DEFLECTION OF BEAMS 115 
dx _Mdx 
7 H and K, and 0= RE” Hence the change in the slope between 
the points «=/ and x=2, is equal wf ne =O, and if the beam is 
horizontal where x=/, then the slope at the point where «=~, is 4,. 
_ 125. Stiffness of a Beam.—tThe ratio of the maximum deflection of 
a beam to its 5 is called the stiffness of the beam. ‘The stiffness may 
_ be denoted by! = where » varies from 1000 to 2000 for steel girders of 
large span, ‘i from 500 to 700 for short spans. For timber beams, 
n should not be less than 360. 
126. General Method of Determining Deflection from Bending 
Moment Diagram.—The analytical method of finding the deflection of a 
beam used in the preceding Articles is simple in simple cases, but in 
many cases in practice it becomes difficult and complicated. The method 
now to be discussed will be 
found to be comparatively 
simple in cases where the 
- analytical method would be 
_ troublesome. 
In what follows, the beam , 
or cantilever is assumed to 
be of uniform cross section. 
Consider first the case of 
: 
> 
j 
5 
& 
4 
= 
a cantilever AB (Fig. 156) 
under any system of loads. 
Let AHKB be the bending 
‘moment diagram, and let 
A,B, be the curve in which 
the cantilever bends. Take Fig. 156. 
two points L and N on the : 
cantilever near to one another, their distance apart being s. If the 
distance s be small enough, the bending moment M may be considered 
as uniform over the length LN. Let R be the radius of curvature of 
LN, then ona and @ the angle between the radii to the centre of 
 eurvature from L, and N, will be the change in the slope of the beam in 
passing from L, to N,. But 0=<, therefore on, which shows that 
_ the change in the slope of the cantilever in passing from L, to N, is 
- equal to the area of the vertical strip of the bending moment diagram 
_ over LN divided by EI. Hence if the bending moment diagram over 
_ the portion BP be divided into a large number of vertical strips, it follows 
_ that the total change in the slope of the cantilever between B and P will 
be equal to the sum of the areas of these strips divided by EI, that is, 
_ equal to the area of the part of’ the bending moment diagram lying 
between the verticals through B and P divided by EI. Hence if a 
_ tangent P,O be drawn to the curve A,P,B,, at P, it will be inclined to 
_ the horizontal B,C at an angle a, whose tangent or circular measure, 
- the angle being very small, is equal to the area of the figure PHKB 
~ 
6 
