Axisymmetric Buckling of Pressure‐Loaded Spherical Caps

Abstract

The axisymmetric buckling behavior of clamped shallow spherical shells under uniform pressure is investigated using Marguerre's shallow‐shell equations of 1939. The deflected shape of the shell as well as the stress function are described by linear combinations of Bessel functions and modified Bessel functions that satisfy all the relevant boundary and continuity conditions. The buckling characteristics of these caps are examined using a fully nonlinear Galerkin solution procedure, a classical bifurcation analysis, and a reduced‐stiffness bifurcation analysis. This allows the elucidation of the imperfection sensitivity and nonlinear behavior of this important class of shell structures. A systematic parametric analysis highlights the interplay between these contrasting approaches, and demonstrates the lower boundedness of the reduced‐stiffness analytical procedure for predicting imperfection‐sensitive elastic‐collapse pressures and its potential importance as a design methodology. A simple closed‐form solution is given for this analytical lower bound. This compares favorably with a large collection of available experimental results but demonstrates the variability and possible inadequacies in some of the existing design rules for spherical shell structures.