622 PROCEEDINGS OF SECTION H. 
We have since the bending moment is proportional to 
the change of curvature 
ar — 8 yet n)— «(5 —°) (4) 
T = Ex 2x f= 2Et xe(u + sy) (5) 
where 2¢ is the thickness of the arch ring at P (not 
necessarily uniform) and f is the elongation of the element 
ater: 
oS aees 
Tl = #t 
E = Young’s modulus for the material 
u == normal displacement, outwards, at P 
» = tangential displacement towards the right at P. 
dv 
From (4) and (5) calling z= + v we have 
oy 
i Meise ine HS be) Ogee Ne 
TAPING SEAS SY ) =} oi 
From (1) & (2) eliminating 7 we have 
aoa 
yet la-s (8) 
A Particular Integral of this is (Forsyth, Diffl. 
Equations, pp. 86-7) 
To = fcp)= cosy fy f (-*) dosdeuy (9) 
So that the complete solution is (A and B being 
arbitrary constants) 
T=Acos~y+ Bsnd+ fi) (10) 
From (1) & 10) L= Asiny — Beosy — f’ WW) (11) 
From (3) & (11) M= fe dy 
K 
= A|siny ds— B| cos ds — | f” (~) ds + C 
or, M =~ Ay—Br+C—f f'(p)ds (12) 
