Nonlinear axisymmetric elastic analysis of deep imperfect spherical shells

The particular problem that is addressed in this thesis is the nonlinear elastic analysis of incomplete thin deep spherical shells, where the displacements on the boundary surface are fully restrained and the shell is loaded by a uniform pressure load. The shell may contain initial axisymmetric geometric or stress imperfections, and the initial response of the shell, the fundamental path, is axisymmetric. Solution techniques are developed and presented for the nonlinear fundamental path, and the nonlinear eigenvalue problem that yields the location of the points of the axisymmetric or periodic secondary paths that bifurcate from the axisymmetric fundamental path and the initial mode shapes of the secondary paths at the point of bifurcation of the secondary paths from the nonlinear axisymmetric fundamental path. Using the nonlinear elastic deep shell theory developed in this thesis, the behaviour of perfect and imperfect incomplete spherical shells with fully restrained boundary surface displacements under uniform pressure loading is examined. This thesis also attempts to derive and present the equations and solution methods used in the nonlinear elastic analysis of thin spherical caps in as general a way as possible, in order that the solution methods developed in the thesis may be applied to shell structures composed of one or more incomplete thin elastic shells of arbitrary shape under arbitrary loading. By extending the derivation of the strain-displacement expressions for shells to include the terms that are quartic in displacements, the limitations present in linear thin shell theory may be lifted, and partial differential equations governing the nonlinear elastic behaviour of thin shells of arbitrary shape may be developed. The methods used to develop the total potential energy functional for pressure loaded spherical shells may be extended and used to express the total potential energy of shell structures composed of one or more incomplete shells of arbitrary shape under arbitrary loading.