Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Stress concentrations also occur at other features commonly found in circular shafts,
such as oil holes and keyways used for attaching pulleys and gears to the shaft. Each of
these discontinuities requires special consideration during the design process.
185
STRESS CONCENTRATIONS
IN CIRCuLAR SHAFTS uNdER
TORSIONAL LOAdINgS
ExAmpLE 6.11
A stepped shaft has a 3 in. diameter for one-half of its length and a 1.5 in.
diameter for the other half. If the maximum shear stress in the shaft
must be limited to 8,000 psi when the shaft is transmitting a torque of
4,400 lb · in., determine the minimum fillet radius r needed at the junction
between the two portions of the shaft.
Plan the Solution
The maximum shear stress produced in the smaller diameter (i.e., minordiameter)
segment of the shaft will be determined. From this shear stress and the
allowable shear stress, the maximum allowable stress-concentration factor K can be
determined. With the allowable K and the other parameters of the shaft, Figure 6.18 can
be used to determine the minimum permissible fillet radius.
SOLUTION
The maximum shear stress produced by the 4,400 lb · in. torque in the minor-diameter
shaft segment is
τ
nom
Tc (4,400 lb⋅in.)(0.75 in.)
= =
= 6,639.7 psi
J π
32 (1.5 in.) 4
Since the maximum shear stress in the fillet between the two portions of the shaft must be
limited to 8,000 psi, the maximum permissible value for the stress-concentration factor K,
on the basis of the nominal shear stress in the minor-diameter section, is
K
τ max
8,000 psi
= ∴ K ≤ = 1.20
τ
6,639.7 psi
nom
The stress-concentration factor K depends on two ratios: D/d and r/d. For the 3 in. diameter
shaft with the 1.5 in. diameter reduced section, the ratio D/d = (3.00 in.)/(1.50 in.) = 2.00.
From the curves in Figure 6.18, a ratio r/d = 0.238 together with a ratio D/d = 2.00 will produce
a stress-concentration factor K = 1.20. Thus, the minimum permissible radius for the
fillet between the two portions of the shaft is
r
d
≥ 0.238 ∴r
≥ 0.238 (1.50 in.) = 0.357 in.
Ans.
3 in.
r
1.50 in.
4,400 lb.in.
pRoBLEmS
p6.49 A stepped shaft with a major diameter D = 20 mm and a
minor diameter d = 16 mm is subjected to a torque of 25 N · m. A
full quarter-circular fillet having a radius r = 2 mm is used to transition
from the major diameter to the minor diameter. Determine the
maximum shear stress in the shaft.
p6.50 A fillet with a radius of 1/2 in. is used at the junction of a
stepped shaft where the diameter is reduced from 8.00 in. to 6.00 in.
Determine the maximum torque that the shaft can transmit if the
maximum shear stress in the fillet must be limited to = ksi.
p6.51 A stepped shaft with a major diameter D = 2.50 in. and a
minor diameter d = 1.25 in. is subjected to a torque of 1,200 lb · in. If the
maximum shear stress must not exceed 4,000 psi, determine the minimum
radius r that may be used for a fillet at the junction of the two shaft
segments. The fillet radius must be chosen as a multiple of 0.05 in.