Barlow's Formula and Hoop Stress for Sample Probe Tube Design
First-principles derivation of Barlow's formula for hoop stress in thin-walled probe tubes. Worked examples for 316L, Hastelloy C276, and Inconel 625 across the standard OD and wall thickness range.
Barlow's Formula in One Line
σ = (P × D) / (2 × t)
Where σ is the hoop (circumferential) stress in the tube wall, P is the internal pressure, D is the outside diameter, and t is the wall thickness. This is the workhorse equation for sizing the wall of any sample probe tube.
When Barlow Applies
Barlow is the thin-wall approximation. It is accurate when D/t ≥ 20. Most commercial probe tubes (0.250"-1.000" OD, 0.020"-0.083" wall) sit comfortably inside that envelope.
For thicker walls (D/t < 20), use Lamé's equations for the exact stress distribution through the wall. The configurator does this automatically when the user enters a non-standard wall thickness.
Derivation in 30 Seconds
Cut a thin-walled tube longitudinally and consider a unit length. The internal pressure pushes outward on a projected area D × 1. That force is resisted by two strips of wall, each t × 1, in tension. Force balance:
P × D = 2 × t × σ → σ = P × D / (2 × t)
This is the simple form. The ASME B31.3 form adds correction factors for material allowable, joint efficiency, and corrosion allowance:
t = (P × D) / (2 × (S × E × W + P × Y))
Where S = allowable stress, E = quality factor (1.0 for seamless), W = weld joint strength reduction factor, Y = temperature coefficient.
Worked Example: 316L Probe at Room Temperature
A 0.500" OD × 0.065" wall probe in seamless 316L. Allowable stress at 100 °F = 16,700 psi (per ASME B31.3 Table A-1).
Maximum allowable working pressure (no corrosion allowance):
P = (2 × t × S) / D = (2 × 0.065 × 16,700) / 0.500 = 4342 psi
Apply a 12.5% manufacturing tolerance on wall thickness (t_min = 0.0569") and the same calculation gives 3800 psi MAWP. The configurator uses the t_min value as the conservative default.
Worked Example: Hastelloy C276 at 600 °F
Same 0.500" × 0.065" geometry. Allowable stress for C276 at 600 °F = 21,000 psi.
P = (2 × 0.065 × 21,000) / 0.500 = 5460 psi
Note that C276's higher allowable at temperature gives it a 40% MAWP advantage over 316L in the same geometry — a major reason it dominates in mid-temperature aggressive service.
Worked Example: Inconel 625 at 1000 °F
Same geometry. Inconel 625 allowable at 1000 °F = 17,400 psi.
P = (2 × 0.065 × 17,400) / 0.500 = 4524 psi
Inconel 625 retains usable strength at temperatures where 316L is essentially structural metal. The deeper material discussion is in Hastelloy vs Inconel.
Hydrogen Service Derating
For hydrogen probes, apply the ASME B31.12 hydrogen material performance factor (HMPF) to the allowable stress before running Barlow. A typical HMPF for 316L at 5000 psig H2 is 0.85, which directly reduces the calculated MAWP by 15%.
End-of-Life Consideration
A probe whose wall has thinned by 10% from corrosion or erosion has lost roughly 10% of its hoop-stress margin (Barlow is linear in t). The cycle life and replacement criteria blog treats wall loss > 10% as a hard replacement threshold for exactly this reason.
In the Configurator
Every change to OD, wall thickness, material, or process pressure in the SPA Configurator re-runs Barlow + the ASME B31.3 corrections in the background. If the resulting MAWP falls below the entered process pressure, the wizard highlights the offending dimension and suggests the next-thicker stock wall.