A general approximate technique for the finite element shakedown and limit analysis of axisymmetrical shells. Part 1: Theory and fundamental relations

This paper describes the theory and the fundamental relations for the development of a displacement formulation for the Þnite element shakedown and limit analysis of axi-symmetrical shells. The material is assumed to be elasticÐperfectly plastic. The technique is developed based upon an upper bound approach using a reformulated kinematic shakedown theorem for a shell with piecewise linear yield conditions. The solution of the problem is obtained by discretizing the shell into Þnite elements. A consistent relationship between the kinematically admissible velocity Þelds and the pure plastic strain rate Þelds during collapse needs to be enforced. Such requirement is satisÞed by using the theory of conjugate approximations to minimize the residual of the two independent descriptions of the plastic strain increments. The discretized problem is then reduced to a minimization problem and solved by linear programming. The class of displacement Þelds chosen assumes plastic hinge lines forming at nodal points and only meridional and circumferential plastic strains occurring within the elements with no change in curvature. Examples of the application of the method are given in the accompanying paper