CHAP. XTV.] THE BEACED ARCH. 269 



Thus, for a load of w per unit of horizontal length, reaching 

 from left end to a point whose angle from vertical through 

 crown is ft (Fig. 92), a being the angle subtended by the half 

 span, we have * 



H = ^? [X sin 13 - ., p + & 3 + sin 8 /3], 



where R is radius of arch, and 



7 2 sin a , sin a cos a , sin 2 a 



k = a + sm a cos a - . \ = -- h s , 



a a 2 



y sin a 7 sin a r -\ 



K* = , fa = ; a sin a cos a , and 

 4a 4a L 



^ = /3 -|- 2 /3 sin 2 ^ + 3 sin ft cos . 



/ 



For V we have 



y_ R T cos a sin 8 ft sin ft K 3 cos a cos 8 a . "I 



L2 (a sin a COS a) 2 a sin a COS a 6 (a sin a COS a) J' 



,, cosjS 8 sin 8 cos s y9 

 where K 



. 



For a concentrated load P for any point [Fig. 92], we havef 



or, more correctly, 



_ _ p a ff sin a cos a sin ft cos /3 + 2 cos a sin ff 



2 (a sin a cos a) 



For the semi-circle, this becomes 



_ _ p TT - 2 ft 2 sin ft cos ft 



27T 



For H we have 



_ 2 sin a [cos /3 cos a + (1 + K) /3 sin /3] (1 -f- K) a (sin* a + sin* /3) 

 2 [(1 4- K) a (a + sin a COS a) 2 sin 2 a] 



* Taken from Capt. Bads' Report to the llUnois and St. Louis Bridge Co., 

 May, 1868. 

 f Die Lehre wn der Elasticitdt und Festigkeit. Winkler. Prag. 1867. 



