BRIDGE. 



.107 



given base and weight, is found by adding. the lialf 

 pier to tlnvo fourths of the arrli, multiplying by the 

 breadth of the base, and dividing by the horizontal 

 thrust. 



M,p = 



2ht 

 b 



i 

 =. 



; 



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 cons-icier the weight of the pier as indepen- 

 dent of its base. Now, though it may be said with 

 propriety, that the weight ot the pier cannot be 

 known until we know its thickness, which is the 

 very thing sought, yet a little consideration will 

 shew, 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 struc- 

 ture, the gravity of their materials, or the weight of 

 solid matter over them, wr may therefore, when the 

 base is given, apply the weight necessary to keep 

 the pier in equilibrio, 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 prac- 

 tice very usual in the ancient structures : or, 2d, By 

 tapering the pier towards the springing of the arches, 

 or by making eacli pier only a row of pill irs in the 

 line of the stream, arching them together at top; a 

 Diode which may perhaps 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 Pcrronet at the Pont St Maxence. 



When pier- indeed are to be exceedingly high, as 

 in the columns which are sometimes employed in sup- 

 porting a lofty aqueduct, the best way is to make 

 them hollow, 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 

 way up to the road ; the spandrel-walls are carried 

 back so far as to unite with those of the neighbour- 

 ing arch, are locked together by a .cross wall just 

 ever the middle of the pier, having also walls longi- 

 tudinally, 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 pro- 

 ceed 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 somewh.it more ; for 

 the sterlings or breakwaters, at each end, will add 

 something to its stability ; and this will be still fur- 

 ther 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 colla- Tnrary. 

 teral advantages, we shall consider the whol" as rec- 

 tangulir, and then the stability may be found in the 



longitudinal section. We have already It =- < 



and in the case of a parallelogram */>= \fi(fi-\-c). c 

 being the height from springing to the roadway. By 

 : ution t liere arises \ li* (h + c) + \ a b=ht ; 

 and by resolving this quadratic equation, we have 

 ' 1 2hT' 



for- 



or thus, b = 



mula for the thickness of solid piers to support equi- 

 librated arches ; and it must be observed, that if the 

 arch be understood to act otherwise than a' \ the 

 thickness cf the pier, this coefficient may be altered 

 accordingly. 



As an example of the use of the above, take an F.xplana- 

 arch of 100 feet span, six feet thick at the crown t" ln ^ 

 and semicircular. The horizontal thrnst is G x 50 formu 

 = '!00 cubic feet ; and let us take the weight of the 

 half arch as = 1200 at a medium, since, on account of 

 the open spandrel, it may be considerably varied. Sup- 

 pose the arch sprung at 18 feet high, then /i-f-c=74 

 r'ii 2^18. 300 _ 

 F+c ~ ~ 7t 



3.1200* 



J =147.93, 



and V/146+ 14.7.93= 17.1*, 



from which subtract TTT-T r =12.17, we have 4.97, 



*v." + c j 



or 5 feet nearly, for the thickness of the pier, 

 which is not one-twentieth of the span. In an ex- 

 ample nearly the same as this, l:i feet has been 

 given by an eminent mathematician for the thickness 

 of the pier ; but the reason is, that the stability 

 which the pier derives from the superincumbent arch, 

 has not been taken into consideration ; an oversight 

 the more extraordinary, since it is evident, that unless 

 this weight did bear completely on the pier, it could 

 have no tendency whatever to overturn it. 



Suppose that -c in the above formula is =0, or, 

 what is the same thing, that the pier is carried no 

 higher than the springing, 



And in an arch of the above dimensions, 

 2<=600, |f = LJ| = 50 

 =2500 



lj r , or about 

 \Ve see there - 



^310050=55.6850=5.68 nearly 

 a seventh part more tl a:> the former, 

 fore how little the stability may depend on the mere 

 weight of the pier. 



We may have a proof of the accuracy of this de- 

 termination, by comparing it with the formula first 



given for the thickness of piers, viz. />= , ht, 

 or the overturning force, will be iiuux it>=5400. 



