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Levy type Fourier analysis of thick cross-ply doubly curved panels
Composite Structures 80 (2007) 475–488 www.elsevier.com/locate/compstruct
Levy type Fourier analysis of thick cross-ply doubly curved panels Ahmet Sinan Oktem a, Reaz A. Chaudhuri a
b,*
Department of Naval Architecture and Marine Engineering, Yildiz Technical University, 34349 Besiktas-Istanbul, Turkey b Department of Materials Science and Engineering, University of Utah, 1225 S. Central Campus Drive, Room 304, Salt Lake City, UT 84112-0560, United States Available online 2 August 2006
Abstract A hitherto unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doubly curved 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 general cross-ply shell analysis, with the SS2-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of a moderately thick cross-ply spherical panel. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, lamina material property, and thickness and curvature effects, as well as their interactions. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Thick laminate; Higher order shear deformation theory (HSDT); Doubly curved panel; Spherical shell; Cylindrical shell; Cross-ply; Analytical solution; Boundary discontinuous double Fourier series; Boundary constraints
1. Introduction 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 flat/curved panels because of many beneficial properties, such as higher strength-to-weight and stiffness-to-weight ratios (resulting in fuel economy), longer inplane fatigue (including sonic fatigue) life and stealth characteristics (of military aircraft, e.g., stealth fighter, F-117A Nighthawk and B-2 bomber), enhanced corrosion resistance, and so on. Since the matrix material is of relatively low shearing stiffness as compared to the fibers, polymeric composite shell type structures are highly prone to transverse shear related fatigue failures. A reliable prediction of the response of these laminated shells or doubly curved panels must account for transverse shear deformation. Additionally, a solution to the problem of *
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 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.05.020
deformation of laminated shells of finite dimensions must satisfy the prescribed 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 analysis of a model 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). Examples of CLT-based analysis of laminated cylindrical and doubly curved shells/panels include Bert and Reddy [1], Chaudhuri et al. [2] and Chaudhuri and Kabir [3], while Chaudhuri and Abu-Arja [4,5], Chaudhuri and Kabir [6–8], Kabir and Chaudhuri [9,10], and Kabir et al. [11], have presented double Fourier series based analytical solutions to moderately thick (FSDT-based) composite shell boundary-value problems. 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
476
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
assumes a uniform transverse shear strain distribution through the thickness, which violates equilibrium conditions at the top and bottom surfaces of the panel. Noor and Burton [12] have presented an extensive survey on various computational models relating to laminated shells. Approximate thick shell theories can be classified into two categories: (a) discrete layer approach (see e.g., [13] and the references contained therein), and (b) continuous inplane displacement through thickness. The former approach appears to be more suitable for numerical methods, such as the finite element methods (FEM). Additionally, post-processing type methods used in conjunction with a discrete layer approach have yielded highly accurate interlaminar shear stress distribution through the thickness of symmetric and unsymmetric laminated shells [14,15]. Basset [16] 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 surface-parallel displacements through the thickness of thick laminated shells, have been developed as special cases of the above (see e.g., [17,18] and the references contained therein). In what follows, a hitherto unavailable HSDT-based boundary-discontinuous double Fourier series solution to the boundary value problem of general cross-ply doubly curved panels, with the SS3 type simply supported boundary condition, prescribed at two opposite edges, while the remaining two edges are subjected to the SS2 type simply supported one is derived. 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 [19,20]. The numerical accuracy of the present solution is ascertained by studying its convergence characteristics, and also by comparison with the available FSDT-based analytical solution. Numerical results are presented to understand the complex deformation behavior of symmet-
Fig. 1. Geometry of a laminated doubly curved panel.
ric and antisymmetric cross-ply cylindrical and spherical panels. 2. Statement of the problem Let (n1, n2, n3 = f) 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 [21,22] 1 1 g u1;1 þ g1;2 u2 þ 1 u3 ; e1 ðn1 ; n2 ; n3 Þ ¼ ð1aÞ g2 R1 1 þ Rn31 g1 1 1 g u2;2 þ g2;1 u1 þ 2 u3 ; e2 ðn1 ; n2 ; n3 Þ ¼ ð1bÞ g1 R2 1 þ Rn32 g2 e3 ðn1 ; n2 ; n3 Þ ¼ u3;3 ;
1 g e4 ðn1 ; n2 ; n3 Þ ¼ u3;2 2 u2 þ u2;3 ; n3 R 2 1 þ R2 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
ð1cÞ ð1dÞ
ð1eÞ
ð1fÞ
where ei (i = 1, 2, 3, 4, 5, 6) represents the components of the strain tensor, and ui (i = 1, 2, 3) denotes the components of the displacement vector along the (n1, n2, n3 = f) coordinates at a point, (n1, n2, n3 = f). 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 neglect of the geodesic curvatures of the coordinate lines. The surfaceparallel displacements can be expanded in power series of n3 = f as suggested by Basset [16]. 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 top and bottom surfaces of the shell, yields f 4f3 1 u1 ¼ 1 þ u1 þ f/1 2 /1 þ u3;1 ; ð2aÞ R1 g1 3h
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
u2 ¼
f 4f3 1 1þ u2 þ f/2 2 /2 þ u3;2 ; R2 g2 3h
u3 ¼ u3 ;
ð2bÞ ð2cÞ
where ui (i = 1, 2, 3) 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
resultants), and higher-order stress couples (resultants of the higher moment of stress). Qi, i = 4, 5, represents the transverse shear stress resultants, while Ki, i = 4, 5 denotes higher-order shear stress resultants. They are written as follows: N i ¼ Aij e0j þ Bij j0j þ Eij j2j ;
ð6aÞ
þ
f2 j21 Þ;
ð3aÞ
M i ¼ Bij e0j þ Dij j0j þ F ij j2j ; P i ¼ Eij e0j þ F ij j0j þ H ij j2j
e2 ¼ e02 þ fðj02 þ
f2 j22 Þ;
ð3bÞ
Q1 ¼ A5j e0j þ D5j j1j ;
e1 ¼
e01
þ
fðj01
e4 ¼ e04 þ f2 j14 ;
ð3cÞ
e5 ¼ e05 þ f2 j15 ;
ð3dÞ
e6 ¼ e06 þ fðj06 þ f2 j26 Þ;
ð3eÞ
in which e01 ¼ u1;1 þ
u3 ; R1
ð4aÞ
j01 ¼ /1;1 ; 4 j21 ¼ 2 ð/1;1 þ u3;11 Þ; 3h u3 0 e2 ¼ u2;2 þ ; R2 j02 ¼ /2;2 ; 4 j22 ¼ 2 ð/2;2 þ u3;22 Þ; 3h e04 ¼ u3;2 þ /2 ; 4 j14 ¼ 2 ð/2 þ u3;2 Þ; h e05 ¼ u3;1 þ /1 ; 4 j15 ¼ 2 ð/1 þ u3;1 Þ; h e06 ¼ u2;1 þ u1;2 ;
ð4bÞ
j06
ð4lÞ
¼ /2;1 þ /1;2 ; 4 j26 ¼ 2 ð/2;1 þ /1;2 þ 2u3;12 Þ: 3h
ð4cÞ
Q2 ¼
A4j e0j
K1 ¼
D5j e0j
þ
D4j j1j ;
þ
F 5j j1j
ði; j ¼ 1; 2; 6Þ
ð6bÞ ð6cÞ ð6dÞ ð6eÞ
ðj ¼ 4; 5Þ
K 2 ¼ D4j e0j þ F 4j j1j ;
ð6fÞ ð6gÞ
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 fk X ðkÞ Qij ð1; f; f2 Þ df; ð7aÞ ðAij ; Bij ; Dij Þ ¼ k¼1
ð4dÞ
477
ðEij ; F ij ; H ij Þ ¼
ð4eÞ
N X k¼1
fk1
Z
fk
ðkÞ
Qij ðf3 ; f4 ; f6 Þ df
ð7bÞ
fk1 ðkÞ
ð4fÞ ð4gÞ ð4hÞ ð4iÞ ð4jÞ ð4kÞ
ð4mÞ
for (i, j = 1, 2, 4, 5, 6), where Qin denotes the reduced elastic stiffnesses of the kth lamina [23]. The stress resultants, moment resultants and higherorder moment and shear resultants in terms of components of displacement and rotation can now be written as follows: 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 ; N 2 ¼ A12 u1;1 þ a6 u3 þ A22 u2;2 þ a3 /1;1 þ a7 /2;2
ð8aÞ
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 ;
ð8bÞ ð8cÞ
M 1 ¼ B11 u1;1 þ b1 u3 þ B12 u2;2 þ b2 /1;1 þ b3 /2;2
The equilibrium equations for a moderately shallow doubly curved panel can be written as shown below:
b4 u3;11 b5 u3;22 ; M 2 ¼ B12 u1;1 þ b6 u3 þ B22 u2;2 þ b3 /1;1 þ b7 /2;2
ð8dÞ
N 1;1 þ N 6;2 ¼ 0; ð5aÞ N 6;1 þ N 2;2 ¼ 0; ð5bÞ 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; ð5cÞ R1 R 2 4 4 ð5dÞ M 1;1 þ M 6;2 Q1 þ 2 K 1 2 ðP 1;1 þ P 6;2 Þ ¼ 0; h 3h 4 4 M 6;1 þ M 2;2 Q2 þ 2 K 2 2 ðP 6;1 þ P 2;2 Þ ¼ 0; ð5eÞ h 3h
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 ;
ð8eÞ ð8fÞ
where q is the distributed transverse load, and Ni, Mi, Pi, i = 1, 2, 6 denote stress resultants, stress couples (moment
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 ; P 2 ¼ E12 u1;1 þ b20 u3 þ E22 u2;2 þ b13 /1;1 þ b16 /2;2
ð8gÞ
b15 u3;11 b17 u3;22 ; P 6 ¼ E66 u2;1 þ E66 u1;2 þ b18 /2;1 þ b18 /1;2 b19 u3;12 ;
ð8hÞ ð8iÞ
Q1 Q2 K1 K2
¼ d 2 /1 þ d 2 u3;1 ; ¼ d 1 /2 þ d 1 u3;2 ; ¼ d 4 /1 þ d 4 u3;1 ; ¼ d 3 /2 þ d 3 u3;2 :
ð8jÞ ð8kÞ ð8lÞ ð8mÞ
478
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
The constants ai, bi, di, referred to in Eqs. (8) are given in Appendix A. Substitution of Eqs. (8) in equilibrium equations, given by Eqs. (5), supplies the following five highly coupled fourth-order governing partial differential equations:
þ f3 u3;122 þ A66 u1;22 þ a9 /1;22 ¼ 0;
ð9aÞ
A66 u2;11 þ f1 u1;12 þ a9 /2;11 þ f2 /1;12 þ f3 u3;112 þ A22 u2;22 þ a6 u3;2 þ a7 /2;22 a8 u3;222 ¼ 0; f4 /1;1 þ f5 /2;2 þ f6 u3;11 þ f7 u3;22 þ a4 u1;111 þ f8 u2;112
ð9bÞ
þ f9 /1;111 þ f10 /2;112 f11 u3;1111 þ f12 u3;1122 þ f8 u1;122
0 < x1 < a; 0 6 x2 6 b; 1 X 1 X u2 ¼ V mn sinðax1 Þ cosðbx2 Þ; 0 < x1 < a; 0 6 x2 6 b; 1 X 1 X u3 ¼ W mn sinðax1 Þ sinðbx2 Þ;
ð12aÞ
ð12bÞ
m¼1 n¼1
0 6 x1 6 a; 0 6 x2 6 b; 1 X 1 X X mn cosðax1 Þ sinðbx2 Þ; /1 ¼ 0 6 x1 6 a; 0 6 x2 6 b;
ð9cÞ
a2 u1;11 þ e1 u2;12 þ e2 u3;1 þ e3 /1;11 þ e4 /2;12 þ e5 /1;22
ð12cÞ
/2 ¼
1 X
1 X
ð12dÞ
Y mn sinðax1 Þ cosðbx2 Þ;
m¼1 n¼0
ð9dÞ
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;
U mn cosðax1 Þ sinðbx2 Þ;
m¼0 n¼1
m¼0 n¼1
þ a8 u2;222 þ f10 /1;122 þ f13 /2;222 f14 u3;2222 a1 u1;1
þ e6 u3;111 þ e7 u3;122 þ a9 u1;22 þ e8 /1 ¼ 0;
1 X 1 X
m¼1 n¼0
A11 u1;11 þ a1 u3;1 þ f1 u2;12 þ a2 /1;11 þ f2 /2;12 a4 u3;111
a6 u2;2 þ f15 u3 ¼ q;
u1 ¼
ð9eÞ
in which ai, ei, fi are constants, which are given in Appendix A. In what follows, a double Fourier series solution is sought for the following mixed boundary condition. The SS2 type simply supported boundary condition, prescribed at the edges x1 = 0, a, is given as follows [18]: u3 ð0; x2 Þ ¼ u3 ða; x2 Þ ¼ 0;
ð10aÞ
M 1 ð0; x2 Þ ¼ M 1 ða; x2 Þ ¼ 0;
ð10bÞ
/2 ð0; x2 Þ ¼ /2 ða; x2 Þ ¼ 0;
ð10cÞ
u1 ð0; x2 Þ ¼ u1 ða; x2 Þ ¼ 0;
ð10dÞ
N 6 ð0; x2 Þ ¼ N 6 ða; x2 Þ ¼ 0;
ð10eÞ
P 1 ð0; x2 Þ ¼ P 1 ða; x2 Þ ¼ 0:
ð10fÞ
At edges x2 = 0, b, the SS3 type simply supported condition is prescribed as per the dictate of the Levy type solution u3 ðx1 ; 0Þ ¼ u3 ðx1 ; bÞ ¼ 0;
ð11aÞ
M 2 ðx1 ; 0Þ ¼ M 2 ðx1 ; bÞ ¼ 0;
ð11bÞ
/1 ðx1 ; 0Þ ¼ /1 ðx1 ; bÞ ¼ 0;
ð11cÞ
N 2 ðx1 ; 0Þ ¼ N 2 ðx1 ; bÞ ¼ 0;
ð11dÞ
u1 ðx1 ; 0Þ ¼ u1 ðx1 ; bÞ ¼ 0;
ð11eÞ
P 2 ðx1 ; 0Þ ¼ P 2 ðx1 ; bÞ ¼ 0:
ð11fÞ
3. Method of solution The displacement functions (particular solution) are assumed to be in the form [18,20,24]:
0 6 x1 6 a; 0 6 x2 6 b;
ð12eÞ
where a¼
mp ; a
b¼
np : b
ð13Þ
It may be noted that the assumed solution functions, given by Eqs. (12), satisfy the SS3 type simply supported condition, given by Eqs. (11), at the edges, x2 = 0, b, a priori. The total number of unknown Fourier coefficients introduced in Eqs. (12) numbers 5mn + 2m + 2n. The next operation is comprised of partial differentiation of the assumed particular solution functions. The procedure for differentiation of these functions is based on Lebesgue integration theory that introduces boundary Fourier coefficients arising from discontinuities (complementary boundary constraints [19,20]) of the particular solution functions at the edges x1 = 0, a. As has been noted by Chaudhuri [20], 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 [19,20], and will not be further discussed here in the interest of brevity of presentation. The partial derivatives of these functions are obtained as follows: 1 X 1 X u1 ¼ U mn cosðax1 Þ sinðbx2 Þ; ð14aÞ m¼0 n¼1
u1;1 ¼
1 X 1 X
aU mn sinðax1 Þ sinðbx2 Þ:
ð14bÞ
m¼1 n¼1
The above function u1 and its first partial derivative, u1,1, are not satisfied a priori at the edges, at x1 = 0, a, thus violating the boundary constraints and complementary boundary constraints, respectively, at these edges. Therefore, for further differentiation, u1,1 is first
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
expanded in double Fourier series, in the form suggested by Chaudhuri [19,20] in order to satisfy the complementary boundary constraint (inequality), while u1 is forced to vanish at these edges (boundary constraints). The second partial derivative is then obtained as follows [20]: u1;11 ¼
1 1 X 1 X X 2 1 a U mn þ cm an þ wm bn an sinðbx2 Þ þ 2 m¼1 m¼1 n¼1
cosðax1 Þ sinðbx2 Þ:
ð15Þ
The partial derivative of u2 are given as follows: 1 1 u2;1 ¼ c0 þ 4 2
1 X
cn cosðbx2 Þ þ
n¼1
þ wm d0 cosðax1 Þ þ
1 2
1 X 1 X
½aV m0 þ cmc0
½aV mn þ cmcn ð16Þ
ð0; 1Þ; m ¼ odd;
ð17Þ
ð1; 0Þ; m ¼ even;
n , bn , cn and and where the boundary Fourier coefficients a dn , are as defined in Appendix B. The remaining particular solution functions or their partial derivatives do not have any boundary discontinuities, and therefore, can be differentiated term by term. This step generates additional 4n + 2 unknowns. Introduction of the displacement functions and their appropriate derivatives into the governing partial differential equations will supply 5mn + 2m + 2n equations as given below: 1 X 1 X
cosðax1 Þ sinðbx2 Þ
m¼1 n¼1
a2 A11 b2 A66 U mn 3
m¼1 n¼1
m¼1 n¼1
ð18bÞ
þ a6 b þ a8 b3 þ f8 a2 b V mn f6 a2 f15 þ f7 b2 þ f11 a4 f12 a2 b2 þ f14 b4 W mn f4 a f9 a3 f10 ab2 X mn f5 b b13 b3 f10 a2 b Y mn a4 aðcm an þ w m bn Þ 1 X 1 X qmn sinðax1 Þ sinðbx2 Þ; ð18cÞ ¼ m¼1 n¼1
m¼1 n¼1
a9 a2 þ a7 b2 V mn þ e12 b e7 a2 b e10 b3 W mn e4 abX mn e5 a2 e11 þ e9 b2 Y mn aa9 cmcn þ wm dn ¼ 0;
sinðbx2 Þ
ð18eÞ
A66 2 A11 an ¼ 0; b U 0n þ a9 b2 X 0n 2 2
ð19aÞ
ð19bÞ ha i a2 9 sinðbx2 Þ b2 U 0n ðe8 e5 b2 ÞX 0n an ¼ 0; ð19cÞ 2 2 n¼1 1 ha i X a9 9 sinðax1 Þ a2 V m0 ðe11 e5 a2 ÞY m0 þ aðcmc0 þ wm d0 Þ ¼ 0: 2 2 m¼1 1 X
ð19dÞ
The remaining equations are supplied by the geometric and natural boundary conditions. u3 and /2 geometric boundary conditions given by Eqs. (10a) and (10c), respectively, at the edges, x1 = 0 and a, are satisfied a priori. Satisfying the geometric boundary conditions given by Eq. (10d) such that u1 should vanish at the edges, x1 = 0, a, and equating the coefficients of sin(ax1), yield the following algebraic equations: For all values of n = 1, 2, . . . 1 X wm U mn ¼ 0; ð20aÞ m¼1
2
abf1 V mn þ a2 a f3 ab2 þ a4 a W mn a9 b2 þ a2 a X mn abf2 Y mn þ A11 ðcm ð18aÞ an þ w m bn Þ ¼ 0; 1 X 1 X sinðax1 Þ cosðbx2 Þfabf1 U mn A66 a2 þ A22 b2 V mn þ a6 b f3 a2 b þ a8 b3 W mn f2 abX mn a9 a2 þ a7 b2 Y mn aA66 ðcmcn þ wm dn Þ ¼ 0; 1 X 1 X
sinðax1 Þ sinðbx2 Þ a1 a þ a4 a3 þ f8 ab2 U mn
ð18dÞ
A66 2 A66 2 a V m0 þ a9 a Y m0 þ aðcmc0 þ wm d 0 Þ ¼ 0; sinðax1 Þ 2 2 m¼1
m¼1
in which ðcm ; wm Þ ¼
abe1 V mn e6 a3 þ e7 ab2 e2 a W mn þ e8 e3 a2 e5 b2 X mn e4 abY mn þa2 ðcm an þ wm bn Þ ¼ 0; 1 X 1 X sinðax1 Þ cosðbx2 Þfabe1 U mn
n¼1 1 X
m¼1 n¼1
cosðax1 Þ sinðbx2 Þ a2 a2 þ b2 a9 U mn
m¼1 n¼1
1 X
1 X
þ wm dn cosðax1 Þ cosðbx2 Þ;
1 X 1 X
479
U 0n þ
1 X
cm U mn ¼ 0:
ð20bÞ
m¼1
The rest of the equations are obtained from satisfying the natural boundary conditions. The natural boundary conditions M1 = 0 and P1 = 0 at x1 = 0, a, are satisfied a priori. Satisfying N6 = 0 at these edges yield the following equations: ( 1 1 X X A66 A66 cn þ A66 bU 0n þ a9 bX 0n þ cosðbx2 Þ aV mn H 2 2 n¼1 m¼1 1 1 1 X X X þA66 bU mn H þ A66 ½cmcn þ wm dn H þ aY mn a9 H m¼1 m¼1 m¼1 ) 1 1 X X þ bX mn a9 H abW mn a10 H ¼ 0; ð21aÞ m¼1
1
X A66 m¼1
2
m¼1
½aV m0 þ cmc0 þ wm d0 H þ aY m0 a9 H
þ
A66 c0 ¼ 0; 4 ð21bÞ
480
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
START
Calculate Deflections Moments etc. -Number of plies -Thickness of plies -Lamination Angles, etc.
-Calculate Stiffness (A, B, D, E, F, H Matrices) -Calculate other constants
Find Umn, Vmn, Wmn, Xmn, Ymn NO Check Convergence
Substitute Umn, Vmn, etc. in Boundary Equations and Solve Linear Algebraic Equations Find Boundary Fourier Coefficients
YES
STOP
Solve Umn, Vmn, Wmn, Xmn, Ymn in terms of Boundary Fourier Coefficients
Number of Terms for Calculation
( an , bn , etc.) Fig. 2. Flow chart for numerical solution. m
in which H ¼ 1 and H ¼ ð1Þ , represent the condition of N6 = 0, at the edges, x1 = 0, a, respectively. In a more useful form, Eqs. (21a) and (21b) can be written as follows: For all values of n = 1, 2, . . . 1 X
fA66 ½aV mn þ bU mn þ aY mn a9 þ bX mn a9
m¼1;3;5...
abW mn a10 þ A66 dn wm ¼ 0; ð22aÞ 1 X fA66 ½aV mn þ bU mn þ aY mn a9 þ bX mn a9 abW mn a10
m¼2;4;6...
A66 þA66cn gcm þ ðcn þ bU 0n Þ þ bX 0n a9 ¼ 0; ð22bÞ 2
1 X A66 ½aV m0 þ cmc0 þ wm d0 þ aY m0 a9 wm ¼ 0; 2 m¼1;3;5...
1 X A66 ½aV m0 þ cmc0 þ wm d0 þ aY m0 a9 wm 2 m¼2;4;6... þ
A66 c0 ¼ 0: 4
ð22cÞ
ð22dÞ
This step generates 4n + 2 equations. On equating the coefficients of sin (ax1) sin(bx2), sin(ax1), etc., the above operations, result in, in total, 5mn + 2m + 6n + 2 linear algebraic equations in as many
unknowns. In the interest of the computational efficiency, Eqs. (18) and (22) are solved for Umn, Vmn, Wmn, Xmn, Ymn and U0n, X0n, Vm0 and Ym0 in terms of the boundary Fourier coefficients an , bn , cn and dn , in a manner outlined in Fig. 2. These coefficients are then substituted in geometric boundary equations, given by Eqs. (20) and (22). Resulting equations are then solved for an , bn , cn and dn . This useful step will reduce the size of problems under consideration by an order or more of magnitude. 4. Numerical results and discussion To illustrate the validity of the analytical procedure presented in the preceding section, the present study investigates cross-ply type laminated shells (curved panels) of rectangular plan-form. In what follows, numerical results pertaining to displacements and moments of symmetric ([0°/90°/0°] and [0°/90°/90°/0°]) and antisymmetric [0°/ 90°] cross-ply spherical as well as cylindrical (either R1 = 1 or R2 = 1) panels of square plan-form, subjected to uniformly distributed transverse loads are presented. Two different types of material properties are used in these numerical calculations. These are given as follows: Material type I: E1 = 175.78 GPa (25,000 ksi), E1/E2 = 25, G12/E2 = G13/E2 = 0.5, G23/E2 = 0.2, m12 = 0.25.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Material type II: E1 = 105.47 GPa (15,000 ksi), E1/E2 = 15, G12/E2 = G13/E2 = 0.4286, G23/E2 = 0.3429, m12 = 0.40. Here E1 and E2 are the surface-parallel Young’s moduli of a 0° lamina in x1 and x2 coordinate directions, respectively, and G12 denotes surface parallel shear modulus. G13 and G23 are transverse shear moduli in the x1–x3 and x2–x3 planes, respectively, whereas m12 is the major Poisson’s ratio on the x1–x2 surface. The following quantities are used to normalize the results: u3 ¼
103 E2 h3 u3 ; q0 a4
M 1 ¼
103 M 1; q0 a2
in which a is assumed equal to 812.8 mm (32 in.) and q0 denotes the uniformly distributed transverse load and equals to 689.5 kPa (100 psi). u3 and M 1 are computed at the center of the panel with the exceptions of Figs. 19 and 20. 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 [24]. Fig. 3 displays the convergence (with m = n) of normalized transverse displacement (deflection), u3 and moment, M 1 , of a moderately thick (a/h = 10) and moderately deep (R1/a = R2/b = 10) antisymmetric cross-ply [0°/ 90°] spherical (R1 = R2) panel of square (a = b) plan-form, computed using the present HSDT with the material type I. Rapid and more or less monotonic convergence is observed for the normalized central deflection, u3 . Although the convergence plot of the central moment, M 1 , exhibits an initially oscillatory behavior, the oscillations die down very rapidly, rendering the convergence plot practically monotonic for m, n P 10 (Fig. 3). These converged results are in full agreement with their FSDT-based counterparts. Material Type I 0/90 Lamination Spherical Panel a/h=10; R/a=R/b=10 180 160 140 120 100 u3* M*1
80 60 40 20 0 0
10
20 m=n
30
40
Fig. 3. Convergence of normalized central deflection, u3 and moment M 1 , of an antisymmetric cross-ply [0°/90°] spherical panel.
481
Table 1 Comparison of u3 , based on FSDT and HSDT, of symmetric and antisymmetric cross-ply moderately deep (R/a = 10) spherical panels for various a/h ratios and material types I and II a/h %Error ¼ HSDTFSDT 100 HSDT
[0°/90°]
3 5 10 20
[0°/90°/0°]
[0°/90°/90°/0°]
I
II
I
II
I
II
3.740 2.650 1.020 0.200
4.710 2.790 0.930 0.220
18.050 17.660 10.550 4.360
11.950 9.610 4.440 1.550
20.510 18.810 11.430 4.760
12.880 9.160 3.890 1.290
Table 2 Comparison of M 1 , based on FSDT and HSDT, of symmetric and antisymmetric cross-ply moderately deep (R/a = 10) spherical panels for various a/h ratios and material types I and II a/h %Error ¼ HSDTFSDT 100 HSDT
[0°/90°]
3 5 10 20
[0°/90°/0°]
[0°/90°/90°/0°]
I
II
I
II
I
II
6.940 1.930 0.210 0.200
3.230 0.810 0.060 0.100
8.710 6.230 2.800 0.410
2.710 2.350 1.090 0.190
11.890 9.680 4.600 1.050
0.300 1.510 0.980 0.220
Tables 1 and 2 present the variation of central deflection, u3 , and moment, M 1 , of antisymmetric [0°/90°] and symmetric ([0°/90°/0°] as well as [0°/90°/90°/0°]) cross-ply moderately deep (R/a = 10) spherical panels computed on the basis of FSDT, and HSDT shell theories, with respect to a/h ratio for the aforementioned two sets of material properties, respectively. The shear correction factors, K 21 ¼ K 21 ¼ 5=6 are assumed in the present FSDT computations. The percentage of error is defined by HSDT FSDT 100: %Error ¼ HSDT Relative inaccuracy of the FSDT with respect to the HSDT is self-evident in the case of thick (a/h < 10) symmetric cross-ply panels, while the former may, in general, be considered acceptable for thick antisymmetric [0°/90°] panels (Table 1). The FSDT may also be considered acceptable for moderately thick (10 6 a/h 6 20) antisymmetric [0°/90°] and intermediate modulus (material type II) symmetric laminates (Table 1). Table 1 further shows that the FSDT based u3 values for high modulus (material I) symmetric laminates are in substantial error (more than 10%) with respect to their HSDT counterparts even for a/h = 10. The difference between two theories increases with the increase of a/h ratio (for thick and very thick regime) and as well as the number of layers increase as seen in Tables 1 and 2. The reason lies in the fact that the effect of interlaminar shear (unmatched properties through the thickness) increases with the increase of the number of layers. It is further noteworthy from Table 1 that a far more pronounced thickness shear effect is observed in the computed central deflection of symmetric ([0°/90°/0°] and
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[0°/90°/90°/0°]) shells, as compared to their antisymmetric [0°/90°] counterparts in the very thick regime. For example, u3 values computed using the FSDT are in relative errors of only 3.74% and 4.71% with respect to their HSDT counterparts for very thick (a/h = 3) antisymmetric [0°/90°] shells constructed of materials I and II, respectively. In contrast, u3 values computed using the FSDT are in relative errors of 18.05% and 11.95% with respect to their HSDT counterparts for very thick (a/h = 3) symmetric [0°/90°/0°] shells constructed of materials I and II, respectively. The same trend continues with very thick (a/h = 3) [0°/90°/90°/0°] shells, in which case u3 values computed using the FSDT are in relative errors of 20.51% and 12.88% with respect
to their HSDT counterparts for materials I and II, respectively. Additional results are shown in Table 1. There is reason to believe [8,25] that effect of thickness is compensated, to a certain extent, by the bending–stretching coupling effect, a characteristic of antisymmetric laminates. Same conclusion can be inferred for the difference between the two theories with regard to the computed central moment, M 1 , as shown in Table 2. These results confirm the earlier assertion that effect of thickness is compensated, to a certain extent, by the bending–stretching coupling effect, a characteristic of antisymmetric laminates [8,25]. Figs. 4 and 5 present the variations of central deflection, u3 , of [0°/90°] spherical panels, with a/h and R/a ratios, Spherical Panel Material Type I a/h=10
Material Type I 0/90 Lamination 20
30 R/a=20 R/a=60 R/a=100 Plate
25
16
u*3
u3*
20
18
0/90 0/90/0 0/90/90/0
14
15
12
10
10 8
5 0
10
20
30
40
0
50
a/h Fig. 4. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different R/a ratios.
20
40
60
80
100
R/a Fig. 6. Variation of normalized central deflection, u3 of a spherical panel with R/a ratio for different laminations.
Material Type I 0/90 Lamination Cylindrical Panel (R1=∞)
Material Type I 0/90 Lamination 30
25
25
u3*
20 15 10
a/h=5 a/h=20 a/h=50 a/h=100
20
u*3
a/h=5 a/h=20 a/h=40 a/h=100
15
5
10 0
0 0
20
40
60 R/a
80
100 u3 ,
Fig. 5. Variation of normalized central deflection, with R/a ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different a/h ratios.
20
40
60 R2 / a
80
100
Fig. 7. Variation of normalized central deflection, u3 , with R2/a ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different a/h ratios.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
respectively. In both the plots, membrane action due to the effect of curvature plays an important role, particularly for the ratio R/a 6 40. As can be seen in Fig. 6, this membrane action has a complex interaction with a ‘‘beam-column/tiebar’’ effect caused by the bending–stretching coupling present in the antisymmetric [0°/90°] panels, as has been explained earlier. Figs. 7 and 8 plot the variations of central deflection, u3 , of antisymmetric cross-ply [0°/90°] cylindrical panels, (R1 ! 1) and (R2 ! 1), with respect to the R/a ratio. Figs. 9 and 10 present similar variations with respect to the a/h ratio. The interaction of the cylindrical curvature
with the surface-parallel boundary constraint is particularly visible in these figures. While the cylindrical shell with R2 ! 1 exhibits a response characteristic similar to its spherical counterpart (Fig. 10), the other (Fig. 9) shows a different behavior. Moreover, for the cylindrical shell with R1 ! 1 type, membrane action has no effect in the thick and moderately thick regimes as shown in Fig. 7. Plots of central moment, M 1 , of spherical panels with respect to R/a and a/h ratios for different types of laminations, are displayed in Figs. 11 and 12, respectively. The normalized central moment is virtually independent of R/ a, for R/a > 20 in case of both symmetric and antisymmetric spherical panels. However, the normalized central
Material Type I Cylindrical Panel (R2=∞) 0/90 Lamination 30
30
25
25
15 10
Material Type I 0/90 Lamination Cylindrical Panel (R2=∞)
20
u3*
u*3
20 a/h=5 a/h=20 a/h=50 a/h=100
483
R/a=10 R/a=50 R/a=100
15 10
5
5
0
0 0
20
40
60
80
100
0
10
20
R1 / a
30
40
50
a/h
Fig. 8. Variation of normalized central deflection, u3 , with R1/a ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different a/h ratios.
Fig. 10. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different R/a ratios.
Material Type I 0/90 Lamination Cylindrical Panel (R1=∞)
Material Type I Spherical Panel a/h=10 150
27
140
23
130
R/a=10 R/a=50 R/a=100
u3*
21 19
M 1*
25
0/90 0/90/0 0/90/90/0
120 110
17 100
15
90
13 0
10
20
30
40
50
a/h Fig. 9. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different R/a ratios.
0
20
40
60 R/a
80
100
Fig. 11. Variation of normalized central moment, M 1 , with R/a ratio, of a spherical panel for different laminations.
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Material Type I 0/90/0 Lamination Spherical Panel
Material Type I Spherical Panel R/a=10
140 140
120 120 R/a=10 R/a=50 Plate
100
0/90 0/90/0 0/90/90/0
80
M *1
M*1
100
80
60 60
40
0
40 10
20
30
40
Fig. 12. Variation of normalized central moment, spherical panel for different laminations.
20
30
40
50
a/h
a/h M 1 ,
10
50
with a/h ratio, of a
moment is greatly influenced by the a/h ratio, as has been shown in Fig. 12. Variations of the central moment, M 1 , of [0°/90°], [0°/90°/0°] and [0°/90°/90°/0°] spherical panels with the a/h ratio for different R/a ratios are individually displayed in Figs. 13–15, respectively. Figs. 16 and 17 present the variations of normalized central deflections, u3 , of moderately thick (a/h = 10) antisymmetric [0°/90°] and symmetric [0°/90°/0°] cross- ply (one spherically and two cylindrically) curved panels with respect to R/a ratio, respectively. It should be noted that the response of the [0°/90°] spherical panel is almost identical to its cylindrical counterpart with R2 ! 1 in the entire range of the R/a ratio. A similar trend is also observed for symmetric [0°/90°/0°] laminates, the difference
Fig. 14. Variation of normalized central moment, M 1 , with a/h ratio, of a symmetric cross-ply [0°/90°/0°] spherical panel for different R/a ratios.
Material Type I 0/90/90/0 Lamination Spherical Panel 140
120
100 R/a=10 R/a=50 Plate
M*1
0
80
60
40
Material Type I 0/90 Lamination Spherical Panel
0
160
10
20
30
40
50
a/h 140
Fig. 15. Variation of normalized central moment, M 1 , with a/h ratio, of a symmetric cross-ply [0°/90°/90°/0°] spherical panel for different R/a ratios.
M 1*
120 R/a=10 R/a=50 Plate
100 80 60 40 0
10
20
30
40
50
a/h
Fig. 13. Variation of normalized central moment, M 1 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different R/a ratios.
between the spherical and R2 ! 1 type cylindrical geometries being negligible, which is, as expected, especially true in the flatter regime (R/a > 40). Fig. 18 presents the variation of normalized central deflections, u3 , of moderately deep (R/a = 10) antisymmetric [0°/90°] cross-ply (one spherically and two cylindrically) curved panels with respect to a/h. Again, the response of the [0°/90°] spherical panel is almost identical to its cylindrical counterpart with R2 ! 1 in the entire range of the a/h ratio. Variations response quantities of of the normalized 3
3
2
3
interest, u1 u1 ¼ 10q Ea24h u1 , u3 , /1 /1 ¼ 10q Ea23h /1 , and 0
0
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
485
Material Type I 0/90 Lamination R/a=10
Material Type I 0/90 Lamination a/h=10 20
30
19
25
18
u*3
Sphere R1 = Inf R2 = Inf
16
u3*
20
17
15
15
10
14
5
13
Sphere R1 = Inf R2 = Inf
0 0
12 0
20
40
60
80
10
20
30
40
50
a/h
100
R/a u3 ,
Fig. 16. Variation of normalized central deflection, with R/a ratio, of moderately thick [0°/90°] spherical and cylindrical panels.
Fig. 18. Variation of normalized central deflection, u3 , with a/h ratio, of moderately deep [0°/90°] spherical and cylindrical panels.
Material Type I 0/90 Lamination Spherical Panel a/h=5 R/a=50
Material Type I 0/90/0 Lamination a/h=10 11
5 4
10.8
u1 =5*u1* u3 =u3* /10 M1 =M1* /30 φ 1*
3 10.6
2 1
10.4
u3*
Sphere R1 = Inf R2 = Inf
10.2
0 -1 0
0.2
0.4
0.6
0.8
1
-2
10
-3 9.8
-4 -5
9.6 0
20
40
60
80
100
x1 / a
R/a Fig. 17. Variation of normalized central deflections, u3 , with R/a ratio, of moderately thick [0°/90°/0°] spherical and cylindrical panels.
Fig. 19. Variations of displacements, rotation, and moment along the center line, x2 = b/2, of a thick (a/h = 5) very shallow (R/a = 50) [0°/90°] spherical panel.
M 1 , computed at x2 = b/2 and along x1 = 0 to a, of a thick (a/h = 5) antisymmetric relatively flat (R/a = 50) spherical panel, and its symmetric counterpart, are shown in Figs. 19 and 20, respectively. In all of these plots, the deflection, u3 , assumes, as expected, its maximum magnitude at the center of the panel, where the surface-parallel displacement, u1 , and rotation, /1 , vanish. In the case of antisymmetric panels, /1 reaches its maxima at the appropriate edges as given by the boundary condition (i.e., at x1 = 0 and a), where u1 , u3 , and M 1 vanish. It is noteworthy that while the spatial variations of u1 , u3 , /1 , and M 1 of an antisymmetric [0°/90°] spherical panel are similar to their flat
plate counterparts, the same is not true for the spherical panel of symmetric [0°/90°/0°] construction. This is mainly because of bending–stretching coupling effect prevalent in antisymmetric panels. Furthermore, the magnitudes of u1 and /1 of the spherical panel of [0°/90°] lamination are greater than their [0°/90°/0°] counterparts. Additionally, these plots clearly indicate absence of any anomaly in the matter of exactly satisfying the boundary conditions, as dictated by the boundary discontinuous double Fourier series approach used in the present analysis. The effect of the lamina material orthotropy (E1/E2) on the normalized central deflection of one antisymmetric and
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Material Type I 0/90/0 Lamination Spherical Panel a/h=5 R/a=50 3 2.5
u1 =20*u1* u3 =u3*/10 M1 =M1* /40 φ1*
2 1.5 1 0.5 0 -0.5
0
0.2
0.4
0.6
0.8
1
-1 x1 / a Fig. 20. Variations of displacements, rotation, and moment along the center line, x2 = b/2, of a thick (a/h = 5) very shallow (R/a = 50) [0°/90°/ 0°] spherical panel.
Material Type I Spherical Panel a/h=5 R/a=50 40
u3*
35 0/90 0/90/0 0/90/90/0
30
25
20 0
5
10
15
20
25
E1 / E2 Fig. 21. Variation of normalized central deflection, u3 , with E1/E2 ratio, of a thick (a/h = 5) very shallow (R/a = 50) spherical panel for different laminations.
two symmetric cross-ply thick and very shallow spherical panels is illustrated in Fig. 21. These plots clearly show that bending–stretching coupling prevalent in a [0°/90°] type laminate has a softening effect of the beam-column type, which consequently, increases the normalized central deflection. 5. Summary and conclusions A heretofore unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional
general cross-ply thick doubly curved panel of rectangular plan-form is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach that assures well-posedness of the Fourier formulation and existence of the Fourier series solution, is used to solve a system of five highly coupled linear partial differential equations, with the SS2type simply supported boundary condition. Numerical results presented here on cross-ply laminates demonstrate fast convergence, and comparison with the available FSDT-based analytical solution testifies to the accuracy and efficiency of the method presented. The key conclusions that emerge from the numerical results can be summarized as follows: (i) The CLT and the FSDT (with appropriate shear correction factor incorporated) under-predict the computed normalized deflections as compared to their HSDT counterparts in the thicker panel regime (a/ h 6 10), while the opposite is true in the case of computed normalized moments. (ii) The FSDT may, in general, be considered acceptable for moderately thick unsymmetric and intermediate modulus symmetric cross-ply panels, 10 6 a/h 6 20 and beyond. The FSDT based u3 values for high modulus symmetric cross-ply panels are, in contrast, in substantial error (more than 10%) with respect to their HSDT counterparts for a/h = 10. (iii) Bending–stretching coupling prevalent in a [0°/90°] type laminate has a softening effect of the beam-column type, which consequently increases the normalized central deflection. (iv) The effect of the transverse shear deformation is compensated to a certain extent by the bending–stretching coupling effect – a characteristic of unsymmetric laminates. (v) The bending–stretching type coupling has a highly pronounced interaction with the type of surfaceparallel boundary restraint, imposed by the SS2 at two opposite edges. For example, the interaction of the membrane action due to the cylindrical curvature with the surface-parallel boundary constraint is quite noticeable. While the cylindrical shell with R2 ! 1 exhibits a response characteristic similar to its spherical counterpart, the other (R1 ! 1) shows a different behavior. Moreover, for the cylindrical shell with R1 ! 1 type, membrane action has no effect in the thick and moderately thick regimes. (vi) The radius-to-length ratio R/a has a pronounced effect on the response of curved cross-ply panels. This effect is progressively more pronounced for R/a < 40 for a given a/h. (vii) The membrane action due to the effect of curvature has a complex interaction with the bending–stretching type coupling effect, caused by the asymmetry of lamination.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Appendix A. Definition of certain constants A11 A12 4E11 4E12 þ ; a2 ¼ B11 2 ; a3 ¼ B12 2 ; R1 R2 3h 3h 4E11 4E12 A12 A22 a4 ¼ ; a5 ¼ ; a6 ¼ þ ; R1 R2 3h2 3h2 4E22 4E22 4E66 a7 ¼ B22 2 ; a8 ¼ ; a9 ¼ B66 2 ; 3h 3h2 3h 8E66 ; ðA1aÞ a10 ¼ 3h2 B11 B12 4F 11 þ ; b2 ¼ D11 2 ; b1 ¼ R1 R2 3h 4F 12 4F 11 4F 12 b3 ¼ D12 2 ; b4 ¼ ; b5 ¼ ; 2 3h 3h 3h2 B12 B22 4F 22 4F 22 þ ; b7 ¼ D22 2 ; b8 ¼ ; b6 ¼ R1 R2 3h 3h2 4F 66 8F 66 E11 E12 b9 ¼ D66 2 ; b10 ¼ ; b11 ¼ þ ; 2 R1 R2 3h 3h 4H 11 4H 12 4H 11 b12 ¼ F 11 ; b13 ¼ F 12 ; b14 ¼ ; 2 2 3h 3h 3h2 4H 12 4H 22 4H 22 b15 ¼ ; b16 ¼ F 22 ; b17 ¼ ; 2 2 3h 3h 3h2 4H 66 8H 66 E12 E22 ; b19 ¼ ; b20 ¼ þ ; ðA1bÞ b18 ¼ F 66 2 2 R1 R2 3h 3h 4D44 4D55 4F 44 d 1 ¼ A44 2 ; d 2 ¼ A55 2 ; d 3 ¼ D44 2 ; h h h 4F 55 ðA1cÞ d 4 ¼ D55 2 ; h 4E12 4E66 e1 ¼ B12 þ B66 2 2 ; 3h 3h 4d 4 4b11 4b12 e 2 ¼ b1 d 2 þ 2 2 ; e 3 ¼ b2 2 ; h 3h 3h 4b13 4b18 4b18 e 4 ¼ b3 þ b9 2 2 ; e 5 ¼ b9 2 ; 3h 3h 3h 4b14 4b15 4b19 e6 ¼ b4 þ 2 ; e7 ¼ b5 b10 þ 2 þ 2 ; 3h 3h 3h 4d 4 4b16 4b17 e8 ¼ d 2 þ 2 ; e9 ¼ b7 2 ; e10 ¼ b8 þ 2 ; h 3h 3h 4d 3 4d 3 4b20 ðA1dÞ e11 ¼ d 1 þ 2 ; e12 ¼ b6 d 1 þ 2 2 ; h h 3h f1 ¼ A12 þ A66 ; f 2 ¼ a3 þ a9 ; f 3 ¼ a10 a5 ; 4d 4 a2 a3 f4 ¼ d 2 2 ; R1 R2 h 4d 3 a3 a7 f5 ¼ d 1 2 ; R1 R2 h 4d 4 4b11 a4 a5 f6 ¼ d 2 2 þ 2 þ þ ; R1 R2 h 3h 4d 3 4b20 a5 a8 f7 ¼ d 1 2 þ 2 þ þ ; R1 R2 h 3h 4E12 8E66 4b12 4b13 8b18 f8 ¼ þ 2 ; f 9 ¼ 2 ; f 10 ¼ 2 þ 2 ; 3h2 3h 3h 3h 3h 4b14 8b15 8b19 4b16 f11 ¼ 2 ; f 12 ¼ 2 2 ; f 13 ¼ 2 ; 3h 3h 3h 3h 4b17 a1 a6 ðA1eÞ f14 ¼ 2 ; f 15 ¼ : R1 R2 3h
a1 ¼
487
Appendix B. Definition of boundary Fourier coefficients The unknown boundary Fourier coefficients are defined as follows: Z b 4 an ¼ ½u1;1 ða; x2 Þ u1;1 ð0; x2 Þ sinðbx2 Þ dx2 ; ðB1aÞ ab 0 Z b bn ¼ 4 ½u1;1 ða; x2 Þ þ u1;1 ð0; x2 Þ sinðbx2 Þ dx2 ; ðB1bÞ ab 0 Z b 4 cn ¼ ½u2 ða; x2 Þ u2 ð0; x2 Þ cosðbx2 Þ dx2 ; ðB1cÞ ab 0 Z b 4 ½u2 ða; x2 Þ þ u2 ð0; x2 Þ cosðbx2 Þ dx2 : ðB1dÞ dn ¼ ab 0 References [1] Bert CW, Reddy VS. Cylindrical shells of bimodulus composite materials. ASCE J Eng Mech 1982;108:675–88. [2] Chaudhuri RA, Balaraman K, Kunukkasseril VX. Arbitrarily laminated anisotropic cylindrical shells under uniform pressure. AIAA J 1986;24:1851–8. [3] Chaudhuri RA, Kabir HRH. A boundary-continuous-displacement based Fourier analysis of laminated doubly-curved panels using classical shallow shell theories. Int J Eng Sci 1992;30:1647–64. [4] Chaudhuri RA, Abu-Arja KR. Exact solution of shear-flexible doubly curved anti-symmetric angle-ply shells. Int J Eng Sci 1988;26:587–604. [5] Chaudhuri RA, Abu-Arja KR. Static analysis of moderately thick anti-symmetric angle-ply cylindrical panels and shells. Int J Solids Struct 1991;28:1–16. [6] Chaudhuri RA, Kabir HRH. On analytical solutions to boundaryvalue problems of doubly-curved moderately-thick orthotropic shells. Int J Eng Sci 1989;27:1325–36. [7] Chaudhuri RA, Kabir HRH. Sensitivity of the response of moderately thick cross-ply doubly-curved panels to lamination and boundary constraint, Part-I: Theory, Part-II: Application. Int J Solids Struct 1992;30:273–86. [8] Chaudhuri RA, Kabir HRH. Effect of boundary constraint on the frequency response of moderately thick doubly curved cross-ply panels using mixed Fourier solution functions. J Sound Vib 2005;283:263–93. [9] Kabir HRH, Chaudhuri RA. Free vibrations of anti-symmetric angle-ply finite doubly curved shells. Int J Solids Struct 1991;28 :17–32. [10] Kabir HRH, Chaudhuri RA. On Gibbs-phenomenon-free Fourier solution for finite shear-flexible laminated clamped curved panels. Int J Eng Sci 1994;32:501–20. [11] Kabir HRH, Khaleefi AM, Chaudhuri RA. Frequency response of a moderately thick anti-symmetric cross-ply cylindrical panel with mixed type of Fourier solution functions. J Sound Vib 2003;259: 809–26. [12] Noor AK, Burton WS. Assessment of computational models for multilayered composite shells. Appl Mech Rev 1990;43:67–97. [13] Seide P, Chaudhuri RA. Triangular finite element for analysis of thick laminated shells. Int J Numer Meth Eng 1987;24:1563–79. [14] Chaudhuri RA, Seide P. An approximate method for prediction of transverse shear stresses in a laminated shell. Int J Solids Struct 1987;23:1145–61. [15] Chaudhuri RA. A semi-analytical approach for prediction of interlaminar shear stresses in laminated general shells. J Solids Struct 1990;26:499–510. [16] Basset AB. On the extension and flexure of cylindrical and spherical thin elastic shells. Philos Trans Roy Soc, London, Ser A 1890;181: 433–80.
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[17] Librescu L, Khdeir AA, Frederick D. A shear deformable theory of laminated composite shallow shell-type panels and their response analysis I, free vibration and buckling. Acta Mech 1989;76:1–33. [18] Chaudhuri RA, Kabir HRH. Fourier solution to higher-order theory based laminated shell boundary-value problem. AIAA J 1995;33: 1681–8. [19] Chaudhuri RA. On boundary-discontinuous double Fourier series solution to a system of completely coupled P.D.E.’s. Int J Eng Sci 1989;27:1005–22. [20] Chaudhuri RA. On the roles of complementary and admissible boundary constraints in Fourier solutions to boundary-value problems of completely coupled rth order P.D.E.’s. J Sound Vib 2002;251: 261–313.
[21] Seide P. Small elastic deformations of thin shells. Leyden, The Netherlands: Noordhoff International Publishing; 1975. [22] Chaudhuri RA. A degenerate triangular shell element with constant cross-sectional warping. Comput Struct 1988;28:315–25. [23] Jones RM. Mechanics of composite materials. 2nd ed. Philadelphia, PA: Taylor and Francis; 1999. [24] Oktem AS, Chaudhuri RA. Levy type analysis of cross-ply plates based on higher-order theory. Compos Struct, in press, doi:10.1016/ j.compstruct.2005.09.012. [25] Abu-Arja KR, Chaudhuri RA. Influence of transverse shear deformation on scaling of cross-ply cylindrical shells. J Compos Mater 1989;23:673–94.
Levy type Fourier analysis of thick cross-ply doubly curved panels Ahmet Sinan Oktem a, Reaz A. Chaudhuri a
b,*
Department of Naval Architecture and Marine Engineering, Yildiz Technical University, 34349 Besiktas-Istanbul, Turkey b Department of Materials Science and Engineering, University of Utah, 1225 S. Central Campus Drive, Room 304, Salt Lake City, UT 84112-0560, United States Available online 2 August 2006
Abstract A hitherto unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doubly curved 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 general cross-ply shell analysis, with the SS2-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of a moderately thick cross-ply spherical panel. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, lamina material property, and thickness and curvature effects, as well as their interactions. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Thick laminate; Higher order shear deformation theory (HSDT); Doubly curved panel; Spherical shell; Cylindrical shell; Cross-ply; Analytical solution; Boundary discontinuous double Fourier series; Boundary constraints
1. Introduction 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 flat/curved panels because of many beneficial properties, such as higher strength-to-weight and stiffness-to-weight ratios (resulting in fuel economy), longer inplane fatigue (including sonic fatigue) life and stealth characteristics (of military aircraft, e.g., stealth fighter, F-117A Nighthawk and B-2 bomber), enhanced corrosion resistance, and so on. Since the matrix material is of relatively low shearing stiffness as compared to the fibers, polymeric composite shell type structures are highly prone to transverse shear related fatigue failures. A reliable prediction of the response of these laminated shells or doubly curved panels must account for transverse shear deformation. Additionally, a solution to the problem of *
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 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.05.020
deformation of laminated shells of finite dimensions must satisfy the prescribed 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 analysis of a model 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). Examples of CLT-based analysis of laminated cylindrical and doubly curved shells/panels include Bert and Reddy [1], Chaudhuri et al. [2] and Chaudhuri and Kabir [3], while Chaudhuri and Abu-Arja [4,5], Chaudhuri and Kabir [6–8], Kabir and Chaudhuri [9,10], and Kabir et al. [11], have presented double Fourier series based analytical solutions to moderately thick (FSDT-based) composite shell boundary-value problems. 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
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assumes a uniform transverse shear strain distribution through the thickness, which violates equilibrium conditions at the top and bottom surfaces of the panel. Noor and Burton [12] have presented an extensive survey on various computational models relating to laminated shells. Approximate thick shell theories can be classified into two categories: (a) discrete layer approach (see e.g., [13] and the references contained therein), and (b) continuous inplane displacement through thickness. The former approach appears to be more suitable for numerical methods, such as the finite element methods (FEM). Additionally, post-processing type methods used in conjunction with a discrete layer approach have yielded highly accurate interlaminar shear stress distribution through the thickness of symmetric and unsymmetric laminated shells [14,15]. Basset [16] 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 surface-parallel displacements through the thickness of thick laminated shells, have been developed as special cases of the above (see e.g., [17,18] and the references contained therein). In what follows, a hitherto unavailable HSDT-based boundary-discontinuous double Fourier series solution to the boundary value problem of general cross-ply doubly curved panels, with the SS3 type simply supported boundary condition, prescribed at two opposite edges, while the remaining two edges are subjected to the SS2 type simply supported one is derived. 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 [19,20]. The numerical accuracy of the present solution is ascertained by studying its convergence characteristics, and also by comparison with the available FSDT-based analytical solution. Numerical results are presented to understand the complex deformation behavior of symmet-
Fig. 1. Geometry of a laminated doubly curved panel.
ric and antisymmetric cross-ply cylindrical and spherical panels. 2. Statement of the problem Let (n1, n2, n3 = f) 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 [21,22] 1 1 g u1;1 þ g1;2 u2 þ 1 u3 ; e1 ðn1 ; n2 ; n3 Þ ¼ ð1aÞ g2 R1 1 þ Rn31 g1 1 1 g u2;2 þ g2;1 u1 þ 2 u3 ; e2 ðn1 ; n2 ; n3 Þ ¼ ð1bÞ g1 R2 1 þ Rn32 g2 e3 ðn1 ; n2 ; n3 Þ ¼ u3;3 ;
1 g e4 ðn1 ; n2 ; n3 Þ ¼ u3;2 2 u2 þ u2;3 ; n3 R 2 1 þ R2 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
ð1cÞ ð1dÞ
ð1eÞ
ð1fÞ
where ei (i = 1, 2, 3, 4, 5, 6) represents the components of the strain tensor, and ui (i = 1, 2, 3) denotes the components of the displacement vector along the (n1, n2, n3 = f) coordinates at a point, (n1, n2, n3 = f). 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 neglect of the geodesic curvatures of the coordinate lines. The surfaceparallel displacements can be expanded in power series of n3 = f as suggested by Basset [16]. 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 top and bottom surfaces of the shell, yields f 4f3 1 u1 ¼ 1 þ u1 þ f/1 2 /1 þ u3;1 ; ð2aÞ R1 g1 3h
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
u2 ¼
f 4f3 1 1þ u2 þ f/2 2 /2 þ u3;2 ; R2 g2 3h
u3 ¼ u3 ;
ð2bÞ ð2cÞ
where ui (i = 1, 2, 3) 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
resultants), and higher-order stress couples (resultants of the higher moment of stress). Qi, i = 4, 5, represents the transverse shear stress resultants, while Ki, i = 4, 5 denotes higher-order shear stress resultants. They are written as follows: N i ¼ Aij e0j þ Bij j0j þ Eij j2j ;
ð6aÞ
þ
f2 j21 Þ;
ð3aÞ
M i ¼ Bij e0j þ Dij j0j þ F ij j2j ; P i ¼ Eij e0j þ F ij j0j þ H ij j2j
e2 ¼ e02 þ fðj02 þ
f2 j22 Þ;
ð3bÞ
Q1 ¼ A5j e0j þ D5j j1j ;
e1 ¼
e01
þ
fðj01
e4 ¼ e04 þ f2 j14 ;
ð3cÞ
e5 ¼ e05 þ f2 j15 ;
ð3dÞ
e6 ¼ e06 þ fðj06 þ f2 j26 Þ;
ð3eÞ
in which e01 ¼ u1;1 þ
u3 ; R1
ð4aÞ
j01 ¼ /1;1 ; 4 j21 ¼ 2 ð/1;1 þ u3;11 Þ; 3h u3 0 e2 ¼ u2;2 þ ; R2 j02 ¼ /2;2 ; 4 j22 ¼ 2 ð/2;2 þ u3;22 Þ; 3h e04 ¼ u3;2 þ /2 ; 4 j14 ¼ 2 ð/2 þ u3;2 Þ; h e05 ¼ u3;1 þ /1 ; 4 j15 ¼ 2 ð/1 þ u3;1 Þ; h e06 ¼ u2;1 þ u1;2 ;
ð4bÞ
j06
ð4lÞ
¼ /2;1 þ /1;2 ; 4 j26 ¼ 2 ð/2;1 þ /1;2 þ 2u3;12 Þ: 3h
ð4cÞ
Q2 ¼
A4j e0j
K1 ¼
D5j e0j
þ
D4j j1j ;
þ
F 5j j1j
ði; j ¼ 1; 2; 6Þ
ð6bÞ ð6cÞ ð6dÞ ð6eÞ
ðj ¼ 4; 5Þ
K 2 ¼ D4j e0j þ F 4j j1j ;
ð6fÞ ð6gÞ
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 fk X ðkÞ Qij ð1; f; f2 Þ df; ð7aÞ ðAij ; Bij ; Dij Þ ¼ k¼1
ð4dÞ
477
ðEij ; F ij ; H ij Þ ¼
ð4eÞ
N X k¼1
fk1
Z
fk
ðkÞ
Qij ðf3 ; f4 ; f6 Þ df
ð7bÞ
fk1 ðkÞ
ð4fÞ ð4gÞ ð4hÞ ð4iÞ ð4jÞ ð4kÞ
ð4mÞ
for (i, j = 1, 2, 4, 5, 6), where Qin denotes the reduced elastic stiffnesses of the kth lamina [23]. The stress resultants, moment resultants and higherorder moment and shear resultants in terms of components of displacement and rotation can now be written as follows: 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 ; N 2 ¼ A12 u1;1 þ a6 u3 þ A22 u2;2 þ a3 /1;1 þ a7 /2;2
ð8aÞ
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 ;
ð8bÞ ð8cÞ
M 1 ¼ B11 u1;1 þ b1 u3 þ B12 u2;2 þ b2 /1;1 þ b3 /2;2
The equilibrium equations for a moderately shallow doubly curved panel can be written as shown below:
b4 u3;11 b5 u3;22 ; M 2 ¼ B12 u1;1 þ b6 u3 þ B22 u2;2 þ b3 /1;1 þ b7 /2;2
ð8dÞ
N 1;1 þ N 6;2 ¼ 0; ð5aÞ N 6;1 þ N 2;2 ¼ 0; ð5bÞ 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; ð5cÞ R1 R 2 4 4 ð5dÞ M 1;1 þ M 6;2 Q1 þ 2 K 1 2 ðP 1;1 þ P 6;2 Þ ¼ 0; h 3h 4 4 M 6;1 þ M 2;2 Q2 þ 2 K 2 2 ðP 6;1 þ P 2;2 Þ ¼ 0; ð5eÞ h 3h
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 ;
ð8eÞ ð8fÞ
where q is the distributed transverse load, and Ni, Mi, Pi, i = 1, 2, 6 denote stress resultants, stress couples (moment
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 ; P 2 ¼ E12 u1;1 þ b20 u3 þ E22 u2;2 þ b13 /1;1 þ b16 /2;2
ð8gÞ
b15 u3;11 b17 u3;22 ; P 6 ¼ E66 u2;1 þ E66 u1;2 þ b18 /2;1 þ b18 /1;2 b19 u3;12 ;
ð8hÞ ð8iÞ
Q1 Q2 K1 K2
¼ d 2 /1 þ d 2 u3;1 ; ¼ d 1 /2 þ d 1 u3;2 ; ¼ d 4 /1 þ d 4 u3;1 ; ¼ d 3 /2 þ d 3 u3;2 :
ð8jÞ ð8kÞ ð8lÞ ð8mÞ
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A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
The constants ai, bi, di, referred to in Eqs. (8) are given in Appendix A. Substitution of Eqs. (8) in equilibrium equations, given by Eqs. (5), supplies the following five highly coupled fourth-order governing partial differential equations:
þ f3 u3;122 þ A66 u1;22 þ a9 /1;22 ¼ 0;
ð9aÞ
A66 u2;11 þ f1 u1;12 þ a9 /2;11 þ f2 /1;12 þ f3 u3;112 þ A22 u2;22 þ a6 u3;2 þ a7 /2;22 a8 u3;222 ¼ 0; f4 /1;1 þ f5 /2;2 þ f6 u3;11 þ f7 u3;22 þ a4 u1;111 þ f8 u2;112
ð9bÞ
þ f9 /1;111 þ f10 /2;112 f11 u3;1111 þ f12 u3;1122 þ f8 u1;122
0 < x1 < a; 0 6 x2 6 b; 1 X 1 X u2 ¼ V mn sinðax1 Þ cosðbx2 Þ; 0 < x1 < a; 0 6 x2 6 b; 1 X 1 X u3 ¼ W mn sinðax1 Þ sinðbx2 Þ;
ð12aÞ
ð12bÞ
m¼1 n¼1
0 6 x1 6 a; 0 6 x2 6 b; 1 X 1 X X mn cosðax1 Þ sinðbx2 Þ; /1 ¼ 0 6 x1 6 a; 0 6 x2 6 b;
ð9cÞ
a2 u1;11 þ e1 u2;12 þ e2 u3;1 þ e3 /1;11 þ e4 /2;12 þ e5 /1;22
ð12cÞ
/2 ¼
1 X
1 X
ð12dÞ
Y mn sinðax1 Þ cosðbx2 Þ;
m¼1 n¼0
ð9dÞ
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;
U mn cosðax1 Þ sinðbx2 Þ;
m¼0 n¼1
m¼0 n¼1
þ a8 u2;222 þ f10 /1;122 þ f13 /2;222 f14 u3;2222 a1 u1;1
þ e6 u3;111 þ e7 u3;122 þ a9 u1;22 þ e8 /1 ¼ 0;
1 X 1 X
m¼1 n¼0
A11 u1;11 þ a1 u3;1 þ f1 u2;12 þ a2 /1;11 þ f2 /2;12 a4 u3;111
a6 u2;2 þ f15 u3 ¼ q;
u1 ¼
ð9eÞ
in which ai, ei, fi are constants, which are given in Appendix A. In what follows, a double Fourier series solution is sought for the following mixed boundary condition. The SS2 type simply supported boundary condition, prescribed at the edges x1 = 0, a, is given as follows [18]: u3 ð0; x2 Þ ¼ u3 ða; x2 Þ ¼ 0;
ð10aÞ
M 1 ð0; x2 Þ ¼ M 1 ða; x2 Þ ¼ 0;
ð10bÞ
/2 ð0; x2 Þ ¼ /2 ða; x2 Þ ¼ 0;
ð10cÞ
u1 ð0; x2 Þ ¼ u1 ða; x2 Þ ¼ 0;
ð10dÞ
N 6 ð0; x2 Þ ¼ N 6 ða; x2 Þ ¼ 0;
ð10eÞ
P 1 ð0; x2 Þ ¼ P 1 ða; x2 Þ ¼ 0:
ð10fÞ
At edges x2 = 0, b, the SS3 type simply supported condition is prescribed as per the dictate of the Levy type solution u3 ðx1 ; 0Þ ¼ u3 ðx1 ; bÞ ¼ 0;
ð11aÞ
M 2 ðx1 ; 0Þ ¼ M 2 ðx1 ; bÞ ¼ 0;
ð11bÞ
/1 ðx1 ; 0Þ ¼ /1 ðx1 ; bÞ ¼ 0;
ð11cÞ
N 2 ðx1 ; 0Þ ¼ N 2 ðx1 ; bÞ ¼ 0;
ð11dÞ
u1 ðx1 ; 0Þ ¼ u1 ðx1 ; bÞ ¼ 0;
ð11eÞ
P 2 ðx1 ; 0Þ ¼ P 2 ðx1 ; bÞ ¼ 0:
ð11fÞ
3. Method of solution The displacement functions (particular solution) are assumed to be in the form [18,20,24]:
0 6 x1 6 a; 0 6 x2 6 b;
ð12eÞ
where a¼
mp ; a
b¼
np : b
ð13Þ
It may be noted that the assumed solution functions, given by Eqs. (12), satisfy the SS3 type simply supported condition, given by Eqs. (11), at the edges, x2 = 0, b, a priori. The total number of unknown Fourier coefficients introduced in Eqs. (12) numbers 5mn + 2m + 2n. The next operation is comprised of partial differentiation of the assumed particular solution functions. The procedure for differentiation of these functions is based on Lebesgue integration theory that introduces boundary Fourier coefficients arising from discontinuities (complementary boundary constraints [19,20]) of the particular solution functions at the edges x1 = 0, a. As has been noted by Chaudhuri [20], 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 [19,20], and will not be further discussed here in the interest of brevity of presentation. The partial derivatives of these functions are obtained as follows: 1 X 1 X u1 ¼ U mn cosðax1 Þ sinðbx2 Þ; ð14aÞ m¼0 n¼1
u1;1 ¼
1 X 1 X
aU mn sinðax1 Þ sinðbx2 Þ:
ð14bÞ
m¼1 n¼1
The above function u1 and its first partial derivative, u1,1, are not satisfied a priori at the edges, at x1 = 0, a, thus violating the boundary constraints and complementary boundary constraints, respectively, at these edges. Therefore, for further differentiation, u1,1 is first
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
expanded in double Fourier series, in the form suggested by Chaudhuri [19,20] in order to satisfy the complementary boundary constraint (inequality), while u1 is forced to vanish at these edges (boundary constraints). The second partial derivative is then obtained as follows [20]: u1;11 ¼
1 1 X 1 X X 2 1 a U mn þ cm an þ wm bn an sinðbx2 Þ þ 2 m¼1 m¼1 n¼1
cosðax1 Þ sinðbx2 Þ:
ð15Þ
The partial derivative of u2 are given as follows: 1 1 u2;1 ¼ c0 þ 4 2
1 X
cn cosðbx2 Þ þ
n¼1
þ wm d0 cosðax1 Þ þ
1 2
1 X 1 X
½aV m0 þ cmc0
½aV mn þ cmcn ð16Þ
ð0; 1Þ; m ¼ odd;
ð17Þ
ð1; 0Þ; m ¼ even;
n , bn , cn and and where the boundary Fourier coefficients a dn , are as defined in Appendix B. The remaining particular solution functions or their partial derivatives do not have any boundary discontinuities, and therefore, can be differentiated term by term. This step generates additional 4n + 2 unknowns. Introduction of the displacement functions and their appropriate derivatives into the governing partial differential equations will supply 5mn + 2m + 2n equations as given below: 1 X 1 X
cosðax1 Þ sinðbx2 Þ
m¼1 n¼1
a2 A11 b2 A66 U mn 3
m¼1 n¼1
m¼1 n¼1
ð18bÞ
þ a6 b þ a8 b3 þ f8 a2 b V mn f6 a2 f15 þ f7 b2 þ f11 a4 f12 a2 b2 þ f14 b4 W mn f4 a f9 a3 f10 ab2 X mn f5 b b13 b3 f10 a2 b Y mn a4 aðcm an þ w m bn Þ 1 X 1 X qmn sinðax1 Þ sinðbx2 Þ; ð18cÞ ¼ m¼1 n¼1
m¼1 n¼1
a9 a2 þ a7 b2 V mn þ e12 b e7 a2 b e10 b3 W mn e4 abX mn e5 a2 e11 þ e9 b2 Y mn aa9 cmcn þ wm dn ¼ 0;
sinðbx2 Þ
ð18eÞ
A66 2 A11 an ¼ 0; b U 0n þ a9 b2 X 0n 2 2
ð19aÞ
ð19bÞ ha i a2 9 sinðbx2 Þ b2 U 0n ðe8 e5 b2 ÞX 0n an ¼ 0; ð19cÞ 2 2 n¼1 1 ha i X a9 9 sinðax1 Þ a2 V m0 ðe11 e5 a2 ÞY m0 þ aðcmc0 þ wm d0 Þ ¼ 0: 2 2 m¼1 1 X
ð19dÞ
The remaining equations are supplied by the geometric and natural boundary conditions. u3 and /2 geometric boundary conditions given by Eqs. (10a) and (10c), respectively, at the edges, x1 = 0 and a, are satisfied a priori. Satisfying the geometric boundary conditions given by Eq. (10d) such that u1 should vanish at the edges, x1 = 0, a, and equating the coefficients of sin(ax1), yield the following algebraic equations: For all values of n = 1, 2, . . . 1 X wm U mn ¼ 0; ð20aÞ m¼1
2
abf1 V mn þ a2 a f3 ab2 þ a4 a W mn a9 b2 þ a2 a X mn abf2 Y mn þ A11 ðcm ð18aÞ an þ w m bn Þ ¼ 0; 1 X 1 X sinðax1 Þ cosðbx2 Þfabf1 U mn A66 a2 þ A22 b2 V mn þ a6 b f3 a2 b þ a8 b3 W mn f2 abX mn a9 a2 þ a7 b2 Y mn aA66 ðcmcn þ wm dn Þ ¼ 0; 1 X 1 X
sinðax1 Þ sinðbx2 Þ a1 a þ a4 a3 þ f8 ab2 U mn
ð18dÞ
A66 2 A66 2 a V m0 þ a9 a Y m0 þ aðcmc0 þ wm d 0 Þ ¼ 0; sinðax1 Þ 2 2 m¼1
m¼1
in which ðcm ; wm Þ ¼
abe1 V mn e6 a3 þ e7 ab2 e2 a W mn þ e8 e3 a2 e5 b2 X mn e4 abY mn þa2 ðcm an þ wm bn Þ ¼ 0; 1 X 1 X sinðax1 Þ cosðbx2 Þfabe1 U mn
n¼1 1 X
m¼1 n¼1
cosðax1 Þ sinðbx2 Þ a2 a2 þ b2 a9 U mn
m¼1 n¼1
1 X
1 X
þ wm dn cosðax1 Þ cosðbx2 Þ;
1 X 1 X
479
U 0n þ
1 X
cm U mn ¼ 0:
ð20bÞ
m¼1
The rest of the equations are obtained from satisfying the natural boundary conditions. The natural boundary conditions M1 = 0 and P1 = 0 at x1 = 0, a, are satisfied a priori. Satisfying N6 = 0 at these edges yield the following equations: ( 1 1 X X A66 A66 cn þ A66 bU 0n þ a9 bX 0n þ cosðbx2 Þ aV mn H 2 2 n¼1 m¼1 1 1 1 X X X þA66 bU mn H þ A66 ½cmcn þ wm dn H þ aY mn a9 H m¼1 m¼1 m¼1 ) 1 1 X X þ bX mn a9 H abW mn a10 H ¼ 0; ð21aÞ m¼1
1
X A66 m¼1
2
m¼1
½aV m0 þ cmc0 þ wm d0 H þ aY m0 a9 H
þ
A66 c0 ¼ 0; 4 ð21bÞ
480
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
START
Calculate Deflections Moments etc. -Number of plies -Thickness of plies -Lamination Angles, etc.
-Calculate Stiffness (A, B, D, E, F, H Matrices) -Calculate other constants
Find Umn, Vmn, Wmn, Xmn, Ymn NO Check Convergence
Substitute Umn, Vmn, etc. in Boundary Equations and Solve Linear Algebraic Equations Find Boundary Fourier Coefficients
YES
STOP
Solve Umn, Vmn, Wmn, Xmn, Ymn in terms of Boundary Fourier Coefficients
Number of Terms for Calculation
( an , bn , etc.) Fig. 2. Flow chart for numerical solution. m
in which H ¼ 1 and H ¼ ð1Þ , represent the condition of N6 = 0, at the edges, x1 = 0, a, respectively. In a more useful form, Eqs. (21a) and (21b) can be written as follows: For all values of n = 1, 2, . . . 1 X
fA66 ½aV mn þ bU mn þ aY mn a9 þ bX mn a9
m¼1;3;5...
abW mn a10 þ A66 dn wm ¼ 0; ð22aÞ 1 X fA66 ½aV mn þ bU mn þ aY mn a9 þ bX mn a9 abW mn a10
m¼2;4;6...
A66 þA66cn gcm þ ðcn þ bU 0n Þ þ bX 0n a9 ¼ 0; ð22bÞ 2
1 X A66 ½aV m0 þ cmc0 þ wm d0 þ aY m0 a9 wm ¼ 0; 2 m¼1;3;5...
1 X A66 ½aV m0 þ cmc0 þ wm d0 þ aY m0 a9 wm 2 m¼2;4;6... þ
A66 c0 ¼ 0: 4
ð22cÞ
ð22dÞ
This step generates 4n + 2 equations. On equating the coefficients of sin (ax1) sin(bx2), sin(ax1), etc., the above operations, result in, in total, 5mn + 2m + 6n + 2 linear algebraic equations in as many
unknowns. In the interest of the computational efficiency, Eqs. (18) and (22) are solved for Umn, Vmn, Wmn, Xmn, Ymn and U0n, X0n, Vm0 and Ym0 in terms of the boundary Fourier coefficients an , bn , cn and dn , in a manner outlined in Fig. 2. These coefficients are then substituted in geometric boundary equations, given by Eqs. (20) and (22). Resulting equations are then solved for an , bn , cn and dn . This useful step will reduce the size of problems under consideration by an order or more of magnitude. 4. Numerical results and discussion To illustrate the validity of the analytical procedure presented in the preceding section, the present study investigates cross-ply type laminated shells (curved panels) of rectangular plan-form. In what follows, numerical results pertaining to displacements and moments of symmetric ([0°/90°/0°] and [0°/90°/90°/0°]) and antisymmetric [0°/ 90°] cross-ply spherical as well as cylindrical (either R1 = 1 or R2 = 1) panels of square plan-form, subjected to uniformly distributed transverse loads are presented. Two different types of material properties are used in these numerical calculations. These are given as follows: Material type I: E1 = 175.78 GPa (25,000 ksi), E1/E2 = 25, G12/E2 = G13/E2 = 0.5, G23/E2 = 0.2, m12 = 0.25.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Material type II: E1 = 105.47 GPa (15,000 ksi), E1/E2 = 15, G12/E2 = G13/E2 = 0.4286, G23/E2 = 0.3429, m12 = 0.40. Here E1 and E2 are the surface-parallel Young’s moduli of a 0° lamina in x1 and x2 coordinate directions, respectively, and G12 denotes surface parallel shear modulus. G13 and G23 are transverse shear moduli in the x1–x3 and x2–x3 planes, respectively, whereas m12 is the major Poisson’s ratio on the x1–x2 surface. The following quantities are used to normalize the results: u3 ¼
103 E2 h3 u3 ; q0 a4
M 1 ¼
103 M 1; q0 a2
in which a is assumed equal to 812.8 mm (32 in.) and q0 denotes the uniformly distributed transverse load and equals to 689.5 kPa (100 psi). u3 and M 1 are computed at the center of the panel with the exceptions of Figs. 19 and 20. 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 [24]. Fig. 3 displays the convergence (with m = n) of normalized transverse displacement (deflection), u3 and moment, M 1 , of a moderately thick (a/h = 10) and moderately deep (R1/a = R2/b = 10) antisymmetric cross-ply [0°/ 90°] spherical (R1 = R2) panel of square (a = b) plan-form, computed using the present HSDT with the material type I. Rapid and more or less monotonic convergence is observed for the normalized central deflection, u3 . Although the convergence plot of the central moment, M 1 , exhibits an initially oscillatory behavior, the oscillations die down very rapidly, rendering the convergence plot practically monotonic for m, n P 10 (Fig. 3). These converged results are in full agreement with their FSDT-based counterparts. Material Type I 0/90 Lamination Spherical Panel a/h=10; R/a=R/b=10 180 160 140 120 100 u3* M*1
80 60 40 20 0 0
10
20 m=n
30
40
Fig. 3. Convergence of normalized central deflection, u3 and moment M 1 , of an antisymmetric cross-ply [0°/90°] spherical panel.
481
Table 1 Comparison of u3 , based on FSDT and HSDT, of symmetric and antisymmetric cross-ply moderately deep (R/a = 10) spherical panels for various a/h ratios and material types I and II a/h %Error ¼ HSDTFSDT 100 HSDT
[0°/90°]
3 5 10 20
[0°/90°/0°]
[0°/90°/90°/0°]
I
II
I
II
I
II
3.740 2.650 1.020 0.200
4.710 2.790 0.930 0.220
18.050 17.660 10.550 4.360
11.950 9.610 4.440 1.550
20.510 18.810 11.430 4.760
12.880 9.160 3.890 1.290
Table 2 Comparison of M 1 , based on FSDT and HSDT, of symmetric and antisymmetric cross-ply moderately deep (R/a = 10) spherical panels for various a/h ratios and material types I and II a/h %Error ¼ HSDTFSDT 100 HSDT
[0°/90°]
3 5 10 20
[0°/90°/0°]
[0°/90°/90°/0°]
I
II
I
II
I
II
6.940 1.930 0.210 0.200
3.230 0.810 0.060 0.100
8.710 6.230 2.800 0.410
2.710 2.350 1.090 0.190
11.890 9.680 4.600 1.050
0.300 1.510 0.980 0.220
Tables 1 and 2 present the variation of central deflection, u3 , and moment, M 1 , of antisymmetric [0°/90°] and symmetric ([0°/90°/0°] as well as [0°/90°/90°/0°]) cross-ply moderately deep (R/a = 10) spherical panels computed on the basis of FSDT, and HSDT shell theories, with respect to a/h ratio for the aforementioned two sets of material properties, respectively. The shear correction factors, K 21 ¼ K 21 ¼ 5=6 are assumed in the present FSDT computations. The percentage of error is defined by HSDT FSDT 100: %Error ¼ HSDT Relative inaccuracy of the FSDT with respect to the HSDT is self-evident in the case of thick (a/h < 10) symmetric cross-ply panels, while the former may, in general, be considered acceptable for thick antisymmetric [0°/90°] panels (Table 1). The FSDT may also be considered acceptable for moderately thick (10 6 a/h 6 20) antisymmetric [0°/90°] and intermediate modulus (material type II) symmetric laminates (Table 1). Table 1 further shows that the FSDT based u3 values for high modulus (material I) symmetric laminates are in substantial error (more than 10%) with respect to their HSDT counterparts even for a/h = 10. The difference between two theories increases with the increase of a/h ratio (for thick and very thick regime) and as well as the number of layers increase as seen in Tables 1 and 2. The reason lies in the fact that the effect of interlaminar shear (unmatched properties through the thickness) increases with the increase of the number of layers. It is further noteworthy from Table 1 that a far more pronounced thickness shear effect is observed in the computed central deflection of symmetric ([0°/90°/0°] and
482
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
[0°/90°/90°/0°]) shells, as compared to their antisymmetric [0°/90°] counterparts in the very thick regime. For example, u3 values computed using the FSDT are in relative errors of only 3.74% and 4.71% with respect to their HSDT counterparts for very thick (a/h = 3) antisymmetric [0°/90°] shells constructed of materials I and II, respectively. In contrast, u3 values computed using the FSDT are in relative errors of 18.05% and 11.95% with respect to their HSDT counterparts for very thick (a/h = 3) symmetric [0°/90°/0°] shells constructed of materials I and II, respectively. The same trend continues with very thick (a/h = 3) [0°/90°/90°/0°] shells, in which case u3 values computed using the FSDT are in relative errors of 20.51% and 12.88% with respect
to their HSDT counterparts for materials I and II, respectively. Additional results are shown in Table 1. There is reason to believe [8,25] that effect of thickness is compensated, to a certain extent, by the bending–stretching coupling effect, a characteristic of antisymmetric laminates. Same conclusion can be inferred for the difference between the two theories with regard to the computed central moment, M 1 , as shown in Table 2. These results confirm the earlier assertion that effect of thickness is compensated, to a certain extent, by the bending–stretching coupling effect, a characteristic of antisymmetric laminates [8,25]. Figs. 4 and 5 present the variations of central deflection, u3 , of [0°/90°] spherical panels, with a/h and R/a ratios, Spherical Panel Material Type I a/h=10
Material Type I 0/90 Lamination 20
30 R/a=20 R/a=60 R/a=100 Plate
25
16
u*3
u3*
20
18
0/90 0/90/0 0/90/90/0
14
15
12
10
10 8
5 0
10
20
30
40
0
50
a/h Fig. 4. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different R/a ratios.
20
40
60
80
100
R/a Fig. 6. Variation of normalized central deflection, u3 of a spherical panel with R/a ratio for different laminations.
Material Type I 0/90 Lamination Cylindrical Panel (R1=∞)
Material Type I 0/90 Lamination 30
25
25
u3*
20 15 10
a/h=5 a/h=20 a/h=50 a/h=100
20
u*3
a/h=5 a/h=20 a/h=40 a/h=100
15
5
10 0
0 0
20
40
60 R/a
80
100 u3 ,
Fig. 5. Variation of normalized central deflection, with R/a ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different a/h ratios.
20
40
60 R2 / a
80
100
Fig. 7. Variation of normalized central deflection, u3 , with R2/a ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different a/h ratios.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
respectively. In both the plots, membrane action due to the effect of curvature plays an important role, particularly for the ratio R/a 6 40. As can be seen in Fig. 6, this membrane action has a complex interaction with a ‘‘beam-column/tiebar’’ effect caused by the bending–stretching coupling present in the antisymmetric [0°/90°] panels, as has been explained earlier. Figs. 7 and 8 plot the variations of central deflection, u3 , of antisymmetric cross-ply [0°/90°] cylindrical panels, (R1 ! 1) and (R2 ! 1), with respect to the R/a ratio. Figs. 9 and 10 present similar variations with respect to the a/h ratio. The interaction of the cylindrical curvature
with the surface-parallel boundary constraint is particularly visible in these figures. While the cylindrical shell with R2 ! 1 exhibits a response characteristic similar to its spherical counterpart (Fig. 10), the other (Fig. 9) shows a different behavior. Moreover, for the cylindrical shell with R1 ! 1 type, membrane action has no effect in the thick and moderately thick regimes as shown in Fig. 7. Plots of central moment, M 1 , of spherical panels with respect to R/a and a/h ratios for different types of laminations, are displayed in Figs. 11 and 12, respectively. The normalized central moment is virtually independent of R/ a, for R/a > 20 in case of both symmetric and antisymmetric spherical panels. However, the normalized central
Material Type I Cylindrical Panel (R2=∞) 0/90 Lamination 30
30
25
25
15 10
Material Type I 0/90 Lamination Cylindrical Panel (R2=∞)
20
u3*
u*3
20 a/h=5 a/h=20 a/h=50 a/h=100
483
R/a=10 R/a=50 R/a=100
15 10
5
5
0
0 0
20
40
60
80
100
0
10
20
R1 / a
30
40
50
a/h
Fig. 8. Variation of normalized central deflection, u3 , with R1/a ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different a/h ratios.
Fig. 10. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different R/a ratios.
Material Type I 0/90 Lamination Cylindrical Panel (R1=∞)
Material Type I Spherical Panel a/h=10 150
27
140
23
130
R/a=10 R/a=50 R/a=100
u3*
21 19
M 1*
25
0/90 0/90/0 0/90/90/0
120 110
17 100
15
90
13 0
10
20
30
40
50
a/h Fig. 9. Variation of normalized central deflection, u3 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] cylindrical panel for different R/a ratios.
0
20
40
60 R/a
80
100
Fig. 11. Variation of normalized central moment, M 1 , with R/a ratio, of a spherical panel for different laminations.
484
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Material Type I 0/90/0 Lamination Spherical Panel
Material Type I Spherical Panel R/a=10
140 140
120 120 R/a=10 R/a=50 Plate
100
0/90 0/90/0 0/90/90/0
80
M *1
M*1
100
80
60 60
40
0
40 10
20
30
40
Fig. 12. Variation of normalized central moment, spherical panel for different laminations.
20
30
40
50
a/h
a/h M 1 ,
10
50
with a/h ratio, of a
moment is greatly influenced by the a/h ratio, as has been shown in Fig. 12. Variations of the central moment, M 1 , of [0°/90°], [0°/90°/0°] and [0°/90°/90°/0°] spherical panels with the a/h ratio for different R/a ratios are individually displayed in Figs. 13–15, respectively. Figs. 16 and 17 present the variations of normalized central deflections, u3 , of moderately thick (a/h = 10) antisymmetric [0°/90°] and symmetric [0°/90°/0°] cross- ply (one spherically and two cylindrically) curved panels with respect to R/a ratio, respectively. It should be noted that the response of the [0°/90°] spherical panel is almost identical to its cylindrical counterpart with R2 ! 1 in the entire range of the R/a ratio. A similar trend is also observed for symmetric [0°/90°/0°] laminates, the difference
Fig. 14. Variation of normalized central moment, M 1 , with a/h ratio, of a symmetric cross-ply [0°/90°/0°] spherical panel for different R/a ratios.
Material Type I 0/90/90/0 Lamination Spherical Panel 140
120
100 R/a=10 R/a=50 Plate
M*1
0
80
60
40
Material Type I 0/90 Lamination Spherical Panel
0
160
10
20
30
40
50
a/h 140
Fig. 15. Variation of normalized central moment, M 1 , with a/h ratio, of a symmetric cross-ply [0°/90°/90°/0°] spherical panel for different R/a ratios.
M 1*
120 R/a=10 R/a=50 Plate
100 80 60 40 0
10
20
30
40
50
a/h
Fig. 13. Variation of normalized central moment, M 1 , with a/h ratio, of an antisymmetric cross-ply [0°/90°] spherical panel for different R/a ratios.
between the spherical and R2 ! 1 type cylindrical geometries being negligible, which is, as expected, especially true in the flatter regime (R/a > 40). Fig. 18 presents the variation of normalized central deflections, u3 , of moderately deep (R/a = 10) antisymmetric [0°/90°] cross-ply (one spherically and two cylindrically) curved panels with respect to a/h. Again, the response of the [0°/90°] spherical panel is almost identical to its cylindrical counterpart with R2 ! 1 in the entire range of the a/h ratio. Variations response quantities of of the normalized 3
3
2
3
interest, u1 u1 ¼ 10q Ea24h u1 , u3 , /1 /1 ¼ 10q Ea23h /1 , and 0
0
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
485
Material Type I 0/90 Lamination R/a=10
Material Type I 0/90 Lamination a/h=10 20
30
19
25
18
u*3
Sphere R1 = Inf R2 = Inf
16
u3*
20
17
15
15
10
14
5
13
Sphere R1 = Inf R2 = Inf
0 0
12 0
20
40
60
80
10
20
30
40
50
a/h
100
R/a u3 ,
Fig. 16. Variation of normalized central deflection, with R/a ratio, of moderately thick [0°/90°] spherical and cylindrical panels.
Fig. 18. Variation of normalized central deflection, u3 , with a/h ratio, of moderately deep [0°/90°] spherical and cylindrical panels.
Material Type I 0/90 Lamination Spherical Panel a/h=5 R/a=50
Material Type I 0/90/0 Lamination a/h=10 11
5 4
10.8
u1 =5*u1* u3 =u3* /10 M1 =M1* /30 φ 1*
3 10.6
2 1
10.4
u3*
Sphere R1 = Inf R2 = Inf
10.2
0 -1 0
0.2
0.4
0.6
0.8
1
-2
10
-3 9.8
-4 -5
9.6 0
20
40
60
80
100
x1 / a
R/a Fig. 17. Variation of normalized central deflections, u3 , with R/a ratio, of moderately thick [0°/90°/0°] spherical and cylindrical panels.
Fig. 19. Variations of displacements, rotation, and moment along the center line, x2 = b/2, of a thick (a/h = 5) very shallow (R/a = 50) [0°/90°] spherical panel.
M 1 , computed at x2 = b/2 and along x1 = 0 to a, of a thick (a/h = 5) antisymmetric relatively flat (R/a = 50) spherical panel, and its symmetric counterpart, are shown in Figs. 19 and 20, respectively. In all of these plots, the deflection, u3 , assumes, as expected, its maximum magnitude at the center of the panel, where the surface-parallel displacement, u1 , and rotation, /1 , vanish. In the case of antisymmetric panels, /1 reaches its maxima at the appropriate edges as given by the boundary condition (i.e., at x1 = 0 and a), where u1 , u3 , and M 1 vanish. It is noteworthy that while the spatial variations of u1 , u3 , /1 , and M 1 of an antisymmetric [0°/90°] spherical panel are similar to their flat
plate counterparts, the same is not true for the spherical panel of symmetric [0°/90°/0°] construction. This is mainly because of bending–stretching coupling effect prevalent in antisymmetric panels. Furthermore, the magnitudes of u1 and /1 of the spherical panel of [0°/90°] lamination are greater than their [0°/90°/0°] counterparts. Additionally, these plots clearly indicate absence of any anomaly in the matter of exactly satisfying the boundary conditions, as dictated by the boundary discontinuous double Fourier series approach used in the present analysis. The effect of the lamina material orthotropy (E1/E2) on the normalized central deflection of one antisymmetric and
486
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Material Type I 0/90/0 Lamination Spherical Panel a/h=5 R/a=50 3 2.5
u1 =20*u1* u3 =u3*/10 M1 =M1* /40 φ1*
2 1.5 1 0.5 0 -0.5
0
0.2
0.4
0.6
0.8
1
-1 x1 / a Fig. 20. Variations of displacements, rotation, and moment along the center line, x2 = b/2, of a thick (a/h = 5) very shallow (R/a = 50) [0°/90°/ 0°] spherical panel.
Material Type I Spherical Panel a/h=5 R/a=50 40
u3*
35 0/90 0/90/0 0/90/90/0
30
25
20 0
5
10
15
20
25
E1 / E2 Fig. 21. Variation of normalized central deflection, u3 , with E1/E2 ratio, of a thick (a/h = 5) very shallow (R/a = 50) spherical panel for different laminations.
two symmetric cross-ply thick and very shallow spherical panels is illustrated in Fig. 21. These plots clearly show that bending–stretching coupling prevalent in a [0°/90°] type laminate has a softening effect of the beam-column type, which consequently, increases the normalized central deflection. 5. Summary and conclusions A heretofore unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional
general cross-ply thick doubly curved panel of rectangular plan-form is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach that assures well-posedness of the Fourier formulation and existence of the Fourier series solution, is used to solve a system of five highly coupled linear partial differential equations, with the SS2type simply supported boundary condition. Numerical results presented here on cross-ply laminates demonstrate fast convergence, and comparison with the available FSDT-based analytical solution testifies to the accuracy and efficiency of the method presented. The key conclusions that emerge from the numerical results can be summarized as follows: (i) The CLT and the FSDT (with appropriate shear correction factor incorporated) under-predict the computed normalized deflections as compared to their HSDT counterparts in the thicker panel regime (a/ h 6 10), while the opposite is true in the case of computed normalized moments. (ii) The FSDT may, in general, be considered acceptable for moderately thick unsymmetric and intermediate modulus symmetric cross-ply panels, 10 6 a/h 6 20 and beyond. The FSDT based u3 values for high modulus symmetric cross-ply panels are, in contrast, in substantial error (more than 10%) with respect to their HSDT counterparts for a/h = 10. (iii) Bending–stretching coupling prevalent in a [0°/90°] type laminate has a softening effect of the beam-column type, which consequently increases the normalized central deflection. (iv) The effect of the transverse shear deformation is compensated to a certain extent by the bending–stretching coupling effect – a characteristic of unsymmetric laminates. (v) The bending–stretching type coupling has a highly pronounced interaction with the type of surfaceparallel boundary restraint, imposed by the SS2 at two opposite edges. For example, the interaction of the membrane action due to the cylindrical curvature with the surface-parallel boundary constraint is quite noticeable. While the cylindrical shell with R2 ! 1 exhibits a response characteristic similar to its spherical counterpart, the other (R1 ! 1) shows a different behavior. Moreover, for the cylindrical shell with R1 ! 1 type, membrane action has no effect in the thick and moderately thick regimes. (vi) The radius-to-length ratio R/a has a pronounced effect on the response of curved cross-ply panels. This effect is progressively more pronounced for R/a < 40 for a given a/h. (vii) The membrane action due to the effect of curvature has a complex interaction with the bending–stretching type coupling effect, caused by the asymmetry of lamination.
A.S. Oktem, R.A. Chaudhuri / Composite Structures 80 (2007) 475–488
Appendix A. Definition of certain constants A11 A12 4E11 4E12 þ ; a2 ¼ B11 2 ; a3 ¼ B12 2 ; R1 R2 3h 3h 4E11 4E12 A12 A22 a4 ¼ ; a5 ¼ ; a6 ¼ þ ; R1 R2 3h2 3h2 4E22 4E22 4E66 a7 ¼ B22 2 ; a8 ¼ ; a9 ¼ B66 2 ; 3h 3h2 3h 8E66 ; ðA1aÞ a10 ¼ 3h2 B11 B12 4F 11 þ ; b2 ¼ D11 2 ; b1 ¼ R1 R2 3h 4F 12 4F 11 4F 12 b3 ¼ D12 2 ; b4 ¼ ; b5 ¼ ; 2 3h 3h 3h2 B12 B22 4F 22 4F 22 þ ; b7 ¼ D22 2 ; b8 ¼ ; b6 ¼ R1 R2 3h 3h2 4F 66 8F 66 E11 E12 b9 ¼ D66 2 ; b10 ¼ ; b11 ¼ þ ; 2 R1 R2 3h 3h 4H 11 4H 12 4H 11 b12 ¼ F 11 ; b13 ¼ F 12 ; b14 ¼ ; 2 2 3h 3h 3h2 4H 12 4H 22 4H 22 b15 ¼ ; b16 ¼ F 22 ; b17 ¼ ; 2 2 3h 3h 3h2 4H 66 8H 66 E12 E22 ; b19 ¼ ; b20 ¼ þ ; ðA1bÞ b18 ¼ F 66 2 2 R1 R2 3h 3h 4D44 4D55 4F 44 d 1 ¼ A44 2 ; d 2 ¼ A55 2 ; d 3 ¼ D44 2 ; h h h 4F 55 ðA1cÞ d 4 ¼ D55 2 ; h 4E12 4E66 e1 ¼ B12 þ B66 2 2 ; 3h 3h 4d 4 4b11 4b12 e 2 ¼ b1 d 2 þ 2 2 ; e 3 ¼ b2 2 ; h 3h 3h 4b13 4b18 4b18 e 4 ¼ b3 þ b9 2 2 ; e 5 ¼ b9 2 ; 3h 3h 3h 4b14 4b15 4b19 e6 ¼ b4 þ 2 ; e7 ¼ b5 b10 þ 2 þ 2 ; 3h 3h 3h 4d 4 4b16 4b17 e8 ¼ d 2 þ 2 ; e9 ¼ b7 2 ; e10 ¼ b8 þ 2 ; h 3h 3h 4d 3 4d 3 4b20 ðA1dÞ e11 ¼ d 1 þ 2 ; e12 ¼ b6 d 1 þ 2 2 ; h h 3h f1 ¼ A12 þ A66 ; f 2 ¼ a3 þ a9 ; f 3 ¼ a10 a5 ; 4d 4 a2 a3 f4 ¼ d 2 2 ; R1 R2 h 4d 3 a3 a7 f5 ¼ d 1 2 ; R1 R2 h 4d 4 4b11 a4 a5 f6 ¼ d 2 2 þ 2 þ þ ; R1 R2 h 3h 4d 3 4b20 a5 a8 f7 ¼ d 1 2 þ 2 þ þ ; R1 R2 h 3h 4E12 8E66 4b12 4b13 8b18 f8 ¼ þ 2 ; f 9 ¼ 2 ; f 10 ¼ 2 þ 2 ; 3h2 3h 3h 3h 3h 4b14 8b15 8b19 4b16 f11 ¼ 2 ; f 12 ¼ 2 2 ; f 13 ¼ 2 ; 3h 3h 3h 3h 4b17 a1 a6 ðA1eÞ f14 ¼ 2 ; f 15 ¼ : R1 R2 3h
a1 ¼
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Appendix B. Definition of boundary Fourier coefficients The unknown boundary Fourier coefficients are defined as follows: Z b 4 an ¼ ½u1;1 ða; x2 Þ u1;1 ð0; x2 Þ sinðbx2 Þ dx2 ; ðB1aÞ ab 0 Z b bn ¼ 4 ½u1;1 ða; x2 Þ þ u1;1 ð0; x2 Þ sinðbx2 Þ dx2 ; ðB1bÞ ab 0 Z b 4 cn ¼ ½u2 ða; x2 Þ u2 ð0; x2 Þ cosðbx2 Þ dx2 ; ðB1cÞ ab 0 Z b 4 ½u2 ða; x2 Þ þ u2 ð0; x2 Þ cosðbx2 Þ dx2 : ðB1dÞ dn ¼ ab 0 References [1] Bert CW, Reddy VS. Cylindrical shells of bimodulus composite materials. ASCE J Eng Mech 1982;108:675–88. [2] Chaudhuri RA, Balaraman K, Kunukkasseril VX. Arbitrarily laminated anisotropic cylindrical shells under uniform pressure. AIAA J 1986;24:1851–8. [3] Chaudhuri RA, Kabir HRH. A boundary-continuous-displacement based Fourier analysis of laminated doubly-curved panels using classical shallow shell theories. Int J Eng Sci 1992;30:1647–64. [4] Chaudhuri RA, Abu-Arja KR. Exact solution of shear-flexible doubly curved anti-symmetric angle-ply shells. Int J Eng Sci 1988;26:587–604. [5] Chaudhuri RA, Abu-Arja KR. Static analysis of moderately thick anti-symmetric angle-ply cylindrical panels and shells. Int J Solids Struct 1991;28:1–16. [6] Chaudhuri RA, Kabir HRH. On analytical solutions to boundaryvalue problems of doubly-curved moderately-thick orthotropic shells. Int J Eng Sci 1989;27:1325–36. [7] Chaudhuri RA, Kabir HRH. Sensitivity of the response of moderately thick cross-ply doubly-curved panels to lamination and boundary constraint, Part-I: Theory, Part-II: Application. Int J Solids Struct 1992;30:273–86. [8] Chaudhuri RA, Kabir HRH. Effect of boundary constraint on the frequency response of moderately thick doubly curved cross-ply panels using mixed Fourier solution functions. J Sound Vib 2005;283:263–93. [9] Kabir HRH, Chaudhuri RA. Free vibrations of anti-symmetric angle-ply finite doubly curved shells. Int J Solids Struct 1991;28 :17–32. [10] Kabir HRH, Chaudhuri RA. On Gibbs-phenomenon-free Fourier solution for finite shear-flexible laminated clamped curved panels. Int J Eng Sci 1994;32:501–20. [11] Kabir HRH, Khaleefi AM, Chaudhuri RA. Frequency response of a moderately thick anti-symmetric cross-ply cylindrical panel with mixed type of Fourier solution functions. J Sound Vib 2003;259: 809–26. [12] Noor AK, Burton WS. Assessment of computational models for multilayered composite shells. Appl Mech Rev 1990;43:67–97. [13] Seide P, Chaudhuri RA. Triangular finite element for analysis of thick laminated shells. Int J Numer Meth Eng 1987;24:1563–79. [14] Chaudhuri RA, Seide P. An approximate method for prediction of transverse shear stresses in a laminated shell. Int J Solids Struct 1987;23:1145–61. [15] Chaudhuri RA. A semi-analytical approach for prediction of interlaminar shear stresses in laminated general shells. J Solids Struct 1990;26:499–510. [16] Basset AB. On the extension and flexure of cylindrical and spherical thin elastic shells. Philos Trans Roy Soc, London, Ser A 1890;181: 433–80.
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