\fdO'^ - Old Forge Coal Mines

STRENGTH OF MATERIALS. 229
F^. Choose the pole P, and draw the rays P 0, P 1, P 2.
Draw a b between the left support and F^ parallel X.o P 0\
b c between F^ and F^ parallel to P i, and c d parallel to
P 2, between F^ and the right support. Through P draw a
line parallel to the closing line a d. 0-1 = 1-2; hence, the
reactions of the supports are equal, and are each equal to 1
ton. The shear between the left reaction and F^ is negative,
and equal to F^-=\ ton. Between the left and the
right support it is 0, and between the latter and F^ it is positive
and equal to 1 ton. The bending moment is constant
and a maximum between the supports. To the scale of
forces /* i = 2 tons = 4,000 lb., and to the scale of distances
a f= 30 in. Hence, the maximum bending is 4,000 X 30
= 120,000 in. -lb. Ans.
{b) Using formula 74,
^ /
J/=-^ - = 120,000. 5, = 38,000; /= 6.
^. / 120,000X6 3G0 ,^ ,
^^^^' 7 =
nd*
38,000
-, / 64 :zd'
But. - = —J- = -—c
d 32
Hence, —-— = 19, or rf* =
'
32
= 19- = 1^' """""^y-
3.1416*
d^V% 'Im! = ''''' ^-
(608) Since the deflections are directly as the cubes of
the lengths, and inversely as the breadths and the cubes
of the depths, their ratio in this case is
18' 12» 27 9 ,^
•
2 X 6' 3 X 8" 2 8
or -- : :r = 12.
That is, the first beam deflects 12 times as much as the
second. Hence, the required deflection of the second beam
is .3 ^12 = .025'. Ans.