"The reader has already frequently seen, that the ultimate pressure may, in every ease, be reduced to two others, viz., the weight of the semi-arch above, and the horizontal thrust. In the equilibrated arch, this pressure is directed perpen dicularly to the joints of the sections ; and these being usually drawn at right angles to the curve, the pressure is in the direction of the tangent to the arch. Hence, we have often called it the tangential pressure. Upon this principle, how ever, when the curve springs at right angles to the horizon, an infinite pressure is required in the vertical direction,—a supposition which cannot have place in practice. We must accordingly call in the assistance of friction in that case ; force which may be set in opposition to the horizontal thrust, and which, increasing with the superincumbent weight, very fortunately keeps pace also with what it is intended to oppose.
"Granting, then, that the friction is so contrived, upon the principles already explained, that there is no danger of any slide at the horizontal or springing-joint ; it will be readily admitted, that no slide is likely to take place in any horizon tal course below that, till we arrive at the foundation ; for the disturbing force is constant, but the friction increases as we descend.
"Figure 15.—Our principal care then must be, that the pier does not overset, by turning on the farther joint, E, 0 its base, as a fulcrum. Take a in the horizontal joint, A a as the centre of pressure ; draw a v to represent the weight of the semi-arch, and v T the horizontal thrust ; then T a is the ultimate pressure; and if, when produced, it falls within the base of the pier, it is perfectly obvious that it can never overturn it. And this is altogether independent of the weight of the pier : for if that were a mass of ice, immersed to the springing in water, the case would be exactly the same.
"But the pier itself has a considerable stability, arising from its own weight; and even though the direction of the ultimate pressure of the arch alone pass out of the base, the tendency to overturn the pier may be balanced by its weight. This weight may be supposed concentrated in the centre of gravity of the pier, and of course to act in the ver tical line which bisects it.
"Its effect will be nearly found by laying off in that line from the point q, where the direction of the ultimate pressure of the arch intersects it, q r = the weight of the pier, and taking q s = the ultimate pressure = a T, and completing the parallelogram, the diagonal drawn from q will represent the direction and magnitude of the united pressure of the arch and pier. This is not strictly accurate; it would be so if a and q coincided, which is the case with a single arch stand ing on a pillar : but in general, the ultimate pressure is still more favourable than this. Its direction at any point is in the tangent of a curve, which approaches the vertical as wo descend, since the proportion arising from of the pier increases with its height.
"In order to find analytical expressions for these forces, let the horizontal thrust of the arch = t. The weight of the
half-arch = a, and that of the pier = p, the height of the pier to the springing of the arch = h, the breadth at the base = b.
"1. Then the horizontal thrust acting in A a, tends to overturn the pier, and its force round the fulcrum, E, will be represented by multiplying it by the perpendicular distance A h x t.
"2. The weight of the pier acts in the direction a c, and its effect will be represented by multiplying it by the leverage C E, p X b.
"3. The arch acts with the leverage E s, which is not equal to the breadth of the pier, by the part it D=A a, say one-half of the depth of the joint at the springing. This will never exceed one-fourth of the breadth, when two different rings of arch-stones rise from the same pier, unless the pier widen below. Call E therefore, = 3- b.
" We have now ht=lbp whence, ht 4 ht" 1st, b = p+la= 2p ± 3 a, and consequently, to find the least breadth of the pier at its base, divide the horizontal thrust by half the pier added to three-fourths of the half-arch. Multiply the height of the pier by the quotient.
"2d it = b (1„ p that is, the height of a pier to the springing, having a given base and weight, is found by adding the half-pier to three-fourths of the arch, multiplying by the breadth of the base, and dividing by the horizontal thrust.
ht—lba "3d, p= T11a; b or the weight of the pier cannot be less than the excess of the horizontal thrust multiplied by twice the height of the pier, and divided by the base, above one and a 'half times the semi-arch.
"In the above determination it may be observed, that we consider the weight of the pier as independent of its base. Now, though it may be said with propriety, that the weight of the pier cannot be known until we know its thickness, which is the very thing sought, yet a little consideration will show, that we may give different magnitudes to piers which have equal bases, and that, either by altering the outline of their sides, the density of their structure, the gravity of their materials, or the weight of solid matter over them, we may therefore, when the base is given, apply the weight necessary to keep the pier in equilibria, provided this does not require the pier to be any more than a solid mass up to the roadway. Should the base assumed admit of the pier being much less than the solid parallelopiped, we may diminish it in various ways : as, 1st, By opening arches over the pier, where, in case of floods, we will procure an addition to the waterway; a practice very usual in the ancient structures : or 52d, By tapering the pier towards the springing of the arches, or by making each pier only a row of pillars in the line of the stream, arching them together at top ; a mode which may perhaps be objectionable in a water-way, but which would have a very striking and light effect in land-arches. Some thing of this kind has been done by Perronet, at the Pont St. Maxence.