MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
S = sin mx 
(A' + C') sinh my + (B' + D') cosh my 
+ ‘Imy (A' cosh my + B' sinh my) 
r (E' + G') sinh my ~ (H' + F') cosh my '| 
+ cos mx < . > 
-j- 2my (E' cosh my — F' sinh my) 
• (^ 8 )> 
\’fhere A' = ~ ^ B' = — B E' = -- FF+b F' = ™ F1±jA b 
2 V + 2/a ’ 2 V + 2/a ’ 2 V + 2/a ’ 2 V + 2/a ’ 
C' = 'lymC, D' = 2fjLmD, G' = 2/AmG, H' = 2 /awH, and the expressions for the 
mean displacements come out to be 
1 r V + 3 /a 
El = sin mx 
Mfl (_ A. "f" /A 
- (A' cosh ??2 ^ + B' sinh my)-\-^' cosh my + D'sinh my 
-h cos mx 
+ ^ - y (A' sinh my -f- B^ cosh my) 
1^ 
I ^7“^^ (E^ cosh m y — F' sinh my) + G' cosh my — sinh my j 
V 
+ ■ (E' sinh my — F' cosh my) 
(29). 
V= cos mx 
2 — (A'sinhmy + B'coshmy)— C'sinhTiiy — D'cosh97iy| 
+ sin mx 
V 
— ■ (A' cosh my + B' sinh my) 
^ni I A'F- (“E^sinh7^^y-hF'coshmy)AG'8inh7?^y —H'cosh?7iy I 
+ F ^B' cosh my — F' sinh my) 
- ^ * 
(30). 
§ 6. Determination of the Arhitrary Constants from the Stress Conditions over the 
Faces y = di 
We shall sujopose that the mean stresses Q and S are given arbitrarily over 
the top and bottom surfaces y = Aii>- Expanding these in Fourier series, we 
have, say : 
[Q]y=+J = “o + COS mx + Sy,, sin mx 
[Ql/=-j = /3o + cos mx + SS„ sin mx 
[S]^=,+i Co + FG cos 7)ix + S/<-„ sin mx 
= ^0 "F cos mx + sin 7nx 
where a„, y„, S„, C», k,,, v„ are known constants, and m = niTja where n is any 
positive integer. 
Now, if we take expressions (27) and (28) and equate them, for y = d: &, to the 
