96 APPLIED MECHANICS ~ 
a distance b from W,, then c=a+b, W=W,+W., Wa=W,¢, and 
Wb= Wye. . 
When both loads are to one side of any section D, which is at a dis- 
tance « from A, the bending moment at D will evidently increase as the 
loads move towards D. Hence the bending moment at D will be a 
maximum, either when W, is at D, or when W, is at D, or when W, 
and W, are on opposite sides of D, It will now be shown that when W, 
and W, are on opposite sides of D, the bending moment at D has a value 
which fies between the values of the bending moment at D when W, and 
W, are in turn placed at D. 
At (a) W, is placed at D, at (+) W, is placed at D, and at (c) W, 
and W, are placed on opposite sides of D, the distance of W, from 
D being #, in the latter case. Let M,, My, and M, be the bending 
moments at D, corresponding to the positions of the loads shown at 
(a), (6), and (c) respectively, It is easy to show that M, eur -«%-G), 
My = V7(0- a+) - Wb, and My 0-2 aay) = Wy 
Wa W. 
If M,>M,, then > (L-w-a+2,) - Wy, >—-(l- 2-4), 
therefore ul >W,. 
We W. 
M,-M,= “yp (la +b) —Wh= (l= ata) + Wye 
l 
Since a, is less than a+, the quantity a+b—«, is positive, and if 
= (y = Wi )(a+ b—2,). 
Wa Wa ; fas 
M,>M,, > Wy therefore the quantity > —W, is positive. Hence 
M, — M, is positive when M,>Mj,, that is, MJ>M, when M,>M,. In 
like manner, if My>M,, My>Mg._ Therefore My lies between M, and 
M,, and the bending moment at D is a maximum either when W, is at 
D or when W, is at D. 
If M, >My, then ™"(7—a +b) -~wh>"*(1-2-a), 
l Z 
therefore > or n> y, and if M,; > Mg, then n< or @< ues 
Hence if the span be divided into two parts, the one to the left, AE, 
having a length =~ and the other to the right, EB ~ the 
maximum bending moment at any section in AE will occur when W, is 
at that section, and the maximum bending moment at any section in EB 
will occur when W, is at that section, AE may be called the field of W,, 
and EB the field of Wy. 
M, = Wid -2-0). M,=0 when w=0 or w=/-a=AH. The 
