from an existing lam-inated plate theory. To this end, the stress (force) andmoment resultants of the beam theory are obtained byignoring certain stress and moment resultants in the consti-tutive law of the laminated plates. This way the character-istic couplings, mentioned earlier, are not lost in the beamtheory. The process, however, demands the inversion ofcertain matrices which can be an inconvenience as far asdeveloping more advanced laminated beam theories areconcerned. This approach has been used in [19] forsymmetric beams and in [2026] for generally laminatedbeams.

It is well known that the equivalent single-layer (ESL)

theories provide a sufficiently accurate description of theglobal laminate response (e.g. maximum transverse deflec-tion, fundamental vibration frequency, critical bucklingload, etc.). However, these theories are often inadequatefor determining the three-dimensional (3-D) stress field atthe ply level. The most deficiency of the ESL theories inmodeling composites is that the transverse strain compo-nents in such theories are continuous across interface ofdissimilar materials and, thus, predicting discontinuoustransverse stress components at the layer interfaces. Unlikethe ESL theories, the layerwise theories (LWTs) assumeseparate displacement field expansions within each material

layer that exhibits onlyC0

-continuity through the laminatethickness. The resulting strain field is kinematically correctin that the in-plane strains are continuous through thethickness while the transverse strains are discontinuousthrough the thickness, thereby allowing for the possibilityof continuous transverse stresses as the number of numer-ical layers is increased. Therefore, in this paper, to obtainaccurate 3-D stress field in the beams a full layerwise theorywill be used.

It is the intention of the present work to develop a newlayerwise laminated beam theory to overcome the short-comings present in the two approaches discussed above.That is, the displacement field will be modified so thatthe constitutive law of a laminated beam can be obtainedin a straightforward manner as in most laminated plate

and shell theories. The resulting equations of motion willbe valid for generally laminated beams. Numerical resultswill be developed for the bending and free vibration ofcross-ply and angle-ply laminated beams. These results willbe compared with the existing 3-D elasticity solutions forcross-ply laminates in cylindrical bending [27] and withanother beam layerwise theory that will be developed froman existing layerwise laminated plate theory. The approachadopted in the present work will be demonstrated withinthe framework of a layerwise theory. However, the ideais straightforward and general and can readily be used indeveloping simpler theories such as classical laminationtheory and shear deformation theories. In this study, in

order to utilize a layerwise theory for laminated beams thatpossesses a full 3-D capability, a full layerwise theory willbe used.

2. Mathematical formulations

In what follows two layerwise laminated beam theoriesand a layerwise formulation for the analysis of laminatedplates subjected to cylindrical bending will be derived.First, a layerwise laminated plate theory is used to derivebeam layerwise theory 1 (BLWT1). Next, a new beam lay-erwise theory (BLWT2) will be developed. Finally, a layer-

wise laminated plate theory will be presented to analyzecylindrical bending (CB) of laminated plates.

2.1. Beam layerwise theory 1 (BLWT1)

Consider a rectangular (a b) laminated compositeplate of total thickness h. The geometry of the plate withpositive set of coordinate axes is shown in Fig. 1. In thisstudy, a full layerwise laminated plate theory is used in

x

y

a

1st Layer

y

z

2nd Layer

kth Layer

Nth Layer

1

2

3

k

k+1

N

N+1

1h2

h

kh

hNq(x,y,t)

z

b

h

Fig. 1. Laminate geometry and coordinate system.

536 M. Tahani / Composite Structures 79 (2007) 535547


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deriving plate equations of motion. The displacement fieldcan be represented as (e.g., see[2830]):

u1x;y;z; t Ukx;y; tUkzu2x;y;z; t Vkx;y; tUkz; k1; 2;. . .;N1u3x;y;z; t Wkx;y; tUkz

1

where, for the sake of brevity, the Einstein summation con-vention has been introduced a repeated index indicatessummation over all values of that index. In Eqs. (1) u1,u2, and u3 are the displacement components in the x, y,and z directions, respectively, of a material point initiallylocated at (x,y, z) in the undeformed laminate, Uk(x,y),Vk(x,y), andWk(x,y) (k= 1,2, . . . , N+ 1) are the displace-ment components of all points located on the kth plane inthe undeformed laminate, and Uk(z) are continuous func-tions of the thickness coordinate z (global interpolationfunctions). Also Ndenotes the total number of numerical(or mathematical) layers considered in a laminate.

It is noted that in the layerwise theory the accuracy ofthe displacement field in Eqs. (1) depends on the shapefunctionsUk(z) and the number of surfaces in the laminate.Here, Uk(z) are assumed to be linear interpolation func-tions. On the other hand, the number of surfaces may beincreased by subdividing each physical layer into a numberof numerical layers. The local Lagrangian linear interpola-tion functions within, say, the kth layer are defined asfollows:

/1kzk1z

hk; /2k

zzkhk

2

where hk is the thickness of the kth numerical layer (see

Fig. 1) and zkdenotes the z-coordinate of the bottom ofthekth numerical layer. This way, the global interpolationfunctions Uk(z) may be presented as (see[2830]):

Ukz

0; z6 zk1;

/2k1z; zk1 6 z6 zk;/1kz; zk6 z6 zk1;0; zP zk1;

8>>>>>>>:

k1; 2;. . .;N1.

3Upon substitution of Eqs.(1) into the linear straindis-

placement relations [31] of elasticity, the following results

will be obtained:

ex oUkox

Uk; ey oVkoy

Uk; ezWkU0k;

cyzVkU0koWk

oy Uk; cxzUkU0k

oWk

ox Uk;

cxy oUk

oyoVk

ox

Uk 4

with a prime indicating an ordinary derivative with respectto the independent variable z.

Using the Hamilton principle[31], 3(N+ 1) equations ofmotion corresponding to 3(N+ 1) unknowns Uk, Vk, and

Wkcan be shown to be:

dUk:oMkxox

oMkxy

oy QkxIkj Uj

dVk:oMkxy

ox oM

ky

oy Qky Ikj Vj

dWk:oRkx

oxoRky

oyNk

zIkj Wj

q

x;y; t

dk

N

1

5

where dk(N+1) is the Kronecker delta and q(x,y, t) is thetransverse load (per unit area) that is applied on the topsurface of the laminate (see Fig. 1). In Eqs.(5)a dot overdisplacement components indicates partial differentiationwith respect to temporal variable t. Also the generalizedstress resultants in Eqs. (5) are defined as:

Nkz;Qky;Qkx Z h=2

h=2rz;ryz;rxzU0kdz

Mkx;M

ky;M

kxy;R

ky;R

kx

Z h=2

h=2rx;ry;rxy; ryz;rxz

Ukdz

6

and the mass terms Ikj are defined as:

Ikj Z h=2

h=2qUkUj dz 7

where q(x,y, z) denotes the mass density of the materialpoint located at (x,y, z). For a laminated plate with a rect-angular platform, the boundary conditions in the layerwiseplate theory at an edge parallel to y axis involves the spec-ification ofUkorM

kx,VkorM

kxy, andWkorR

kx. Similarly, at

an edge parallel to x axis, we can specify the required

boundary conditions.In the full layerwise theory, instead of the plane stressassumption, the 3-D constitutive law [32]is used:

frgk Ckfegk 8Here, the matrixCk is called the off-axis stiffness matrixof the kth layer. On substitution of Eqs. (4) into Eqs. (8)and the subsequent results into Eqs. (6), the generalizedstress resultants are obtained which can be presented asfollows:

Nkz;Mkx;Mky;Mkxy

Bjk13;Dkj11;Dkj12;Dkj16oUjox

Bjk23;Dkj12;Dkj22;Dkj26oVjoy

Akj33;Bkj13;Bkj23;Bkj36Wj Bjk36;Dkj16;Dkj26;Dkj66 oUj

oy oVj

ox

Qky;Rky Akj45;Bkj45Uj Akj44;Bkj44Vj

Bjk45;Dkj45oWj

ox Bjk44;Dkj44

oWj

oy

Qkx;Rkx Akj55;Bkj55Uj Akj45;Bkj45Vj

B

jk55;D

kj55

oWj

ox B

jk45;D

kj45

oWj

oy

9

M. Tahani / Composite Structures 79 (2007) 535547 537


4/13

where the rigidity terms Akjpq, Bkjpq, and D

kjpq are given by

AkjpqXNi1

Z zi1zi

CipqU0kU

0j dz;

Bkjpq XN

i

1 Z

zi1

zi

CipqUkU0j dz;

DkjpqXNi1

Z zi1

zi

CipqUkUj dz

10

Using Eq.(3), by carrying out the integrations in Eqs.(10)the rigidity terms will be found (see [29]).

Here, plate equations of motion are adapted to obtainbeam equations of motion. It is more reasonable for abeam to let Mky be equal to zero. It is noted that thisassumption is similar to the ones assumed in the ESL the-ories for obtaining laminated beam theories from lami-nated plate theories (see, e.g., [20,21,25]). First thisassumption is invoked, yielding:

Mky Dkj12 oUjox

Dkj22 oVjoy

Bkj23WjDkj26 oUjoy

oVjox

011

Solving for oVj/oyand substituting the result into(9), witho

oy0, yields:

Nkz;Mkx;Mkxy Bjk13;Dkj11;Dkj16oUj

ox Bjk36;Dkj16;Dkj66

oVj

ox Akj33;Bkj13;Bkj36Wj

Qky;Qkx;Rkx Akj45;Akj55;Bkj45Uj Akj44;Akj45;Bkj55Vj Bjk45;Bjk55;Dkj55

oWj

ox

12where the reduced rigidity terms Akjpq, B

kjpq, and D

kjpq(p, q= 1,

3, 6) are defined inAppendix A. It is also assumed that allthe stress resultants are functions of coordinate x and timet only. Hence, Eqs. (5) are simplified as follows:

dUk:oMkxox

Qkx Ikj Uj

dVk:oMkxy

ox Qky Ikj Vj

dWk:oRkxox

Nkz Ikj Wjqx; tdkN1

13

Upon substitution of Eqs.(12)into Eqs.(13)the followinggoverning equations of motion are obtained:

Dkj

11

o2Uj

ox2Akj55UjDkj16

o2Vj

ox2Akj45Vj Bkj13Bjk55

oWj

ox

Ikj UjD

kj

16

o2Uj

ox2Akj45UjDkj66

o2Vj


oWj

ox

Ikj VjBkj55Bjk13

oUj

ox Bkj45Bjk36

o2Vj

ox2Dkj55

o2Wj

ox2 Akj33Wj

Ikj Wjqx; tdkN114

2.2. Beam layerwise theory 2 (BLWT2)

A generally laminated beam is considered here with atotal thickness h, width b in the lateral (y-) direction, andlength a in the longitudinal (x-) direction. A full layerwiselaminated beam theory is used to obtain accurate 3-D

stress field in the beam. In this theory, it is assumed thatthe displacement field of the beam may be represented as:

u1x;y;z; t Ukx; tUkzu2x;y;z; t Vkx; tUkz; k1; 2;. . .;N 1u3x;y;z; t Wkx; tUkz

15

Hence, the strain components are:

ex oUkox

Uk; ey0; ezWkU0k; cyzVkU0k;

cxzUkU0koWk

ox Uk; cxy

oVk

ox Uk 16

As far as the stress components are concerned, it is seenfrom Eqs. (16) that only ry are needed to be assumed tovanish. That is:

ry0 17Here, the equations of motion are derived using Hamiltonsprinciple[31]as:

dUk:oMkxox

Qkx Ikj Uj

dVk:oMkxy

ox Qky Ikj Vj

dWk:

oRkxoxN

k

z Ikj

Wjqx; tdkN1

18

where q(x, t) is the applied transverse load at z= h/2. Thegeneralized stress resultants and the mass terms in Eqs.(18) are defined as Eqs. (6) and (7), respectively. Theboundary conditions in this theory involve the specificationofUkor M

kx, Vkor M

kxy, and Wkor R

kx.

Next, in order to find the governing equations ofmotion, it is assumed that the beam is laminated of ortho-tropic laminae with arbitrary fiber direction in the xyplane with respect to the x-axis. The constitutive law ofthe kth lamina with respect to the global xyzcoordinatesystem is[32]:

ex

ey

ez

cyz

cxz

cxy

8>>>>>>>>>>>>>>>>>:

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

k

S11 S12 S13 0 0 S16

S12 S22 S23 0 0 S26

S13 S23 S33 0 0 S36

0 0 0 S44 S45 0

0 0 0 S45 S55 0

S16 S26 S36 0 0 S66

26666666664

37777777775

k rx

ry

rz

ryz

rxz

rxy

8>>>>>>>>>>>>>>>>>:

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

k

19where the matrixSk is called the off-axis compliance ma-trix of thekth layer. Next, invoking the assumption(17)in

(19)results in:

http://-/?-http://-/?-


5/13

ex

ey

cxy

8>:

9>=>;

k

S11 S13 S16

S13 S33 S36

S16 S36 S66

2664

3775

krx

ry

rxy

8>:

9>=>;

k

;

cyz

cxz( )

k

S44 S45

S45 S55" #

kryz

rxz( )

k

20

Inverting the relations in(20)results in:

rx

ry

rxy

8>:

9>=>;

k

C11 C13 C16

C13 C33 C36

C16 C36 C66

2664

3775

kex

ey

cxy

8>:

9>=>;

k

;

ryz

rxz

( )k C44 C45

C45 C55

" #kcyz

cxz

( )k21

where Cijs (i,j = 4,5) are the off-axis stiffness coefficients

[32] and Cijs (i,j= 1,3,6) are the off-axis reduced stiff-nesses given by

C11 C13 C16

C13 C33 C36

C16 C36 C66

2664

3775

S11 S13 S16

S13 S33 S36

S16 S36 S66

264

375

1

22

Now substituting Eqs.(16)into Eqs.(21)and the subse-quent results into Eqs.(6), the generalized stress resultantsare obtained which can be presented as follows:


ox Bjk36;Dkj16;Dkj66

oVj

ox

Akj33;Bkj13;Bkj36WjQky;Qkx;Rkx Akj45;Akj55;Bkj45Uj Akj44;Akj45;Bkj55Vj

Bjk45;Bjk55;Dkj55oWj

ox

23where

AkjpqXNi1

Z zi1zi

CipqU0kU

0j dz;

Bkjpq X

N

i1 Z zi1

zi

CipqUkU0j dz; p; q

1; 3; 6

DkjpqXNi1

Z zi1zi

CipqUkUj dz;

24

and Akjpq, Bkjpq, and D

kjpq (p, q = 4,5) are the same as those

appearing in BLWT1 (i.e., Eqs.(10)). Finally, the govern-ing equations of motion are obtained by substituting(23)into(18):

Dkj11

o2Uj

ox2Akj55Uj Dkj16

o2Vj

ox2Akj45Vj Bkj13 Bjk55

oWj

oxIkj Uj

Dkj16

o2Uj

ox2A

kj45Uj

D

kj66

o2Vj

ox2A

kj44Vj

B

kj36

B

jk45

oWj

ox Ikj Vj

Bkj55 Bjk13oUj

ox Bkj45 Bjk36

o2Vj

ox2Dkj55

o2Wj

ox2Akj33Wj

Ikj Wj qx; tdkN1

2.3. Cylindrical bending (CB)

Consider a plate of thickness h, length a in the x-direc-tion, and infinite extent in the y-direction. Assume furtherthat the plate is subjected to transverse loading q(x, t)which acts normally and upwards on its top surface,z=h/2. Since the laminate is long, it may safely beassumed that a state of plane strain exists. Hence, in orderto obtain a layerwise formulation for the analysis of thisplate, the displacement field in(15)are assumed. Therefore,the equations of motion are similar to those presented inEqs. (18). Next, substituting strain components (16) intothe constitutive law in (8) and the subsequent results intothe definition of the generalized stress resultants yields:


ox Akj33;Bkj13;Bkj36Wj

Bjk36;Dkj16;Dkj66oVj

ox

Qky;Qkx;Rkx Akj45;Akj55;Bkj55Uj Akj44;Akj45;Bkj45Vj

Bjk45;Bjk55;Dkj55oWj

ox

26where the rigidity terms are defined as those appearing inEqs. (10). Upon substitution of Eqs. (26) into Eqs. (18)

the following governing equations of motion are obtained:

Dkj

11

o2Uj

ox2Akj55UjDkj16

o2Vj


oWj

oxIkj Uj

Dkj

16

o2Uj

ox2Akj45UjDkj66

o2Vj


oWj

oxIkj Vj

Bkj55Bjk13oUj

ox Bkj45Bjk36

o2Vj

ox2Dkj55

o2Wj

ox2Akj33Wj

Ikj Wj qx; tdkN127

3. Analytical solutions

The objective of this section is to solve analytically Eqs.(14), (25), and (27). As far as BLWT1, BLWT2, and CB areconcerned, any arbitrarily laminate admits analytical solu-tion for any combinations of edge conditions at x = 0 andx= a. Before the procedure adopted for solving theseequations is discussed, it is appropriate to indicate herethat in the beam layerwise theory two types of simple sup-ports at the ends of the beam (i.e., x= 0, a) may be classi-fied, namely:

S3: MkxVkWk0 28a

S4:

M

k

xMk

xyWk0 28b



6/13

also two types of clamped supports may be classified,namely:

C1: UkVkWk0 29aC2: UkMkxy Wk0 29b

furthermore, the traction-free conditions are defined as:

F: Mkx Mkxy Rkx0 30

It is to be noted that these types of boundary conditions(i.e. S3, S4, C1, and C2) are defined similar to the defini-tions in the ESL theories. For simplicity, the boundaryconditions of a composite beam may be represented in aconcise rule. For example, a beam with the edge conditionsC1 at x= 0 and Fat x=a may be called C1F.

In the following sections, analytical solutions of staticbending and free vibration of laminated beams will be

developed. Because the solution procedure for Eqs. (14),(25), and (27)are completely analogous to each other, forthe sake of brevity, only solution for Eqs. (14) will bediscussed.

3.1. Static bending analysis

Here, the analytical solution for static version of Eqs.(14) subjected to transverse load q(x) are discussed. Inorder to obtain analytical solutions of Eqs. (14), witho

ot0, it must be noted that the numerical results indicate,

however, that there exist repeated zero roots (or eigen-values) in the characteristic equation of the set of equations

in(14). To enhance the solution scheme of these equations,some small artificial terms will be added to these equationsso that the characteristic roots become all distinct (formore complete descriptions of the present method see[29,30]). Therefore, Eqs. (14)rewritten as follows:

Dkj

11

d2Ujdx2

Akj55UjDkj16d2Vjdx2

Akj45Vj Bkj13 Bjk55dWjdx

akjUj

Dkj

16

d2Ujdx2

Akj45UjDkj66d2Vjdx2

Akj44Vj Bkj36 Bjk45dWjdx

akjVj

B

kj

55

B

jk

13

dUj

dx B

kj

45

B

jk

36

d2Vj

dx2D

kj

55

d2Wj

dx2A

kj

33Wj

qxdkN1 akjWj 31

where

akj aZ h=2

h=2UkUj dz 32

with a being a prescribed number such that akjs in Eqs.(31)are relatively small compared to the numerical valuesof stiffnesses Akj55, A

kj

44, and Akj

33 (see [29,30]). Next, in orderto solve Eqs. (31), the following state space variables are

introduced:

X1xf g Uxf g; X2xf g dUdx

dX1

dx

X3xf g Vxf g; X4xf g dVdx

dX3

dx

X5xf g Wxf g; X6xf g dWdx

dX5dx

33

where, for example, {X1}T = [U1, U2, . . . , UN+1]. Substitu-

tion of Eqs. (33) into Eqs. (31) results in a system of6(N+ 1) coupled first-order ordinary differential equationswhich, on the other hand, may be presented as:

dX

dx

AfXg fFg 34

with {X}T = [{X1}T,{X2}

T, . . .,{X6}T]. In Eq. (34) the

coefficient matrix [A] and vector {F} are presented inAppendix A. The general solutions of Eq. (34) are givenby (e.g. see[33]):

Xf g UQkxfKg UQkxZQkx1U1fFgdx

35with Qkx diagek1x; ek2x;. . . ; ek6N1x and {K} being6(N+ 1) arbitrary unknown constants of integration tobe found by imposing the boundary conditions. Here, [U]and kk(k= 1,2, . . . , 6(N+ 1)) are, respectively, the matrixof eigenvectors and eigenvalues of the coefficient matrix[A] which, in general, must be regarded to have complexvalues.

The solution presented in(35)is completely general for

every loading function q(x). For a special case of cross-plybeams subjected to transverse loading qx q0sin pxa ,where q0 is magnitude of sinusoidal loading, with theboundary conditions S3, the other form of solution canbe obtained. It is noted that the boundary conditions S3in(28a)will identically be satisfied if the following expres-sions for the displacement components are assumed:

UjU1jcospx

a ; Vj0; WjW1jsin

px

a 36

where U1j and W1j (j= 1,2, . . . , N+ 1) are coefficients to be

determined. Upon substitution of Eqs.(36)into static ver-sion of Eqs. (14) the following algebraic equations are

obtained:

Dkj

11

p2

a2U1jAkj55U1j Bkj13Bjk55

p

aW1j 0

Bkj55Bjk13p

aU1jDkj55

p2

a2W1jAkj33W1j q0dkN1

37

Eqs. (37) can be solved for coefficients U1j and W1j

(j= 1,2, . . . , N+ 1).

3.2. Free vibration analysis

In order to obtain the natural frequencies of the beam

we consider a solution as:

http://-/?-http://-/?-


7/13

Ujx; tVjx; tWjx; t

8>:

9>=>;

Unj xVnj xWnj x

8>:

9>=>;Tnt 38

where Tn eixnt with i ffiffiffiffiffiffiffi1p and xn is the natural fre-quency of the beam. Substitution of (38) into Eqs. (14),

with q(x, t) = 0, yields:

Dkj

11

d2Unjdx2

Akj55UnjDkj16d2Vnjdx2

Akj45Vnj Bkj13Bjk55dWnj

dx

x2nIkjUkj

Dkj

16

d2Unjdx2

Akj45UnjDkj66d2Vnjdx2

Akj44Vnj Bkj36Bjk45dWnj

dx

x2nIkjVnj

Bkj55Bjk13dUnjdx

Bkj45Bjk36d2Vnjdx2

Dkj55d2Wnj

dx2 Akj33Wnj

x2nI

kjWnj

39Solution of Eqs.(39), subjected to homogeneous boundaryconditions, results in the natural frequencies xn andthe eigenfunctions Unj , V

nj , and W

nj . Next, for the sake of

convenience, the following state space variables areintroduced:

Xn1x Unxf g; Xn2x dUndx

dX

n1

dx

Xn3x

Vnxf g; Xn4x

dVn

dx

dX

n3

dx

Xn5x Wnxf g; Xn6x dWndx dX

n

5

dx

40

where, for example,fXn1gT Un1;Un2;. . . ;UnN1. Substitu-tion of Eqs. (40)into Eqs.(39)results in:

dXn

dx

AnfXng 41

withfXngT fXn1gT; fXn2gT;. . . ;fXn6gT. In Eq. (41) thecoefficient matrix [An] is presented in Appendix A. It isnoted that the natural frequency xn is yet an unknown.

Actually xnis found in a trial and error procedure. To thisend, we assume a value for xnand solve Eq.(41). The gen-eral solution of Eq.(41)is given by (e.g. see[33]):

Xnf g UnQnkxfKng 42By imposing 6(N+ 1) boundary conditions at x= 0 andx= a on the solution given by Eq. (42), a homogeneoussystem of algebraic equations can be found:

MnfKng f0g 43For non-trivial solutionjMnj= 0. If this condition is satis-fied, then the value was guessed for xn is a correct value.Otherwise, another value forxnmust be guessed. However,

sincekis are in general complex numbers,jMn

jwill also be

a complex number. For this reason, the above procedure ismodified slightly. To this end, we note from Eq. (42) atx= 0 that we have:

fXn0g UnfKng 44That is,

fKng Un1fXn0g 45Substitution of Eq.(45)into Eq.(43)results in:

MnUn1fXn0g f0g 46Now, for non-trivial solution the following conditionshould be satisfied which always be a real number:

jMnj=jUnj 0 47

4. Numerical results and discussion

The effectiveness of the present BLWT1 and BLWT2 aredemonstrated through examples of static bending and freevibration. The assessment of the accuracy of the presentbeam theories for the case of bending of cross-ply lami-nates will be obtained by comparison with the exact 3-Delasticity solution [27]. Also the results of BLWT1 andBLWT2 for the case of bending of angle-ply beams andfree vibration will be compared with those obtained by uti-lizing the commercial finite element package of ANSYS[34]. In the latter method, the mesh is refined till no signif-icant change in stress distributions and the natural frequen-cies are obtained. In order to compare the results of finite

element with those of the beam theories, the laminatedplate is assumed to have free edges at y= 0 and y= b.Therefore, by reducing the width of the plate in they-direc-tion (i.e., by decreasingb) the laminated plate reduces to alaminated beam. This way the accuracy of BLWT1 andBLWT2 may be evaluated by comparison with the finiteelement results.

As previously mentioned, in the layerwise theory eachactual physical layer in a laminate can be treated as manynumerical (or mathematical) layers with the same fiberdirection as the actual layer. Clearly, as the number ofnumerical layers is increased, the accuracy of the resultsis also increased. Tahani and Nosier [30] showed that ina four-layer composite laminate, six numerical layers ineach physical lamina results in accurate local effects. Con-sequently, for obtaining highly accurate results, 24 numer-ical layers across the entire laminate thickness areconsidered in all examples. For all results, the interlaminarstresses are computed by integrating the local equations ofequilibrium.

In what follows, static bending and free vibration ofcomposite beams with general laminations will be consid-ered. It is to be noted that the solution procedure outlinedin this paper is completely general and can be used for anyarbitrarily lamination and end conditions at x= 0 and

x= a.

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4.1. Static bending problems

In this section, to test the validity and accuracy of thepresent method, first numerical examples are presentedfor simply supported laminated cross-ply beams [0/90]and [0/90/0] in bending under transverse sinusoidal

loading qx q0sinpx

a on the top surfaces of the beams.These problems have the exact 3-D elasticity [27] and theCLT solutions. Each lamina is assumed to be of the samethickness and is idealized as a homogeneous orthotropicmaterial with the following material properties in the prin-cipal material coordinate system (see [27]):

EL 25 106 psi172:36 GPa; ET 106 psi6:894 GPaGLT 0:5 106 psi3:447 GPaGTT 0:2 106 psi1:379 GPa; mLT mTT 0:25

48where the subscripts L and T signify the direction parallel

to the fibers and the transverse direction, respectively.All numerical results shown in what follows are pre-

sented by means of the following normalized quantitiesas used by Pagano[27]:

w100ETh3w

q0a4

; rx;rz;rxz rx;rz;rxzq0

; Sah

49

Figs. 2 and 3 illustrate the distributions of normalizedin-plane stress rx at x= a/2 and normalized interlaminarshear stress rxz at x= 0 through the thickness in a [0/90] beam for length-to-thickness ratio of 4 (i.e., S= 4).These values are an example of a very thick beam with high

stiffness ratioEL/ET.Figs. 2 and 3also show the validity ofthe proposed methods for simply supported cross-plybeams. All stress distributions predicted by the CLT showconsiderable error for this thick beam whereas excellentagreement between the layerwise solutions and the exact3-D elasticity solutions[27]is found.

The variation of maximum normalized transverse dis-placement with various S for a [0/90/0] beam is showninFig. 4. It is seen that the layerwise theories and Paganos

solution[27]are in close agreement with each other for anyarbitrary S. Also as is expected, CLT underestimates max-imum transverse deflection and gives a poor estimate espe-cially for relatively low values ofS.

Figs. 5 and 6illustrate the variations of normalized in-plane stress rx at x=a/2 and normalized interlaminarshear stress rxz at x= 0 through the thickness in a [0/90/0] beam forS= 4. It is seen that the present solutionsare in excellent agreement with the 3-D elasticity solutions,whereas the CLT solutions have significant error.

The numerical results presented above indicate that forhomogeneous 0- and 90-layered beams in static bendingthe results of BLWT1, BLWT2, and CB become, as analyt-ically anticipated, identical.

Next, in order to test the correctness and accuracy of thepresent layerwise methods for angle-ply beams, static bend-ing of a [30/0/30] beam under uniform transverse load(q0) is considered. The assessment of the accuracy of thepresent beam theories is obtained by comparison with

those obtained by utilizing the finite element package of

x

h/z

-30 -20 -10 0 10 20 30-0.5

-0.25

0

0.25

0.5

3-D Elasticity [27]BLWT1BLWT2CBCLT

Fig. 2. Distribution of normalized in-plane stressrx through the thickness

at x =a/2 of a [0/90] laminate under sinusoidal transverse load.

h/

z

0 1 2 3-0.5

-0.25

0

0.25

0.5

3-D Elasticity [27]

BLWT1

BLWT2

CB

CLT

xz

Fig. 3. Distribution of normalized transverse shear stress rxzthrough thethickness at x = 0 of a [0/90] laminate under sinusoidal transverse load.

S=a/h

w

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3-D Elasticity [27]

BLWT1

BLWT2

CB

CLT

Fig. 4. Variation of maximum normalized transverse deflection w versuslength-to-thickness ratio S of a [0/90/0] laminate under sinusoidaltransverse load.



9/13

ANSYS[34]. A simply supported laminated plate has beenmodeled in ANSYS by using 3-D 20-node layered struc-tural solid elements which is proper to model thick lami-nates. It is assumed that the laminated plate has freeedges at y= 0 and y= b. Therefore, by decreasing b thelaminated plate reduces to a laminated beam. It is furtherassumed that S = 10 in all the theories and b/h= 3, b/

h= 2, and b/h= 1 in finite element method (FEM). Forthe remaining of this paper the mechanical properties ofthe layers are taken to be those of a typical high-modulusgraphite/epoxy lamina[32]:

EL132 GPa; ET10:8 GPaGLT5:65 GPa; GTT3:38 GPamLT0:24; mTT0:59; q1540 kg=m3

50

The variation of normalized in-plane stress rxat x= a/2in the [30/0/30] laminate is displayed in Fig. 7. AlsoFigs. 8 and 9 illustrate distributions of normalized stresscomponents rxz and ryz, respectively, at x= a/4 in the

[30/0/30] laminate. The numerical results are gener-

ated through the thickness of the laminate. It is observedfrom these figures that the results of BLWT2 agree wellwith those obtained from finite element method with b/h= 1. Also it is seen that the results of cylindrical bendingfor rx and ryz disagree with those of the finite elementresults. Therefore, the cylindrical bending of laminatedplates differs from the bending of laminated beams as faras angle-ply laminations are concerned. It is to be notedthat the laminates ofFigs. 29are purposefully chosen inorder to obtain a better assessment of the inaccuracy ofBLWT2. Many numerical results are generated, but not

shown here, for laminates with various stacking sequencesand boundary conditions. In all cases, it is seen that thenew beam layerwise theory, BLWT2, is quite accurate forboth angle-ply and cross-ply laminates. From Figs. 79itis seen that in general the accuracy of BLWT1 is less thanBLWT2. Also BLWT1 demands the inversion of matrix[D22] which can be an inconvenience as far as developingadvanced laminated beam theories are concerned.

4.2. Free vibration studies

In this section, the fundamental natural frequency ofvarious cross-ply and angle-ply beams and plates areobtained according to the two beam theories (BLWT1and BLWT2) as well as cylindrical bending and FEM.The beam is assumed to have simple supports at x= 0and x= a with the boundary conditions given in (28a).The plate is also assumed to have the same support condi-tions atx = 0 andx = a and free edges aty = 0 andy = b.Numerical values of FEM are generated for various ratiosof width (b) of the plate to its thickness (h) (i.e., b/h= 3,b/h= 2, and b/h= 1). Also the results are obtained fora/h= 10, a/h= 15, and a/h = 20. It is to be noted thatthe numerical results of the two beam theories and thecylindrical bending problem will be compared with those

of FEM for the case b/h= 1. This is justified here by the

h

/z

-100 -50 0 50 100-0.5

-0.25

0

0.25

0.5 FEM (b/h=3)

FEM (b/h=2)

FEM (b/h=1)

BLWT1

BLWT2

CB

x

Fig. 7. Distribution of normalized in-plane stressrxthrough the thicknessat x=a/2 of a [30/0/30] laminated beam under uniform transverseload.

h/z

0 0.5 1 1.5 2-0.5

-0.25

0

0.25

0.5

3-D Elasticity [27]

BLWT1

BLWT2

CB

CLT

xz

Fig. 6. Variation of normalized transverse shear stress rxz through thethickness at x= 0 of a [0/90/0] laminate under sinusoidal transverseload.

h/z

-20 -10 0 10 20-0.5

-0.25

0

0.25

0.5

3-D Elasticity [27]BLWT1BLWT2

CBCLT

x

Fig. 5. Variation of normalized in-plane stressrx through the thickness at

x=a/2 of a [0/90/0] laminate under sinusoidal transverse load.



10/13

fact that no assumption made in FEM whereas BLWT1assumes Mky0 and BLWT2 assumes ry= 0.

The fundamental natural frequency of various cross-plybeams and plates according to the two beam theories(BLWT1 and BLWT2), cylindrical bending, and FEMare presented in Table 1. It is seen that the results ofBLWT1, BLWT2, and CB are almost identical. In fact,for cross-ply beams presented in Table 1 the difference

between the results of BLWT1 and FEM is less than 5%,between the new beam theory BLWT2 and FEM is lessthan 1.6%, and between CB and FEM is less than 1.8%.Also the averages of differences for various cross-ply lami-nates and length-to-thickness ratios are 2.15% for BLWT1,0.37% for BLWT2, and 0.55% for CB. The results pre-sented in Table 1 indicate that the cylindrical bending ofcross-ply laminates and bending of cross-ply beams areactually identical. Also BLWT1 and BLWT2 are able toaccurately estimate the natural frequencies of cross-plybeams.

The fundamental natural frequency of [h/h] laminatedbeams for various values ofh are presented in Table 2. It

is seen fromTable 2that BLWT1 and BLWT2 are, exceptfor the [15/15] and [30/30] laminates, accurate forall values of h and a/h ratios. The maximum differencebetween BLWT1 and FEM and also between BLWT2and FEM occur for the [15/15] laminated beam andis about 13% and 14%, respectively. Also the maximum dif-ference between CB and FEM occurs for the [45/45]laminated beam and is about 46%. It should be noted thatalthough the difference for BLWT2 is grater than BLWT1with respect to FEM, but the averages of differences are7.3% for BLWT1, 4.8% for BLWT2, and 18.8% for CB.

Furthermore, the natural frequency of [h/0/h] lami-nates are presented in Table 3. The maximum differencebetween BLWT1 and FEM is about 14%, between BLWT2and FEM is about 15%, and between CB and FEM isabout 40%. Also the averages of differences are 6.3% forBLWT1, 5% for BLWT2, and 16.3% for CB.

It should be noted from Table 1that for homogeneous0- and 90-layered beams the results of BLWT1, BLWT2,CB, and FEM are almost identical and the accuracy lost isinsignificant. Finally, the results presented in Tables 13indicate that BLWT2 are more accurate than BLWT1

h/z

0 1 2 3 4 5-0.5

-0.25

0

0.25

0.5FEM (b/h=3)

FEM (b/h=2)

FEM (b/h=1)

BLWT1

BLWT2

CB

xz

Fig. 8. Distribution of normalized transverse shear stress rxzthrough thethickness at x=a/4 of a [30/0/30] laminated beam under uniformtransverse load.

yz

h/z

-0.7 -0.35 0 0.35 0.7-0.5

-0.25

0

0.25

0.5

FEM (b/h=3)

FEM (b/h=2)

FEM (b/h=1)

BLWT1

BLWT2

CB

Fig. 9. Distribution of normalized transverse shear stress ryzthrough thethickness at x=a/4 of a [30/0/30] laminated beam under uniformtransverse load.

Table 1

Non-dimensional fundamental frequency of [0/90], [0/90/0], and [0/90/0/90] laminated beams according to finite elements, BLWT1, BLWT2, andCB; xxa2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq=ELh

2q

a/h Laminate FEM BLWT1 BLWT2 CB

b/h= 3 b/h= 2 b/h= 1

10 [0/90] 1.36872 1.36872 1.36872 1.43226 1.36516 1.36832[0/90/0] 2.41339 2.41312 2.41285 2.41996 2.45204 2.45689[0/90/0/90] 1.78128 1.78098 1.78026 1.79781 1.79273 1.79671

15 [0/90] 1.39322 1.39361 1.39269 1.46493 1.39397 1.39724[0/90/0] 2.61680 2.61825 2.61604 2.58735 2.62724 2.63292[0/90/0/90] 1.87247 1.87216 1.87140 1.87741 1.87380 1.87810

20 [0/90] 1.40496 1.40483 1.40442 1.47699 1.40456 1.40787[0/90/0] 2.69551 2.69524 2.69483 2.65526 2.69866 2.70471

[0

/90

/0

/90

] 1.90622 1.90595 1.90514 1.90810 1.90512 1.90954



11/13

and also CB is not accurate in estimating the naturalfrequency of laminated composite beams with generallaminations. In fact, because BLWT2 is simple andstraightforward and its numerical results are accurateenough, this theory can be used for modeling laminated

composite beams.

5. Conclusions

Within a layerwise laminate theory, a new laminatedbeam theory with general lamination is developed. Theapproach adopted in the derivation of the equations of

motion in the new beam theory is direct and straightfor-

Table 2Non-dimensional fundamental frequency of [h/h] laminated beams according to finite elements, BLWT1, BLWT2, and CB; xxa2


2q

a/h h () FEM BLWT1 BLWT2 CB

b/h= 3 b/h= 2 b/h= 1

10 0 2.56809 2.56792 2.56225 2.52618 2.56259 2.5680015 1.9518 1.95098 1.95071 2.20149 2.22380 2.31649

30 1.38189 1.37602 1.37378 1.51444 1.56625 1.8872145 1.00947 1.00411 1.00221 0.96277 1.04770 1.4322260 0.84724 0.84585 0.84537 0.83659 0.84849 1.0329275 0.80191 0.80211 0.80163 0.87277 0.80361 0.8298790 0.79627 0.79658 0.79624 0.89504 0.79769 0.79953

15 0 2.71338 2.71323 2.70964 2.66611 2.70986 2.7159215 2.01562 2.01494 2.01463 2.27261 2.29988 2.4301530 1.40155 1.39834 1.39719 1.55028 1.59184 1.9569245 1.01590 1.01316 1.01224 0.97348 1.06163 1.4667460 0.85320 0.85252 0.85229 0.84503 0.85723 1.0475875 0.81228 0.81221 0.81220 0.88305 0.81170 0.8386090 0.80533 0.80533 0.80533 0.90633 0.80579 0.80768

20 0 2.77030 2.77030 2.76799 2.72121 2.76808 2.7744215 2.03924 2.03870 2.03856 2.30367 2.32802 2.47487

30 1.40836 1.40632 1.40564 1.56394 1.59925 1.9844345 1.01799 1.01636 1.01583 0.97735 1.06668 1.4799560 0.85213 0.85050 0.84766 0.84806 0.86038 1.0529275 0.81385 0.81378 0.81367 0.88676 0.81460 0.8417490 0.80833 0.80832 0.80833 0.91041 0.80870 0.81060

Table 3Non-dimensional fundamental frequency of [h/0/h] laminated beams according to finite elements, BLWT1, BLWT2, and CB; xxa2


2q

a/h h () FEM BLWT1 BLWT2 CB

b/h= 3 b/h= 2 b/h= 1

10 0 2.56598 2.56581 2.56225 2.52618 2.56259 2.56800

15 1.97636 1.97575 1.97555 2.22231 2.25540 2.3334130 1.39865 1.39444 1.39285 1.56688 1.58452 1.9072445 1.10394 1.10038 1.09909 1.07792 1.15629 1.4892560 0.98657 0.98555 0.98518 0.97297 0.98832 1.1424675 0.95532 0.95522 0.95518 1.00349 0.95210 0.9741690 0.94989 0.94985 0.94985 1.02245 0.94748 0.94972

15 0 2.71086 2.71079 2.70964 2.66611 2.70986 2.7159215 2.04242 2.04181 2.04151 2.31988 2.33025 2.4469130 1.41972 1.41735 1.41651 1.60390 1.62927 1.9731245 1.11386 1.11203 1.11149 1.09192 1.17354 1.5232560 0.99445 0.99399 0.99376 0.98436 1.00029 1.1594775 0.96361 0.96361 0.96353 1.01614 0.96317 0.9857890 0.95925 0.95925 0.95925 1.03581 0.95845 0.96073

20 0 2.77030 2.76758 2.76758 2.72121 2.76808 2.77442

15 2.06693 2.06653 2.06625 2.35756 2.36548 2.4911530 1.42695 1.42546 1.42505 1.61770 1.64118 1.9983045 1.11728 1.11618 1.11471 1.09699 1.17980 1.5359560 0.99860 0.99830 0.99721 0.98847 1.00461 1.1656775 0.96788 0.96785 0.96687 1.02070 0.96715 0.9899790 0.96358 0.96358 0.96263 1.04064 0.96240 0.96469



12/13

ward similar to the ones used in developing laminated plateand shell theories. The ideas developed in the present workmay readily be used in developing simpler theories such asCLT and shear deformation beam theories for generallylaminated composite beams. Based on analytical solutionsnumerical results are generated for natural frequencies and

in-plane and interlaminar stresses of a variety of laminatedbeams. The results are obtained according to the new lam-inated beam theory (BLWT2) developed in the presentwork as well as a laminated beam theory developed froman existing laminated plate theory (BLWT1). The numeri-cal results clearly indicate the accuracy of BLWT2. TheBLWT1 is also shown to be accurate enough. However,because of the existing complexities in its derivation, it isbelieved that the ideas presented here could be used indeveloping accurate beam theories. Finally, from the newbeam theory it is analytically shown that the displacementfield often used for cross-ply beams in the literature is aproper displacement field.

Appendix A

The coefficients appearing in Eqs. (12)are defined as:

A33 A33 B23TD221B23B13 B13 D12D221B23B36 B36 D26D221B23D11 D11 D12D221D12D16 D16 D12D221D26

D66 D66 D26D221

D26The coefficient matrix [A] and vector {F} in Eq.(34)are

defined as:

A

0 I 0 0 0 0a1 0 a2 0 0 a30 0 0 I 0 0b1 0 b2 0 0 b30 0 0 0 0 I0 c1 0 c2 c3 0

26666666664

37777777775

; fFg

f0gf0gf0gf0gf0gfc4g

8>>>>>>>>>>>>>>>>>:

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

where [0] and [I] are (N+ 1) (N+ 1) square zero andidentity matrices, respectively, and {0} is a zero vector withN+ 1 rows. The remaining matrices and vectors in theabove equations are as follows:

a1 d11d2;a2 d11d3;a3 d11d4b1 D661A45 D16a1b2 D661A44 D16a2 a

b3 D661

B45T

B36 D16a3

c1 D551B13T B55c2 D551B36T B45c3 D551A33 afc4g D551fqxdkN1gwith

d1 D111 D16D661D16d2 A55 D16D661A45 ad3 A45 D16D661A44 ad4 B55T B13 D16D661B36 B45T

The coefficient matrix [An] appearing in Eq. (41) isdefined as:

An

0 I 0 0 0 0an1 0 an2 0 0 a3

0

0

0

I

0

0

b1 0 bn2 0 0 b30 0 0 0 0 I0 c1 0 c2 cn3 0

2

666666664

3

777777775where

an1 d11dn2;an2 d11dn3bn2 D661A44 D16a2 x2nIcn3 D551A33 x2nIwith

dn2 A55 D16D661A45 x2nIdn3 A45 D16D661A44 x2nIIin the above equations is the matrix of mass moments ofinertia defined in Eq.(7).

References

[1] Kapania RK, Raciti S. Recent advances in analysis of laminatedbeams and plates, part I. AIAA J 1989;27:92334.

[2] Kapania RK, Raciti S. Recent advances in analysis of laminatedbeams and plates, part II. AIAA J 1989;27:93546.

[3] Miller AK, Adams DF. An analytic means of determining the flexural

and torsional resonant frequencies of generally orthotropic beams. JSound Vib 1975;41:43349.

[4] Teoh LS, Huang CC. The vibration of beams of fiber reinforcedmaterial. J Sound Vib 1977;51:46773.

[5] Abarcar RB, Cunnif PF. The vibration of cantilever beams of fiberreinforced materials. J Compos Mater 1972;6:50417.

[6] Heyliger PR, Reddy JN. A higher order beam element for bendingand vibration problems. J Sound Vib 1988;126:30926.

[7] Abramovich H, Livshits A. Free vibrations of non-symmetric cross-ply laminated composite beams. J Sound Vib 1994;176:597612.

[8] Gordaninejad F, Ghazavi A. Effect of shear deformation on bendingof laminated composite beams. J Pressure Vessel Tech 1989;11:15964.

[9] Ghazavi A, Gordaninejad F. Nonlinear bending of thick beamslaminated from bimodular composite materials. Compos Sci Tech

1989;36:28998.



13/13

[10] Shimpi RP, Ghugal YM. A new layerwise trigonometric sheardeformation theory for two-layered cross-ply beams. Compos SciTech 2001;61:127183.

[11] Shimpi RP, Ainapure AV. A beam finite element based on layerwisetrigonometric shear deformation theory. Compos Struct 2001;53:15362.

[12] Icardi U. A three-dimensional zig-zag theory for analysis of thicklaminated beams. Compos Struct 2001;52:12335.

[13] Matsunaga H. Interlaminar stress analysis of laminated compositebeams according to global higher-order deformation theories. Com-pos Struct 2002;55:10514.

[14] Arya H, Shimpi RP, Naik NK. A zigzag model for laminatedcomposite beams. Compos Struct 2002;56:214.

[15] Abramovich H. Shear deformation and rotary inertia effects ofvibrating composite beams. Compos Struct 1992;20:16573.

[16] Chandrashekhara K, Krishnamurthy K, Roy S. Free vibration ofcomposite beams including rotary inertia and shear deformation.Compos Struct 1990;14:26979.

[17] Bhimaraddi A, Chandrashekhara A. some observations on themodeling of laminated composite beams with general lay-ups.Compos Struct 1991;19:37180.

[18] Sheinman I. On the analytical closed-form solution of high-orderkinematic models in laminated beam theory. Int J Numer Meth Eng2001;50:91936.

[19] Chen AT, Yang TY. Static and dynamic formulation of a symmet-rically laminated beam finite element for microcomputer. J ComposMater 1985;19:45975.

[20] Kapania RK, Raciti S. Nonlinear vibrations of unsymmetricallylaminated beams. AIAA J 1989;27:20110.

[21] Singh G, Rao GV, Iyengar NG. Analysis of the nonlinear vibrationsof unsymmetrically laminated composite beams. AIAA J 1991;29:172735.

[22] Chandrashekhara K, Bangera KM. Free vibration of compositebeams using a refined shear flexible beam element. Compos Struct1992;43:71927.

[23] Bangera KM, Chandrashekhera K. Nonlinear vibration of moder-ately thick laminated beams using finite element method. FiniteElements Anal Des 1991;9:32133.

[24] Krishnaswamy S, Chandrashekhara K, Wu WZB. Analytical solu-tions to vibration of generally layered composite beams. J Sound Vib1992;159:8599.

[25] Obst AW, Kapania RK. Nonlinear static and transient finite elementanalysis of laminated beams. Compos Eng 1992;2:37589.

[26] Teboub Y, Hajela P. Free vibration of generally layered compositebeams using symbolic computations. Compos Struct 1995;33:12334.

[27] Pagano NJ. Exact solutions for composite laminates in cylindricalbending. J Compos Mater 1969;3:398411.

[28] Nosier A, Kapania RK, Reddy JN. Free vibration analysis oflaminated plates using a layerwise theory. AIAA J 1993;31(12):233546.

[29] Tahani M, Nosier A. Free edge stress analysis of general cross-plycomposite laminates under extension and thermal loading. ComposStruct 2003;60:91103.

[30] Tahani M, Nosier A. Accurate determination of interlaminar stressesin general cross-ply laminates. Mech Adv Mater Struct 2004;11(1):6792.

[31] Fung YC. Foundations of solid mechanics. Englewood Cliffs,NJ: Prentice-Hall; 1965.

[32] Herakovich CT. Mechanics of fibrous composites. New York: JohnWiley & Sons; 1998.

[33] Gopal M. Modern control system theory. 2nd ed. New AgeInternational (P) Limited; 1993.

[34] Ansys Release 9. UP20041104, SAS IP, Inc., 2004.


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