CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
439 
we obtain 
and 
Uhl 
cb 
= ^2 
/dH\ _ 
\du^l - 
d-v 
d? 
d^vj 
Ju 
div 
- ? 
dz 
dwU 
dr j 
dvs-\ 
dr ) 
A, 
m + n 
dv\ V 
dr) a 
1 dw 
— -J 
a d(p 
dw 
dr 
A 
m + n 
- EK 
( 8 ), 
dho 
dz dr 
dma 
dr 
—+E "n 
ru + n dz dz 
+ ^- 
1 
_(fA 
a{m + n) d<^ a d(f> 
— E (\ “b 
[■ 
J 
(9). 
5 . We can now obtain the equations of motion in terms of the sectional stresses. 
Let dS be an element of the middle surface whose coordinates are [a, z, (f)), and 
dS' an element of a layer of the shell whose coordinates are (a + h', z, </>) ; then 
dS' = (1 + h'/a) dS. If we consider a small element of volume bounded by the two 
external surfaces of the shell, and the four planes passing through the sides of dS, 
which are perpendicular to the middle surface, we obtain by resolving parallel 
to OA, 
I (T^a S(jr) Sz + ^ (Ml 82 ) Scf> = p dS f ^ {u' - X) (1 + It/a) dli ( 10 ). 
But 
-h 
u' = u h' \ It 
du 
dhi 
dw- 
accordingly if we substitute the values of {dujdr) and {d^ujdr^) from ( 8 ) and (9) and 
recollect that all terms which vanish with It may be omitted when multiplied by h^, 
the right hand side of ( 10 ) becomes 
prfs{2A(«_X) + iA3Ef_|;^}. 
Resolving parallel to OB, OC, and then taking moments about OA, OB, OC, we 
shall obtain in a similar way five other equations, which, together with ( 10 ), may be 
written 
, 1 _ 
— ^ ... 
dz 
+ - = p i 24 (m - X) + |/i3E ^ J- 
a d(j) ^ L ^ ^ (i;^ 
1 dT^ , 
a dcf) a dz 
^ 1 _ Tj 
dz a dcf) a 
1 (ZGi f/Hi 
I , dK 21d (■• dw 
= p^24(*-Y) + -E- + g^^b-;^ 
.... 
= p\2h{iv- Z) + g; - EK 
a d(f) 
+ -*■ + N. = 
2ph^ /d^ 
3a \d(fr 
— 2 'y + Y 
1 f/H, -T o 7 Q . X 
T H-TT — No = — 4 pAO T-+ “ 
dz a d(f) \dz a a 
(11). 
(M.o-Mi)a-Ho = 0. 
