Arched Ribs of Uniform Section. 381 



verse sections, or, as it will be called hereafter, the axis of the 

 rib ; it is plain that this supposition, which is exactly true for 

 the weight of the rib, will in other cases produce no consider- 

 able error. Also let the stress on any section of the rib be 

 resolved into a shear F, a thrust H, and a bending moment M ; 

 and let it moreover be supposed that the shear and thrust may 

 be considered as concentrated at the centre of gravity of the 

 section. These suppositions being made, we have for equilibrium 

 of the small portion, by resolving tangentially and normally and 

 taking moments, 



B.d(f)=d¥+pds > 



¥d<j> +dK = qds, 



dM=¥.ds, 



where d(f> is an element of the angle which the section consi- 

 dered makes with a given section ; ds an element of the axis ; p 

 the normal pressure estimated per unit of length of the axis and 

 reckoned positive when acting inwards ; q the tangential pres- 

 sure per unit of length of the axis, reckoned positive when tend- 

 ing to increase <£. If p be the radius of curvature of the axis, 

 these equations may be written, 



d$ +P P = R > . 



dK _, 



dM 



d<f> 



p. Pp. 



Differentiating the first equation and adding it to the second, 

 we have 



whence 



F = K + F . cos (f> + H . sin cf>, 

 where F , H are arbitrary constants, 



TT ^F , «K TT I -O • f 



d6 +PP = PP+ ~m + ° ' C0S 0~~ °* sm ^ 



