Navier (1826) identified the three, and only three, groups of equations that can be formulated to analyse a structure. Foremost are the equations of equilibrium, which relate the internal forces to the given externally applied loads. If these equations alone determine the internal forces, then the structure is, by definition, statically determinate.

In general, a structure is hyperstatic, and the other two sets of equations must be used in order to solve the prime structural problem, that of finding the internal forces. Statements must be made about how the internal forces are related to internal deformations – a ‘stress–strain’ relationship must be specified, and, until the advent of plastic methods, this relationship was usually taken to be linear-elastic. Other material properties may also come into play in calculating the internal deformations – for example, strains due to temperature. Finally, the equations of compatibility are used to make geometrical statements; the members are constrained to fit together, internal deformations must be related to external movements of the structure, and the structure as a whole is constrained by its attachment to its environment.

Hambly's paradox

Hambly (1985) posed a pedagogic problem to illustrate the difficulties of design of a hyperstatic structure:

The stool is supposed to be symmetrical, the milkmaid sits at the centre of the seat, and so on. The answer to the question is, of course, 200 N.

The same milkmaid now sits on a square stool with four legs, one at each corner, and again the stool and the loading are symmetrical.