Optimal stress fields and load capacity ratios of perfectly plastic bodies and structures

A mathematical setting that applies to the analysis of strength of perfectly plastic bodies and structures is presented. Specifically, the worst case analysis under the loads in any vector subspace of the space of all loadings is performed. It is shown that there is a number C, depending only on the geometry of the body, such that the body will not collapse plastically under any applied boundary load t, as long as ∥t∥≤syC independently of the distribution of t, where sY is the yield stress of the material. We give examples for computations of C, the load capacity ratio of the body, and examples of computations of worst case loadings. The analysis also shows that perfectly plastic materials are optimal in the following sense. Without specifying a constitutive relation, let Σ_f denote the collection of all stress fields σ that are in equilibrium with a given loading f and consider s_f^opt=inf _σ∈Σf{sup_x|σ(x)|} - the optimal maximal stress. Then, for perfectly plastic materials at the limit state, the optimum is attained where s_f^opt=sY. The abstract mathematical setting may be described as follows. For a norm preserving linear mapping ε:W→S, we consider the optimal solution of the under-determined equation f=ε*(σ), i.e., we look for s _f^opt=inf{∥σ∥|σ∈ε *¹{f}}. Next, for a mapping β*:M*→W*, an expression is obtained for the worst case factor K=sup S _β*(t)^opt/∥t∥ t∈M*.