Dynamic analysis of laminated composite plates subjected ...web.iitd.ac.in/~bppatel/publicatios/Dynamic analysis of laminated... · Dynamic analysis of laminated composite plates

Dynamic analysis of laminated composite plates subjected tothermal/mechanical loads using an accurate theory

D.P. Makhecha, M. Ganapathi *, B.P. Patel

G.M. Faculty, Institute of Armament Technology (Deemed University), Girinagar, Pune 411 025, India

Abstract

This paper deals with the application of a new higher-order theory that accounts for the realistic variation of in-plane and

transverse displacements through the thickness for the dynamic response analysis of thick multi-layered composite plates. The

solutions are obtained employing the ®nite element procedure based on a C0 eight-noded serendipity quadrilateral element. The

importance of various higher-order terms in the present model is highlighted through the numerical study of mechanical and thermal

loads. A detailed study is also carried out considering the in¯uences of ply angle, aspect ratio, number of layers and thermal co-

e�cients on the global response of thick laminates. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Dynamic response; Higher-order; Finite element; Laminates; Thermal and mechanical loads

1. Introduction

Advanced composite materials are widely used inaircraft and space systems due to their advantages ofhigh sti�ness- and strength-to-weight ratios. However,the analysis of multi-layered structures is a complex taskcompared with conventional single layer metallic struc-tures due to the exhibition of coupling among mem-brane, torsion and bending strains; weak transverseshear rigidities; and discontinuity of the mechanicalcharacteristics along the thickness of the laminates.More accurate analytical/numerical analysis based onthree-dimensional models may be computationally in-volved and expensive. Hence, among researchers, thereis a growing appreciation of the importance of devel-oping new kinematics for the evolution of accurate two-dimensional theories for the analysis of thick laminateswith high orthotropic ratio, leading to less expensivemodels. In this context, the applications of analytical/numerical methods based on various higher-order the-ories, not only for the vibrations of thick laminates, butalso for the high frequency vibrations of thin compositeplates, has recently attracted the attention of severalinvestigators/researchers.

Various structural theories proposed for evaluatingthe characteristics of composite laminates under di�er-ent loading situations have been reviewed and assessedby Noor and Burton [1,2], Tauchert [3], Kapania andRaciti [4], Reddy [5] and, more recently, by Mallikarj-una and Kant [6] and Varadan and Bhaskar [7]. It maybe concluded from the literature that the analysis ofcomposite plates under thermal environment is generallybased on classical lamination theory and ®rst-ordershear deformation theory. Furthermore, the assumptionof displacements as linear functions of the coordinate inthe thickness direction has proved to be inadequate forpredicting the response of thick laminates. Higher-orderdisplacement ®elds yielding quadratic variations oftransverse shear strains have been attempted by manyresearchers [8±15] for better accuracy, but the applica-tion of higher-order theory for the investigation of thickmulti-layered plates under thermal load seems to bescarce in literature compared to the analysis of me-chanically loaded laminates [6]. Three-dimensionalelasticity analysis carried out by Bhaskar et al. [16] forthick laminates subjected to thermal loads reveals thenon-linear variation of in-plane displacements throughthe thickness and abrupt discontinuity in slope at anyinterface and thickness-stretch/contraction e�ects in thetransverse displacement. Although higher-order theoriesbased on the discrete layer approach [17±21] account forslope discontinuity at the interfaces, the number of un-knowns to be solved increases with increase in thenumber of layers. Recently, Ali et al. [22] have proposed

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Composite Structures 51 (2001) 221±236

* Corresponding author. Tel.: +91-020-4389550; fax: +91-020-

4389509.

E-mail addresses: [email protected], [email protected] (M.

Ganapathi).

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 2 3 ( 0 0 ) 0 0 1 3 3 - 1

a new higher-order plate theory based on the globalapproximation approach for the static analysis of multi-layered symmetric composite laminates under thermal/mechanical loading, incorporating realistic through-the-thickness approximations of the in-plane andtransverse displacements based on the work given in[16]. This formulation has proved to give very accurateresults for the static analysis of symmetric cross-plylaminates, and this excellent performance of the theoryfor thick laminates motivated the present extension ofthe formulation for the dynamic analysis of thermally/mechanically loaded general composite laminatesthrough the ®nite element procedure.

Here, a C 0 eight-noded quadrilateral serendipity plateelement with thirteen degrees of freedom per node hasbeen developed by extending the theory employed forsymmetric laminates [22] to general composite laminatedplates. The e�cacy of the present formulation, forthe dynamic response analysis of laminates subjectedto thermal/mechanical loads, has been brought outthrough numerical studies. The solutions are obtainedusing NewmarkÕs direct integration technique. All theinertia terms due to the part given by ®rst-order theory,that arising from the higher-order displacement functionand that resulting from the coupling between the dif-ferent order displacements are included in the formula-tion. The e�ect of various terms in the higher-orderdisplacement model on the response characteristics isbrought out considering di�erent parameters.

2. Formulation

A composite plate with arbitrary lamination is con-sidered with the coordinates x, y along the in-planedirections and z along the thickness direction. Thein-plane displacements uk and vk, and the transversedisplacement wk for the kth layer, are assumed as

uk�x; y; z; t� � u0�x; y; t� � zhx�x; y; t� � z2bx�x; y; t�� z3/x�x; y; t� � Skwx�x; y; t�;

vk�x; y; z; t� � v0�x; y; t� � zhy�x; y; t� � z2by�x; y; t�� z3/y�x; y; t� � Skwy�x; y; t�;

wk�x; y; z; t� � w0�x; y; t� � zw1�x; y; t� � z2C�x; y; t�:

�1�

The terms with even powers of z in the in-plane dis-placements and odd powers of z occurring in the ex-pansion for wk correspond to stretching problems.However, the terms with odd powers of z in the in-planedisplacements and the even ones in the expression for wk

represent ¯exure problems. u0, v0 and w0 are the dis-placements of a generic point on the reference surface; hx

and hy are the rotations of the normal to the referencesurface about the y and x axes, respectively; w1, bx, by ,

C, /x and /y are the higher-order terms in the Taylor'sseries expansions, de®ned at the reference surface. wx

and wy are generalized variables associated with thezigzag function, Sk. The zigzag function, Sk, as given in[23], is de®ned by

Sk � 2�ÿ1�kzk=hk; �2�

where zk is the local transverse coordinate with its originat the center of the kth layer and hk is the correspondinglayer thickness. Thus, the zigzag function is piecewiselinear with values of )1 and 1 alternately at the di�erentinterfaces. The ÔzigzagÕ function, as de®ned above, takescare of the inclusion of the slope discontinuities of uand v at the interfaces of the laminate as observed inexact three-dimensional elasticity solutions of thicklaminated composite structures. The use of such afunction is more economical than a discrete layer ap-proach of approximating the displacement variationsover the thickness of each layer separately. Althoughboth these approaches account for slope discontinuityat the interfaces, in the discrete layer approach, thenumber of unknowns increases with the increase inthe number of layers, whereas it remains constant in thepresent approach.

The strains in terms of mid-plane deformation, ro-tations of the normal and higher-order terms associatedwith displacements for kth layer are as

ef g � ebm

es

� �ÿ f�etg: �3�

The vector febmg includes the bending and membraneterms of the strain components and vector fesg containsthe transverse shear strain terms. These strain vectorscan be de®ned as

ebm

es

� ��

exx

eyy

ezz

exy

cxz

cyz

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

9>>>>>>>=>>>>>>>;�

uk;x

vk;y

wk;z

uk;y � vk

;x

uk;z � wk

;x

vk;z � wk

;y

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

9>>>>>>>>=>>>>>>>>;� ��Z�f e0 e1 e2 e3 e4 c0 c1 c2 c3 gT

;

�4a�where

��Z� ��I1� z�I1� z2�I1� z3�I1� Sk�I1� �O� �O� �O� �O��O�T �O�T �O�T �O�T �O�T �I2� z�I2� z2�I2� Sk

;z�I2�

" #:

�4b�

[I1] and [I2] are identity matrices of size 4� 4 and 2� 2,respectively, and [O] is a null matrix of size 4� 2.

222 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236

fe0g �

u0;x

v0;y

w1

u0;y � v0;x

8>>>><>>>>:

9>>>>=>>>>;; fe1g �

hx;x

hy;y

2C

hx;y � hy;x

8>>>><>>>>:

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

fe2g �

bx;x

by;y

0

bx;y � by;x

8>>>><>>>>:

9>>>>=>>>>;; fe3g �

/x;x

/y;y

0

/x;y � /y;x

8>>>><>>>>:

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

fe4g �

wx;x

wy;y

0

wx;y � wy;x

8>>>><>>>>:

9>>>>=>>>>;; �4c�

fc0g �hx�w0;x

hy �w0;y

( ); fc1g �

2bx�w1;x

2by �w1;y

( );

fc2g �3/x�C;x

3/y �C;y

( ); fc3g �

wx

wy

( ):

�4d�

The subscript comma denotes partial derivative withrespect to the spatial coordinate succeeding it.

The thermal strain vector f�etg is represented as

f�etg �

�exx

�eyy

�ezz

�exy

�exz

�eyz

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

9>>>>>>>=>>>>>>>;� DT

ax

ay

az

axy

0

0

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

9>>>>>>=>>>>>>;; �4e�

where DT is the rise in temperature and is generallyrepresented as a function of x, y, z and time t. ax; ay ; az

and axy are thermal expansion coe�cients in the platecoordinates and can be related to to the thermal coe�-cients (a1, a2 and a3) in the material principal directions.

The constitutive relations for an arbitrary layer k inthe laminate (x, y, z) coordinate system can be expressedas

frg � f rxx ryy rzz sxy sxz syz gT

� � �Qk� ebm

es

� ��ÿ f�etg

�; �5�

where the terms of the � �Qk� matrix of the kth ply arereferred to as the laminate axes and can be obtainedfrom the [Qk] corresponding to the ®ber directions withthe appropriate transformation, as outlined in the lit-erature [24]. frg; feg and f�etg are stress, strain andthermal strain vectors due to rise in temperature, re-spectively. The superscript T refers the transpose of amatrix/vector.

The governing equations are obtained by applyingthe Lagrangian equations of motion given by

d

dt�o�T ÿ UT�=o _di� ÿ �o�T ÿ UT�=odi� � 0; i � 1 to n;

�6�where T is the kinetic energy and UT is the total po-tential energy consisting of strain energy contributionsdue to the in-plane and transverse stresses and workdone by the externally applied mechanical loads, re-spectively. fdg � fd1; d2; . . . ; di; . . . ; dngT

is the vector ofthe degrees of freedom/generalized coordinates. A dotover a variable represents partial derivative with respectto time.

The kinetic energy of the plate is given by

T �d� �1

2

Z Z Xn

k�1

Z hk�1

hk

qkf _uk _vk _wk gf _uk _vk _wk gTdz

" #dxdy;

�7�where qk is the mass density of the kth layer. hk and hk�1

are the z-coordinates of the laminate corresponding tothe bottom and top surfaces of the kth layer.

Using the kinematics given in Eq. (1), Eq. (7) can berewritten as

T �d� � 1

2

Z Z Xn

k�1

Z hk�1

hk

qkf _degT�Z�T�Z�f _degdz

" #dxdy;

�8�where

f _degT �f _u0 _v0 _w0

_hx_hy _w1

_bx_by

_C _/x_/y

_wx_wy g

and

�Z� �1 0 0 z 0 0 z2 0 0 z3 0 Sk 00 1 0 0 z 0 0 z2 0 0 z3 0 Sk

0 0 1 0 0 z 0 0 z2 0 0 0 0

24 35:The potential energy function UT is given by,

UT�d� � 1

2

Z Z Xn

k�1

Z hk�1

hk

frgTfegdz

" #dxdy

ÿZ Z

qwdxdy; �9�

where q is the distributed force acting on the top surfaceof the plate.

Substituting the constitutive relation, Eq. (5), inEq. (9), one can write UT as

D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 223

UT�d� � 1

2

Z Z Xn

k�1

Z hk�1

hk

�febmesgT� �Q�febmesg"

ÿ 2febmesgT� �Q�f�etg � f�etgT� �Q�f�etg�dz

#dxdy

ÿZ Z

qwdxdy: �10�

For obtaining the element-level governing equations, thekinetic and the total potential energies may be conve-niently written as

T �de� � 1

2f _degT�M e�f _deg; �11�

UT�de� � 1

2fdegT�Ke�fdeg ÿ fdegTfF e

Tg ÿ fdegTfF eMg

� 1

2

Z Z Xn

k�1

Z hk�1

hk

f�etgT� �Q�f�etgdz

" #dxdy:

�12�The elemental mass and sti�ness matrices and thermal/mechanical load vectors involved in Eqs. (11) and (12)can be de®ned as

�M e� �Z Z Xn

k�1

Z hk�1

hk

qkfHgT�Z�T�Z�fHgdz

" #dxdy;

�Ke� �Z Z Xn

k�1

Z hk�1

hk

�B�T��Z�T� �Qk��Z��B�dz

" #dxdy;

�F eT� �

Z Z Xn

k�1

Z hk�1

hk

�B�T��Z�T� �Qk�f�etgdz

" #dxdy;

�F eM� �

Z ZfHwgTqdxdy;

where fdeg is the vector of the elemental degrees offreedoms/generalized coordinates and [H] and [B] arethe interpolation and strain matrices pertaining to theelement, respectively.

Substituting Eqs. (11) and (12) in Eq. (6), one obtainsthe governing equation for the element as

�M e�f�deg � �Ke�fdeg � fF eTg � fF e

Mg: �13�The coe�cients of the mass and sti�ness matrices andthe load vectors involved in governing equation (13) canbe rewritten as the product of the term having thicknesscoordinate z alone and the term containing x and y. Inthe present study, while performing the integration,terms having thickness coordinate z are explicitly inte-grated, whereas the terms containing x and y are eval-uated using full integration with 3� 3 point Gaussintegration rule.

Following the usual ®nite element assembly proce-dure, the governing equation for the forced response ofthe laminate are obtained as

�M �f�dg � �K�fdg � fFTg � fFMg; �14�where [M] and [K] are the global mass and sti�nessmatrices. fFTg and fFMg are the global thermal andmechanical load vectors, respectively.

The solutions of Eq. (14) can be obtained usingNewmarkÕs direct integration method.

3. Element description

In the present work, a simple, C0 continuous, eight-noded serendipity quadrilateral shear ¯exible plate el-ement with 13 nodal degrees of freedom (u0, v0, w0, hx,hy , w1, bx, by , C, /x, /y , wx and wy : 13-DOF) isdeveloped.

Table 1

Alternative eight-noded ®nite element models considered for para-

metric study

Finite element model Degrees of freedom per node

Q8-HSDT13 (present) u0, v0, w0, hx, hy , w1, bx, by , C, /x, /y ,

wx, wy

Q8-HSDT11a u0, v0, w0, hx, hy , bx, by , /x, /y , wx, wy

Q8-HSDT11b u0, v0, w0, hx, hy , w1, bx, by , C, /x, /y

Q8-HSDT7 u0, v0, w0, hx, hy , /x, /y

Q8-FSDT u0, v0, w0, hx, hy

Table 2

Non-dimensional displacement of simply supported three-layered cross-ply square laminates �E1=E2 � 25;E3 � E2;G12=E2 � G13=E2 � 0:5,

G23=E2 � 0:2; m12 � m23 � m13 � 0:25� due to thermal and mechanical loading for di�erent aspect ratios, S � a=h

S Theory Thermal loading �T0 sinpx=a sinpy=b� Mechanical loading �q0 sinpx=a sinpy=b��u�0; b=2; h=2� �v�a=2; 0; h=2� �w�a=2; b=2; h=2� �u�0; b=2; h=2� �v�a=2; 0; h=2� �w�a=2; b=2; h=2�

4 Present 17.83 81.23 42.33 0.959 2.265 2.102

Elasticity [16] 18.11 81.23 42.69 0.9694 2.281 2.006

10 Present 16.59 31.92 17.37 0.738 1.108 0.753

Elasticity [16] 16.61 31.95 17.39 0.7351 1.099 0.753

20 Present 16.16 20.34 12.11 0.693 0.795 0.517

Elasticity [16] 16.17 20.34 12.12 0.6926 0.7944 0.5164

50 Present 16.02 16.72 10.50 0.680 0.697 0.445

Elasticity [16] 16.02 16.71 10.50 0.6799 0.6967 0.4451

��u; �m� � �u; v�=�ha1T0S� or �u; v��100E2=q0hS3� and �w � w=�ha1T0S2� or w�100E2=q0hS4�.


Table 3

Non-dimensional fundamental frequencies � �x � x��qh2=E2

p � 10� of simply supported cross-ply square plates with S � 5�E1=E2 � 40;E3 � E2;

G12=E2 � G13=E2 � 0:6;G23=E2 � 0:5; m12 � m23 � m13 � 0:25�No. of layers, N Model E1=E2

3 10 20 30 40

2 Q8-HSDT13 2.4935 2.7886 3.0778 3.2940 3.4638

Elasticity [25] 2.5031 2.7938 3.0698 3.2705 3.4250

4 Q8-HSDT13 2.6029 3.2488 3.7677 4.0841 4.3001

Elasticity [25] 2.6182 3.2578 3.7622 4.0660 4.2719

6 Q8-HSDT13 2.6264 3.3478 3.9219 4.2686 4.5035

Elasticity [25] 2.6440 3.3657 3.9359 4.2783 4.5091

10 Q8-HSDT13 2.6390 3.4018 4.0093 4.3770 4.6279

Elasticity [25] 2.6583 3.4250 4.0337 4.4011 4.6498

Fig. 1. Transverse displacement versus time curve for angle-ply laminates subjected to thermal loading (aspect ratio S� 5).


The ®nite element represented as per the kinematics,for instance see Eq. (1), is referred to as Q8-HSDT13with cubic variation. Four more alternate discretemodels are proposed to study the in¯uence of higher-order terms in the displacement functions, whose dis-placement ®elds are deduced from the original elementby deleting the appropriate degrees of freedom (w1 andC� 0; or w� 0; or z2 terms, w, w1 and C� 0; or droppingall the higher-order terms). These alternate models, andthe corresponding degrees of freedom, are shown inTable 1.

4. Results and discussion

The study, here, has been focussed mainly on thedynamic response behavior of composite laminatessubjected to thermal and mechanical loads of in®niteduration. Since higher-order theory, in general, is re-quired for accurate analysis for thick laminates, theemphasis in the present work is on the thick, multi-layered, anisotropic plate considered for the numericalstudy. The in¯uence of various higher-order terms in thedisplacement ®elds on the response characteristics of

Fig. 2. In-plane displacement versus time curve for angle-ply laminates subjected to thermal loading (aspect ratio S� 5).


laminates is highlighted, assuming di�erent values forply angle, aspect ratio, coe�cient of thermal expansion,number of layers, etc.

Based on progressive mesh re®nement, an 8� 8 gridmesh is found to be adequate to model the full lami-nated plate for the present analysis. Before proceedingto the detailed analysis of the dynamic response

characteristics of the laminates, the formulation de-veloped herein is validated considering the static andfree vibration analyses of laminated cross-ply plates.Table 2 shows the de¯ections of the laminate alongwith the exact solutions of the three-dimensionalelasticity theory [16]. For free vibration analysis, thefundamental frequencies obtained by varying the

Fig. 3. (a) Response of a two-layered angle-ply laminate subjected to thermal loading (S� 10). (b) Response of an eight-layered unsymmetric

angle-ply laminate subjected to thermal loading (S� 10).


degree of orthotropicity of the layers are given inTable 3 along with the three-dimensional elasticityresults [25]. It is observed from these tables that thepresent results obtained employing Q8-HSDT13 modelagree very well with existing literature. The materialproperties, unless speci®ed otherwise, assumed in thepresent analysis are

E1=E2 � 40; G12=E2 � G13=E2 � 0:6;

G23=E2 � 0:5; m12 � m23 � m13 � 0:25;

a2=a1 � a3=a1 � 1125;

q � 800 kg=m3; E2 � E3 � 1010 N=m

2;

a1 � 1� 10ÿ6=°C;

�15�

Fig. 4. Response of a two-layered cross-ply laminate subjected to thermal loading (S� 5).


where E, G and m are YoungÕs modulus, shear modulusand PoissonÕs ratio, respectively. Subscripts 1, 2 and 3refer to the principal material directions. The ®rst layercorresponds to the bottom-most layer, the ply angle ismeasured from the x axis in an anti-clockwise directionand all the layers are of equal thickness.

The simply supported boundary conditions consid-ered here are

v0 � w0 � hy � w1 � C � by � /y � wy � 0; at x � 0;a;

u0 � w0 � hx � w1 � C � bx � /x � wx � 0; at y � 0;b;

where a and b refer to the length and width of the plate,respectively.

For the dynamic response study, all the initial con-ditions are assumed to be zero. The critical time step of a

Fig. 5. Response of an eight-layered cross-ply laminate subjected to thermal loading (S� 5).


conditionally stable ®nite di�erence is introduced as aguide [26]. Subsequently, a convergence study is con-ducted to select a time step which yields a stable andaccurate solution. The laminated plate assumed here is asquare simply supported one. The in-plane and thetransverse displacements u, v and w presented here withrespect to time correspond to the (x, y, z) locations of(0, b=2, h=2), (a=2, 0, h=2) and (a=2, b=2, h=2), respec-tively. The (x, y) locations of transverse shear stresses sxz

and syz are (0, b=2) and (a=2, 0), respectively. The spatialdistributions of the two types of loading considered hereare:

for mechanical: q � q0 sin�px=a� sin�py=b�;for thermal: DT � T0�2z=h� sin�px=a� sin�py=b�.Next, through the present model (Q8-HSDT13), the

variation of transverse and in-plane dynamic responsesof two-, three- and eight-layered angle-ply laminates�45°=ÿ 45°; 45°=ÿ 45°=45°; �45°=ÿ 45°�4� having as-

Fig. 6. Response of a two-layered cross-ply laminate subjected to thermal loading (S� 10).


pect ratio S (� a=h; thickness h� 1 cm) equal to 5 sub-jected to thermal load (T0� 1), with respect to time isevaluated and given in Figs. 1 and 2. These ®gures alsoshow the e�ect of higher-order terms pertaining to thepresent model on the response characteristics, obtainedthrough di�erent ®nite element models considered in thepresent study. Since the in¯uence of the zigzag functionw in the in-plane displacements is negligible on thetransverse displacement (i.e., the results obtained using

model Q8-HSDT11b almost coincide with that shownfor Q8-HSDT13), for clarity, the response curves cor-responding to the Q8-HSDT11b model are not shown inthe ®gures. It is observed from Fig. 1 that the third-order theory, considered here (Q8-HSDT7), has littlee�ect on the response in comparison with the ®rst-ordertheory (Q8-FSDT). Furthermore, the signi®cance of thez2 term in the in-plane displacement, in addition to lessimportant z3 term, has been highlighted through the

Fig. 7. Response of an eight-layered cross-ply laminate subjected to thermal loading (S� 10).


Q8-HSDT7 and Q8-HSDT11a models. However, thepredominant contribution to the global response arisesfrom the presence of the z2 term (C) in the transversedisplacement function. With an increase in the numberof layers, as seen from Fig. 1, the in¯uence of the higherorder z2 term (C) in the w expression increases signi®-cantly. It is also further noticed that the in¯uence of thez2 terms in the in-plane displacements is dictated by

bending±stretching coupling due to lay-up, and itsin¯uence on the responses for symmetric laminate isinsigni®cant, i.e., the response predicted by the Q8-HSDT11a and Q8-HSDT7 models are the same. It maybe opined here that the ®rst- and third-order theories ofQ8-HSDT7 grossly under predict transverse displace-ment with time. For in-plane response evaluation, z3 andz2 terms, in general, are important, as seen from Fig. 2

Fig. 8. E�ect of thermal coe�cient on transverse displacement for an eight-layered cross-ply laminate (S� 5).


(Q8-HSDT11a and Q8-HSDT7), depending on lay-up,and more so for the C term in w. The e�ect of the zigzagfunction w in the in-plane displacements is hardly no-ticeable at the peak values with increase in responsetime, but is not shown in the ®gure, for the purpose ofclarity, by limiting the response time. A similar study iscarried out for the aspect ratio a=h � 10 and the eval-uated transient responses are presented in Fig. 3 for two-and eight-layered angle-ply laminates. The responsebehaviors are, in general, qualitatively same as those ofa=h � 5. However, the e�ectiveness of the presence ofhigher-order terms in the present model in predictingglobal responses decreases. Further, although the in¯u-ence of C decreases at faster rate compared to z3 termsas noticed from the models Q8-HSDT11a andQ8-HSDT7, the inclusion of C (Q8-HSDT13) in thedisplacement leads to considerable contribution to theover-all transverse response behavior. In general, onecan draw the conclusions that the higher term C in wand the z3 terms in the in-plane displacements are es-sential irrespective of the lay-up, whereas the z2 terms inin-plane displacements are necessary for unsymmetriclaminates, for accurately predicting the global responsesof the thick laminates considered here.

For cross-ply cases, the results considering two andeight layers are plotted in Figs. 4 and 5 for aspect ratioa=h � 5. It is inferred from Figs. 4 and 5 that the vari-ation of transverse displacement is similar to those ofangle-ply laminates and the e�ectiveness of the higher-order terms here is like that of the angle-ply case.However, for the in-plane behaviors, the presence of Cdepends on the in-plane directions for the two-layeredlaminate case (i.e., for a 0°=90° laminate, the variationof the in-plane displacement u predicted by modelsQ8-HSDT13 and Q8-HSDT11a is almost similar), andthe z2 term in the in-plane displacement functions is verysigni®cant. But the z3 term has slightly less e�ect com-pared to the angle-ply cases. Similar investigation is alsomade with a=h � 10, and the response characteristics arehighlighted in Figs. 6 and 7. It is again inferred fromthese studies that, with increase in aspect ratio, the e�ectof higher-order theory decreases, as expected. It may beopined from the investigation of cross- and angle-plycases, that higher-order terms are predominant withincrease in layers and decrease in aspect ratio.

The in¯uence of the thermal coe�cient of expansionis also investigated and is shown in Fig. 8 for two ratiosof a2=a1. The C term in w becomes more signi®cant with

Fig. 9. Thickness distribution of displacements and stresses for a three-layered cross-ply laminate subjected to thermal loading at time� 2:2� 10ÿ5 s.


increase in a2=a1, as seen from the comparison of resultscorresponding to the Q8-HSDT13 and Q8-HSDT11amodels. The displacements and transverse shear stressdistributions through the thickness are drawn in Fig. 9for a three-layered cross-ply laminate with S� 5. Thetransverse shear stresses satisfy the stress conditions atthe top and bottom surfaces of the laminate. Thesestresses are evaluated integrating the three-dimensionalstress equations of equilibrium through the thickness. Itis brought out that the displacements, in general, vary innon-linear fashion.

For mechanical load (q0� 106 N/m2), a similar dy-namic response analysis is carried out for fairly thick(a=h � 10) cross-ply three- and eight-layered laminates�0°=90°=0°; �0°=90°�4�, and the results are shown inFigs. 10 and 11. Since the in¯uence of C on responsehistory is negligible, unlike in the thermal environment,the results obtained from model Q8-HSDT11a are notdepicted in these ®gures. However, the z3 term is veryimportant for accurately predicting the transverse dis-placement of symmetric laminates. Like in the thermalcase, the z2 term in in-plane displacement has in¯uenceon the displacement functions while analyzing unsym-

metric laminates. The in¯uence of w is initially felt onthe peak values and then more pronounced on the re-sponse with time. In general, it may be concluded thatthe present complete model Q8-HSDT13 accuratelypredicts the results for both mechanical and thermalloads. Third-order theory Q8-HSDT7 is not su�cientfor thick laminates.

5. Conclusions

A displacement-based C 0 continuous isoparametric,eight-noded quadrilateral plate element has been pre-sented here based on a realistic model. The accuracy ande�ectiveness of the present model over the ®rst- andother higher-order theories for dynamic analysis ofcomposite laminates have been demonstrated consider-ing thick laminates subjected to thermal/mechanicalloading. The in¯uences of the thickness stretching termsin the transverse displacement ®elds and slope discon-tinuity in thickness direction for in-plane displacements,and various other high-order terms, on the dynamicresponse characteristics have been highlighted, and they

Fig. 10. Response of a three-layered cross-ply laminate subjected to mechanical loading (S� 10).


mainly depend on the ply angle, lamination scheme,aspect ratio and thermal coe�cients. For the combinedthermo-mechanical loading situation, the present for-mulation (Q8-HSDT13) will be more accurate comparedto other available models.

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