Load capacity of perfectly plastic bodies and structures

For a statically indeterminate structure we examine the class of internal forces that are in equilibrium with a given external loading/. We define the optimal stress φ ^opt as the smallest possible magnitude of any equilibrating internal force distribution. The stress sensitivity k = max _f{φ _f ^opt/||f||}, a purely geometric property of structure, is a measure of the sensitivity of the structure to variable external loading. Using the result for optimal stresses, an expression for the stress sensitivity factor is obtained in terms of the structure's kinematic interpolation mapping. For a given structure made of a perfectly plastic material with a yield stress σ _y, we consider the load capacity ratio of the structure: the largest positive number C, depending only on the geometry of the structure, which satisfies the following property. For any loading distribution / on the structure whose maximum is /max, the structure will not undergo plastic collapse as long as f _max ≤ σ _yC, independently of the distribution of the load. The paper presents the mathematical aspects, related mechanical notions, algorithms and examples corresponding to load capacity ratios of structures. These notions, the corresponding theoretical results, and a simple implementation to finite element models are presented using linear and conic programming.