Chap. XXV.

ON STONE BRIDGES.

1503

determine the relation which ought to exist between the weights and the directionof the levers, in order that the equilibrium may be maintained. Let us consider thehalf of the arch divided into two symmetrical parts, by the axis E C, and let ussuppose, first, that the point D is fixed : call p the weight applied in M. The systemwill not be changed in substituting for it two other weights, the one applied in E,FQ EF

and represented by MjVq* the other applied in D, and represented by jTq* The point E

FQ

will then be loaded with a weight equal to 2 M-£-q» an d there will result from it in the

FQ ED

sense of the lever E D a pressure represented by jTq* an d which, if the point D

were really fixed, would be destroyed by its resistance, as well as the effect of the vertical

EF . ...

force, MjTq* which is applied at the same point: but this point D being situated at the ex-tremity of another lever, the point of support of which is in K, and which is loaded at thepoint N with a new weight, represented by 7 , we must for the equilibrium take the mo-mentum of these different forces with relation to the point K as null, which will give theequation,

F Q, ED E1 „ _ Tro

"eq'eq kv= '“eq kr +" ks >

K V being a perpendicular lowered from the point K on the line E D prolonged, andKS and KR being the horizontal distances from the point K to the point N and to thepoint D. We have,

TrxT KU.DQ-DU.EQ,

K y_ ED

and in substituting this value in the preceding equation it becomes,

4 to ku ^ kr+ " KS;

and if we desire that the system have stability we must have,

The value of the thrust, and consequently the thickness to be given to the abutments,augments with the value of F Q, that is to say, when the centre of gravity of the upperparts of the vault approaches the summit; it is the same with the horizontal pressure whichthe two parts of the arch exert on each other, in the case of the equilibrium, as is indi-cated by the form of its expression, which is

FQ D Q^EQ' EQ’

It is to be remarked, with M. Boistard, that there has been a mistake hitherto, inendeavouring to make the sinking of an arch depend on the diminution of the length of thecurve of the intrados : this sinking evidently depends on the shortening of the curve D Erf,fig. 2737., which unites the points by which the voussoirs carry each other near thejoints of rupture, and which may also be considered as the place of the points of supportof the intermediate voussoirs : the nature of this curve is not easy to be determined, but wemay in the application regard it, without any great error, as being of the same kind withthe curve of the intrados.

Application of the Theory to the determining of the Thickness of the Abutments. — Thisequation

, 41’sri ,i “ = '‘‘ KR+ ’' KS

contains all that is necessary to resolve this question ; and it is now only required to find avalue of 13 K which shall satisfy this equation, or rather which shall render the secondmember a little larger than the first, in order that the arch should have the necessary sta-bility ; this is supposing that the positions of the points of rupture D ,d is known a priori,which is not generally the case ; we must then begin by determining it. And we shall ob-serve that these points must be so placed that the momentum of the force which tends tooverthrow the lower part be the greatest possible with regard to that of the forces whichtend to retain it in its position. We must then seek the value of the arc B D, which corre-sponds to the maximum of the expression

F Q D Q.

^ EQ’ E Q* wp Kll + vKS