A simple formula for buckling load was derived from the asymptotic analysis of nonlinear behavior of a thin spherical shell. Firstly, two asymptotic cases were studied: the initial post-buckling regime of a perfect structure with small (compared to shell thickness) deflections and equilibrium states with large deflections. Two asymptotic formulae were jointed to obtain the solution for the entire range of deflection amplitude. Then the solution was modified for an imperfect shell. Initial deflections were introduced by only one parameter: the slope of the load–deflection diagram at small pressure. This minimal information was enough to predict the buckling load of the structure with localized imperfections. The suggested asymptotic result was validated by the finite element method and by comparison with experimental data.

A numerical study (ANSYS software) deals with deformation and buckling of two series of elastic isotropic shallow closed shells subjected to external pressure: 1) circular cones and 2) spherical segments. All the shells have the same thickness and dimensions. Fixation of edges is periodically discrete. Movable supports are alternative with hinged supports. The fixation causes periodically non-uniform stress-strain state in the circumferential direction. Periodicity of prebuckling stress-strain state is equal to a number of fixed sections. Buckling behaviour of cones and spheres is different. The phenomenon of “static resonance” is found for conical shells. The essence of the effect is as follows. The minimum limit pressure of geometrically nonlinear analysis corresponds to the periodicity of stress-strain state of a conical shell which coincides with a half-sum of periodicity of the first eigenvibration mode of an unloaded shell and the first eigenmode of a continuously hinged-supported shell subjected to external pressure.