THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 
This equation gives the result for an areal distribution X,Y, Zon z=2. 
181 
For 
traction on z==th replace 2 by +h ; for a volume distribution X , Y , Z replace 
hi 
Z by | LZ(a , y, @)de! 
=i 
(B,D) wy 4 
*\ da * dy Y dx 
and so on, 
27. Extensional strain. 
Ti d he 
' , 1 IW ae ’ 
[2X@.y, 2a + [2X9 eae 
Differential equations of the principal mode. 
The unambiguous extensional terms of (83) remain to be considered. 
= 1 r , Pe: , 
Write Esa ahi [xe y’, Z)x(R)dx'dy’. 
Then for an areal distribution X , these terms are 
y= -4(E-4$2y°E) + 4(42? +3) VE 
, 99 2 +5, 5- 9 FETE Sao ~ 
= —(E- fey" agra} (= Bap = a + a+ Ih ) vk 
; as 2 aon, oo a , 
g'= -(E-#'E) + oy (SP +g ) vE 
The second parts of these expressions give 
"9 d? Oe PD ") 9 . 
w= (a—3) (42?- fh?) Tay'E + 4(2? + 3h) VE | 
2 
; Cate 
v=(a-3) (2-0) paVE | 
w=0 
The first parts give 
(ID ONS ys ae ace aa 
ge ge | a ae Se) 
ad) Gun a@=s dd? _5 
O— tay” dedy = = «2 dedy'™ 
d _» 
= = ee E 
w (3 -—a) Cir Ni 
: 3 1 
K= ern Oo , , ,! sp 
If now further we write _ 32h | i Y(@’, y, 7)x(R)da'dy 
the corresponding displacements for a distribution of Y force on z=z’ can at once be 
written down from symmetry. The results for X and Y force combined cannot con- 
veniently be expressed in terms of one function, as in the case of the flexural mode, 
and the best plan is probably to put everything in terms 
u,v, namely, 
ee ak CK 
U= -(atl)-5 -8 4 - 
ce ae dy? Cae oie dy 
aE aE PK 
V=—-(a+1)——- +8 
ps: ii eles 
dady  dady Oe yap 
of the principal values of 
aK 
dx dy 
aK 
= ete 
dx? 
(88) 
