492 



BBIDGE. 



Theory, the crown. If the vertical pressure be less than the 

 -v ' tension, the extrados falls below the horizontal line ; 

 if greater, it will rise above it. 



Mathematicians finding the circle and other com- 

 mon curves so little adapted to the arch of a bridge, 

 which has a horizontal roadway, have, in the next 

 place, endeavoured to solve the converse of the pro- 

 blem, and give a rule for finding the intrados or figure 

 of the arch, which have the exterior curve a horizon- 

 tal line. 



This problem can only be resolved by calling the 

 fluxionary calculus to our aid. It is a case of the 

 more general one to find the intrados, when the ex- 

 trados is given ; and being the most useful case of that 

 problem, fortunately admits of a solution compara- 

 tively easy. 



Method of We have already seen, that the load DC is inversely 

 finding the as rad. curv. X sine * inclination to vertical. Calling 

 intrados therefore, as usual, theabcissa VE=.r,CE=y, VC=z, 

 when the /. , j f~- f 



extrados is we have cf=x,fe=y, Cc=z ; and since Cc : c/ :: 



given. rad. : sin. inclin. at C ; therefore the load DC is inverse- 



LXXX ty as ra d- curv. X 4 But, as is well known, the 



Fig. 8. *' 



We subjoin a Table, calculated by Dr Hutton Theorj 

 from this formula, for an arch of 100 feet span and ' f 

 40 feet rise, the thickness of the crown being taken 

 at 6 feet. It is nearly of the same dimensions as the 

 middle arch of Blackfriar's Bridge, and which may 

 answer for any arch where these dimensions are simi- 

 larly related to each other. 



rad. of curvature=-r- 



-TT ; therefore, by multi- 



y x x y 



plication, DC is inversely as T = -.^-, that is, di- 



y x x y 



rectly as y a T* ^ , or as d( 4- -=- y, ) and is equal 



yi -\X I 



to rf(''4- X , ) where C is a constant quantity, found 



by taking the real value of DC at the vertex V of the 

 curve. ' 



Now, in the present case, calling AV=, we have 



DC=a+x,(AV+VE,)thereforea+a:=-^ Xflux. of 



t t/ 



-. Talie =:Uy and by integrating, we have 



* 9 y 

 v= ^ T , and therefore 



y ( = jy'Cx ^===^ whence by integration 

 \u J \/"2a xxx* 



H^ + B. 

 At the vertex :t=0, therefore 



And consequently the ordinate 



Lastly, to find the value of ^/C, we take some point 

 of the extrados, where the ratio of .r and y is known. 

 For example, if the span = 2S, and height =/j are given, 



4 * 

 wehaveS=V'CxLo. 



finally J/=SX" 



Lo. a( 



The curve of Fig. 8. is accurately drawn to these 

 dimensions, and may give an idea of the form of an 

 equilibrated arch. It is not destitute of grace, and 

 is abundantly roomy for craft. 



Such, then, is the analytical theory of equilibra- 

 tion : for a practical subject it docs, we confess, ap- 

 pear abstruse. 



Those who have already studied the theory, will 

 observe, that we have greatly simplified the investi- 

 gation. The construction we have given for circular 

 arches we shall probably find useful hereafter. We 

 could with pleasure have prosecuted the subject far- 

 ther, not only as it affords borne good general views of 

 the equilibration of arches, but exhibits also several 

 beautiful examples of the application of the higher cal- 

 culus. Yet we must repeat, with all due respect to the 

 learned and eminent men who have turned their atten- 

 tion to it, that we fear their speculations have been of 

 little value. In saying this, we do not mean to sur- 

 mise, that their deductions are any way erroneous ; 

 they are legitimate consequences from the principles 

 assumed. But it appears to us, that the writers on 

 equilibration, like many others who have hastily ap- 

 plied analysis to physics, have taken too narrow a 

 view of their subject to comprehend all the variety 

 of practice. Setting out with one leading principle, 

 best adapted, perhaps, to the application oi" calculus, 

 they neglect the numerous circumstances by which 

 it may be modified, and which are too important to 

 be overlooked in drawing practical inferences from 

 such an investigation. 



Their principal care respects the figure of the sof- 

 fit, a thing which the practical engineer knows may 

 admit of the greatest variety. As to the thickness 

 of arcastones, side v.-aii, and piers, the horizontal 

 section or ground plan of the bridge, the manner of 

 filling up its haunches, of forming the joints, of con- 

 necting it with the abutiv.u.ts, wing wulk, &c we 

 arc still left in the dark. 



The analytical writers have assumed one leading 



Pi.tre 

 I.XXX 



Fig. 8. 



