116 APPLIED MECHANICS 
divided by ET, and therefore a= A , where A is the area of the figure 
EI 
PHKB. 
In measuring the area of the bending moment diagram for the 
purpose of finding the change of slope, the unit of area is a rectangle, 
whose base is the unit of length, and whose height is the unit of bending 
moment. 
Tf on AB as base a curve BFD be constructed such that the ordinate 
PF at any point P in AB is equal to the area of the figure PHKB divided 
by EI, then the ordinate of this curve at any point in AB will represent 
the slope of the bent cantilever at that point, and this curve may be 
called the curve of slope. 
Now let the mean distance of LN from P be denoted by 2, and let w 
be the deflection at P, due to the curvature of the part LN, then 
u=20= Ts where a’ is the area of the strip of bending moment 
diagram over LN. The total deflection P,Q at P will be the sum of all 
Er between B and P, therefore P,Q ~~ where & is 
the distance of G, the centre of gravity of the figure PHKB from PH. 
This simple rule may therefore be used for constructing the curve A,P,B,, 
the work being done graphically, or in part graphically and in part by 
calculation. 
Again, P,Q=OQ tan a, but since a is a small angle, tan a may be 
taken equal to a, therefore P}Q=OQ*a. But dm and PQ=a5 
A& 
hence ogs. =75;, and therefore OQ=%. This shows that the tangent — 
El EI 8 
to the curve A,P,B, at P, meets BC at a point vertically under G, the 
centre of gravity of the figure PHKB. 
The deflection A,C at the free end of the cantilever is obviously 
equal to ar, where A, is the area of the whole diagram AHKB, and 
such quantities as 
#, is the horizontal distance of the centre of gravity of the figure 
AHKB from A. Also, the tangent to the curve A,P,B, at A, will 
meet B,C at a point vertically under the centre of gravity of the figure 
AHKB. 
Suppose that the area of the figure PHKB, on a scale drawing, is A’ 
square inches. Let the scale for the base PB be 1 inch to m inches, and 
let the scale for the ordinates or bending moments be 1 inch to m inch- 
pounds, then A= A’mn. Z must be measured with the scale 1 inch to 
m inches. Then if E is in lbs. per square inch and I is in inch-units, the 
deflection P,Q = 4 will be in inches. 
If the bending moment diagram AHKB (Fig. 156) be considered as 
a diagram, showing the intensity of a load distributed over the canti- 
lever, and if the cantilever be fixed at the end A instead of at the end 
B, as shown in Fig. 157, then at any section P of the cantilever 
(Fig. 157) the shearing force is proportiona: to the area of the figure 
PHKB, and therefore the ordinate PF of the curve BFD in Fig. 156, 
