Page 141 - Shigley's Mechanical Engineering Design

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116 Mechanical Engineering Design
where r is the radius to the stress element under consideration, ρ is the mass density,
and ω is the angular velocity of the ring in radians per second. For a rotating disk, use
r i = 0 in these equations.

3–16 Press and Shrink Fits
When two cylindrical parts are assembled by shrinking or press fitting one part upon
another, a contact pressure is created between the two parts. The stresses resulting from
this pressure may easily be determined with the equations of the preceding sections.
Figure 3–33 shows two cylindrical members that have been assembled with a shrink
fit. Prior to assembly, the outer radius of the inner member was larger than the inner radius
of the outer member by the radial interference δ. . After assembly, an interference contact
pressure p develops between the members at the nominal radius R, causing radial stresses
σ r =−p in each member at the contacting surfaces. This pressure is given by 12
δ
p = (3–56)
1 r + R 1 R + r i
2 2 2 2
o
R + ν o + 2 − ν i
2
2
E o r − R 2 E i R − r i
o
where the subscripts o and i on the material properties correspond to the outer and
inner members, respectively. If the two members are of the same material with
E o = E i = E,ν o = v i , the relation simplifies to
2
2
2
2
Eδ (r − R )(R − r )
o
i
p = 3 2 (3–57)
2
2R r − r i
o
For Eqs. (3–56) or (3–57), diameters can be used in place of R, r i , and r o , provided δ is
the diametral interference (twice the radial interference).
With p, Eq. (3–49) can be used to determine the radial and tangential stresses in
each member. For the inner member, p o = p and p i = 0, For the outer member, p o = 0
and p i = p. For example, the magnitudes of the tangential stresses at the transition
radius R are maximum for both members. For the inner member
R + r
2 2
i
=−p (3–58)
2 2
(σ t ) i
r=R R − r i
Figure 3–33
Notation for press and shrink
r
fits. (a) Unassembled parts; o
R
(b) after assembly.
r i

(a) (b)

12 Ibid, pp. 348–354.