Galileo's concern was with the breaking strength of a cantilever beam. The behaviour of such a structure is determined by the equations of statics and by the strength of the material; there is only one internal force system in equilibrium with the applied loads and, for the bending problem, collapse will occur when the value of the largest internal bending moment reaches the moment of resistance of the cross-section. Thus the problem of finding the actual state of a statically determinate structure and the problem of calculating its strength are, effectively, one and the same.

This is, of course, not so far the hyperstatic structure. Historically, three types of hyperstatic structure were examined (and the theories have been described in previous chapters) – the (masonry) arch, the continuous beam and the trussed framework. It is of interest that the early (eighteenth-century) work on arches did not concentrate on the ‘actual’ state – rather, limiting states were examined in order to determine the value of one of the main structural parameters, the abutment thrust. This approach continued through the nineteenth century until Castigliano applied his elastic energy theorems to both iron and masonry arches in order to calculate the same structural parameter. Thereafter, arch analysis was seen to fall within the mainstream techniques for the elastic design of hyperstatic structures.

Similarly, specialized elastic techniques were developed for redundant beam systems. Statics alone did not give enough information; the second-order differential equation of bending introduced the elastic properties of the sections; and the boundary conditions (clamped ends, rigid supports) provided the geometrical information leading finally to sufficient equations to solve the problem.