Thin-Walled Pressure Vessels

Pressure vessels are enclosed containers designed to hold gases or liquids at a pressure substantially different from the ambient pressure. A vessel is classified as "thin-walled" when its wall thickness (tt) is small compared to its inner radius (rr), generally when t/r<0.1t/r < 0.1 (or D/t>20D/t > 20). For such structures, we can reasonably assume that the internal stress distribution across the thickness of the wall is uniform, greatly simplifying the analysis.

Stresses in Thin-Walled Vessels

Stresses in Cylindrical Vessels

When a cylindrical tank is subjected to internal fluid pressure pp, two types of stresses are developed in the wall:

Tangential (Hoop) Stress (σt\sigma_t)

This stress acts along the circumference of the cylinder. It tends to split the cylinder into two troughs.
σt=pD2tη\sigma_t = \frac{pD}{2t\eta}

Longitudinal Stress (σl\sigma_l)

This stress acts along the length of the cylinder. It tends to blow off the ends of the cylinder.
σl=pD4tη\sigma_l = \frac{pD}{4t\eta}
where:
  • pp = internal pressure (Pa, psi).
  • DD = inside diameter.
  • tt = wall thickness.
  • η\eta = Joint Efficiency (a dimensionless factor ≤1.0\le 1.0 accounting for the strength reduction at welds or riveted joints. Use η=1.0\eta = 1.0 for seamless pipes).
Design Note
The tangential stress is twice the longitudinal stress (σt=2σl\sigma_t = 2\sigma_l). Therefore, design is typically governed by the tangential stress (Hoop Stress).

Stresses in Spherical Vessels

A spherical vessel is the most efficient shape for resisting internal pressure. Due to symmetry, the stress is the same in all tangential directions.
σ=pD4tη\sigma = \frac{pD}{4t\eta}
Note that for the same diameter and pressure, the stress in a sphere is half the hoop stress in a cylinder.

Thick-Walled Vessels and Failure

While thin-walled equations (σt=pr/t\sigma_t = pr/t) assume a uniform stress distribution across the thickness, thick-walled cylinders (where t/r>0.1t/r > 0.1) cannot use this simplification.

Lame's Equations (Thick-Walled Cylinders)

For thick-walled cylinders under internal pressure pip_i and external pressure pop_o, the radial stress (σr\sigma_r) and tangential stress (σt\sigma_t) at a radius rr are given by Lame's Equations:
σr=piri2−poro2ro2−ri2−ri2ro2(pi−po)r2(ro2−ri2)\sigma_r = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} - \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2 (r_o^2 - r_i^2)}
σt=piri2−poro2ro2−ri2+ri2ro2(pi−po)r2(ro2−ri2)\sigma_t = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} + \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2 (r_o^2 - r_i^2)}
where rir_i and ror_o are the inner and outer radii. Notice that unlike thin-walled vessels, radial stress (σr\sigma_r) is not assumed to be zero, and the maximum hoop stress occurs at the inner surface (r=rir = r_i).
Important: The assumption of uniform stress in thin-walled vessels implies a maximum error of roughly 5%. As the wall thickens, this error grows exponentially, necessitating Lame's thick-wall formulation for safety and material efficiency.

Autofrettage

Autofrettage (from French for "self-hooping") is a manufacturing technique used to increase the pressure capacity and fatigue life of thick-walled cylinders, particularly in high-pressure applications like cannon barrels, fuel injection systems, and deep-sea vessels.
According to Lame's Equations, the highest hoop stress always occurs at the inner wall (rir_i). If the internal pressure is increased until this inner layer yields plastically, while the outer layers remain elastic, a permanent deformation is created.
When the massive internal pressure is removed, the outer elastic layers try to shrink back to their original size, but they are resisted by the permanently expanded inner plastic layer. This creates a state of residual compressive stress at the inner wall and residual tensile stress at the outer wall.
In service, when the cylinder is pressurized again, the operational tensile hoop stress must first overcome the residual compressive stress at the inner wall before it can cause yielding or fatigue crack initiation. This dramatically increases the effective elastic limit and fatigue life of the vessel.

Failure Theories and Yield Criteria

When designing thin-walled pressure vessels, we analyze a biaxial state of stress (longitudinal and tangential stress). For ductile materials, we must ensure the stress state doesn't exceed yield thresholds using advanced failure criteria.
  • Maximum Principal Stress Theory (Rankine): Fails if maximum principal stress equals or exceeds yield stress (σmax=σt≥Sy\sigma_{max} = \sigma_t \ge S_y). Simplest, but not conservative for ductile materials.
  • Maximum Shear Stress Theory (Tresca): Fails if maximum shear stress equals or exceeds yield shear stress (τmax≥Ssy\tau_{max} \ge S_{sy}). Commonly used in pressure vessel codes (ASME Section VIII Div 1).
  • Von Mises Yield Criterion (Distortion Energy): Fails if distortion energy exceeds the yield threshold. Uses an effective equivalent stress (σvm=σ12−σ1σ2+σ22\sigma_{vm} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}). Typically the most accurate for ductile metals like steel.

Interactive Tool: Pressure Vessel Calculator

Compare the stresses in Cylindrical and Spherical vessels by adjusting the pressure and geometry below.

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.
Key Takeaways
  • Thin-Walled Assumption (t/r<0.1t/r < 0.1): Valid for most tanks and pipes. Stress is assumed uniform across the thickness.
  • Hoop Stress (σt=pD/2tη\sigma_t = pD/2t\eta): Always the governing stress in a cylinder (twice the longitudinal stress). It acts to split the pipe along its length.
  • Longitudinal Stress (σl=pD/4tη\sigma_l = pD/4t\eta): Acts to separate the ends.
  • Spherical Stress (σ=pD/4tη\sigma = pD/4t\eta): The most efficient shape; stress is half that of a cylinder's hoop stress.
  • Joint Efficiency (η\eta): Accounts for the reduced strength of welded or riveted joints.
  • Thick-Walled vs. Thin-Walled: Thin-walled assumes uniform stress; thick-walled assumes varying stress (Lame's Equations). In thick-walled vessels, the highest stresses occur at the inner surface.
  • Autofrettage: A technique where a thick-walled cylinder is intentionally yielded to create beneficial residual compressive stresses at the inner wall, drastically increasing its pressure capacity and fatigue life.
  • Failure Theories: Use Von Mises or Tresca yield criteria to accurately predict failure under biaxial stress states in ductile materials.
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