214 



Tllli CIVIL ENGIx\EH.R AND ARCHITECT'S JOURNAL. 



[July. 



though felt, and to « certain degree acted upon, has not hitherto admitted 

 uf a clear explanalioo. 



The annexed diagrams, Hgs. 18 and 19, exhibit forms of arches, supposed 

 to be loaded w.th a material of equal weight with the aroh, and show the 

 abiitn.euts necessary to sustaiu them. In these, it will be observed that 



Fig. 18. Fig. 19. 



the division between the arch and the abutment is not made where the 

 arch, so called, commences. The point of division adopted, has not been 

 chosen on account of the result as to the required thickness of abutment 

 being materially affected by it ; but as being the place at which rupture 

 would ensue, if the abutments yielded. In fact, the effective part of an 

 arch is only so much of it as would not stand unless the arch were entire. 

 So much of the arch as lies below this point, would stand of itself, and is 

 practically a part of the pier or abutmt-nt, curved out for the arch to spring 

 from. This point of division also permits a readier means of computing 

 the thickness of abutments; as with the exception of the small projection 

 at the springing, the weighi of which may be omitted, the abutment is a 

 rectangle, when the roadway is horizontal. 



In this manner the following formula is derived, for ascertaining the 

 thickness of the abutments necessary to support a given arch ; the height 

 of the abutment being given. 



KVC 



Kig. 20. 



If B EF (lig. 20) represent the half arch with its backing, and G is the 



centre of gravity— assuming ;; and ;/ to be two points of application of 

 the pressure, or points in the curve of equal horizontal thrust in the arch — 

 let;;/=d, kf=h, p q — m, p r = n, A U = a, «• = weight or area 

 of arch and backing, and x ^ D C the width of abutment sought ; then 

 assuming equal areas to produce equal weights, ax will be the weight of 

 the abutment ; and when x is such, that the curve of equal horizontal 

 thrust will meet the base of the abutment at the extreme point D, we 

 have 



w (j- + n) wd 



2 m + 



_ «• + 



Vc"-) "+(-:)' 



Also to find the thickness of abutment when any given additional load 

 is placed on the arch, let U be the load, expressed in terms of the area of 

 the arch and backing, and s its horizontal distance from the point p; 



then 



w + 



/ f (Bs + ivd)m > 2 /w \' 



Id like manner, when the arch and abutment are given, we can find the 

 extreme load which may be placed on the half arch. 



Ueferring again to fig. 20, let G K be a vertical line passing through the 

 centre of gravity of the arch and abutment, and p'l — H, D K = D, 

 •S = horizontal distance of load from the back of the abutment, and 

 W = weight or area of arch and pier ; then using the same letters as 

 before lor the other dimensions — 



\V I) , ^S _ u-d Bs 



H "•" "H " X "*" h 

 hWV — Hwd 



or B = — ;-; 7 



H J — A t> 

 In these cases, the two half arches are assumed to be loaded alike ; 

 hence, when the load is at the crown, the result must be doubled, to give 

 the entire load. As regards the positions of the assumed points /ip', it is 

 sufficient, in an arch of large dimensions, to take them in the centre of the 

 thickness. Though the extreme limit, theoretically, in an arch turned in 

 one ring of voussoirs, is when the points p p' are in the line of resist- 

 ance. 



In offering the foregoing as a practical outline of the laws which govern 

 the equilibrium of arches, it must be observed, that the most simple condi- 

 tions consistent with practice, have been assumed as data, and so far as 

 these conditions can be fulfilled, there is no doubt that the principles here 

 set forth will be fully borne out in actual execution. How much further 

 the inquiry might be carried with advantage, it is difficult to say : but there 

 appears to be much connected with the unequal loading of arches, which 

 has not hitherto bei'U the subject of investigation. 



Moseley has introduced in his researches, the effect of the adhesion of 

 cements; but he has accompanied it with the remark, that " that structure 

 (being of large dimensions) which would not stand without cement, would 

 assuredly be a perilous one," a remark which applies very properly to 

 arches of masonry; but in brick arches, turned in numerous rings, the ad- 

 hesion of the cement undoubtedly becomes an element, materially affecting 

 the stability of the structure. Upon such subjects, and upon the varying 

 conditions in which arches are placed, it is in vain to attempt to bring 

 theory to bear. They are considerations which must and ought, at all 

 times, to be left to the skill and judgment of the engineer. 



In practice, an arch will exert more pressure and resist less than theory 

 would denote ; because the conditions of unyielding materials and mathe- 

 matical adjustment of the joints, are incompatible with practice. Even in 

 arches of the hardest stone, and with the best workmanship, the lines of 

 resistance and impression must not be brought too near the extremities of 

 the voussoirs ; and in brick arches, particularly those turned in separate 

 rings, a much greater latitude must be allowed. It must be evident, how- 

 ever, that it is desirable to form brick arches as much as possible in one 

 bonded mass, using the best cement. 



Id abutments, a still greater variety of considerations will arise. To 

 render this part of the subject tangible by theory, the abutment must be 

 assumed as standing alone, the foundations being perfect, and the point of 

 rupture being at the base of the abutment. In practice, they are rarely, if 

 ever, without earth behind them, aiding more or less in their support. 

 Some cases, such as in arches under embankments, the force acting to push 

 in the abutment, exceeds the horizontal thrust of the arch, and a tendency 

 has been frequently exhibited in arches so situated, to rise in the crown. 



The foundations, the wing walls, the spandril walls, the backing, the 

 nature of the materials employed, and many other practical considerations. 



