Uniqueness criterion for incremental variation of steady state and symmetry limit

Abstract

A sufficient condition for uniqueness of the incremental variation of steady state is established for a cantilever beam-column subjected to completely reversed tip-deflection cycling. The critical steady state, at which the sufficient condition for uniqueness is first broken, is found on the sequence of symmetric steady states generated under a continuously increasing tip-deflection amplitude. In a previous paper ( Uetani K. and Nakamura T., J. Mech. Phys. Solids 31,449, 1983), a symmetry limit has been defined as the critical steady state, at which transition from a symmetric steady state to an asymmetric steady state occurs first, and its theoretical solution has been obtained. The uniqueness limit turns out to coincide with this symmetry limit. Though the previous theory for symmetry limits was constructed on the basis of five hypotheses introduced without proof, the present theory involves no such hypothesis.