BEAM OF IvECTANGULAli CliOSS-SECTJOA' UA'DEii ANY SYiSTEM OF LOAi>. 
131 
but il we iJiit ill tJiese values and then proceed to make a inlinite, certain | 
the expressions for U and V do not give finite integrals in the limit. 
Ihis is due to tlie tact tliat the conditions of rigid ecpnlihriiim reipiire 
L6;2a at the two ends (fig, viii,). 
arts of 
shea.rs 
Lb 
Fig. viii. 
ihese shears Ijbj'la will produce a deflection due to bending alone, which, 
calculated from the Eulej-Beriioulli formula, conies to 
+ VT 
\ [^ A/ "h 
(for a; > 0), 
and when a is made very large, tliis gives 
y _ Eil ^ 1 I ^ 
32 ^ A' + 
(104) 
for the bending deflection produced by the end shears at large distances a;, wliich, 
liowever, are still finite compared witli a. If, therefore, we allow the beam to bend 
fieely under these end loads, in sucli a Avay that each of tliese produces its projiei’ 
hendmg deflection and no more, tlie constant A must be adjusted so tliat, for large 
values of a;, V tends to the value (104). 
This implies that A must have an infinite part, Avhich will exactly cancel the 
infinite part of V. It is easily found that the value 
A 
+ jji 
where A' is finite, will introduce terms in both U and V which will make these 
quantities remain finite in the limit when a is infinite. 
A e then find, ])utting in for B the value found and proceeding to the limit, 
s 2 
