This document discusses stresses in pressure vessels and thin-walled approximations. It covers stresses in spherical and cylindrical pressure vessels. For thin-walled vessels with a ratio of inner radius to wall thickness over 10:1, the maximum normal stress is within 5% of the average normal stress. Cylindrical vessels experience hoop/tangential, axial/longitudinal, and radial stresses. Rotating rings experience similar tangential and radial stresses as thick-walled cylinders due to inertial forces. Press and shrink fits between cylinders create contact pressure at the interface.

1. Stresses in Pressure
Vessels
MAE 322 Review

2. Spherical Pressure Vessel
• Considering a free-body diagram of one half of the
sphere, force equilibrium (ΣF=0) requires that
+ = 0
ΣF=0 =

3. Thin Walled Vessels (spherical)
• If the ratio of the inside radius to the wall thickness
is greater than 10:1 (ri/t ≥ 10) , it can be shown that
the maximum normal stress is no more than 5
percent greater than the average normal stress.
• Therefore, a vessel can be classified as thin walled if
the ratio of the inside radius to the wall thickness is
greater than about 10:1 (ri/t ≥ 10)

4. Stress in Thin-walled Spherical
Pressure Vessel
Inside of vessel
Outside of vessel

5. Cylindrical Pressure Vessel Stresses
Tangential stress = Hoop stress
Longitudinal Stress = Axial Stress
Radial Stress (pressure)
σt
σt
σt

6. Stresses in Pressurized Cylinders
• Cylinder with inside radius ri, outside radius ro, internal
pressure pi, and external pressure po
• Tangential and radial stresses,
Fig. 3−31

7. Stresses in Pressurized Cylinders
• Special case of zero outside pressure, po = 0
Fig. 3−32
Eq. 3−50

8. Stresses in Pressurized Cylinders
• If ends are closed, then longitudinal stresses also exist
Shigley’s Mechanical
Engineering Design
ri/t ≥ 10

9. Thin-Walled Vessels
• Cylindrical pressure vessel with wall thickness 1/10 or
less of the radius
• Radial stress is quite small compared to tangential
stress
• Average tangential stress
• Maximum tangential stress
• Longitudinal stress (if ends are closed)
Shigley’s Mechanical
Engineering Design
ri/t ≥ 10
𝜎𝑡 =
𝑝𝑟
𝑡

10. Example 3-14
Shigley’s Mechanical Engineering Design

11. Example 3-14
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12. Stresses in Rotating Rings
• Rotating rings, such as flywheels, blowers, disks, etc.
• Tangential and radial stresses are similar to thick-walled
pressure cylinders, except caused by inertial forces
• Conditions:
• Outside radius is large compared with thickness (>10:1)
• Thickness is constant
• Stresses are constant over the thickness
• Stresses are
Shigley’s Mechanical
Engineering Design

13. Press and Shrink Fits
• Two cylindrical parts are assembled with radial
interference d
• Pressure at interface
• If both cylinders are of the same material Fig. 3−33
Eqn. (3−57)

14. Press and Shrink Fits
• Eq. (3-49) for pressure cylinders applies
• For the inner member, po = p and pi = 0
• For the outer member, po = 0 and pi = p
Shigley’s Mechanical
Engineering Design