The Effect of Shear on the Plastic Bending of Beams

Abstract

Abstract The end-loaded cantilever beam of perfectly plastic material has been studied in considerable detail but many questions remain unanswered. As a first step in extension to plates, the concept is explored of an interaction curve relating limiting values of shearing force and bending moment for perfectly plastic beams. Simple illustrations demonstrate that, far more than in the elastic range, such interaction is not just a local matter but depends upon the geometry and loading of the entire beam. Useful interaction curves are obtained, nevertheless, with the aid of the upper and lower bound techniques of limit analysis, choosing the maximum-shearing-stress criterion of yielding for convenience. It is shown, in particular, that although a small amount of shear produces but a second-order reduction in the limit moment of beams, a small moment reduces the limiting shear value by a first-order term.