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

Maximum Circumferential Stress in the tube
The smaller cylinder is subjected to external pressure only. Using an external pressure of
p o = p c = 29.30 MPa in Equation (14.26) with values of a = 40 mm and b = 80 mm for the
tube, we calculate the circumferential stress on the inner surface of the tube at r = 40 mm:
2
b po
σ θ =−
b − a
2 2
⎛ a
1 +
⎝
⎜
r
2
2
⎞
⎠
⎟
2
(80 mm) (29.30 MPa) ⎡ (40 mm)
=−
1
2 2
(80 mm) − (40 mm)
⎢ +
⎣ (40 mm)
2
2
⎤
⎥ =− 78.125 MPa =−78.1 MPa
⎦
Ans.
Maximum Circumferential Stress in the Jacket
The larger cylinder is subjected to internal pressure only. Using an internal pressure of p i =
p c = 29.30 MPa in Equation (14.24) with values of a = 80 mm and b = 120 mm for the
jacket, we calculate the circumferential stress on the inner surface of the tube at r = 80 mm:
σ =
θ
=
2
a pi
b − a
2 2
⎛ b
1 +
⎝
⎜
r
2
2
⎞
⎠
⎟
2
(80 mm) (29.30 MPa) ⎡ (120 mm)
1
2 2
2
(120 mm) − (80 mm)
⎢ +
⎣ (80 mm)
2
⎤
⎥ 76.172 MPa 76.2 MPa
⎦
= =
Ans.
ExAmpLE 14.5
Determine the maximum tensile stress in the compound cylinder of Example 14.4 after an
internal pressure of 170 MPa is applied. Sketch a figure showing the variation of the circumferential
stresses in the compound tube and jacket before and after the pressure is applied.
Plan the Solution
Stresses in the compound cylinder after the internal pressure is applied are due partly to
the contact pressure (from the shrink-fit operation) and partly to the applied pressure. To
determine the final stresses, the contact pressure results will be combined with the stresses
found in the complete assembly subjected to internal pressure only.
SolutioN
The maximum tensile stress will be a circumferential stress; however, it may occur either
at the inside surface of the tube (i.e., the smaller cylinder) or at the inside surface of the
jacket (i.e., the larger cylinder), depending on the magnitudes of the stresses associated
with the two pressure loadings.
Maximum Circumferential Stress in the tube
For the tube, the circumferential stress on its inner surface due to the shrink-fit pressure
alone was calculated in Example 14.4 as σ θ = –78.125 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 = 40 mm:
612
σ =
θ
=
2
2
a pi
⎛ b ⎞
1 +
2 2 2
b − a ⎝
⎜
r ⎠
⎟
2
(40 mm) (170 MPa)
(120 mm) − (40 mm)
2 2
⎡ (120 mm)
⎢1
+
2
⎣ (40 mm)
2
⎤
⎥ 212.500 MPa
⎦
=