Numerical analysis of complex instability behaviour using incremental-iterative strategies Original Research Article

The paper describes how quasi-static, conservative instability problems can be analysed in a multi-parametric space, using generalised path-following procedures for augmented equilibrium problems. The general formulation of such augmented equilibrium problems is discussed in some detail. The focus is set on two classes of generalised 1D paths: basic equilibrium paths and fold lines, i.e. critical subset paths. The solution methods are seen as extensions to common incremental-iterative strategies, allowing the computation of subsets of equilibrium states which also fulfil some auxiliary conditions, e.g. criticality. In this context, some emphasis is also given to the evaluation of the properties of the problem, at a certain state; the tangential stiffness is here used to evaluate – possibly multidimensional – tangent spaces, and in the isolation of special states, i.e. vanishing variables, turning points and exchanges of stability, being important aspects of instability analyses. A set of carefully chosen numerical examples demonstrate on one hand the ability of the numerical procedures to deal with complex instability phenomena, including coincident or near coincident buckling modes, modal interaction, secondary bifurcations, and, on the other hand, their versatility in performing parameter sensitivity analyses. Finally, comparisons with alternative techniques, based on asymptotic strategies, are also put forth.