The Theory of Deformation of Non-hardening Rigid-Plastic Plates under Transverse Load

Abstract

Only recently has the study of the bending of thin plates been made on the basis of the incremental theory of plasticity. The first approach leading to the exact solution of problems of this type was made by Hopkins and Prager [1] who discussed the quasi-static yielding of a circular plate subjected to transverse load under conditions of rotational symmetry. The plate material was taken to be non-hardening rigid-plastic and to obey the yield condition and flow rule of Tresga. The basic assumptions made by these authors were similar to those of the conventional engineering elastic theory of thin plates (see, for example, Timoshenko [2]) and the effect of transverse shear strain-rate was neglected. A notable feature was that, within the framework of the theory, exact solutions were obtained to certain important problems such as, for example, those involving either uniformly-distributed or concentrated loads. The solution of similar problems for other yield conditions, in particular that of von Mises, has been studied by Hopkins and Wang [3], and for combined loads has been studied by Drucker and Hopkins [4]. The problem of the limits of economy of material has been studied by Hopkins and Prager [5]. The above studies all refer to quasi-static problems, and the extension of the fundamental analysis to dynamic problems was first made by Hopkins and Prager [6], the effect of rotatory inertia being neglected. Again it was notable that exact solutions could be obtained to certain problems. Thus the complete solution was obtained for a simply-supported circular plate subjected to a uniformly-distributed load which is suddenly applied and, after a certain time, suddenly removed. The solution of certain impulsive load problems for simply-supported and built-in edge conditions has been given by Wang [7] and by Wang and Hopkins [8], respectively. The problems treated theoretically by Hopkins and Prager [1] were first investigated experimentally by Haythornethwmte [9]; for subsequent experimental work see the references cited by Prager [10]. An extended summary of the greater part of the work contained in [1–9] is given by Prager [10]. Finally, Prager [11] has given a method for solving statical problems when the plate material is work-hardening rigid-plastic.