- Email: [email protected]

Postbuckling of shear deformable cross-ply laminated cylindrical shells under combined external pressure and axial compression Hui-Shen Shen School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, People’s Republic of China Received 8 August 2000; received in revised form 10 April 2001

Abstract A postbuckling analysis is presented for a shear deformable cross-ply laminated cylindrical shell of ,nite length subjected to combined loading of external pressure and axial compression. The governing equations are based on Reddy’s higher order shear deformation shell theory with von K4arm4an–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the e7ects of nonlinear prebuckling deformations, large de8ections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under combined loading cases. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unsti7ened or sti7ened, moderately thick, antisymmetric and symmetric cross-ply laminated cylindrical shells for di7erent values of load-proportional parameters. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Postbuckling; Composite laminated cylindrical shell; Higher order shear deformable shell theory; Boundary layer theory of shell buckling; Singular perturbation technique

1. Introduction In the recent years, ,ber-reinforced composite laminated shell structures have been widely used in the aerospace, marine, automobile and other engineering industries. There has been considerable interest in the structural instability of relatively thick composite shells. E-mail address: [email protected] (H.-S. Shen). 0020-7403/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 1 ) 0 0 0 5 8 - 3

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

Nomenclature Aij ; Bij ; Dij d1 ; d2 E11 ; E22 e1 ; e2 Eij ; Fij ; Hij B F F; G12 ; G13 ; G23 L ns ; nr NB i ; MB i ; PB i

extensional, bending–extension coupling and bending sti7ness, see Eq. (A:4) distance between centers of sti7eners elastic moduli of a single ply sti7ener eccentricity higher order sti7ness, see Eq. (A:4) stress function and its dimensionless form, see Eq. (8) shear modulus of a single ply length of a shell number of stringer and ring sti7eners in-plane stress resultants, stress couples and higher order stress couples, N tk+1 = i (1; Z; Z 2 ) dZ, as de,ned in Ref. [24] tk

R t UB ; VB WB ; W ZB x ; "p ; "q % &∗ &p ; &p∗ &q ; &q∗ ' (12 ; (21 )B x ; )B y )x ; )y

mean radius of a shell thickness of a shell displacement components in the X and Y directions de8ection of shell and its dimensionless form, see Eq. (8) geometric parameter of shell, = L2 =Rt end-shortening and its two dimensionless forms, see Eq. (8) a small perturbation parameter, see Eq. (8) imperfection sensitivity parameter dimensionless forms of axial compressive load, see Eq. (8) dimensionless forms of external pressure, see Eq. (8) imperfection parameter, see beneath Eq. (21) Poisson’s ratios of a single ply rotations of the normals about the X - and Y -axis dimensionless forms of )B x and )B y , see Eq. (8)

k=1

Many linear buckling studies have been made for moderately thick laminated cylindrical shells or thick cylinders under pure axial compression, uniform external pressure or their combinations, see, for example, Refs. [1–11]. In these analyses only perfectly initial con,gurations were assumed. However, investigations involving the application of the shear deformation shell theory to the postbuckling analysis are limited in number. Iu and Chia [12] developed a ,rst order shear deformation theory to study nonlinear vibration and postbuckling of imperfect, moderately thick, cross-ply laminated cylindrical shells. However, in the ,rst order shear deformation theory the conditions of zero shear stress on the top and bottom surfaces of the shell are not met, and this requires a shear correction to the transverse shear sti7nesses. Simitses and Anastasiadis [13] developed a higher order shear deformation theory, which included both geometric nonlinearity and initial geometric imperfections, but their numerical results were only for linear buckling

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

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loads of perfect, moderately thick, symmetrically laminated cylindrical shells. Reddy and Savoia [14] used a layer-wise shell theory to study the postbuckling response of imperfect laminated cylindrical shells, which can produce much more accurate results but the boundary conditions cannot be imposed accurately in their solutions. Recently, Eslami et al. [15] and Eslami and Shariyat [16] developed, respectively, a layer-wise and a higher order shear deformation theory to study dynamic buckling and postbuckling of thick laminated cylindrical shells and their solution was sought, in both papers [15,16], on the basis of numerical methods. It is noted that in most of the foregoing studies, a membrane prebuckling state is assumed. To the best of the author’s knowledge, there is no literature covering the postbuckling response of shear deformable laminated cylindrical shells subjected to combined loading of external pressure and axial compression. This is the problem studied in the present paper. It has been shown in Shen and Chen [17,18] that in shell buckling, there is a boundary layer phenomenon where prebuckling and buckling displacement vary rapidly. They suggested a boundary layer theory of shell buckling, which includes the e7ects of nonlinear prebuckling deformations, large de8ections in the postbuckling range, and initial geometric imperfections of the shell. Based on this theory, postbuckling analyses for perfect and imperfect, unsti7ened and sti7ened, isotropic and multilayered cylindrical shells under various loading cases have been performed by Shen and Chen [19], Shen et al. [20], and Shen [21–23]. It should be noted that in the above studies the shells are considered as being relatively thin and therefore the transverse shear deformation is usually not accounted for. The present study extends the previous work to the case of shear deformable laminated cylindrical shells under combined loading cases. The shell may be reinforced by sti7eners and the “smeared sti7ener” approach is adopted for the beam sti7eners. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account but, for simplicity, the form of initial geometric imperfection is assumed to be the same as the initial buckling mode of the shell. 2. Theoretical development Consider a circular cylindrical shell with mean radius R, length L and thickness t, which consists of N plies, subjected to two loads combined out of a uniform external pressure q and axial load P0 . The shell is referred to a coordinate system (X; Y; Z), in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward B The normal to the middle surface, the corresponding displacement designated by UB , VB and W. origin of the coordinate system is located at the end of the shell. The shell is assumed to be relatively thick, geometrically imperfect and may have stringer and=or ring sti7eners. Letting nr be the number of ring sti7eners, then L = (nr + 1)d2 . The nS stringer sti7eners have spacing d1 and eccentricity e1 . Both d2 and e2 are de,ned analogously to d1 and e1 , but are for the ring sti7eners. Generally, nS ; nr ; d1 ; d2 ; e1 and e2 can be di7erent. Denoting the initial de8ection ∗ B Y ) be the stress function for by WB (X; Y ), let WB (X; Y ) be the additional de8ection and F(X; the stress resultants de,ned by NB x = F;B yy ; NB y = F;B xx and NB xy = − F;B xy , where a comma denotes partial di7erentiation with respect to the corresponding coordinates.

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Attention is focused on the case of moderately thick cross-ply laminated cylindrical shells, from which solution for moderately thick isotropic or orthotropic cylindrical shells follows as a limiting case. Reddy and Liu [24] developed a simple higher order shear deformation shell theory, in which the transverse shear strains are assumed to be parabolically distributed across the shell thickness and which contains the same dependent unknowns as in the ,rst order shear deformation theory. Based on Reddy’s higher order shear deformation theory, and using von K4arm4an–Donnell-type kinematic relations, governing di7erential equations are derived and can be expressed in terms B two rotations )B x and )B y , and transverse displacement WB , along with of a stress function F, ∗ initial geometric imperfection WB . They are B − 1 F;B xx = L( ˜ WB + WB ∗ ; F) B + q; L˜ 11 (WB ) − L˜ 12 ()B x ) − L˜ 13 ()B y ) + L˜ 14 (F) (1) R B + L˜ 22 ()B x ) + L˜ 23 ()B y ) − L˜ 24 (WB ) + 1 W; B xx = − 1 L( ˜ WB + 2WB ∗ ; WB ); L˜ 21 (F) (2) R 2 B = 0; L˜ 31 (WB ) + L˜ 32 ()B x ) − L˜ 33 ()B y ) + L˜ 34 (F)

(3)

B = 0; L˜ 41 (WB ) − L˜ 42 ()B x ) + L˜ 43 ()B y ) + L˜ 44 (F)

(4)

˜ ) are de,ned in Appendix A. where linear operators L˜ ij ( ) and nonlinear operator L( The two end edges of the shell are assumed to be simply supported or clamped, so that the boundary conditions are X = 0; L: WB = )B y = 0;

MB x = PB x = 0

WB = )B x = )B y = 0 0

2,R

(simply supported);

(5a)

(clamped);

(5b)

NB x dY + 2,Rtx + ,R2 qa = 0;

(5c)

where a = 0 and 1 for lateral and hydrostatic pressure loading case, respectively, and x is the average axial compressive stress, MB x is the bending moment and PB x is higher order moment, as de,ned in Ref. [24]. Also we have the closed (or periodicity) condition 2,R B @V dY = 0 (6a) @Y 0 or 2,R 2 B 2 B 4 ∗ @)B x 4 ∗ @)B y ∗ @ F ∗ @ F ∗ ∗ A22 2 + A12 2 + B21 − 2 E21 + B22 − 2 E22 @X @Y 3t @X 3t @Y 0 4 − 2 3t

2 B 2 B ∗ @ W ∗ @ W E21 + E 22 @X 2 @Y 2

WB 1 + − R 2

@WB @Y

2

@WB @WB − @Y @Y

∗

dY = 0:

(6b)

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Because of Eq. (6), the in-plane boundary condition VB = 0 (at X = 0; L) is not needed in Eq. (5). The average end-shortening relationship is 2,R L B 1 @U x =− dX dY L 2,RL 0 0 @X 2,R L 2 B 2 B 1 4 ∗ @)B x ∗ @ F ∗ @ F ∗ A11 2 + A12 2 + B11 − 2 E11 =− 2,RL 0 @Y @X 3t @X 0

4 ∗ ∗ + B12 − 2 E12 3t 1 − 2

@WB @X

2

@)B y 4 − 2 @Y 3t

@WB @WB − @X @X

∗

2 B 2 B ∗ @ W ∗ @ W E11 + E 12 @X 2 @Y 2

dX dY:

(7)

Eqs. (1)–(7) are the governing equations describing the required large de8ection response of the shear deformable cross-ply laminated cylindrical shell. 3. Analytical method and asymptotic solutions Having developed the theory, we now try to solve Eqs. (1) – (4) with boundary conditions (5). Before proceeding, it is convenient ,rst to de,ne the following dimensionless quantities [with /ijk in Eqs. (13), (15) and (16) below are de,ned in Appendix B] x = ,X=L;

y = Y=R;

0 = L=,R;

ZB = L2 =Rt;

∗ ∗ ∗ ∗ ∗ 1=4 D22 A11 A22 ] ; (W; W ∗ ) = %(WB ; WB )=[D11

∗ ∗ ∗ ∗ 1=4 % = (,2 R=L2 )[D11 D22 A11 A22 ] ;

∗ ∗ 1=2 B F = %2 F=[D 11 D22 ] ;

∗ ∗ ∗ ∗ 1=4 D22 A11 A22 ] ; ()x ; )y ) = %2 ()B x ; )B y )(L=,)=[D11 ∗ ∗ 1=2 =D11 ] ; /14 = [D22

/24 = [A∗11 =A∗22 ]1=2 ;

/5 = − A∗12 =A∗22 ;

∗ (/31 ; /41 ) = (L2 =,2 )(A55 − 8D55 =t 2 + 16F55 =t 4 ; A44 − 8D44 =t 2 + 16F44 =t 4 )=D11 ; ∗ ∗ ∗ ∗ ∗ 1=4 [D11 D22 A11 A22 ] ; (Mx ; Px ) = %2 (MB x ; 4PB x =3t 2 )L2 =,2 D11 ∗ ∗ D22 =A∗11 A∗22 ]1=4 ; &p = x =(2=Rt)[D11

"p = (

&p∗ = x (R=t)[3(1 − (12 (21 )=E11 E22 ]1=2 ;

"∗p = (

∗ ∗ 3=8 &q = q(3)3=4 LR3=2 [A∗11 A∗22 ]1=8 =4,[D11 D22 ] ;

"q = (

∗ ∗ ∗ ∗ 1=4 x =L)=(2=R)[D11 D22 A11 A22 ] ;

3=4 1=2 ∗ ∗ ∗ ∗ 3=8 x =L)(3) LR =4,[D11 D22 A11 A22 ] ;

x =L)(R=t)[3(1

− (12 (21 )]1=2 ;

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

&q∗ = q(3)3=2 LR3=2 (1 − (12 (21 )3=4 =,t 5=2 [2E11 E22 ]1=2 ; "∗q = ( x =L)(3)3=2 LR1=2 (1 − (12 (21 )3=4 =,t 3=2 (2)1=2 : The nonlinear equations (1) – (4) may then be written in dimensionless form as

(8)

%2 L11 (W ) − %L12 ()x ) − %L13 ()y ) + %/14 L14 (F) − /14 F;xx = /14 02 L(W + W ∗ ; F) + /14 43 (3)1=4 &q %3=2 ;

(9)

L21 (F) + /24 L22 ()x ) + /24 L23 ()y ) − %/24 L24 (W ) + /24 W;xx = − 12 /24 02 L(W + 2W ∗ ; W );

(10)

%L31 (W ) + L32 ()x ) − L33 ()y ) + /14 L34 (F) = 0;

(11)

%L41 (W ) − L42 ()x ) + L43 ()y ) + /14 L44 (F) = 0;

(12)

where L11 ( ) = /110

4 @4 @4 2 4 @ + 2/ 0 + / 0 ; 112 114 @x4 @x2 @y2 @y4

L12 ( ) = /120

3 @3 2 @ + / 0 ; 122 @x3 @x @y2

L13 ( ) = /131 0 L14 ( ) = /140 L21 ( ) =

3 @3 3 @ 0 ; + / 133 @x2 @y @y3

4 @4 @4 2 4 @ + 2/ 0 + / 0 ; 142 144 @x4 @x2 @y2 @y4

4 @4 @4 2 4 @ + 2/ 0 + / 0 ; 212 214 @x4 @x2 @y2 @y4

L22 ( ) = /220

3 @3 2 @ + / 0 ; 222 @x3 @x @y2

L23 ( ) = /231 0 L24 ( ) = /240 L31 ( ) = /31

3 @3 3 @ 0 ; + / 233 @x2 @y @y3

4 @4 @4 2 4 @ + 2/ 0 + / 0 ; 242 244 @x4 @x2 @y2 @y4

@ @3 @3 ; + /310 3 + /312 02 @x @x @x @y2

L32 ( ) = /31 − /320

2 @2 2 @ − / 0 ; 322 @x2 @y2

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

L33 ( ) = /331 0

2499

@2 ; @x @y

L34 ( ) = L22 ( ); L41 ( ) = /41 0

@ @3 @3 + /411 0 2 + /413 03 3 ; @y @x @y @y

L42 ( ) = L33 ( ); L43 ( ) = /41 − /430

2 @2 2 @ − / 0 ; 432 @x2 @y2

L44 ( ) = L23 ( ); L( ) =

@2 @2 @2 @2 @2 @2 − 2 : + @x2 @y2 @x @y @x @y @y2 @x2

(13)

Because of the de,nition of % given in Eq. (8), for most of the composite materials ∗ D∗ A∗ A∗ ]1=4 = (0:24 − 0:3)t, hence when ZB = (L2 =Rt) ¿ 2:96, we have % ¡ 1. Specially, [D11 22 11 22 √ for isotropic cylindrical shells, we have % = ,2 = ZB B 12, where ZB B = (L2 =Rt)[1 − (2 ]1=2 is the Batdorf shell parameter, which should be greater than 2.85 in the case of classical linear buckling analysis (see Ref. [25]). In practice, the shell structure will have ZB ¿10, so that we always have %1. When %¡1, then Eqs. (9) – (12) are the equations of the boundary layer type, from which nonlinear prebuckling deformations, large de8ections in the postbuckling range, and initial geometric imperfections of the shell can be considered simultaneously. The boundary conditions of Eq. (5) become x = 0; ,: W = )y = 0;

Mx = Px = 0

W = )x = )y = 0 1 2,

0

2,

02

(simply supported);

(clamped);

(14a) (14b)

2 @2 F dy + 2&p % + (3)1=4 &q %3=2 a = 0 2 @y 3

(14c)

and the closed condition becomes 2, 2 2 2 @)y @F @)x @2 W [email protected] F [email protected] W − %/24 /240 2 + /622 0 − /5 0 + /24 /220 + /522 0 @y @x2 @y2 @x @x @y2 0 1 + /24 W − /24 02 2

@W @y

2

@W ∗ − /24 0 dy = 0: @y @y 2 @W

(15)

It has been shown [17,18] that the e7ect of the boundary layer on the solution of a pressurized shell is of the order %3=2 and on the solution of a shell in compression is of the order %1 , hence

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

the unit end-shortening relationship may be written in two dimensionless forms as 2, , @)y (3)3=4 @2 F @2 F @)x "q = − 2 %−3=2 /224 02 2 − /5 2 + /24 /511 + /233 0 8, /24 @y @x @x @y 0 0 2 @2 W 1 @W 2 @W @W ∗ [email protected] W − /24 − %/24 /611 2 + /244 0 − /24 d x dy; @x @y2 2 @x @x @x

2,

,

@2 F /224 02 2

@2 F − /5 2 @y @x

@)y @)x + /24 /511 + /233 0 @x @y 0 0 2 @2 W 1 @W 2 @W @W ∗ [email protected] W − /24 − /24 d x dy: − %/24 /611 2 + /244 0 @x @y2 2 @x @x @x

1 "p = − 2 %−1 4, /24

(16a)

(16b)

Applying Eqs. (9)–(16), the postbuckling behavior of perfect and imperfect, shear deformable cross-ply laminated cylindrical shells subjected to combined loading of external pressure and axial compression is determined by a singular perturbation technique. The essence of this procedure, in the present case, is to assume that W = w(x; y; %) + W˜ (x; 4; y; %) + Wˆ (x; &; y; %); ˜ 4; y; %) + F(x; ˆ &; y; %); F = f(x; y; %) + F(x; )x =

x (x; y; %)

+ )˜ x (x; 4; y; %) + )ˆ x (x; &; y; %);

)y =

y (x; y; %)

+ )˜ y (x; 4; y; %) + )ˆ y (x; &; y; %);

(17)

where % is a small perturbation parameter [see Eq. (13)] and w(x; y; %); f(x; y; %); x (x; y; %); ˜ (x; 4; y; %), F(x; ˜ 4; y; %); y (x; y; %) are called outer solutions or regular solutions of the shell, W ˆ &; y; %); )ˆ x (x; &; y; %), )ˆ y (x; &; y; %) are the bound)˜ x (x; 4; y; %); )˜ y (x; 4; y; %) and Wˆ (x; &; y; %); F(x; ary layer solutions near the x = 0 and , edges, respectively, and 4 and & are the boundary layer variables, de,ned as √ 4 = x= %;

√ & = (, − x)= %:

(18)

(This means for isotropic cylindrical shells the width of the boundary layers is of the order √ Rt.) In Eq. (17), the regular and boundary layer solutions are taken in the form of perturbation expansions as w(x; y; %) = % j=2 wj=2 (x; y); f(x; y; %) = % j=2 fj=2 (x; y); j=1 x (x; y; %) =

j=1

j=0

%

j=2

( x )j=2 (x; y);

y (x; y; %) =

j=1

% j=2 (

y )j=2 (x; y);

(19a)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

W˜ (x; 4; y; %) =

% j=2+1 W˜ j=2+1 (x; 4; y);

˜ 4; y; %) = F(x;

j=0

)˜ x (x; 4; y; %) =

2501

% j=2+2 F˜ j=2+2 (x; 4; y);

j=0

%( j+3)=2 ()˜ x )( j+3)=2 (x; 4; y);

)˜ y (x; 4; y; %) =

j=0

% j=2+2 ()˜ y )j=2+2 (x; 4; y);

j=0

(19b) Wˆ (x; &; y; %) =

% j=2+1 Wˆ j=2+1 (x; &; y);

ˆ &; y; %) = F(x;

j=0

)ˆ x (x; &; y; %) =

% j=2+2 Fˆ j=2+2 (x; &; y);

j=0

%( j+3)=2 ()ˆ x )( j+3)=2 (x; &; y);

)ˆ y (x; &; y; %) =

j=0

%j=2+2 ()ˆ y )j=2+2 (x; &; y):

j=0

(19c) The initial buckling mode is assumed to have the form w2 (x; y) = A(2) 11 sin mx sin ny

(20)

and the initial geometric imperfection is assumed to have a similar form W ∗ (x; y; %) = %2 a∗11 sin mx sin ny = %2 'A(2) 11 sin mx sin ny;

(21)

where ' = a∗11 =A(2) 11 is the imperfection parameter. Substituting Eqs. (17)–(19) into Eqs. (9) – (12), collecting the terms of the same order of %; three sets of perturbation equations are obtained for the regular and boundary layer solutions, respectively. As argued earlier, the e7ect of the boundary layer on the solution of a pressurized shell is of the order %3=2 and on the solution of a shell in compression is of the order %1 . Thus, in the case of combined loading, two kinds of loading conditions should be considered. Case (1) high values of external pressure combined with relatively low axial load. Let P0 = b1 ,R2 q

(22a)

or 2&p % 4 1=4 3=2 3 (3) &q %

=

b1 2

in this case, the boundary condition of Eq. (14c) becomes 2 1 2, 2 @2 F 0 dy + (3)1=4 &q %3=2 (a + b1 ) = 0: 2 2, 0 @y 3

(22b)

(23)

For convenience we replace (a + b1 ) with a1 in Eq. (25) below, by using Eqs. (20) and (21) to solve these perturbation equations of each order, and matching the regular solutions with the boundary layer solutions at each end of the shell, so that the asymptotic solutions satisfying the

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

clamped boundary conditions are constructed as

x x x (3=2) (3=2) (3=2) (3=2) 3=2 W =% A00 − A00 a01 cos : √ + a10 sin : √ exp −; √ % % % ,−x ,−x ,−x (3=2) (3=2) (3=2) √ √ √ + a10 sin : a01 cos : exp −; − A00 % % % 3 (3) + %2 [A(2) 11 sin mx sin ny] + % [A11 sin mx sin ny] (4) (4) 5 (24) + %4 [A(4) 00 + A20 sin 2mx + A02 cos 2ny] + O(% );

1 (0) 2 2 y2 1 (1) 2 2 y2 F = − B00 0 x + a1 + % − B00 0 x + a1 2 2 2 2

1 (2) 2 2 y2 (2) 2 + % − B00 0 x + a1 + B11 sin mx sin ny 2 2

x x x (3=2) (5=2) (5=2) 5=2 √ √ √ + b10 sin : +% A00 b01 cos : exp −; % % % ,−x ,−x ,−x (3=2) (5=2) (5=2) + A00 b01 cos : √ + b10 sin : √ exp −; √ % % %

1 (3) 2 2 y2 + %3 − B00 0 x + a1 2 2

1 (4) 2 2 y2 (4) (4) (4) 4 + % − B00 0 x + a1 + B11 sin mx sin ny + B20 cos 2mx + B02 cos 2ny 2 2

+ O(%5 );

x x x (1) (2) (2) 2 exp −; √ )x = % C11 cos mx sin ny + c01 cos : √ + c10 sin : √ % % % ,−x ,−x ,−x (2) (2) + c01 exp −; √ cos : √ + c10 sin : √ % % %

(3) (4) (4) + %3 C11 cos mx sin ny + %4 C11 cos mx sin ny + C20 sin 2mx

(25)

+ O(%5 );

(2) (3) sin mx cos ny + %3 D11 sin mx cos ny )y = %2 D11

(4) (4) 4 + % D11 sin mx cos ny + D02 sin 2ny + O(%5 ):

(26)

(27)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2503

Note that all of the coeQcients in Eqs. (24) – (27) are related and can be written as functions of A(2) 11 , but for the sake of brevity the detailed expressions are not shown, whereas ; and : are given in detail in Appendix C. Next, substituting Eqs. (24) – (27) into the boundary condition (23) and into Eq. (16a), the postbuckling equilibrium paths can be written as 2 2 &q = &q(0) + &q(2) (A(2) 11 % ) + · · ·

(28)

(2) (2) 2 2 "q = "(0) q + "q (A11 % ) + · · ·

(29)

and

2 in Eqs. (28) and (29), (A(2) 11 % ) is taken as the second perturbation parameter relating to the dimensionless maximum de8ection. If the maximum de8ection is assumed to be at the point (x; y) = (,=2m; ,=2n), then 2 2 A(2) 11 % = Wm − ?1 Wm + · · ·

and the dimensionless maximum de8ection of the shell is written as WB 1 t % ∗ ∗ ∗ ∗ 1=4 + ?2 : Wm = C3 [D11 t D22 A11 A22 ]

(30a)

(30b)

All symbols used in Eqs. (28)–(30) and Eqs. (37)–(39) below are also described in detail in Appendix C. Case (2) high values of axial compression combined with relatively low external pressure. Let ,R2 q = b2 P0

(31a)

or 4 1=4 3=2 3 (3) &q %

2&p %

= 2b2

in this case, the boundary condition of Eq. (14c) becomes 1 2, 2 @2 F 0 dy + 2&p %(1 + ab2 ) = 0: 2, 0 @y2

(31b)

(32)

Similarly, by taking a2 = 2b2 =(1 + ab2 ) and by using a singular perturbation procedure, the asymptotic solutions satisfying the clamped boundary conditions are obtained as

x x x (1) (1) (1) √ √ √ W = % A(1) + a − − A a cos : sin : exp ; 00 00 01 10 % % % ,−x ,−x ,−x (1) (1) (1) exp −; √ − A00 a01 cos : √ + a10 sin : √ % % %

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

(2) + % A(2) 11 sin mx sin ny + A02 cos 2ny 2

(2) − (A02 cos 2ny)

(2) − (A02 cos 2ny)

x a(1) 01 cos : √

+

%

,− a(1) 01 cos : √

x

%

x a(1) 10 sin : √

A(3) 02 cos 2ny

+

+ A(4) 13 sin mx sin 3ny

A(4) 04 cos 4ny

x exp −; √ % %

+

,− a(1) 10 sin : √

+ %3 A(3) 11 sin mx sin ny

%

x

,−x exp −; √ %

(4) (4) + %4 A(4) 00 + A20 sin 2mx + A02 cos 2ny

+

+ O(%5 );

(33)

1 (0) 1 (1) 2 2 2 2 2 2 F = − B00 (a2 0 x + y ) + % − B00 (a2 0 x + y )

2

2

1 (2) (2) + % − B00 (a2 02 x2 + y2 ) + B11 sin mx sin ny 2

2

x + exp −; √ % % % ,−x ,−x ,−x (1) (2) (2) + A00 b01 cos : √ + b10 sin : √ exp −; √ % % %

1 (3) (3) + %3 − B00 (a2 02 x2 + y2 ) + B02 cos 2ny 2 x x x (3) (3) (2) exp −; √ + (A02 cos 2ny) b01 cos : √ + b10 sin : √ % % % ,−x ,−x ,−x (2) (3) (3) + (A02 cos 2ny) b01 cos : √ + b10 sin : √ exp −; √ % % % 1 (4) (4) (4) (4) + %4 − B00 (a2 02 x2 + y2 ) + B11 sin mx sin ny + B20 cos 2mx + B02 cos 2ny 2

+ A(1) 00

x b(2) 01 cos : √

x b(2) 10 sin : √

(4) sin mx sin 3ny + O(%5 ); + B13

x x ,−x ,−x (1) (3=2) (1) (3=2) 3=2 A00 c10 sin : √ exp −; √ + A00 c10 sin : √ exp −; √ )x = % % % % %

(34)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

+% +

2

(1) C11 cos mx sin ny

+%

5=2

,− (5=2) (A(2) 02 cos 2ny)c10 sin : √

+%

3

(3) C11 cos mx sin ny

,−x exp −; √ % %

x (5=2) (A(2) 02 cos 2ny)c10 sin : √

x

x exp −; √ % %

2505

(4) (4) (4) + %4 C11 cos mx sin ny + C20 sin 2mx + C13 cos mx sin 3ny + O(%5 );

(2) sin mx cos ny )y = %2 D11

(2) − (A02 2n0 sin 2ny)

(2) − (A02 2n0 sin 2ny)

(3) (3) + %3 D11 sin mx cos ny + D02 sin 2ny

x d(3) 01 cos : √

%

(35)

+

,− d(3) 01 cos : √

x

%

x d(3) 10 sin : √

%

+

x exp −; √ %

,− d(3) 10 sin : √

%

x

,−x exp −; √ %

(4) (4) (4) + %4 [D11 sin mx cos ny + D02 sin 2ny + D13 sin mx cos 3ny] + O(%5 ):

(36)

Next, substituting Eqs. (33) – (36) into the boundary condition (32) and into Eq. (16b), the postbuckling equilibrium paths can be written as 1 (2) (2) &p = &(0) − &p(2) (A11 %)2 + &p(4) (A11 %)4 + · · · (37) 1 + ab2 p and 2 (4) (2) 4 "p = "p(0) + "p(2) (A(2) 11 %) + "p (A11 %) + · · ·

(38)

in Eqs. (37) and (38), similarly, (A(2) 11 %) is taken as the second perturbation parameter in this case, and we have 2 A(2) 11 % = Wm − ?3 Wm + · · ·

and the dimensionless maximum de8ection of the shell is written as WB t 1 Wm = + ?4 : ∗ D∗ A∗ A∗ ]1=4 t C3 [D11 22 11 22

(39a)

(39b)

Eqs. (28)–(30) and (37)–(39) can be employed to obtain numerical results for the postbuckling load-shortening or load-de8ection curves of moderately thick cross-ply laminated cylindrical shells subjected to combined loading of external pressure and axial compression. Buckling under external pressure alone and buckling under axial compression alone follow as two limiting

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

cases. By increasing b1 and b2 , respectively, the interaction curve of a laminated cylindrical shell under combined loading can be constructed with these two lines. Note that since b2 = 1=b1 , only one load-proportional parameter should be determined in advance. The initial buckling load ∗ of a perfect shell can readily be obtained numerically, by setting WB =t = 0 (or ' = 0), while taking WB =t = 0 (note that Wm = 0). In this case, the minimum buckling load is determined by considering Eq. (28) or Eq. (37) for various values of the buckling mode (m; n), which determine the number of half-waves in the X -direction and of full waves in the Y -direction. Note that because of Eqs. (24) and (33), the nonlinear prebuckling deformation of the shell has been shown. 4. Numerical results and comments A postbuckling analysis has been presented for shear deformable laminated cylindrical shells subjected to combined loading of external pressure and axial compression. Numerous examples were solved to illustrate their application to the performance of perfect and imperfect, unsti7ened or sti7ened, antisymmetric and symmetric cross-ply laminated cylindrical shells, where the outmost layer is the ,rst mentioned orientation. For these examples (except for Tables 1 and 2) all plies are of equal thickness and the material properties adopted are: E11 = 130 GPa, E22 = 7 GPa, G12 = G13 = 6 GPa, G23 = 4:2 GPa and (12 = 0:28. Typical results are presented in dimensionless graphical form in which &p∗ , "∗p , &q∗ and "∗q [see Eq. (8)] are used. It should be ∗ appreciated that in all of the ,gures WB =t and WB =t mean the dimensionless forms of, respectively, the maximum initial geometric imperfection and maximum additional de8ection of the shell. As part of the validation of the present method, the buckling loads for clamped, 6-ply (0=90=0)S and (90=0=90)S symmetric cross-ply laminated cylindrical shells under pure axial compression, pure lateral pressure and their combinations are compared in Table 1 with the results obtained by Anastasiadis et al. [11] based on a higher order shear deformation theory (HOSD) along with the ,rst order shear deformation theory (FOSD) and the classical laminate theory (CLT). The material properties adopted here are E11 = 206:844 GPa, E22 = 18:6159 GPa, G12 = G13 = 4:48162 GPa, G23 = 2:55107 GPa and (12 = 0:21. Note that in Ref. [11] the boundary condition was chosen to be completely ,xed and the solution was chosen in the form of a double in,nite trigonometric series as WB = m=0 n=2 (Wmn sin n(Y=R) + Wmn cos n(Y=R)) × (cos m,(X=L) − cos (m + 2),(X=L)). It can be observed that all the results presented (HSDT) are de,nitely lower than those of CLT solutions. It can also be seen that the present results agree well with those of HOSD solutions for the action of axial compression. In contrast, for the action of lateral pressure, the present results are lower than those of Anastasiadis et al. [11]. The di7erences between HSDT and HOSD solutions may be partly caused by di7erent forms of de8ection WB chosen by di7erent authors. In addition, the dimensionless buckling loads (qcr =E22 ) for simply supported, single-layer orthotropic cylindrical shells under combined loading case (1) are compared in Table 2 with 3-D solutions of Soldatos and Ye [7], for di7erent values of sti7ness ratio E11 =E22 and radius-tothickness ratio R=t shown. The computing data adopted here are G12 =E22 = G13 =E22 = 0:6, G23 =E22 l = 0:5, (12 = 0:25 and L=R = 5. In Table 2, b1 = 0 indicates the loading case of pure uniform

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2507

Table 1 Comparisons of buckling loads (Nxcr ; qcr ) for perfect symmetric cross-ply laminated cylindrical shells under combined axial compression and lateral pressure (E11 = 206:844 GPa; E22 = 18:6159 GPa, G12 = G13 = 4:48162 GPa, G23 = 2:55107 GPa, (12 = 0:21, L=R = 5 and R = 0:1905 m) Lay-up (0=90=0)S

R=t 60

30

15

(90=0=90)S

60

30

15

a

Present HSDT

Anastasiadis et al. [11] HOSD FOSD

CLT

(0.940, 0.0) (0.626, 0.299) (0.313, 0.361) (0.0, 0.420) (3.717, 0.0) (2.457, 1.517) (1.223, 1.890) (0.0, 2.235) (14.604, 0.0) (9.741, 7.303) (4.615, 9.692) (0.0, 11.66)

(0.944, 0.0) (0.629, 0.427) (0.314, 0.483) (0.0, 0.517) (3.732, 0.0) (2.488, 1.965) (1.244, 2.413) (0.0, 2.758) (14.54, 0.0) (9.690, 11.31) (4.845, 14.06) (0.0, 15.58)

(0.946, 0.0) (0.630, 0.427) (0.315, 0.483) (0.0, 0.517) (3.748, 0.0) (2.498, 1.979) (1.249, 2.413) (0.0, 2.758) (14.69, 0.0) (9.795, 11.38) (4.898, 14.48) (0.0, 15.99)

(0.953, 0.0) (0.635, 0.434) (0.318, 0.483) (0.0, 0.517) (3.844, 0.0) (2.563, 2.020) (1.281, 2.441) (0.0, 2.827) (15.64, 0.0) (10.43, 11.38) (5.213, 15.86) (0.0, 32.60)

(0.934, 0.0) (0.615, 0.547) (0.305, 0.629) (0.0, 0.707) (3.673, 0.0) (2.380, 2.840) (1.201, 3.316) (0.0, 3.772) (14.163, 0.0) (9.135, 14.98) (4.617, 17.32) (0.0, 19.71)

(0.932, 0.0) (0.621, 0.745) (0.310, 0.841) (0.0, 0.931) (3.600, 0.0) (2.403, 4.240) (1.200, 4.620) (0.0, 4.960) (13.67, 0.0) (9.110, 18.48) (4.560, 22.90) (0.0, 26.88)

(0.932, 0.0) (0.621, 0.751) (0.310, 0.841) (0.0, 0.931) (3.630, 0.0) (2.420, 4.240) (1.210, 4.650) (0.0, 5.030) (13.93, 0.0) (9.290, 18.55) (4.640, 23.10) (0.0, 27.20)

(0.946, 0.0) (0.630, 0.752) (0.315, 0.848) (0.0, 0.931) (3.767, 0.0) (2.510, 4.410) (1.256, 4.860) (0.0, 5.240) (15.10, 0.0) (10.10, 19.20) (5.030, 24.30) (0.0, 29.30)

a

In N=m × 10−6 and Pa × 10−6 .

lateral pressure. It can be seen that the present results are in good agreement with 3-D solutions of Soldatos and Ye [7] for moderately thick cylindrical shells. In contrast, for very thick cylinders (R=t = 5) the present results are higher than those of Soldatos and Ye [7] and discrepancies between HSDT and 3-D solutions become quite remarkable. Fig. 1 shows the interaction between &p∗ and &q∗ for 4-ply (0=90)S and (0=90)2T cross-ply laminated cylindrical shells with R=t = 30 and 15 under combined loading cases. The results calculated show that for the (0=90)S shell with R=t = 30 buckling occurs due to axial compression alone with a buckling mode (m; n) = (3; 4) while due to lateral pressure alone the buckling mode (m; n) = (1; 4). In contrast, for the (0=90)2T shell with R=t = 30 buckling occurs due to axial compression alone with a buckling mode (m; n) = (5; 4) while due to lateral pressure alone the buckling mode (m; n) = (1; 3). Changes in buckling mode are clearly observed by increasing the load-proportional parameter b2 (or b1 ), i.e. m = 3 (or 5) becomes m = 1. The interaction curve consists of two lines (by increasing b2 and b1 , respectively), and the transition from one to another is smooth, so that they seem to be as one line. Clearly, the shape of the interaction

2508

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523 Table 2 Comparisons of dimensionless buckling loads (×10−3 ) for perfect single-layer orthotropic cylindrical shells under combined lateral pressure and axial compression (G12 =E22 = G13 =E22 = 0:6, G23 =E22 = 0:5, (12 = 0:25 and L=R = 5) E11 =E22

R=t

b1

Present HSDT

Soldatos and Ye [7] 3-D

5

20

0.0 0.1 10.0 0.0 0.2 20.0 0.0 0.4 40.0

0:1465 (1; 3)a 0:1462 (1; 3) 0:1202 (1; 3) 0:8347 (1; 3) 0:8296 (1; 3) 0:5003 (1; 2) 4:749 (1; 2) 4:666 (1; 2) 1:614 (1; 2)

0:1359 (1; 3) 0:1355 (1; 3) 0:1073 (1; 3) 0:7910 (1; 3) 0:7880 (1; 3) 0:4594 (1; 3) 3:997 (1; 2) 3:997 (1; 2) 1:027 (1; 2)

0.0 0.1 10.0 0.0 0.2 20.0 0.0 0.4 40.0

0:1644 (1; 3) 0:1641 (1; 3) 0:1377 (1; 3) 0:9238 (1; 3) 0:9188 (1; 3) 0:5759 (1; 2) 5:190 (1; 2) 5:107 (1; 2) 1:806 (1; 2)

0:1607 (1; 3) 0:1604 (1; 3) 0:1271 (1; 3) 0:8428 (1; 3) 0:8395 (1; 3) 0:5571 (1; 3) 4:687 (1; 2) 4:687 (1; 2) 1:216 (1; 2)

0.0 0.1 10.0 0.0 0.2 20.0 0.0 0.4 40.0

0:1870 (1; 3) 0:1867 (1; 3) 0:1595 (1; 3) 1:036 (1; 3) 1:031 (1; 3) 0:6679 (1; 3) 5:788 (1; 2) 5:704 (1; 2) 2:075 (1; 2)

0:1885 (1; 4) 0:1875 (1; 4) 0:1567 (1; 3) 0:9303 (1; 3) 0:9263 (1; 3) 0:6155 (1; 3) 5:545 (1; 3) 5:545 (1; 3) 1:476 (1; 2)

10 5

10

20 10 5

25

20 10 5

40

20

0.0 0.1 10.0 10 0.0 0.2 20.0 5 0.0 0.4 40.0 a Numbers in parentheses indicate

0:1975 (1; 4) 0:1952 (1; 4) 0:1972 (1; 4) 0:1940 (1; 4) 0:1694 (1; 3) 0:1709 (1; 3) 1:091 (1; 3) 0:9801 (1; 3) 1:086 (1; 3) 0:9758 (1; 3) 0:7123 (1; 3) 0:6488 (1; 3) 6:139 (1; 2) 5:722 (1; 3) 6:052 (1; 2) 5:722 (1; 3) 2:232 (1; 2) 1:624 (1; 2) the buckling mode (m; n).

buckling curves for (0=90)S and (0=90)2T laminated cylindrical shells di7ers substantially. Also, it can be seen that radius-to-thickness ratio R=t has a small e7ect on the shape of the interaction buckling curves.

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2509

Fig. 1. E7ect of radius-to-thickness ratio on the interaction buckling curves of moderately thick laminated cylindrical shells: (a) (0=90)S ; (b) (0=90)2T .

Fig. 2 shows the e7ect of shell geometric parameter (Z = 30; 120 and 375) on the interaction buckling curves of (0=90)2T laminated cylindrical shells. Then Fig. 3 shows the e7ect of total number of plies (N = 2; 4 and 10) on the interaction buckling curves of antisymmetric cross-ply laminated cylindrical shells. It can be seen that shell geometric parameter Z or total number of plies N has a signi,cant e7ect on the shape of interaction buckling curves. Figs. 4 and 5 show, respectively, the postbuckling load-shortening and load-de8ection curves of perfect (W ∗ =t = 0) and imperfect (W ∗ =t = 0:1), (0=90)2T and (0=90)S laminated cylindrical shells under combined loading case (2), with load-proportional parameter b2 = 0:0 and 0.04 (or 0.01). These results show that the buckling load is decreased by increasing the load-proportional parameter b2 and that postbuckling equilibrium path becomes signi,cantly lower as b2 increases. In Figs. 4 and 5, only weak ‘snap-through’ behavior could be seen. The elastic limit load can be achieved only for very small imperfections and in such a case imperfection sensitivity can be predicted. When the magnitude of the initial imperfection amplitude becomes larger (say W ∗ =t = 0:1), the postbuckling path is stable and the shell structure becomes imperfection-insensitive.

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

Fig. 2. E7ect of shell geometric parameter on the interaction buckling curves of (0=90)2T laminated cylindrical shells.

Fig. 3. E7ect of total number of plies on the interaction buckling curves of (0=90)2T laminated cylindrical shells.

Fig. 6 shows the postbuckling load-shortening and load-de8ection curves for a 40-ply (0=90)10S laminated cylindrical shell reinforced by outside stringer sti7eners (nS = 20) under combined loading case (2), with load-proportional parameter b2 = 0:0 and 0.01. Then Fig. 7 shows the postbuckling load-shortening and load-de8ection curves for the same cylindrical shell reinforced by inside ring sti7eners (nr = 9) under combined loading case (1), with load-proportional parameter b1 = 0:0 and 2.0. The cylindrical shell has the following geometric parameters: R = 0:3 m, L=R = 5 and total thickness t = 0:01 m. The geometric and material properties of the beam sti7eners adopted here are E1 = E2 = 210 GPa, G1 = G2 = 80:8 GPa, A1 = A2 = 0:12 × 10−3 m2 , I1 = I2 = 0:72 × 10−7 m4 , J1 = J2 = 0:4 × 10−10 m4 , d1 = d2 = 0:094 m, and e1 = − e2 = − 0:035 m. Note that now the coupling sti7nesses Bij∗ are present, even though the skin of the shell is symmetric. In Fig. 6, the usual ‘snap-through’ behavior can be found. In contrast, in

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2511

Fig. 4. Postbuckling of (0=90)2T laminated cylindrical shells under combined axial compression and external pressure: (a) load-shortening; (b) load-de8ection.

Fig. 5. Postbuckling of (0=90)S laminated cylindrical shells under combined axial compression and external pressure: (a) load-shortening; (b) load-de8ection.

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

Fig. 6. Postbuckling of stringer sti7ened (0=90)10S laminated cylindrical shells under combined axial compression and external pressure: (a) load-shortening; (b) load-de8ection.

Fig. 7. Postbuckling of ring sti7ened (0=90)10S laminated cylindrical shells under combined external pressure and axial compression: (a) load-shortening; (b) load-de8ection.

Fig. 7, an increase in pressure is usually required to obtain an increase in deformation, in such a case the postbuckling equilibrium path is stable and the shell structure is imperfectioninsensitive.

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2513

Fig. 8. Comparisons of imperfection sensitivities of moderately thick laminated cylindrical shells under axial compression and external pressure: (a) unsti7ened; (b) sti7ened.

Fig. 8 shows curves of imperfection sensitivity for unsti7ened (0=90)2T and outside stringer sti7ened (0=90)10S laminated cylindrical shell with small initial geometric imperfections. &∗ is the maximum value of x for the imperfect shell, made dimensionless by dividing by the critical value of x for the perfect shell. These results show that the imperfection sensitivity is decreased by increasing the load-proportional parameter b2 for both unsti7ened and stringer sti7ened cylinders, but it has a small e7ect. They also show that the imperfection sensitivity can only be predicted by a very small imperfection (say W ∗ =t 6 0:012) for unsti7ened shells. When W ∗ =t ¿ 0:012 no elastic limit loads could be found and the shell structures become virtually imperfection-insensitive. 5. Concluding remarks A fully nonlinear postbuckling analysis is presented for shear deformable cross-ply laminated cylindrical shells based on Reddy’s higher order shear deformation theory with von K4arm4an–Donnell type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of shear deformable laminated cylindrical shells subjected to combined loading of external pressure and axial compression. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. After comparing the present solutions with the existing ones for some limiting cases available in the literature, the interaction buckling curves and postbuckling response of perfect and imperfect, unsti7ened or sti7ened, moderately thick, antisymmetric and symmetric cross-ply laminated cylindrical shells have been investigated. The results show that the shape of interaction buckling curves depend signi,cantly on the laminate stacking sequence, shell geometric parameter,

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

and total number of plies and the postbuckling characteristics additionally depend signi,cantly upon the load-proportional parameter b2 (or b1 ). The results reveal that in combined loading case (2) the postbuckling equilibrium path is unstable for perfect cylindrical shells, but virtually imperfection-insensitive for imperfect ones. In contrast, in combined loading case (1) the postbuckling equilibrium path is stable for both perfect and imperfect cylinders and the shell structure is imperfection-insensitive.

Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 59975058. The author is grateful for this ,nancial support.

Appendix A In Eqs. (1) – (4),

4 4 4 @ @ @ 4 ∗ ∗ ∗ ∗ ∗ L˜ 11 ( ) = 2 F11 4 + (F12 + F21 + 4F66 ) 2 2 + F22 4 ; 3t @X @X @Y @Y

4 ∗ @3 4 @3 ∗ ∗ ∗ ∗ ∗ ˜ + (D + 2D ) − (F + 2F ) ; L12 ( ) = D11 − 2 F11 12 66 12 66 3t @X 3 3t 2 @X @Y 2

L˜ 13 ( ) =

∗ (D12

+

∗ 2D66 )

4 @3 4 ∗ @3 ∗ ∗ ∗ − 2 (F21 + 2F66 ) ; + D22 − 2 F22 3t @X 2 @Y 3t @Y 3

4 4 @4 ∗ @ ∗ ∗ ∗ ∗ @ + (B + B − 2B ) + B ; L˜ 14 ( ) = B21 11 22 66 12 @X 4 @X 2 @Y 2 @Y 4

@4 @4 @4 L˜ 21 ( ) = A∗22 4 + (2A∗12 + A∗66 ) 2 2 + A∗11 4 ; @X @X @Y @Y

4 ∗ @3 4 @3 ∗ ∗ ∗ ∗ ∗ L˜ 22 ( ) = B21 − 2 E21 + (B − B ) − (E − E ) ; 11 66 11 66 3 2 3t @X 3t @X @Y 2

4 ∗ @3 4 ∗ @3 ∗ ∗ ∗ ∗ ˜ + B12 − 2 E12 ; L23 ( ) = (B22 − B66 ) − 2 (E22 − E66 ) 3t @X 2 @Y 3t @Y 3

4 4 4 @4 ∗ @ ∗ ∗ ∗ ∗ @ + (E + E − 2E ) + E ; L˜ 24 ( ) = 2 E21 11 22 66 12 3t @X 4 @X 2 @Y 2 @Y 4

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2515

16 8 @ ˜ L31 ( ) = A55 − 2 D55 + 4 F55 t t @X

3 4 ∗ @ 4 @3 4 ∗ ∗ ∗ ∗ ∗ + 2 F11 − 2 H11 + (F + 2F ) − (H + 2H ) ; 21 66 66 3t 3t @X 3 3t 2 12 @X @Y 2

8 8 ∗ 16 16 ∗ @2 ∗ ˜ L32 ( ) = A55 − 2 D55 + 4 F55 − D11 − 2 F11 + 4 H11 t t 3t 9t @X 2

8 ∗ 16 ∗ @2 ∗ − D66 − 2 F66 + 4 H66 ; 3t 9t @Y 2

16 ∗ 4 @2 ∗ ∗ ∗ ∗ ∗ ∗ ˜ L33 ( ) = (D12 + D66 ) − 2 (F12 + F21 + 2F66 ) + 4 (H12 + H66 ) ; 3t 9t @X @Y

L˜ 34 ( ) = L˜ 22 ( );

16 8 @ L˜ 41 ( ) = A44 − 2 D44 + 4 F44 t t @Y

3 4 4 @3 4 ∗ @ ∗ ∗ ∗ ∗ ∗ + 2 (F12 + 2F66 ) − 2 (H12 + 2H66 ) ; + F22 − 2 H22 2 3t 3t @X @Y 3t @Y 3

L˜ 42 ( ) = L˜ 33 ( );

16 16 ∗ @2 8 8 ∗ ∗ − 2 F66 + 4 H66 L˜ 43 ( ) = A44 − 2 D44 + 4 F44 − D66 t t 3t 9t @X 2

16 ∗ @2 8 ∗ ∗ − D22 − 2 F22 + 4 H22 ; 3t 9t @Y 2

L˜ 44 ( ) = L˜ 23 ( ); 2 2 2 @2 @2 @2 ˜ )= @ @ − 2 @ : L( + @X 2 @Y 2 @X @Y @X @Y @Y 2 @X 2

(A.1)

In the above equations [A∗ij ], [Bij∗ ], [Dij∗ ], [Eij∗ ], [Fij∗ ] and [Hij∗ ] (i; j = 1; 2; 6) are reduced sti7ness matrices, de,ned as A∗ = A−1 ;

B∗ = − A−1 B;

H∗ = H − EA−1 E

D∗ = D − BA−1 B;

E∗ = − A−1 E;

F∗ = F − EA−1 B; (A.2)

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H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

and

A11 A = A12 0 B11 B = B12 0

A12 A22 0 B12 B22 0

0 E1 A1 =d1 0 0 0 + 0 E2 A2 =d2 0 ; A66 0 0 0 0 E1 A1 e1 =d1 0 0 0 E2 A2 e2 =d2 0 ; 0 + B66 0 0 0

D11 D12 0 0 D = D12 D22 0 0 D66 E1 (I1 + A1 e12 )=d1 + 0 0

0 E2 (I2 + A2 e22 )=d2 0

0 0 ; (G1 J1 =d1 + G2 J2 =d2 )=4

(A.3)

where E1 A1 ; E2 A2 ; G1 J1 and G2 J2 are the extensional and torsional rigidities of the beam sti7eners in the longitudinal and transverse directions; I1 and I2 are the moments of inertia of the beam sti7ener cross sections about their centroidal axes; e1 and e2 refer to beam sti7ener eccentricities; and Aij , Bij ; etc., are the laminate sti7nesses, de,ned by (Aij ; Bij ; Dij ; Eij ; Fij ; Hij ) =

N k=1

(Aij ; Dij ; Fij ) =

N k=1

tk

tk − 1

tk

tk − 1

(QB ij )k (1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 ) dZ

(QB ij )k (1; Z 2 ; Z 4 ) dZ

(i; j = 4; 5)

(i; j = 1; 2; 6);

(A.4a)

(A.4b)

and QB ij are the transformed elastic constants, de,ned by B Q11 c4 B 2 2 Q12 c s B Q22 s4 = B c3 s Q16 3 QB 26 cs

QB 66

c 2 s2

2c2 s2

s4

c 4 + s4

c 2 s2

2c2 s2

c4

cs3 − c3 s

−cs3

c3 s − cs3

−c3 s

−2c2 s2

c 2 s2

4c2 s2

Q11 Q 4c2 s2 12 2 2 Q −2cs(c − s ) 22 2cs(c2 − s2 ) Q66 −4c2 s2

(c2 − s2 )2

(A.5a)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

and

QB 44 c2 QB = −cs 45 s2 QB 55

where Q11 =

s2

Q44 cs ; Q55 2 c

E11 ; (1 − (12 (21 )

Q44 = G23 ;

Q22 =

Q55 = G13 ;

E22 ; (1 − (12 (21 )

2517

(A.5b)

Q12 =

(21 E11 ; (1 − (12 (21 )

Q66 = G12 ;

(A.5c)

in which E11 ; E22 ; G12 ; G13 ; G23 ; (12 and (21 have their usual meanings and c = cos D;

s = sin D;

(A.5d)

where D is lamination angle with respect to the shell X -axis. Appendix B In Eqs. (13)–(15), ∗ ∗ ∗ ∗ ∗ ∗ ; (F12 + F21 + 4F66 )=2; F22 ]=D11 ; (/110 ; /112 ; /114 ) = (4=3t 2 )[F11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − 4F11 =3t 2 ; (D12 + 2D66 ) − 4(F12 + 2F66 )=3t 2 ]=D11 ; (/120 ; /122 ) = [D11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ + 2D66 ) − 4(F21 + 2F66 )=3t 2 ; D22 − 4F22 =3t 2 ]=D11 ; (/131 ; /133 ) = [(D12 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1=4 ; (B11 + B22 − 2B66 )=2; B12 ]=[D11 D22 A11 A22 ] ; (/140 ; /142 ; /144 ) = [B21

(/212 ; /214 ) = (A∗12 + A∗66 =2; A∗11 )=A∗22 ; ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1=4 − 4E21 =3t 2 ; (B11 − B66 ) − 4(E11 − E66 )=3t 2 ]=[D11 D22 A11 A22 ] ; (/220 ; /222 ) = [B21 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1=4 − B66 ) − 4(E22 − E66 )=3t 2 ; B12 − 4E12 =3t 2 ]=[D11 D22 A11 A22 ] ; (/231 ; /233 ) = [(B22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1=4 ; (E11 + E22 − 2E66 )=2; E12 ]=[D11 D22 A11 A22 ] ; (/240 ; /242 ; /244 ) = [E21 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − 4H11 =3t 2 ; (F21 + 2F66 ) − 4(H12 + 2H66 )=3t 2 ]=D11 ; (/310 ; /312 ) = (4=3t 2 )[F11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − 8F11 =3t 2 + 16H11 =9t 4 ; D66 − 8F66 =3t 2 + 16H66 =9t 4 )=D11 ; (/320 ; /322 ) = (D11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + D66 ) − 4(F12 + F21 + 2F66 )=3t 2 + 16(H12 + H66 )=9t 4 ]=D11 ; /331 = [(D12 ∗ ∗ ∗ ∗ ∗ ∗ ∗ + 2F66 ) − 4(H12 + 2H66 )=3t 2 ; F22 − 4H22 =3t 2 ]=D11 ; (/411 ; /413 ) = (4=3t 2 )[(F12 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − 8F66 =3t 2 + 16H66 =9t 4 ; D22 − 8F22 =3t 2 + 16H22 =9t 4 )=D11 ; (/430 ; /432 ) = (D66 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1=4 − 4E11 =3t 2 ; B22 − 4E22 =3t 2 )=[D11 D22 A11 A22 ] ; (/511 ; /522 ) = (B11 ∗ ∗ ∗ ∗ ∗ ∗ 1=4 (/611 ; /622 ) = (4=3t 2 )(E11 ; E22 )=[D11 D22 A11 A22 ] :

(B.1)

2518

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

Appendix C In Eqs. (28)–(30), 1 ?1 = C3 /24

1 1 − a1 /5 &q(2) ; 2

1 1 − a1 /5 &q(0) ; 2

/24 m2 g05 + (1 + ')g07 /24 m4 + % C1 (1 + ')g06 C1 (1 + ')2 g06

1 g05 (1 + ')g07 − '(2 + ')g05 2 g08 + /14 /24 % + /14 C1 (1 + ') g06 (1 + ')2

g05 ' g05 − 1 + % (1 + ')2 /14 m2 C1 (1 + ')m2

g05 (1 + ')g07 + g05 g08 + /14 /24 (2 + ') %3 ; g06 (1 + ')2

&q(0)

1 = (3)3=4 %−3=2 4

&q(2)

1 m4 n2 02 = (3)3=4 %−3=2 16 g06

"(0) q

1 ?2 = − /24

1 /24 g06 g13 (1 + 2') 4 n2 02 C1

1 a1 m2 /24 n2 02 g06 2 − 2(1 + ') + (1 + 2') C1 (1 + ')g06 − 2a1 m6 g10 2 C1 (1 + 2')g06 + 8m4 g10 (1 + ') + (2 + ') ; g06 2/24 (1 + ')(2 + ') +

2 /5 1 2 1 (5=2) (5=2) 1=2 1 − a1 /5 (;b01 − :b10 )% &q a1 / − /5 + 2 24 , /24 2 2 1 b11 1 + 1 − a1 /5 % &q2 ; 2 ,(3)3=4 /224 ;

1 = /24

"(2) q =

2 g05 1 3=4 2 −3=2 −1=2 1=2 m (1 + 2')% − 2g05 % + 2% ; (3) m 32

1 C1 = n2 02 + a1 m2 ; 2

C3 = 1 −

g05 % m2

(C.1)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

and in Eqs. (37)–(39)

1 2(/5 − a2 ) (2) m4 (1 + ') −1 m2 g11 ?3 = /14 /24 % − / / + & ; 14 24 p C3 16n2 02 g09 g06 32n2 02 g09 /24 ?4 = &p(0)

&p(2)

1 = C2 2

/24 m2 g05 + (1 + ')g07 %−1 + /24 (1 + ')g06 (1 + ')2 g06

1 g05 (1 + ')g07 − '(2 + ')g05 + g08 + /14 /24 % /14 (1 + ')m2 g06 (1 + ')2

g05 ' g05 − 1+ % (1 + ')2 /14 m4 (1 + ')m2

g05 (1 + ')g07 + g05 g08 + /14 /24 (2 + ') %2 ; g06 (1 + ')2

m6 (2 + ') −1 1 = C2 /14 /224 % 2 8 2g09 g06

m4 g05 1 1 (2 + ') g07 + /14 /24 (1 + ') + g12 (1 + ') − + g11 2g09 g06 g06 1 + ' g06 2 (1 + ')

2 1 1 g07 2 2 m g11 g05 − /24 m g13 (1 + 2')% + /14 /24 − − g12 % 4 2g09 g06 1 + ' g06

2 ' g05 2(1 + ')2 − (1 + 2') 2 m g05 + /14 /24 g14 + (2 + ')% 2g09 g06 2(1 + ')2 1 + ' g06 m2 n4 04 S2 + 24 % ; g06 S1

&p(4) = "(0) p

2(/5 − a2 ) (0) &p ; /24

1 m10 (1 + ') S3 −1 % ; C2 /214 /324 2 g3 128 S13 g09 06

(1 + ab2 ) 2 /5 (/5 − a2 ) (2) (2) 1=2 2 = (/24 − a2 /5 ) − (;b01 − :b10 )% &p /24 , /24

b11 (a2 − /5 )2 2 1=2 + (1 + ab ) % &p2 ; 2 2,; /224

2519

2520

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

2 g05 1 2 2 3 m (1 + 2')% − 2g05 % + 2 % ; 16 m 2 8 (1 + ')2 1 b m S4 11 "(4) %3 ; /214 /224 4 4 2 2 %−3=2 + m2 n4 04 (1 + ')2 p = 128 32,; S1 n 0 g09 g06

"(2) p =

S1 = g06 (1 + ') − 4m2 C2 g10 ; S2 = g06 [(4 + 9' + 4'2 ) + C2 (1 + 2')] + 8m4 (1 + ')(2 + ')g10 ; S3 = g136 [C9 (1 + 3' + '2 ) + C5 (4 + 2') + (1 + ')] + g06 [C5 (6 + 8' + 2'2 ) − (2' + 3'2 + '3 )]; S4 = g06 (1 + 2') + 8m4 (1 + ')g10 ; S13 = g136 C9 − g06 (1 + '); C2 =

m2 ; m2 + a2 n2 02

C5 =

m2 + 5a2 n2 02 ; m2 + a2 n2 02

C9 =

m2 + 9a2 n2 02 m2 + a2 n2 02

in the above equations g00 = (/31 + /320 m2 + /322 n2 02 )(/41 + /430 m2 + /432 n2 02 ) − /2331 m2 n2 02 ; g01 = (/41 + /430 m2 + /432 n2 02 )(/220 m2 + /222 n2 02 ) − /331 n2 02 (/231 m2 + /233 n2 02 ); g02 = (/31 + /320 m2 + /322 n2 02 )(/231 m2 + /233 n2 02 ) − /331 m2 (/220 m2 + /222 n2 02 ); g03 = (/31 + /320 m2 + /322 n2 02 )(/41 − /411 m2 − /413 n2 02 ) − /331 m2 (/31 − /310 m2 − /312 n2 02 );

g04 = (/41 + /430 m2 + /432 n2 02 )(/31 − /310 m2 − /312 n2 02 ) − /331 n2 02 (/41 − /411 m2 − /413 n2 02 );

g05 = (/240 m4 + 2/242 m2 n2 02 + /244 n4 04 ) +

m2 (/220 m2 + /222 n2 02 )g04 + n2 02 (/231 m2 + /233 n2 02 )g03 ; g00

g06 = (m4 + 2/212 m2 n2 02 + /214 n4 04 ) + /14 /24

m2 (/220 m2 + /222 n2 02 )g01 + n2 02 (/231 m2 + /233 n2 02 )g02 ; g00

(C.2)

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

g07 = (/140 m4 + 2/142 m2 n2 02 + /144 n4 04 ) −

m2 (/120 m2 + /122 n2 02 )g01 + n2 02 (/131 m2 + /133 n2 02 )g02 ; g00

g08 = (/110 m4 + 2/112 m2 n2 02 + /114 n4 04 ) +

m2 (/120 m2 + /122 n2 02 )g04 + n2 02 (/131 m2 + /133 n2 02 )g03 ; g00

g10 = 1 + /14 /24 /2220

4m2 ; /31 + /320 4m2

g12 =

/244 (/41 + /432 4n2 02 ) + /233 (/41 − /413 4n2 02 ) ; /214 (/41 + /432 4n2 02 ) + /14 /24 /2233 4n2 02

∗ g12 =

/214 (/41 − /413 4n2 02 ) − /14 /24 /233 /244 4n2 02 ; /214 (/41 + /432 4n2 02 ) + /14 /24 /2233 4n2 02

∗ g09 = /114 + /133 g12 + /14 /24 /144 g12 ;

g13 =

/41 + /432 4n2 02 ; /214 (/41 + /432 4n2 02 ) + /14 /24 /2233 4n2 02

g14 = −

/144 (/41 + /432 4n2 02 ) − /133 /233 4n2 02 ; /214 (/41 + /432 4n2 02 ) + /14 /24 /2233 4n2 02

g11 = g14 (1 + 2') + 2

g05 ; g06

g130 = (/31 + /320 m2 + /322 9n2 02 )(/41 + /430 m2 + /432 9n2 02 ) − /2331 9m2 n2 02 ; g131 = (/41 + /430 m2 + /432 9n2 02 )(/220 m2 + /222 9n2 02 ) − /331 9n2 02 (/231 m2 + /233 9n2 0)2 ;

g132 = (/31 + /320 m2 + /322 9n2 02 )(/231 m2 + /233 9n2 02 ) − /331 m2 (/220 m2 + /222 9n2 02 ); g136 = (m4 + 18/212 m2 n2 02 + /214 81n4 04 ) + /14 /24

m2 (/220 m2 + /222 9n2 02 )g131 + 9n2 02 (/231 m2 + /233 9n2 02 )g132 ; g130

g15 = /220 (/310 + /120 ) − /320 (/140 + /240 );

2521

2522

H.-S. Shen / International Journal of Mechanical Sciences 43 (2001) 2493–2523

g16 = (/320 + /14 /24 /2220 )(/320 /110 − /310 /120 ) + /14 /24 (/320 /140 − /120 /220 )(/320 /240 − /310 /220 );

/14 /24 /2320 b= g16

1=2

;

; = [(b − c)=2]1=2 ;

c = /14 /24 /320

g15 ; 2g16

: = [(b + c)=2]1=2 ;

g17 =

(/310 + /14 /24 /220 /240 )b − /14 /24 /220 ; (/310 + /14 /24 /220 /240 )b + /14 /24 /220

g19 =

/320 (2;2 g17 − c) /310 /220 − /320 /240 + ; 2 b2 /320 + /14 /24 /220 /320 + /14 /24 /2220

g20 = − −

/320 (2:2 g17 + c) 2 /320 g17 + 2 2 b b /320 + /14 /24 /2220 /320 + /14 /24 /220 2/310 /320 − (/310 /220 g17 − /320 /240 )[(/310 + /14 /24 /220 /240 )b + /14 /24 /220 ] ; (/320 + /14 /24 /2220 )[(/310 + /14 /24 /220 /240 )b + /14 /24 /220 ]

(3=2) a(1) 01 = a01 = 1

(3=2) a(1) 10 = a10 =

(5=2) b(2) 01 = b01 = /24 g19 ;

; g17 ; :

; (5=2) b(2) 10 = b10 = /24 g20 ; :

1 4 2 2 4 b11 = [(a(1) )2 :2 b + a(1) 10 2;:c + (2; − ; : + : )]: b 10

(C.3)

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