Abstract In this paper, double-hinge modes are introduced to analyse the response of a stepped or bent cantilever beam subjected to step loading at its free end. The material of the beams is assumed to be rigid-perfectly plastic, and complete solutions are constructed for four cases, which are summarized in tables. An analogy is found between the responses of stepped and bent cantilevers. For the bent cantilevers, the results herein provide a replacement for the ‘unstable hinge’ solutions previously derived, which violate the yield criterion. The results obtained are also useful for estimating the effects of imperfections in beams during dynamic loading.