Page 424 - Shigley's Mechanical Engineering Design

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Shafts and Shaft Components 399
The pressure p generated at the interface of the interference fit, from Eq. (3–56)
converted into terms of diameters, is given by
δ
p = (7–39)
2
2
d d + d 2 d d + d 2
o i
+ ν o + − ν i
2
d − d 2 2 2
E o E i d − d
o i
or, in the case where both members are of the same material,
2
2
2
2
Eδ ( (d − d )(d − d ) )
o
i
p = (7–40)
2
2d 3 d − d 2
o i
where d is the nominal shaft diameter, d i is the inside diameter (if any) of the shaft,
d o is the outside diameter of the hub, E is Young’s modulus, and v is Poisson’s ratio, with
subscripts o and i for the outer member (hub) and inner member (shaft), respectively.
The term δ is the diametral interference between the shaft and hub, that is, the differ-
ence between the shaft outside diameter and the hub inside diameter.
(7–41)
δ = d shaft − d hub
Since there will be tolerances on both diameters, the maximum and minimum pres-
sures can be found by applying the maximum and minimum interferences. Adopting the
notation from Fig. 7–20, we write
(7–42)
δ min = d min − D max
(7–43)
δ max = d max − D min
where the diameter terms are defined in Eqs. (7–36) and (7–38). The maximum inter-
ference should be used in Eq. (7–39) or (7–40) to determine the maximum pressure to
check for excessive stress.
From Eqs. (3–58) and (3–59), with radii converted to diameters, the tangential
stresses at the interface of the shaft and hub are
2
d + d i 2
σ t, shaft =−p (7–44)
2
d − d 2
i
2
d + d 2
o
σ t, hub = p (7–45)
2
d − d 2
o
The radial stresses at the interface are simply
σ r, shaft =−p (7–46)
σ r, hub =−p (7–47)
The tangential and radial stresses are orthogonal, and should be combined using a
failure theory to compare with the yield strength. If either the shaft or hub yields during
assembly, the full pressure will not be achieved, diminishing the torque that can be trans-
mitted. The interaction of the stresses due to the interference fit with the other stresses in
the shaft due to shaft loading is not trivial. Finite-element analysis of the interface would
be appropriate when warranted. A stress element on the surface of a rotating shaft will
experience a completely reversed bending stress in the longitudinal direction, as well as
the steady compressive stresses in the tangential and radial directions. This is a three-
dimensional stress element. Shear stress due to torsion in shaft may also be present. Since
the stresses due to the press fit are compressive, the fatigue situation is usually actually
improved. For this reason, it may be acceptable to simplify the shaft analysis by ignoring