Introduction

One of the most powerful aspects of the theory of plasticity lies in its ability to easily predict approximate values for the collapse load in a very wide range of applications. This comes about through two theorems called the upper bound theorem and the lower bound theorem. As their names imply, the theorems provide bounds, or limiting values, for the collapse load. Often any usage of the theorems is referred to as limit analysis.

The business of predicting collapse loads is totally concerned with finding the loads that will bring the structure or body to an imminent state of collapse. We are not concerned with what happens before or after in the sense of trying to analyse elastic strains or plastic flow. Also, we must not confuse the collapse load with the yield load. In some instances they will be the same and yield will immediately lead to collapse, but in other cases yield may happen well before collapse. As an example, yield precedes collapse by a significant margin in the shallow foundation problem where localised yielding may happen immediately near the edges of a rigid footing, well in advance of the collapse load. There are restrictions on the applicability of both theorems. A key factor in the development of limit theorems rests with the normality relationship between the yield surface and its associated plastic strain rate vector. For either rigid–perfectly plastic or elastic–perfectly plastic materials, the limit theorems can be proved rigorously (see Appendix H).