Obviously, in our quadrupedal bridge, the superstructure does not terminate (as it did in our former diagram) at the two points of support, but it extends beyond them at each end, carrying the head at one end and the tail at the other, upon a pair of projecting arms or “cantilevers” (Fig. [346]).
In a typical cantilever bridge, such as the Forth Bridge (Fig. [345]), a certain simplification is introduced. For each pier carries, in this case, its own double-armed cantilever, linked by a short connecting girder to the next, but so jointed to it that no weight is transmitted from one cantilever to another. The bridge in short is cut into separate sections, practically independent of one another; at the joints a certain amount of bending is not precluded, but shearing strain is evaded; and each pier carries only its own load. By this arrangement the engineer finds that design and construction are alike simplified and facilitated. In the case of the horse, it is obvious that the two piers of the bridge, that is to say the fore-legs and the hind-legs, do not bear (as they do in the Forth Bridge) separate and independent loads, but the whole system forms a continuous structure. In this case, the calculation of the loads will be a little more difficult and the corresponding design of the structure a little more complicated. We shall accordingly simplify our problem very considerably if, to begin with, we look upon the quadrupedal skeleton as constituted of two separate systems, that is to say of two balanced cantilevers, one supported on the fore-legs and the other on the hind; and we may deal afterwards with the fact that these two cantilevers are not independent, but are bound up in one common field of force and plan of construction.
In the horse it is plain that the two cantilever systems into which we may thus analyse the quadrupedal bridge are unequal in magnitude and importance. The fore-part of the animal is much bulkier than its hind quarters, and the fact that the fore-legs carry, as they so evidently do, a greater weight than the hind-legs has long been known and is easily proved; we have only to walk a horse onto a weigh-bridge, weigh first his fore-legs and then his hind-legs, to discover that what we may call his front half weighs {695} a good deal more than what is carried on his hind feet, say about three-fifths of the whole weight of the animal.
The great (or anterior) cantilever then, in the horse, is constituted by the heavy head and still heavier neck on one side of the pier which is represented by the fore-legs, and by the dorsal vertebrae carrying a large part of the weight of the trunk upon the other side; and this weight is so balanced over the fore-legs that the cantilever, while “anchored” to the other parts of the structure, transmits but little of its weight to the hind-legs, and the amount so transmitted will vary with the position of the head and with the position of any artificial load[627]. Under certain conditions, as when the head is thrust well forward, it is evident that the hind-legs will be actually relieved of a portion of the comparatively small load which is their normal share.
Our problem now is to discover, in a rough and approximate way, some of the structural details which the balanced load upon the double cantilever will impress upon the fabric.
Working by the methods of graphic statics, the engineer’s task is, in theory, one of great simplicity. He begins by drawing in outline the structure which he desires to erect; he calculates the stresses and bending-moments necessitated by the dimensions and load on the structure; he draws a new diagram representing these forces, and he designs and builds his fabric on the lines of this statical diagram. He does, in short, precisely what we have seen nature doing in the case of the bone. For if we had begun, as it were, by blocking out the femur roughly, and considering its position and dimensions, its means of support and the load which it has to bear, we could have proceeded at once to draw the system of stress-lines which must occupy the field of force: and to precisely these stress-lines has nature kept in the building of the bone, down to the minute arrangement of its trabeculae.
The essential function of a bridge is to stretch across a certain span, and carry a certain definite load; and this being so, the {696} chief problem in the designing of a bridge is to provide due resistance to the “bending-moments” which result from the load. These bending-moments will vary from point to point along the girder, and taking the simplest case of a uniform load supported at both ends, they will be represented by points on a parabola. If the girder be of uniform depth, that is to say if its two flanges,
Fig. 340. A, Span of proposed bridge. B, Stress diagram, or diagram of bending-moments[628].