THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER A407 
and on substituting in (18) and (19) 
M? = wr?(1 — 1-2728 sin 6), 
T? = wr?(15708 — 1:2728 cos 6 — 8). 
This makes M,=0 when sin O= a5g = 7850: 1.e. when 0=51°43’, and makes 
T,=0 when 0= 22°40’, and again when 0=90°. M, is a maximum when er =: 
ue. when cos 6=0, and therefore at the supports. T, is a maximum when ae 0; .€. 
de 
when sin 9= ‘7850, or when 0= 51°43’. 
Writing 0, = - in (21’) and substituting for T, and M,, the deflection at the centre 
ig given by 
___ 47272 , 053 
Y(centre) = 17 His Boo 
Experimental Verification. — An experimental semicircular girder, 10°1 ins. 
radius, loaded so as to give w= ‘0636 lb. per inch run, gave the following results :— 
Deflection (calculated) =°306 in. : (measured) = ‘310 in. 
A comparison of a straight encastré girder with a semicircular bow-girder of the 
same length / (=77r), each being loaded with w lbs. per foot run, shows the following 
results :— 
Bending Moment at | Bending Moment at | Distance from Centre, of 
Supports. Centre. the Point where BM =0. 
Bow-Girdr . . . 1014? — -0282wi? 2121 
Straight Girder . : : 0833 wil? — 04172? ‘2891 
§ 6. CrrcuLaR-Anc Bow-GirpER, SUBTENDING AN ANGLE (180-2¢)’, 
BuILT IN AT THE ENDS aND CarryING A UnirorMLyY LoapED PLATFORM. 
Let w lb. per unit area be the load on the platform whose area will be 
> a —2h—sin 2h . Imagine the latter to be divided into a series of strips parallel 
to AB, each of these strips transmitting its load to the girder at its ends. The length 
of the particular strip resting on the girder at points distant 6 from A and B, is 
2r cos (0+ ¢) (fig. 10). If this strip covers a length ds=7rd6 of the girder, its width 
is 700 cos (8+¢), and the load on it is 2wr? cos’ (0+ f)00. 
Its moment about AB = 2wr? cos’ (6+ ){sin(9+ ) — sin p} 88, 
Tv 
7? 
.. Moment of whole load, about AB = 2w7? ee (06+ $){sin (6+ ) - sin f} 80 
= Qaors { ORS SP (we — 24 — sin 24) \ 
TRANS. ROY. SOC. EDIN, VOL. XLVIII. PART II. (NO. 20). 61 
