This appendix gives a brief sketch of various theorems in structural mechanics which are used in several parts of the book. These theorems apply to small deflections of elastic structures in the absence of buckling or other ‘geometrychange’ effects. In this appendix, for the sake of brevity and simplicity, they are described with reference to a simple plane pin-jointed truss which can be discussed in terms of a few, discrete, variables; but they can readily be translated into more general forms relevant to continuous structures (see appendix 2 on the idea of corresponding forces and displacements, etc.). The following description is restricted to frameworks whose members are made from weightless linear-elastic material, and which are stress-free in the initial configuration; but there is no difficulty in extending the scope of the theorems to include nonlinear elasticity and problems involving initial stress.

The description begins with the principle of virtual work, which makes a connection between the two distinct sets of conditions describing statical equilibrium and geometric compatibility of the various parts of the structure, respectively. This principle holds irrespective of the mechanical properties (or ‘constitutive law’) of the material from which the structure is made; and all of the various elastic theorems are derived directly from it by incorporation of the elastic material properties. (The theorems of the plastic theory of structures, which are used in chapter 18, are also derived directly from the principle of virtual work; but they are not proved here (see Calladine, 1969a).)