Dynamic buckling of a simple geometrically imperfect frame using Catastrophe Theory

This paper deals with nonlinear static and dynamic buckling of a geometrically imperfect two-bar frame due to initially crooked bars, which is subjected to an eccentrically applied load at its joint. The analysis is facilitated by considering the frame (being a continuous system) as one degree-of-freedom (1-DOF) system with generalized coordinate unknown the column axial force and then by employing catastrophe theory. Through a local analysis via Taylor's expansion of the nonlinear equilibrium equation of the frame, one can classify the total potential energy (TPE) function of the frame to the canonical form of the corresponding TPE function of the seven elementary Thom's catastrophes. Using energy criteria static catastrophes are extended to the corresponding dynamic catastrophes of undamped frames under step loading (autonomous systems) by conveniently determining the dynamic singularity and bifurcational sets. Numerical examples associated with static and dynamic fold catastrophes demonstrate the efficiency and reliability of the present approach.