Catastrophe Analysis of Structures by Discretization and Modal Transforms

Abstract

This paper is concerned with the catastrophe of static instability of multi-degree-of-freedom systems representing typical civil engineering structures like beams, columns, arches, plates and stiffened plates. The proposed method makes use of discretization methods, such as a finite element method and a simplified element method, and also some diffeomorphic transformations similar to modal analysis. The main interest of the study is the classification of the catastrophe of the structures through the evaluation of certain derivatives of potential in the light of the Thorn and Thompsons' theories. Numerical illustrations were performed on the structures, including elastically supported columns, plates, stiffened plates, lateral buckling of beams and shallow arches. Among new findings, the unstable symmetric buckling for lateral buckling, and hyperbolic umbilics for the simultaneous buckling of stiffened plates and of elastically supported columns are particularly noted.