Plastic Instability Pressure of Toroidal Shells

This paper considers the determination of plastic instability pressure in toroidal shells under internal uniform pressure. Analytical and numerical approaches, as well as verification by experiments, are presented. This work is inspired by Mellor’s treatment (1983, Engineering Plasticity, Ellis Horwood Ltd., Chichester; 1960, “The Ultimate Strength of Thin-Walled Shells and Circular Diaphragms Subjected to Hydrostatic Pressure,” Int. J. Mech. Sci., 1, pp. 216–228; 1962, “Tensile Instability in Thin-Walled Tubes,” J. Mech. Eng. Sci., 4(3), pp. 251–256), which assumed that plastic instability occurs at the maximum load. A closed-form formula of plastic instability condition is derived analytically. This expression for toroidal shells turns out to be the general case of spherical and cylindrical shells given by Mellor. Then the corresponding pressure is obtained by semi-analytical analysis for a material with the strain hardening characteristic, $σ=A(B+ε)n$⁠. For the numerical approach, plastic instability pressure is the maximum pressure at which a small pressure increment causes a very large deformation. This is identified by the slope of pressure—change of volume curve approaching zero. Both approaches predict the onset of instability at the inner equator point. Experimental results of two nominally identical stainless steel toroidal shells correlated well to both approaches in terms of the magnitude of pressure and failure location.