Chap. XXV.

ON STONE BRIDGES.

1505

some point of the height of the pier, and they have in consequence endeavoured to deter-mine what form it ought to have to present everywhere an equal resistance; the solution of* this problem is difficult, on account of the calculations to which it would lead ; it isbesides of no practical value.

We have hitherto supposed that the rupture of arches could not take place unless theabutments should turn round their exterior edge; there might, however, happen a horizontaldisjunction, when the upper part would slide over the lower. The resistance which theabutment opposes to this movement depends on the weight of the upper parts, and on themanner in which the beds are united- to each other ; the friction and the adhesion of themortars being here the two principal means of solidity, the value of which effect has beencalculated. *

M. Boistard published in 1804 some experiments on this subject, the results of whichwere that the adhesion of the mortar is proportioned to the surface; that the time afterwhich the stones are detached has little influence on the value of this adhesion, whichis almost as great after the first month as after years. It may be valued as 6960 kilo-grammes for a square metre for mortar composed of lime and sand, and 3700 for mortarof lime and cement; these values only being considered as proximate results becausethey necessarily vary, on account of the quality of the material of which the mortar is com-posed. The great superiority of mortar made with sand over that of cement, which at theend of a year is increased, does not exist when employed under water; in this latter casethe cement contracts quickly into a strong consistence, which is not the same with theother.

With regard to the friction, M. Boistard has equally endeavoured to discover its effects,and found that its relation to the pressure is constant, and that in taking the least value givenby these experiments, this relation for a stone indented or picked, gliding on a similarstone, or what is the same, on a surface of mortar hardened by the air, was equal to 0*76,or about

Considering the equilibrium of arches in this point of view, we find for the abut-ments thicknesses different to those which are indicated in the preceding table ; the equationof equilibrium is then

/* °’76 ( p + v) + 6960 R R.

The following table contains these new results applied to arches extradossed to the level, theopening of which is 20 metres, and the thickness of the key one metre : we have supposedthat the disjunction always took place at the level of the springing, and no account hasbeen taken of the vertical pressure resulting from the weight of the upper part of the vaults,which reduces the preceding equation toFG DO

M EQ ' ]3Q =0 ' 76 + 6960 KR '

The masonry is supposed to be 2600 kilogrammes per cube metre.

Thickness of the Position of the

Abutments. Points of Rupture.

Metres. Degrees.

Semicircular arches - - - - 1*32 14

Arches struck from three centres as before - 1 ’62 32

depressed one-fourth - - 2'24 41

Arc of a circle of 60 degrees ... 3 09 0

These last results are again below the dimensions which engineers have generally adopted,but they, however, approach nearer to them than those of the former table. Accordingto this manner of calculating the equilibrium of arches, there is less of hypothesis than in thefirst, and if it be observed that the values of the second table suppose that the arches haveremained long enough on their centres for the mortar to have acquired a tenacity similarto that which it presented in the experiments just given, which scarcely ever happens, asit dries very slowly in the interior of the masses of masonry, we shall be convinced of thenecessity of increasing them still more, and of thus approaching nearer to the results ofpractice.

We have allowed a little more thickness to arches than is generally given, which tendsto favour the acting power : no account is taken of the beds of sand or earth, or of thepavement with which the arches are .generally charged, nor of the difference of specificweights of the various kinds of masonry; it would be easy to introduce these objects indetail in the calculations, but we are convinced we should not then obtain for the thick-ness of the abutments values sensibly different from those which have been given.

The arches of the arc of a circle require the thickness of the abutments much moreconsiderable than others, and these thicknesses would be still greater, if the arch, whichwe have supposed described by an arc of 60 degrees, was more flattened, as in manybridges. The first engineers who raised arches of this form opposed to their great thrus

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