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

Cylinder shrink fitted on a solid shaft: Shrink fitting or press fitting a gear, disk, or jacket
to a solid shaft can be considered by setting a = 0 in Equation (14.33). With this revision,
the contact pressure between a cylinder and a solid shaft can be expressed as
611
INTERFERENCE FITS
p
c
=
2 2
Eδ
( c − b )
2
4bc
(14.34)
Tube and jacket of different materials: If the tube and the jacket are of different materials,
then the contact pressure between the two cylinders can be expressed as
p
c
=
⎡
2 2
1 ⎛ b + a ⎞ 1 c b
b − νT
E
2 2
T ⎝
⎜
b − a ⎠
⎟ + ⎛ +
⎢
EJ
⎝
⎜
⎣
c − b
δ
2 2
2 2
⎞ ⎤
+ νJ
⎠
⎟ ⎥
⎦
(14.35)
If the inner cylinder is a solid shaft (i.e., a = 0), then the contact pressure between a solid
shaft of one material and an outer cylinder consisting of a second material is
p
c
=
b
E
T
δ
b ⎛ c
(1 − νT
) +
E ⎝
⎜
c
J
+ b
− b
2 2
2 2
⎞
+ νJ
⎠
⎟
(14.36)
ExAmpLE 14.4
A steel [E = 200 GPa, ν = 0.3] cylindrical jacket having an external diameter of 240 mm
is shrink fitted over a steel tube having internal and external diameters of 80 mm and
160 mm, respectively. The radial interference is 0.050 mm. Calculate
(a) The pressure between the tube and the jacket.
(b) The maximum circumferential stress in the tube.
(c) The maximum circumferential stress in the jacket.
Plan the Solution
From the cylinder dimensions and the radial interference, calculate the contact pressure
created by shrinking the jacket onto the tube. The contact pressure will be an external
pressure for the tube and an internal pressure for the jacket. Once the contact pressure is
established, the circumferential stresses in both the tube and the jacket can be calculated.
SolutioN
The radii for the combined tube and jacket are a = 40 mm, b = 80 mm, and c = 120 mm.
Contact Pressure
The radial interference of δ = 0.050 mm creates a contact pressure p c on the interface
between the two cylinders:
p
c
2 2 2 2
Eδ
( c − b )( b − a )
=
3 2 2
2 b ( c − a )
2 2 2 2
(200,000 MPa)(0.050 mm)[(120 mm) − (80 mm) ][(80 mm) − (40 mm) ]
=
3 2 2
2(80 mm) [(120 mm) − (40 mm) ]
= 29.30 MPa
Ans.