A CONSISTENT EQUILIBRIUM IN A CROSS-SECTION OF AN ELASTIC-PLASTIC BEAM

A phenomenon of inequality of equilibrium and constitutive internal fo rces in a cross-section of elastic-plastic beams is common to many fin ite element formulations. It is here discussed in a rate-independent, elastic-plastic beam context, and a possible treatment is presented. T he starting point of our discussion is Reissner's finite-strain beam t heory, and its finite element implementation. The questions of the con sistency of interpolations for displacements and rotations, and the re lated locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function O f the problem. Approximate equilibrium equations are derived by the us e of the distribution theory in conjunction with the collocation metho d. The novelty of our formulation is an inclusion of a balance functio n that ''measures'' the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present appro ach is that the locations, where the balance of equilibrium and consti tutive moments should be satisfied, can be prescribed in advance. In o rder to minimize the error, explicit analytical expressions are used f or the constitutive forces; for a rectangular cross-section and biline ar constitutive law, they are given in Appendix A.-The comparison betw een the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordi nate system is transformed such to follow high gradients of curvature. (C) 1998 Elsevier Science Ltd. All rights reserved.