Pipe stress analysis
How much does the line grow, and is the leg long enough? Free thermal growth, the restrained axial stress, the guided-cantilever bending stress and the ASME B31.3 allowable stress range, with the leg length that would satisfy the code.
| Quantity | Value | Unit |
|---|---|---|
| Free thermal growth | 25.20 | mm |
| Restrained axial stress | 504.0 | MPa |
| Restrained anchor force | 1814.6 | kN |
| Guided-cantilever bending | 25.4 | MPa |
| Guided anchor load | 0.7 | kN |
| Sustained stress S_L | 5.92 | MPa |
| Allowable S_A | 204.9 | MPa |
| Required leg length | 3.52 | m |
S_A = f(1.25Sc + 0.25Sh) from ASME B31.3 §302.3.5.
Δ = α·L·ΔT · Mean expansion coefficient over the temperature range.
S = 3·E·D·Δ / L² · Classic screening formula for an L or Z leg.
S_A = f(1.25 S_c + 0.25 S_h) · ASME B31.3 §302.3.5, reduced when S_L > S_h.
A clear path from inputs to results.
Every hot line grows. The first question is how much, and the second is whether the leg that absorbs it is long enough. This screening check gives both, against the ASME B31.3 allowable stress range, and shows what a fully restrained line would do instead.
How to use
- Pick the line size and the developed leg length.
- Set the installation and design temperatures and the pressure.
- Read the free growth, the governing stress and the allowable, then compare the utilisation.
The formula
Reference: ASME B31.3 §302.3.5 allowable stress range; guided-cantilever screening formula.
Worked example
NPS 6, 10 m leg, 20 to 220 °C
- OD 168.3 mm, wall 7.11 mm, α 12 × 10⁻⁶, ΔT 200 K.
- 1
Δ = 12e-6 × 10,000 × 200 = 24 mm. Guided bending S = 3 × 200,000 × 168.3 × 24 / 10,000² = 24.2 MPa. S_A = 206.9 MPa.
24 mm growth, 24 MPa against 207 MPa allowable — 12% utilisation.
Interpreting the result
Restrained, the same line would develop 480 MPa of axial stress, far above the allowable. That is why a line is never fully restrained without expansion joints or loops.
Assumptions & limitations
- One leg absorbs the movement, guided at both ends.
- Constant modulus and expansion coefficient.
- Pressure stress treated as thin-wall longitudinal.
- No support friction, weight or nozzle loads.
Frequently asked questions
When do I need a real flexibility analysis?
When the utilisation is above about 0.8, when there are nozzles or rotating equipment, or when the code requires it.
What is f?
The stress range reduction factor for fatigue: 1.0 up to 7,000 equivalent cycles, falling to 0.6 at 100,000.
Why does stress fall with the square of the leg?
Bending stiffness of a cantilever scales with 1/L³ and the deflection with L, so the stress scales with 1/L².
Does insulation change the growth?
Only through the metal temperature; the insulation itself does not restrain the pipe.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.