Inelastic Instability and Incremental Theories of Plasticity

A most troublesome paradox has existed for a number of years with respect to buckling in the plastic range. Theoretical considerations and all direct experimental evidence show conclusively that an incremental or flow type of mathematical theory of plasticity is valid. However, the results of plastic buckling tests are well correlated by a simple total or deformation theory 1 4 and bear no resemblance to published predictions of incremental theory. ' The suggestion was made that initial imperfections of shape or loading might well explain this most peculiar result. However, subsequent investigations by several authors seem to have given the impression that excessively large imperfections would be needed and that the answer would be overly sensitive to the magnitude of such imperfections.^' I t is the purpose of this paper to demonstrate that extremely small, and therefore unavoidable, imperfections of shape do account for the paradox in a simple manner. The buckling load is shown to be extremely insensitive to the amount of imperfection. The example chosen is a simplified version of the long rectangular plate hinged along one edge and free on the other under uniform compressive stress at the ends. This is the equivalent of the case of the cruciform column, which has been so disturbing in the past because incremental theory applied to a perfect cruciform column did lead to an entirely incorrect result.