This paper examines the problem of assigning economical sections to frames which derive their strength from a bending action. The theory of plastic collapse is used to determine the strength of a design, and a virtual work approach is used to derive the equations governing the problem. These equations show that a linear function whose variables are constrained by linear inequalities has to be minimized. In order to illustrate the meaning of these equations a geometrical analogue is described and this analogue is then used to prove a few general theorems. The paper concludes with a demonstration of the way in which these theorems can be applied to the design of a simple practical structure.