A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross!section of elastic plastic beams is common to many _nite element formulations[ It is here discussed in a rate!independentelastic plastic beam context\ and a possible treatment is presented[ The starting point of our discussion is Reissner|s _nite!strain beam theory\ and its _nite element implementation[ The questions of the consistency of interpolations for displacements and rotations\ and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem[ Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method[ The novelty of our formulation is an inclusion of a balance function that {{measures|| the error between the equilibrium and constitutive bending moments in a cross!section[ An advantage of the present approach is that the locations\ where the balance of equilibrium and constitutive moments should be satis_ed\ can be prescribed in advance[ In order to minimize the error\ explicit analytical expressions are used for the constitutive forces^ for a rectangular cross!section and bilinear constitutive lawthey are given in Appendix A[ The comparison between the results of the two _nite element formulationsthe 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 plasti_ed region is also discussed and solved by using an adapted collocation method\ in which the coordinate system is transformed such to follow high gradients of curvature[ 0887 Elsevier Science Ltd[ All rights reserved