CES" ee ee 
= 
> 
DEFLECTION OF BEAMS 133 
For a beam of length L supported at its ends and leaded with a weight 
W at an intermediate point di- 
viding L into two parts a and 4, 
the bending moment diagram 
(Fig. 176) is a triangle ACB and 
the curves AC’ and BC’, whose 
ordinates are the squares of the 
ordinates of the bending moment 
diagram, are semi-parabolas Acai fu : 
whose axes are vertical and whose Kr~~~ @ ~~ se ----- b ------4 
vertices are at A and B re- Pea. 176 
spectively. Hence the area of Aah 
2p2 iq? 
ACB =, (a+) =, and the work done in deflecting the 
- Wah? ‘ : 
beam is LET’ But the work done is equal to $W6, where 4 is the 
deflection at the load. Hence 3=Ner , a result which was proved in 
another way in Art. 127. 
The work done in deflecting a beam of length L may be found 
analytically as follows. Let «=mean distance of LN (Fig. 175) from A, 
and let d=s, then work done in deflecting beam = ser : [ Mae. 
0 
o2EL 2EI 
Applying this to the case of a beam supported at the ends and carrying 
a uniformly distributed load of w per unit of length, M= (Ln - x”), and 
work done 
L 
1 (woe 5 mune eel ) 
oer a ee ae Ke dee — Madde + atde 
_ wi (LS 21s a ob wLs WL 
4 ~ 240EL~ 240EI’ 
SEI\3 4 +5 
where W = wL = total load. 
Exercises VIII. 
1. A pitch pine beam rests on supports 15 feet apart, and carries a uniformly 
distributed load of 2 tons per foot run. The cross section is a rectangle 15 inches 
deep, and the maximum stress is 3000 Ibs. per square inch at every cross section. 
The breadth 6 of the section is to vary so that the beam will bend to a circular 
arc. Find bat the centre of the span, and at 2 feet and 4 feet from the centre. 
Find also the deflection at the centre, and the radius of curvature of the neutral 
surface. E=1,900,000 Ibs. per square inch. 
2. A cast-iron cantilever, 54 inches long, carries a load of 3000 Ibs. at its 
outer end. The cross section is a rectangle 2 inches broad. At the fixed end 
the depth is 8 inches. The depth at other points is to be such that the lever 
will bend to a circular arc, and the lever is to be symmetrical about the neutral 
surface. Find the depth at 9, 27, and 45 inches from the fixed end. Find also 
the deflection at the free end, and the radius of curvature of the neutral surface 
when the lever is bent. What are the maximum stresses at the fixed end and 
at the other sections mentioned ? E=17,000,000 lbs. per square inch. 
3. A beam, instead of being straight when free from bending moment, is curved 
