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12. THE EQUILIBRIUM of THE CIRCULAR ARCH UNDER THE CONDITIONs which OBTAIN IN IT.s ACTUAL CONSTRUCTION.
The condition, taken as the basis of the conclusions arrived at in the last section, “that the resultant pressure P of the opposite semi-arch (see Fig. 15) is applied to that point in the depth AD of the key-stone which corresponds to its minimum value,” true under an hypothetical perfection of the masonry, does not obtain as a practical condition.
It supposes a mathematical adjustment of the contiguous surfaces of the stones to one another, an immoveability of the abutments, and an unyielding quality of the arch-stones and cement, which have no practical existence.
Every arch, on the striking of the centers which have supported it whilst it was built up, sinks at the crown.
The effect of this sinking or settlement is to cause the voussoirs about the crown to separate slightly from one another at their lower edges, somewhat like the leaves of a book, and thus to throw the whole of their pressure, upon one another, on their upper edges.
However skilful may be the masonry of an arch, and however small comparatively may be its first settlement, some settlement always perceptibly takes place; and there can be little doubt that in every arch a transfer of the whole pressure upon the voussoirs at the crown to these upper edges, from the first, obtains.
Moreover it is certain, from numerous experiments of Gauthey and others, that when an arch is in the state bordering upon rupture by the yielding of its abutments, the direction of its pressure is through the superior edges of its voussoirs at the crown, and through the inferior edges of the voussoirs at its points of rupture, in the haunches. Now the great practical question is to determine the conditions of the pressure under those possible circumstances, which are most unfavourable to the stability of the arch; circumstances which manifestly occur in the state bordering upon its rupture. This question necessarily then supposes a direction of the pressure, and therefore of the line of resistance, touching the extrados at the crown, and the intrados at the haunches; and, this being supposed, all those conditions of the equilibrium which depend upon the nearer approach of the voussoirs after the first striking of the center, or which arise from the long continued pressure, or from the influence of changes in the temperature, are eliminated.
inoo oo, “-one o tan *
a cOs G cOt So - sin # cot3 COS ä tan 2"
Substituting this value for the first member of the preceding equa
tion; multiplying by the denominator of the fraction in the second member, and neglecting powers of a above the first,
Hence, by Lagrange's theorem, we have from equation (60), neglecting powers of a above the first,
Where Y, is taken to represent the value of Y when a = 0," so that by equations (60) and (61),
* By Lagrange's Theorem, if y = z + a py, then, neglecting powers of a above the first, and representing by fy any function of y,
If therefore FY = FY,+ a FY and FY be any other function of Y, then, neglecting powers of a above the first,