1912-13.] 
Plane Strain in a Wedge. 
293 
(2) It will at once be seen that, for the type of loading considered, the 
shear on the plane of xy vanishes everywhere and we have 
= e 2 * = e Z2 = 0 ; 
and thus 
yz = zl = 0 . 
The general stress equations of equilibrium are 
_* x . + _^ + pX = °, ^ 
^xi) + ^ + P Y = 0,\ 
which, if the body forces are omitted, are identically satisfied if we put 
xx = 
¥ 
&X 
dp 
d\ 
( 2 ) 
(3) 
(4) 
where y is some function of x and y only. 
The stress equations in cylindrical co-ordinates are 
i a. ^ S / 
_ ( rn .) + _ ( r £) — = o , 
r or r 00 r 
1 ^(ee) =o, 
r 2 dr v r dO 
j 
and it has been shown * that the stress components rr, 00, and rO, when 
expressed in terms of a single stress function y, are 
1 0y 1 0 2 y 
rr rdr + r i d6 i ’ 
dr\r dO 
(5) 
(3) Solutions of equation (1) which will be used are as follows : — 
y = A sin 2# -I- B cos 20 + C0 + D . \ 
y = r[( A + B0) sin 0 + (C + D0) cos 0] . 
y = r 2 [A sin 20 + B cos 20 + CO + D] . 
y = r n [A sin nO + B cos nO + C sin ( n - 2)0 + D cos (n - 2)0 ] . , 
(4) To represent a moment of + M (see fig. 2) at the angle of the wedge 
( 6 ) 
put 
y = - M(sin 20-2 cos 2a0)/2(sin 2a - 2a cos 2a) 
= - M(sin 20 - 2 cos 2a0)/2m , say , . . . . 
■ (O 
giving stresses 
rr = + 2Mr -2 sin 20/m , ^ 
00 = 0 everywhere , 
• (8) 
rO= + Mr~ 2 (cos 20 — cos 2a)/m.J 
1 
* J. H. Michell, Proc. Lond. Math. Soc., vol. xxxi. p. 111. 
