A central notion in the concept of virtual work (appendix 1) is that both the external force and displacement quantities and the internal tension and elongation quantities are related to each other in the sense that the product of corresponding variables represents a quantity of work. If a single force P acts at a joint, the ‘corresponding’ measure of displacement of the joint is the component of the displacement in the (positive) direction of the line of action of the force. More generally, if the components of a force are specified, say U, V, W, in mutually perpendicular directions, the ‘corresponding’ displacements are the components of displacement u, v, w in the same directions; and the appropriate (scalar) work product is simply Uu + Vv + Ww.

We are not, however, limited to discussion of loads on structures in terms of force as such. A structure may be loaded by a couple, for which the corresponding displacement is an angle of rotation (measured in radians); or a pressure, for which the corresponding displacement is a ‘swept volume’; or a uniform line load, for which the corresponding displacement is a ‘swept area’.

In relation to internal variables we saw in appendix 1 that we must multiply the tension in a bar by the elongation in order to obtain the appropriate work quantity. For a uniform bar of length L and cross-sectional area A, precisely the same quantity would be obtained by evaluating σ∈V, where σ = T/A is the tensile stress, ∈ = e/L is the tensile strain and V = AL is the volume of the bar.