150 
MiL L. X. G. FILOX OX AX APl’KOXlMATE SOLUTIOX FOR BEXDIXG A 
In the above terms in A^, correspond to a uniform tension along x, the terms 
to a rigid l)ody rotation, the terms B^to a solution for a pure bending couple, and the 
/I) 
terms ( ^ + I>ij to a solution for bending under a uniform shear. These various 
constants can be adjusted according to the conditions at the ends a; = i 
If, for instance, the total pressure over the ends and the total bending moment are 
to lie zero, the load '2qa lieing balanced by the shear at these ends, \ve have 
:iAi 
-j- C] — 0, 
■? + T), = 0, 
4 
and we then have 
Q 
%// 
45 
B, = + 
qa~ 
17^ 
+ f 
'1 
I 
3 I I 3 
4 i- 2lf ^ 53 
gi7 
4/d 
V = -(|, 
a" I ^ \ g / o ‘■’N I 
+ eG-w-) + 
1653 
A 6aV , 4 
^ -v^ + y 
1 
E 
^ (129). 
-/ 
This is the solution for a beam uniformly loaded on the top over a length '2a and 
held up by shears over its terminal cross-sections. In this way the case which 
occurred in the general solution, and of which the consideration was postponed in ^ 9, 
namely Wq =p /S^, is seen to lead to a fairly simple solution in finite terms. 
§ 42. Remarks on the above Solution. 
The above values (129) for U, V, P, Q, S in the case of a beam carrying a uniform 
load, lead to the following remarks :— 
(1) There is no “Neutral Axis” properly so-called; i.e., although the tension 
vanishes for y = 0, t is not strictly pro|)ortional to y, a cubic term being introduced. 
But if [a^ — laige, which is the case for any beam whose length is large 
compared with its height, the proportional efiect of the terms introduced will 
be small. 
