BRIDGE. 



want of a rufficicnt weight ov«v the portion A m n, and an 

 equal portion on the other fide of the vertex, would be in 

 eonftaiit danger of rifing in the haunches. But a bridge 

 formed of hollow iron voufloirs would be abundantly ftrong, 

 with far Icfj thickncfs over the crown, as B j ; and then the 

 true extrados e i i// would, in every part, have a proper 

 flope for a road-way ; wliile, at the fame time, tlie ftrufture 

 is in no danger of being dcflroycd for the want of an equili- 

 brium in all its parts. 



Wc have mentioned under the article Arch, what kind of 

 arches ought to be preferred in the eie£\ion of bridges ; and 

 have fiiewii whieh are ftrongell : we may here obfervc, that 

 if there be two arches of the fame kind, with an equilibrated 

 load over each of them, the llrength of the one will be to 

 the (Irength of the other reciprocally, as the radii of curva- 

 ture at the vertices of the two arches : hence, an elliptical 

 arch, ftanding on its (hoiter anh, will be (Iron^er than a fe- 

 micircular arch of tlic fame fpan ; and the femicircular arch 

 of equilibration will be llronger than a flat clilptical arch of 

 the fame fpan. As to the efftia of an additional weight 

 over any part of an arch, it will vary in proportion of the 

 horizontal dillance* from the extremities of the arch. 

 Hence, the greatcft danger ariling from an additional weight, 

 is when it lies over the crown of the arch ; for then the pro- 

 duct of the horizontal dillances from the abutments is equal 

 to the fquare of the femi-fpan, and is the greatcft it can be. 



Since in anv arch of equilibration, the preffin-e arifing 

 from the incumbent weight at any point is reduced to the 

 direction of the tangent at that point, we have in any fuch 

 arch VB,/-. 4. PI. XXXVIIl.of yi';vto-c7(.')v,the weight 

 «f the piirtVBEA, the prelTurc along the tangent FB, and 

 the horizontal prefTure in dirtiTtion DB, refpeftively as the 

 lines FD, BF, and BD, or as the conefponding lines in a 

 triangle, whofe fides are feverally perpendicular to thofc in 

 BDF. Hence, it is tafy to find the area of the portion 

 AEVB, thus : make c v parallel and equal to CV, the ra- 

 dius of curvature at the veitex ; and draw c b perpendicular 

 to the tangent BF, meeting v b the perpendicular to f -u in 

 b ; then in the triangle c v b, c v correfponds to DB, and 



V b ta DF ; and the area of the parallelogram a it, having 



V t =^ VE, is equal to the area of ABVE : in like manner, 

 by drawing <:^ perpendicular to GI, the tangent at G, we 

 Ciould have the parallelogram /j b equal to the portion HB 

 over the part GB of the arch. The area of the fpace 

 HEVG, between the arch and the road-way, being thus 

 afcertalned, its weight of courfe becomes known, and, con- 

 fequcntly, its horizontal preflure againll the abutment, as at 

 G : for it will be, as the line -vg : vc :: the weight over the 

 fcmi-areh : the horizontal thrull againft the abutment, or a 

 pitr, at G. 



But in cllimating the thruft againft the piers, &c. it is 

 molt common to afcertain the pofition of the centre of gra- 

 vity of the load above the arch. Now, in cafes of equili- 

 bration, this may fometimes be cfFeded without much diffi- 

 culty : for it is well known, that if a heavy body be fuf- 

 (aincd by two forces, their diredions muft meet, either at 

 the centre of gravity of that body, or in a vertical line 

 wh.ich pa^Tes through it ; therefore, fince the whole incum- 

 bent weight, over a properly balanced arch, is fuftained in 

 equilibrio by two forces, ailing in the direilion of the tan- 

 gents to the extreme points of the curve, the centre of gra- 

 vity of the materials upon the arch will be in the vertical 

 line which paQls through the interfeflion of thefe tangents : 

 and, in moft cafes occurring in pratlice, the centre of gra- 

 »itv will be nearly eqni-dillant from the e?;trados and intra- 

 dos of the equilibrated arch. Thus, in the curve AVE, load- 

 4^ to the equibbriuni, _/ff. 5. Fl. X.XXVIIl.of ^/rf/^i/ciSMrs, 



the centre ofgravity of the fuperincumbent mafsism the ve> 

 tical line D if, palling through the interfeftion of the tan- 

 gents AD, and BD. And the centre of gravity of the 

 materials AVHK, between the crown and the abutment, 

 is about the middle of the vertical line E e, paffing through 

 the interfcftion of the tangents AD and V i. If the arch 

 be part of a circle, (V is the tangent of half the arch AV, 

 which, fubtrafted from half the fpan, leaves AG = fine of 

 A V — tangent of half AV : and fince G e = verfed fine of 

 arc AV — verfed fine of arc e V, we fhall, by adding A Ea 

 to G e, have the ahitude of the centre of gravity, from AC 

 the horizontal fine. If AV be a parabola, AG = | AC ; 

 but if it be an equilibrated curve, with a horizontal extra- 



dos, then AG = ^/ — TnTirCV"' ^"^"'^ ^^ '* '"^ radius 

 of curvature of the arch at the crown. When the arch is- 

 not julliy equilibrated, otlier methods of finding the centre 

 of gravity of the mafs fupported muft be had recourfe to, 

 See Hutton on bridges, p. 49 — ^S- It may be worth 

 while, however, to dcfcribe here an eafy praClical method, 

 accurate enough for moft purpofes : namely, to drav.' on a. 

 piece of card paper, a plan of the arch, and its loa-d ; then 

 to cut out half of it as DABC,/^. 6. P/. XXXVIII._ of 

 ylirhheclure, and to determine experimentally the point K in 

 the piece cut out, on which, when fupported, the whole wilt- 

 reft ; for this point will manifelUy correfpond with the centre 

 of gravity. 



The place of the centre of gravity being determined, we 

 mav now (liew how to afcertain the thicknefs of a pier, ne- 

 ccffiiry to fuj)port a given arch. Let ABCD, fig. 6. P!, 

 XXXVIII. reprefenL the mafs over half the ai-ch;DEFG the 

 pier. From the centre of gravity K of the mafs, draw KL, 

 perpendicular to the horizon : then the weight of the arch, 

 in tire direftion KL, wiH be to the horizontal pufli, or lateral 

 preffnre at A, in the dirtftion LA, as KL to LA. For the 

 weight of the arch in the direction KL, the horizontal pufti in 

 the direelion LA, and the oblique pufh in the direftion KA, 

 will be as the three fides KL, LA, KA. So that if A 



LA 

 denote the weight or area of the arch, then-p-p. A, will be 



LA 

 its force at A in the direftion LA ; and^^r^XGAxA, its 



effeft on the lever GA, to overfet the pier, or to turn it 

 about the point F. Again, the weight of the pier will be 

 as its area EFxFG, and, fuppofing the load over the ai'ch, 

 and the pier to be of fimilar materials, EFxGFX = FG or 

 ^EFkFG% is the eil'eft on the lever |FG to prevent tho 

 pier from bein)j overfet. Here it is luppofed, that the 

 length of the pier, from point to point, is the lame as the. 

 thiekntis of the arch, and that the centre of gravity of the 

 pier falls in the vertical plane bifefting FG. Now, that the, 

 pier and the arch- may be in equilibrio, the two efFeds jult 



LA 

 ftatcd muft be equal: therefore, we have |EF.FG" — -r-j-X 



G.\XA, from which it follows, that the thicknefs of the 



. . ^ 2GA-AL 

 pier is FG = v^£p — ^kT'^'^' 



In the above inveftigation, it is fuppofed, that the whole 

 of the pier is out of water i but if any part of it be im- 

 mcrfed in water, tJiat part will lofe fo much of its weight aa 

 !.? equal to its bulk of water, if the water can get below the 

 pier or into the joints. This, however, may eafily be 

 brought into the calculation. By applying the above the- 

 orem to the feveral cafes which m.ay arife, the thicknefs of 

 the pier may be found, fo that it fhall _////? balance the fpread 



