Sensitivity of the response of thick cross-ply doubly curved panels to edge clamping

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Composite Structures 87 (2009) 293–306 www.elsevier.com/locate/compstruct

Sensitivity of the response of thick cross-ply doubly curved panels to edge clamping Ahmet Sinan Oktem a, Reaz A. Chaudhuri b,* a

Department of Naval Architecture and Marine Engineering, Yildiz Technical University, 34349 Besiktas-Istanbul, Turkey b Department of Materials Science and Engineering, University of Utah, Salt Lake City, UT 84112-0560, USA Available online 12 February 2008

Abstract A heretofore unavailable analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doublycurved panel of rectangular plan-form, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach, is used to solve a system of five highly coupled linear partial differential equations, generated by the HSDT-based laminated shell analysis, with the C3-type clamped boundary condition prescribed at all four edges. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of a cross-ply spherical panel. The primary focus of the present study is to investigate the effect of edge clamping on the response of a thick laminated doubly-curved panel, while keeping the surface-parallel edge constraints unaltered. Important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, material property, thickness and curvature effects, and edge clamping as well as their interactions. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Thick cross-ply laminate; Higher order shear deformation theory (HSDT); Spherical shell; Cylindrical shell; Boundary discontinuous double fourier series; Edge clamping

1. Introduction Curved panels (open shells) are common load-bearing structural elements in aerospace, hydrospace, nuclear and other industrial applications. Recent years have witnessed an increasing use of advanced composite materials (e.g., graphite/epoxy, boron/epoxy, Kevlar/epoxy, graphite/ PEEK, etc.) which are replacing metallic alloys in the fabrication of such panels because of such beneficial properties as higher strength-to-weight ratios, longer (surface-parallel) fatigue (including sonic fatigue) life, better stealth characteristics, enhanced corrosion resistance, and so forth. The latest example is the Boeing 787 Dreamliner, the first large transport jet to be produced largely from carbon fiber reinforced plastic materials. The advantages that accrue from these properties are, however, not attainable without *

Corresponding author. Tel.: +1 (801) 581 6282; fax: +1 (801) 581 4816. E-mail address: [email protected] (R.A. Chaudhuri).

0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.01.014

paying for the complexities that are introduced by various coupling effects. Furthermore, since the matrix material is of relatively low shearing stiffness as compared to the fibers, a reliable prediction of the response of these laminated shells must account for transverse shear deformation. Additionally, a solution to the problem of the deformation of laminated shells and panels of finite dimensions must satisfy the boundary conditions, which introduce additional complexities into the analysis. The present study is intended to capture some of these intricacies of the response of laminated composite structural components through double Fourier series based analysis of a model thick laminated shell boundary-value problem. The majority of the investigations on laminated shells utilize either the classical lamination theory (CLT) or the first-order shear deformation theory (FSDT). A detailed literature review on CLT-based analyses is available in, e.g., Bert and Francis [1], and Chaudhuri et al. [2], while that relating to their FSDT counterparts is available in,

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A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

e.g., Abu Arja and Chaudhuri [3,4], and Chaudhuri and Abu Arja [5]. Noor and Burton [6] have presented an extensive survey on various computational models relating to laminated shells. The development of the theoretical framework of linear elastic shells has received considerable attention by numerous authors, who have employed a variety of approximations (with respect to the three-dimensional elasticity theory of curved deformable bodies); see Chaudhuri and Kabir [7] for a brief review. In comparison, the development of general analytical techniques for solving boundary value problems of these shells has been more conspicuous by its relative absence. This is, in part, due to the rapid growth experienced by such popular numerical (approximate) methods as, e.g., finite difference, and most notably, finite elements, but primarily due to the formidable difficulty posed by the system of highly coupled partial differential equations (PDE) to be solved in conjunction with arbitrary admissible boundary conditions. The present study is intended to fill in this critical analytical gap. The present solution technique utilizes the double Fourier series approach, first expounded by Hobson [8] in the context of ordinary Fourier series, and used by Goldstein [9] to solve the stability problem of fluid flow. Winslow [10], following Hobson’s [8] lead, has discussed the mathematical conditions of differentiation, of stress functions and their partial derivatives represented by ordinary Fourier series, in the presence of ordinary discontinuities and has concluded that unless additional conditions, imposed by term-wise differentiation are fulfilled, the hypothetical representation by Fourier series may not have sufficient generality to satisfy all the required conditions and furnish a solution. The underlying mathematical principle is concerned with well-posedness or lack thereof of the Fourier-type formulation, and existence of the resulting series solution. This kind of ill-posedness can be removed by addition of appropriate mathematical ‘‘structures” to the formulation, which is accomplished through introduction of certain constraints [11,12]. Green [13], Green and Hearmon [14], Whitney [15,16], Whitney and Leissa [17], Kabir et al. [18], and Chaudhuri et al. [19] have presented double Fourier series based analytical solutions to thin (CLTbased) homogeneous and laminated anisotropic plate boundary-value problems. Double Fourier series solutions to CLT-based homogeneous and laminated shell boundary value problems are available in, e.g., Kabir and Chaudhuri [20], and Chaudhuri and Kabir [21–23]. Chaudhuri and Kabir [24–27], and Kabir and Chaudhuri [28] have presented double Fourier series based analytical solutions to various FSDT-based laminated anisotropic plate boundary-value problems, while their shell counterparts are available in Chaudhuri and Abu-Arja [29,30], Chaudhuri and Kabir [7,31,32], Kabir and Chaudhuri [33,34], and Kabir et al. [35]. It may be remarked here that although the boundary-discontinuous Fourier series method has been applied previously by earlier investigators, such as Goldstein [9], Green [13], Green and Hearmon [14], Whitney

[15,16], and Whitney and Leissa [17], the criteria for determining when the boundary Fourier series are needed or not needed had never been clearly spelled out. A clear exposition of this important issue for completely coupled systems of second-order and nth-order PDE’s, subjected to completely coupled admissible boundary conditions, is available in Chaudhuri [11,12]. Superiority of the FSDT over the CLT in prediction of the transverse deflection of a moderately thick panel notwithstanding, the former theory requires incorporation of a shear correction factor, due to the fact that the FSDT assumes a uniform transverse shear strain distribution through the thickness, which violates equilibrium conditions at the top and bottom surfaces of the panel. Basset [36] appears to have been the first to suggest that the displacements can be expanded in power series of the thickness coordinate, n3 = f (see Fig. 1). Following Basset’s lead, second- and higher-order shear deformation theories (HSDT), involving continuous in-plane/surface-parallel displacements through the thickness of thick laminated plates/shells, have been developed as special cases of the above by Nelson and Lorch [37], Levinson [38], Bert [39], Librescu and Khdeir [40] among others to account for the afore-mentioned shear correction factor. A detailed review of the literature and a boundary-discontinuous type double Fourier series solution to the HSDT-based crossply laminated shell (of which laminated plate is a special case), subjected to the SS2 type simply supported boundary condition are available in Chaudhuri and Kabir [41]. More recently, a detailed review of the literature and Levy type boundary-discontinuous type double Fourier series solutions to the problems of HSDT-based cross-ply laminated plates and shells, subjected to the SS2 type simply supported and C4 type (rigidly) clamped boundary condition

Fig. 1. Geometry of a laminated doubly curved panel.

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

on two opposite edges, are available in Oktem and Chaudhuri [42–45]. In Levy type problems, SS3 type simply supported boundary conditions are invariably prescribed on the remaining two edges. HSDT-based boundary-discontinuous Fourier solutions to the class of problems of general cross-ply plates, subjected to SS1 and SS4-type simply supported as well as C3-type clamped boundary conditions prescribed at all four edges, have been presented by Oktem and Chaudhuri [46,47]. The precise mathematical premises of the boundary-discontinuous type double Fourier series approach to solution of completely coupled system of partial differential equations subjected to admissible general boundary conditions are available in Chaudhuri [11,12]. In what follows, a self-adjoint differential system of five highly coupled fourth order linear partial differential equations arising out of the afore-mentioned HSDT-based formulation for general cross-ply doubly curved panels, and subjected to the C3 (clamped edge 3) conditions prescribed at all four edges, is solved analytically using the previously mentioned boundary discontinuous double Fourier series technique. The primary focus of the present study is to investigate the effect of clamping boundary constraints on the response of a thick laminated shell/panel, while keeping the surface-parallel edge constraints unaltered. The C3 and SS3 boundary conditions are selected here because (i) the latter admits Navier type solution that can be readily computed without much computational resources, and (ii) C3 and SS3 boundary conditions represent the two extremes of the rotational edge constraint, the latter ensuring complete rotational freedom, while the former enforcing complete fixity. A review of the literature suggests an absence of such investigations in the context of HSDT-based laminated shell analyses. The numerical accuracy of the present solution is ascertained by studying its convergence characteristics. Numerical results are presented to understand the complex deformation behavior of clamped thick cylindrical and spherical panels of symmetric and antisymmetric crossply constructions. 2. Statement of the problem

e3 ðn1 ; n2 ; n3 Þ ¼ u3;3 ;

  1 g  u3;2  2 u2 þ u2;3 ; e4 ðn1 ; n2 ; n3 Þ ¼  R2 1 þ Rn32 g2   1 g1   u3;1  u1 þ u1;3 ; e5 ðn1 ; n2 ; n3 Þ ¼ R1 1 þ Rn31 g1   1 1  u2;1  g1;2 u1 e6 ðn1 ; n2 ; n3 Þ ¼  g2 1 þ Rn31 g1   1 1  u1;2  g1;2 u1 ; þ g2 1 þ Rn32 g2

295

ð1cÞ ð1dÞ

ð1eÞ

ð1fÞ

where ei ði ¼ 1; 2; 3; 4; 5; 6Þ represents the components of i ði ¼ 1; 2; 3Þ denotes the compothe strain tensor, and u nents of the displacement vector along the ðn1 ; n2 ; n3 ¼ fÞ coordinates at a point, ðn1 ; n2 ; n3 ¼ fÞ. The principal radii of normal curvature of the reference (middle) surface are denoted by R1 and R2, while g1 and g2 are the first fundamental form quantities of the shell reference (middle) surface for lines of curvature coordinates. In order to model the kinematic behavior of the shell, an additional set of simplifying assumptions are invoked: (i) transverse inextensibility, (ii) moderate shallowness (in regards to the normal curvatures), and (iii) negligibility of geodesic curvature. For a cylindrical shell, the lines of principal curvature coincide with the surface-parallel coordinate lines, while for a spherical shell, the same can be assumed upon negligence of the geodesic curvatures of the coordinate lines. The surface-parallel displacements can be expanded in power series of n3 = f as suggested by Basset [36]. The basic equations for a HSDT-based cross-ply shell are provided in Appendix A. Substitution of Eqs. (A7) into equilibrium equations given by Eqs. (A5) in the Appendix A supplies the following five highly coupled fourth-order governing partial differential equations: A11 u1;11 þ a1 u3;1 þ f1 u2;12 þ a2 /1;11 þ f2 /2;12  a4 u3;111 þ f3 u3;122 þ A66 u1;22 þ a9 /1;22 ¼ 0;

ð2aÞ

A66 u2;11 þ f1 u1;12 þ a9 /2;11 þ f2 /1;12 þ f3 u3;112 þ A22 u2;22 Let ðn1 ; n2 ; n3 ¼ fÞ denote the orthogonal curvilinear coordinates as shown in Fig. 1. The n1 and n2 curves are lines of curvature on the shell mid-surface, n3 = f = 0, while n3 = f is a straight line normal to the mid-surface. The cross-ply shell under consideration is composed of a finite number of orthotropic layers of uniform thickness. Strain-displacement relations from the theory of elasticity in curvilinear coordinates are given by [48,49]:   1 1 g1    ð1aÞ u1;1 þ g1;2  u2 þ  u3 ; e1 ðn1 ; n2 ; n3 Þ ¼ R1 g2 1 þ Rn31 g1   1 1 g2   e2 ðn1 ; n2 ; n3 Þ ¼  ð1bÞ u2;2 þ g2;1  u1 þ  u3 ; R2 g1 1 þ n3 g R2

2

þ a6 u3;2 þ a7 /2;22  a8 u3;222 ¼ 0;

ð2bÞ

f4 /1;1 þ f5 /2;2 þ f6 u3;11 þ f7 u3;22 þ a4 u1;111 þ f8 u2;112 þ f9 /1;111 þ f10 /2;112  f11 u3;1111 þ f12 u3;1122 þ f8 u1;122 þ a8 u2;222 þ f10 /1;122 þ f13 /2;222  f14 u3;2222  a1 u1;1  a6 u2;2 þ f15 u3 ¼ q;

ð2cÞ

a2 u1;11 þ e1 u2;12 þ e2 u3;1 þ e3 /1;11 þ e4 /2;12 þ e5 /1;22 þ e6 u3;111 þ e7 u3;122 þ a9 u1;22 þ e8 /1 ¼ 0;

ð2dÞ

a9 u2;11 þ e1 u1;12 þ e5 /2;11 þ e4 /1;12 þ e7 u3;112 þ a7 u2;22 þ e12 u3;2 þ e9 /2;22 þ e10 u3;222 þ e11 /2 ¼ 0;

ð2eÞ

where ai, ei and fi are constants and given in Appendix B.

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A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

In what follows, the C3 type clamped boundary condition is prescribed at all four edges: u2 ð0; x2 Þ ¼ u2 ða; x2 Þ ¼ u1 ðx1 ; 0Þ ¼ u1 ðx1 ; bÞ ¼ 0;

ð3aÞ

u3 ð0; x2 Þ ¼ u3 ða; x2 Þ ¼ u3 ðx1 ; 0Þ ¼ u3 ðx1 ; bÞ ¼ 0;

ð3bÞ

u3;1 ð0; x2 Þ ¼ u3;1 ða; x2 Þ ¼ u3;2 ðx1 ; 0Þ ¼ u3;2 ðx1 ; bÞ ¼ 0;

ð3cÞ

/1 ð0; x2 Þ ¼ /1 ða; x2 Þ ¼ /2 ðx1 ; 0Þ ¼ /2 ðx1 ; bÞ ¼ 0;

ð3dÞ

/2 ð0; x2 Þ ¼ /2 ða; x2 Þ ¼ /1 ðx1 ; 0Þ ¼ /1 ðx1 ; bÞ ¼ 0;

ð3eÞ

N 1 ð0; x2 Þ ¼ N 1 ða; x2 Þ ¼ N 2 ðx1 ; 0Þ ¼ N 2 ðx1 ; bÞ ¼ 0:

ð3fÞ

The SS3 type simply supported boundary condition is obtained by replacing the conditions on Eqs. (3c) and (3d), which are given as follows: P 1 ð0; x2 Þ ¼ P 1 ða; x2 Þ ¼ P 2 ðx1 ; 0Þ ¼ P 2 ðx1 ; bÞ ¼ 0

ð3c Þ

M 1 ð0; x2 Þ ¼ M 1 ða; x2 Þ ¼ M 2 ðx1 ; 0Þ ¼ M 2 ðx1 ; bÞ ¼ 0:

ð3d Þ

3. Method of solution The particular solution to the boundary-value problem of a HSDT-based cross-ply shell, given by Eqs. (2) and (3), is assumed as follows: u1 ¼

1 X 1 X

U mn cosðax1 Þ sinðbx2 Þ;

m¼0 n¼1

0 < x1 < a; 0 6 x2 6 b u2 ¼

1 X

1 X

ð4aÞ

V mn sinðax1 Þ cosðbx2 Þ;

on Lebesgue integration theory that introduces boundary Fourier coefficients arising from discontinuities (complementary boundary constraints; see Chaudhuri [11,12]) of the particular solutions at the edges x1 ¼ 0; a. As has been noted by Chaudhuri [12], the boundary Fourier coefficients serve as complementary solution to the problem under investigation. The procedure imposes certain boundary constraints in the form of equalities and complementary boundary constraints in the form of inequalities, the details of which are available in Chaudhuri [11,12]. The function /1 in Eq. (4d) and the first partial derivative of the transverse displacement, u3,1 obtained by termwise differentiation (see below), are two rotational quantities associated with the HSDT, which are not satisfied at the edges, x1 ¼ 0; a, thus violating the boundary constraints. Therefore, these two rotational quantities are forced to vanish at these edges (boundary constraints). Their next partial derivatives, /1,1 and u3,11, which represent changes of curvature, are seen to vanish at the edges, x1 ¼ 0; a, thus violating the complementary boundary constraints (boundary discontinuities) at these edges. Therefore, for further differentiation, /1,11 and u3,111 are expanded in double Fourier series, in order to satisfy the complementary boundary constraint (inequality); see Chaudhuri [11,12]. A similar argument holds for obtaining the partial derivatives of the rotational quantities, /2 and u3,2 obtained by term-wise differentiation (see below), which are not satisfied at the edges, x2 ¼ 0; b. The partial derivatives of interest can be written as follows:

m¼1 n¼0

0 < x1 < a; 0 6 x2 6 b u3 ¼

1 X 1 X

ð4bÞ

1 X 1 X

W mn sinðax1 Þ sinðbx2 Þ;

0 6 x1 6 a; 0 6 x2 6 b

u3;11 ¼ 

ð6aÞ

1 X 1 X

ð4cÞ u3;111 ¼

X mn cosðax1 Þ sinðbx2 Þ;

0 6 x1 6 a; 0 6 x2 6 b 1 X 1 X

1 X 1 X

a2 W mn sinðax1 Þ sinðbx2 Þ;

ð6bÞ

m¼1 n¼1

m¼0 n¼1

/2 ¼

aW mn cosðax1 Þ sinðbx2 Þ;

m¼1 n¼1

m¼1 n¼1

/1 ¼

u3;1 ¼

ð4dÞ

1 1X an sinðbx2 Þ 2 n¼1 1 X 1 X  3  þ a W mn þ cm an þ wm bn m¼1 n¼1

 cosðax1 Þ sinðbx2 Þ;

Y mn sinðax1 Þ cosðbx2 Þ;

m¼1 n¼0

0 6 x1 6 a; 0 6 x2 6 b

ð4eÞ

u3;2 ¼

1 X

1 X

bW mn sinðax1 Þ cosðbx2 Þ;

ð6cÞ ð6dÞ

m¼1 n¼1

where mp ; a¼ a

np b¼ : b

ð5Þ

It is important to note that the assumed solution functions, given by Eqs. (4), satisfy the SS3 type simply supported condition at the edges a priori (Navier’s solution). The same is, however, not true for the C3 boundary condition prescribed at all four edges. The total number of unknown interior or panel Fourier coefficients introduced in Eqs. (4) is 5mn þ 2m þ 2n. The next operation is comprised of partial differentiation of the assumed particular solution functions. The procedure for differentiation of these functions is based

u3;22 ¼ 

1 X 1 X

b2 W mn sinðax1 Þ sinðbx2 Þ;

ð6eÞ

m¼1 n¼1

u3;222 ¼

1 1X cm sinðax1 Þ 2 m¼1 1 X 1 X  3  þ b W mn þ cncm þ wn dm m¼1 n¼1

 sinðax1 Þ cosðbx2 Þ; /1;1 ¼ 

1 X 1 X m¼1 n¼1

aX mn sinðax1 Þ sinðbx2 Þ;

ð6fÞ ð7aÞ

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

/1;11 ¼

1 X 1 X

1 1X en sinðbx2 Þ 2 n¼1 1 X 1 X  2  þ a X mn þ cmen þ wm f n

/2;2 ¼ 

1 X 1 X

 abe1 V mn þ ðe6 a3  e7 ab2 þ e2 aÞW mn g ð7bÞ

þ ðe8  e3 a2  e5 b2 ÞX mn  e4 abY mn þ e3 ðcmen þ w f n Þ þ e6 ðcm an þ w bn Þg ¼ 0; m

bY mn sinðax1 Þ sinðbx2 Þ;

ð7cÞ

1 X 1 X

m¼1 n¼1

/2;22

ð7dÞ

n ;  in which the boundary Fourier coefficients a bn ; cm ; dm ; en ;   f n ; gm and hm are as defined in Appendix C, and  ð0; 1Þ; n ¼ odd; ðcn ; wn Þ ¼ ð8Þ ð1; 0Þ; n ¼ even: The remaining partial derivatives are obtained by term by term differentiation. The above step generates additional 4m þ 4n unknown (boundary) Fourier coefficients. Introduction of the displacement functions and their appropriate derivatives into the governing partial differential Eqs. (2) will yield the following: cosðax1 Þ sinðbx2 Þfða2 A11 þ b2 A66 ÞU mn

m¼1 n¼1

 abf1 V mn þ ða2 a  f3 ab2 þ a4 a3 ÞW mn  ða9 b2 þ a2 a2 ÞX mn  abf2 Y mn þ a2 ðcmen þ wm f n Þ n Þg ¼ 0;  a4 ðcm  an þ w m b 1 X 1 X

ð9aÞ

cosðax1 Þ sinðbx2 Þfabf1 U mn

m¼1 n¼1

 ðA66 a2 þ A22 b2 ÞV mn þ ða6 b  f3 a2 b þ a8 b3 ÞW mn hm Þ  f2 abX mn  ða9 a2 þ a7 b2 ÞY mn  a7 ðcn gm þ wn   þ a8 ðcncm þ wn d m Þg ¼ 0; 1 X

1 X

ð9bÞ

sinðax1 Þ sinðbx2 Þfða1 a þ a4 a3 þ f8 ab2 ÞU mn

m¼1 n¼1

þ ða6 b þ a8 b3 þ f8 a2 bgÞV mn þ ðf6 a2 þ f15  f7 b2  f11 a4 þ f12 a2 b2  f14 b4 ÞW mn þ ðf4 a þ f9 a3 þ f10 ab2 ÞX mn bn Þ þ ðf5 b þ b13 b3 þ f10 a2 bÞY mn þ f11 aðcm  an þ w m     f9 aðcmen þ w f n Þ  f13 bðcn gm þ w hm Þ m

þ f14 bðcncm þ wn dm Þg 1 X 1 X qmn sinðax1 Þ sinðbx2 Þ; ¼ m¼1 n¼1

sinðax1 Þ cosðbx2 Þfabe1 U mn  ða9 a2 þ a7 b2 ÞV mn

þ ðe12 b  e7 a2 b  e10 b3 ÞW mn  e4 abX mn

m¼1 n¼1

 sinðax1 Þ cosðbx2 Þ;

ð9dÞ

m

m¼1 n¼1

1 1X gm sinðbx2 Þ ¼ 2 m¼1 1 X 1 X  2  hm þ b Y mn þ cn gm þ wn 

1 X 1 X

cosðax1 Þ sinðbx2 Þfða2 a2 þ b2 a9 ÞU mn

m¼1 n¼1

m¼1 n¼1

 cosðax1 Þ sinðbx2 Þ;

297

n

þ ðe5 a2 þ e11  e9 b2 ÞY mn þ e9 ðcn gm þ wn hm Þ þ e10 ðcncm þ wn dm Þg ¼ 0; ð9eÞ  1 X A66 2 a2 a4 b U 0n  a9 b2 X 0n þ en   sinðbx2 Þ  an ¼ 0; 2 2 2 n¼1 1 X m¼1

ð10aÞ  A66 2 a5 a6 a V m0  a7 a2 Y m0 þ gm  cm ¼ 0; sinðax1 Þ  2 2 2 ð10bÞ

n a

 7 sinðbx2 Þ  b2 U 0n þ e8  e5 b2 X 0n 2 n¼1 e3 e6 o þ en þ an ¼ 0; 2 2 1 n a X 7 sinðax1 Þ  a2 V m0 þ ðe11  e5 a2 ÞY m0 2 m¼1 e9 e10 o cm ¼ 0: þ gm þ 2 2

1 X

ð10dÞ

Equating the coefficients of the trigonometric functions such as sin (ax1) sin(bx2), sin(ax1), etc. of both sides of Eqs. (9) and (10) yields 5mn þ 2m þ 2n linear algebraic equations. The remaining equations are supplied by the boundary conditions. The natural boundary conditions, given by Eq. (3f), relating to vanishing of N1 at x1 ¼ 0; a, and of N2, at x2 ¼ 0; b, are satisfied a priori. In addition, the geometric boundary conditions, given by Eqs. (3a) and (3b), relating to vanishing of displacements, u3 at all edges, of u1 at x2 ¼ 0; b, and of u2 at x1 ¼ 0; a, are also satisfied a priori. Same is true for rotations, /1 at x2 ¼ 0; b, and /2 at x1 ¼ 0; a. In order to match the number of unknown constant coefficients, the remaining equations are obtained from satisfying the geometric boundary conditions relating to vanishing of the rotational quantities /1 and u3,1 at x1 ¼ 0; a, as well as /2 and u3,2 at x2 ¼ 0; b. Equating the coefficients of sin(ax1), sin(bx2), etc., on both sides of these equations yields the following algebraic equations as shown below: For all values of n ¼ 1; 2; . . . 1 X

ð9cÞ

ð10cÞ

m¼1;3;5;...

wm X mn ¼ 0;

X 0n þ

1 X m¼2;4;6;...

cm X mn ¼ 0;

ð11aÞ

298

a

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306 1 X

wm W mn ¼ 0;

1 X

a

m¼1;3;5;...

cm W mn ¼ 0:

ð11bÞ

¼ G13 =E2 ¼ 0:5;

m¼2;4;6;...

For all values of m ¼ 1; 2; . . . 1 X

wn Y mn ¼ 0;

Y 0n þ

n¼1;3;5;...

b

1 X

cn Y mn ¼ 0;

ð11cÞ

n¼2;4;6;...

1 X

wn W mn ¼ 0;

n¼1;3;5;...

b

1 X

E1 ¼ 175:78 GPað25; 000 ksiÞ;

cn W mn ¼ 0:

ð11dÞ

n¼2;4;6;...

This step generates additional 4m þ 4n equations for the solution. It may be remarked here that introduction and satisfaction of the boundary equations, given by Eq. (3c), is a necessary component for solution of clamped shell boundary-value problems based on the higher order shear deformation theory (HSDT), and the present solution goes a long way in filling a large void in the existing literature. Finally, the above operations result in, in total, 5mnþ 6m þ 6n linear algebraic equations in as many unknowns. In the interest of computational efficiency, Eqs. (9) and (10) are solved for the panel Fourier coefficients, Umn, Vmn, Wmn, Xmn, Ymn, and U0n, Vm0, X0n, Ym0, respectively in terms of the boundary Fourier coefficients,  bn ; cm ; dm ; en ; f n ; gm and  hm .These are, in turn, substian ;  tuted into the boundary equations, given by Eqs. (11). This step reduces the size of the solution matrix from ð5mn þ 6m þ 6nÞ  ð5mn þ 6m þ 6nÞ to ð4m þ 4nÞ  ð4m þ 4nÞ, which is then solved using the MATLAB code developed for the solution. 4. Numerical results and discussion For illustrative purposes, numerical results for antisymmetric [0°/90°] as well as symmetric [0°/90°/0°] and [0°/90°/ 90°/0°] cross-ply square (a = b) clamped, cylindrical (either R1 = 1 or R2 = 1), and spherical panels of square planform, subjected to uniformly distributed transverse loads, are presented. The following material property is assumed:

E1 =E2 ¼ 25;

G23 =E2 ¼ 0:2;

G12 =E2

m12 ¼ 0:25;

in which E1 and E2 are the surface-parallel Young’s moduli in x1 and x2 coordinate directions, respectively, while G12 denotes surface-parallel shear modulus. G13 and G23 are transverse shear moduli in the x1–x3 and x2–x3 planes, respectively, while m12 is major Poisson’s ratio on the x1– x2 plane. The following normalized quantities are defined: u3 ¼

103 E2 h3 u3 ; q0 a4

M 1 ¼

103 M 1; q0 a2

ð12a; bÞ

u1 ¼

103 E2 h3 u1 ; q0 a4

/1 ¼

102 E2 h3 /1 : q0 a3

ð12c; dÞ

in which ‘a’ is assumed equal to 812.8 mm (32 in.). q0 denotes the uniformly distributed transverse load. For all the numerical results presented here, the displacement, u3, and moment, M1, are computed at the center of the panel with the obvious exception of Fig. 9. The results obtained for the clamped 3 (C3) boundary condition are compared with their simply supported 3 (SS3) counterparts obtained by regenerating the SS3 solution with vanishing of the boundary Fourier coefficients, an ; bn ; cm ; dm ; en ; f n ; gm and hm . Before presenting numerical results for spherical and cylindrical panels, those pertaining to flat (R1 ? 1 and R2 ? 1) symmetric and antisymmetric cross-ply plates with the same boundary conditions have been reproduced first [47] including those pertaining to convergence. Fig. 2 plots the convergence characteristics of the normalized central transverse displacement (deflection), u3 , and moment, M 1 , of a thick (a/h = 5) square (a = b) symmetric [0°/90°/0°] moderately deep (R1/a = R2/b = 10) spherical panel. The central deflection, u3 , shows relatively rapid and monotonic convergence. In contrast, M 1 shows a less rapid convergence. Convergence is relatively slow when compared to the SS2 boundary conditions investigated

Spherical Panel a/h=5 45

00/900/00 Spherical Panel a/h=5 R/a=10

40

40

35

35 30

M1* (0/90)

25

M1* (0/90/0)

30 u3* M1*

25 20

M1* (0/90/90/0) 20

15

15

10

10

5

5

0

u3* (0/90) u3* (0/90/0) u3* (0/90/90/0)

0

0

10

20 m=n

30

40

Fig. 2. Convergence of normalized central deflection, u3 , and moment, M 1 , of a thick (a/h = 5) moderately deep (R/a = 10) [0°/90°/0°] spherical laminated panel.

0

10

20

30

40

50

a/h

Fig. 3. Variation of normalized central deflection, u3 , and moment, M 1 , with a/h ratio, of moderately deep (R/a = 10) spherical panels for different laminations.

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

earlier by Chaudhuri and Kabir [41]. This is possibly due to the presence of u3,n discontinuities (complementary boundary constraints) at all four edges, xn = constant, in the case of clamped 3 type boundary condition under investigation [47]. Fig. 3 presents the variations of the normalized central deflection, u3 , and moment, M 1 , of moderately deep (R/ a = 10) symmetric and antisymmetric cross-ply panels with respect to a/h ratio. The normalized central deflections of the two symmetric cross-ply panels are almost same in the entire range of the length-to-thickness ratio, a/h, considered. In contrast, the normalized central deflection, u3 , of the antisymmetric cross-ply panel is higher than its symmetric counterparts, although the difference progressively decreases as the aspect ratio, a/h, decreases from thin to moderately thick range. This difference for the values a/ h P 10 can be attributed to the ‘‘beam-column/tie-bar” effect caused by the bending-stretching coupling present in the antisymmetric laminate. However, the ‘‘beam-column/tie-bar” effect is countered by the transverse shear

299

deformation effect in the thick laminate regime, which explains the progressive decrease in difference between the two as the ratio, a/h, decreases from 15 to 10. The membrane action due to the effect of curvature has a complex three-way interaction with the bending-stretching type coupling effect, caused by the asymmetry of lamination as well as the type of surface-parallel boundary constraint. This interaction is stronger in the case of surface-parallel displacement boundary constraint, i.e., un = 0 prescribed at an edge xn = constant (e.g., SS2, SS4, C2 or C4) as compared to lack of it, i.e., Nn = 0 (e.g., SS1, SS3, C1, C3). For example, the bending-stretching type coupling has a highly pronounced interaction with the type of surface-parallel boundary constraint, imposed by e.g., C4 as compared to C3 type clamped boundary conditions, prescribed at all four edges. That is why unlike the other boundary conditions investigated earlier by the present author(s) [41,43,45], the effect of transverse shear deformation is not that pronounced even in the thick shell regime (a/h < 10) for both central deflection and moment.

Spherical Panel a/h=5

o

o

0 /90 /0

35

o

Spherical Panel

42

30

40

25 20 15

38

M 1*

u3* (0/90) u3* (0/90/0) M1* (0/90) M1* (0/90/0)

R/a=10 Plate

36 34 32

10

30

5

0

0

20

40

60

80

10

20

100

30

40

50

a/h

R/a

Fig. 4. Variation of normalized central deflection, u3 , and moment, M 1 , with R/a ratio, of thick (a/h = 5) spherical panels for different laminations.

Fig. 6. Variation of normalized central moment, M 1 , with a/h ratio, of [0°/ 90°/0°] spherical panels for different R/a ratios.

o

o

0 /90 Spherical & Cylindrical Panels a/h=10 o

0 /90

o

Spherical Panel

7

12 11

6.5

10

u 3*

8 7

6

Sphere R1=Inf R2=Inf

u 3*

a/h=5 a/h=10 a/h=20 a/h=50

9

5.5

6 5

5

4 3

4.5 0

20

40

60

80

100

R/a

Fig. 5. Variation of normalized central deflection, u3 , with R/a ratio, of [0°/90°] spherical panels for different a/h ratios.

0

20

40

60

80

100

R/a

Fig. 7. Variation of normalized central deflection, u3 , with R/a ratio, of moderately thick (a/h = 10) [0°/90°] spherical and cylindrical panels.

300

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306 o

o

o

0 /90 /0 Spherical & Cylindrical Panels a/h=10 4.58 4.575 4.57

u 3*

4.565

Sphere R1=Inf

4.56

R2=Inf 4.555 4.55 4.545 4.54 0

20

40

60

80

100

R/a

Fig. 8. Variation of normalized central deflection, u3 , with R/a ratio, of moderately thick (a/h = 10) [0°/90°/0°] spherical and cylindrical panels.

The radius-to-length ratio, R/a, exhibits a much less pronounced effect, to the point of being negligible, on both central moment and deflection for the C3 boundary condition (Nn = 0), investigated here, as can be seen in Figs. 4 and 5, the sole exception being very thin (e.g., a/h = 50) and moderately deep to deep (R/a < 20) [0°/90°] panels (see Fig. 5). Again, the membrane action due to the curvature effect is very sensitive to the type of surface-parallel boundary constraint, i.e., un = 0 or Nn = 0, prescribed at an edge xn = constant. This is also visible in Fig. 6. A moderately deep spherical (R/a = 10) and a flat [0°/90°/0°] panel show very close moment characteristics except in the thinner laminate regimes. Figs. 7 and 8 present the variations of normalized central deflections, u3 , of moderately thick (a/h = 10) antisymmetric [0°/90°] as well as symmetric [0°/90°/0°] cross-ply (one spherically and two cylindrically) curved panels with respect to R/a ratio, respectively. Unlike what has already been observed in the case of the SS3-SS2 and SS3-C4

o

boundary conditions [43,45], the normalized response, u3 , of the [0°/90°] cylindrical panel, with R2 ? 1, is higher (softening effect) than its spherical counterpart in the entire range of the R/a ratio considered. In contrast, u3 , of the [0°/90°] cylindrical panel, with R1 ? 1, is lower (hardening effect) than its spherical counterpart in the entire range of the R/a ratio considered. The beam-column/tie-bar effect due to bending-stretching coupling has a complex interaction with the membrane action due to curvature. In the former case, the membrane effect in the x1 direction acts in opposition to the bending-stretching coupling producing a beam-column effect, and hence the net softening. In the latter case, the membrane effect in the x2 direction acts synergistically with the bending-stretching coupling producing a tie-bar effect, and hence the net hardening. This is a very important factor that should be considered particularly at the initial conceptual design stages of composite shells. The differences in normalized deflections are almost negligible for spherical and cylindrical geometries for the values of R/a > 30. The normalized responses, u3 , of the R1 ? 1 and R2 ? 1 type cylindrical panels of symmetric [0°/90°/0°] construction are, however, almost identical in the same range of R/a ratio considered (Fig. 8). Variations of u1 ; u3 ; /1 , and M 1 of an antisymmetric thick (a/h = 5) and relatively flat (R/a = 50) spherical panel, computed at x2 = b/2 and plotted along x1 = 0 to a, are displayed in Fig. 9. The normalized deflection, u3 , assumes its maximum magnitude at the center of the panel where surface-parallel displacement, u1 and rotation /1 vanish. /1 and u1 reach their maximum values at x1 = a/ 4 and x1 = 3a/4. The central moment, M 1 , reaches its maximum value at the edges x1 = 0 and a, in order to provide the zero rotation (slope) there as dictated by the prescribed boundary condition Eqs. (3c) and (3d). Other boundary conditions imposed for the solution are also fully satisfied at the edges as can be seen in those plots.

o

0 /90 Spherical Panel a/h=5 R/a=50 4

2

7.5 6.5

u 3*

1 0 -1 0

8.5

u1=2*u1* u3=u3*/4 M1=M1*/10 Phi1=2*Phi1*

3

0.2

0.4

0.6

0.8

1

Spherical Panel a/h=20 R/a=50

9.5

0/90 0/90/0 0/90/90/0

5.5 4.5

-2

3.5

-3

2.5

-4

1.5 0

-5 x1 / a

Fig. 9. Variation of displacements, rotation, and moment along the panel center line, x2 = b/2, of a thick (a/h = 5) relatively flat (R/a = 50) [0°/90°] spherical panel.

5

10

15

20

25

30

E1 / E2

Fig. 10. Variation of normalized central deflection, u3 , with E1/E2 ratio, of a relatively thin (a/h = 20) and relatively flat (R/a = 50) spherical panel for different laminations.

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

Spherical Panel

Spherical Panel R/a=10

500 (0/90) a/h=5

400 350

(0/90/0) a/h=5

300

(0/90/90/0) a/h=5

250

(0/90) a/h=50

200 150

(0/90/0) a/h=50

100

(0/90/90/0) a/h=50

50

Relative Difference (Central Moment)

300

450 Relative Difference (Central Deflection)

301

250 200 150

0/90

100

0/90/90/0

0/90/0

50

0 0

20

40 60 R/a

80

100

0 0

10

20

30

40

50

a/h

Fig. 11. Variation of relative difference (central deflection), with R/a ratio, of spherical panels for different laminations and a/h ratios.

Spherical Panel

500

Relative Difference (Central Deflection)

450 400 350 300 250 200

(0/90) R/a=10

150

(0/90/0) R/a=10 (0/90) R/a=50

100

(0/90/0) R/a=50

50 0

10

20

30

40

50

a/h

Fig. 12. Variation of relative difference (central deflection), with a/h ratio, of spherical panels for different laminations and for different R/a ratios.

Spherical Panel a/h=5 Relative Difference (Central Moment)

260 240 220

0/90 0/90/0 0/90/90/0

200 180 160 140 120 100 0

20

40

60

80

100

R/a

Fig. 13. Variation of relative difference (central moment), with R/a ratio, of thick (a/h = 5) spherical panels for different laminations.

Fig. 14. Variation of relative difference (central moment), with a/h ratio, of moderately deep (R/a = 10) spherical panels for different laminations.

The degree of lamina material orthotropy,E1/E2, has a pronounced effect on the normalized central deflection of one antisymmetric and two symmetric cross-ply relatively thin (a/h = 20) and almost flat (R/a = 50) spherical panels as is illustrated in Fig. 10. These plots clearly show that this effect is significantly abetted by the afore-mentioned beamcolumn effect in the case of an antisymmetric laminate. This is because of the fact that the bending-stretching coupling prevalent in an antisymmetric laminate has a softening effect of the beam-column type, which consequently, increases the normalized central deflection. Variations of the relative differences, in central deflection, u3 , and moment, M 1 , with respect to both R/a and a/h ratios, of both antisymmetric [0°/90°] and symmetric [0°/90°/0°], [0°/90°/90°/0°] cross-ply spherical panels, between two different boundary conditions, SS3 and C3, signifying the effects of end clamping, are shown in Figs. 11–14. For the calculation of relative difference (% difference) between the two boundary conditions, the following formula is used: SS3  C3  100 ð13Þ % Difference ¼ C3 Fig. 11 shows that the relative difference in normalized central deflection of a thick (a/h = 5) spherical panel is virtually unaffected by membrane action due to curvature even in the deeper shell regime when compared to its thin (a/h = 50) counterpart. The relative difference in central deflection, between C3 and SS3 boundary conditions, of a thin (a/h = 50) cross-ply spherical panel monotonically decreases with the membrane action due to curvature. However, the effect of curvature is more pronounced in the case of a [0°/90°] spherical panel in comparison to its symmetric laminate counterparts. Fig. 13 shows that the relative differences in normalized central moments of thick (a/h = 5) symmetric spherical panels are virtually unaffected by membrane action due to curvature even in the

302

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

deeper shell regime. The same of an antisymmetric spherical panel decreases very gradually in the deeper shell regime. It is further interesting to observe from Fig. 12 that the relative difference in central deflection of a moderately deep (R/a = 10) antisymmetric cross-ply spherical panel first increases with a/h ratio from the very thick to moderately thick shell regime, and then decreases from moderately thick to thin shell regime. In contrast, the relative difference in central deflection of a relatively flat (R/a = 50) antisymmetric cross-ply spherical panel increases monotonically with a/h ratio in the entire range of thickness considered. Furthermore, the relative differences in central deflections of the two sets of symmetric cross-ply spherical panels increase monotonically and more drastically with a/h ratio in the entire range of thickness considered (Fig. 12). This is because the effect of the transverse shear deformation (in the thick shell regime) is compensated to a certain extent by the bending-stretching coupling effect that characterizes antisymmetric laminates [3,32]. A somewhat opposite behavior appears to be true in the case of variation of the relative difference in central moment (Fig. 14). The relative difference in central moment of an antisymmetric cross-ply spherical panel decreases monotonically and more drastically with a/h ratio than its symmetric counterparts. 5. Summary and conclusions A new boundary discontinuous Fourier solution to the problem of deformation of a finite-dimensional general cross-ply thick rectangular shell is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, with the C3-type clamped boundary condition prescribed at all four edges. Unlike the conventional Navier and Levy type approaches which can provide only particular solutions, the present method is general enough to provide the complete (particular as well as the complementary) solution for any arbitrary combination of admissible boundary conditions with relative ease. Due to its inherent advantages over the classical lamination theory (CLT) and the first order shear deformation theory (FSDT), which under-predicts the deflections as compared to their HSDT counterparts in the thicker shell regime (a/h 6 10), a simple higher order shear deformation theory (HSDT) is considered in the formulation of thick shell boundary-value problems, and subsequent derivation of their boundary-discontinuous double Fourier series solutions. The primary focus of the present study is the effect of edge clamping on the response of thick cross-ply shells, while keeping the surface-parallel edge constraints unaltered. Numerical results presented here on symmetric and antisymmetric cross-ply laminates demonstrate convergence characteristics of deflections and moments. The C3 and SS3 boundary conditions are selected because (i) the

latter admits Navier type solutions that are readily available in the literature, and (ii) represent the two extremes of the rotational edge constraint, the latter ensuring complete rotational freedom, while the former enforcing complete fixity. Important conclusions drawn from the numerical results may be summarized as follows:

(i) The central deflection (transverse displacement) shows a rapid and monotonic convergence, while the central moment shows an initially oscillatory convergence. The latter is possibly due to the presence of a discontinuity (complementary boundary constraint) in the derivative of the displacement in the expression for the moment. In addition, because of the same reason, the computed solutions for clamped boundary conditions (e.g., C3) converge less rapidly than their simply supported counterparts. (ii) The normalized central deflections of antisymmetric cross-ply spherical panels are higher than their symmetric counterparts, though the difference progressively decreases as the length-to-thickness ratio, a/h, decreases from thin to moderately thick range. The difference becomes negligible in the thick shell regime (a/h < 10). This difference for the values a/h P 10 can be attributed to the ‘‘beam-column” effect caused by the bending-stretching coupling present in the antisymmetric laminate. (iii) The membrane action due to the effect of curvature has a complex three-way interaction with the bending-stretching type coupling effect, caused by the asymmetry of lamination as well as the type of surface-parallel boundary constraint. This interaction is stronger in the case of surface-parallel displacement boundary constraint, i.e., un = 0 prescribed at an edge xn = constant (e.g., SS2, SS4, C2 or C4) as compared to a lack of it, i.e., Nn = 0 (e.g., SS1, SS3, C1, C3). For example, the bending-stretching type coupling has a highly pronounced interaction with the type of surface-parallel boundary constraint, imposed by e.g., C4 as compared to C3 type clamped boundary conditions, prescribed at all four edges. That is why unlike the other boundary conditions investigated earlier by the present author(s) [41,43,45], the effect of transverse shear deformation is not that pronounced even in the thick shell regime (a/h < 10) for both central deflection and moment. (iv) The radius-to-length ratio, R/a, exhibits a much less pronounced effect, to the point of being negligible, on both central moment and deflection for the C3 boundary condition (Nn = 0), investigated here, the sole exception being very thin (e.g., a/h = 50) and moderately deep to deep (R/a < 20) [0°/90°] panels. Again, the membrane action due to the curvature effect is very sensitive to the type of surface-parallel boundary constraint, i.e., un = 0 or Nn = 0, prescribed at an edge xn = constant.

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

(v) Unlike what has already been observed in the case of the SS3-SS2 and SS3-C4 boundary conditions [43,45], the normalized response, u3 , of the [0°/90°] cylindrical panel, with R2 ? 1, is higher (softening effect) than its spherical counterpart in the entire range of the R/a ratio considered. In contrast, u3 , of the [0°/90°] cylindrical panel, with R1 ? 1, is lower (hardening effect) than its spherical counterpart in the entire range of the R/a ratio considered. The beam-column/tie-bar effect due to bending-stretching coupling has a complex interaction with the membrane action due to curvature. In the former case, the membrane effect in the x1 direction acts in opposition to the bending-stretching coupling producing a beam-column effect, and hence the net softening. In the latter case, the membrane effect in the x2 direction acts synergistically with the bending-stretching coupling producing a tie-bar effect, and hence the net hardening. The differences in normalized deflections are almost negligible for spherical and cylindrical geometries for the values of R/a > 30. The normalized responses, u3 , of the R1 ? 1 and R2 ? 1 type cylindrical panels of symmetric [0°/90°/0°] construction are, however, almost identical in the same range of R/a ratio considered. (vi) The lamina material orthotropy (E1/E2) has a pronounced effect on the normalized central deflection of both symmetric and antisymmetric cross-ply relatively thin (a/h = 20) and almost flat (R/a = 50) spherical panels investigated here. Furthermore, this effect is significantly abetted by the afore-mentioned beam-column effect in the case of an antisymmetric laminate. This is because of the fact that the bending-stretching coupling prevalent in an antisymmetric laminate has a softening effect of the beam-column type, which consequently, increases the normalized central deflection. (vii) Clamped C3 type boundary condition decreases the deflections by more than a factor of 4 in the case of a thin (a/h = 50) relatively flat (R/a = 50) antisymmetric cross-ply [0°/90°] spherical panel when compared to its SS3 counterpart. The corresponding reduction factor for a thin (a/h = 50) relatively flat (R/a = 50) symmetric spherical panel is of the order of 5.5. This ratio is greater than 2 for a moderately thick (a/h = 10) and moderately deep (R/a = 10) symmetric cross-ply spherical panel, and drops to a value of 1.75 (approx.) for a thick (a/h = 5) and moderately deep (R/a = 10) symmetric cross-ply spherical panel. The corresponding reduction factor for a thick (a/h = 5) and moderately deep (R/a = 10) antisymmetric cross-ply spherical panel is 2.3 (approx). The difference between the effects of clamping on the two symmetric cross-ply panels varies from being small (a/h = 50) to negligible (a/h = 5). The comparisons provided between the SS3 and C3

303

boundary conditions here clearly show the severe effect of edge clamping on both the deflections and moments.

Appendix A. Basic equations Only keeping the cubic terms and satisfying the conditions of transverse shear stresses (and hence strains) vanishing at a point (n1, n2, ±h/2) on the outer (top) and inner (bottom) surfaces of the shell, yields



 f 4f3 1 u1 þ f/1  2 /1 þ u3;1 ; 1þ R1 g1 3h

  3 f 4f 1 u2 ¼ 1 þ u2 þ f/2  2 /2 þ u3;2 ; R2 g2 3h u1 ¼

u3 ¼ u3 ;

ðA1aÞ ðA1bÞ ðA1cÞ

where ui ði ¼ 1; 2; 3Þ denotes the displacements of a point on the middle surface, while /1 and /2 are the rotations at f = 0 with respect to the n2 and n1 axes, respectively. The corresponding kinematic relations are given by e1 ¼ e01 þ fðj01 þ f2 j21 Þ;

ðA2aÞ

f2 j22 Þ;

ðA2bÞ

e2 ¼

e02

þ

fðj02

þ

e4 ¼ e04 þ f2 j14 ; e5 ¼

e05

þ

ðA2cÞ

f2 j15 ;

e6 ¼ e06 þ fðj06 þ

ðA2dÞ f2 j26 Þ;

ðA2eÞ

in which e01 ¼ u1;1 þ

u3 ; R1

j01 ¼ /1;1 ;

j21 ¼ 

e02 ¼ u2;2 þ

u3 ; R2

j02 ¼ /2;2 ;

j22 ¼ 

4 ð/1;1 þ u3;11 Þ; 3h2 ðA3a–cÞ 4 ð/2;2 þ u3;22 Þ; 3h2 ðA3d–fÞ

4 ð/2 þ u3;2 Þ; h2 4 e05 ¼ u3;1 þ /1 ; j15 ¼  2 ð/1 þ u3;1 Þ; h e06 ¼ u2;1 þ u1;2 ; j06 ¼ /2;1 þ /1;2 ; 4 j26 ¼  2 ð/2;1 þ /1;2 þ 2u3;12 Þ: 3h e04 ¼ u3;2 þ /2 ;

j14 ¼ 

ðA3g;hÞ ðA3i; jÞ

ðA3k–mÞ

For a general cross-ply (symmetric [0°/90°. . .]S and antisymmetric [0°/90°/0°/90°. . .] being two special cases) shell, the following elastic rigidities (integrated stiffnesses) are zero: A16 ¼ A26 ¼ B12 ¼ B16 ¼ B26 ¼ D16 ¼ D26 ¼ D45 ¼ A45 ¼ E12 ¼ E16 ¼ E26 ¼ F 16 ¼ F 26 ¼ F 45 ¼ H 16 ¼ H 26 ¼ 0: ðA4Þ

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A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

The equilibrium equations are written as shown below [45]:

P 2 ¼ E12 u1;1 þ b19 u3 þ E22 u2;2 þ b13 /1;1 þ b16 /2;2

N 1;1 þ N 6;2 ¼ 0;

ðA5aÞ

P 6 ¼ E66 u2;1 þ E66 u1;2 þ b18 /2;1 þ b18 /1;2  b19 u3;12 ;

ðA7iÞ

N 6;1 þ N 2;2 ¼ 0;

ðA5bÞ

Q1 ¼ d 2 /1 þ d 2 u3;1 ;

ðA7jÞ

Q2 ¼ d 1 /2 þ d 1 u3;2 ;

ðA7kÞ

K 1 ¼ d 4 /1 þ d 4 u3;1 ;

ðA7lÞ

K 2 ¼ d 3 /2 þ d 3 u3;2 ;

ðA7mÞ

4 4 Q1;1 þ Q2;2  2 ðK 1;1 þ K 2;2 Þ þ 2 ðP 1;11 þ P 2;22 þ 2P 6;12 Þ h 3h N1 N2  ¼ q; ðA5cÞ  R1 R 2 4 4 M 1;1 þ M 6;2  Q1 þ 2 K 1  2 ðP 1;1 þ P 6;2 Þ ¼ 0; ðA5dÞ h 3h 4 4 ðA5eÞ M 6;1 þ M 2;2  Q2 þ 2 K 2  2 ðP 6;1 þ P 2;2 Þ ¼ 0; h 3h where q is the distributed transverse load, and Ni, Mi, Pi, i ¼ 1; 2; 6 denote stress resultants, stress couples (moment resultants), and higher-order stress couples (resultants of the higher moment of stress). Qi, i ¼ 4; 5, represents the transverse shear stress resultants, while Ki, i ¼ 4; 5 denotes higher-order transverse shear stress resultants. They are written as follows: ðN i ; M i ; P i Þ ¼

N Z X

xk3

xk1 3

k¼1

ðkÞ

ri ð1; x3 ; x33 Þ dx3 ;

ðQ1 ; K 1 Þ ¼

k¼1

ðQ2 ; K 2 Þ ¼

Z

xk1 3

N Z X k¼1

xk3

xk3

xk1 3

ðkÞ

r5 ð1; x23 Þ dx3 ; ðkÞ

r4 ð1; x23 Þ dx3 :

in which Aij, Bij, Dij are the laminate rigidities (integrated stiffnesses) shared by all laminated shell/plate theories, such as CLT, FSDT and HSDT, while rigidities Eij, Fij, Hij arise out of the higher order shear terms specific to the present HSDT. These are given as follows: N Z xk3 X ðkÞ Qij ð1; x3 ; x23 Þ dx3 ; ðA8aÞ ðAij ; Bij ; Dij Þ ¼ k¼1

ðEij ; F ij ; H ij Þ ¼

xk1 3

N Z X k¼1

xk3

xk1 3

ðkÞ

Qij ðx33 ; x43 ; x63 Þ dx3 ;

ðA8bÞ

for ði; j ¼ 1; 2; 4; 5; 6Þ,

Appendix B. Definition of certain constants ðA6cÞ

N 1 ¼ A11 u1;1 þ a1 u3 þ A12 u2;2 þ a2 /1;1 þ a3 /2;2  a4 u3;11  a5 u3;22 ;

ðA7aÞ

N 2 ¼ A12 u1;1 þ a6 u3 þ A22 u2;2 þ a3 /1;1 þ a7 /2;2  a5 u3;11  a8 u3;22 ; N 6 ¼ A66 u2;1 þ A66 u1;2 þ a9 /2;1 þ a9 /1;2  a10 u3;12 ;

ðA7bÞ ðA7cÞ

M 1 ¼ B11 u1;1 þ b1 u3 þ B12 u2;2 þ b2 /1;1 þ b3 /2;2  b4 u3;11  b5 u3;22 ;

ðA7dÞ

M 2 ¼ B12 u1;1 þ b6 u3 þ B22 u2;2 þ b3 /1;1 þ b7 /2;2  b5 u3;11  b8 u3;22 ; M 6 ¼ B66 u2;1 þ B66 u1;2 þ b9 /2;1 þ b9 /1;2  b10 u3;12 ;

where Qij denotes the reduced elastic stiffnesses of the kth lamina [50]. The constants ai, bi, di, referred to in Eqs. (A7) above, are given in Appendix B.

ðA6bÞ

These stress resultants, moment resultants and higherorder moment and shear resultants, on application of Hooke’s law, and integration through the laminated shell thickness (in conjunction with Eq. (A4)), in terms of components of displacement and rotation can now be written as follows:

ðA7eÞ ðA7fÞ

P 1 ¼ E11 u1;1 þ b11 u3 þ E12 u2;2 þ b12 /1;1 þ b13 /2;2  b14 u3;11  b15 u3;22 ;

ðA7hÞ

ðkÞ

ði; j ¼ 1; 2; 6Þ ðA6aÞ

N X

 b15 u3;11  b17 u3;22 ;

ðA7gÞ

A11 A12 4E11 4E12 þ ; a2 ¼ B11  2 ; a3 ¼ B12  2 ; R1 R2 3h 3h 4E11 4E12 A12 A22 ; a5 ¼ ; a6 ¼ þ ; a4 ¼ R1 R2 3h2 3h2 4E22 4E22 4E66 ; a9 ¼ B66  2 ; a7 ¼ B22  2 ; a8 ¼ 2 3h 3h 3h 8E66 ; ðB1aÞ a10 ¼ 3h2 B11 B12 4F 11 4F 12 þ ; b2 ¼ D11  2 ; b3 ¼ D12  2 ; b1 ¼ R1 R2 3h 3h 4F 11 4F 12 B12 B22 ; b5 ¼ ; b6 ¼ þ ; b4 ¼ R1 R2 3h2 3h2 4F 22 4F 22 4F 66 ; b9 ¼ D66  2 ; b7 ¼ D22  2 ; b8 ¼ 2 3h 3h 3h 8F 66 E11 E12 4H 11 b10 ¼ ; b11 ¼ þ ; b12 ¼ F 11  ; 2 R1 R2 3h 3h2 4H 12 4H 11 4H 12 ; b14 ¼ ; b15 ¼ ; b13 ¼ F 12  2 2 3h 3h 3h2 4H 22 4H 22 4H 66 b16 ¼ F 22  ; b17 ¼ ; b18 ¼ F 66  ; 2 2 3h 3h 3h2 8H 66 b19 ¼ ; ðB1bÞ 3h2 4D44 4D55 4F 44 d 1 ¼ A44  2 ; d 2 ¼ A55  2 ; d 3 ¼ D44  2 ; h h h 4F 55 d 4 ¼ D55  2 ; ðB1cÞ h a1 ¼

A.S. Oktem, R.A. Chaudhuri / Composite Structures 87 (2009) 293–306

e1 ¼ B12 þ B66  4b12 ; 3h2 4b18 e 5 ¼ b9  2 ; 3h

e 3 ¼ b2 

e7 ¼ b5  b10 þ e 9 ¼ b7 

4b16 ; 3h2

4E12 4E66  2 ; 3h2 3h

e 2 ¼ b1  d 2 þ

e 4 ¼ b3 þ b 9  e6 ¼ b4 þ

4d 4 4b11  2; h2 3h

4b13 4b18  2; 3h2 3h

4b14 ; 3h2

4b15 4b19 þ 2; 3h2 3h e10 ¼ b8 þ

e8 ¼ d 2 þ 4b17 ; 3h2

4d 3 4b20  2; h2 3h f1 ¼ A12 þ A66 ; f 2 ¼ a3 þ a9 ;

4d 4 ; h2

e11 ¼ d 1 þ

e12 ¼ b6  d 1 þ

4d 3 ; h2 ðB1dÞ

f 3 ¼ a10  a5 ;

4d 4 a2 a3 4d 3 a3 a7   ; f 5 ¼ d1  2   ; 2 R 1 R2 R1 R2 h h 4d 4 4b11 a4 a5 f 6 ¼ d2  2 þ 2 þ þ ; R1 R2 h 3h 4d 3 4b20 a5 a8 4E12 8E66 þ 2 ; f 7 ¼ d1  2 þ 2 þ þ ; f 8 ¼ R1 R2 h 3h 3h2 3h 4b12 4b13 8b18 4b14 f 9 ¼ 2 ; f 10 ¼ 2 þ 2 ; f11 ¼ 2 ; 3h 3h 3h 3h 8b15 8b19 4b16 4b17 f 12 ¼  2  2 ; f 13 ¼ 2 ; f 14 ¼ 2 ; 3h 3h 3h 3h a1 a6 ðB1eÞ f 15 ¼   : R1 R2 f 4 ¼ d2 

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