429. The same principles that we have employed in the construction of the bridge of [Fig. 58] may be extended further, as shown in the diagram of [Fig. 59].
Fig. 59.
We have here two horizontal rods, 48" × 0"·5 × 0"·5, each end being secured to the supports; one of these rods is shown in the figure. It is divided into five equal parts in the points b, c, c´, b´. We support the rod in these four points by struts, the other extremities of which are fastened to the framework. The points b, c, c´, b´ are fixed, as they are sustained by the struts: hence a weight suspended from p, which is to break the bridge, must be sufficiently strong to break a piece c c´, which is secured at the ends; the rod a a´ would have been broken with 38 lbs., hence 190 lbs. would be necessary to break c c´. There is a similar beam on the other side of the bridge, and therefore to break the bridge 380 lbs. would be necessary, but this force must be applied exactly at the centre of c c´; and if the weights be spread over any considerable length, a heavier load will be necessary. In fact, if I were to distribute the weight uniformly over the distance c c´, it appears from [Art. 408] that double the load would be necessary to produce fracture.
430. We shall now break this model. I place 18 stone upon it ranged uniformly, and the cathetometer tells me that the bridge only deflects 0"·1, and that its elasticity is not injured. Placing the tray in position, and loading the bridge by this means, I find with a weight of 2 cwt. that there is a deflection of 0"·15; with 4 cwt. the deflection amounts to 0'·72. We therefore infer that the bridge is beginning to yield, and the clamps give way when the load is increased to 500 lbs.
A BRIDGE WITH TWO TIES.
431. It might happen that circumstances would not make it convenient to obtain points of support below the bridge on which to erect the struts. In such a case, if suitable positions for ties can be obtained, a bridge of the form represented in [Fig. 60] may be used.
a d is a horizontal rod of pine 40" × 0"·5 × 0"·5; it is trisected in the points b and c, from which points the ties b e and c f are secured to the upper parts of the framework. a d is then supported in the points b and c, which may therefore be regarded as fixed points. Hence, for the reasons we have already explained, the strength of the bridge should be increased nearly threefold. Remembering that the bridge has two beams we know it would require about 70 lbs. or 80 lbs. to produce fracture without the ties, and therefore we might expect that over 200 lbs. would be necessary when the beams were supported by the ties. I perform the experiment, and you see the bridge yields when the load reaches 194 lbs.: this is somewhat less than the amount we had calculated; the reason being, I think, that one of the clamps slipped before fracture.
Fig. 60.