Nonlinear shallow shell equations are derived for a thin shell of revolution having the shape of a narrow segment of a toroidal shell centered at the equator. The equations are derived by considering a cylindrical shell, described by nonlinear Donnell theory, with an initial radial deformation. Linear buckling equations are obtained by perturbing the nonlinear shell equations. The buckling equations are solved for the case of a simple supported equatorial segment of a spherical shell loaded in the axial direction by its own weight. Plots are presented which compare a critical thickness parameter with the results of an elementary approach. The elementary approach assumes that the shell will buckle if the maximum compressive stress is greater than the critical compressive stress for a complete sphere loaded by uniform external pressure. / M.S.

This problem has been selected from the mathematical theory of elasticity. We consider a rectangular plate of thickness h, length a, and width b. The plate is subjected to compressive forces. These forces act in the neutral plane and give the plate a tendency to buckle. However, this problem differs from other plate problems in that it is assumed that there are two intermediate supports located on the edges of the plate parallel to the compressive forces.

The theory is based on an assumed displacement field, in which the surface displacements are expanded in powers of the thickness coordinate up to the third order. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory accounts for small strains but moderately large displacements (i.e., von Karman strains). Exact solutions for certain cross-ply shells and finite-element models of the theory are also developed. The finite-element model is based on independent approximations of the displacements and bending moments (i.e., mixed formulation), and therefore only C°-approximations are required. Further, the mixed variational formulations developed herein suggest that the bending moments can be interpolated using discontinuous approximations (across inter-element boundaries). The finite element is used to analyze cross-ply and angle-ply laminated shells for bending, vibration, and transient response. Numerical results are presented to show the effects of boundary conditions, lamination scheme (i.e., bending-stretching coupling and material anisotropy) shear deformation, and geometric nonlinearity on deflections and frequencies. Many of the numerical results presented here for laminated shells should serve as references for future investigations. / Ph. D.

University of Macau / Faculty of Science and Technology / Department of Civil and Environmental Engineering