Upper bounds, lower bounds and ‘exact' yield-line design: structural concrete in plane stress

Plane-stress plasticity, at its simplest, assumes rigid-plastic materials and inherits the bound theorems of classic rigid-plastic theory. There are, again, upper-bounds, lower-bounds and ‘exact' solutions: (a) an upper bound must display a detailed kinematically correct collapse mechanism while (b) a lower-bound must satisfy equilibrium everywhere and (c) an ‘exact' solution is unique, certainly as to collapse load and often, but not always, as to the collapse mechanism. Exact solutions tend to flow from upper bounds because of the mechanism condition. Solely lower-bound solutions are necessarily arbitrary. As in other areas of structural plasticity, there is a tendency to focus on lower-bound solutions thereby avoiding the rigour required in formulating a kinematically correct collapse mechanism. In the case of plane stress, this has resulted in disproportionate attention being paid to the strut-and-tie method. This paper sets out to demonstrate that upper-bound and ‘exact' solutions are not largely unknown or unknowable and seeks to restore balance between upper-, lower-bound and ‘exact' solutions. This author is working on other papers to exhibit novel and little-known ‘exact' solutions for plane-stress problems. One purpose of this paper is to extract and summarise the general discussion and theory so that those other papers can focus on problem solving.