Large deflection of composite laminated plates under transverse and in-plane loads and resting on elastic foundations

Composite Structures 45 (1999) 115±123

Large de¯ection of composite laminated plates under transverse and in-plane loads and resting on elastic foundations Hui-Shen Shen School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China

Abstract A large de¯ection analysis is presented for a simply supported, composite laminated thin plate subjected to combined uniform lateral pressure and compressive edge loading and resting on a two-parameter (Pasternak-type) elastic foundation. The formulations are based on the classical laminated plate theory, and including the plate-foundation interaction. The analysis uses a perturbation technique to determine the load-de¯ection curves and load-bending moment curves. Numerical examples are presented that relate to the performances of antisymmetrically angle-ply and symmetrically cross-ply laminated plates subjected to combined loading and resting on two-parameter elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The in¯uence played by a number of e€ects, among them foundation sti€ness, the plate aspect ratio, the total number of plies, ®ber orientation and initial compressive load is studied. Typical results are presented in dimensionless graphical form. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Large de¯ection; Composite laminated plate; Pasternak-type elastic foundation; Combined loading; Perturbation method

1. Introduction Composite laminated structures are being widely used in aerospace, automotive, marine and other technical applications. Their components are often subjected to combinations of lateral pressure and edge loads, and may be supported by an elastic foundation. In reality, many structures are subjected to high load levels that may result in nonlinear load-de¯ection relationships due to large deformations of the plate. Therefore, there is a need to understand the large de¯ection of composite laminated plates resting on elastic foundations. Such solutions may ®nd important applications in analyzing the face plate behavior of certain types of foam-®lled sandwich panels. Many linear bending studies for isotropic and anisotropic plates subjected to transverse loads and with or without elastic foundations are available in the literature, see, for example, Turvey [1]. Numerous studies for the large de¯ection of multilayered composite plates can be found in Refs. [2±9] using the classical plate theory (CPT) or the shear deformation plate theory (SDPT). However, published literature on the large de¯ection of isotropic and anisotropic plates resting on elastic foundations has received very little attention. Sinha [10] studied the large de¯ection behavior of uniformly loaded isotropic plates on a Winkler elastic foundation

by using Berger's approximate approach [11]. Yang [12] calculated the load-central de¯ection curves of uniformly loaded isotropic plates on a Winkler elastic foundation by using the ®nite element method (FEM). Ghosh [13] calculated the load-bending moment curves and load-shear force curves of uniformly loaded isotropic plates on elastic foundation of Pasternak type. Chia [14] gave the analytical formulation for the large de¯ection of uniformly loaded orthotropic plates on a Winkler elastic foundation. All these investigations [10± 14] are concerned with the plate simply supported at four edges with no in-plane displacement. The only large de¯ection solutions available for isotropic and anisotropic thin plates subjected to combined uniform lateral pressure and compressive edge loading are those due to Levy et al. [15], Aalami and Chapman [16], Brown and Harvey [17], and Prabhakara and Chia [18,19]. To the best of the author's knowledge, there is no literature covering the large de¯ection of composite laminated thin plates subjected to combined transverse and inplane compressive loads and resting on two-parameter elastic foundations. A recent approach for the postbuckling analysis of isotropic and anisotropic plates with or without elastic foundations was developed [20±22] by using a perturbation technique. This method is extended here to determine the required load-de¯ection curves and load-bending

0263-8223/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 0 7 - 0

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H.-S. Shen / Composite Structures 45 (1999) 115±123

moment curves. The formulations are based on the classical laminated plate theory, i.e. neglecting transverse shear deformation e€ect, and including the plate-foundation interaction. Numerical examples are presented that relate to the performances of anti-symmetrically angle-ply and symmetrically cross-ply laminated plates subjected to combined loading and resting on two-parameter elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. 2. Analytical formulation Consider a thin rectangular plate of length a, width b and thickness t which consists of N plies, simply supported at four edges and rests on a two-parameter elastic foundation. The plate is subjected to a uniform lateral pressure q combined with in-plane compressive loads Px in the X-direction and Py in the Y-direction. Let U , V and W be the plate displacements parallel to a righthand set of axes (X, Y, Z), where X is longitudinal and Z is perpendicular to the plate. The foundation is assumed to be an attached foundation in which the plate cannot separate from the elastic medium. The load-displacement relationship of the foundation is assumed to be p ˆ K 1 W ÿ K 2 r2 W , where p is the force per unit area, K 1 is the Winkler foundation sti€ness and K 2 is a constant showing the e€ect of the shear interactions of the vertical elements, and r2 is the Laplace operator in X and Y. Let W …X ; Y † and F …X ; Y † be the de¯ection and the stress function for the stress resultants, and denoting di€erentiation by a comma, so that N x ˆ F ;yy , N y ˆ F ;xx and N xy ˆ ÿF ;xy . Attention is con®ned to the two cases of: (1) antisymmetrically angle-ply laminated plates; and (2) symmetrically cross-ply laminated plates from which solutions for isotropic and orthotropic plates follow as a limiting case. From the classical laminated plate theory, including the plate-foundation interaction, the governing di€erential equations are 2

L1 …W † ‡ L3 …F † ‡ K 1 W ÿ K 2 r W ˆ L…W ; F † ‡ q;

…1†

1 …2† L2 …F † ÿ L3 …W † ˆ ÿ L…W ; W †; 2 where o4 o4 o4 L1 … † ˆ D11 4 ‡ 2…D12 ‡ 2D66 † 2 2 ‡ D22 4 ; oX oX oY oY 4 4 4 o o o L2 … † ˆ A22 4 ‡ …2A12 ‡ A66 † 2 2 ‡ A11 4 ; oX oX oY oY o4 o4     ; L3 … † ˆ …2B26 ÿ B61 † 3 ‡ …2B16 ÿ B62 † oX oY oX oY 3 o2 o2 o2 o2 o2 o2 ‡ ÿ 2 ; L… † ˆ oX 2 oY 2 oX oY oX oY oY 2 oX 2 o2 o2 ‡ 2 r2 … † ˆ 2 oX oY

in these equations ‰Aij Š, ‰Bij Š and ‰Dij Š…i; j ˆ 1; 2; 6† are reduced sti€ness matrices and details of which can be found in Appendix A. Because all the edges are assumed to be simply supported, the boundary conditions are X ˆ 0; a; W ˆ 0; N xy ˆ 0; 2

…3a† 2

2

oW oW oF M x ˆ ÿD11 ÿ D12 ‡ B61 ˆ 0; oX 2 oY 2 oX oY Z b N x dY ‡ rx tb ˆ 0:

…3b† …3c†

0

Y ˆ 0; b; W ˆ 0; N xy ˆ 0; 2

…3d† 2

2

oW oW oF M y ˆ ÿD12 ÿ D22 ‡ B62 ˆ 0; 2 2 oX oY oX oY Z a N y dX ‡ rx ta ˆ 0;

…3e† …3f†

0

where rx and ry are average stresses and M x and M y are the bending moments per unit width and per unit length of the plate. Eqs. (1)±(3) are the governing equations describing the required large de¯ection response of the plate.

3. Analytical method and asymptotic solutions Introducing the dimensionless quantities (in which the alternative forms k1 and k2 are not needed until the numerical examples are considered) x ˆ pX =a; y ˆ pY =b; W ˆW c12 ˆ

…D12

‡

2D66 †=D11 ;

c22 ˆ …A12 ‡ A66 =2†=A22 ; c6 ˆ

b ˆ a=b;

1=4 =‰D11 D22 A11 A22 Š ;

F ˆ F =‰D11 D22 Š c14 ˆ

1=2

;

‰D22 =D11 Š1=2 ;

c24 ˆ ‰A11 =A22 Š

1=2

;

ÿD12 =D11 ;

…c31 ; c33 ; c361 ; c362 † ˆ …2B26 ÿ B61 ; 2B16 ÿ B62 ; B61 ; B62 †=‰D11 D22 A11 A22 Š1=4 ; 1=4

…Mx ; My † ˆ …M x ; M y †a2 =p2 D11 ‰D11 D22 A11 A22 Š

;

…K1 ; k1 † ˆ …a4 ; b4 †K 1 =p4 D11 ; …K2 ; k2 † ˆ …a2 ; b2 †K 2 =p2 D11 ; kq ˆ qa4 =p4 D11 ‰D11 D22 A11 A22 Š1=4 ; …kx ; ky † ˆ …rx b2 ; ry a2 †t=4p2 ‰D11 D22 Š

1=2

;

…4†

enables the nonlinear Eqs. (1) and (2) to be written in dimensionless form as L1 …W † ‡ c14 L3 …F † ‡ K1 W ÿ K2 r2 W ˆ c14 b2 L…W ; F † ‡ kq ;

…5†

H.-S. Shen / Composite Structures 45 (1999) 115±123

1 L2 …F † ÿ c24 L3 …W † ˆ ÿ c24 b2 L…W ; W †; 2 where

…6†

117

…1†

W ˆ e‰A11 sin x sin yŠ …3†

…3†

‡ e3 ‰A13 sin x sin 3y ‡ A31 sin 3x sin yŠ ‡ O…e4 †:

4 o4 o4 2 2 4 o ‡ 2c b ‡ c b 12 14 ox4 ox2 oy 2 oy 4 4 4 o o o4 L2 … † ˆ 4 ‡ 2c22 b2 2 2 ‡ c224 b4 4 ; ox ox oy oy 4 4 o o L3 … † ˆ c31 b 3 ‡ c33 b3 ox oy oxoy 3 o2 o2 o2 o2 o2 o2 ‡ ‡ 2 2; L… † ˆ 2 2 ÿ 2 ox oy oxoy oxoy oy ox 2 2 o o r2 … † ˆ 2 ‡ b2 2 : ox oy

L1 … † ˆ

…3†

…3†

‡ e3 ‰B13 cosx cos 3y ‡ B31 cos3x cos yŠ ‡ O…e4 † …11† and …12† kq ˆ ek1 ‡ e3 k3 ‡ O…e4 †: Note that all coecients in Eqs. (10)±(12) are related …1† and can be written as functions of A11 , so that Eqs. (10) and (12) can be rewritten as

The boundary conditions of Eq. (3) become

…1†

x ˆ 0; p; W ˆ 0; F;xy ˆ 0; 2

…7a† 2

2

oW oW oF ˆ 0; Mx ˆ ÿ 2 ÿ c6 b2 2 ‡ c14 c361 b ox oy oxoy Z p 1 o2 F b2 2 dy ‡ 4kx b2 ˆ 0; p 0 oy y ˆ 0; p; W ˆ 0; F;xy ˆ 0; 2

2

2

oW oW oF ˆ 0; My ˆ ÿc6 2 ÿ c214 b2 2 ‡ c14 c362 b ox oy oxoy Z p 2 1 oF dx ‡ 4ky ˆ 0: p 0 ox2

…7b† …7c† …7d† …7e† …7f†

Applying Eqs. (5)±(7), the large de¯ection of a simply supported composite laminated thin plate under combined loading and resting on a two-parameter elastic foundation is now determined by a perturbation technique. The essence of this procedure, in the present case, is to assume that X X ej wj …x; y†; F …x; y; e† ˆ ej fj …x; y†; W …x; y; e† ˆ jˆ1

X kq ˆ e j kj ;

…10†

2 2 …0† y …0† x …1† F ˆ ÿB00 ÿ b00 ‡ e‰B11 cosx cos yŠ 2 2 …2† …2† ‡ e2 ‰B20 cos2x ‡ B02 cos2yŠ

…1†

3

W ˆ W …1† …x; y†…A11 e† ‡ W …3† …x; y†…A11 e† ‡


and …1†

…1†

3 …3† kq ˆ k…1† q …A11 e† ‡ kq …A11 e† ‡




…13† …14†

From Eqs. (13) and (14) the load-central de¯ection relationship can be written as    3 qa4 …1† W …3† W ˆ A ‡


…15† ‡ A w w D11 t t t Similarly, the bending moment-central de¯ection relationships can be written as    3 M x a2 W W …1† …3† ˆ AMx ‡ AMx ‡


…16†  D11 t t t    3 M y a2 W W …1† …3† ˆ AMy ‡ AMy ‡


…17† D11 t t t in Eqs. (15)±(17) all coecients are given in detail in Appendix B.

jˆ0

…8†

jˆ1

where e is a small perturbation parameter and the ®rst term of wj …x; y† is assumed to have the form …1†

w1 …x; y† ˆ A11 sin x sin y:

…9†

Substituting Eq. (8) into Eqs. (5) and (6), collecting the terms of the same order of e, gives a set of perturbation equations. By using Eq. (9) to solve these perturbation equations of each order, the amplitudes of the terms wj …x; y† and fj …x; y† are determined step by step, and kj can be determined by the Galerkin procedure. As a result up to 3rd-order asymptotic solutions can be obtained as

Fig. 1. Comparison between present solution and other isotropic thin plate solutions.

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H.-S. Shen / Composite Structures 45 (1999) 115±123

Fig. 2. Comparisons of load-de¯ection curves for (0/90/0) laminated thin square plate.

Eqs. (15)±(17) can be employed to obtain numerical results for the load-de¯ection curves and load-bending moment curves of initially compressed composite laminated plates subjected to uniform lateral pressure and resting on two-parameter elastic foundations. Usually, the compressive loads in the X- and Y-directions vary proportionally, so that ry ˆ a rx , where a is a constant. The load-de¯ection curves and load-bending moment curves for such plates under any combination of rx and ry can be obtained simply by varying a. The relationships between lateral load and in-plane stress resultants (N x and N y ) can easily be found by applying Eq. (11). As expected, there are two special cases: (1) if Py ˆ Px ˆ 0, Eqs. (15)±(17) reduce to the solutions of composite laminated plates subjected to uniform lateral pressure alone and resting on two-parameter elastic foundations; and (2) if K2 ˆ 0, Eqs. (15)±(17) reduce to the solutions of initially compressed composite laminated plates subjected to uniform lateral pressure and resting on Winkler elastic foundations. 4. Numerical examples and discussion A large de¯ection analysis has been presented for simply supported, composite laminated thin plates subjected to the combined action of uniform lateral pressure and in-plane compressive loads and resting on two-parameter elastic foundations. In the numerical analysis, asymptotic solutions up to 3rd-order were used. A number of examples were solved to illustrate their application to the performance of antisymmetrically angle-ply and symmetrically cross-ply laminated plates resting on Winkler or two-parameter elastic foundations. For all of the examples b=t ˆ 100:0, all plies are of equal thickness and material properties were:

Fig. 3. E€ect of foundation sti€ness on the large de¯ection behavior of …452 †T laminated square plate: (a) load-de¯ection; (b) load-bending moment. 2

2

E11 ˆ 1:303  105 MN=m , E22 ˆ 9:377  103 MN=m , 2 G12 ˆ 4:502  103 MN=m and m ˆ 0:33 (except for Figs. 1 and 2). Typical results are presented in dimensionless graphical form. It is mentioned that in all ®gures W =t and M x b2 =E22 t4 mean the dimensionless forms of the central de¯ection and bending moment of the plate, i.e. at the point …x; y† ˆ …p=2; p=2†. As part of the validation of the present method, the load-de¯ection curves of an isotropic square plate …m ˆ 0:316† subjected to uniform lateral pressure alone and without an elastic foundation are compared in Fig. 1 with solutions of Levy [23] and Prabhakara and Chia [18] using the Fourier series method, and of Yamaki [24] using the Galerkin method, and of Bauer et al. [25] using the interaction method, and of Brown and Harvey [17] using the ®nite deference method. In addition the load-de¯ection curves of 3-ply (0/90/0) symmetrically cross-ply laminated square plates subjected to uniform lateral pressure alone and without an elastic foundation are compared in Fig. 2 with the SDPT solution of Savithri and Varadan [9] using their material

H.-S. Shen / Composite Structures 45 (1999) 115±123

119

Fig. 4. E€ect of foundation sti€ness on the large de¯ection behavior of (0/90)S laminated square plate: (a) load-de¯ection; (b) load-bending moment.

Fig. 5. E€ect of initial compressive load on the large de¯ection behavior of …452 †T laminated square plate: (a) load-de¯ection; (b) loadbending moment.

properties, i.e. material I with E11 =E22 ˆ 40, G12 =E22 ˆ 0:6, m12 ˆ 0:25 and material II with E11 =E22 ˆ 25, G12 =E22 ˆ 0:5, m12 ˆ 0:25. These two comparisons show that for the isotropic thin plate case the present solution is very closed to that of Levy and the di€erences between these solutions may be partly caused by the di€erent sets of in-plane boundary conditions. Figures 3 and 4 give, respectively, the load-de¯ection curves and load-bending moment curves of 4-ply …452 †T antisymmetrically angle-ply and …0=90†S symmetrically cross-ply laminated square plates subjected to uniform lateral pressure combined with initial compressive load Px and either resting on Winkler or twoparameter elastic foundations or without an elastic foundation. The dimensionless uniaxial compression is de®ned by Px =Pcr in which Pcr is the critical buckling load for the plate under uniaxial compression in the X-direction, as previously given in Shen [21]. The sti€nesses are …k1 ; k2 † ˆ …2:0; 0:2† for the two-parameter elastic

foundation, and …k1 ; k2 † ˆ …2:0; 0:0† for the Winkler elastic foundation and …k1 ; k2 † ˆ …0:0; 0:0† for the plate without an elastic foundation. It can be seen that the foundation sti€ness has a signi®cant e€ect on the large de¯ection behavior of the plate. Figures 5 and 6 give the load-de¯ection curves and load-bending moment curves for the same two plates under the di€erent values of initial compressive load Px shown, when the plate is subjected to uniform lateral pressure and supported by a two-parameter elastic foundation. Clearly the initial compressive stress a€ects the large de¯ection behavior of the plate signi®cantly. Figure 7 shows the e€ect of plate aspect ratio b…ˆ 1:0; 0:75† on the large de¯ection behavior of initially compressed …452 †T laminated plates resting on a two-parameter elastic foundation. It can be found that the central de¯ection is decreased by decreasing the plate aspect ratio, whereas the rectangular plate has higher bending moment when the lateral pressure is suciently large.

120

H.-S. Shen / Composite Structures 45 (1999) 115±123

Fig. 6. E€ect of initial compressive load on the large de¯ection behavior of (0/90)S laminated square plate: (a) load-de¯ection; (b) loadbending moment.

Fig. 7. E€ect of the plate aspect ratio on the large de¯ection behavior of …452 †T laminated plates: (a) load-de¯ection; (b) load-bending moment.

Figure 8 shows the e€ect of the total number of plies N …ˆ 4; 10† on the large de¯ection behavior of initially compressed antisymmetrically laminated square plates resting on a two-parameter elastic foundation. It can be seen that the central de¯ection is decreased, but the bending moment is increased, by increasing the total number of plies N. Figure 9 shows comparisons between the load-de¯ection curves and load-bending moment curves of initially compressed …452 †T , …302 †T and …152 †T laminated square plates under uniform lateral pressure, when they are supported by a two-parameter elastic foundation. It can be seen that the …152 †T plate has higher bending moment, but lower central de¯ection in this case. Figure 10 shows the e€ect of biaxial load ratio a…ˆ 0:0; 0:5; 1:0† on the large de¯ection behavior of initially compressed …452 †T laminated square plate under combined loading case and resting on a two-parameter elastic foundation. As expected, these results show that

the central de¯ection and bending moment are increased by increasing the ratio between the biaxial stresses. In Figs. 3, 4 and 7±10, the initial compressive load Px =Pcr ˆ 0:25; and in Figs. 3±9, the biaxial load ratio a ˆ 0:0; and in Figs. 5±10, the two-parameter elastic foundation sti€ness is characterized by …k1 ; k2 † ˆ …2:0; 0:2†. 5. Conclusions Large de¯ection behavior of a simply supported composite laminated thin plate subjected to the combined action of uniform lateral pressure and in-plane compressive loads and resting on a two-parameter elastic foundation has been studied by a perturbation method. The large de¯ection of such plates subjected to uniform lateral pressure alone is treated as a limiting case. The numerical examples presented relate to the performance of antisymmetrically angle-ply and

H.-S. Shen / Composite Structures 45 (1999) 115±123

121

Fig. 8. E€ect of the total number of plies on the large de¯ection behavior of laminated plates: (a) load-de¯ection; (b) load-bending moment.

Fig. 9. E€ect of ®ber orientation on the large de¯ection behavior of laminated plates: (a) load-de¯ection; (b) load-bending moment.

symmetrically cross-ply laminated plates resting on Winkler or two-parameter elastic foundations. They show that the characteristics of large de¯ection are signi®cantly in¯uenced by foundation sti€ness, the plate aspect ratio, the total number of plies, ®ber orientation and the amount of initial compressive load.

and Qij are the transformed elastic constants, de®ned by 2 3 Q11 6Q 7 6 12 7 6 7 6 Q22 7 6 7 6Q 7 6 16 7 6 7 4 Q26 5 Q66

Appendix A

2

‰Aij Š,

‰Bij Š

‰Dij Š…i; j

and In Eqs. (1) and (2) reduced sti€ness matrices, de®ned as 

ÿ1

ÿ1





ˆ 1; 2; 6† are

ÿ1

…A:1† A ˆ A ; B ˆ ÿA B; D ˆ D ÿ BA B; where Aij , Bij and Dij are the plate sti€nesses, de®ned by N Z tk X …Qij †k …1; Z; Z 2 †dZ …i; j ˆ 1; 2; 6†; …Aij ; Bij ; Dij † ˆ kˆ1

c4 6 c2 s2 6 6 4 6 s ˆ6 6 c3 s 6 6 3 4 cs c 2 s2

s4 c 2 s2

2c2 s2 cs3 ÿ c3 s

c4 ÿ cs3

c3 s ÿ cs3

ÿ c3 s

ÿ 2c2 s2

c 2 s2

4c2 s2 ÿ 4c2 s2

3

7 2Q 3 11 7 76 7 6 Q12 7 4c2 s2 7 76 7; ÿ 2cs…c2 ÿ s2 † 7 7 4 Q22 5 7 2cs…c2 ÿ s2 † 5 Q66 …c2 ÿ s2 †

2

…A:3†

tkÿ1

…A:2†

2c2 s2 c 4 ‡ s4

where

122

H.-S. Shen / Composite Structures 45 (1999) 115±123 …1†

AMx ˆ p2 H41 ; …3†

AMx ˆ p2 ‰…H41 ÿ H42 †a313 ‡ …H41 …1†

AMy ˆ p2 H51 ;

…B:1†

…3†

AMy ˆ p2 ‰…H51 ÿ H52 †a313 ‡ …H51 in which S11 ˆ H11 ‡ K1 ‡ K2 …1 ‡ b2 † H11 ˆ …1 ‡ 2c12 b2 ‡ c214 b4 † ‡ c14 c24

2

b2 …c31 ‡ c33 b2 † ; 1 ‡ 2c22 b2 ‡ c224 b4

H2 ˆ …1 ‡ c224 b4 †; a313 ˆ

1 c14 1 ; 16 c24 g13

a331 ˆ

1 b4 ; c14 c24 g31 16

g13 ˆ S13 ÿ

Px …1 ‡ 9ab2 † S11 ; Pcr …1 ‡ ab2 †

g31 ˆ S31 ÿ

Px …9 ‡ ab2 † S11 ; Per …1 ‡ ab2 †

S13 ˆ H13 ‡ K1 ‡ K2 …1 ‡ 9b2 †; S31 ˆ H31 ‡ K1 ‡ K2 …9 ‡ b2 †; H13 ˆ …1 ‡ 18c12 b2 ‡ 81c214 b4 † ‡ c14 c24 2

Fig. 10. E€ect of biaxial load ratio on the large de¯ection behavior of …452 †T laminated square plate: (a) load-de¯ection; (b) load-bending moment.

E11 E22 ; Q22 ˆ ; …1 ÿ m12 m21 † …1 ÿ m12 m21 † m21 E11 ; Q66 ˆ G12 : Q12 ˆ …A:4a† …1 ÿ m12 m21 † E11 , E22 , G12 , m12 and m21 have their usual meanings and Q11 ˆ

c ˆ cosh; s ˆ sin h …A:4b† where h is lamination angle with respect to the plate Xaxis. Appendix B In Eqs. (15)±(17)   p6 Px 2 …1† Aw ˆ S11 1 ÿ …1 ‡ ab † ; 16 Pcr    6 p Px 2 …3† …a313 ‡ a331 †S11 1 ÿ …1 ‡ ab † Aw ˆ 16 Pcr  1 c14 t2 H2 ‡ 1=2 16 c24 ‰D11 D22 A11 A22 Š ;



9b2 …c31 ‡ 9c33 b2 † ; 1 ‡ 18c22 b2 ‡ 81c224 b4

H31 ˆ …81 ‡ 18c12 b2 ‡ c214 b4 † ‡ c14 c24 2



9b2 …9c31 ‡ ‡c33 b2 † ; 81 ‡ 18c22 b2 ‡ c224 b4

H41 ˆ …1 ‡ c6 b2 † ÿ c361 c14 c24

b2 …c31 ‡ c33 b2 † ; 1 ‡ 2c22 b2 ‡ c224 b4

H42 ˆ …1 ‡ 9c6 b2 † ÿ c361 c14 c24 H43 ˆ …9 ‡ c6 b2 † ÿ c361 c14 c24

9b2 …c31 ‡ 9c33 b2 † ; 1 ‡ 18c2 2b2 ‡ 81c224 b4

9b2 …9c31 ‡ c33 b2 † ; 81 ‡ 18c22 b2 ‡ c224 b4

H51 ˆ …c6 ‡ c214 b2 † ÿ c362 c14 c24

b2 …c31 ‡ c33 b2 † ; 1 ‡ 2c22 b2 ‡ c224 b4

H52 ˆ …c6 ‡ 9c214 b2 † ÿ c362 c14 c24

9b2 …c31 ‡ 9c33 b2 † ; 1 ‡ 18c22 b2 ‡ 81c224 b4

H53 ˆ …9c6 ‡ c214 b2 † ÿ c362 c14 c24

9b2 …9c31 ‡ c33 b2 † ; 81 ‡ 18c2 2b2 ‡ c224 b4 …B:2†

H.-S. Shen / Composite Structures 45 (1999) 115±123

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