The position of the joints is usually given in a differ ent way from that which we have just considered. In circular arches they are generally formed by producing the radii from the centre ; and in others they are com monly drawn perpendicular to the curve. Now, though we have just shewn, that this is by no means necessary to the equilibrium, yet, as it is in reality the most con venient in practice, it may be of importance to attend to the effects likely to be produced by this modifica tion.
NVe see, in Plate LXXX. Fig. 10. that the tangents on the horizontal line rapidly increase as we pass out ward, and we should therefore increase, in the same proportion, the weight of our sections. \Ve cannot in crease the base as proposed above, for that is necessarily given by the position of the joints, but, as we are still able either to increase the height or the breadth of the sections, we may consider the effect of both these modes.
Let it be required, then, to equilibrate a circular arch, where the stones being all of equal thickness, with joints equally distant, and drawn all to one centre, we are only at liberty to increase the width of the roadway, or length of the horizontal courses.
Considering each course of arch stones as a prism of a given base, a supposition sufficiently accurate, it is evi dent, that its magnitude or weight increases with the length only. But this weight must, from the principles already laid down, be as the difference of the tangents of its abutments ; the length therefore must be in that ratio. Accordingly, we find the breadth at different distances from the vertex in the same way with the weights of the sections : the breadth at 45° must be double, and at 55° must be about triple of that at the crown, and will increase still more rapidly afterwards. Proportions such as these may answer well in the short flight of steps for a flying staircase, but are quite unfit for our present purpose. When we recollect, however, that in a bridge, the extraordinary expansion towards the haunches is materially corrected by the increased pressure of the incumbent mass in that part, we are encouraged to proceed a little farther, and consider the effect of the second mode of effecting the equilibrium.
The pressure of matter upon each section has already been stated as proportional to tvXsw ;•Plate LXXXI.
Fig. 2. but tv is the difference of the sines of the angu lar distances of the successive abutments from the ver tex, and sw is the mean versed sine added to the given thic kness at the crown, when the roadway is horizontal. We have therefore the pressure as the difference of the sines x (mean versed sine + thickness at vertex.) But
these pressures are also, from the theory, as the differ ence of the tangents of these angular distances. In the present case, where the angles of abutment, and con sequently, where the difference of their sines and tan gents are known, and where the mean versed sine may also be readily thrilled, it will not be difficult to state the conditions of equilibrium for an arch of any dimensions.
In the common mode of building, we must give the arch a sufficient thickness at the keystone, to resist the horizontal thrust, ensure stability, and bear the loads likely to come upon it. We must also cover this part with a certain thickness of gravel, or other matter, so as to form a roadway. The varying pressure of the wheels of a loaded carriage, when it is propagated through this stratum of gravel, will be so far diffused as not to disturb the stone immediately below it, nor injure the bridge by splintering away its corners. This thick ness is made as small as possible, that the bridge may not be unnecessarily elevated, and the roadway is pre. served nearly horizontal. The other courses of archstones too, do not often differ much in thickness from that at the crown. But although these things are pretty con stant, there is a considerable degree of latitude in filling up the space between the back of the arch and the road way. It may be done with substances varying in density, from the lightest charcoal or pumice, open shiver or chalk, to closely rammed clay, or even solid masonry ; and it is not uncommon to make, in various ways, open spaces in the masonry of the spandrel, covering them above, so as still to support the roadway.
It will therefore be proper for us to enquire, what is the density requisite over every section of an arch, where the thickness of the crown is given, the roadway hori zontal, the arch of uniform thickness, and the angles of abutment of the several sections constant, that is, all drawn from the same centre ; or, what is the same thing, let us suppose the structure built up to the horizontal roadway with parallel sides, and then enquire, what is the proportion between the pressure borne by each sec tion, in this way and the pressure of equilibrium ; we shall thereby discover the ratio in which the density of the backing must, if needful, be diminished ; and the quantity of expansion necessary towards the springing of the arch, that the advantages of equilibration may be preserved, even in this state of things.