THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 203 
(iii) Permanent flexural mode. 
The solution is given by (101) with 
3 1 
= STeple zea - work diffee. from O= —d=zlogp. 
Z=0 
pp = 2m(a + 1)z/p? 
= -d=21 ’ == a+1)z 
In the system Baa’ ae ie 
Hence the solution balancing u and % at p= is 
~ 
u= (a+ 1)zp/a? zp = 
pp = 2u(7 — a)z/a? 
w= —(a+ 1)p?/2a2 + (a — 3)22/a? 
In the balanced solution, at the edge 
w=(a+1)(loga—4)+(a—3)2?/a?; pp =16p2/a? 
The constant term in the value of w will disappear since | * dz=0. Thus 
ye eee | Fe(a~ 3) - Bu \ ae 
pam oDane Ty 5 On 5 
and 
U(py  %) = = %P, eGR ony ; : 
t0(P1 5%) = 5 Pr + : Set rea} “1 ey aa a 
(iv) Transitory flecural modes. 
The solution is given by (108) with 
= J okpy . : = Goxp sinh xz 
ins om T)x@h(cosh 2x = 1) work diffce. from the system pe 2 Goel aod: 
Tn this system 
= «Gy kp{(a — cosh 2xh)sinh xz+2x«z cosh xz} 
= KGoxp{ - (a+ cosh 2h) cosh «z+ 2x«z sinh xz} 
= G) «p{(1 — cosh 2«h) cosh xz + 2«z sinh Kz} 
II 
Go«p{(cosh 2«h — 3) sinh xz — 2«z cosh xz} 
eee G, «p{ (a — cosh 2«h) sinh xz + 2«z cosh xz} 
Kp 
The system balancing u and zp at p=a is 
¢=- pane J xp sinh xz, 9= —cosh 2xh- ¢. 
0 K 
In the balanced system, at the edge 
w { — (a+ cosh 2h) cosh «z+ 2x«z sinh xz i 
Fe ad, Ka | 
an = Te { (cosh 2«h — 3) sinh xz — 2«z cosh xz ; 
ap a 0 ka J 
Hence for the free solution with edge values wu =u, , = =Z 
a9 
1 Jax, he 4, 2) Je 
F, = oKPy Lape : 
2 ~ 2(a+ 1)h(cosh 2xh — 1) ial Duh a.+ cosh 2«h cosh xz + 2«z sinh xz) hy. 
-n \ +x«u,(3 — cosh 2«h sinh xz + 2xz cosh xz) 
If the given values of Z,, u, are the same as the edge values of z , u, in one of the 
particular solutions, then clearly this particular solution by itself is the solution, and the 
