Chap. 11^

OX STONE.

707

the voussoirs are exposed, and the degree of resistance of the stone, deserve enquiry ; it isnecessary to direct our particular attention to these objects when we are projecting archesof considerable span.

Suppose Q the horizontal thrust of the arch, a thepressure that the two halves exert against eachother, and which is the greatest possible force itwill be necessary to apply horizontally in X to pre-vent a portion, m n and X 31, of the arch turning fromtop to bottom on the edge m.

The co-ordinates A (), N O, of the point M,counted from the point A, are designated by ab , andthe length of the joint M X by c. The co-ordinatesAp, pm from the point m, are named x and y, thelength mn of the joint z, and the angle which thisjoint forms with the vertical, 0. We represent byG the weight of the portion of the arch m n X M, aswell as the parts of the construction which it sup-ports, and by cc the horizontal distance from the centre of gravity of this weight to thepoint A. The value of the horizontal thrust at Q being known, we shall easily ascertainthe pressure exerted perpendicularly against any joint, mn; in effect, this pressure can beno other than the result of the forces QG, to which is submitted the portion of the vaulttnn NM, discomposed perpendicularly at mn ; that is to say, it is expressed by

G Sin. 0 + Q Cosin. 0

It is not, however, right to consider the horizontal thrust Q a force applied to the arrisN, as the voussoirs at the summits of vaults necessarily lean against each other through thewhole or a portion of the height of the superficies of the joint; the pressure then beingextended over a certain space below the arris N, it acts to prevent the descent of the portionof the vault m n 31 N, with an arm of a lever less than is supposed, when we consider it asapplied in N ; consequently we find, as the effect of this supposition, the value of Q smallerthan it really should be.

The value of the normal pressure exercised on each joint should be exactly known, or wecannot deduce from it the efforts supported by the stone, since we are ignorant of the man-ner in which this pressure is spread over the surface of the joint; far from being able toadmit that the pressure is equally distributed over all that surface, we know on the contrarythat in all the arches, with the exception of a small number of joints, the pressure is prin-cipally exercised near one of the edges: this circumstance takes place above all at the summit,when the pressure exerts itself near the upper edge; at the joints of rupture, placed in thehaunches, when it exercises itself near the bottom edge; and at the inferior joints belowthe springing, when the pressure exerts itself at the exterior edge. We suppose here, con-formably to what has so often taken place, that the inferior parts of the vault have a ten-dency to thrust outwards.

The manner in which the pressure is spread over the surface of the joints is besides moreuncertain, as it depends on the precautions with which the voussoirs are cut and placed,on the disposition of the wedges, on the consistency of the mortar, and the settlement of thevault, according to which the joints are more or less open.

Fig. 594.