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

The circumferential stress on the inner surface of the tube is found by superimposing the
stress due to the contact pressure and the stress caused by the internal pressure:
σ = σ (contact pressure) + σ (compound cylinderwith p )
θ θ θ
=− 78.125 MPa + 212.500 MPa
= 134.375 MPa
i
Maximum Circumferential Stress in the Jacket
For the jacket, the circumferential stress on its inner surface due to the shrink-fit
contact pressure alone was calculated in Example 14.4 as σ θ = 76.172 MPa. Next,
the circumferential stress created in the compound assembly by an internal pressure
of p i = 170 MPa is calculated from Equation (14.24) with a = 40 mm, b = 120 mm,
and r = 80 mm:
σ =
θ
=
2
a pi
b − a
2 2
⎛ b
1 +
⎝
⎜
r
2
(40 mm) (170 MPa)
(120 mm) − (40 mm)
2
2
⎞
⎠
⎟
2 2
⎡ (120 mm)
⎢1
+
2
⎣ (80 mm)
2
⎤
⎥ 69.063 MPa
⎦
=
The circumferential stress on the inner surface of the jacket is found by superimposing the
stress from the contact pressure and the stress from the internal pressure:
σ = σ (contact pressure) + σ (compound cylinderwith p )
θ θ θ
= 76.172 MPa + 69.063 MPa
= 145.234 MPa
i
Maximum tensile Stress in the Compound Cylinder
The maximum tensile stress occurs on the inner surface of the jacket:
( σ θ ) max = 145.2 MPa
Ans.
Variation of Circumferential Stresses in Compound tube and Jacket
The variation of circumferential stresses in the compound cylinder is shown in
Figure 14.14.
Take a closer look at the plot of the circumferential stress distribution due to the 170 MPa
internal pressure in the figure. Notice that the material close to the inner surface of the
compound cylinder is highly stressed while the stresses at the larger radii are relatively
low. This effect becomes more pronounced as the ratio of the outside diameter to the inside
diameter increases. From this observation, we can conclude that the use of a single
huge cylinder to contain very high pressure is an inefficient use of material. By using a
compound cylinder, stresses are more broadly distributed in the material. When the tube
and jacket are first assembled, the shrink-fit contact pressure causes the inner cylinder to
be in compression and the outer cylinder to be in tension. When the tensile circumferential
stresses due to the 170-MPa internal pressure are superimposed on the stresses created
by the contact pressure, the final distribution shows that circumferential stresses are
spread to more of the cross section.
613