82 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
u = - 
2&3E 
3.(/ q 
86 ^ 
[ 3X’ + 4//. if 
]_ Qjx (X + fx) Ir 
3M 1 
\ 13X' + 
86 1 
[ 10^ (X' + fx) 
+ 4/ii 
\0jJb + fx) 
+ 
V 
f Q^dx 
Jo 
X' 
2/x (X' + fx) h~ 
( 60 ), 
J 
dropping a constant in V. 
The stresses P, Q, S might be directly deduced from the equations (60) by 
differentiation. But here we require to he extremely careful, for, y and x being of 
different orders of magnitude, differentiation with regard to y will not give a term of 
the same order as differentiation with regard to x. The criterion to he used in this 
case is this ; The series L = — dS/dx is of order 0 in m, and is therefore among the 
terms which we have agreed to neglect. Similarly for the series Q. In consequence, 
every time L and Q appear owing to differentiation, they should be neglected if we 
keep the same order of approximation for the stresses as for the displacements. It 
will then be found that some terms disappear whose effect is felt in the displacements, 
as it were, by accumulation. 
Keeping this rule in mind, we obtain easily 
p _ SMy -1 
^ “ 25^ j 
Q = o 
s = IS - ,f) 
(61). 
Now these are the stresses we should have obtained had we treated that part of the 
bar as free, but subject to a bending moment M and a total shear S, transmitted 
from a distant terminal. Hence we see that, to a first approximation the stress at 
each point of a bar, whatever the manner of its transverse loading, depends only upon 
the total bending moment at the section and upon the total shear at the section, and 
will be given in terms of these by the same formulae which are valid for a free bar 
subjected to a given couple and shear at its extremities. Similar conclusions follow 
from the formulae found by Professor Pochhammer in the paper quoted previously. 
§ 12. Analysis of the Apjrroximate Expressions for the Disp)lacenients. 
Shearing Deflection. 
Now if we look at the values (60) we see easily that they are composed of three 
parts. 
(i.) The parts —i U and - ;p — [ [ Mt/x” of A". 
2o E Jq Lo’E JqJq 
These are what we may call the “ Euler-Bernoulli ” terms. They correspond to a 
