Studies on strength and stability of toroidal shell forms for containment applications

Shells find applications in many engineering disciplines (Zingoni 1997), with containment shells of revolution being among the most important (Zingoni 2001, 2015). Searching for the most efficient geometrical forms remains one of the most important goals of shell research. Associated with that is the need to develop appropriate analytical tools for novel shell forms. Complete toroidal shells are not widely used shells of revolution owing to their geometries and associated complexities in the theory of the shells. They can offer certain structural and functional advantages over conventional shells and are mainly used for fluid containment. Toroids have also been proposed for nuclear fusion reactors, rocket fuel tanks, medical hyperbaric treatment units and applications in aerospace and underwater fields. Any desired cross-sectional forms of the shells can be developed theoretically, suggesting the possibility of adopting toroidal shells in many engineering applications if the behaviour of the shells can be understood and quantified. The design analysis of thin-walled structures is mostly based on strength and stiffness considerations. Based on linear elastic theory, Zingoni, Enoma & Govender (2015) presented an elegant theoretical solution for the non-shallow bending of an elliptic toroid, while Enoma and Zingoni (2017) have investigated toroidal shells having the same type of multi-shell cross-section as was first proposed by Zingoni (2001) for novel sludge digester shells. On the basis of classical elastic shell theory and numerical modelling, this thesis attempts to provide a framework upon which complete toroidal shells of any cross-sectional profile can be analysed, and investigates the state of stress and buckling of selected toroidal shell forms including unconventional ones under axisymmetric pressure loading, i.e. when each of the shells is used as a pressure vessel and a storage tank. Following the general strategy developed for shells of revolution by Zingoni (1997) over the past 20 years, reasonably accurate results for shell stresses are derived by combining the membrane solution with an approximate solution of the bending-theory equations for toroidal shells, instead of attempting to solve the exact differential equations, which is extremely difficult. The developed formulations are applied to various cross-sectional types of toroidal vessels under both uniform pressure and hydrostatic pressure loading, and the accuracy of the formulation is verified in each of the cases through numerical examples with finite-element analysis. For the buckling considerations, governing stability equations of toroidal shells of any cross section are presented. These are specialised for the problem of a multi-shell toroid under uniform external pressure, and approximately solved to obtain the critical buckling solution for this geometry. The proposed solution approach provides accurate failure loads of pressurised multi-shell toroids when compared with those from a finite-element analysis. Finite element modelling is then used to study the nonlinear effects on the buckling response, post-bifurcation behaviour and geometric imperfection sensitivity of this type of vessel, as well as two other cross-sectional geometries (parabolic-ogival and circular-elliptic). Extensive parametric studies on each of these toroids reveal significant aspects of shell behaviour. This thesis represents a significant extension of the work of the group of Prof. Zingoni at the University of Cape Town, and provides much-needed information on the design of new forms of toroidal vessels. The simplified theory developed for the determination of stresses and buckling behaviour has facilitated the investigation of the effects of the various geometric parameters, which in turn has led to new insights on the behaviour of the toroidal shell. It has been found that perfect toroidal vessels under external pressure loading can generally have stable post-buckling behaviour and may, therefore, be able to resist further load beyond the elastic bifurcation loads. The imperfection sensitivity of each of the toroids investigated is seen to vary from shell to shell