84 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
IP we assume de Saint-Vex ant’s fixing condition, we find, for the additional 
terms in V corresponding to (iii.)> 
•s 
ixb 
A 1 
W (/ - s;) if 
YJj^ 
for a: > 0, and 
8 ..7. > 
W (I + x) ^ 
7] -— for X < 0. 
The terms are therefore identical in this and in the true solution, but the first 
term which represents the additional defiection of the central axis of the beam, and 
which is sometimes spoken of as the shearing deflection, is less than in the true 
solution, being (13V + 16/r)/20 (X + p-), that is (42X + 32p)/(60X + 40p) of that 
given by the double cantilever solution. This fraction comes to be 74 for uni-con¬ 
stant isotropy. 
If we assume what I have called Love’s flxing condition, the shearing deflection 
disappears entirely. 
The true solution shows us, therefore, that it is permissible in this case to use the 
double cantilever as an artifice to obtain the solution, 2 ^rovided we adopt, at the 
section of fictitious severance, a fixing condition intermediate between those of Love 
and DE Saint-Venant, but nearer to the latter. In other words, a central isolated 
load does actually introduce a sharp bend. 
§ 13. Value of the Deflection when h is not small and the Beam is Doubly 
Sup 2 ')orted. 
Suppose the beam rests on two knife-edge supports A, B (fig. i.) at a distance 2l 
apart, and a weight W is borne by another knife-edge which presses on the upper 
part of the beam at C. 
, s,l 
SJC 
A 
ZCL 
^_ 
\ 
w' 
L 
' 
Fig. i. 
Then we have 
ao = — 
{n O), / 3 q — ct-Q, fn — 
Oj 
nirl 
- cos —. 
a a 
The central deflection of the elastic line (what de Saint-Venant calls “la 
lleche de flexion”) is then given by _/’= V,r=?j,=o ~ '^^=oy=o i substituting for a’s- 
;ind yS's in (5G), we find 
