10. THE CIRCULAR ARCH.

Let the intrados and extrados be circular cylindrical surfaces;

=p{X sin 0+ P cos 0+ (R2 — r2)(0 – →) sin 0}......(46).

=r{X sin + Pcos + (R2 - 2)(¥ − →) sin Y}......(47).

} (R − r) sin ¥ = r{Xcos-Psin ¥ + §(R2 — r2) (¥ − →) cos¥ + (R2 −72) sin ¥ };

= {Xr+\r(R2 − )( − )} sin¥-Xx+}(R3 − r3) (cos-cos)......(49).

Also, by equation (48),

Xr +{r(R2 − r2) (¥ − 0) = {Pr + {r3a2 (2a + 3)} tan ;

.. P{p―r cos¥} = {Pr+}r3a2 (2a+3)} tan sin

¥

−Xx + r3a (}a2 + a + 1)(cos - cos →);

11.

THE EQUILIBRIUM OF THE CIRCULAR ARCH, THE MATERIAL
BEING SUPPOSED UNYIELDING AND THE CONTIGUOUS SURFACES
MATHEMATICALLY ADJUSTED.

Let now the force P be supplied by the opposite pressure of an equal semi-arch, then on the hypothesis made, P is a minimum function of Y.

a(‡a2

. . Xxx + PP + a(} a2 + a + 1) cos 0} = √ {2 22 + a (§a + 1)}a (a + 2);

, P, p are thus completely determined, and all the circumstances of the equilibrium of the circular arch, thus loaded, are known.

If there be no loading, and the two semi-arches be parts of the same continuous cylindrical mass, X = 0, and = 0.

Therefore, by equation (52), ¥ = 0.

In this case, therefore, the point of rupture is in the crown of the arch (Fig. 18), at the intrados, also by equation (53),

Substituting these values of P and p in equation (46), we obtain for the equation to the line of resistance in the unloaded circular arch,

Let

be the angular distance from the crown, at which the line

of resistance meets the extrados, (Fig. 17), as

is that at which it

meets the intrados. Therefore, by the preceding equation,

.. (1 + a)( a + 1) y sin ↓ + (2 + 2a − 1 a3) cos ↓ = a + 2.......................(56).

determined from this equation will measure the greatest semi-arch, which being unloaded, can be made to stand.

To determine the inclination of the resultant P, to the vertical, corresponding to the angle e, we have

P1 sin horizontal force on segment =

P1cos

=

P

= r2 {a}a3},

= mass of segment = r2 {a2 + 2a} 0;

Suppose the arch to be supported upon upright piers of a given breadth a, (Fig. 18), and let it be required to determine what is the greatest height (≈) to which they can be carried.

It is evident, from equation (50), that as X is increased increases; that is, the points of rupture descend continually upon the arch as it is more loaded. The experiments of Professor Robison on chalk models are explained by this fact*,

* Having constructed chalk models of the voussoirs of a circular arch, and put them together, he loaded the arch upon its crown, increasing the load until it fell. The first tendency in the chalk to crush was observed at points of the intrados, equidistant from the crown on either side, but near it; these points were manifestly those where the line of resistance first touched the intrados. As the load was increased, the tendency to crush exhibited itself continually at points more distant from the crown—that is, the points of rupture descended,

12.

THE EQUILIBRIUM OF THE CIRCULAR ARCH UNDER THE CONDI-
TIONS WHICH OBTAIN IN ITS 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