134 APPLIED MECHANICS 
to aradius r, On applying a bending moment the radius of curvature is altered — 
from r to R. Show that Raat (“2) and therefore if x is large compared 
F > 
ret Mh 1% 
with y, he +E (§ 2 nearly. 
4. The cross section of a cantilever is a circle of diameter d. Length of lever, 
4 feet. Load at free end, 5000 lbs. Maximum stress, 9000 lbs. per square inch, 
E= 29,000,000 lbs. per square inch. Find d and the deflection at the free end. 
5. A cylindrical cantilever is £0 inches long, and 5 inches in diameter. There 
is a load at the free end which causes a maximum stress of 2500 Ibs. per square 
inch. Taking E at 1,800,000 lbs. per square inch, what is the deflection at the 
free end? 2 
6. A vertical mild steel tube of 6 inches external diameter, and 4 inch thick, 
is securely bedded in the ground. Its height above ground is 10 feet, and it is 
subjected at the upper end to a horizontal pull of 1500 Ibs. Calculate the 
maximum stress at the ground section and the deflection at the top. (Take E 
as 30,000,000 lbs. per square inch.) [Inst.C.E. ] 
7. A beam 12 feet long, 1 foot deep, and 5 inches wide rests on supports 
at its ends, and carries a load of W Ibs. at its centre. The maximum stress being 
2000 lbs. per square inch, find W and the deflection at the centre. E=1,800,000 lbs. 
per square inch. 
8. Referring to the beam of the preceding exercise, what load, in pounds, 
distributed uniformly over the length, will cause a deflection at the centre equal 
to 1-500th of the span ? 
9. A steel joist, 10 inches deep and 10 feet long, is supported at the ends. 
The joist has equal flanges 5 inches wide and 0°54 inch thick, and a web 0:35 
inch thick. The weight of the joist is 29 lbs. per foot. What central load, in 
addition to its own weight, will this joist carry when its deflection at the 
centre is 1-1000th of the span, and what will then be the maximum stress? 
E=30,000,000 lbs. per square inch. 
10. A wooden plank, 12 inches wide and 3 inches deep in section, rests freely 
on two supports, in the same horizontal level, which are 20feet apart. A man 
weighing 12 stone stands in the middle of this plank carrying on his shoulder 
a hod of bricks which weighs 84 lbs. Find:—(a) The maximum stress at the 
central section due to this load, and the weight of the plank. (1 cubic foot of 
wood weighs 46 lbs.) (5) The deflection in the centre, if Young’s modulus of 
elasticity, is 1,600,000 lbs. per square*inch. [B.E.] 
11. In connection with a contract for the supply of cast-iron pipes, certain 
bending tests were specified on bars (cast at the same time) 40 inches long, 
2 inches deep, and 1 inch thick. The following results were obtained when one 
of these bars was tested on edge on a 36-inch span :— 
- 
Load at centre of beam;\) 109 | 400 | 800 | 1200 | 1600 | 2000 | 2400 
pounds , ! : : 
ees at centre of beam, \| 9.979 | 0-048 | 0-098 | 0°150 | 0-204 | 0-256 | 0:314 
(a) Plot on squared paper a curve to show the relation between the load at the 
centre of the beam and the deflection at the centre of the beam. (6) From your 
curve determine the load which will be required at the centre of the beam in order 
to give a deflection of one-eighth of an inch. (c) Calculate in Ibs. per square inch 
Young’s modulus of elasticity for this cast-iron. (d) Calculate in inch-pounds 
the total work done in bending this beam up to a load of 2400 lbs. in the centre 
of the span. (¢e) The beam eventually broke with a load of 3200 Ibs. in the 
centre. Assuming that the ordinary beam formula holds up to the breaking 
point in cast-iron beams, what was:the maximum intensity of tensile stress in 
the metal at the instant of rupture? - [B.E.] 
12. A steel girder, having a uniform depth of 13 feet, rests on piers which 
are 150 feet apart, and carries a uniformly distributed load. Find the deflection 
at the centre in inches ; (a) when the area of the flanges is proportioned so that 
there is a uniform flange stress of 6} tons per square inch; (6) when the girder 
