Instability of semi-ellipsoidal shells

Abstract

This paper deals with the analysis of axisymmetric elastic buckling of semi-ellipsoidal shells under external pressure. Reissner's non-linear equations for the axisymmetric deformations of shells of revolution are specialized here for ellipsoidal shells and used in this analysis. The geometric singularity of these equations at the apex of shells is avoided through further modifications by imposing the conditions of continuity of the shell parameters and other variables across the apex. A multisegment method of integration, as developed by Kalnins and Lestingi, has been used to solve the non-linear equations of ellipsoidal shells. The critical pressure of a shell is interpreted from the fact that the mode of primary deformation along the fundamental equilibrium path of a structure cannot change without a change in its status of stability. The appearance of a secondary deformation on the fundamental equilibrium path is always hinted by a substantial increase in the deformation rate with respect to the load parameter. Specifically, for shells at the critical equilibrium, any further increase in loading, however small, causes the shell enormous deformation indicating that the state of deformation of the shell which corresponds to the lowest potential energy is far from that at critical pressure. Extensive numerical results on the buckling of semi-ellipsoidal shells with completely restrained edges and for varying thickness ratios have been obtained. It is observed that the critical pressure increases with an increase in the ratio of minor to major axes of the ellipsoidal shell and decreases with an increase in its thickness ratio.