COS. ¥=μ(1+a)2 {(1 +ẞ) vers. ¥¥ sin. Y

Р

Р

(474).

Substituting this value in equation (455), making Yæ-Xy=0, solving that equation in respect to and making 1+^, we have

{ba2+a-tu(1+a) ysin. - {a+a+‡a3—μ(1+a)2 (1+3)} vers.

+vers. Y

..(475)

If, instead of supposing the pressure of the water to be borne by the extrados, we suppose it to take effect upon the intrados, tending to blow up the arch, and if 3 represent the height of the water above the crown of the intrados, we shall obtain precisely the same expressions for X and Y as before, except that must be substituted for (1+a)r, and X and Y must be taken case, therefore, sin. F

negatively; in this

-{(13) vers. Y

Y

X

Cos. Y=

sin. Y}; whence, by substitution in

equation (455), and reduction,

2

P (a+a+u) sin. - {a+a2 + 1a3 + μ (1+)} vers. Y

.(476)

=0; differentiating equa

Now by note, page 438,

tions (475) and (476), therefore, and reducing, we have,

which equation applies to both the cases of the pressure of a fluid upon an arch with equal voussoirs; that in which its pressure is borne by the extrados, and that in which it is borne by the intrados; the constant A representing in the — a2 + } a3 — μ ( 1 + B) (1 + a)* and in the

— a2+a—1μ(1+a)2 a2 + fa2 +μ (++3) If the line of resistance Za2+a+2μ

pass through the summit of the key-stone, λ must be taken=a.

If it pass along the inferior edge of the key-stone,

Ꮠ

>=0. In this second case, tan. -sin. ¥} =0, therefore,

2

=0; so that the point of rupture is at the crown of the arch. For this value of equations (475) and (476) become vanishing fractions, whose values are determined by known methods of the differential calculus to be, when the pressure is on the extrados,

It is evident that the line of resistance thus passes through the inferior edge of the key-stone, in that state of its equilibrium which precedes its rupture, by the ascent of its crown. The corresponding equation to the line of resistance is deter

P

mined by substituting the above values of in equation (454). In the case in which the pressure of the water is sustained by the intrados, we thus obtain, observing that

3

(ta2+a+tu) e sin. + (a−‡a3 +μ) cos. 8—μ(1+6)

α

If for any value of in this equation, less than the angle of the semi-arch, the corresponding value of p exceed (1+ar, the line of resistance will intersect the extrados, and the arch will blow up.

THE EQUILIBRIUM OF AN ARCH, THE CONTACT OF WHOSE

VOUSSOIRS IS GEOMETRICALLY ACCURATE.

342. The equations (459) and (456) completely determine

the value of P, subject to the first of the two conditions stated in Art. 333., viz. that the line of resistance passing through a given point in the key-stone, determined by a given value of λ, shall have a point of geometrical contact with the intrados. It remains now to determine it subject to the second condition, viz. that its point of application P on the key-stone shall be such as to give it the least value which it can receive subject to the first condition. It is evident that, subject to this first condition, every different value of A will give a different value of; and that of these values of

that which gives the least value of P, and which corresponds to a positive value of not greater than a, will be the true angle of rupture, on the hypothesis of a mathematical adjustment of the surfaces of the voussoirs to one another. To determine this minimum value of P, in respect to the variation of dependent on the variation of λ or of p, let it be observed that does not enter into equation (456); let that equation, therefore, be differentiated in respect to P and Y,

dP

and let be assumed 0, and Y constant, we shall thence dv

obtain the equation

P

= {3a(a+2) cos. 'T-a (2a+3)} (483);

and by eliminating sec. between equations (457) and (481), and reducing,

The value of given by this equation determines the actual direction of the line of resistance through the key-stone, on the hypothesis made, only in the case in which it is a positive quantity, and not greater than a; if it be negative, the line of resistance passes through the bottom of the key-stone, or if it be greater than a, it passes through the top.

Such a mathematical adjustment of the surfaces of contact of the voussoirs as is supposed in this article is, in fact, supplied by the cement of an arch. It may therefore be considered to involve the theory of the cemented arch, the influence on the conditions of its stability of the adhesion of its voussoirs to one another being neglected. In this settlement, an arch is liable to disruption in some of those directions in which this adhesion might be necessary to its stability. That old principle, then, which assigns to it such proportions as would cause it to stand firmly did no such adhesion exist, will always retain its authority with the judicious engineer.

APPLICATIONS OF THE THEORY OF THE ARCH.

343. It will be observed that equation (459) or (472) determines the angle of rupture in terms of the load Y, and the horizontal distance of its centre of gravity from the centre C of the arch, its radius r, and the depth ar of its voussoirs; moreover, that this determination is wholly independent of the angle of the arch, and is the same whether its are be the half or the third of a circle; also, that if the angle of the semi-arch be less than that given by the above equation as the value of Y, there are no points of rupture, such as they have been defined, the line of resistance passing through the springing of the arch and cutting the intrados there.

The value of being known from this equation, P is determined from equation (456), and this value of P being substituted in equation (454), the line of resistance is completely determined; and assigning to the value ACB (p. 437.), the corresponding value of p gives us the position of the point Q, where the line of resistance intersects the lowest voussoir of the arch, or the summit of the pier. Moreover, P is evidently equal to the horizontal thrust on the top of the pier, and the vertical pressure upon it is the weight of the arch and load: thus all the elements are known, which determine the conditions of the stability of a pier or buttress (Arts. 293. and 312.) of given dimensions sustaining the proposed arch and its loading.

Every element of the theory of the arch and its abutments is involved, ultimately, in the solution in respect to of equation (459) or equation (472). Unfortunately this solution presents great analytical difficulties. In the failure of any direct means of solution, there are, however, various methods by which the numerical relation of and Y may be arrived at indirectly. Among them, one of the simplest is this:

Let it be observed that that equation is readily soluble in respect to Y; instead, then, of determining the value of for an assumed value of Y, determine conversely the value of Y for a series of assumed values of Y. Knowing the distribution of the load Y, the values of a will be known in respect to these values of Y, and thus the values of Y may be numerically determined, and may be tabulated. From such tables may be found, by inspection, values of corresponding to given values of Y.

The values of Y, P, and r are completely determined by equations (482, 483, 484), and all the circumstances of the equilibrium of the circular arch are thence known, on the hypothesis, there made, of a true mathematical adjustment of the surfaces of the voussoirs to one another; and although this adjustment can have no existence in practice when the voussoirs are put together without cement, yet may it obtain in the cemented arch. The cement, by reason of its yielding qualities when fresh, is made to enter into so intimate a contact with the surfaces of the stones between which it is interposed that it takes, when dry, in respect to each joint (abstraction being made of its adhesive properties), the character of an exceedingly thin voussoir, having its surfaces mathematically adjusted to those of the adjacent voussoirs; so that if we imagine, not the adhesive properties