On buckling of toroidal shells under external pressure

Abstract

The paper presents results of a numerical study into the buckling resistance of geometrically perfect and imperfect steel toroidal shells with closed cross-sections. Elastic and elastic-plastic buckling analyses of shells subjected to uniform external pressure were carried out for a range of geometries, boundary conditions and material properties. Toroids with circular and elliptical cross-sections were investigated. Elastic-plastic analyses carried out for toroids with circular cross-sections allowed identifying shell configurations, which fail by bifurcation buckling or by collapse. An appropriate equation is proposed for identification of configurations for which either bifurcation or collapse governs the shell's stability. The proposed equation supplements Jordan's long-standing design formula, which is applicable to elastic buckling only. The obtained results show that toroids with an elliptical cross-section can be much stronger than shells with a circular cross-section. Calculations identified geometries possessing the largest load carrying capacity. It was found numerically that on both sides of ‘the peak performance geometry’ there is a different failure mechanism which in turn, as it is discussed in the paper, leads to different imperfection sensitivity. A simple design curve is provided for the separation of these two regions. Within each region, the paper provides simple design equations for the elastic buckling strength of geometrically perfect toroids with elliptical cross-sections. Imperfection sensitivities of elastic and elastic-plastic buckling loads to initial, localised and global shape deviations from perfect geometry are given for typical shells with circular and elliptical cross-sections. Initial geometric imperfections in the form of the eigenmode, ‘a single wave’ extracted from the eigenmode and inward dimple modelled by a cosine function in both R-circumferential and r-meridional directions are studied.