68 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
are the mean values of the two tractions and of the shear in the plane xy —taken, for 
any values of x, y, across the breadth of the beam. These will in future he referred 
to as the mean stresses and often, for shortness, as the stresses. 
We obtain, therefore, the equations 
r7P , dS 
7 "^ 7 — 
(U ay ^ 
■ . . ( 6 ). 
Now consider equations (1), namelj’ 
XX = \ 
(IQ 
d r + ^ 
( 0 - 
yu 
\ 
[dn dv\ dll div 
^ A'-' ■ ■ ■ ■ 
• • • (8), 
fdu , dy\ dv dw 
(*+ + ■ • • ■ 
. . . (9), 
/du . dv\ , ^ \ dio 
• ■ ■ ■ 
. . . (10), 
'—' fdu dv\ 
- 0 [dg + ,u) . 
. . . (11). 
If we integrate (8), (9), and (11) with regard to 2 from — c to + c, we have 
Q = \ 
clc (ly 
cW 
fw+c — W-Q 
_ /dU cll\ 
( 12 ), 
( 13 ). 
(I’l). 
1 1 
where U = udz, V = vdz are the mean displacements in the plane of 
-C J Ic J _c 
xy taken across tlie Ijreadth of the beam for any point [x, y). They will be referred 
to as tbe mean displacements. Besides these there is a variable {u'+c — u'_c)!'2c 
which has to be eliminated somehow. 
One way of doing this is by integrating (lO) in the same way. We obtain 
1 
to. 
VJ 
Now if, as explained in the last section, 22 may be treated as small, so that its 
mean value across the breadth of the beam may be neglected, we have 
_x _ /du w\ 
X + 2^ \di.' cly / 
Substituting for {u\c — w_Qj/'2c, the equations for P and Q become 
