i( 8 2. The weight of the pier acts in the direction b c, and1„ effect will be represented by multiplying it by the,f a geCE, y\z.,p X %b.

6(1 ”■ The arch acts with the leverage ek, which is noto,J to l * ie breadth of the pier, by the part k d = a h, sayMil ^ Hie depth of the joint at the springing. This(Jig, never exceed one-fourth of the breadth, when twothe r ? nt rings of arch-stones rise from the same pier, unless, P‘ ar widen below. Call e k, therefore, zz\b.e have now ht — \b p 4- \ b a ; whence,

^ J = 2^h’ “ d

^ eas * btfi.dth of the pier at it9 base, divide the

H b =

of ,L° nta l thrust

„»;h1f-„eh.

by half the pier added to three-fourthsMultiply the height of the pier by the

-2<U = *(i£±l£> thatfa,th e v, - 1

5m ° e 'gbt of a pier to the springing, having a given baseis found by adding the half-pier to three-fourths

^idi

the arch.

multiplying by the breadth of the base, and

ln g by the horizontal thrust.

3d,

P =

h t — f b a _ 2 h t

— 1

i o,

n.

, e height of the pier cannot be less than the excess ofWdf l0l ''zontal thrust multiplied by twice the height of thetl, e ’ an d divided by the base, above one and a half times

^nai-arch.

tOf! . , n Hie above determination it may be observed, that weer weight of the pier as independent of its base,it may be said with propriety, that the weightM lic] ® Pier cannot be known until we know its thickness,V 18 the very thing sought, yet a little consideration willt lat we may give different magnitudes to piers whichl ^ir 6 ^ Ua i bases, and that, either by altering the outline ofthe density of their structure, the gravity of theirl Vef' a s> or tbe weight of solid matter over them, we mayL ° re ’ when the base is given, apply the weight necessary. P the pier in equilibrio, provided this does not requireSliojji*! 1 ' to be any more than a solid mass up to the roadway,'kju ^ Hie base assumed admit of the pier being much less

tys, e solid parallelopiped, we may diminish it in various1st, By opening arches over the pier, where, in

Of

,{ tcti

08.

fl j J r .° w — ~ ...

.floods, we will procure an addition to the water-way;

p ei , ,ce very usual in the ancient structures : or 2d, By

' ,n g the

bo

pier towards th§ springing of the arches, or by

Maf 6ac * 1 P' er only a row of pillars in the line of theNli a ’ ar ching them together at top; a mode which mayJe objectionable in a water-way, but which wouldAig 0 V „ ei y striking and light effect in land-arches. Some-At., ^'' a hind has been done by Perronet, at the Pont

c ."$J eace ;

^a ln ; en Piers indeed are to be exceedingly high, as in the,H|- Which are sometimes employed in supporting a loftyk'^'litv ’k^ e ^ est wa y' s to ma he them hollow, and give themb y enlarging the base. They will, in that case,v be ** 0I } Ibe^foundations, be less expensive, and they

Uj^^Htly stiffened by hooping, _ e d, it is not usual to make nie

the y ee<1, * 8 not usual to make piers solid all the way uplj te ’ ^ le spandrel-walls are carried back so far as to^ a c 1 those of the neighbouring arch, are locked togetherwall just over the middle of the pier, having also°t» s H&'tudinally, and the whole arched or flagged over-'i^drel to spandrel just under the roadway.

e rtheless, as the case of solidity will enable us to

assign a limit to the breadth of piers, which it may be properto be acquainted with, we shall proceed in that investigation.

“ The weight of the pier in that case will be as the rect-angle under its height and thickness, expressing the weightof arch and pier by the cubic feet of stone. The pier indeedwill be somewhat more ; for the sterlings, or breakwaters, ateach end, will add something to its stability ; and this willbe still farther increased in proportion to the horizontal push,if the whole bridge be wider at the foundation than at top,as is very common. Excluding these collateral advantages,we shall consider the whole as rectangular, and then thestability may be found in the longitudinal section. We

have already b = —, and in the case of a parallelo-

gram J p = 1 b (h -f c), c being the height from springingto the roadway. By substitution, there arises ^ b* {h + c)+ \ ab = h t; and by resolving this quadratic equation,

, , J 2 h t / 3 a \ !

we have 6 — v ^^-- » —

h + c \4 (h + c V

3 a

4 (h 4- n\

or thus, b

J 2 (h + c)/it+jah + c

as a formula

for the thickness of solid piers to support equilibrated arches;and it must be observed, that if the arch be understood toact otherwise than at three-fourths the thickness of the pier,this co-efficient may be altered accordingly.

“ When the arch is a segment less than a semicircle, agreater thickness of pier becomes necessary. For, the spancontinuing the same, we must either make the arch a part ofa circle of greater radius, which would increase the horizontalthrust, or we must, in order to obviate that, diminish thethickness at the crown. In either case the weight of the archis diminished, and with it the assistance which it gives to thestability of the pier.

“ There is an interesting subject of inquiry, which mightnot be unappropriated noticed here; we mean the lowestversed sine that can be used for arches in proportion to thespan. We conceive this, however, as in a great measurea practical question. We have already given some idea ofthe greatest possible arch of stone, or brick ; a segmentof that circle may, of course, be employed in any situation,but tbe piers (if the arch be of considerable span and heightto the springing) must be made very great. Indeed, theinvestigation depends intimately on the thickness of piers.We ought to know the dimensions of the largest pier that canbe trusted, and this, we conceive, depends chiefly on the careof tbe mason ; for stone, and especially cement, is a com-pressible substance: and when an arch is very flat, a verysmall yielding at the springing produces an enormous depres-sion at the crown, insomuch that there may be reason todread, lest the arch pass down below the horizontal line, andfall to pieces before the stability of the abutments can beacted upon. A compression in the joints is equivalentto a yielding at the abutments, and appears equally difficultof remedy.

“ In great horizontal thrusts, where the segment is flat,the immersion of the pier in water comes to have an impor-tant effect. On the weight of the pier, in those cases, thestability chiefly depends, and a deduction from that of two-fifths must be compensated by enlarging the thickness.

“ But, indeed, the immersion of the pier, if it be very tall,that is, if the depth of water be great in proportion to thespan, will demand attention, although the arch should not bevery flat. In such a case, the stability arising from the pieris often as great as that which is derived from the weight