| A complete Fourier solution to the boundary value problem of static response under transverse load of general cross-ply doubly curved shells of rectangular planform is investigated. Plates are treated a special case of shells. A higher order shear deformation theory (HSDT) based formulation pertaining to laminated shell boundary-value problems is selected because of its inherent accuracy over the well-established classical lamination theory (CLT) and first order shear deformation theory (FSDT). A boundary-discontinuous double Fourier series approach is used to solve a system of five partial differential equations and associated admissible boundary conditions, generated by the HSDT. Unlike the conventional Navier and Levy type approaches, which can provide only particular solutions, the present method is general enough to provide the complete (particular as well as the complementary) solution for any arbitrary combination of admissible boundary conditions with almost equal ease.; Five specific combinations of admissible boundary conditions, (i, ii) SS3 (prescribed on two opposite edges) plus SS2 or C4 (prescribed on the remaining two), and (iii, iv, v) SS1 or SS4 or C4 (prescribed on all four edges), are investigated to illustrate the theory. The first two cases are studied to test the validity of the present approach by applying it to the Levy type boundary conditions. The effect of surface-parallel (or in-plane) boundary conditions on the out-of-plane response of the laminated structure is investigated in the cases (iii) and (iv), while the case (v) is studied to determine the effect of end clamping. A single set of displacement functions is used to model the five unknown displacements and rotations. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of cross-ply spherical panels. Hitherto unavailable numerical results presented include sensitivity of the predicted response quantities of interest to (cross-ply) lamination asymmetry (bending-extensional coupling), surface-parallel boundary constraint, material properties, thickness (or transverse shear) and curvature (or membrane) effects, as well as their interactions. These numerical results should serve as bench-mark solutions for comparison with those obtained by approximate numerical techniques such as finite element, boundary element, Raleigh-Ritz, etc. |
