Postbuckling of shear deformable laminated plates under biaxial compression and lateral pressure and resting on elastic foundations

International Journal of Mechanical Sciences 42 (2000) 1171}1195

Postbuckling of shear deformable laminated plates under biaxial compression and lateral pressure 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 Received 3 November 1998; received in revised form 8 March 1999

Abstract Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to biaxial compression combined with uniform lateral pressure and resting on an elastic foundation. The lateral pressure is "rst converted into an initial de#ection and the initial geometrical imperfection of the plate is also taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and including the plate-foundation interaction. The analysis uses a perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates under combined loading and resting on Pasternak-type or softening nonlinear elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The e!ects played by foundation sti!ness, transverse shear deformation, plate aspect ratio, total number of plies, "ber orientation, the biaxial load ratio and initial lateral pressure are studied. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Structural stability; Postbuckling; Composite laminated thick plate; Higher-order shear deformation plate theory; Combined loading; Elastic foundation; Perturbation technique

1. Introduction Composite laminated structures are widely used in aerospace, automotive, marine and other technical applications. Their components are often subjected to combinations of lateral pressure E-mail address: [email protected] (H.-S. Shen) 0020-7403/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 4 4 - 2

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and in-plane compressive loads. These plates may be supported by an elastic foundation and may have signi"cant and unavoidable initial geometrical imperfections. In such a case two di!erent kinds of problems should be considered. When the edge compression is relatively small and the lateral pressure exceeds high levels, the large de#ection pattern appears and a nonlinear bending problem should be solved. In contrast, when the lateral pressure is relatively small, the postbuckling caused by an increase in edge compression should be considered. This is the problem studied in the present paper for the case when all four edges of the plate are simply supported. Many initial buckling studies for composite laminated thick plates with or without elastic foundations subjected to compressive edge loads are available in the literature, see, for example, Refs. [1}11]. However, investigations involving the application of the shear deformation plate theory to the postbuckling analysis are limited in number. The postbuckling of a perfect transversely isotropic thick plate was studied by Chen and Doong [12] and of an imperfect transversely isotropic symmetrically laminated plate was performed by Librescu and Stein [13] using the Galerkin method. Minguet et al. [14] studied the postbuckling behavior of symmetrically laminated or sandwich plates using a direct energy minimization technique. Carrera and Villani [15,16], Noor and Peters [17] and Sundaresan et al. [18] calculated the postbuckling response of perfect laminated thick plates using the "nite element method. These analyses [12}18] were based on the "rst-order shear deformation plate theory (FSDPT). Moreover, the postbuckling analysis was performed by Librescu and Stein [19] for transversely isotropic symmetrically laminated thick plates under uniaxial compression, and by Bhimaraddi [20] for orthotropic and symmetrically cross-ply laminated thick plates under equal biaxial compression, according to a higher-order shear deformation plate theory (HSDPT). For elastic foundation, Naidu et al. [21] and Jayachandran and Vaidyanathan [22] calculated the postbuckling response of an equally biaxially compressed, isotropic square thin plate resting on a Winkler elastic foundation by the "nite element method. Recently, Shen [23] and Shen and Williams [24,25] analyzed the postbuckling of imperfect, composite laminated thin plates under in-plane compression and resting on Pasternak-type or softening nonlinear elastic foundations from which results for Winkler elastic foundations follow as a limiting case. Moreover, the postbuckling solutions available for isotropic thin plates subjected to combined lateral pressure and compressive edge loading are those due to Levy et al. [26], Yosiki et al. [27], Yamamoto et al. [28], Shen [29] and Ye [30]. Recently, Shen [31] gave the postbuckling solutions of composite laminated thin plates subjected to combined lateral pressure and compressive edge loads and resting on Pasternak-type elastic foundations, using the classical laminated plate theory (CPT), i.e. neglecting transverse shear deformation e!ects of the plate. Also recently, Librescu and Lin [32] analyzed the postbuckling of imperfect, shear deformable transverse isotropic, #at and curved panels subjected to combined loading and resting on Winkler or nonlinear elastic foundations. As mentioned by Leissa [33] and Qatu and Leissa [34], unsymmetrically laminated plates do not usually remain #at in the cases of: (1) simply supported antisymmetrically cross-ply laminated plates subjected to uniaxial or biaxial loading; (2) antisymmetrically cross-ply laminated plates having uniaxially loaded edges simply supported or clamped and the other edges free or elastically supported; (3) antisymmetrically angle-ply laminated plates having one or more unloaded edges free; and (4) simply supported antisymmetrically angle-ply laminated plates loaded in shear.

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Therefore, the postbuckling response of unsymmetrically laminated plates is considerably more complex and requires careful attention. As is well known, in the case of composite laminates, transverse shear deformation is considerably more important than that in classical plates. Therefore, the present study extends the previous work to the case of simply supported, perfect and imperfect, shear deformable laminated plates subjected to biaxial compression combined with lateral pressure and resting on elastic foundations. The formulation is based on the Reddy's higher-order shear deformation plate theory (see Ref. [35]), and including plate-foundation interaction. The analysis uses a perturbation technique to determine the required buckling loads and postbuckling equilibrium paths. The lateral pressure is "rst converted into an initial de#ection and the initial geometrical imperfection of the plate is also taken into account but, for simplicity, its form is taken as the initial buckling mode of the plate.

2. Governing equations Consider a rectangular thick plate of length a, width b and thickness t which consists of N plies, simply supported at four edges and rests on an elastic foundation. The plate is subjected to a uniform lateral pressure q combined with in-plane compressive loads P in the X-direction and x P in the >-direction, as shown in Fig. 1. Let ;M , , Z), where X is longitudinal and Z is perpendicular to the plate. (M and (M are the mid-plane rotations of the normals about the >- and X-axis, respectively. The x y foundation is assumed to be an attached foundation, that means no part of the plate lifts o! the foundation in the postbuckled regime. The load}displacement relationship of the foundation is assumed to be p"KM = M !KM +2= M !KM = M 3, where p is the force per unit area, +2 is the Laplace 1 2 3 operator in X and >, and KM , KM and KM are the Winkler, Pasternak and softening nonlinear 1 2 3 elastic foundation sti!nesses, respectively. If KM "0 a Pasternak-type foundation is under consid3 eration, and if KM "0 the foundation is a softening nonlinear elastic foundation, as used for 2

Fig. 1. A rectangular plate subjected to combined action of biaxial compression and lateral pressure and resting on an elastic foundation.

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imperfect columns by Amazigo et al. [36]. Denoting the initial geometrical imperfection by = M H(X, >), let = M (X, >) be the additional de#ection and FM (X, >) be the stress function for the stress resultants, and denoting di!erentiation by a comma, so that N "FM , , N "FM , and x yy y xx N "!FM , . xy xy Attention is con"ned to the two cases of: (1) simply supported antisymmetrically angle-ply laminated plates and; (2) simply supported symmetrically cross-ply laminated plates, from which solutions for isotropic and orthotropic plates follow as a limiting case. Note that for all these cases the plate remains #at up to the bifurcation point unless there is an initial geometrical imperfection. From the Reddy's higher-order shear deformation plate theory and including the plate}foundation interaction, the KaH rmaH n-type nonlinear large de#ection equations of such plates can be written as ¸ (= M )!¸ ((1 )!¸ ((1 )#¸ (FM )#KM = M !KM +2= M !KM = M 3"¸(= M #= M H, FM )#q, 11 12 x 13 y 14 1 2 3 (1) ¸ (FM )#¸ ((1 )#¸ ((1 )!¸ (= M )"!1¸(= M #2= M H, = M ), 21 22 x 23 y 24 2

(2)

¸ (= M )#¸ ((1 )!¸ ((1 )#¸ (FM )"0, 31 32 x 33 y 34

(3)

¸ (= M )!¸ ((1 )#¸ ((1 )#¸ (FM )"0, 41 42 x 43 y 44

(4)

where operators ¸ ( ) and ¸( ) are de"ned in Appendix A. ij Because all the edges are assumed to be simply supported, the boundary conditions are X"0, a: = M "(1 "0, y

(5a)

N "M M "PM "0, xy x x

(5b)

P

(5c)

b

0

N d>#p tb"0, x x

>"0, b: = M "(1 "0, x

(5d)

N "M M "PM "0, xy y y

(5e)

P

(5f)

a

0

N dX#p ta"0, y y

where p and p average stresses and M M and M M are the bending moments and PM and PM are the x y x y x y higher-order moments, respectively. Eqs. (1)}(4) and (5a)}(5f) are the governing equations describing the required large de#ection postbuckling response of the shear deformable laminated plate.

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3. Analytical methods and asymptotic solutions Introducing dimensionless quantities (in which the alternative forms k , k and k are not needed 1 2 3 until the numerical examples are considered) (=H, =)"(= M H, = M )/[DH DH AH AH ]1@4, 11 22 11 22 F"FM /[DH DH ]1@2, (( , ( )"((M , (M )a/p[DH DH AH AH ]1@4, 11 22 x y x y 11 22 11 22 c "[DH /DH ]1@2, c "[AH /AH ]1@2, c "!AH /AH , 14 22 11 24 11 22 5 12 22 (c , c , c )"(4/3t2)[FH , (FH #FH #4FH )/2, FH ]/DH , 110 112 114 11 12 21 66 22 11 (c , c )"[DH !4FH /3t2, (DH #2DH )!4(FH #2FH )/3t2]/DH , 120 122 11 11 12 66 12 66 11 (c , c )"[(DH #2DH )!4(FH #2FH )/3t2, DH !4FH /3t2]/DH , 131 133 12 66 21 66 22 22 11 (c , c )"(2BH !BH , 2BH !BH )/[DH DH AH AH ]1@4, 62 11 22 11 22 61 16 141 143 26 (c , c )"(AH #AH /2, AH )/AH , 212 214 12 66 11 22 (c , c )"[(BH !BH )!4(EH !EH )/3t2, BH !4EH /3t2]/[DH DH AH AH ]1@4, 221 223 26 61 26 61 16 16 11 22 11 22 H H H H H H (c , c )"[B !4E /3t2, (B !B )!4(E !E )/3t2]/[DH DH AH AH ]1@4, 230 232 26 26 16 62 16 62 11 22 11 22 H H H H H H H H (c , c )"(4/3t2)(2E !E , 2E !E )/[D D A A ]1@4, 241 243 26 61 16 62 11 22 11 22 (c , c )"(a2/p2)(A !8D /t2#16F /t4, A !8D /t2#16F /t4)/DH , 31 41 55 55 55 44 44 44 11 (c , c )"(4/3t2)[FH !4HH /3t2, (FH #2FH )!4(HH #2HH )/3t2]/DH , 310 312 11 11 21 66 12 66 11 (c , c )"(DH !8FH /3t2#16HH /9t4, DH !8FH /3t2#16HH /9t4)/DH , 320 322 11 11 11 66 66 66 11 c "[(DH #DH )!4(FH #FH #2FH )/3t2#16(HH #HH )/9t4]/DH , 331 12 66 12 21 66 12 66 11 (c , c )"(4/3t2)[(FH #2FH )!4(HH #2HH )/3t2, FH !4HH /3t2]/DH , 411 413 12 66 12 66 22 22 11 (c , c )"(DH !8FH /3t2#16HH /9t4, DH !8FH /3t2#16HH /9t4)/DH , 430 432 66 66 66 22 22 22 11 H H (K , k )"(a4, b4)KM /p4D , (K , k )"(a2, b2)KM /p2D , 1 1 1 11 2 2 2 11 H H H H H (K , k )"(a4, b4)KM [D D A A ]1@2/p4D , 3 3 3 11 22 11 22 11 (M , M , P , P )"(MM , M M , 4PM /3t2, 4PM /3t2)a2/p2DH [DH DH AH AH ]1@4, x y x y x y x y 11 11 22 11 22 j "qa4/p4DH [DH DH AH AH ]1@4, (j , j )"(p b2, p a2)t/4p2[DH DH ]1@2 q 11 11 22 11 22 x y x y 11 22 enables Eqs. (1)}(4) to be written in dimensionless form as x"pX/a, y"p>/b, b"a/b,

¸ (=)!¸ (( )!¸ (( )#c ¸ (F)#K =!K +2=!K =3 11 12 x 13 y 14 14 1 2 3 H "c b2¸(=#= , F)#j , 14 q ¸ (F)#c ¸ (( )#c ¸ (( )!c ¸ (=)"!1 c24b2¸(=#2=H, =), 21 24 22 x 24 23 y 24 24 2 ¸ (=)#¸ (( )!¸ (( )#c ¸ (F)"0, 31 32 x 33 y 14 34 ¸ (=)!¸ (( )#¸ (( )#c ¸ (F)"0, 41 42 x 43 y 14 44

(6)

(7) (8) (9) (10)

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where L4 L4 L4 ¸ ( )"c #2c b2 #c b4 , 11 110 Lx4 112 Lx2Ly2 114 Ly4 L3 L3 ¸ ( )"c #c b2 , 12 120 Lx3 122 LxLy2 L3 L3 #c b3 , ¸ ( )"c b 133 Ly3 13 131 Lx2Ly L4 L4 #c b3 , ¸ ( )"c b 143 LxLy3 14 141 Lx3Ly L4 L4 L4 #c b4 , ¸ ( )" #2c b2 212 Lx2Ly2 214 Ly4 21 Lx4 L3 L3 ¸ ( )"c b #c b3 , 22 221 Lx2Ly 223 Ly3 L3 L3 #c b2 , ¸ ( )"c 232 Lx Ly2 23 230 Lx3 L4 L4 #c b3 , ¸ ( )"c b 243 Lx Ly3 24 241 Lx3Ly L3 L3 L #c #c b2 , ¸ ( )"c 310 Lx3 312 Lx Ly2 31 31 Lx L2 L2 !c b2 , ¸ ( )"c !c 322 Ly2 32 31 320 Lx2 L2 ¸ ( )"c b , 33 331 Lx Ly ¸ ( )"¸ ( ), 34 22 L L3 L3 ¸ ( )"c b #c b #c b3 , 41 41 Ly 411 Lx2 Ly 413 Ly3 ¸ ( )"¸ ( ), 42 33 L2 L2 !c b2 , ¸ ( )"c !c 432 Ly2 43 41 430 Lx2 ¸ ( )"¸ ( ), 44 23

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L2 L2 L2 L2 L2 L2 ¸( )" !2 # , Lx2 Ly2 LxLy LxLy Ly2 Lx2 L2 L2 +2( )" #b2 . Lx2 Ly2 In Eq. (6) [AH], [BH], [DH], [EH], [FH] and [HH] (i, j"1, 2, 6) are reduced sti!ness matrices ij ij ij ij ij ij the details of which can also be found in Appendix A. The boundary conditions of Eqs. (5a)}(5f) becomes x"0, p: ="( "0, y F, "M "P "0, xy x x 1 p L2F b2 dy#4j b2"0. x Ly2 p 0 y"0, p:

P

(11a) (11b) (11c)

="( "0, (11d) x F, "M "P "0, (11e) xy y y 1 p L2F dx#4j "0. (11f) y p Lx2 0 Applying Eqs. (7)}(10) and (11a)}(11f), the postbuckling of a simply supported shear deformable laminated plate under combined loading and resting on an elastic foundation is determined by a perturbation technique. The lateral pressure is applied "rst. Then in-plane compressive edge loading is added. Let

P

="= #= , ( "( #( , ( "( #( , F"F #F (12) L C x xL xC y yL yC L C the solution of Eqs. (7)}(10), in which = is an initial de#ection due to lateral pressure q, = is an L C additional de#ection due to inplane compression. ( , ( and F are rotations and stress function xL yL L corresponding to = . ( , ( and F are de"ned analogously to ( , ( and F , but are for = . L xC yC C xL yL L C If lateral pressure, q, is not too large, let j "ej , where e is a small perturbation parameter, then q 1 = is a small de#ection, so that = ,( ,( and F are of the order e1, hence the small quantity of L L xL yL L a higher order in Eqs. (7) and (8) can be omitted. Then = , ( , ( and F satisfy the small L xL yL L de#ection equations ¸ (= )!¸ (( )!¸ (( )#c ¸ (F )#K = !K +2= "j , 11 L 12 xL 13 yL 14 14 L 1 L 2 L q ¸ (F )#c ¸ (( )#c ¸ (( )!c ¸ (= )"0, 21 L 24 22 xL 24 23 yL 24 24 L ¸ (= )#¸ (( )!¸ (( )#c ¸ (F )"0, 31 L 32 xL 33 yL 14 34 L ¸ (= )!¸ (( )#¸ (( )#c ¸ (F )"0. 41 L 42 xL 43 yL 14 44 L

(13) (14) (15) (16)

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The small de#ection solutions of Eqs. (13)}(16) can be obtained as = "e(A(1) sin mx sin ny), L L F "e(B(1) cos mx cos ny), L L ( "e(C(1) cos mx sin ny), xL L ( "e(D(1) sin mx cos ny) yL L

(17) (18) (19) (20)

and j "j(1)(A(1)e). q q L From Eqs. (17) and (21) the initial de#ection and load relationship can be written as

(21)

A B

qa4 = M L"A(1) (22) W DH t t 11 in which A(1) is given in detail in Appendix B. Then = , ( , ( and F satisfy nonlinear W C xC yC C equations ¸ (= )!¸ (( )!¸ (( )#c ¸ (F )#K = !K +2= !K =3 11 C 12 xC 13 yC 14 14 C 1 C 2 C 3 C "c b2¸(= #=H, F ), 14 C T C ¸ (F )#c ¸ (( )#c ¸ (( )!c ¸ (= )"!1 c b2¸(= #2=H, = ) C T C 21 C 24 22 xC 24 23 yC 24 24 C 2 24 ¸ (= )#¸ (( )!¸ (( )#c ¸ (F )"0, 31 C 32 xC 33 yC 14 34 C ¸ (= )!¸ (( )#¸ (( )#c ¸ (F )"0 41 C 42 xC 43 yC 14 44 C in which the total initial de#ection =H"= #=H. T L To construct an asymptotic solution of Eqs. (23)}(26), it is necessary to assume that

(23) (24) (25) (26)

= (x, y, e)" + ejw (x, y), F (x, y, e)" + ej f (x, y), C j C j j/1 j/0 ( (x, y, e)" + ejt (x, y), ( (x, y, e)" + ejt (x, y). xC xj yC yj j/1 j/1 The "rst term of w (x, y) is assumed to have the form j w (x, y)"A(1) sin mx sin ny 1 11 and the initial geometrical imperfection is assumed to have a similar form =H(x, y, e)"eaH sin mx sin ny 11 then the total initial de#ection can be written as

(27)

(28) (29a)

=H(x, y, e)"ekA(1) sin mx sin ny, (29b) T 11 where k"(aH #A(1))/A(1) is the imperfection parameter. 11 L 11 Usually, the compressive loads P and P vary proportionally, so that p "ap and hence x y y x j "ab2j , where a is a constant. Substituting Eq. (27) into Eqs. (23)}(26), gives a set of perturbay x tion equations. By using Eqs. (28), (29a) and (29b) to solve these perturbation equations of each order, the amplitudes of the terms w (x, y), f (x, y), t (x, y) and t (x, y) are determined step by j j xj yj

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step. As a result up to 4th-order asymptotic solutions can be obtained as = "e[A(1) sin mx sin ny]#e3[A(3) sin mx sin 3ny#A(3) sin 3mx sin ny C 11 13 31 #A(3) sin 3mx sin 3ny]#e4[A(4) sin 2mx sin 2ny#A(4) sin 2mx sin 4ny 33 22 24 #A(4) sin 4mx sin 2ny]#O(e5), 42 x2 y2 F "!B(0) !b(0) #e[B(1) cos mx cos ny] 00 2 11 C 00 2

C

(30)

D

y2 x2 #e2 !B(2) !b(2) #B(2) cos 2mx#B(2) cos 2ny 00 2 00 2 20 02

#e3[B(3) cos mx cos 3ny#B(3)cos 3mx cos ny#B(3) cos 3mx cos 3ny] 13 31 33 y2 x2 #e4 !B(4) !b(4) #B(4) cos 2mx#B(4) cos 2ny 00 2 00 2 20 02

C

#B(4) cos 2mx cos 2ny#B(4) cos 4mx#B(4) cos 4ny#B(4) cos 2mx cos 4ny 22 40 04 24

D

#B(4)cos 4mx cos 2ny #O(e5), 42

(31)

( "e[C(1) cos mx sin ny]#e2[C(2) sin 2ny]#e3[C(3) cos mx sin 3ny xC 11 02 13 #C(3)cos 3mx sin ny#C(3) cos 3mx sin 3ny]#e4[C(4) sin 2ny#C(4) cos 2mx sin 2ny 31 33 02 22 #C(4) sin 4ny#C(4) cos 2mx sin 4ny#C(4) cos 4mx sin 2ny]#O(e5), (32) 04 24 42 ( "e[D(1) sin mx cos ny]#e2[D(2) sin 2mx]#e3[D(3) sin mx cos 3ny yC 11 20 13 #D(3)sin 3mx cos ny#D(3) sin 3mx cos 3ny]#e4[D(4) sin 2mx#D(4) sin 2mx cos 2ny 31 33 20 22 #D(4) sin 4mx#D(4) sin 2mx cos 4ny#D(4) sin 4mx cos 2ny]#O(e5). (33) 40 24 42 Note that all coe$cients in Eqs. (30)}(33) are related and can be written as functions of A(1) but, 11 for the sake of brevity, the detailed expressions are not shown. Note that in Eqs. (30)}(33) the coe$cients A(3), B(3), C(3) and D(3) are all zero-valued for the case of Pasternak-type elastic 33 33 33 33 foundation. Next, substituting Eq. (31) into boundary condition (11c), the postbuckling equilibrium path can be written as j "j(0)#j(2)=2 #j(4)=4 #2 (34) x x x m x m in which = is the dimensionless form of = , and is assumed to be at the point (x, y)" m C (n/2m, n/2n), and j(0), j(2) and j(4) are also given in detail in Appendix B. x x x Eq. (34) can be employed to obtain numerical results for the post-buckling load}de#ection curves of simply supported shear deformable laminated plates subjected to biaxial compression combined with lateral pressure and resting on elastic foundations. As is mentioned, the compressive loads in the X- and >-directions vary proportionally, so that the load}de#ection curves for such plates under any combination of P and P can be obtained simply by varying a. If lateral x y pressure q"0, the buckling load of perfect plates can readily be obtained numerically, by setting

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k"0 (or = M H/t"0), while taking = "0 (or = M /t"0). In such a case, the minimum buckling load m is determined by considering Eq. (34) for various values of the buckling mode (m, n), which determine the number of half-waves in the X- and >-directions. Note that when K "0, the 2 buckling loads for Winkler and nonlinear elastic foundations are identical.

4. Numerical examples and discussion A postbuckling analysis has been presented for simply supported, shear deformable laminated plates subjected to biaxial compression combined with uniform lateral pressure and resting on elastic foundations. In the numerical analysis, asymptotic solutions up to fourth order were used. A number of examples were solved to illustrate their application to the performance of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates resting on Pasternak-type or softening nonlinear elastic foundations. For all of the examples (except for Tables 1}3, and Fig. 2) the material properties were: E "1.303]105 MN/m2, 11 Table 1 Comparisons of initial buckling load p b2/E t2 for perfect ($45) x 22 T laminated square plates with di!erent values of b/t subjected to uniaxial compression (E /E "40, G /E "G /E "0.6, 11 22 12 22 13 22 G /E "0.5 and l "0.25) 23 22 12 b/t

100 50 25 20 12.5 10 5

Present HSDPT

Senthilnathan (1987) HSDPT

21.666(1, 1)! 21.539(1, 1) 21.046(1, 1) 20.6915(1, 1) 19.2868(1, 1) 18.154(1, 1) 12.270(1, 1) 10.8807(2, 1)

21.666 21.539 21.046 20.691 19.286 18.154 12.270

CPT

21.7089

!The numbers in brackets indicate the buckling mode (m, n). Table 2 Comparisons of initial buckling load p b2/E t2 for perfect symmetrically cross-ply x 22 laminated square plates with di!erent values of b/t subjected to equal biaxial compression Layer-up

b/t

5

10

15

20

100

(0/90/0)

Present Khdeir (1988)

5.4636 5.416

9.9754 10.079

12.0500 12.117

13.0549 13.124

14.6283 *

(0/90) S

Present Nair (1996)

3.5583 *

7.0880 7.2121

9.0112 *

10.0113 10.1546

11.6649 11.6795

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Table 3 Comparisons of initial buckling load p b2/E t2 for perfect (0/90/0) x 22 laminated plates with or without elastic foundations subjected to equal biaxial compression (E /E "40, G /E "G /E " 11 22 12 22 13 22 0.6, G /E "0.5 and l "0.25) 23 22 12 Present HSDPT

Xiang (1996) FSDPT

(0.0, 0.0) (0.32, 0.0) (0.32, 0.31) (0.0, 0.0) (0.32, 0.0) (0.32, 0.31)

14.7035 16.7299 26.7299 9.9754 12.0018 22.0018

14.7036 16.7300 26.7300 10.2024 12.2288 22.2288

(0.0, 0.0) (0.32, 0.0) (0.32, 0.31) (0.0, 0.0) (0.32, 0.0) (0.32, 0.31)

3.6760 10.9929 20.9930 3.2637 9.3743 19.3743

3.6760 10.9930 20.9930 3.2868 9.5904 19.5904

a/b

b/t

(k , k ) 1 2

1.0

1000

10

2.0

1000

10

Fig. 2. Comparisons of postbuckling load}de#ection curves for a single-ply orthotropic square plate under equal biaxial compression.

E "9.377]103 MN/m2, G "G "4.502]103 MN/m2, G "1.724]103 MN/m2 and 22 12 13 23 l "0.33; all plies are of equal thickness, e.g. t/4 for Fig. 3 and b"50 mm. Typical results are 12 presented in dimensionless graphical form in which jH"p (b/t)2/E . On all "gures = M H/t and = M /t x x 22 mean the dimensionless forms of, respectively, the maximum initial geometrical imperfection and additional de#ections of the plate.

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H.-S. Shen / International Journal of Mechanical Sciences 42 (2000) 1171}1195

Fig. 3. Postbuckling load}de#ection curves for initially pressurized square plates under equal biaxial compression and resting on elastic foundations.

As part of the validation of the present method, the buckling loads for perfect, 2-ply ($45) T antisymmetrically angle-ply laminated square plates with di!erent values of b/t, without an elastic foundation subjected to uniaxial compression (a"0.0) alone are compared in Table 1 with HSDPT results of Senthilnathan et al. [5], using their material properties, i.e. E /E "40, 11 22 G /E "G /E "0.6, G /E "0.5 and l "0.25. Note that because of taking wrong value 12 22 13 22 23 22 12 of the buckling mode (m, n)"(1, 1), the result calculated by Senthilnathan et al. [5] is found to be invalid for the plate with b/t"5. The buckling loads for perfect, (0/90/0) and (0/90) symmetrically S cross-ply laminated square plates with di!erent values of b/t, without an elastic foundation subjected to equal biaxial compression (a"1.0) alone are compared in Table 2 with existing results of Khdeir and Librescu [6], and Nair et al. [8]. The buckling loads for perfect, (0/90/0) symmetrically cross-ply laminated plates with or without elastic foundations and subjected to equal biaxial compression alone are compared in Table 3 with FSDPT results of Xiang et al. [10]. In addition, the postbuckling load}de#ection curves of single-ply orthotropic square plates without an elastic foundation and subjected to equal biaxial compression alone are compared in Fig. 2 with HSDPT results given by Bhimaraddi [20]. In Fig. 2 j "jH/j , where j is the initial buckling load of p x C C a perfect plate corresponding to CPT. These four comparisons reveal the good accuracy of the method presented. Fig. 3 gives the postbuckling load}de#ection curves of 4-ply ($45 ) antisymmetrically angle2T ply and (0/90) symmetrically cross-ply laminated square plates subjected to equal biaxial compresS sion combined with lateral pressure (q "5 MN/m2) and resting on either Pasternak-type or 0

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Winkler or softening nonlinear elastic foundations. The sti!nesses are (k , k )"(2.0, 1.0) for the 1 2 Pasternak-type elastic foundation, (k , k )"(2.0, 0.0) for the Winkler elastic foundation and 1 2 (k , k )"(2.0, 2.0) for the softening nonlinear elastic foundation. It can be seen that the foundation 1 3 sti!ness has a signi"cant e!ect on the postbuckling response of the plate. As expected, the results show that the initial de#ections for Winkler and nonlinear elastic foundations are identical, but that the postbuckling responses are quite di!erent. They also show that the postbuckling equilibrium path changes from stable to unstable as the nonlinear elastic foundation sti!ness k increases, 3 enabling the nature of the imperfection sensitivity to be predicted. Fig. 4 gives the postbuckling load}de#ection curves of initially pressurized ($45 ) square 2T plates with di!erent thickness ratio b/t ("20.0, 10.0) subjected to equal biaxial compression, when they are supported by a Pasternak-type elastic foundation. It can be seen that the initial de#ection of the plate with b/t"20.0 is larger and is considerably greater than that of the plate with b/t"10.0. It can also be seen that the plate thickness ratio b/t a!ects the initial postbuckling behavior of the plate signi"cantly, but it has a small e!ect when the de#ection is su$ciently large. Fig. 5 gives the postbuckling load}de#ection curves for the same plate under di!erent values of initial lateral pressure q ("0, 5, 10 MN/m2), when the plate is subjected to equal biaxial compres0 sion and resting on a Pasternak-type elastic foundation. As can be seen in the case of initially pressurized plates, the de#ections deviate greatly from those of a plate without any lateral pressure, and at higher postbuckling loads the net de#ection for initially pressurized plates is smaller than that for a plate without any lateral pressure. The load}de#ection curves for the initially pressurized plate is almost asymptotic to that for a plate without any lateral pressure. Fig. 6 shows the e!ect of plate aspect ratio b ("1.0, 1.5) on the postbuckling behavior of initially pressurized ($45 ) laminated plates under equal biaxial compression and resting on a 2T

Fig. 4. E!ect of plate thickness ratio b/t on the postbuckling of initially pressurized square plates under equal biaxial compression and resting on a Pasternak-type elastic foundation.

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Fig. 5. E!ect of initial lateral pressure on the postbuckling of a square plate under equal biaxial compression and resting on a Pasternak-type elastic foundation.

Fig. 6. E!ect of plate aspect ratio on the postbuckling of initially pressurized plates under equal biaxial compression and resting on a Pasternak-type elastic foundation.

Pasternak-type elastic foundation. Then Fig. 7 shows the e!ect of the total number of plies N ("4, 10) on the postbuckling response of initially pressurized antisymmetrically angle-ply laminated square plates under equal biaxial compression and resting on a Pasternak-type elastic

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Fig. 7. E!ect of total number of plies on the postbuckling of initially pressurized square plates under equal biaxial compression and resting on a Pasternak-type elastic foundation.

foundation. As expected, these results show that the postbuckling strength is increased by increasing the total number of plies N or by decreasing the plate aspect ratio b. Fig. 8 shows the e!ect of biaxial load ratio a ("0.0, 0.5, 1.0) on the postbuckling behavior of an initially pressurized ($45 ) laminated square plate under combined loading case and resting on 2T a Pasternak-type elastic foundation. It can be found that the postbuckling strength is decreased by increasing the parameter a and that the postbuckling equilibrium path becomes signi"cantly lower as the parameter a increases. Note that in Fig. 8 under the loading cases of uniaxial compression (a"0.0) and biaxial compression (a"0.5) the plate has buckling mode (m, n)"(2, 1), whereas under equal biaxial compression (a"1.0) the plate buckles with (m, n)"(1, 1). The postbuckling load}de#ection curves for imperfect shear deformable laminated plates have been plotted, along with the perfect plate results, in Figs. 3}8. It can be found that when lateral pressure is present, the plate will buckle at the onset of edge compression, i.e. deviations from #atness, and the shape of the load}de#ection curves of perfect plates appears similar to that for plates with initial geometrical imperfections. In Figs. 3, 4 and 6}8, the initial lateral pressure q "5 MN/m2; and in Figs. 3, 5}8, the plate thickness ratio b/t"10.0; and in Figs. 4}8, the elastic 0 foundation sti!ness is characterized by (k , k )"(2.0, 1.0). 1 2 Figs. 9}12 show the postbuckling results for initially pressurized shear deformable laminated plates analogous to the results of Figs. 5}8, but are for the case of softening nonlinear elastic foundation with (k , k )"(2.0, 2.0). Note that now the postbuckling equilibrium paths of the 1 3 plates are unstable. In Fig. 12 the plate has buckling mode (m, n)"(2, 1) only under the loading cases of uniaxial compression. Otherwise Figs. 9}12 lead to broadly the same conclusions as do Figs. 5}8.

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Fig. 8. E!ect of biaxial stress ratio on the postbuckling of initially pressurized square plates resting on a Pasternak-type elastic foundation.

Fig. 9. E!ect of initial lateral pressure on the postbuckling of a square plate under equal biaxial compression and resting on a nonlinear elastic foundation.

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Fig. 10. E!ect of plate aspect ratio on the postbuckling of initially pressurized plates under equal biaxial compression and resting on a nonlinear elastic foundation.

Fig. 11. E!ect of total number of plies on the postbuckling of initially pressurized square plates under equal biaxial compression and resting on a nonlinear elastic foundation.

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Fig. 12. E!ect of biaxial stress ratio on the postbuckling of initially pressurized square plates resting on a nonlinear elastic foundation.

5. Concluding remarks Postbuckling behavior of simply supported, shear deformable laminated plates subjected to biaxial compression combined with uniform lateral pressure and resting on an elastic foundation has been studied by a perturbation method. The postbuckling of such plates subjected to biaxial compression alone is treated as a limiting case. A parametric study of perfect and imperfect, anti-symmetrically angle-ply and symmetrically cross-ply laminated plates resting on either Pasternak-type or Winkler or softening nonlinear elastic foundations has been carried out. The results presented herein show that plates subjected to combined loading have substantially di!erent postbuckling behavior from that of plates without any lateral pressure. They also con"rm that the characteristics of postbuckling are signi"cantly in#uenced by foundation sti!ness, "ber orientation, the transverse shear deformation, the plate aspect ratio, the total number of plies, the amount of lateral pressure and an initial geometrical imperfection as well as the proportional combination of biaxial loads. Unlike the plate resting on a Winkler or Pasternak-type elastic foundation, which has a stable postbuckling equilibrium path, the initially pressurized shear deformable laminated plates resting on a softening nonlinear elastic foundation has an unstable postbuckling equilibrium con"guration. For such a case, the plate is an imperfection-sensitive structure that exhibits all of the interesting features of such structures.

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Acknowledgements This work was supported in part by the Science & Technology Foundation of Shanghai, China under Grant 98JC14032. The author is grateful for the "nancial support.

Appendix A In Eqs. (1)}(4)

C

D

4 L4 L4 L4 ¸ ( )" FH #(FH #FH #4FH ) #FH , 11 11 12 21 66 22 3t2 LX4 LX2L>2 L>4

C C

D

C

D

L3 L3 4 4 ¸ ( )" DH ! FH # (DH #2DH )! (FH #2FH ) , 12 11 3t2 11 LX3 12 66 66 LXL>2 3t2 12

D

C

D

L3 L3 4 4 ¸ ( )" (DH #2DH )! (FH #2FH ) # DH ! FH , 13 12 22 3t2 22 L>3 66 66 LX2L> 3t2 21 L4 L4 #(2BH !BH ) , ¸ ( )"(2BH !BH ) 14 16 26 62 LXL>3 61 LX3L> L4 L4 L4 #(2AH #AH ) #AH , ¸ ( )"AH 12 66 LX2L>2 11L>4 21 22 LX4

C C

D

C

D

4 4 L3 L3 # BH ! EH , ¸ ( )" (BH !BH )! (EH !EH ) 26 61 61 LX2L> 16 3t2 16 L>3 22 3t2 26

D

C

D

L3 L3 4 4 # (BH !BH )! (EH !EH ) , ¸ ( )" BH ! EH 23 26 3t2 26 LX3 16 62 62 LXL>2 3t2 16

C

D

4 L4 L4 ¸ ( )" (2EH !EH ) #(2EH !EH ) , 24 26 61 16 62 3t2 LX3L> LXL>3

C

D

L 8 16 ¸ ( )" A ! D # F 55 t2 55 t4 55 LX 31

C

A

B

D

4 4 L3 4 L3 # (FH ! HH ) # (FH #2FH )! (HH #2HH ) , 11 3t2 11 LX3 21 66 66 LXL>2 3t2 3t2 12

C

D C D

D

8 16 L2 8 16 ¸ ( )" A ! D # F ! DH ! FH # HH 55 t2 55 t4 55 32 11 3t2 11 9t4 11 LX2

C

8 16 L2 ! DH ! FH # HH , 66 3t2 66 9t4 66 L>2

C

D

L2 4 16 , ¸ ( )" (DH #DH )! (FH #FH #2FH )# (HH #HH ) 33 12 66 21 66 LXL> 66 3t2 12 9t4 12

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H.-S. Shen / International Journal of Mechanical Sciences 42 (2000) 1171}1195

¸ ( )"¸ ( ), 34 22

C

D

8 16 L ¸ ( )" A ! D # F 41 44 t2 44 t4 44 L> 4 # 3t2

CA

¸ ( )"¸ ( ), 42 33

C

B

D

4 4 L3 L3 (FH #2FH )! (HH #2HH ) #(FH ! HH ) , 12 66 66 LX2L> 22 3t2 22 L>3 3t2 12

D C D

D

L2 8 16 8 16 ¸ ( )" A ! D # F ! DH ! FH # HH 43 44 t2 44 t4 44 66 3t2 66 9t4 66 LX2

C

L2 8 16 ! DH ! FH # HH , ¸ ( )"¸ ( ), 23 22 3t2 22 9t4 22 L>2 44 L2 L2 L2 L2 L2 L2 ¸( )" !2 # , LX2 L>2 LXL> LXL> L>2 LX2

(A.1)

In the above equations [AH], [BH], [DH], [EH], [FH] and [HH] (i, j"1, 2, 6) are reduced sti!ness ij ij ij ij ij ij matrices, de"ned as AH"A~1, BH"!A~1B, DH"D!BA~1B, EH"!A~1E, FH"F!EA~1B, HH"H!EA~1E,

(A.2)

where A , B etc., are the plate sti!nesses, de"ned by ij ij tk (QM ) (1, Z, Z2, Z3, Z4, Z6) dZ (i, j"1, 2, 6), (A , B , D , E , F , H )" + ij k ij ij ij ij ij ij k/1 tk~1 tk (A , D , F )" + (QM ) (1, Z2, Z4) dZ (i, j"4, 5), ij ij ij ij k k/1 tk~1 where QM are the transformed elastic constants, de"ned by ij QM c4 2c2s2 s4 4c2s2 11 QM c2s2 c4#s4 c2s2 !4c2s2 Q 12 11 QM s4 2c2s2 c4 4c2s2 Q 22 " 12 QM c3s cs3!c3s !cs3 !2cs(c2!s2) Q 16 22 QM Q cs3 c3s!cs3 !c3s 2cs(c2!s2) 26 66 QM c2s2 !2c2s2 c2s2 (c2!s2)2 66 and

P

P

CDC

C DC D QM c2 s2 44 QM " !cs cs 45 QM s2 c2 55

DC D

C D

Q 44 , Q 55

(A.3a) (A.3b)

(A.4a)

(A.4b)

H.-S. Shen / International Journal of Mechanical Sciences 42 (2000) 1171}1195

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where E E l E 11 22 Q " , Q " , Q " 21 11 , 11 (1!l l ) 22 (1!l l ) 12 (1!l l ) 12 21 12 21 12 21 Q "G , Q "G , Q "G 44 23 55 13 66 12

(A.4c)

and c"cos h, s"sin h,

(A.4d)

where h is the lamination angle with respect to the plate X-axis.

Appendix B In Eq. (22) 16 A(1)" w p6mnS 11 in which

(B.1)

S "# #K #K (m2#n2b2), 11 11 1 2 g g # "g #c c m2n2b2 05 07 11 08 14 24 g 06 and in Eq. (34) 1 (S , S , S ), (j(0), j(2), j(4))" 0 2 4 x x x 4b2c C 14 11 where 1 1 S # , S " C (C !C ), S " 11 , S " 2 16(1#k) 2 4 256 11 24 44 0 (1#k)

A

B

m4 n4b4 # "c c # (1#k) (1#2k)!9K , 2 14 24 c 3 c 6 7 4n2b2 4m2 , c "c2 #c c c2 , c "1#c c c2 7 24 14 24 223 c #c 4n2b2 6 14 24 230 c #c 4m2 31 322 41 322 C "(m2#an2b2), C "(m2#9an2b2), C "(9m2#an2b2), 11 13 31 b K b 13 # 31! 3 , C "2# 24 2 J J J 31 33 13 b d 3K2 b d C " 13 13# 31 31# 3 , 44 J J J 31 33 13

A

A

B

B

(B.2)

(B.3)

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m4 b "c c (1#k) (1#2k) !3K , 13 14 24 3 c 7 n4b4 !3K , b "c c (1#k) (1#2k) 3 31 14 24 c 6 m4 d "c c [2(1#k)2#(1#2k)] !9K , 3 13 14 24 c 7 n4b4 !9K , d "c c [2(1#k)2#(1#2k)] 3 31 14 24 c 6 J "S C (1#k)!S C , 13 13 11 11 13 J "S C (1#k)!S C , 31 31 11 11 31 J "S C (1#k)!9S C , 33 33 11 11 11 S "# #K #K (m2#9n2b2), 13 13 1 2 S "# #K #K (9m2#n2b2), 31 31 1 2 S "81# #K , 33 11 1 g g # "g #c c 9m2n2b2 135 137, 13 138 14 24 g 136 g g # "g #c c 9m2n2b2 315 317, 31 318 14 24 g 316 g "(c #c m2#c n2b2) (c #c m2#c n2b2)!c2 m2n2b2, 00 31 320 322 41 430 432 331 g "(c #c m2#c n2b2) (c m2#c n2b2)!c n2b2(c m2#c n2b2), 01 31 320 322 230 232 331 221 223 g "(c #c m2#c n2b2) (c m2#c n2b2)!c m2(c m2#c n2b2), 02 41 430 432 221 223 331 230 232 g "(c #c m2#c n2b2) (c !c m2!c n2b2) 03 31 320 322 41 411 413 !c m2(c !c m2!c n2b2), 331 31 310 312 g "(c #c m2#c n2b2) (c !c m2!c n2b2) 04 41 430 432 31 310 312 !c n2b2(c !c m2!c n2b2), 331 41 411 413 (c m2#c n2b2)g #(c m2#c n2b2)g 232 03 221 223 04, g "(c m2#c n2b2)# 230 05 241 243 g 00 g "(m4#2c m2n2b2#c n4b4) 06 212 214 m2(c m2#c n2b2)g #n2b2(c m2#c n2b2)g 230 232 01 221 223 02, #c c 14 24 g 00

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1193

(c m2#c n2b2)g #(c m2#c n2b2)g 133 01 120 122 02, g "(c m2#c n2b2)! 131 07 141 143 g 00 g "(c m4#2c m2n2b2#c n4b4) 08 110 112 114 m2(c m2#c n2b2)g #n2b2(c m2#c n2b2)g 120 122 04 131 133 03, # g 00 g "(c #c m2#c 9n2b2) (c #c m2#c 9n2b2)!c2 9m2n2b2, 130 31 320 322 41 430 432 331 g "(c #c m2#c 9n2b2) (c m2#c 9n2b2) 131 31 320 322 230 232 !c 9n2b2(c m2#c 9n2b2), 331 221 223 g "(c #c m2#c 9n2b2) (c m2#c 9n2b2)!c m2(c m2#c 9n2b2), 132 41 430 432 221 223 331 230 232 g "(c #c m2#c 9n2b2) (c !c m2!c 9n2b2) 133 31 320 322 41 411 413 !c m2(c !c m2!c 9n2b2), 331 31 310 312 g "(c #c m2#c 9n2b2) (c !c m2!c 9n2b2) 134 41 430 432 31 310 312 !c 9n2b2(c !c m2!c 9n2b2), 331 41 411 413 (c m2#c 9n2b2)g #(c m2#c 9n2b2)g 232 133 221 223 134, g "(c m2#c 9n2b2)# 230 135 241 243 g 130 g "(m4#18c m2n2b2#c 81n4b4) 136 212 214 m2(c m2#c 9n2b2)g #9n2b2(c m2#c 9n2b2)g 230 232 131 221 223 132, #c c 14 24 g 130 (c m2#c 9n2b2)g #(c m2#c 9n2b2)g 133 131 120 122 132, g "(c m2#c 9n2b2)! 131 137 141 143 g 130 g "(c m4#18c m2n2b2#c 81n4b4) 138 110 112 114 m2(c m2#c 9n2b2)g #9n2b2(c m2#c 9n2b2)g 120 122 134 131 133 133, # g 130 g "(c #c 9m2#c n2b2) (c #c 9m2#c n2b2)!c2 9m2n2b2, 310 31 320 322 41 430 432 331 g "(c #c 9m2#c n2b2) (c 9m2#c n2b2)!c n2b2(c 9m2#c n2b2), 311 31 320 322 230 232 331 221 223 g "(c #c 9m2#c n2b2) (c 9m2#c n2b2)!c 9m2(c 9m2#c n2b2), 312 41 430 432 221 223 331 230 232 g "(c #c 9m2#c n2b2) (c !c 9m2!c n2b2) 313 31 320 322 41 411 413 !c 9m2(c !c 9m2!c n2b2), 331 31 310 312 g "(c #c 9m2#c n2b2) (c !c 9m2!c n2b2) 314 41 430 432 31 310 312 !c n2b2(c !c 9m2!c n2b2), 331 41 411 413

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(c 9m2#c n2b2)g #(c 9m2#c n2b2)g 232 313 221 223 314, "(c 9m2#c n2b2)# 230 315 241 243 g 310 g "(81m4#18c m2n2b2#c n4b4) 316 212 214 9m2(c 9m2#c n2b2)g #n2b2(c 9m2#c n2b2)g 230 232 311 221 223 312, #c c 14 24 g 310 (c 9m2#c n2b2)g #(c 9m2#c n2b2)g 133 311 120 122 312, g "(c 9m2#c n2b2)! 131 317 141 143 g 310 g "(81c m4#18c m2n2b2#c n4b4) 318 110 112 114 9m2(c 9m2#c n2b2)g #n2b2(c 9m2#c n2b2)g 120 122 314 131 133 313. # g 310 g

References [1] Noor AK. Stability of multilayered composite plates. Fibre Sciences and Technology 1975;8:81}9. [2] Reddy JN, Phan ND. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. Journal of Sound and Vibration 1985;98:157}70. [3] Putcha NS, Reddy JN. Stability and natural vibration analysis of laminated plates by using a mixed element based on a re"ned plate theory. Journal of Sound Vibration 1986;104:285}300. [4] Owen DRJ, Li ZH. A re"ned analysis of laminated plates by "nite element displacement methods * II: vibration and stability. Computers and Structures 1987;26:915}23. [5] Senthilnathan NR, Lim SP, Lee KH, Chow ST. Buckling of shear-deformable plates. AIAA Journal 1987;25:1268}71. [6] Khdeir AA, Librescu L. Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory: part II * buckling and free vibration. Composite Structures 1988;9:259}77. [7] Singh G, Sadasiva Rao YVK. Stability of thick angle-ply composite plates. Computers and Structures 1989;29:317}22. [8] Nair S, Singh G, Rao GV. Stability of laminated composite plates subjected to various type of in-plane loading. International Journal of Mechanical Sciences 1996;38:191}202. [9] Wang WJ, Tseng YP, Lin KJ. Stability of laminated plates using "nite strip method based on a higher-order plate theory. Composite Structures 1996;34:65}76. [10] Xiang Y, Kitipornchai S, Liew KM. Buckling and vibration of thick laminates on Pasternak foundations. ASCE Journal of Engineering Mechanics 1996;122:54}63. [11] Tarn JQ. Elastic buckling of multilayered anisotropic plates. Journal of the Mechanics and Physics of Solids 1996;44:389}411. [12] Chen LW, Doong JL. Postbuckling behavior of a thick plate. AIAA Journal 1983;21:1157}61. [13] Librescu L, Stein M. Postbuckling of shear deformable composite #at panels taking into account geometrical imperfections. AIAA Journal 1992;30:1352}60. [14] Minguet PJ, Dugundji J, Lagace P. Postbuckling behavior of laminated plates using a direct energy-minimization technique. AIAA Journal 1989;27:1785}92. [15] Carrera E, Villani M. Large de#ection and stability FEM analysis of shear deformable compressed anisotropic #at panels. Composite Structures 1994;29:433}44. [16] Carrera E, Villani M. E!ects of boundary conditions on postbuckling of compressed symmetrically laminated thick plates. AIAA Journal 1995;33:1543}6. [17] Noor AK, Peters JM. Finite element buckling and postbuckling solutions for multilayered composite panels. Finite Elements in Analysis and Design 1994;15:343}67.

H.-S. Shen / International Journal of Mechanical Sciences 42 (2000) 1171}1195

1195

[18] Sundaresan P, Singh G, Rao GV. Buckling and postbuckling analysis of moderately thick laminated rectangular plates. Computers and Structures 1996;61:79}86. [19] Librescu L, Stein M. A geometrically nonlinear theory of transversely isotropic laminated composite plates and its use in the post-buckling analysis. Thin-Walled Structures 1991;11:177}201. [20] Bhimaraddi A. Buckling and post-buckling behavior of laminated plates using the generalized nonlinear formulation. International Journal of Mechanical Sciences 1992;34:703}15. [21] Naidu AR, Raju KK, Rao GV. Post-buckling of a square plate resting on elastic foundation under biaxial compression. Computers and Structures 1990;37:343}5. [22] Jayachandran SA, Vaidyanathan CV. Post critical behaviour of biaxially compressed plates on elastic foundation. Computers and Structures 1995;54:239}46. [23] Shen HS. Postbuckling analysis of composite laminated plates on two-parameter elastic foundations. International Journal of Mechanical Sciences 1995;37:1307}16. [24] Shen HS, Williams FW. Biaxial buckling and post-buckling of composite laminated plates on two-parameter elastic foundations. Computers and Structures 1997;63:1177}85. [25] Shen HS, Williams FW. Postbuckling analysis of imperfect composite laminated plates on non-linear elastic foundations. International Journal of Non-Linear Mechanics 1995;30:651}9. [26] Levy S, Goldenberg D, Zibritosky G. Simply supported long rectangular plate under combined axial load and normal pressure. NACA Tech. Note No. 949, Washington, DC, 1944. [27] Yosiki M, Yamamoto Y, Kondo H. Buckling of plates subjected to edge thrusts and lateral pressure. Journal of Society of Naval Architects of Japan 1965;118:249}58. [28] Yamamoto Y, Matsubara N, Murakami T. Buckling of plates subjected to edge thrusts and lateral pressure. Journal of Society of Naval Architects of Japan 1970;127:171}9. [29] Shen HS. Postbuckling behaviour of rectangular plates under combined loading. Thin-Walled Structures 1989;8:203}16. [30] Ye JQ. Postbuckling analysis of plates under combined loads by a mixed "nite-element and boundary-element method. ASME Journal of Pressure Vessel and Technology 1993;115:262}7. [31] Shen HS. Postbuckling of composite laminated plates under biaxial compression combined with lateral pressure and resting on elastic foundations. Journal of Strain Analysis for Engineering Design 1998;33:253}61. [32] Librescu L, Lin W. Postbuckling and vibration of shear deformable #at and curved panels on a non-linear elastic foundation. International Journal of Non-Linear Mechanics 1997;32:211}25. [33] Leissa AW. Conditions for laminated plates to remain #at under inplane loading. Composite Structures 1986;6:261}70. [34] Qatu MS, Leissa AW. Buckling or transverse deformation of unsymmetrically laminated plates subjected to in-plane loads. AIAA Journal 1993;31:189}94. [35] Reddy JN. A re"ned nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984;20:881}96. [36] Amazigo JC, Budiansky B, Carrier GF. Asymptotic analysis of the buckling of imperfect columns on nonlinear elastic foundation. International Journal of Solids and Structures 1975;6:1341}56.