100 
MR. L. X. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
P 
= — irh 
'{F„ - G.) - I i (F^ - Gs) 
Trh 
•391 - - 2-005 
u~ 
_ AV 
= ~ ~ irh 
1-444 - f. 3745 
we have therefore at the origin a horizontal tension and vertical pressure. These 
vanish when x = ± -1955 and x = ± •386/> respectively, assuming that for these 
values of x tlie first two terms are a sufficient approximation, which is certainly true 
_ - 195 ^^ only roughly true for a- = •386/>, as it amounts to neglecting 
terms of order about y compared with I‘44. It will, however, be sufficient foi a 
roimli estimate. 
The actual stresses at the origin are :— 
p _ . ('249), or about y of the tension due to W acting along the horizontal, 
'2h ^ ' 
Q _ _ ^ (-920), or about iV'hs of the pressure due to W acting along the 
horizontal. 
If we had used the expressions Pj, which hold for an Infinite solid, we should 
2AV At" 
find; at the origin, P = 0, Q = — = — — (U273). • 
If we correct tlie last hy Stokes’ empirical rule, we have to add — y [0 + (stre.ss 
at liottom of beam as given liy the formula for an Infinite solid)]. 
This will o-ive 0=—'’-=—— (-955). The error in the vertical stre.ss, 
^ 27rC 2a 
calculated from this amended formula, is therefore only (-035) W/2/>, or only about 
37 per cent. 
With regard to tlie correction for the horizontal tension, Boussinesq finds, for a 
span 21 and depth 2h, 
P = 
AV 
2h 
Sjf 3 Q/ - h ) j 
- + 21^ 
i _ 2A _p 
TT ITU 
where y' is measured from the point (0, h) as l^efore. 
The terms {y' — h) I correspond to the liending moment wliicli we have 
removed. 
We have left therefore P = 
AV |~4 _ 2 / 
2h 
'4 
TT irli 
, so that, at the origin, when y' =■ h, 
p — = — (‘filS), and this gives a tension whlcli is greater than the actual one 
2li TT 2h ^ ’ 
liy only (•009) W/25, 
