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SAMPLE PROBLEM 3.1 0.9 m 0.7 m
d
0.5 m A
120 mm
Shaft BC is hollow with inner and outer diameters of 90 mm and 120 mm, respectively. Shafts AB and CD are solid and of diameter d. For the loading shown, determine (a) the maximum and minimum shearing stress in shaft BC, (b) the required diameter d of shafts AB and CD if the allowable shearing stress in these shafts is 65 MPa.
d TA 6 kN · m
B
TB 14 kN · m
C TC 26 kN · m
D
TD 6 kN · m
TA 6 kN · m
SOLUTION Equations of Statics. Denoting by TAB the torque in shaft AB, we pass a section through shaft AB and, for the free body shown, we write
A
TAB
T
16 kN ? m2 2 TAB 5 0
©M x 5 0:
x
AB
5 6 kN ? m
We now pass a section through shaft BC and, for the free body shown, we have
TA 6 kN · m
©M x 5 0:
TB 14 kN · m
T
16 kN ? m2 1 114 kN ? m2 2 TBC 5 0
BC
5 20 kN ? m
a. Shaft BC. For this hollow shaft we have A
J5 TBC
B
x
p 4 p 1c2 2 c41 2 5 3 10.0602 4 2 10.0452 4 4 5 13.92 3 1026 m 4 2 2
Maximum Shearing Stress.
On the outer surface, we have
120 kN ? m2 10.060 m2 TBC c2 5 J 13.92 3 1026 m4
tmax 5 t2 5 2 1
c1 45 mm c2 60 mm
Minimum Shearing Stress. We write that the stresses are proportional to the distance from the axis of the shaft. c1 tmin 5 tmax c2
mm 86.2t MPa 5 45 60 mm min
tmin 5 64.7 MPa b
b. Shafts AB and CD. We note that in both of these shafts the magnitude of the torque is T 5 6 kN ? m and tall 5 65 MPa. Denoting by c the radius of the shafts, we write
6 kN · m
A
tmax 5 86.2 MPa b
6 kN · m
B
t5
Tc J
65 MPa 5 16 kNp ? m2c
c4 2 c3 5 58.8 3 1026 m 3 c 5 38.9 3 1023 m d 5 2c 5 2138.9 mm2 d 5 77.8 mm b
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