THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 187 
The resultant is a couple in the plane wy, of magnitude 87uh, and we observe that 
the solution occurs with coefficient — Q,0,/87uh. 
IJ. The stresses in the general flexural solution (94) are 
aaah pe” 
2p dp 
pe. w 5 
Da a F : : : , (95) 
# = 92-12) 4 oR 
2m 
LU. i pp _a—T. p Gael a+) Do \ a= 
(i) et 8 210g 4 ( 6 23 — 2hz |p 
po = 0 
PE = (2 - H2)p} | 
The resultant is a force along Oz, of magnitude —“s7pi 
II. (11) eRR es sl ~~ zp"! cos w — (2 an oY ®— ahs )p —3 cos w 
2p 2 6 
pn = ae zp! sin w — eee - 2n?2\p-8 sinw + 
PF = 2(2—h*)p~ cos w 
2p. 
The resultant is a couple about Oy, of magnitude 
| i {2(pp COS w — pa Sin w) — p COS wpz}pdwdz, taken over the cylinder p, 
32 
= = mph 
f I. (11) PP ee zp} sin w — i = 2102) 3 sin w 
2p 2 6 
| be = avl cOs w + oe 9 3 2122) COs w 
2p 2 6 
oe = 2(2—-h?)p“* sin w. 
13 
The resultant is a couple about Ox, of magnitude 
[ [4 -2esin w+ pw COS w) +p sin w+ p2}p dw dz= — mpl 
31. Conditions for the existence of a solution with finite potential energy. 
Elastic equivalence of statically equipollent loads. 
The corresponding results for any distribution of body force, or of traction on the 
faces of the plate, may be deduced at once from the above by integration with respect 
ee, 2 OF p,, ©, with 2; == h. 
If the region within which the force is applied be entirely enclosed by a cylinder 
p=az, the results are valid for all pots exterior to this cylinder. 
tl Seer ee = 
