Elastoplastic Deflections for Prismatic and Nonprismatic Beams

Abstract

Analytical solutions are presented for deflections beyond the elastic limit for prismatic and nonprismatic beams (beams with linearly varying depth) restrained and loaded in any way. The analytical solutions provide by differentiation the rotation, curvature, moment, and the shearing force. Simply supported and built‐in beams with uniform cross section and with linear variations in depth symmetrically with respect to midspan, loaded at midspan, are analyzed; and the results are given in a dimensionless form. The displacement occurring just prior to the collapse mechanism is 2.243 of that occurring at the limit elastic behavior for simple and built‐in uniform beams. For ratios between the minimum and the maximum depths of the beam less than 0.41, the collapse mechanism occurs with plastic hinges located at midspan and along the beam. Five elastoplastic behaviors for built‐in beams with linearly varying depth are established, and the limit ratios between the maximum and the minimum depths of the beam separating these behaviors are determined.