396 PROFESSOR A. H. GIBSON ON 
§ 4. CrrcutaR-Arc GIRDER, Bott 1n at Two Enpbs, wits Sincere Loap W. 
Let the arc subtend an angle (7—2¢), and let O (fig. 4) be its centre; AB the 
line of supports; AOW=a; BOW=6; R, and R, the vertical reactions at A and B; 
M, and M,, T, and T, the bending and twisting moments at the supports A and B, 
the axes of these moments being respectively parallel to and perpendicular to OA 
and OB. 
Fie. 4. 
The bending and twisting moments at any point between A and W, distant 0 
from OA, are now given by 
Me=M,cos6—R,rsind+T,siné . 3 : : ; . (4) 
Te=T,cos6+R,r(1 -—cos6)—-M,sin6@ . : : . : « ©) 
while the moments at a point between B and W, distant @ from OB, are given by 
similar expressions, with suffix b taking the place of sufiix a. 
Before these moments can be calculated for any particular case, the values of 
the six unknowns, M,, M,, T,, T,, Ry, Ry, are to be ascertained; and for this, six 
relationships between these unknowns are necessary. 
Taking moments about B, of the forces and couples acting in a vertical plane we 
have, for equilibrium :— 
R,,(2r cos ¢) — T,, cos # — M, sin d — Wr{cos ¢ + cos (a+ ¢)} + T, cos$+ M,sind=O0 
k= 1 4 Sa ys SS ; : 
aD) a ate: tir or a 8) 
while 
Wi = cos (a+¢) T, ia ae M,- M, 
B= 3-4 1 ere } + = + = tan : ; F re) 
Again, taking moments about the line AB, 
(M,+M,) cos @—(T,+T,)sin@d=Wr{sin(a+¢)-sing}  . : : . <8) 
