BEAM OF RECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 67 
then the mean values TJ, V taken across the breadth of the beam of the displace¬ 
ments u, V in the plane xy are found to satisfy two dilferential equations of the same 
form as the equations of elasticity wdien the displacements are independent of 2 
and IV = 0, with this change, that the elastic constant k is replaced by another 
constant /V. The mean stresses in the plane of xy are found by differentiation from 
U and^ V by similar formula to those giving S, in terms of u, v for two- 
dimensional strain. 
Now the distribution of such mean stresses inside the beam is independent of the 
ratio \ :ix. This has been shown by Mr. J. H. Michell (‘London Mathematical 
Society s Proceedings, vol. 31, pp. 100-124). It liad been previously pointed out by 
Stokes (‘ Phil. Mag.,’ Ser. V., vol. 32, p. 503). The equations being of the same form 
in problems (a) and (b), there follows this curious result, that the distribution of stress 
inside the beam, consequent upon a given distribution of stress upon the upper and 
lovmr faces (this latter distribution being uniform wdth regard to the breadth of the 
beam) is the same when this breadth is very small and when it is very large. 
§ 3. EstahJishrnent of the Equations. 
The centre of the rectangular beam being the origin, let its axis, which is supposed 
horizontal, be taken as axis of x. The axis of y will be vertical and the axis of 2 ; 
horizontal. The bounding surfaces of the beam are x = ^ a, y z = ^ c. 
Using the notation explained in § 1, equations (2) may be written 
dxx dxy dxz 
dx dy dz 
= 0 
• . . (3), 
do:y dyy dyz 
dx dy ' dz 
= 0 
( 4 ), 
dxz , dx/z . dzz 
7 ' + .7 ■ + = 0 
dx dy dz 
(5). 
Integrate eqmtions (3) and (4) with regard to 2 from - c to + c. Then, noting 
that are both zero, owing to the surface conditions at the side of 
the beam, and also that integration with regard to 2 and differentiation with regard 
to X and y are independent, we find 
= 0 , 
= 0 . 
d 
C + c ^ 
d r 
*-J-C -V 
dx 
I 
1 
+ 
^ ; ' 
t - 
xy dz 
d 
dx 
~r+c^ —^ 
xy dz 
+ 
I 
I 
1-- 
'r+c^ 
j yydz 
^xxdz — 2cP, I yy dz = 2cQ, | xy dz= 2cS, then P, Q, S 
yy 
K 2 
Now* if we write 
