BEAM OF RECTANGULAR CROSS-SECTION UNDER ANA SYSTEM OF LOAD. 
69 
where V = 2X/r/(\ + 2/x). Putting these into (6) and (7), we have 
A' 4- I _L 
+ ixV~JJ = 0 
(V J_ „) IfiE . 
(15) , 
(16) , 
(17) , 
(18) . 
(15), (16), (17), and (18) are precisely of the same form as the stress-strain relations 
and the body equations of equilibrium for two-dimensional elastic strain, with the 
exception that X' is written for X. They will in fact be found to be identical with 
the equations satisfied liy the displacements of an elastic plate under thrust in its 
owm plane, as obviously they should be, since, when the beam is made indefinitely 
thin, the mean displacements U, V coincide with the actual displacements u, v. 
§ 4. General Soitition of the Equations in Arhitranj Functions. 
If we 
dY 
write —- 
dx 
. . d 
d 
and 
d 
“h ^ ~r 
— 0 41 
- -J , 
dy 
dr] 
dx 
find 
^ + m = t X — iij — 77 , where i = 1, so that 
■ d d - . 
multiply (18) by ^ and add to (17), we 
“ t + 
Multiply (18) by i and subtract from (17) 
2 (y fx) fxV' (U — lY) = 0. 
But = 4 and if T = U + tV, W = U - iV, then 
Hence 
YU ^ 
dx dy 
<f 
(U + iV) + (U - V) 
d^ + dy • 
