Catastrophes of Elastic Column Structures

Abstract

The elasto-static instability phenomena of structures can be typically classified by the catastrophe theory in the local form near a certain singular point of their total potential energy. Then, the imperfection sensitivity of the structures can be evaluated in terms of the bifurcation set, mapping the singular points of the equilibrium space to the control space. In ordinary structural problems, this consists of a simple loading parameter and some imperfection parameters. In this paper, several simplified column models are studied by the catastrophe theory. They include both continuous and discrete models exhibiting stable symmetric, unstable symmetric, asymmetric and their compound buckling. From a comparison of the results of the two-degree-of-freedom systems, namely the continuous systems and the finite element discrete systems, the following conclusions are drawn : (i) The discrete analysis can be shown to realize the instability phenomena, predicted by the continuous analysis. The results of the discrete analysis are shown to converge to those of the continuous analysis generally, as the number of discrete finite elements increases. (ii) The imperfection sensitivity of the structures can be evaluated qualitatively and quantitatively by means of the bifurcation set in the control space through the catastrophe theory. (iii) For a legitimate evaluation of the “cusp and dual cusp catastrophe”, the 4th order non-linear terms must be earnestly considered in expressions for both the strain energy and the external work. Then, the stable symmetric buckling model is shown to indicate a typical cusp catastrophe. (iv) The unstable symmetric buckling model can be shown to indicate a typical dual cusp catastrophe for a relatively small stiffness of the elastic foundation, the cusp catastrophe for a relatively large stiffness and the compound double cusp catastrophe at a certain critical stiffness value. (v) The asymmetric buckling model can be shown to indicate the typical fold catastrophe for a relatively small stiffness of the inclined elasic foundation, the dual cusp catastrophe for a relatively large stiffness and the compound hyperbolic umbilic catastrophe at a certain critical stiffness value.