Elastic post-buckling and imperfection sensitivity of symmetric structural systems.

This thesis is concerned with multiple equilibrium path generation and imperfection sensitivity of elastic conservative systems, the behaviour of which may be described using a finite number of discrete displacement coordinates and which, in the ideal case, exhibit compound or m-fold branching points. A review of previous work has been made, including that on specific systems, distinct critical points, compound critical points, optimisation and stochastic approaches. An appreciation of the role of Catastrophe Theory in problems of elastic stability has been included. The literature relevant to two particular mechanical systems, examined in detail later in the Thesis, is discussed. Simple criteria, which govern the existence of post-buckling equilibrium paths passing through a symmetric type of m-fold branching point, have been developed. Explicit expressions for the initial curvatures and initial post-buckling stiffnesses of these paths, where they exist, have been found. Quantitative means for assessing the statical stability of paths have been devised. The equilibria and critical equilibria of ideal and imperfect two-fold doubly-symmetric branching points have been examined in great detail. A classification of this type of system has been suggested, which includes a distinction between non-hysteresis and hysteresis behavior. General algebraic forms for the critical load-displacement and imperfection sensitivity surfaces have been derived and their qualitative appearances suggested. A non-approximate numerical technique for calculating the equilibria, deriving the complete critical load-displacement and imperfection sensitivity surfaces of two-fold doubly-symmetric branching systems has been applied to two particular mechanical systems, namely: an axially loaded pin-ended strut on a linear Winkler-type elastic foundation; and a thin simply-supported rectangular plate subjected to proportional bi-axial in-plane compression. The strut problem was modelled by means of a two-mode Rayleigh-Ritz procedure applied to a suitable higher-order total potential energy expression. The plate problem was modelled using a two-mode Galerkin integral technique applied to the von Karman large deflection equations. The role played by the two major imperfections and the imperfection sensitivity surface in dynamic mode-jumping and buckling load degradation respectively was elucidated. The general conclusions reached in this Thesis are broadly supported by other analytical, numerical and experimental work from the literature