Non-local behavior of plastic softening beams

Abstract

This paper deals with the modelling of a plastic beam experiencing softening. This kind of behaviour is observed in steel or reinforced concrete structural members undergoing large rotation amplitudes, which may occur typically for civil engineering structures in the seismic area. The homogeneous cantilever beam loaded by a concentrated force at its extremity is considered. This simple structural problem with gradient bending moment allows an analytical treatment of the evolution problem. It is shown that a local plastic softening model makes the evolution problem ill-posed. Moreover, if we require the plastic curvature to be a continuous variable of the spatial coordinate, Wood’s paradox is encountered. A non-local gradient plastic model is developed in order to overcome this paradox. However classical gradient plastic models may not eliminate the ill-posedness since the beam response may not be continuous with respect to the loading parameter. The new gradient plastic model, presented in this paper, is similar to previous classical gradient models. The main difference is the yield moment considered as a non-local material parameter. This permits to ensure continuity between the elastic and the plastic regions during the loading process. These solutions are controlled by the ratio between the material length and the geometrical length of the beam. The new evolution problem may remain ill-posed as it possesses a finite number of solutions (which can be unique). Closed-form solutions of the unknown deflection are given.