Buckling and post-buckling of an elastica under a lateral restraining force

Abstract

Buckling of structures under lateral load restraint has been found to differ from the conventional bifurcation problems and cannot be handled with conventional analysis. Towards a better understanding of this type of unconventional buckling problems, this work investigates the upheaval buckling of an elastica resting on a smooth surface and restrained by a lateral constant force. Analytical solutions for the Mode-1 and Mode-2 post-buckling responses and configurations of the elastica are established. Stability of the post-buckling configurations under force-control are investigated via the energy method. An energy barrier is found to exist and the buckling load depends on the restraining force and perturbation energy. The imperfection sensitivity of the system is different from the conventional buckling ones, and a characteristic imperfection is found to exist that can transform a snap-through instability into a bifurcation buckling. As a first attempt to address the problem of flange wrinkling under a constant blank-holding force during the cup-drawing process, the fundamental solutions are extended to study the wrinkling of a shrinking elastic annulus under a constant lateral restraining force. The post-wrinkling responses are established and an energy barrier is found to exist for each wrinkling mode. A characteristic imperfection is also found to exist for each mode, which eliminates the energy barrier and enables bifurcation wrinkling. The linear relationship between the critical restraining force that prevents the wrinkling of the elastic annulus and the imperfection amplitude is established, which is also verified by numerical simulations. This study contributes to a better understanding of load-restrained buckling problems and is expected to shed light on the elastoplastic buckling of plates and shells under lateral load restraints in general, and on the wrinkling of a thin sheet during cup-drawing in particular.