Stresses in Pressure VesselsMAE 322 Review

Spherical Pressure Vessel

• Considering a free-body diagram of one half of the sphere, force equilibrium (ΣF=0) requires that

+ = 0

ΣF=0 =

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)

Stress in Thin-walled Spherical Pressure Vessel

Inside of vessel

Outside of vessel

Cylindrical Pressure Vessel Stresses

Tangential stress = Hoop stress

Longitudinal Stress = Axial Stress

Radial Stress (pressure)

σtσt

σt

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

Stresses in Pressurized Cylinders• Special case of zero outside pressure, po = 0

Fig. 3−32

Eq. 3−50

Stresses in Pressurized Cylinders• If ends are closed, then longitudinal stresses also exist

Shigley’s Mechanical Engineering Design

ri/t ≥ 10

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

Example 3-14

Shigley’s Mechanical Engineering Design

Example 3-14

Shigley’s Mechanical Engineering Design

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

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)

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