Theorems of Limit Analysis

Theorems of Limit Analysis

In Previous Section , it was relatively easy to do a complete study of the plastic behav-ior of a thick ring because symmetry reduces the equilibrium requirement to one separable, first-order differential equation. However, most structures do not have symmetry, simple loadings, or ideal supports. In these situations even with mod-ern computational power, a full range elastic-plastic analysis can be nearly impossible or of dubious value.

Fortunately, there is an alternative approach where we forget about obtaining information between first yield and collapse and concentrate on finding the col-lapse state directly. This strategy, called limit analysis, is reasonable if, as is usu-ally the case, we are primarily interested in the strength of a structure. We may lose all information about deformations or even the stress distribution in the plastic range but, after all, most structures are, or should be, designed to operate under normal working loads in the elastic range well below yield. All we really need is a good estimate of the “collapse mechanism” and the load at which it occurs to allow design for safety under extreme loading (earthquake, hurricane, collision, etc). For these rare events, we usually want to also control the failure mode to prevent sudden collapse and to localize residual plastic deformations, whatever they may be, so that the structure can be repaired.

Application of the so-called “bound theorems” and in particular the “upper-bound theorem” of limit analysis provide a way to do this with a straightforward strategy. Limit analysis from either a lower-bound or upper-bound perspective is an intuitive and interactive process. We shall find that in most cases we are free to specify certain variables arbitrarily, make edu-cated guesses as to failure modes or stress distribution, and use trial-and-error methods to converge on good answers. It is this freedom and the rela-tively simple calculations involved that makes limit analysis so valuable. In fact, graphical analysis is very often the best method giving the simplest approach and greatest insight to the interplay of variables. In an important sense, limit analysis is more of an art than elastic analysis. Limit analysis requires, and in turn develops, a physical feel for structural behavior that may be even more important for a creative engineer than the simplicity of the method itself.