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.

The problem of the breaking strength of a beam continued to be visualized in the form stated by Galileo, namely that of a cantilever beam encastred at its left-hand end and loaded by a single weight at the free end. From this formulation was abstracted the ‘cleaner’ problem of the calculation of the breaking resistance of the cross-section adjacent to the support, since clearly this was the critical section of the beam.

In calculating the moment of resistance of the beam, Galileo considered only one of the three statical equations (or four, since Persy's contribution of 1834 must be included), namely that the moment of the forces acting at the cross-section must equal the moment of the applied load. He did not write the equation of longitudinal equilibrium (Parent (1713)) which helps to determine the location of the neutral axis of bending, nor did he resolve forces vertically, which leads to the idea of a shearing action on the critical section.

As has been seen, Coulomb (1773) did realise that the forces acting on the critical section must have vertical components in order to balance the load applied to the tip of the cantilever. Indeed two of Coulomb's four problems (the strength of columns, the thrust of soil) are concerned with shear fractures, and he tried to test his (stone) cantilever beam in pure shear by applying the load as close as he could to the encastred end. The experimental technique was not good, but Coulomb measured to his own reasonable satisfaction the strength of stone in pure tension and in pure shear, and related these two strengths by ‘Coulomb's equation’, involving two physical parameters, cohesion and friction.

As was mentioned in Chapter 2, Coulomb's memoir of 1773 made contributions to each of the four major problems of civil engineering in the eighteenth century – the strength of beams, the strength of columns, the thrust of soil and the thrust of arches. In all four of these topics Coulomb made advances by considering closely the basic equations of equilibrium, both for the structure overall and at imaginary internal cuts. The work on the fracture of beams has been summarized in Chapter 2.

For the next two problems, columns and soil, Coulomb studies failure planes along which slip is occurring, resisted by the cohesion and friction of the material. That is, just as for the beam problem, solutions are obtained from equilibrium equations combined with a knowledge of material properties. These solutions, and their relation to previous work, are described further in Heyman (1972). By contrast, Coulomb's solutions for arches make only marginal reference to the strength of the material (masonry), and his exploration of the stability of the arch is based solely on considerations of equilibrium, coupled (as is made explicit in the title of his memoir) by principles of maximum and minimum. Indeed the arch seems to have been regarded as a problem separate from other studies in the development of structural mechanics, at least until the end of the nineteenth century.

Navier

In section 4 of Navier's 1826 Leçons he tackles the problem of the redundant truss. The example used is that of a weight II supported from the ground by a number of bars, fig. 7.1, and the problem is to determine the forces in the bars. Navier states that if the number of bars is more than two in the same plane, or more than three not in the same plane, then the equations of equilibrium do not determine the values of the bar forces. Navier shows how the problem may be solved, using the three-bar plane-truss example of fig. 7.1.

There is first a short digression in which Navier attempts to estimate limits within which the bar forces must lie. If, for example, all bars are removed from a plane truss of the type sketched in fig. 7.1, except for the two necessary to carry the load, then the forces in those two bars may be found from the equations of statics. By considering different arrangements of bars to produce such statically determinate trusses, a greatest load may be found for a particular bar; the stability of that bar against buckling may then be checked, using the ‘Euler’ theory of a previous section of the Leçons. (These observations are, of course, incorrect. Even in the absence of the load II, a turnbuckle tightened in bar A'C will produce compression in the two outer bars, and buckling of one or the other will eventually occur. The ability of a redundant truss to sustain self-stress was certainly known to Maxwell (1864).)

Navier then lays out clearly the three groups of equations required for the elastic solution of the problem.

It was clear to Galileo that a beam resting on three supports (which, in modern terminology, would be hyperstatic) could be subjected to forces not envisaged by the engineer. That is, an accidental imperfection (and Galileo used the word accidente), such as decay of one of the end supports, could lead to a set of forces that would break the beam. He was equally clear that no such accident could happen to a beam on two supports; if the supports sink then the beam follows – the statics of a statically determinate beam are unique.

However, it does not seem that Galileo was concerned with any concepts that might stem from the consideration of what is now known as the hyperstatic structure. His objective was, as has been described, to calculate the breaking strength of beams, and for this purpose he determined the greatest value of bending moment in a beam, whether that beam were simply supported or a simple cantilever. The value of bending moment having been found, the problem then became one of the strength of materials, and the historical notes given in Chapter 2 are concerned with the correct way of calculating the moment of resistance of a cross-section.

Girard 1798

It was noted in passing in Chapter 2 that Mariotte made tests on fixed-ended beams, and that he concluded from his experiments that their strengths were twice those of corresponding simply supported beams. Mariotte gave no theoretical explanation for this result, and the problem of the strength of the hyperstatic beam seems to have remained unexplored throughout the eighteenth century.

As has been seen, Galileo's problem was the determination of the ultimate moment of resistance of a member (wooden, stone, metal, glass) in bending. The problem was posed by reference to a cantilever beam, acted upon by a tip load, or its self-weight, or both; the value of the breaking load(s) was sought. Static equilibrium requires that the moment of the applied load(s) at the root of the cantilever must equal the moment of resistance of the cross-section; since the problem is statically determinate, a problem in the theory of structures is transformed into a problem of strength of materials. Galileo, and later scientists, did not of course think in this way; in particular, the notion of hyperstatic structures, for example the beam on three supports or, later, the propped cantilever or the fixed-ended beam, is not made explicit. These last two more complex structures were in fact discussed in 1798 by Girard, and ‘correct’ solutions were found (see Chapter 6); the solutions were, however, specific for the problems, and Girard does not make general statements about statical indeterminacy. Such ideas became formalized a quarter of a century later; the date of 1826, when Navier published his Leçons, is a convenient marker, and indeed it was not until over a century after that date that the straitjacket imposed by Navier on structural design was finally loosened.

Girard starts the introduction to his book (on the strength of materials and on solids of equal resistance) by stating that his subject consists of something more than rigid-body statics.