THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 403 
Knowing M, and 'T,, the deflection at the weight may be obtained by substituting 
these values in equation (14). 
Hapervmental Verification.—The experimental girder (fp =30°,7=10°10 inches) 
was loaded with a central weight, and the deflection measured. Under these conditions 
the measured deflection for a weight of 1 lb., and the deflection calculated from (14) 
with M,=°3145Wr; T,=°045W7, were as follows :— 
Deflection (measured) = °0747 inch ; (calculated) = 0743 inch. 
§ 5. Crrcutar-Arc Bow-Girper, Buitr in At Born Enps, witH 
Unirorm Loaping—w ups. PER Unit LENGTH. 
Let 7 — 2«p be the angle subtended by the arc (fig. 4). The total load = wi(a — 2¢) lbs. 
2, = lt, = w= - #). 
The centre of gravity of the load is at a distance from the line of supports given by— 
; 7 aval on one-one 
p 
Tv m 
oes Feo) 
a zi 
Let M,, M,, T., T,, have the same meanings as before. Then, from symmetry, M,=M,, 
T,=T,; and, on taking moments about the line AB :— 
2M, cos ¢ - 2T, sin $= 2wr? \ COS b — G- -¢)sin d \ 
°, M,=wr? 1 =e —p-—4 tan | b le 
Taking the origin of ¢ at the supports, 
Me=M, cos 6— R,rsin 0+T,, sin 6 + wr?(1 —cos 6)* 
= (M, — wr?) cos 6— (R,v — T,) sin 6 + wr? : : : (18) 
T,=T, cos 6+ R,7(1 —cos ae M_ sin 6 — wr?(6 - sin @)* 
= (T, — R,r) cos 6 - (M, — wr?) sin 6+ Ry — wr? : : 3 : (19) 
If the girder is fixed horizontally at the ends, 
8; 
rh 
(3), = =5 a M, cos (6, — 0)d6+ ae sin (6, — 0)d0 
0 
and, on substituting for M, and T, from (18) and (19), this gives 
a (Me — wr?) {, cos 6, + sin 6,} - (Rur —'T,)6, sin 6, + 27? sin 6s | 
(aa), | : 
dO/e, ie (T. — Ryr)0, sin 6, - (M, — wr?) {sin 6, - 6, cos 6, } ; ap 
2CJ + 2R,r(1 - cos 6,) — 2wr?(6, — sin 6,) 
* The last terms, representing the moments due to the portion of the load between A and 8, being obtained as at 
the beginning of § 3. 
