The non-axisymmetric postbuckling behaviour of shallow spherical domes

Abstract

The non-axisymmetric postbuckling behaviour of an elastic, shallow spherical dome, which is rigidly clamped along its contour and loaded with a uniform transverse pressure, is considered. The solution of the problem is constructed using the Rayleigh-Ritz method based on the Marguerre equations in which the displacements in the circumferential direction are approximated by a Fourier series, and the radial displacements by Bessel functions. The resulting system of non-linear algebraic equations is solved by prolongation methods. It is shown for the first time that a shell has postbuckling, non-axisymmetric equilibrium states with loads which are significantly less than the upper critical load as well as the loads corresponding to bifurcation points. It is suggested that, taking into account the forms of these equilibrium states as the initial inaccuracies of a spherical dome should enable one to model the spread in its experimentally obtained critical loads.