Thin-wall pressure membrane stress — SWBPIPE verification

Part VI · VerificationVI–C4

Part VI

Verification

C. Stress recovery · Thin-wall pressure membrane stress

A thin cylindrical wall of radius 3.0 m and thickness 0.5 m holds an internal pressure of 100.0 Pa. What hoop and longitudinal membrane stresses does the pressure set up in the wall? The pressure, radius and thickness are entered directly as the pressure inputs; the solver does not work them out from a pipe size.

Read with

The formulas use the mean radius of a thin wall; code forms such as pD_o/4t differ slightly. On straight pipe the program today leaves the longitudinal pressure stress out of its results (Status).

Fig. VI–C4.—Thin-wall pressure membrane stress

1.Inputs.

Illustrative values, taken from no standard and chosen so the arithmetic can be followed by hand. Most do not describe a real pipe; read them in any consistent set of units.

Quantity	Symbol	Value
Pressure	p	100.0 Pa
Membrane radius	r	3.0 m
Wall thickness	t	0.5 m

2.Method.

The thin-wall membrane results are used. The hoop stress balances the pressure across a cut along the axis and equals pr/t. The longitudinal stress balances the pressure on a closed end against the wall ring and is half the hoop stress.

σ_hoop = p r / t = 100.0 × 3.0 / 0.5 = 600.0 Pa(1)

σ_long = σ_hoop / 2 = 600.0 / 2.0 = 300.0 Pa(2)

3.Results.

Expected membrane stresses
Quantity	Expected
Hoop membrane stress, σ_hoop	600.0 Pa
Longitudinal membrane stress, σ_long	300.0 Pa

The tests check that:

The stress recovery completes without being blocked.
The hoop stress matches 600.0 Pa.
The longitudinal stress matches 300.0 Pa.

What it shows. Compared with an independent hand calculation.

Path exercised. The benchmark calls the solver’s components directly: elements, loads, frame solver and stress recovery. It does not go through the program’s own model-to-solve path.

Agreement. Each recovered stress must match the hand calculation to within an absolute difference of 1.0 × 10⁻⁹ Pa. Yes-or-no conditions, such as whether the result is blocked, are checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

Why is the longitudinal stress exactly half the hoop stress? Cut the cylinder along its axis: per unit length, the pressure pushes on a projected width 2r and two wall sections of thickness t resist it. Cut it across the axis: the pressure pushes on the end area πr² and the ring 2πrt resists it. The first gives pr/t, the second pr/2t.

4.Run it yourself.

cd projects/chirality-piping
cargo test --manifest-path validation/benchmarks/stress/Cargo.toml recovers_pressure_membrane_fixture

Hand calculation: validation/hand_calcs/stress/pressure_membrane.md. Test record, with the recorded run of 2026-07-10: stress-pressure-membrane-original.md.