132 APPLIED MECHANICS 
spans. The Forth Bridge, one of the greatest achievements of the 
engineer, is a cantilever bridge. This bridge, which crosses the Firth of 
- Forth, consists of three great double cantilevers, one at each end and one 
in the centre, the centre cantilever being connected to the others by inde- 
pendent girders. Fig. 174 is a skeleton diagram of one of the end canti- 
+ ----680------ 44145 se - == -- 680------- a4 --350--- 4 
I 1 
| 1 
A G H 
! 
| 
! 
Fig. 174. 
levers ABG, and the bridging girders GH in the centre of one of the 
large spans. As the arm BG has to carry half the weight of the central 
girders GH and of the train loads which may be passing over them the 
arm AB is made heavier than the arm BG, and at the extremity A there 
is an additional weight sufficient to counterpoise with an excess of 200 
tons half the weight of GH when carrying a full train load. 
138. Resilience of a Beam.—Consider a very short portion LN of 
length s of a beam, and let M be the mean bending moment over LN, 
also let 6 be the change of slope of the beam in passing from L to N. 
The work done in bending LN is equal to $M@. But by Art. 126, 
2 
Ms : : M?s 
O= Er’ therefore work done in bending LN = ORT 
Referring to Fig. 175, let ACB be the bending moment diagram for ~ 
a beam. Construct another curve 
AC’B on the same base AB, the 
ordinates of AC’B being equal 
to the squares of the correspond- 
ing ordinates of ACB. The work 
done in bending the portion of 
the beam lying between A and 
B will evidently be equal to the 
area of the figure AC’B divided Fic. 175. 
by 2EI. In measuring the area 
of the figure AC’B the unit of area is a rectangle, whose base is the 
unit of length, and whose height is the unit of bending moment. 
In the simple case where a beam of length L is subjected to a uniform 
2 
bending moment M, the work done in bending it is obviously ats oot 
J, is the greatest stress at the elastic limit, and y, the distance at which 
it acts from the neutral axis, then M ~o and the resilience of the beam 
1 
{ul 
2Ey} 
is 
