Bounding Techniques in Shakedown and Ratchetting

A review of Shakedown and Ratchetting concepts and their extensions is presented in an attempt to recount all the aspects of the problems considered in this research programme. The concept of Stress Concentration Factor was the first to be further investigated, by analysing two representative types of structures operating under severe stress concentration, namely; two-bar structures and cylindrical vessels with variable thickness subjected to cyclic mechanical loads. The material behaviour considered are: elastic-perfectly plastic and isotropic hardening. Such an analytical investigation allowed the assessment of the influence of the Stress Concentration Factor below and above the limit of reversed plasticity. The primary aim of this research was to develop simplified techniques capable of solving thermal loading problems in the presence of steady mechanical loads. A simplified technique was then developed to analyse a tube subjected to a complex thermal loading simulating the fluctuation of level of sodium in Liquid Metal Fast Breeder Reactors (LMFBR). The technique was also able to include a second important aspect of shakedown problems which is cases of multiple mechanical loads. The construction of bi-dimensional Bree type diagrams, from tri-dimensional ones obtained for such cases, allowed an easy assessment of the modes of deformation of the structure. The effects of the temperature on the yield stress were explored. A third aspect of thermal cyclic problems investigated was the experimental verification of the reliability of the extended Upper Bound Theorem proposed in Chapter 2. This was achieved by experimental tests on portal frames at 400°C. Contours representing states of constant of deformation were obtained from the experimental measurements. A fourth aspect of the problem was the development of theoretical technique to estimate the transient plastic deformation in excess of the shakedown limit which allowed the construction of theoretical contours directly comparable with the experimental ones. The fifth and major contribution of this thesis was the development of a general technique for the analysis of axi-symmetric shells based in a displacement formulation for the Finite Element Method. Limit analysis and shakedown problems were reduced to minimization problems by developing a technique to obtain consistent relationship between the displacement field and the plastic strain field. Such a technique, based upon a Galerkin type of approach, consist of minimizing the difference between the two representations of the strain within the element; in terms of nodal displacement and in terms of plastic multipliers. The problem was then solved by Linear Programming. Finally, the conclusions and proposal for future work are presented.