the resultant of all the loads to the 
__ the section BD towards Q the shear- 
____ ing force remains equal to R, so long 
BEAMS AND BENDING 
Reactions at supports of beam=loads at free ends of cantilevers 
=4W. 
er the beam AB. 
M=3Wi(i, — 7). 
M=0 at A and B where 
e=l,. 
M,,=4W/, at the centre 
C where a= 0. 
F=4W between A and C 
and F =-—4W between C 
and B. 
For the cantilever AD. 
= —4Ws2, a straight line 
which is a continuation of the 
bending moment line for AC. 
M=0 at A where z=0. 
M,, = — 4W/, at D where 
g=i, F=3W. 
For the cantilever BE, Fia. 120. 
= —4}Ws2, a straight line 
which is a continuation of the bending moment line for BC. 
M=0Oat Bwherex=0. M,,= —4W/, at Ewherex=1, F=—43W. 
If M,, for the beam=M,, for the cantilevers, then }W/,=}W/, or 
=k. 
103. Shearing Force at a Section where there is a Concentrated 
Load.—Let AC and BD (Fig. 121) be cross sections of a beam on 
ite sides of a concentrated load 
Let R, be the resultant of all Q IR; Q 
the loads to the right of Q, and R, 
left of Q. The shearing force at BD 
is equal to R,, and the shearing force 
at AC is equal to R,. In moving 
as the section is to the right of Q 
and however near it may be to Q. 
Tn like manner, in moving the section 
AC towards Q the shearing force 
remains equal to R, so long as the 
section remains to the left of Q and 
near it may be to Q. The Fic. 121. 
question then is, what is the shear- 
ing force at Q? is it equal to R, or is it equal to R,? The answer is that 
_ itis probably near the algebraical mean of the two. In practice there is 
no such case as a load acting at a point or line. What is called a concen- 
trated load must act over a certain amount of surface, even if it acts 
through what is called a “knife edge.” At (a) (a) (Fig. 121) are shown 
examples of shearing force diagrams as usually drawn in the neighbour- 
hood of a concentrated load. At (2) (b) the diagrams are shown cor- 
7 
