Dynamic buckling of simple two-bar frames using catastrophe theory

Non-linear static and dynamic elastic buckling of simple imperfect two-bar frames, treated as continuous systems, are analyzed with the aid of catastrophe theory using a comprehensive and readily employed procedure. Static catastrophes are extended to the corresponding dynamic catastrophes of undamped frames under step loading (autonomous systems) by properly determining the dynamic singularity and bifurcational sets. Attention is focused on fold and cusp catastrophes. A local analysis based on Taylor's expansion of the non-linear equilibrium equation of the frame allows us: (a) to classify the total potential energy function of the frames to the canonical form of the corresponding universal unfolding of the seven elementary Thom's catastrophes, and (b) to easily obtain static and dynamic buckling loads, critical points (singularity sets) and related imperfection sensitivities (bifurcational sets). An illustrative example associated with a static and dynamic fold catastrophe demonstrates the efficiency and reliability of the methodology proposed herein.