398 PROFESSOR A. H. GIBSON ON 
as the deflection at a point distant 9, from A. On integrating and simplifying, this 
becomes— 
ee 6, sin 6, — (Rar - T,) (sin 8, — 6, cos0 | 
Yo.= (12) 
ale _ Ryr)(sin 8, — 6, cos 6,) + 2Ryr(d, — sin 6,) +M,(6, sin 6, + 2.cos 6, -2)| 
Similarly for a point between B and W, distant 0, from B, 
| 
Jal M 6, sin 6, — (Ryr — T,)(sin 6, — 8, cos 6 | 
Yo, = 7 (13) 
+ yl (To - - R,r)(sin 6, — 6, cos 6,) + 2R,7(A, — sin 6,) + M,(6, sin 6, + 2 cos 6, — 2) | 
At the weight, 6, becomes a in (12) and 8 in (13) and these expressions give 
(A to W)— 
al Mu sin a — (Rr — T,)(sin a — a cos a) | 
yw = ‘ . (14) 
+ aay (Os — R,r)(sin a — a cos a) + 2R,r(a — sin a) + M,(a sin a + 2 cos a — 2) | 
and (B to ww) 
ses: sin 8 —(R,r - T,)(sin B — B cos p) | 
Yw= 5 ae 
yi aes] (" — R,r)(sin B — B cos B) + 2R,r(B — sin B) + M,( sin B + 2 cos B — 2) 
On equating the identities (14) and (15) the final relationship is obtained, and 
from the six equations (6), (7), (8), (9), (1O0=—11), (14=15), the six unknown 
fixing moments and reactions may be determined in any particular case. These 
moments depend somewhat on the relative values of EI and of CJ, except where the 
load is in the middle of the span. Comparatively large differences in the ratio EI : CJ, 
however, only produce small changes in their values, particularly when the angle a is 
large. An increase in this ratio is accompanied by an increase in both bending and 
twisting moments at the ends. The alteration in the fixing bending moment, accom- 
panying any such change in this ratio as is likely to be found between extreme 
cases in practice is very small—in general, not more than 1 per cent. The alteration 
in the fixing torque is more pronounced, but as this is always a comparatively small 
fraction of the bending moment at the supports, the effect on the resultant moment 
is very slight. 
In order to facilitate the application of the results of this analysis, and to make 
it more useful in practice, the foregoing equations have been solved for a series of 
values of a and of #, and the values of the end moments and reactions have been 
calculated for the case where E[=1:24CJ. For purposes of practical design these 
values may be taken as being substantially accurate for any other probable value 
of the ratio. 
