Page 36 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
Brown˙C01
Brown.cls
18
U.S. Customary 12:26 STRENGTH OF MACHINES SI/Metric
Example 2. Determine the shear stress (τ i ) Example 2. Determine the shear stress (τ i )
at the inside surface of a hollow circular shaft, at the inside surface of a hollow circular shaft,
where where
T = 8,000 ft · lb = 96,000 in · lb T = 12,000 N · m
D o = 4in = 2R o D o = 10 cm = 0.1 m = 2R o
D i = 2in = 2R i D i = 5cm = 0.05 m = 2R i
solution solution
Step 1. Calculate the polar moment of inertia Step 1. Calculate the polar moment of inertia
(J) of the shaft using Eq. (1.23). (J) of the shaft using Eq. (1.23).
1 4 4 1 4 4
J = π R − R i J = π R − R i
o
0
2 2
1 4 4 1 4 4
= π((2in) − (1in) ) = π((0.05 m) − (0.025 m) )
2 2
= 23.56 in 4 = 0.00000920 m 4
Step 2. Substitute this value for (J), the torque Step 2. Substitute this value for (J), the torque
(T ), and the inside radius (R i ) into Eq. (1.21). (T ), and the inside radius (R i ) in Eq. (1.21).
TR i (96,000 in · lb)(1in) TR i (12,000 N · m)(0.025 m)
τ i = = τ i = =
J 23.56 in 4 J 0.00000920 m 4
2
2
= 4,074 lb/in = 4.1 kpsi = 32,600,000 N/m = 32.6MPa
Level of Torque Reduction. For the geometry of Example 2, the outside radius (R o )
is twice the inside radius (R i ). It is interesting that reduction in torque carrying capa-
bility of the hollow shaft compared to the solid shaft is small. This is because the
material near the center of the shaft carries very little of the shear stress, or load, pro-
duced by the applied torque (T ). It is instructive to determine the exact value of this
reduction.
Start with the fact that the maximum shear stress (τ max ) will be the same for both the
shafts, because they are made of the same material and the outside radius (R o ) is the same.
This fact is shown mathematically in Eq. (1.21) to both the solid shaft and the hollow shaft.
This fact is shown mathematically in Eq. (1.24) as

T solid R o T hollow R o
= (1.24)
J solid J hollow

The outside radius (R o ) cancels on both sides, so Eq. (1.24) can be rearranged to give
the ratio of the torque carried by the hollow shaft (T hollow ) divided by the torque carried by
the solid shaft (T solid ).

T hollow J hollow
= (1.25)
T solid J solid