Through-Thickness Stress in Pressure Vessels
Radial, hoop and longitudinal stress variation through the wall, inner-wall peak, outer-wall response, stress gradients and the significance for assessment and fatigue.
Technical provenance
Applicable standards / specifications
- ASME BPVC Section VIII Division 1 (2025) — Rules for Construction of Pressure Vessels
- ASME BPVC Section VIII Division 2 (2025) — Alternative Rules for Construction of Pressure Vessels
- EN 13445 — Unfired pressure vessels — Relevant European pressure-vessel code family where specified by the project.
References
- ASME Boiler and Pressure Vessel Code — 2025 edition — Primary code family reference for pressure-vessel design where ASME BPVC is the governing basis.
- Moss, D. R. & Basic, M. — Pressure Vessel Design Manual — Background engineering reference for pressure-vessel load paths, stresses and design checks.
Three stress components through the wall
The stress state in a pressure vessel wall is triaxial: hoop (circumferential) stress, longitudinal (axial) stress and radial (through-thickness) stress. In a thin-walled vessel, the radial stress is neglected and the state is approximated as biaxial. In a thick-walled vessel, all three components are significant and vary through the wall thickness.
Radial stress distribution
The radial stress varies from -p_i at the inner surface to -p_o (or zero) at the outer surface. It is always compressive under positive pressure. The variation is non-linear, following the Lamé equation. The radial stress reduces the equivalent (von Mises) stress at the inner surface because it is compressive, partially offsetting the tensile hoop stress. However, the radial stress does not prevent yielding — it modifies the equivalent stress.
Hoop stress distribution
The hoop stress varies from a maximum at the inner surface to a minimum at the outer surface under internal pressure. The variation is non-linear (from the Lamé equation). For a cylinder with b/a = 1.5, the inner-surface hoop stress is approximately 17% higher than the thin-wall average. For b/a = 2.0, it is approximately 33% higher. The inner-surface peak is the governing stress for fatigue and for yielding assessment.
Longitudinal stress distribution
Unlike the hoop and radial stresses, the longitudinal stress in a closed-ended cylinder is uniform through the wall thickness. It equals (p_i * a^2 - p_o * b^2) / (b^2 - a^2) from the Lamé derivation, which is constant for all r. This means the longitudinal stress does not have a through-thickness gradient — the inner and outer surfaces carry the same longitudinal stress.
Inner-wall peak
The inner wall of a pressure vessel is the highest-stressed location under internal pressure. The hoop stress peaks at the inner surface, the radial stress is most compressive at the inner surface, and stress concentrations from surface roughness or manufacturing defects are most damaging at this location. For fatigue assessment, the inner surface stress range is the governing parameter. Surface finish, inspection and material quality at the bore are therefore critical.
Outer-wall response
The outer wall carries the lowest hoop stress but is the surface visible for inspection. The radial stress at the outer surface is zero (for atmospheric external pressure) or equals the external pressure. The outer surface is where surface cracks from external corrosion or mechanical damage would initiate. The outer surface stress is relevant for environmental cracking assessment and for external load superposition.
Stress gradients and FEA
In FEA, the through-thickness stress gradient must be captured by the mesh. For a thick-walled vessel, at least 4-5 elements through the wall thickness are needed to resolve the stress gradient accurately. Shell elements, which assume a linear stress distribution through the wall, cannot capture the non-linear gradient in a thick wall — solid elements are required. The stress linearisation procedure (membrane plus bending plus peak) is used to extract assessment-quality stresses from the FEA through-thickness distribution.
Three-dimensional stress state
Pressure-vessel stress is inherently three-dimensional through the wall. Hoop, longitudinal and radial components coexist, and local geometry can introduce additional shear. In a thin shell the radial component is often small compared with membrane tension, but in a thick wall it becomes part of the principal stress state and should not be discarded.
Radial stress boundary values
At a pressure boundary, radial stress is constrained directly by the applied normal traction. At the inner wall it is approximately minus the internal pressure; at the outer wall it is approximately minus the external pressure. This provides a direct check on solid-element results and helps distinguish physically required radial compression from numerical artefacts.
Membrane and bending representation
Stress linearisation decomposes a through-thickness distribution into membrane, linear bending and nonlinear peak components along a defined stress-classification line. The decomposition is an interpretation tool; it does not alter the underlying FEA field. The line must be selected to represent a meaningful structural section and oriented consistently with the pressure boundary.
Local peaks and fatigue
A sharp local peak may have little influence on gross plastic collapse but a major influence on fatigue or crack initiation. Conversely, a broad membrane stress may govern primary-load capacity. This is why reporting only the maximum von Mises stress across a pressure model can obscure the actual failure mechanism.
Solid versus shell recovery
Shell elements recover membrane and bending resultants naturally through the thickness. Solid models provide the full stress distribution but require the analyst to define how that distribution is interpreted. Both can be valid; the choice depends on whether local three-dimensional stress and geometry are important.
Through-thickness verification
Plot individual stress components through a nominal smooth wall and compare them with the expected thin-wall or Lamé distribution. This check is more informative than comparing only equivalent stress because it reveals whether pressure direction, wall orientation and end restraint are correct.
Verification point: Plot individual stress components through a nominal smooth wall and compare them with the expected thin-wall or Lamé distribution. This check is more informative than comparing only equivalent stress because it reveals whether pressure direction, wall orientation and end restraint are correct.
Why component stresses matter
Equivalent stress is convenient for yield assessment, but it hides the physical composition of the pressure state. Reviewing hoop, axial and radial components separately reveals whether the solution follows equilibrium and whether local bending is plausible. It also supports fatigue and fracture work, where principal direction, membrane/bending separation and stress range can matter more than a single equivalent-stress contour.