Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells

Abstract

This study aimed to provide a static solution to the boundary value problem presented by symmetric (0°/90°/0°) and antisymmetric (0°/90°) cross-ply composite, moderately thick shallow shells and plates (a special case of the shells) subjected to mixed-type unsolved boundary conditions. The boundary-discontinuous double Fourier series (BDM) method, in which displacements are expressed in trigonometric functions, is employed in a well-established framework. The analytical solution obtained using the BDM is compared with the successful integration of the generalized differential quadrature (GDQ) method for the static analysis of composite shells with a roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. Comprehensive results are presented in order to show the effects of curvature on the deflections and stresses of moderately thick shallow shells made up of symmetric and antisymmetric cross-ply laminated composite materials. The validity of the proposed model is authenticated through the available HSDT-based literature review, and the convergence characteristics are demonstrated. The changing trends of displacements and stresses are explained in detail by investigating the effect of various parameters such as lamination, material properties, the effect of curvature, etc. Based on the results obtained using the proposed static solution, analytical BDM results were found to be in very close agreement with the numerical GDQ method, especially for symmetric lamination. However, the results obtained using the BDM and GDQ methods for antisymmetric lamination show differences, possibly due to the presence of a discontinuity in the derivatives originating from the bending–stretching matrix in antisymmetric lamination. Important numerical results presented include the sensitivity of the predicted response quantities of interest to material properties, lamination, and thickness effects, as well as their interactions. The results presented here may also serve as benchmark comparison points with numerical solutions such as finite elements, boundary elements, etc.

1. Introduction

Laminated composite materials are increasingly preferred in major industries such as aerospace, naval, and construction due to their high strength-to-weight and stiffness-to-weight ratios (reducing fuel consumption), and, more importantly, their ability to be tailored to achieve optimal design through manipulation of the composition of composite materials, stacking ply orientation, etc. Over the years, the mechanical behavior of laminated composites such as glass/epoxy, graphite/epoxy, boron/epoxy, Kevlar-49/epoxy, etc., has been researched by many scientists. However, the anisotropic behavior and bending–stretching coupling of composite materials and the need to satisfy distinct boundary conditions create additional problems for the analysis of those structures.

Over the years, several well-known shell theories with many assumptions have been established in order to investigate the static behavior of plate and shell structures, such as those provided by Love [1], Donnell [2], Sanders [3], Timoshenko [4], and Reissner [5]. A literature review including refined theories of laminated composite plates and shells is given in [6,7,8,9,10,11]. Many approximate numerical techniques and analytical methods have been employed in the literature to analyze the static behavior of composite shells. The Levy method has been the most popular analytical method for the analysis of two edges fixed by simply supported laminated plates and shells, and it is able to analyze cross-ply and antisymmetric angle-ply laminates. Such solutions are based on the well-known method of separation of variables. A view of the comprehensive study on the boundary value problems of laminated plates and shells using analytical methods can be found in the following studies. Pagano [12] constructed three-dimensional elasticity solutions for cross-ply rectangular plates for simply supported boundary conditions. Srinivas and Rao (1970) presented an elasticity solution for the static and dynamic analysis of cross-ply composite plates with Navier-type boundary conditions. Fan and Zhang [13] introduced a similar procedure, using an elasticity solution on doubly curved cross-ply shells with simply supported boundary conditions. In a recent overview, various three dimensional (3D) analytical approaches have been introduced for resolving multilayered panels with arbitrary boundary conditions. Kumari and Kar [14] improved an analytical three-dimensional elasticity solution based on a multi-term EKM along with a mixed formulation for the static bending of single- and multi-layer composite thick, moderately thick, and thin cylindrical shell panels. The proposed technique can yield accurate results with better computational efficiency using just one or two terms and two or three iteration steps in comparison to 3D FE for arbitrary boundary conditions (SS, CS, CF). Atashipour et al. [15] presented exact 3-D Levy-type solutions for the static analysis of thick composite plates based on 3-D elasticity and various shear deformation theories with different types of classical boundary conditions, namely simply supported, clamped, and free edge. Nik and Tahani [16] analyzed the bending of rectangular laminated plates with arbitrary lamination and various combinations of simply supported, clamped, and free-type boundary conditions by employing the Extended Kantorovich method. Analytical solutions may fail to carry out complex analyses such as those of surface-parallel orthotropy/anisotropy, lamination asymmetry, and curvature effect (leading to bending, stretching, and coupling), which usually involve the use of laminated composite shell equations. Therefore, a different methodology should be proposed to enable effective solutions. Methodologies integrating the Fourier series have great potential to solve the problems encountered in conventional analytical solutions. The following studies include these Fourier series-based methodologies.

Whitney and Leissa [17] obtained an exact solution based on the Fourier series as displacement functions for the static, frequency, and buckling analysis of antisymmetric cross-ply and angle-ply laminated plates for different boundary conditions. Reddy and Liu [18] presented an extension of Sanders’s shell theory for laminated doubly curved, symmetric, and antisymmetric cross-ply shells with simply support boundary conditions under sinusoidal, uniform, and point load. Static and natural frequency analysis was achieved with an exact solution, imparting Fourier series to the displacement fields. Huang and Yuan [19] presented a general analytical solution for the static bending problem posed by an anisotropic plate with arbitrary boundary conditions and loading. The numerical results were given only for simply supported-type boundary conditions. Li et al. [20] presented an accurate solution method for the static and vibration analysis of a functionally graded composite plate with general boundary conditions considering the improved Fourier series method. The characteristic equations are easily obtained by substituting admissible displacement functions into the governing equations and general elastic boundary equations. Chaudhuri [21,22] presented an analytical method called the boundary discontinuous double Fourier series approach as a solution to the boundary value problems, satisfying Dirichlet, Neumann, and mixed admissible boundary conditions. According to the proposed method, a cross-ply panel should be solved with only a single set of Fourier series corresponding to each displacement and rotation component. Certain boundary conditions which result in discontinuities in the solution were improved by proposing additional coefficients, so-called the boundary Fourier coefficients. The state of the art for mathematical modeling and solution techniques employed in the analysis of composite shells was discussed in Chaudhuri and Kabir [23], Chaudhuri and Kabir [24,25], Chaudhuri and Abu-Arja [26,27], Oktem and Chaudhuri [28,29,30,31], Mantari et al. [32], and Oktem and Soares [33] for various types of boundary conditions. Investigation of deformation, stresses, and different coupling effects for internally pressurized composite cylinders by considering the mixed type of boundary conditions were presented by [21,34]. Furthermore, the term “roller skate” has been introduced previously in these articles, but the solution for this boundary condition was not provided.

Apart from conventional computational algorithms using the weighted residual method, the mentioned techniques strongly depend on the way field variables are described within the bounds of the computational domain. For instance, generalized differential quadrature (GDQ) has been proven to be an effective and precise technique for numerically assessing higher-order differential equations, particularly when employed for structural analysis. Extensive successful integration of GDQ was carried out in the following articles in order to explore statistics and dynamics of curved structures. Viola et al. [35] investigated the static analysis of doubly curved composite shells using a 2D HSDT. The numerical problems were solved using the GDQ technique. Asadi and Qatu [36] carried out the static analysis of relatively deep, thick composite doubly curved shells using FSDT. Cross-ply, angle-ply, and general lay-up cylindrical shells with six discrete boundary conditions were examined with using GDQ. Kurtaran [37,38] presented a series of geometrically nonlinear transient analyses of moderately thick laminated composite shallow shells, thick and deep laminated composite curved beams, and moderately thick deep functionally graded curved beams with constant curvature, respectively, using the generalized differential quadrature method. It has been demonstrated that a non-uniform grid should be selected. The local form of GDQ has been shown to offer highly accurate results by reducing the number of points used in the solution [39]. Computational grid selection plays a vital role in the easy implementation of boundary conditions, which impacts the structure’s global response [40]. Several works concerning plate and shell structure solutions using the GDQ method can be found in [41,42,43,44]. Since the boundary conditions can easily be adapted and accurate results could be obtained with this technique, this study aims to compare analytical solutions with the numerical method, GDQ.

The provided literature review showed that there is still a gap in the static analysis of moderately thick, cross-ply composite plates and shells with roller skate boundary conditions. The available FSDT- or HSDT-based solutions for cross-ply curved panels are primarily focused on simply supported and clamped boundary conditions. Roller skate-type boundary conditions are of great importance, as they may be encountered in modeling many structures, especially in civil engineering. As stated, a solution for this boundary condition has not been provided in the literature to the best of the authors’ knowledge. Thus, the primary objective of the present study is to provide analytical solutions by applying boundary-discontinuous Fourier analysis to the deformation of symmetric (0°/90°/0°) and antisymmetric (0°/90°) cross-ply composite shells with roller skate-type boundary conditions prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. The virtual work principle and the first-order shear deformation theory are used to obtain the governing equilibrium equations. Partial derivatives in the equilibrium equations are expressed and solved using an analytical or strong-form solution, the so-called boundary discontinuous Fourier series method, and with the numerical method, GDQ. It is also important to study the sensitivity of the predicted response quantities of interest to lamination, material property, thickness, and curvature effects as well as their interactions as a second objective. Numerical results using both methods are compared with the available results in the literature and those obtained using the finite element commercial program ANSYS.

2. Statement of the Problem

A model of the schematic plot and geometric parameters of a curved shallow shell based on an orthogonal curvilinear coordinate (ξ₁, ξ₂, ξ₃ = ξ) system is shown in Figure 1. The orthogonal coordinates (ξ₁, ξ_2, ξ) are placed on the curved shell mid-surface, ξ₃ = ξ = 0, and the ζ coordinate coincides with the vertical direction to the mid-surface. The curved shell has a principal curvature denoted as R₁ and R₂ located on the middle surface (ξ = 0). The shell is made up of N, a finite number, of perfectly bonded layers with uniform thickness. The diverse geometries of the shell are obtained by adjusting the radius of curvature and designating a cylindrical shell (R₁ = R, R₂ = ∞ or R₁ = ∞, R₂ = R), spherical (R₁ = R, R₂ = R), and plate (R₁ = ∞, R₂ = ∞). In this study, the analysis used is a linear static analysis. It is also assumed that there is no penetration and no delamination between the layers and that they are perfectly bonded.

The displacement field addressing first order shear deformation theory may be written as [45]:

$${\overline{u}}_1\left({\xi }_1,{\xi }_2,\xi \right)=u_1\left({\xi }_1,{\xi }_2\right)+\xi ·{\theta }_1\left({\xi }_1,{\xi }_2\right)$$

(1)

$${\overline{u}}_2\left({\xi }_1,{\xi }_2,\xi \right)=u_2\left({\xi }_1,{\xi }_2\right)+\xi ·{\theta }_2\left({\xi }_1,{\xi }_2\right)$$

(2)

$${\overline{u}}_3\left({\xi }_1,{\xi }_2,\xi \right)=u_3\left({\xi }_1,{\xi }_2\right)$$

(3)

where u_i (i = 1,2,3) are the displacement field of a point on the middle-surface of the shell along the ξ₁, ξ₂, and ξ₃ axes. ${\theta }_1$ and ${\theta }_2$ are the rotations around the ξ₂ and ξ₁ axis, respectively. The equivalent single-layer approach based on a first-order shear deformation theory is incorporated into the shell formulation. According to the kinematic relations from the theory of elasticity in curvilinear coordinates, the generalized strain–displacement relations for the doubly curved panels using the displacement fields in Equations (1)–(3) are expressed as below in the case of small elastic deformation:

$${\epsilon }_1^0=\frac{\partial u_1}{\partial {\xi }_1}+\frac{u_3}{{\mathrm{R}}_1}, {\epsilon }_1^1=\frac{\partial {\theta }_1}{\partial {\xi }_1}$$

(4)

$${\epsilon }_2^0=\frac{\partial u_2}{\partial {\xi }_2}+\frac{u_3}{{\mathrm{R}}_2}, {\epsilon }_2^1=\frac{\partial {\theta }_2}{\partial {\xi }_2}$$

(5)

$${\epsilon }_6^0=\frac{\partial u_2}{\partial {\xi }_1}+\frac{\partial u_1}{\partial {\xi }_2}+\frac{1}{2}\left(\frac{1}{{\mathrm{R}}_2}-\frac{1}{{\mathrm{R}}_1}\right)\left(\frac{\partial u_2}{\partial {\xi }_1}-\frac{\partial u_1}{\partial {\xi }_2}\right), {\epsilon }_6^1=\frac{\partial {\theta }_1}{\partial {\xi }_2}+\frac{\partial {\theta }_2}{\partial {\xi }_1}$$

(6)

$${\epsilon }_4^0={\theta }_1+\frac{\partial u_3}{\partial {\xi }_1}-\frac{u_1}{{\mathrm{R}}_1}$$

(7)

$${\epsilon }_5^0={\theta }_2+\frac{\partial u_3}{\partial {\xi }_2}-\frac{u_2}{{\mathrm{R}}_2}$$

(8)

For the sake of brevity, the derivation of the equilibrium equations of the composite doubly curved panel using the virtual work principle is not explained here. Further explanations are given in Reddy [45]. With the use of the virtual work principle, the characteristic equations of the panel, defined by five highly coupled partial differential equations in five unknowns, i.e., three displacements, and two rotations are given as below:

$$\delta u_1: {\mathrm{N}}_{1,1 }+{\mathrm{N}}_{6,2 }+\frac{{\mathrm{Q}}_2}{{\mathrm{R}}_1}-c{\mathrm{M}}_{6,2}=0$$

(9)

$$\delta u_2: {\mathrm{N}}_{6,1 }+{\mathrm{N}}_{2,2 }+\frac{{\mathrm{Q}}_1}{{\mathrm{R}}_2}+c{\mathrm{M}}_{6,1}=0$$

(10)

$$\delta u_3:{\mathrm{Q}}_{1,2 }{+\mathrm{Q}}_{2,1}-\frac{{\mathrm{N}}_1}{{\mathrm{R}}_1}-\frac{{\mathrm{N}}_2}{{\mathrm{R}}_2}=-q$$

(11)

$$\partial {\theta }_1:{\mathrm{M}}_{1,1}{+\mathrm{M}}_{6,2}-{ \mathrm{Q}}_2=0$$

(12)

$$\partial {\theta }_2: {\mathrm{M}}_{6,1}{+\mathrm{M}}_{2,2}-{\mathrm{Q}}_1=0$$

(13)

where q is the distributed transverse load and N_i, M_i, P_i (i = 1, 2, 6) denote stress resultants, stress couples (moment resultants), and second stress couples (resultant from the second moment of stress), are as defined by Reddy and Liu [46]. Qi (i = 4, 5), represents the transverse shear stress resultants. The forces and moments are further elaborated upon to show the individual components as follows:

$$\left\{\begin{array}{c}{\mathrm{N}}_1\\ {\mathrm{N}}_2\\ {\mathrm{N}}_6\\ {\mathrm{M}}_1\\ {\mathrm{M}}_2\\ {\mathrm{M}}_6\end{array}\right\}=\left[\begin{array}{cccccc}{\mathrm{A}}_{11}& {\mathrm{A}}_{12}& {\mathrm{A}}_{16}& {\mathrm{B}}_{11}& {\mathrm{B}}_{12}& {\mathrm{B}}_{16}\\ {\mathrm{A}}_{21}& {\mathrm{A}}_{22}& {\mathrm{A}}_{26}& {\mathrm{B}}_{21}& {\mathrm{B}}_{22}& {\mathrm{B}}_{26}\\ {\mathrm{A}}_{61}& {\mathrm{A}}_{62}& {\mathrm{A}}_{66}& {\mathrm{B}}_{61}& {\mathrm{B}}_{62}& {\mathrm{B}}_{66}\\ {\mathrm{B}}_{11}& {\mathrm{B}}_{12}& {\mathrm{B}}_{16}& {\mathrm{D}}_{11}& {\mathrm{D}}_{12}& {\mathrm{D}}_{16}\\ {\mathrm{B}}_{21}& {\mathrm{B}}_{22}& {\mathrm{B}}_{26}& {\mathrm{D}}_{21}& {\mathrm{D}}_{22}& {\mathrm{D}}_{26}\\ {\mathrm{B}}_{61}& {\mathrm{B}}_{62}& {\mathrm{B}}_{66}& {\mathrm{D}}_{61}& {\mathrm{D}}_{62}& {\mathrm{D}}_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_1^0\\ {\epsilon }_2^0\\ {\epsilon }_6^0\\ {\epsilon }_1^1\\ {\epsilon }_2^1\\ {\epsilon }_6^1\end{array}\right\}$$

(14)

$$\left\{\begin{array}{c}{\mathrm{Q}}_1\\ {\mathrm{Q}}_2\end{array}\right\}=\left[\begin{array}{cc}{\mathrm{A}}_{55}& {\mathrm{A}}_{54}\\ {\mathrm{A}}_{45}& {\mathrm{A}}_{44}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_4^0\\ {\epsilon }_5^0\end{array}\right\}$$

(15)

where ${\mathrm{A}}_{\mathrm{ij}},{\mathrm{B}}_{\mathrm{ij}},$ ${\mathrm{D}}_{\mathrm{ij}}$ are the laminate rigidities. These are given as follows:

$$\left({\mathrm{A}}_{\mathrm{ij}},{\mathrm{B}}_{\mathrm{ij}},{\mathrm{D}}_{\mathrm{ij}}\right)={{\displaystyle \sum }}_{k=1}^N{{\displaystyle \int }}_{{\xi }_{k-1}}^{{\xi }_k}{\mathrm{Q}}_{\mathrm{ij}}^{\left(k\right)}\left(1,\xi ,{\xi }^2\right)d\xi , (\mathrm{i}, \mathrm{j}=1, 2, 6)$$

(16)

The obtained governing equations are expressed as follows:

$${\mathrm{K}}_{\mathrm{ij}}{\delta }_j=f_{\mathrm{i}}\left(\mathrm{i},\mathrm{j}=1, 2, 3, 4, 5\right) \mathrm{and} \left({\mathrm{K}}_{\mathrm{ij}}={\mathrm{K}}_{\mathrm{ji}}\right)$$

(17)

where

$${\left\{{\delta }_{\mathrm{j}}\right\}}^T=\left\{u_1 u_2 u_3,{\theta }_1,{\theta }_2\right\}$$

(18)

$${\left\{f_{\mathrm{j}}\right\}}^T=\left\{0 0-q,0,0 \right\}$$

(19)

The problem is solved in conjunction with the mixed type of simply supported and roller skate boundary conditions, and they are prescribed at the edges as given below:

Roller Skate:

$${\xi }_1=0‥a, u_1=u_2={\theta }_2={\mathrm{M}}_1={\mathrm{Q}}_1=0$$

(20)

Simply Supported:

$${\xi }_2=0‥b, u_1=u_3={\theta }_1={\mathrm{M}}_2={\mathrm{N}}_2=0$$

(21)

Following section explains the solution procedure for the governing partial differential equations given in Equations (9)–(13) subjected to the above boundary conditions.

3. Solution Methodology

3.1. Boundary Discontinues Fourier Series Method

We proposed a boundary discontinuous Fourier series method as the first proposed analytical solution. The functions expressing a displacement solution to the boundary value problem when mixed-type boundary conditions were included into the mathematical formulation initially. These functions, which are adapted for FSDT-based cross-ply panels, are as follows:

$$\left\{u_1,{\theta }_1\right\}={\displaystyle \sum }_{m=0}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\left\{{\mathrm{U}}_{\mathrm{mn}},{\mathrm{X}}_{\mathrm{mn}}\right\}\cos \left(\alpha {\xi }_1\right)\sin \left(\beta {\xi }_2\right), 0<{\xi }_1<a;0<{\xi }_2<b$$

(22)

$$\left\{u_2,{\theta }_2\right\}={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=0}^{\infty }\left\{{\mathrm{V}}_{\mathrm{mn}},{\mathrm{Y}}_{\mathrm{mn}}\right\}\sin \left(\alpha {\xi }_1\right)\cos \left(\beta {\xi }_2\right), 0<{\xi }_1<a;0<{\xi }_2<b$$

(23)

$$\left\{u_3\right\}={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\left\{{\mathrm{W}}_{\mathrm{mn}}\right\}\sin \left(\alpha {\xi }_1\right)\sin \left(\beta {\xi }_2\right), 0\le x_1\le a;0\le x_2\le b$$

(24)

The unknown coefficients (U_mn, V_mn, W_mn, X_mn, Y_mn,) regarding the displacements are represented as 5 mn + 2 m + 2 n. The assumed solution functions in Equations (22)–(24) do not entirely ensure that the prescribed boundary conditions. Thus, whenever the relevant boundary conditions are not satisfied, the assumed solution function or its derivative at the boundaries is forced to satisfy it. The derivatives of the assumed solution functions are achieved by expanding them in a double Fourier series in the form discussed by Hobson [46], Green [47] and Chaudhuri [21,22]. Boundary Fourier coefficients arising from the discontinuities of the assumed functions or their derivatives at the boundaries are handled by utilizing the Lebesgue integration theory. This theory was considered the procedure for the differentiation of these functions. The function u₁ given by Equation (22) and its first partial derivative, $u_{1,1}$, given by Equation (25) and obtained by term-wise differentiation, are not satisfied at the edges, ${\xi }_1$ = 0, a, and thus violate the boundary constraints and complementary boundary constraints, respectively, at these edges. Therefore, u₁ is forced to vanish at these edges, while for further differentiation, $u_{1,11}$ is expanded in a double Fourier series in the form advised by Chaudhuri [21,22], thereby satisfying the complementary boundary constraints (inequality). The partial derivatives are obtained as follows:

$$u_{1,1}=-{\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\alpha U_{mn}\sin (\alpha {\xi }_1)\sin (\beta {\xi }_2)$$

(25)

$$u_{1,11}={\displaystyle \sum }_{m=1}^{\infty }\frac{1}{2}{\overline{a}}_n\sin (\beta {\xi }_2)+{\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\left[-{\alpha }^2{U}_{mn}+{\gamma }_m{\overline{a}}_n+{\psi }_m{\overline{b}}_n\right]\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2),$$

(26)

$$u_{1,22}=-\frac{1}{2}{\displaystyle \sum }_{n=1}^{\infty }{\beta }^2{U}_{0n}\sin (\beta {\xi }_2)-{\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{\beta }^2{U}_{mn}\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2)$$

(27)

In the same way as $u_1, u_{3,1}$ is forced to vanish at the edges, ${\xi }_1=0$ and a. Thus, the derivatives of $u_{3,1}$ and $u_{3,11}$ are obtained as shown in Equations (28) and (29) by expanding them in a double Fourier series as previously discussed.

$$u_{3,1}={\displaystyle \sum }_{m=1}^{\infty }\frac{1}{2}{\overline{c}}_n\sin (\beta {\xi }_2)+{\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\left[\alpha W_{mn}+{\gamma }_m{\overline{c}}_n+{\psi }_m{\overline{d}}_n\right]\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2)$$

(28)

$$u_{3,11}={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }-\alpha \left[\alpha W_{mn}+{\gamma }_m{\overline{c}}_n+{\psi }_m{\overline{d}}_n\right]\sin (\alpha {\xi }_1)\sin (\beta {\xi }_2)$$

(29)

in which

$$\left({\gamma }_n,{\delta }_n\right)=\left\{\begin{array}{cc}\left(0,1\right),& n=\mathrm{odd}\\ \left(1,0\right),& n=\mathrm{even}\end{array}\right.$$

(30)

Other derivatives of the assumed solution functions can be obtained through term-wise differentiation. The details of the proposed method are available in Chaudhuri [21,22]. The above step generates an additional 4 n unknown boundary Fourier coefficients (${\overline{a}}_n$, ${\overline{b}}_n$, ${\overline{c}}_n$, ${\overline{d}}_n$) generated from the differentiation procedure. Boundary Fourier coefficients are defined in Appendix A.

The substitution of the assumed displacement functions and their appropriately obtained derivatives into the governing PDE’s yields 5 mn + 2 m + 2 n linear algebraic equations for equating the coefficients of $\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2),$ sin(β${\xi }_2$), sin(α${\xi }_1$), etc., as shown below:

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2)\{\left(-{\mathrm{A}}_{11}{\alpha }^2-{\mathrm{A}}_{66}{\beta }^2-f_1+f_2{\beta }^2\right)U_{mn}\hfill \\ +\left(-f_2-{\mathrm{A}}_{12}\alpha \beta -{\mathrm{A}}_{66}\alpha \beta \right)V_{mn}+\left(g_1\alpha +f_1\alpha \right)W_{mn}\}\hfill \\ +\left(-{\mathrm{B}}_{11}{\alpha }^2+f_1-\frac{f_2{\beta }^2}{c}\right)X_{mn}+\left(-c{\mathrm{D}}_{66}\alpha \beta \right)Y_{mn}+{\mathrm{A}}_{11}{b}_n+\left(g_1+f_1\right)d_n\hfill \\ =0\hfill \end{array}$$

(31)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}\sin (\alpha {\xi }_1)\cos (\beta {\xi }_2)\{\left(-f_2\alpha \beta -{\mathrm{A}}_{12}\alpha \beta -{\mathrm{A}}_{66}\alpha \beta \right)U_{mn}\hfill \\ +\left(-{\mathrm{A}}_{66}{\alpha }^2-{\mathrm{A}}_{22}{\beta }^2-f_5+f_2{\alpha }^2\right)V_{mn}+f_6{W}_{mn}+\left(c{\mathrm{D}}_{66}\alpha \beta \right)X_{mn}\hfill \\ +f_7{Y}_{mn}\}=0\hfill \end{array}$$

(32)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}\sin (\alpha {\xi }_1)\sin (\beta {\xi }_2)\{\left(f_3\alpha \right)U_{mn}+\left(f_6\right)V_{mn}+\left(f_8-\frac{f_9}{R_y}-f_{10}\right)W_{mn}+\left(f_{11}\right)X_{mn}\\ +\left(f_{12}\right)Y_{mn}-{\mathrm{A}}_{55}\alpha b_n\}={\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}{q}_{mn}\sin (\alpha {\xi }_1)\sin (\beta {\xi }_2)\end{array}$$

(33)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}\cos (\alpha {\xi }_1)\sin (\beta {\xi }_2)\{\left(-{\mathrm{B}}_{11}{\alpha }^2+c{\mathrm{D}}_{66}{\beta }^2+f_1\right)U_{mn}+\left(\frac{-f_2\alpha \beta }{c}\right)V_{mn}\\ +\left(f_{13}\alpha -{\mathrm{A}}_{55}\alpha \right)W_{mn}\}+\left(g_2\right)X_{mn}+\left(-{\mathrm{D}}_{12}\alpha \beta -{\mathrm{D}}_{66}\alpha \beta \right)Y_{mn}+{\mathrm{B}}_{11}{b}_n\\ +\left(f_{13}-{\mathrm{A}}_{55}\right)d_n=0\hfill \end{array}$$

(34)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }}\sin (\alpha {\xi }_1)\cos (\beta {\xi }_2)\{\left(c{\mathrm{D}}_{66}\alpha \beta \right)U_{mn}+\left(-c{\mathrm{D}}_{66}{\alpha }^2-{\mathrm{B}}_{22}{\beta }^2+f_5{R}_y\right)V_{mn}+(f_{14}\beta \\ -{\mathrm{A}}_{44}\beta )W_{mn}+\left(-{\mathrm{D}}_{12}\alpha \beta -{\mathrm{D}}_{66}\alpha \beta \right)X_{mn}+f_{15}{Y}_{mn}\}=0\end{array}$$

(35)

$${\displaystyle \sum }_{n=1}^{\infty }\sin (\beta {\xi }_2)\left\{U_{0n}{f}_4+\frac{{\mathrm{A}}_{11}}{2}{a}_n+\frac{f_3}{2}{c}_n-\frac{c{\mathrm{D}}_{66}{\beta }^2}{2}{X}_{0n}\right\}=0$$

(36)

$${\displaystyle \sum }_{n=1}^{\infty }\sin (\beta {\xi }_2)\left\{\frac{c{\mathrm{D}}_{66}{\beta }^2}{2}{U}_{0n}-\frac{{\mathrm{D}}_{66}{\beta }^2}{2}{X}_{0n}+\left(\frac{-{\mathrm{A}}_{55}}{2}+\frac{{\mathrm{B}}_{11}}{2R_x}\right)c_n+\frac{{\mathrm{B}}_{11}}{2}{a}_n\right\}=0$$

(37)

$${\displaystyle \sum }_{m=1}^{\infty }\sin (\alpha {\xi }_1)\left\{V_{m0}\left({\mathrm{D}}_{66}{c}^2-{\mathrm{A}}_{66}\right)+c{\mathrm{D}}_{66}{Y}_{m0}\right\}=0$$

(38)

$${\displaystyle \sum }_{m=1}^{\infty }\sin (\alpha {\xi }_1)\left\{V_{m0}\left(-c{\mathrm{D}}_{66}\right)-{\mathrm{D}}_{66}{Y}_{m0}\right\}=0$$

(39)

The constants of Equations (31)–(39) can be found in Appendix B. The remaining equations are supplied by the geometric and boundary conditions given in Equation (20). $u_1$ and ${\mathrm{Q}}_1$ are forced to vanish at the edges, ${\xi }_1=0$ and a. Satisfying these geometric boundary conditions generates an additional 4 n linear algebraic equations.

For all values of $n=1,2,\dots$

$${\displaystyle \sum }_{m=1,3,5,\dots \dots }^{\infty }{\delta }_m{U}_{mn}=0, U_{0n}+{\displaystyle \sum }_{m=2,4,6,\dots \dots }^{\infty }{\gamma }_m{U}_{mn}=0,$$

(40)

$${\displaystyle \sum }_{m=1}^{\infty }\sin (\alpha {\xi }_1)\left\{{\mathrm{A}}_{55}\left(\alpha W_{mn}+X_{mn}-\frac{U_{mn}}{R_x}\right)\overline{H}\right\}=0,$$

(41)

in which, at the boundaries ${\xi }_1=0$ and a, $\overline{H}=1$ and $\overline{H}={(-1)}^m$ in Equation (41), respectively.

Finally, the above operations result in, in total, 5 mn + 2 m + 6 n linear algebraic equations with as many unknowns and supply a complete solution to the boundary value problem considered here. The unknown Fourier coefficients (U_mn, V_mn, W_mn, X_mn, Y_mn, U_0n, V_m0, X_0n, and Y_m0) in Equations (31)–(39) are solved in terms of boundary Fourier coefficients (${\overline{a}}_n$, ${\overline{b}}_n$, ${\overline{c}}_n$, and ${\overline{d}}_n$) in the interest of computational efficiency. These are, consequently, substituted into the boundary equations given in Equations (40) and (41). This step reduces the size of the solution matrix from (5 mn + 2 m + 6 n) × (5 mn + 2 m + 6 n) to (4 n) × (4 n), which is solved using the MATLAB code developed for this solution.

3.2. Generalized Differantial Quadrature Method

In this study, the GDQ method introduced by Shu [48] which is the improved form of DQM, is used to convert the partial derivatives of the field variables in equilibrium equations as well as boundary conditions at the grid points into algebraic equations. Before the application of GDQ, the laminated shell is discretized with grid points in order to apply the equilibrium equations and boundary conditions. Gauss–Lobatto points, which increase the accuracy of derivatives in the GDQ method [39], are selected as the grid points for the solution. The Gauss–Lobatto points formulized in Equations (42) and (43) are distributed more extensively at edge points, and they create grid points on the boundaries.

$$x_i=\frac{a}{2}(1-\cos \left(\frac{\left(i-1\right)\pi }{\left(M-1\right)}\right), i=1,2,\dots , n_x+1$$

(42)

$${\mathrm{y}}_j=\frac{b}{2}(1-\cos \left(\frac{\left(j-1\right)\pi }{\left(N-1\right)}\right), j=1,2,\dots , n_y+1$$

(43)

In the GDQ method, the derivative of a solution function with respect to a variable at a given discrete point can be calculated as a weighted linear sum of the function values at all discrete points in the mesh line. The stability and accuracy of the proposed numerical method is highly affected by the calculation of the weight coefficients, location, and count of grid points [40]. According to the GDQ method, the r-th or s-th-order derivative of a function f at x and y or both directions with n_x and n_y discrete grid points can be computed by:

$$\left(\frac{\partial f^r\left(x_i,y_j\right)}{\partial x^r}\right)={\displaystyle \sum }_{k=1}^{n_x}{C}_{ik}^{\left(r\right)}f(x_k,y_j), r=1,2,‥, n_x-1$$

(44)

$$\left(\frac{\partial f^s\left(x_i,y_j\right)}{\partial y^s}\right)={\displaystyle \sum }_{m=1}^{n_y}{C}_{jm}^{\left(r\right)}f(x_i,y_m), s=1,2,\dots , n_y-1$$

(45)

$$\left(\frac{\partial f^{\left(r+s\right)}\left(x_i,y_j\right)}{\partial x^r\partial y^s}\right)={\displaystyle \sum }_{k=1}^{n_x}{C}_{ik}^{\left(r\right)}{\displaystyle \sum }_{m=1}^{n_y}{C}_{jm}^{\left(s\right)}f(x_k,y_m), r=1,2,\dots , n_x-1, s=1,2,\dots , n_y-1$$

(46)

f(x,y) and $C_{ij}^{\left(r\right)}$ are the function values and the weight coefficients for the r-th order derivatives at (x_i, y_j) grid point. Lagrange polynomial functions are chosen to determine the weight coefficients $C_{ij}^{\left(r\right)}.$ Weight coefficients for the first-order derivative, i.e., r = 1, can be written as

$$C_{ik}^{\left(1\right)}=\frac{\varphi \left(x_i\right)}{\left(x_i-x_j\right)\varphi \left(x_j\right)}, i, j=1, 2,\dots , n_x \mathrm{and} i\ne j$$

(47)

where

$$C_{ik}^{\left(1\right)}=\frac{\varphi \left(x_i\right)}{\left(x_i-x_j\right)\varphi \left(x_j\right)}, i, j=1, 2,\dots , n_x \mathrm{and} i\ne j$$

(48)

Recursive relations for higher-order derivatives can be written as

$$C_{ij}^{\left(r\right)}=r\left[C_{ii}^{\left(r-1\right)}{C}_{ij}^{\left(1\right)}-\frac{C_{ij}^{\left(r-1\right)}}{\left(x_i-x_j\right)}\right], {\left(i\ne j\right)}_{ }$$

(49)

$$C_{ii}^{\left(r\right)}=-{\displaystyle \sum }_{j=1, i\ne j }^{n_x}{C}_{ij}^{\left(r\right)}$$

(50)

Following the procedure as explained above, the governing equilibrium equations subjected to mixed-type simply supported and roller skate boundary conditions are written in terms of the function values at the grid points. All of the algebraic equation system can be expressed simply in the matrix form as

$$\mathrm{K}U=\mathrm{F}$$

(51)

The total number of unknown coefficients in terms of unknown displacement values is 5(n_x + 1) (n_y + 1). The number of equations written for the governing equilibrium equations at the internal grid points is 5(n_x − 1) (n_y − 1). The number of equations written for the boundary conditions given in Equations (20) and (21) is 10(n_x + 1) + 10(n_y − 1). The total number of equations is equal to the total number of unknown coefficients. Equation (51) is subsequently solved in order to find unknown displacement values.

4. Numerical Results and Discussion

In this section, the static analysis of symmetric ([0°/90°/0°] and [0°/90°/90°/0°]) and antisymmetric ([0°/90°]) laminated cross-ply spherical (R₁ = R₂ = R), cylindrical (either R₁ = ∞ or R₂ = ∞), and flat panels (R₁ = R₂ = ∞) subjected to simply supported boundary conditions prescribed on two opposite edges and roller skate-type boundary conditions on the remaining two edges is investigated. A computer program for that purpose was written in the MATLAB program in order to implement both proposed solution methodologies, the BDM and GDQ methods. In this part, several examples are carried out parametrically by changing the effect of curvature, thickness-to-length ratio, lamination, and their interactions. In all of the examples except for the validation study, the following material properties were used. This material was chosen to investigate the impact of material strength on the statical behavior of a composite panel structure. The mechanical properties of each layer of laminate are taken as [49]:

E₁ = 175.78 GPa, E₁/E₂ = 25, G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, and ν₁₂ = 0.25

in which E₁ and E₂ are the in-plane Young’s moduli in the x and y coordinate directions, respectively, while G₁₂ denotes in-plane shear modulus. G₁₃ and G₂₃ are transverse shear moduli in the x–z and y–z planes, respectively, while ν₁₂ is a major Poisson’s ratio on the x–y plane.

The non-dimensional displacement, w*, and stress, σ_i*, response of cross-ply laminated shells are:

$$w^{\ast }=-\frac{{10}^3{E}_2{h}^3}{q_0{a}^4}{u}_3, {\sigma }_i{}^{\ast }=-\frac{100\times h}{q_0{a}^2}{\sigma }_i \left(\mathrm{i}=x, y\right)$$

(52)

$w^{\ast }$ and ${\sigma }_i{}^{\ast }$ are computed at the center of the panel. The load term expressions for arbitrary transverse applied distributed load can be defined using a Fourier double series as

$$q\left(x,\mathrm{y}\right)={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{q}_{mn}\sin (\alpha x)\sin (\beta \mathrm{y})$$

(53)

where

$$q_{\mathrm{mn}}=\frac{16q_0}{{\mathsf{\pi }}^{2}mn} (m =\mathrm{odd})$$

(54)

As it was said before, the FSDT theory (also known as the Mindlin–Reissner theory) needs a shear correction factor (ks) in order to eliminate the error in the calculation of constant out-of-plane shear stresses through the thickness direction. This factor connects with several parameters such as the geometrical shape of the structure, the lamina sequence pattern, etc. For the calculations in the proposed solution, correction factor is chosen as ks = 5/6 for a solid rectangular type of cross-section.

4.1. Validation Study

This section is firstly intended to examine the validity of the obtained results. For that purpose, displacement results were validated with thin (a/h = 100) symmetric and anti-symmetric cross-ply laminated panels with well-known Navier-type boundary conditions, as mentioned in Reddy and Liu [18]. The stress results were then validated using the finite element method for the same geometrical and mechanical properties. In this solution, the coefficients b_n = d_n = 0 are taken as zero in order to regenerate the Navier-type boundary conditions. Normalized central deflections and stress results for cross-ply shells are given in

Table 1. Normalized central deflections and stresses of cross-ply laminated shells under distributed load (a/h = 100).

		[0°/90°]			[0°/90°/0°]			[0°/90°/90°/0°]
R/a		BDM	GDQ	Ref. [18]	BDM	GDQ	Ref. [18]	BDM	GDQ	Ref. [18]
10	$w^{\ast }$	5.5428	5.542	5.5428	3.644	3.644	3.6445	3.720	3.720	3.720
20		11.273	11.27	11.273	5.547	5.547	5.5473	5.661	5.661	5.661
50		15.714	15.71	15.714	6.482	6.482	6.4827	6.614	6.614	6.614
100		16.645	16.64	16.645	6.642	6.642	6.6421	6.777	6.777	6.777
Plate		16.979	16.97	16.980	6.697	6.696	6.6970	6.833	6.833	6.842
R/a		BDM	GDQ	FEM	BDM	GDQ	FEM	BDM	GDQ	FEM
10	${\sigma }_x{}^{\ast }$	420.17	420.2	418.98	4688.6	4688.6	4678.5	4961.8	4961.8	4954.4
20		868.116	868.1	866.180	6962.9	6963	6950	7238.2	7238.2	7227
50		1196.10	1196.1	1196	7956.6	7956.7	7985.2	8175.8	8175.8	8209.2
100		1256.62	1256.6	1255.6	8081.2	8081.3	8075	8271	8271	8265.9
Plate		1268.70	1268.7	1267.4	8073	8073	8064	8229.6	8229.5	8222.7

. It is worth noting that almost no difference was detected changing from moderately deep (R/a = 10) to (R/a = Plate) between the BDM and GDQ under the very well-known SS3-type boundary conditions. The relatively small discrepancies between these methodologies and the finite element method in the ANSYS Package program can be predicated to the relationship between strong-form and weak-form formulations. In our proposed solution, the methodologies for BDM and GDQ methods are based on the strong form, which satisfies the compulsory and natural boundary conditions. In contrast to strong form, the finite element method uses weak-form formulation and does not apply natural or force boundary conditions which are imposed on the secondary variable, such as forces and moments.

The proposed methodologies are also compared with the finite element method for the current boundary conditions (SS-RS) in

Table 2. Normalized central deflections of cross-ply [0°/90°] laminated shells under uniform load. With SS–RS boundary conditions (a/h = 20).

	R/a	BDM	GDQ	FEM
	10	14.880	18.315	20.900
$w^{\ast }$	20	20.763	20.796	22.165
	Plate	22.152	19.901	22.603
	10	154.778	194.918	142.100
${\sigma }_x{}^{\ast }$	20	172.361	176.382	129.650
	Plate	136.668	118.459	105.135
	10	1850.1	2323.4	2569.2
${\sigma }_y{}^{\ast }$	20	2612.8	2614.2	2684.3
	Plate	2785.6	2466.0	2690.4

. Comparison results are given for antisymmetric [0°/90°] laminated cross-ply spherical (R₁ = R₂ = R) moderately thick (a/h = 20) panels for different R/a ratios. Since the boundary conditions cannot be fully integrated using the finite element method, there is a difference between the FEM and the intended solution methods.

4.2. Convergence Study

The convergence for displacements, $w^{\ast }$, and stresses, ${\sigma }_x{}^{\ast }$, of a moderately thick (a/h = 20) and symmetric cross-ply (0°/90°/0°) spherical panel (R₁/a = R₂/b = 10) is shown in Figure 2. A rapid and monotonic convergence is observed for the Fourier series of the terms m, n > 10 for displacements and m, n >20 for stresses within the boundary discontinues method. The same convergence characteristics are observed for grid numbers m, n = 13 within the generalized differential quadrature method.

4.3. Parametrice Study

The non-dimensional displacements, $w^{\ast }$, and stresses, ${\sigma }_x{}^{\ast }$, ${\sigma }_y{}^{\ast }$, of symmetric ([0°/90°/0°] and [0°/90°/90°/0°]) and antisymmetric ([0°/90°]) laminated cross-ply plates and shells with different a/h and R/a ratios under uniformly distributed load are presented in

Table 3. Normalized central deflections of cross-ply laminated shells under uniform load.

			[0°/90°]			[0°/90°/0°]			[0°/90°/90°/0°]
R/a	a/h	Method	w*	${\mathit{\sigma }}_{\mathit{x}}{{}^{\ast }}^{ }$	${\mathit{\sigma }}_{\mathit{y}}{}^{\ast }$	w*	${\mathit{\sigma }}_{\mathit{x}}{}^{\ast }$	${\mathit{\sigma }}_{\mathit{y}}{}^{\ast }$	w*	${\mathit{\sigma }}_{\mathit{x}}{}^{\ast }$	${\mathit{\sigma }}_{\mathit{y}}{}^{\ast }$
10	20	BDM	14.880	154.778	1850.1	10.950	1432.6	101.046	10.999	1489.3	110.386
		GDQ	18.315	194.918	2323.4	10.811	1450.8	99.4543	10.864	1510.8	108.887
	50	BDM	4.9204	206.615	1460.8	2.485	1851.6	40.88	3.1181	2394	76.979
		GDQ	8.1924	308.405	2690	2.508	1874.7	41.746	3.1443	2419.0	78.042
	100	BDM	1.3248	159.760	550.63	0.666	1778.9	9.1941	0.9236	2499.9	36.3349
		GDQ	2.3004	201.457	1313.9	0.671	1792.5	9.9637	0.9304	2518.5	37.3247
Plate	20	BDM	22.147	136.554	2785	28.660	944.08	269.434	20.966	769.890	200.210
		GDQ	19.901	118.459	2466.0	27.058	1004.4	252.13	20.190	828.9	191.707
	50	BDM	19.818	399.629	6406.5	19.598	3048.2	436.80	15.786	2680.8	390.017
		GDQ	17.114	338.554	5439.2	19.469	3070.2	458.07	15.853	2681.5	389.6595
	100	BDM	19.193	837.166	1244.5	16.799	6544.3	793.02	14.137	5893.2	703.959
		GDQ	16.588	695.911	1058.2	17.976	6381.6	846.779	14.98	5622.0	741.542

. It can be seen that there is a difference between the range 11.28% (R/a = Plate) and 18.75% (R/a = 10) and for moderately thick (a/h = 20) antisymmetric ([0°/90°]) plates and shell between the GDQ and BDM methods. However very close results were obtained for symmetric ([0°/90°/0°] and [0°/90°/90°/0°]) plates and shells for all R/a and a/h ratios.

Table 4. Normalized central deflections of cross-ply [0°/90°] laminated shells under uniform load.

	R/a	a/h	Method	E1/E2
				3	10	20	30	40	50
w*	10	20	BDM	33.877	23.8006	17.047	13.171	10.655	8.902
			GDQ	34.559	25.9268	20.218	16.769	14.392	12.638
		50	BDM	16.055	9.336	5.860	4.229	3.283	2.668
			GDQ	17.222	12.108	9.130	7.444	6.314	5.494
		100	BDM	5.4233	2.7262	1.6004	1.1292	0.8699	0.705
			GDQ	5.8318	3.665	2.6041	2.0677	1.7306	1.495
	Plate	20	BDM	41.708	31.449	24.464	20.277	17.427	15.346
			GDQ	39.952	28.524	21.942	18.274	15.833	14.059
		50	BDM	37.953	28.482	21.977	18.077	15.429	13.500
			GDQ	36.683	25.320	19.007	15.609	13.396	11.810
		100	BDM	36.995	27.699	21.311	17.487	14.895	13.010
			GDQ	36.123	24.734	18.448	15.089	12.914	11.362
${\sigma }_x{}^{\ast }$	10	20	BDM	408.591	268.165	181.547	134.061	104.293	84.170
			GDQ	420.420	294.297	218.847	175.913	147.473	127.130
		50	BDM	727.107	402.740	247.767	176.540	135.605	109.158
			GDQ	755.659	480.335	346.930	278.613	234.863	203.835
		100	BDM	696.610	332.097	193.171	136.115	104.819	85.020
			GDQ	706.825	363.486	233.464	178.37	146.929	126.241
	Plate	20	BDM	344.536	231.245	159.110	118.960	93.386	75.828
			GDQ	338.443	212.727	139.872	101.943	78.762	63.292
		50	BDM	932.956	647.333	459.655	352.222	282.161	233.055
			GDQ	895.587	582.226	394.657	294.155	231.333	188.603
		100	BDM	1905.6	1336.1	958.7	740.8	597.8	496.9
			GDQ	1808.3	1183.6	807.6	605.1	478	391.2
${\sigma }_y{}^{\ast }$	10	20	BDM	834.2	1380.2	1747.4	1922.3	2009.3	2051.6
			GDQ	858	1525.6	2111.6	2499.8	2776.9	2984.3
		50	BDM	1085.2	1373.3	1453.2	1460.3	1447.6	1428.7
			GDQ	1184	1894	2480.7	2866	3141.9	3350
		100	BDM	731.916	670.952	579.850	528.011	495.387	472.958
			GDQ	811.3	1060.7	1244.8	1374	1474.6	1557
	Plate	20	BDM	967.9	1795.5	2521.2	3006.7	3361.3	3634.6
			GDQ	921.1	1601.7	2225.8	2672.7	3016.9	3293.2
		50	BDM	2259.0	4162.7	5810.8	6906.9	7708.3	8329.4
			GDQ	2166.5	3633	4934.7	5869.5	6600.4	7198.3
		100	BDM	4428	8127	11,303	13,401	14,929	16,112
			GDQ	4285	7123	9611	11,390	12,781	13,921

and

Table 5. Normalized central deflections of cross-ply [0°/90°/0°] laminated shells under uniform load.

	R/a	a/h	Method	E1/E2
				3	10	20	30	40	50
w*	10	20	BDM	32.259	19.215	12.731	9.620	7.755	6.504
			GDQ	31.795	18.864	12.544	9.514	7.692	6.466
		50	BDM	13.260	5.632	3.059	2.091	1.587	1.277
			GDQ	13.344	5.678	3.086	2.111	1.601	1.289
		100	BDM	4.356	1.631	0.834	0.554	0.414	0.329
			GDQ	4.357	1.635	0.839	0.559	0.4177	0.332
	Plate	20	BDM	43.818	34.616	30.074	27.493	25.597	24.062
			GDQ	42.631	32.968	28.419	25.954	24.192	22.785
		50	BDM	37.065	25.627	20.850	18.653	17.263	16.240
			GDQ	37.146	25.559	20.728	18.520	17.132	16.116
		100	BDM	35.240	23.015	18.047	15.883	14.585	13.674
			GDQ	36.173	24.169	19.235	17.042	15.701	14.742
${\sigma }_x{}^{\ast }$	10	20	BDM	875.1	1245.6	1395.2	1459.5	1496.1	1519.9
			GDQ	886.2	1264.1	1413.8	1477.5	1513.6	1537.1
		50	BDM	1550.2	1865.4	1867	1836.2	1809.6	1788.7
			GDQ	1555.8	1883.5	1889.3	1859.8	1833.7	1813.1
		100	BDM	1559	1798.8	1792.4	1765.9	1743.5	1725.8
			GDQ	1554.9	1802.4	1803.4	1781.6	1762.4	1747
	Plate	20	BDM	699.61	927.91	956.36	931.67	895.94	858.31
			GDQ	720.6	972.1	1011.4	991.6	958.5	922.4
		50	BDM	1998.1	2828.2	3033.5	3041.7	2998.1	2936.2
			GDQ	1993.2	2839	3052.9	3065.7	3025	2965.3
		100	BDM	4140.5	5968.8	6481.1	6559.9	6519.6	6434.3
			GDQ	4059.6	5843.3	6328	6388.9	6334.7	6237.9
${\sigma }_y{}^{\ast }$	10	20	BDM	361.159	190.887	119.383	87.688	69.430	57.484
			GDQ	357.635	187.504	117.347	86.424	68.601	56.918
		50	BDM	409.137	127.946	54.639	32.178	22.032	16.464
			GDQ	412.912	129.886	55.701	32.898	22.574	16.899
		100	BDM	265.097	51.627	14.368	6.362	3.601	2.383
			GDQ	264.017	52.361	15.190	7.074	4.208	2.906
	Plate	20	BDM	452.400	334.082	283.944	257.786	239.216	224.367
			GDQ	440.178	316.606	266.128	241.087	223.864	210.326
		50	BDM	993.809	632.001	496.701	439.754	405.647	381.325
			GDQ	992.951	627.821	491.164	433.981	399.974	375.874
		100	BDM	1910.4	1142	859.2	746	681.9	638.5
			GDQ	1947.4	1193.2	913.2	799.1	733.2	687.7

present the effects of the lamina material orthotropy (E₁/E₂) on the non-dimensional displacements, $w^{\ast }$, and stresses, ${\sigma }_x{}^{\ast }$, ${\sigma }_y{}^{\ast }$, of symmetric [0°/90°/0°] and antisymmetric [0°/90°] laminated cross-ply plates and shells. The results are in good agreement for symmetric [0°/90°/0°], whereas BDM underestimates the non-dimensional displacements and stresses for the R/a = 10 ratio. The difference between BDM and GDQ for the ratio of R/a = 10 antisymmetric [0°/90°] shell increases with the increase of (E₁/E₂). The reason for this may be that discontinues being dominant in the bending–stretching matrix (B matrix presence) in antisymmetric lamination.

Variations of normalized central deflection, w*, and stresses, ${\sigma }_x{}^{\ast }$, of spherical (R/a = 10) shell and two cylindrically curved panels for various length-to-thickness, a/h ratios under uniform loading are presented in Figure 3 for antisymmetric cross-ply [0°/90°] and Figure 4 for symmetric [0°/90°/0°] lamination, respectively. The transverse shear deformation curves of the [0°/90°] spherical and cylindrical panel, with R₂ → ∞, are almost the same for the whole range of a/h ratio in the BDM method given in Figure 3a, as has already been observed in the case of the mixed simply supported boundary conditions in Reference [30]. While there is a relative difference between the BDM and GDQ methods for [0°/90°] spherical and cylindrical panels, normalized central deflection, w*, and stress, ${\sigma }_x{}^{\ast }$, curves have similar characteristics for both methods. However, the normalized central deflection, w*, and stress, ${\sigma }_x{}^{\ast }$, curves of symmetric [0°/90°/0°] spherical and cylindrical panels with R₂ → ∞ are almost the same for the whole range of a/h ratio, as is shown in Figure 4. The normalized central deflection and stress results obtained with BDM for symmetric [0°/90°/0°] lamination are exactly matched with GDQ method. Variations of the normalized stress, ${\sigma }_x{}^{\ast }$, of antisymmetric [0^o/90^o] and symmetric [0°/90°/0°] spherical and two cylindrically curved panels for different R/a ratios are individually displayed in Figure 3b and Figure 4b, respectively.

Figure 5 and Figure 6 present the variations of the normalized central deflection, w*, and stress, ${\sigma }_x{}^{\ast }$, of moderately thick (a/h = 20) curved panels (one spherical and two cylindrical) with antisymmetric [0°/90°] and symmetric [0°/90°/0°] laminations with R/a ratio. Based on the present results, the spherical panel response was found to represent similar characteristic to those observed for its cylindrical counterpart, with R₂ → ∞ for each R/a ratio value. Symmetric [0°/90°/0°] laminates also exhibited a similar trend. For symmetric [0°/90°/0°] lamination, non-dimensional displacements, w*, both for the spherical panel and cylindrical counterparts—as (R₂ → ∞)—possess a flatter regime after R/a > 60, while non-dimensional displacement, w*, values in the cylindrical panel under R₁ → ∞ conditions possess a flatter regime after R/a > 20. The normalized central deflection results for symmetric [0°/90°/0°] laminations obtained with BDM are exactly matched with the GDQ solution. Variations of the normalized stress, ${\sigma }_x{}^{\ast }$, of the moderately thick (a/h = 20) antisymmetric [0°/90°] and symmetric ([0°/90°/0°] spherical and two cylindrically curved panels for different R/a ratios are individually displayed in Figure 5b and Figure 6b, respectively.

Figure 7 and Figure 8 present the nondimensional displacement, w*, of [0°/90°] and [0°/90°/0°] spherical panels with the change of R/a and a/h ratios, respectively. The impact of transverse shear deformation and the action of membrane owing to curvature on w* is observable. The curvature effect on transverse displacement, w*, also plays a critical role in the thinner shell regime, specifically beginning from the ratio R/a < 40. Bending–stretching coupling is inherent in antisymmetric laminations, and it directly impacts the interaction of membrane action with the beam–column/tie-bar effect. The membrane action due to curvature has a complicated interaction with the stated mechanism. This interaction must be considered for the prior design of composite shells. Figure 9 and Figure 10 display the variations of the normalized stresses, ${\sigma }_x{}^{\ast }$, with a/h and R/a ratios for antisymmetric [0°/90°] and symmetric [0°/90°/0°] spherical panels, respectively. The membrane action has similar effect in both the thick and the thinner panel regime, particularly for R/a ≤ 40, as mentioned above.

Variations of the non-dimensional displacements, $w^{\ast }$, and stresses, ${\sigma }_x{}^{\ast }$ of a moderately thick (a/h = 20) shell panel (R/a = 10) of antisymmetric [0°/90°] cross-ply spherical panels, computed at the whole surface are presented in Figure 11 and Figure 12, respectively. Non-dimensional displacements, w*, and stresses, ${\sigma }_x{}^{\ast }$, reached their maximum value at the center of the panel. It can be observed on the graphs that the boundary conditions were exactly satisfied by the boundary discontinuous double Fourier series method.

5. Conclusions

In this study, an analytical solution was presented using the boundary discontinuous Fourier series method for the static analysis of thick and shallow cylindrical, spherical panels in addition to plates (a special case of the shells) made of symmetric and antisymmetric cross-ply laminated composite material subjected to unsolved mixed type boundary conditions described on the edges. The governing equilibrium equations for laminated composite panels were obtained using the virtual work principle. Boundary Fourier coefficients arising from the discontinuity of the assumed functions and their derivatives at the boundaries were introduced in order to solve the complementary solution. Analytical solutions obtained with the BDM method were also compared with the successful integration of the generalized differential quadrature (GDQ) method. Comprehensive tabular and graphical results were displayed so as to show the effects of curvature on the deflections and stresses of thick shallow shells. The validity of the proposed model was authenticated with a review of the available literature, and the convergence characteristics were demonstrated. The changing trends of displacements and moments were explained in detail by investigating the effects of various parameters, such as lamination, material properties, curvature, etc. The numerical results could particularly be utilized during the early design stages of such laminated structures and as benchmark solutions for the future comparison of numerical results, such as finite element analysis and boundary element methods. The following are the major findings of the present research.

✓

It is worth noting that almost no difference resulted from changing from moderately deep (R/a = 10) to (R/a = Plate) between BDM and GDQ for the very well-known SS3 type boundary conditions. The relatively small discrepancies between the present methodologies and the finite element method can be predicated to the variation between strong-form and weak-form formulations. Our proposed solution methodologies for BDM and GDQ methods are based on the strong form, as it satisfies the compulsory and natural boundary conditions. In contrast to strong-form, the finite element method uses the weak-form formulation and does not apply natural or force boundary conditions which are imposed on the secondary variable, such as forces and moments.

✓

A rapid and monotonic convergence was observed with the Fourier series for the terms m, n > 10 for displacements and m, n > 20 for stress in the BDM. The stress convergence was higher than the displacement. This is possibly due to the presence of a discontinuity (complementary boundary constraint) in the derivative of the displacement in expression of the stress. The same convergence characteristics were observed for grid numbers m, n = 13 with the generalized differential quadrature method.

✓

A difference was seen when changing from the range 11.28% (R/a = Plate) to 18.75% (R/a = 10) for moderately thick (a/h = 20) antisymmetric [0°/90°] plates and shell with the roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges were subjected to the simply supported constraint between the BDM and GDQ methods. This is possibly due to the presence of a discontinuity in the derivatives which comes from the bending–stretching matrix (B matrix presence) in antisymmetric lamination. However very close results were obtained for symmetric ([0°/90°/0°] plates and shells using the whole range R/a and a/h ratios. It is also important to state that all three methods (BDM, DQM, and FEM) use different formulations, and among them, BDM provides analytical solutions and satisfies boundary conditions exactly, as is shown in Table 2 compared to FEM and DQM.

✓

The difference between BDM and GDQ for an antisymmetric [0°/90°] shell increases with the increase of (E₁/E₂). The reason for this may be the dominant discontinuity in the bending–stretching matrix (B matrix presence) in antisymmetric lamination.

✓

The effect of the modulus ratio, E₁/E₂, was more pronounced in the thin laminates (a/h > 50). Furthermore, this effect was increased by the beam–column effect in the case of an antisymmetric laminate. This is because the bending–stretching coupling is dominant in an antisymmetric laminate and produces a softening effect on the beam–column type, which consequently increases the normalized central deflection.

✓

The effect of the radius-to-length ratio R/a on transverse displacement, w*, also plays a critical role in the thinner shell regime, specifically beginning from the ratio R/a < 40. Bending–stretching coupling is inherited in antisymmetric laminations, and it directly impacts the interaction of membrane action with the beam–column/tie-bar effect. The membrane action due to curvature has a complicated interaction with the stated mechanism. This interaction should be considered during the prior design of composite shells.