1. Then the horizontal thrust acting in AG, tends to overturn the pier, and its force round the fulcrum E will be represented by multiplying it by the perpendi cular distance AD= viz. hxt.
2. The weight of the pier acts in the direction BC, and its effect will be represented by multiplying it by the leverage CE, viz. fix.p.
3. The arch acts with the leverage EK, which is not equal to the breadth of the pier, by the part KD=AH, say $ 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 EK, therefore =-1b. We have now whence, lit 4ht 1st, 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.
b(111-1- la) 2d, h=--, 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.
3d, 2 ht 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 skew, that we may give differ ent 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 there fore, when the base is given, apply the weight neces sary 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 water-way ; a practice very usual in the ancient structures : or, 2d, 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 per haps be objectionable in a water-way, but which would have a very striking and light effect in land arches.
Something of this kind has been done by Perronet at the Pont St Maxence.
When piers indeed are to be exceedingly high, as in the columns which are sometimes employed in support ing a lofty aqueduct, the best way is to make them hol low, and give them stability, by enlarging the base. They will, in that case, press less on the foundations, be less expensive, and they may be greatly stiffened by hooping.
Indeed it is not usual to make piers solid all the up to the road ; the spandrel-walls arc carried back so far as to unite with those of the neighbouring arch, arc locked together by a cross wall just over the middle of the pier, having also walls longitudinally, and the whole arched or flagged over from spandrel to spandrel just under the roadways.
Nevertheless, as the case of solidity will enable us to assign a limit to the breadth of piers, which it may be proper to be acquainted with, we shall proceed in that investigation.
The weight of the pier in that case will be as the rectangle under its height and thickness, expressing the weight of arch and pier by the cubic feet of stone. The pier indeed will be somewhat more ; for the ster lings, or breakwaters, at each end, will add something to its stability ; and this will be still further increased in proportion to the horizontal push, if the whole bridge be wider at the foundation than at top, as is very com mon. Excluding these collateral advantages, we shall consider the whole as rectangular, and then the stability may be found in the longitudinal section. We have al, ht ready b= ip-1-73, and in the case of a parallelogram c being the height from springing to the roadway. By substitution there arises bb20+0-Flab =ht; and by resolving this quadratic equation, we have — b h t I 3 a 2 3 a h+ c k4(11+01 4(h+c)or thus, b= (h+c) h t + 442 —la as a formula h c for the thickness of solid piers to support equilibrated arcoes ; and it must be observed, that if the arch be understood to act otherwise than at 4 the thickness of the pier, this coefficient may be altered accordingly.