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

Transcription

1 Composite Structures 51 (2001) 221±236 Dynamic analysis of laminated composite plates subjected to thermal/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 , 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 C 0 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 coe 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 in aircraft and space systems due to their advantages of high sti ness- and strength-to-weight ratios. However, the analysis of multi-layered structures is a complex task compared with conventional single layer metallic structures due to the exhibition of coupling among membrane, torsion and bending strains; weak transverse shear rigidities; and discontinuity of the mechanical characteristics along the thickness of the laminates. More accurate analytical/numerical analysis based on three-dimensional models may be computationally involved and expensive. Hence, among researchers, there is a growing appreciation of the importance of developing new kinematics for the evolution of accurate twodimensional theories for the analysis of thick laminates with high orthotropic ratio, leading to less expensive models. In this context, the applications of analytical/ numerical methods based on various higher-order theories, not only for the vibrations of thick laminates, but also for the high frequency vibrations of thin composite plates, has recently attracted the attention of several investigators/researchers. * Corresponding author. Tel.: ; fax: addresses: gana@iat.ernet.in, mganapathi@hotmail.com (M. Ganapathi). Various structural theories proposed for evaluating the characteristics of composite laminates under di erent loading situations have been reviewed and assessed by Noor and Burton [1,2], Tauchert [3], Kapania and Raciti [4], Reddy [5] and, more recently, by Mallikarjuna and Kant [6] and Varadan and Bhaskar [7]. It may be concluded from the literature that the analysis of composite plates under thermal environment is generally based on classical lamination theory and rst-order shear deformation theory. Furthermore, the assumption of displacements as linear functions of the coordinate in the thickness direction has proved to be inadequate for predicting the response of thick laminates. Higher-order displacement elds yielding quadratic variations of transverse shear strains have been attempted by many researchers [8±15] for better accuracy, but the application of higher-order theory for the investigation of thick multi-layered plates under thermal load seems to be scarce in literature compared to the analysis of mechanically loaded laminates [6]. Three-dimensional elasticity analysis carried out by Bhaskar et al. [16] for thick laminates subjected to thermal loads reveals the non-linear variation of in-plane displacements through the thickness and abrupt discontinuity in slope at any interface and thickness-stretch/contraction e ects in the transverse displacement. Although higher-order theories based on the discrete layer approach [17±21] account for slope discontinuity at the interfaces, the number of unknowns to be solved increases with increase in the number of layers. Recently, Ali et al. [22] have proposed /01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S ( 0 0 )

2 222 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 a new higher-order plate theory based on the global approximation approach for the static analysis of multilayered symmetric composite laminates under thermal/ mechanical loading, incorporating realistic throughthe-thickness approximations of the in-plane and transverse displacements based on the work given in [16]. This formulation has proved to give very accurate results for the static analysis of symmetric cross-ply laminates, and this excellent performance of the theory for thick laminates motivated the present extension of the formulation for the dynamic analysis of thermally/ mechanically loaded general composite laminates through the nite element procedure. Here, a C 0 eight-noded quadrilateral serendipity plate element with thirteen degrees of freedom per node has been developed by extending the theory employed for symmetric laminates [22] to general composite laminated plates. The e cacy of the present formulation, for the dynamic response analysis of laminates subjected to thermal/mechanical loads, has been brought out through numerical studies. The solutions are obtained using NewmarkÕs direct integration technique. All the inertia terms due to the part given by rst-order theory, that arising from the higher-order displacement function and that resulting from the coupling between the different order displacements are included in the formulation. The e ect of various terms in the higher-order displacement model on the response characteristics is brought out considering di erent parameters. 2. Formulation A composite plate with arbitrary lamination is considered with the coordinates x, y along the in-plane directions and z along the thickness direction. The in-plane displacements u k and v k, and the transverse displacement w k for the kth layer, are assumed as u k x; y; z; t ˆu 0 x; y; t zh x x; y; t z 2 b x x; y; t z 3 / x x; y; t S k w x x; y; t ; v k x; y; z; t ˆv 0 x; y; t zh y x; y; t z 2 b y x; y; t z 3 / y x; y; t S k w y x; y; t ; w k x; y; z; t ˆw 0 x; y; t zw 1 x; y; t z 2 C x; y; t : 1 The terms with even powers of z in the in-plane displacements and odd powers of z occurring in the expansion for w k correspond to stretching problems. However, the terms with odd powers of z in the in-plane displacements and the even ones in the expression for w k represent exure problems. u 0, v 0 and w 0 are the displacements of a generic point on the reference surface; h x and h y are the rotations of the normal to the reference surface about the y and x axes, respectively; w 1, b x, b y, C, / x and / y are the higher-order terms in the Taylor's series expansions, de ned at the reference surface. w x and w y are generalized variables associated with the zigzag function, S k. The zigzag function, S k, as given in [23], is de ned by S k ˆ 2 1 k z k =h k ; 2 where z k is the local transverse coordinate with its origin at the center of the kth layer and h k is the corresponding layer thickness. Thus, the zigzag function is piecewise linear with values of )1 and 1 alternately at the di erent interfaces. The ÔzigzagÕ function, as de ned above, takes care of the inclusion of the slope discontinuities of u and v at the interfaces of the laminate as observed in exact three-dimensional elasticity solutions of thick laminated composite structures. The use of such a function is more economical than a discrete layer approach of approximating the displacement variations over the thickness of each layer separately. Although both these approaches account for slope discontinuity at the interfaces, in the discrete layer approach, the number of unknowns increases with the increase in the number of layers, whereas it remains constant in the present approach. The strains in terms of mid-plane deformation, rotations of the normal and higher-order terms associated with displacements for kth layer are as fgˆ e ebm fe e t g: 3 s The vector fe bm g includes the bending and membrane terms of the strain components and vector fe s g contains the transverse shear strain terms. These strain vectors can be de ned as e bm ˆ e s where e xx e yy u k ;x v k ;y >< e >= >< zz w >= k ;z ˆ e xy u k ;y vk ;x c xz u >: >; ;z wk ;x c yz >: v k ;z >; wk ;y ˆ ZŠf e 0 e 1 e 2 e 3 e 4 c 0 c 1 c 2 c 3 g T ; 4a ZŠ ˆ " # I 1 Š z I 1 Š z 2 I 1 Š z 3 I 1 Š S k I 1 Š OŠ OŠ OŠ OŠ OŠ T OŠ T OŠ T OŠ T OŠ T I 2 Š z I 2 Š z 2 I 2 Š S;z k I : 2Š 4b [I 1 ]and[i 2 ] are identity matrices of size 4 4and2 2, respectively, and [O] is a null matrix of size 4 2.

3 8 u 0;x 9 8 h x;x >< v 0;y >= >< h y;y >= fe 0 gˆ ; fe 1 gˆ ; w 1 2C >: >; >: >; u 0;y v 0;x h x;y h y;x b x;x / x;x >< b >= >< y;y / >= y;y fe 2 gˆ ; fe 3 gˆ ; 0 0 >: >; >: >; b x;y b y;x / x;y / y;x 8 9 w x;x >< w >= y;y fe 4 gˆ ; 4c 0 >: >; w x;y w y;x ( ) ( fc 0 gˆ hx w 0;x ; fc 1 gˆ 2b ) x w 1;x ; h y w 0;y 2b y w 1;y ( fc 2 gˆ 3/ ) ( ) x C ;x ; fc 3 gˆ wx : 3/ y C ;y w y 9 4d The subscript comma denotes partial derivative with respect to the spatial coordinate succeeding it. The thermal strain vector fe t g is represented as 8 e xx e yy >< e >= >< zz fe t gˆ ˆ DT e xy e xz >: >; >: e yz 9 8 a x a y a z a xy >= ; 4e >; where DT is the rise in temperature and is generally represented as a function of x, y, z and time t. a x ; a y ; a z and a xy are thermal expansion coe cients in the plate coordinates and can be related to to the thermal coe cients (a 1, a 2 and a 3 ) in the material principal directions. The constitutive relations for an arbitrary layer k in the laminate (x, y, z) coordinate system can be expressed as frg ˆfr xx r yy r zz s xy s xz s yz g T e ˆ Q k Š bm fe e t g ; 5 s where the terms of the Q k Š matrix of the kth ply are referred to as the laminate axes and can be obtained from the [Q k ] corresponding to the ber directions with the appropriate transformation, as outlined in the literature [24]. frg; feg and fe t g are stress, strain and thermal strain vectors due to rise in temperature, respectively. The superscript T refers the transpose of a matrix/vector. D.P. Makhecha et al. / Composite Structures 51 (2001) 221± The governing equations are obtained by applying the Lagrangian equations of motion given by d dt o T U T =o _ d i Š o T U T =od i Šˆ0; i ˆ 1ton; 6 where T is the kinetic energy and U T is the total potential energy consisting of strain energy contributions due to the in-plane and transverse stresses and work done by the externally applied mechanical loads, respectively. fdg ˆfd 1 ; d 2 ;...; d i ;...; d n g T is the vector of the degrees of freedom/generalized coordinates. A dot over a variable represents partial derivative with respect to time. The kinetic energy of the plate is given by T d ˆ Z Z " 1 X n Z # hk 1 q 2 k f _u k _v k _w k gf _u k _v k _w k g T dz dxdy; kˆ1 h k 7 where q k is the mass density of the kth layer. h k and h k 1 are the z-coordinates of the laminate corresponding to the bottom and top surfaces of the kth layer. Using the kinematics given in Eq. (1), Eq. (7) can be rewritten as Z Z " X T d ˆ1 n Z # hk 1 q 2 k f d _ e g T ZŠ T ZŠf d _ e g dz dxdy; kˆ1 h k 8 where f d _ e g T ˆ f _u 0 _v 0 _w 0 hx _ hy w 1 bx _ by C / _ x _/ y _w x _w y g and ZŠ ˆ z 0 0 z z 3 0 S k z 0 0 z z 3 0 S k 5: z 0 0 z The potential energy function U T is given by, Z Z " X U T d ˆ1 n Z # hk 1 frg T fegdz dxdy 2 kˆ1 h k Z Z qwdxdy; 9 where q is the distributed force acting on the top surface of the plate. Substituting the constitutive relation, Eq. (5), in Eq. (9), one can write U T as

4 224 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 Z Z " X U T d ˆ1 n Z hk 1 fe bm e s g T QŠfe bm e s g 2 kˆ1 h k # 2fe bm e s g T QŠfe t g fe t g T QŠfe t g dz dxdy Z Z qw dx dy: 10 For obtaining the element-level governing equations, the kinetic and the total potential energies may be conveniently written as T d e ˆ1 2 f d _ e g T M e Šf d _ e g; 11 U T d e ˆ1 2 fde g T K e Šfd e g fd e g T ff e T g fde g T ff e M g 1 Z Z " X n Z # hk 1 fe t g T QŠfe t gdz dxdy: 2 kˆ1 h k 12 The elemental mass and sti ness matrices and thermal/ mechanical load vectors involved in Eqs. (11) and (12) can be de ned as Z Z " X n Z # hk 1 M e Šˆ q k fhg T ZŠ T ZŠfHgdz dxdy; kˆ1 h k X n Z hk 1 Z Z " # K e Šˆ BŠ T ZŠ T Q k Š ZŠ BŠdz dxdy; F e T ŠˆZ Z F e M ŠˆZ Z kˆ1 h k X n Z hk 1 " # BŠ T ZŠ T Q k Šfe t gdz dxdy; kˆ1 h k fh w g T qdxdy; where fd e g is the vector of the elemental degrees of freedoms/generalized coordinates and [H] and [B] are the interpolation and strain matrices pertaining to the element, respectively. Substituting Eqs. (11) and (12) in Eq. (6), one obtains the governing equation for the element as Table 1 Alternative eight-noded nite element models considered for parametric study Finite element model Degrees of freedom per node Q8-HSDT13 (present) u 0, v 0, w 0, h x, h y, w 1, b x, b y, C, / x, / y, w x, w y Q8-HSDT11a u 0, v 0, w 0, h x, h y, b x, b y, / x, / y, w x, w y Q8-HSDT11b u 0, v 0, w 0, h x, h y, w 1, b x, b y, C, / x, / y Q8-HSDT7 u 0, v 0, w 0, h x, h y, / x, / y Q8-FSDT u 0, v 0, w 0, h x, h y M e Šfd e g K e Šfd e gˆff e T g ff e Mg: 13 The coe cients of the mass and sti ness matrices and the load vectors involved in governing equation (13) can be rewritten as the product of the term having thickness coordinate z alone and the term containing x and y. In the present study, while performing the integration, terms having thickness coordinate z are explicitly integrated, whereas the terms containing x and y are evaluated using full integration with 3 3 point Gauss integration rule. Following the usual nite element assembly procedure, the governing equation for the forced response of the laminate are obtained as MŠf dg KŠfdg ˆfF T g ff M g; 14 where [M] and [K] are the global mass and sti ness matrices. ff T g and ff M g are the global thermal and mechanical load vectors, respectively. The solutions of Eq. (14) can be obtained using NewmarkÕs direct integration method. 3. Element description In the present work, a simple, C 0 continuous, eightnoded serendipity quadrilateral shear exible plate element with 13 nodal degrees of freedom (u 0, v 0, w 0, h x, h y, w 1, b x, b y, C, / x, / y, w x and w y : 13-DOF) is developed. Table 2 Non-dimensional displacement of simply supported three-layered cross-ply square laminates E 1 =E 2 ˆ 25; E 3 ˆ E 2 ; G 12 =E 2 ˆ G 13 =E 2 ˆ 0:5, G 23 =E 2 ˆ 0:2; m 12 ˆ m 23 ˆ m 13 ˆ 0:25 due to thermal and mechanical loading for di erent aspect ratios, S ˆ a=h S Theory Thermal loading T 0 sinpx=asinpy=b Mechanical loading q 0 sinpx=asinpy=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 Elasticity [16] Present Elasticity [16] Present Elasticity [16] Present Elasticity [16] u; m ˆ u; v = ha 1 T 0 S or u; v 100E 2 =q 0 hs 3 and w ˆ w= ha 1 T 0 S 2 or w 100E 2 =q 0 hs 4.

5 D.P. Makhecha et al. / Composite Structures 51 (2001) 221± Table 3 p Non-dimensional fundamental frequencies x ˆ x qh 2 =E 2 10 of simply supported cross-ply square plates with S ˆ 5 E 1 =E 2 ˆ 40; E 3 ˆ E 2 ; G 12 =E 2 ˆ G 13 =E 2 ˆ 0:6; G 23 =E 2 ˆ 0:5; m 12 ˆ m 23 ˆ m 13 ˆ 0:25 No. of layers, N Model E 1 =E Q8-HSDT Elasticity [25] Q8-HSDT Elasticity [25] Q8-HSDT Elasticity [25] Q8-HSDT Elasticity [25] Fig. 1. Transverse displacement versus time curve for angle-ply laminates subjected to thermal loading (aspect ratio S ˆ 5).

6 226 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 The nite element represented as per the kinematics, for instance see Eq. (1), is referred to as Q8-HSDT13 with cubic variation. Four more alternate discrete models are proposed to study the in uence of higherorder terms in the displacement functions, whose displacement elds are deduced from the original element by deleting the appropriate degrees of freedom (w 1 and C ˆ 0; or w ˆ 0; or z 2 terms, w, w 1 and C ˆ 0; or dropping all the higher-order terms). These alternate models, and the corresponding degrees of freedom, are shown in Table Results and discussion The study, here, has been focussed mainly on the dynamic response behavior of composite laminates subjected to thermal and mechanical loads of in nite duration. Since higher-order theory, in general, is required for accurate analysis for thick laminates, the emphasis in the present work is on the thick, multilayered, anisotropic plate considered for the numerical study. The in uence of various higher-order terms in the displacement 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 ˆ 5).

7 D.P. Makhecha et al. / Composite Structures 51 (2001) 221± laminates is highlighted, assuming di erent values for ply angle, aspect ratio, coe cient of thermal expansion, number of layers, etc. Based on progressive mesh re nement, an 8 8 grid mesh is found to be adequate to model the full laminated plate for the present analysis. Before proceeding to the detailed analysis of the dynamic response characteristics of the laminates, the formulation developed herein is validated considering the static and free vibration analyses of laminated cross-ply plates. Table 2 shows the de ections of the laminate along with the exact solutions of the three-dimensional elasticity theory [16]. For free vibration analysis, the fundamental frequencies obtained by varying the Fig. 3. (a) Response of a two-layered angle-ply laminate subjected to thermal loading (S ˆ 10). (b) Response of an eight-layered unsymmetric angle-ply laminate subjected to thermal loading (S ˆ 10).

8 228 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 degree of orthotropicity of the layers are given in Table 3 along with the three-dimensional elasticity results [25]. It is observed from these tables that the present results obtained employing Q8-HSDT13 model agree very well with existing literature. The material properties, unless speci ed otherwise, assumed in the present analysis are E 1 =E 2 ˆ 40; G 12 =E 2 ˆ G 13 =E 2 ˆ 0:6; G 23 =E 2 ˆ 0:5; m 12 ˆ m 23 ˆ m 13 ˆ 0:25; a 2 =a 1 ˆ a 3 =a 1 ˆ 1125; q ˆ 800 kg=m 3 ; E 2 ˆ E 3 ˆ N=m 2 ; a 1 ˆ = C; 15 Fig. 4. Response of a two-layered cross-ply laminate subjected to thermal loading (S ˆ 5).

9 D.P. Makhecha et al. / Composite Structures 51 (2001) 221± where E, G and m are YoungÕs modulus, shear modulus and PoissonÕs ratio, respectively. Subscripts 1, 2 and 3 refer to the principal material directions. The rst layer corresponds to the bottom-most layer, the ply angle is measured from the x axis in an anti-clockwise direction and all the layers are of equal thickness. The simply supported boundary conditions considered here are v 0 ˆ w 0 ˆ h y ˆ w 1 ˆ C ˆ b y ˆ / y ˆ w y ˆ 0; at x ˆ 0; a; u 0 ˆ w 0 ˆ h x ˆ w 1 ˆ C ˆ b x ˆ / x ˆ w x ˆ 0; at y ˆ 0; b; where a and b refer to the length and width of the plate, respectively. For the dynamic response study, all the initial conditions 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 ˆ 5).

10 230 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 conditionally stable nite di erence is introduced as a guide [26]. Subsequently, a convergence study is conducted to select a time step which yields a stable and accurate solution. The laminated plate assumed here is a square simply supported one. The in-plane and the transverse displacements u, v and w presented here with respect 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), respectively. The (x, y) locations of transverse shear stresses s xz and s yz are (0, b=2) and (a=2, 0), respectively. The spatial distributions of the two types of loading considered here are: for mechanical: q ˆ q 0 sin px=a sin py=b ; for thermal: DT ˆ T 0 2z=h sin px=a sin py=b. Next, through the present model (Q8-HSDT13), the variation of transverse and in-plane dynamic responses of two-, three- and eight-layered angle-ply laminates 45 = 45 ; 45 = 45 =45 ; 45 = 45 Š 4 having as- Fig. 6. Response of a two-layered cross-ply laminate subjected to thermal loading (S ˆ 10).

11 D.P. Makhecha et al. / Composite Structures 51 (2001) 221± pect ratio S (ˆ a=h; thickness h ˆ 1 cm) equal to 5 subjected to thermal load (T 0 ˆ 1), with respect to time is evaluated and given in Figs. 1 and 2. These gures also show the e ect of higher-order terms pertaining to the present model on the response characteristics, obtained through di erent nite element models considered in the present study. Since the in uence of the zigzag function w in the in-plane displacements is negligible on the transverse displacement (i.e., the results obtained using model Q8-HSDT11b almost coincide with that shown for Q8-HSDT13), for clarity, the response curves corresponding to the Q8-HSDT11b model are not shown in the gures. It is observed from Fig. 1 that the thirdorder theory, considered here (Q8-HSDT7), has little e ect on the response in comparison with the rst-order theory (Q8-FSDT). Furthermore, the signi cance of the z 2 term in the in-plane displacement, in addition to less important z 3 term, has been highlighted through the Fig. 7. Response of an eight-layered cross-ply laminate subjected to thermal loading (S ˆ 10).

12 232 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 Q8-HSDT7 and Q8-HSDT11a models. However, the predominant contribution to the global response arises from the presence of the z 2 term (C) in the transverse displacement function. With an increase in the number of layers, as seen from Fig. 1, the in uence of the higher order z 2 term (C) in the w expression increases signi cantly. It is also further noticed that the in uence of the z 2 terms in the in-plane displacements is dictated by bending±stretching coupling due to lay-up, and its in uence on the responses for symmetric laminate is insigni cant, i.e., the response predicted by the Q8- HSDT11a and Q8-HSDT7 models are the same. It may be opined here that the rst- and third-order theories of Q8-HSDT7 grossly under predict transverse displacement with time. For in-plane response evaluation, z 3 and z 2 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 ˆ 5).

13 D.P. Makhecha et al. / Composite Structures 51 (2001) 221± (Q8-HSDT11a and Q8-HSDT7), depending on lay-up, and more so for the C term in w. The e ect of the zigzag function w in the in-plane displacements is hardly noticeable at the peak values with increase in response time, but is not shown in the gure, for the purpose of clarity, by limiting the response time. A similar study is carried out for the aspect ratio a=h ˆ 10 and the evaluated transient responses are presented in Fig. 3 for twoand eight-layered angle-ply laminates. The response behaviors are, in general, qualitatively same as those of a=h ˆ 5. However, the e ectiveness of the presence of higher-order terms in the present model in predicting global responses decreases. Further, although the in uence of C decreases at faster rate compared to z 3 terms as noticed from the models Q8-HSDT11a and Q8-HSDT7, the inclusion of C (Q8-HSDT13) in the displacement leads to considerable contribution to the over-all transverse response behavior. In general, one can draw the conclusions that the higher term C in w and the z 3 terms in the in-plane displacements are essential irrespective of the lay-up, whereas the z 2 terms in in-plane displacements are necessary for unsymmetric laminates, for accurately predicting the global responses of the thick laminates considered here. For cross-ply cases, the results considering two and eight layers are plotted in Figs. 4 and 5 for aspect ratio a=h ˆ 5. It is inferred from Figs. 4 and 5 that the variation of transverse displacement is similar to those of angle-ply laminates and the e ectiveness of the higherorder terms here is like that of the angle-ply case. However, for the in-plane behaviors, the presence of C depends on the in-plane directions for the two-layered laminate case (i.e., for a 0 =90 laminate, the variation of the in-plane displacement u predicted by models Q8-HSDT13 and Q8-HSDT11a is almost similar), and the z 2 term in the in-plane displacement functions is very signi cant. But the z 3 term has slightly less e ect compared to the angle-ply cases. Similar investigation is also made with a=h ˆ 10, and the response characteristics are highlighted in Figs. 6 and 7. It is again inferred from these studies that, with increase in aspect ratio, the e ect of higher-order theory decreases, as expected. It may be opined from the investigation of cross- and angle-ply cases, that higher-order terms are predominant with increase in layers and decrease in aspect ratio. The in uence of the thermal coe cient of expansion is also investigated and is shown in Fig. 8 for two ratios of a 2 =a 1. 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 ˆ 2: s.

14 234 D.P. Makhecha et al. / Composite Structures 51 (2001) 221±236 Fig. 10. Response of a three-layered cross-ply laminate subjected to mechanical loading (S ˆ 10). increase in a 2 =a 1, as seen from the comparison of results corresponding to the Q8-HSDT13 and Q8-HSDT11a models. The displacements and transverse shear stress distributions through the thickness are drawn in Fig. 9 for a three-layered cross-ply laminate with S ˆ 5. The transverse shear stresses satisfy the stress conditions at the top and bottom surfaces of the laminate. These stresses are evaluated integrating the three-dimensional stress equations of equilibrium through the thickness. It is brought out that the displacements, in general, vary in non-linear fashion. For mechanical load (q 0 ˆ 10 6 N/m 2 ), a similar dynamic response analysis is carried out for fairly thick (a=h ˆ 10) cross-ply three- and eight-layered laminates 0 =90 =0 ; 0 =90 4 Š, and the results are shown in Figs. 10 and 11. Since the in uence of C on response history is negligible, unlike in the thermal environment, the results obtained from model Q8-HSDT11a are not depicted in these gures. However, the z 3 term is very important for accurately predicting the transverse displacement of symmetric laminates. Like in the thermal case, the z 2 term in in-plane displacement has in uence on the displacement functions while analyzing unsymmetric laminates. The in uence of w is initially felt on the peak values and then more pronounced on the response with time. In general, it may be concluded that the present complete model Q8-HSDT13 accurately predicts the results for both mechanical and thermal loads. Third-order theory Q8-HSDT7 is not su cient for thick laminates. 5. Conclusions A displacement-based C 0 continuous isoparametric, eight-noded quadrilateral plate element has been presented here based on a realistic model. The accuracy and e ectiveness of the present model over the rst- and other higher-order theories for dynamic analysis of composite laminates have been demonstrated considering thick laminates subjected to thermal/mechanical loading. The in uences of the thickness stretching terms in the transverse displacement elds and slope discontinuity in thickness direction for in-plane displacements, and various other high-order terms, on the dynamic response characteristics have been highlighted, and they

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