It has been noted that James Bernoulli (1694,1695) discussed the problem of finding the moment of resistance of a cross-section in bending. This same paper makes a fundamental contribution to the problem of the elastic flexure of a member. Bernoulli remarks that Galileo had contended (wrongly) that the deflected form of the cantilever was a parabola. Saint-Venant, in his annotated edition of Navier's Leçons, repeats this attribution to Galileo, but in fact there is no such contention to be found in the Dialogues of 1638. The first discussion of an elastic deflected form seems to be that of Pardies (1673), and he indeed asserts that the parabolic form is correct.

Pardies starts his book on Statics with clear and accurate statements of basic laws – the law of the lever, for example (in which his presentation follows exactly that of Galileo, fig. 1.2), moments of forces, the laws of pulleys, the forces in windlasses, gear trains and so on. He then moves on to discuss the question of the shape of a hanging uniform cord, and he establishes the powerful ‘Pardies’ theorem', namely that the tangents at any two points on the cord intersect at a point directly below the centre of gravity of the portion of the cord between the two points. He states that the shape of the hanging cord is not a parabola, and settles finally for the hyperbola (he had, of course no knowledge of the calculus. Leibniz (1691) published the solution of the catenary).

The business of the structural engineer is to make a design to meet some specified brief – for example, a steel-framed factory to house a manufacturing activity; a bridge to span a wide estuary; a gantry to carry an overhead cable for an electric train. Design criteria must first be identified – heavy crane loads may be critical for the factory, wind-induced vibrations for a suspension bridge, accurate location of the cable for the train. To satisfy these criteria the engineer makes calculations, and it has proved convenient, during the last century and a half, whether explicitly recognized or not, to divide the engineer's activity into two parts.

In the first stage, The Theory of Structures is used in order to determine the way in which a structure actually carries its loads. There are many alternative load paths for a (hyperstatic) structure; one of these will be chosen by the structure, and must be discovered by the engineer. This formulation of the problem seems to imply something more than a dispassionate search for truth; the structure seems somehow to have anthropomorphic qualities, and indeed nineteenth (and twentieth) century notions such as those of ‘least work’ may colour the engineer's judgement. For example, in forming such ideas the designer may assume unthinkingly that of course there is an actual state of the structure in which it will be comfortable. It is, however, a matter of fact that the structural equations are extremely sensitive to very small variations in the information used in any structural analysis, and that the structural action, the state in which a structure finds itself to be comfortable, can show enormous variation caused by trivial imperfections in manufacture or construction.

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.

Kazinczy

Kazinczy (1914) tested two steel beams, each about 6 m long, which were embedded at their ends in substantial abutments; the loading, which consisted of increasing numbers of courses of bricks, was uniformly distributed. The steel beams were in fact encased in concrete, but Kazinczy easily dissects out the conclusions that apply to the steel alone. If the ends of the beam in fig. 9.1 (a) were perfectly fixed, then conventional elastic theory gives the bending-moment diagram sketched in fig. 9.1(b); the beam must be designed for a maximum bending moment of value wl2/12. The explicit question asked by Kazinczy, to which the experiments were designed to provide the answer, was whether the end embedment may be taken to be complete and, if not, what degree of fixity may be assumed.

The concrete provided an effective tell-tale to monitor the progress of the experiments. As the loading was increased, cracks in the casing first appeared at the ends of the beams, indicating yield at those points. However, the beams could carry further load, and it was not until a substantially greater weight had been added that deflexions became very large. Upon unloading, each beam was found to have permanent kinking deformation, at the two ends and at the centre. Kazinczy called these kinks ‘hinges’, and he states that a fixed-ended beam cannot collapse (undergo increasing deflexions) until three hinges have formed. Two (end) hinges merely transform the fixed-ended into an effectively pin-ended beam; the third central hinge is necessary for collapse. Moreover, says Kazinczy, the degree of end clamping is irrelevant, provided the embedment is strong enough to allow the hinges to develop.

Stresses in ancient and medieval structures are low. The stone in a Greek temple, in a Gothic cathedral, or in the arch ring of a masonry bridge, is working at a level one or two orders of magnitude below its crushing strength. This is a necessary condition for survival through the centuries; it is not sufficient. It is necessary also that the shape of the structure should be correct, so that structural forces may somehow be accommodated satisfactorily; this is a question of correct geometry. Thus for such structures the calculation of stress is of secondary interest; it is the shape of the structure that governs its stability. All surviving ancient and medieval writings on buildings are concerned precisely with geometrical rules. The architects had, no doubt, an intuitive understanding of forces and resulting stresses, but this understanding was not articulated in a form that would be of use in design; there is no trace in the records, over the two or three millennia for which they exist, of any ideas of this sort.

Instead, the design process would have proceeded by trial and error, by recording past experience, by venturing, more or less timidly, into the unknown, and by the use of models. A large-scale model served several functions – to demonstrate the design to the commissioner, for example, and to solve constructional problems; above all, if the model were stable, so would be the full-scale building, since the model proved that the geometry was correct. All of this experience was recorded, and refined into rules of construction.