. XV.] THE STONE ARCH. 329 



than those obtained by these tables, as it is assumed in them 

 that the point of application of the horizontal thrust is at the 

 upper edge of the abutment. Thus in every respect the con- 

 struction gives results reliable and even more accurate than the 

 tables, as we take the arch as it really is in any given case, while 

 in the tables suppositions are made with reference to spandrel 

 filling, etc., which do not hold good for every case. 



1O. Proper Thickness of Arch at Crown. The proper 

 depth of the arch at the key depends not only upon the rise 

 and sp^n, but also upon the load. The pressure at the extrados 

 at the key, which is, in general, the most exposed part of joint, 

 should not, according to the best authorities, exceed ^V^ 1 the 

 ultimate resisting power of the material. If P is the pressure 

 per unit of surface, H the thrust, and d the depth of key-stone 



O JJ 



joint, then P = , 



since, on the assumption that the curve of pressure does not pass 

 outside the kernel, the maximum pressure is twice the mean 



jj 

 pressure . This mean pressure, then, should not exceed ^th 



d 



the ultimate resistance of the material. In the best works of 

 Rennie and Stevenson the thickness at key varies from -^th to 

 /g-d the span, and from -g^th to -^th the radius of the intrados. 

 The augmentation of thickness at the springing line is made 

 by the Stevenson's from 20 to 40 per cent., by the Rennie's at 

 about 100 per cent. 



Perronet gives for the depth at crown the empirical formula 



d = 0.0694 r + 0.325 meters, 



in which r is the greatest length in meters of the radius of cur- 

 vature of the intrados. 



For arches with radius exceeding 15 meters, this gives too 

 great a thickness. According to Rankine, 



d = 0.346 tfr 

 for circular arches, and 



d = 0.412 |/r, 



where r is the radius of curvature of the intrados at the crown 

 in feet. 



" The London Bridge is in its plan and workmanship per- 

 haps the most perfect work of its kind. The intrados is an 



