88 
MR. L. N. (t. FILON OX AN APPROXIMATI^ SOLUTION FOR BENDING A 
_ ® 2W ('Inirhja) co^Xi'linrhla . ^ i o / 
S = -^—rr sm 2n7rx a sinh ‘imry/a 
1 a sinh ■imrhja + 4:n7rbla 
“ 2W (2«7ry/«) sinh ^nirl a . , , , 
^ — . , ' -;— -i— sin ^UTTX a cosh 2n77^/rt 
1 a sinh ‘^mrnja + 4:mrhja 
-P 
” 2W (2?i + i7r?V«) sinh 2;i+ Xirlija 
° ^ sinh 4?z.+27r?>/a — 47i + 27r?//c 
2W (2h. + Itt/z/a) cosli 2 / 14 - I'rrhja 
sin 2w+ iTrxja cosh 2/1 + 177^/ 
a 
a> 
•< 
*0 « 
sinh 4/1 + 27r?//A — 4?i+ 2TThjn 
sin 2?i + i7ra;/a sinh 2n-\-\7ryla 
(70). 
in 
) 15. Dejinite Integrals to which the expressions of the last Section tend when we 
niahe “a” very large. 
If we make a very large, the S’s in the preceding expressions will become integrals 
the limit. It will be found, however, that certain terms in the last found values 
of U, V, P, Q, S b ecome Infinite when 0 is substituted for hja. In these cases the 
sum may not be directly transformed into an integral. The reason why this occurs 
is that, if Oj be made infinite, an infinite bending moment is introduced at the 
centre of the beam. It is this moment which produces the parts ot the displacements 
and stresses that become infinite when a is infinite. If, however, we apply at the 
two ends pure couples — we get lid of this infinite moment, and we have only 
the terms due to the local efiect, which produces only finite stresses at a finite 
distance from the origin. 
Thus, if in U we add —- v 3 _ 
.Trt" 
0 o 
to the second 2 and -- 2 f ry— prr -.— 
a 0 A + yA Ir (2/1 + l)-7r' 
to the third 2, these 2’s remain finite even when we make a = co, We have, 
however, to introduce negative terms to ha:lance those that have been added. 
TT'f, we see that the part of 
llememhering that ( y—- + 
^ \A + yu- n 
1 \ 4 ” 1 
and 2 7—^ 
/ Ij 0 (2/1 + 
li¬ 
the series in U which becomes infinite, is 
xifhVa 
TJh 
, w 
hich, added to the other infinite 
term in the last line of (fif), gives for the infinite part of U : 
IT - _ 
^0- 4 ■ 
Similarly with V. The terms which have to he added to the second and fourth 2’s 
to make them finite in the limit are 
ri- 
fd 
W® 
a I (2/1 + [_\a' + fx, fj. 
y 
0//AV “ 
"vT ^ (2/1+’ 
l\/_ . (2/i+l)V=t- 
+ .7 ) ( 1 + To -nr-) + — 
a (2/1 + ® ' fA' + /x. 
77 
1 (2/l+l)-7rW 
a- 
7)' 
