THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 409 
As before, if the girder be fixed horizontally at the ends :— 
cal 
d ye ' 
lake EI [ cos (0, — 6)d0 + fn sin (6, — 0)d6, 
and, on substituting the foregoing values of M, and T, and integrating, this gives 
the value of a at any point 6, Thus, 
dé 
. ~M,(6, cos 6, + sin 6,) — (Ry — T,)9, sin 6, 
2EI — = 4 sin 6 if —sin? 6{3 + 6,}) — 6, cos 0,(1 + sin?¢) — ; sin 2 (6, + $) t 
(:2) 4 "('T, — Ryr)6, sin 6, ~ M, (sin 6, — 4, cos 6,) + 2R,,r(1 — cos 6,) mnicys 
LO} 6, 7 erin) le cost oe 2h sin 2g | cos @,cos2p 4 eee : 
+ 3 6 om 9° 
205 | _ ays 
+ 6, cos 6,(cos® ot me = 3) + meee, sin 0, — sin? 6, + cos 6, — 1) 
From symmetry the slope is zero at the centre of the beam where 0, = are and, 
on substituting this value for 9, in (24), and also substituting the values of M, and R, 
as givenon p. 408, and equating to zero, the value of T, may be obtained. 
E.g., Semicircular Girder (= 0). 
In this case, on putting ¢=0 in (24) :— 
“M,{6, cos 6, + sin 6,} — (R,r — T,)0, sin 0, 
- 
5m PA NO | 
PADI +r {TF sin 8, - cos 6,(2 sin 6, +6) | 
fe |. 
Ra = 3 (24) 
db) , | (Ta- Rur)6, sin 6, — M,{sin 6, — 6, cos 6,} 
re 
ieee 
203) + 2R,7(1 — cos 6,) — wr? | 6, — sin 0% - ; cos 6) + aa cos 6, \ 
_o : wr 9 7 
At the centre, where 0, =o the slope is zero, and M.=-~- > R,= wr. vs 
wr ll (© see U 1 aN ee 
ae os GE 5 )9 +5 | tau (ae e-3t 9 [7° 
t.=(G- =.) = = ‘078wr?. 
Orr 
It follows that, on substituting in (22) and (23) :— 
au Easin2 
Me= wr? { 3 — "7074 sin 0 \ 
Ty= wr? | oe 7 _-7074 cos 6 — 
\ 
j 
The deflection at any point 4, is obtained by writing 4, = in (24’) and integrating 
between the limits 9, and 0. Thus, 
Sieg 
