1504

THEORY AND PRACTICE OF ENGINEERING.

Book II.

The calculation is almost impracticable for semicircular arches, on account of thevarious quantities which the nature of the circle introduces in this formula, and it becomestotally so, for the arch composed of several arcs; we must therefore have recourse to an in-direct method, which consists in making different hypotheses on the position of the point D,and in determining for each the corresponding value of B K, from examples of known bridges ,whose form approaches that of the one projected; it is evident, moreover, that the greatestvalue which we could find for BKwill be that which we must take into consideration,and that the positions of the joints of rupture will be determined by the correspondentvalue of the arc B.D.

The following table contains the result of this calculation for those vaults most frequentlyused; we have supposed that the opening of these vaults was 20 metres, that the thickness atthe key was 1 metre, that the upper part was extradossed level.

Thickness of theAbutments.

Metres.

Semicircular arches - - - - -0-45

Arches struck from three centres depressed one-third - 0*66depressed one quarter of the span - - 0’82

Arch of a circle of 60 degrees raised upon piers 5 me-tres high - - - - - - 2 95

Position of thePoints of Rupture.Degrees.

27

45

54

0

The calculation has given for the arches of the three centres flattened a quarter a jointof rupture placed differently to that which experiment has indicated for the bridge ofNeuilly ; this joint of rupture will be here found at the 16th and 17th course of vous-soirs, and we have before observed that it was above the 26th course; this difference de-pends on the supposition in the calculation that the masonry of the haunches was raised,which was not the case with the bridge at Neuilly at the moment of striking the centres,and in which the position of the joint of rupture was decided.

The addition of this masonry necessarily changes the position, and the joints of rupture,as shown in the experiments of M. Boistard, are the more elevated, as the summit ofthe arch is more loaded with relation to the haunches. The results of the calculation agreeexactly with the observations made at the bridge of Nogent, where the masonry of thehaunches was constructed when the centres were struck.

The results comprised in the table are below the ordinary dimensions, and the theoryindicates thicknesses less considerable than those which have hitherto been thought neces-sary for the abutments of bridges.

The preceding calculations suppose that the different portions of the arch form solidmasses, the parts of which are perfectly united together, and cannot undergo any sinking ; itsupposes also that the abutments are established on a base entirely incompressible, and thatin the fall of the arch these abutments would turn, without disjointing, round their exterioredge; these suppositions are in general far removed from the truth. The fall of a bridgecould scarcely happen without some disjunction taking place on the abutments, howevergreat the care with which they may have been constructed; and even should there be none,these abutments could not turn round their external edge, where the stones would neces-sarily be crushed under the effort they would have to support, an effort which should be onthat account spread over a sufficiently great surface: as for the incompressibility of thefoundations, this condition is very difficult to obtain exactly, above all when the masonry ofthe walls is not established on a platform placed on piles; and we cannot doubt that thefall of the greater number of bridges must be attributed to the movements which havemanifested themselves in the points of support on which they were carried; but as the foun-dation approaches to incompressibility, in proportion as the effort which it supports isdistributed on a greater surface, the constructor is obliged, in order to unite physicalcircumstances to analytical hypothesis, to increase the dimensions of the points of support.The difference existing between the methods adopted by practice and theory may be thusaccounted for, and it is easy to conclude that we must augment the points of support: itis not easy to determine the value of this augmentation; it depends evidently on the natureof the materials and the kind of construction used, on the nature and solidity, more or lessgreat, of the foundation, and of the other particular circumstances relative to the projectedbridge. It is prudent to adopt greater thicknesses than those indicated in the preceding table,or than can be calculated by means of the same theory ; but as this theory in its applicationto very flattened arches gives very considerable thicknesses, we have sought and succeededby different means, as will be seen below, to reduce this enormous mass of masonry withoutany loss of its resisting force. The arches may be raised on piers, and then the positionsof the points of rupture change, as well as the thickness of the abutments, which becomesmore considerable; it is indispensable to determine its value in each particular case. Wesuppose then, conformably to the results of the experiments already alluded to, that thepiers form only one single body with the inferior part of the arch, and that in the ruptureit turns round its external edge: some have thought that a rupture might take place in-