6 - Elements of limit analysis

Summary

The object of the chapter In this chapter we are interested in the ruin of perfectly-plastic–elastic structures and we introduce the notions of limit load and of maximum admissible load, the determination of which constitutes the essential object of every engineer's office computations. We shall only attempt an introduction to this type of calculation, which will be illustrated by two examples. Certain minimum principles apply to velocities and to stresses. The static and dynamic methods in the determination of the maximum admissible load are only given in a rough draft.

The notion of limit load

The object of our attention is the notion of limit load and the collapse of perfectly plastic structures under unrestrained plastic strains. What do we mean by that? As certain deformable structures evolve, we observe that the elastoplastic response is produced in three stages. The first phase is elastic, the material being elastic everywhere. This phase lasts until the appearance of the first yielding. But the fact that the criterion of plasticity f(σ) = 0 is reached at one point does not necessarily mean that there is collapse. If the strain rate is still controlled, the plastic strain rate is not unlimited, since it is expressed in terms of έ (Section 4.2). We say then that the plastic strain is still controlled. The second phase of the response corresponds to the appearance and the extension of one or more regions of the structure, generally called plastic zones; in these, the plasticity criterion is satisfied at all the points.