Non-linear buckling of simple models with tilted cusp catastrophe

Non-linear buckling universal solutions of simple multiple-parameter discrete models are discussed via a comprehensive and readily employed procedure using Catastrophe Theory. Attention is focused on perfect models whose total potential energy (TPE) function, upon small disturbance breaking symmetry, reduces to the universal unfolding of the tilted cusp catastrophe. A local analysis based on simple approximations allows us to classify the TPE universal unfolding of any model to one of the seven elementary Thom's catastrophes by defining the corresponding control parameters. Subsequently, using global analyses one can obtain exact results for establishing the non-linear equilibrium paths: (a) of the “perfect” perturbed model (due to the small effect of an extra parameter) with the corresponding “imperfect” bifurcation and limit points(s), and (b) of the imperfect models (resulting after inclusion of the effect of normal imperfection parameters) together with the corresponding to each parameter limit points. Moreover, conditions for the direct evaluation of (non-degenerate) hysteresis points associated with a tilted cusp point in the control parameter plane, are established. Numerical results illustrate the methodology proposed herein.