Module 3: Thin-Walled Pressure Vessels

Designing safe containment for pressurized liquids and gases.
Pressure vessels are closed structures containing liquids or gases under pressure. Examples include water pipes, compressed air tanks, boilers, and submarine hulls. In this module, we analyze the stresses developed in the walls of these vessels due to internal pressure.

Key Concepts

Checklist

Thin-Walled Pressure Vessel

A pressure vessel is considered "thin-walled" if the ratio of its inner radius (rr) to its wall thickness (tt) is greater than or equal to 10 (r/t≥10r/t \ge 10). Under this geometric assumption, the stress distribution throughout the wall thickness is considered uniform, simplifying the analysis significantly.

Internal Pressure

When a fluid is pressurized inside a vessel, it exerts a uniform normal force against all interior surfaces. This internal pressure tends to burst the vessel outwards, creating tensile stresses within the material forming the vessel's walls.

Thin-Walled Pressure Vessel Analysis

Ratio (r/tr/t) = 50.0(Thin-Walled)
Tangential (Hoop) Stress (σh\sigma_h)
100.0 MPa
Maximum Stress
Longitudinal Stress (σL\sigma_L)
50.0 MPa
For a cylinder, the hoop stress is always twice the longitudinal stress. A cylinder will tend to burst along its length before it pulls apart at the ends.

Stresses in Cylindrical Vessels

Hoop and Longitudinal Stresses
A cylindrical pressure vessel (like a typical pipe or a standard hot water tank) experiences two primary types of normal stress when subjected to internal pressure: tangential (hoop) stress and longitudinal stress.

  1. Tangential (Hoop) Stress (σt\sigma_t)

Hoop stress acts around the circumference of the cylinder. It is the stress that resists the bursting of the cylinder along its longitudinal axis (like a hot dog splitting lengthwise).
To find the hoop stress, imagine slicing the cylinder in half longitudinally. The internal gauge pressure (pp) pushes against the projected rectangular area (Diameter×LengthDiameter \times Length), creating a bursting force. This force is resisted by the tensile force in the two cut surfaces of the vessel wall.
The formula for tangential (hoop) stress is:
σt=pD2t \sigma_t = \frac{pD}{2t}
Where:
  • σt\sigma_t = Tangential or Hoop Stress
  • pp = Internal gauge pressure
  • DD = Inner diameter of the cylinder
  • tt = Wall thickness

  1. Longitudinal Stress (σl\sigma_l)

Longitudinal stress acts parallel to the longitudinal axis of the cylinder. It is the stress that resists the vessel being pulled apart end-to-end (like a soda can popping its top off).
To find the longitudinal stress, imagine slicing the cylinder transversely. The internal pressure acts against the circular area of the end cap, creating a longitudinal bursting force. This force is resisted by the tensile force in the circular ring area of the vessel wall.
The formula for longitudinal stress is:
σl=pD4t \sigma_l = \frac{pD}{4t}

Important

Notice that for a cylindrical vessel, the tangential (hoop) stress is exactly twice the longitudinal stress (σt=2σl\sigma_t = 2\sigma_l). Therefore, when designing a cylindrical tank or pipe, the hoop stress is always the critical design factor that dictates the required wall thickness. If it can withstand the hoop stress, it will easily withstand the longitudinal stress.
Key Takeaways
  • In cylinders, tangential (hoop) stress (σt=pD/2t\sigma_t = pD/2t) resists bursting and is the critical design factor.
  • Longitudinal stress (σl=pD/4t\sigma_l = pD/4t) resists end-to-end pulling and is exactly half the hoop stress.

Joint Efficiency

Accounting for welds and riveted joints
Many cylindrical pressure vessels are constructed by rolling steel plates and welding or riveting the longitudinal and circumferential seams. These joints are often the weakest point in the structure.
To account for this, engineers introduce a Joint Efficiency factor (η\eta, a value ≤1\le 1) into the stress equations. The modified formulas become:
σt=pD2tηl \sigma_t = \frac{pD}{2t\eta_l}
σl=pD4tηc \sigma_l = \frac{pD}{4t\eta_c}
Where:
  • ηl\eta_l = Efficiency of the longitudinal joint
  • ηc\eta_c = Efficiency of the circumferential joint
Key Takeaways
  • Joint efficiency accounts for the reduced strength of welded or riveted seams in pressure vessels (η≤1\eta \le 1).

Stresses in Spherical Vessels

Uniform stress distribution
Spherical pressure vessels (like natural gas storage spheres or deep-sea submersibles) are highly efficient structural forms. Due to perfect geometric symmetry, a spherical vessel subjected to internal pressure experiences the exact same normal stress in all tangential directions along its surface.
If we slice a sphere exactly in half, the internal pressure acts on the projected circular area (π4D2\frac{\pi}{4}D^2), creating a bursting force. This force is resisted by the tensile force in the circular ring of the vessel wall (πDt\pi D t). This derivation is mathematically identical to the longitudinal stress derivation for a cylinder.
The formula for uniform stress in a spherical vessel is:
σs=pD4t \sigma_s = \frac{pD}{4t}

Note

A spherical vessel is twice as strong as a cylindrical vessel of the exact same diameter and thickness because its maximum stress (pD4t\frac{pD}{4t}) is exactly half that of the cylinder's maximum hoop stress (pD2t\frac{pD}{2t}). This is why extremely high-pressure containment vessels are almost always spherical, despite the fact that spheres are generally harder and more expensive to manufacture than cylinders.
Key Takeaways
  • Spherical vessels experience uniform normal stress in all tangential directions (σs=pD/4t\sigma_s = pD/4t).
  • A sphere is twice as strong as a cylinder of the same dimensions under internal pressure.

Thin-Walled vs. Thick-Walled Vessels

Defining the boundary for thin-walled assumptions

The Thin-Walled Assumption

The formulas for tangential (σt=pr/t\sigma_t = pr/t) and longitudinal (σl=pr/2t\sigma_l = pr/2t) stresses in pressure vessels assume that the stress distribution across the wall thickness is uniform. This assumption is only valid if the wall is sufficiently thin compared to the radius of the vessel.
A pressure vessel is generally considered thin-walled if the ratio of its wall thickness (tt) to its inner radius (rr) is less than or equal to 0.10.1 (t/r≤0.1t/r \le 0.1). If t/r>0.1t/r > 0.1, the vessel is thick-walled, the stress distribution becomes non-linear, and more complex equations (like Lame's Equations) must be used.
Key Takeaways
  • The thin-walled vessel formulas are only valid when t/r≤0.1t/r \le 0.1.
  • For t/r>0.1t/r > 0.1, the vessel is thick-walled, and stress is no longer uniform across the thickness.
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