Analysis of toroidal–spherical shells under hydrostatic pressure: large displacements and elastic buckling

A novel form of thin-walled structure featuring two center-combined shells consisting of a spherical shell and a toroidal shell is designed for new applications in ocean and offshore engineering. First, the energy functional of the underwater multi-segmented shell system is defined by the surface fundamental form and variational formulation. The discontinuity displacements and slopes at the shell junctions are constrained by using Lagrange multipliers. The nonlinear results in terms of the shell displacements are numerically solved by a nonlinear finite element method (FEM) with an iterative procedure. Second, a three-dimensional buckling analysis of the toroidal–spherical shell is performed using commercial FE software. The numerical results of the stability analysis showed that the minor-to-major radii of the toroid impact the lowest buckling pressure and corresponding buckling shape of the toroidal–spherical shell. Therefore, toroidal–spherical shell buckling behavior must be considered for the design of underwater shell structures. Finally, the results indicate that the relationship between buckling pressures and the number of circumferential waves in toroidal–spherical shells does not follow a consistent pattern and depends on the shell’s geometry.