Optimal Shakedown Design of Circular Plates

Abstract

The optimal design of circular plates of elastic‐perfectly plastic material and subjected to variable repeated loads is studied according to the shakedown criterion. Two different types of design problem are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the maximum shakedown limit load design whose volume is assigned. In both cases the design problem is formulated by means of a statical approach on the grounds of the shakedown lower‐bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper‐bound theorem. The Euler‐Lagrange equations of these problems are found by a variational approach. The equivalence of the two types of design problem is proved and the design optimality condition is shown to constitute an extension to the shakedown context of the well‐known Drucker‐Prager‐Shield‐Rozvany theorem of optimal plastic design; namely, a modified unit cost is envisaged, the sum of the plate unit cost with some energy density, whose gradient with respect to the thickness equals, at the optimum, the analogous gradient of the plate plastic dissipation density. A few numerical applications are presented.