7 
4 
' 
| 
E—————e eS Ee 
_—-Ts-° ee. - 
DEFLECTION OF BEAMS 135 
is of uniform cross section throughout and the maximum flange stress is 6} tons 
per square inch, E=13,400 tons per square inch. 
"13. Show that if a cantilever of length L carries a load W at a point » from 
— 3 
the fixed end, the deflection at the free end is Gee: 
14. A steel shaft AB, 3 inches in diameter, rests on supports at C and D, and 
is loaded at the ends, as shown 
in Fig 177. If the maximum WwW 
stress due to bending is 10,000 i uy 
Ibs. square inch, what is the wane -4------ 4 --.-+---<- 
StF Usiection +, at the centre? t-—30 - Mavens st 
E=30,000,000 Ibs. per square inch. é 
15. A continuous girder, built Fig. 177. 
for crossing two equal spans, has 
a uniform section whose moment of inertia is I, while its uniform weight per foot 
lineal is w. The girder is launched across the spans from one end, and, when its 
centre comes nearly over the central pier, the leading end will droop downwards 
under its unsupported weight. Write the expression for the extreme deflection 
of the leading end, the length of each span being denoted by L. {Inst.C.E.] 
-' 16. A cast-iron pipe, internal diameter 18 inches, and thickness 1 inch, rests 
on supports 40 feet apart. Find the maximum bending stress and the deflection 
at the centre when the pipe is full of water. Take weight of cast-iron=450 lbs. 
- cubic foot, weight of water=62°3 lbs. per cubic foot, and 
=6000 tons per square inch, _ : 
17. If any beam of uniform section deflects 1 inch ina span 
of 100 inches under a central load, what will be the slope of the 
beam at each end? 4 ; [Inst.C.E.] 
18. Suppose that three beams or planks, A, B, and C (Fig. 
178), of the same material, are laid side by side across a span 
L=100 inches, and a load W=600 Ibs, is laid across them at 
the centre of the span so that they must all bend together. Fig. 178. 
The beams are all 6 inches wide, but while A and C have a depth of 3 inches, 
the depth of the middle beam B is twice as great. How much of the weight W 
will be carried by each of the three beams, and what will be the extreme fibre 
stress in each ? g [Inst.C. E.] 
_ 19. A flitch beam is made up of two timbers, each 6 inches wide and 14 inches 
deep, and a steel plate § inch thick and 12 inches deep, as shown in Fig. 179, 
Taking the modulus of elasticity of the steel as 21 times 
that of the timber, find the maximum tensile stress in the 
steel when the maximum tensile stress in the timber is 
1000 lbs. per square inch, Find also the percentage in- ¢ 
crease in the strength of the timber beam to resist bend- 
ing due to the addition of the steel plate, allowing the 
Same stresses. 
20. A beam of uniform section rests on supports 
whose distance apart is L, and carries two loads each 
equal to W, one at a distance a from one support, and 
the other at a distance a from the other support. Show that the deflection 
under each load is equal to We-(3L- 4a), and that the maximum deflection (at 
Wa 
the centre) is equal to aap (3? - 42°). 
21. A single line railway bridge is carried by two main girders, each of 
40 feet span. The total weight 
of a locomotive standing on the 20 TONS. 20TONS. 20TONS. 8 TONS. 
bridge in the position shown in 
Fig. 180 is 68 tons, distributed (#) (*) ) (t) 
L 
upon 4 axles, the leading axle 
carrying 8 tons, and each of the ee Pe TI ae ' i 
others 20 tons. Find the maxi- free? Coie FE he 8 te — 1 
mum deflection of the girders, 
and where it occurs. [U.L.] Fig. 180. 
