## 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  
Figure 11: Variations of the non-dimensional displacements, \(w^{*}\), of a moderately thick (\(a/h=20\)) shell panel (R/\(a=10\)) and of antisymmetric [0\({}^{\circ}\)/90\({}^{\circ}\)] cross-ply spherical panels along the panel surface. Figure 12: Variations of the non-dimensional stresses, \(\sigma_{x}{{}^{*}}\), of a moderately thick (\(a/h=20\)) shell panel (R/\(a=10\)) and of antisymmetric [0\({}^{\circ}\)/90\({}^{\circ}\)] cross-ply spherical panels along the panel surface. 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\({}^{\circ}\)/90\({}^{\circ}\)] 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.