THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 413 
On substituting these values, each of which is given in terms of P, in (27) and 
equating to (26), the resultant expression contains P as the only unknown and enables 
this to be calculated. 
L.g., Sencireular Girder with uniform loading and with two piers at 60° from 
the ends of the span. 
From fig. 7 the values of M,’, and T,’ for substitution in (26) are M,’=wr’; T,’ = 
‘298wr"; while R,’=1'5708wr, and, on substituting, the downward deflection at the 
supports (y=60°) is given by— 
564 oy 
Aol Y OCT 
The values of M,", T,", and R,” for substitution in (27) are, from figs. 5 and 6 :— 
M,” = (M, + M,)o=0, a=eor= (‘588 + -278)Pr = -866Pr. 
T,” =(T,+Ty'9=0, ¢=o0 = (155 + -127)Pr=-282Pr. 
R, =P, 
and, on making these substitutions, 
2) (Pee 
SE SOBs OCF 
Hquating these two expressions for Yo gives 
P = wr] 2640) + oad 
‘539CJ + 035EI |’ 
and taking EI = 1°24CJ, this makes P = 1-05wr. 
The reactions at A and B are then given by 
R= See (5 : 1-05) = iin, 
while the moments M, and M, are given by 
M,=M, = M,'— M,"= wr(1 — 866 x 1:05) = 091 wr?. 
The torques T, and T, are given by— 
T,=T,=T, - T,” = wr? {298 — +281 x 1:05} = -003wr. 
The state of affairs at any point on the girder is thus given by the relations :—- 
Between A and C— 
Mo=M, cos 6 — (R,7 — T,) sin 8 = wr?(-091 cos 6 — -518 sin 6). 
To=(T, — Ru) cos 6+ Ryr — M, sin 6 = wr?(521 — 518 cos 6 — ‘091 sin 6). 
Between C and the centre (9 being measured from OA)— 
Mo =M, cos 6 - (R,r — T,) sin6 + Pr sin (6 — 60°) 
= wr" (007 sin 6 - 819 cos 8). 
T.=(T, — R,r) cos 6 - Ryr - M, sin 6 — Pr{1 — cos (6 — 60°) } 
= wr (-007 cos 6 + °819 sin 0 — 529). 
