How to Perform Stress Linearisation

Step-by-step guide to extracting membrane, bending and peak stress components from a pressure vessel FEA model using stress classification lines.

Pressure & Containment8 min read

pressurehow-tostress linearisationmembranebendingpeakSCLFEA

Step 1 — Define the stress classification line

A stress classification line (SCL) is a line through the wall thickness, perpendicular to both the inner and outer surfaces. At curved geometries such as nozzle corners, the SCL may follow a curved path. Place SCLs at all critical locations.

Step 2 — Extract the stress tensor along the SCL

Extract the full six-component stress tensor at each node or integration point along the SCL, from the inner surface to the outer surface. The stress tensor defines the complete stress state at each point through the wall.

Step 3 — Calculate the membrane stress

The membrane stress is the average of the stress tensor through the wall. Integrate each stress component along the SCL and divide by the wall thickness:

sigma_m = (1/t) * integral_0^t sigma(x) dx

where:
  sigma_m = membrane stress tensor component
  t = wall thickness
  sigma(x) = stress at position x through the wall

Step 4 — Calculate the bending stress

The bending stress is the linearly varying component, obtained from the first moment of the stress distribution about the mid-plane:

sigma_b = (6/t^2) * integral_0^t sigma(x) * (x - t/2) dx

The bending stress varies from tension on one surface
to compression on the other, with zero at the mid-surface.

Step 5 — Calculate the peak stress

The peak stress at each point is the difference between the actual stress and the membrane plus bending approximation:

sigma_peak(x) = sigma(x) - sigma_m - sigma_b(x)

Step 6 — Classify and assess

Classify the membrane, bending and peak components according to the applicable assessment framework. Compare each component with the appropriate allowable. The classification depends on the location and load type — a stress at a nozzle junction may be classified differently from a stress in the remote shell.

Common errors

Placing the SCL at a singularity where the stress does not converge
Using a non-perpendicular SCL at a curved geometry
Failing to extract all six stress components
Comparing peak stress directly with a membrane allowable