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 internal pressure of 100.0 Pa. What hoop and longitudinal membrane stresses follow? Pressure, radius and thickness enter directly. Recovery does not derive these inputs from a pipe size.

This page preserves a recorded benchmark and its expected values. The dated run below is not a new run of the current product.

These calculations recover separate mechanics stress components. Component recovery does not establish a combined stress maximum or code stress.

This benchmark supplies pressure inputs directly to the historical membrane-stress helper. It does not establish pressure regions, closure ownership or pressure mechanics through realized bends.

Independent numerical comparison. This historical thin-wall helper and its test are absent from the current source. Reproduce them only at the evidence revision.

The invented wall-to-radius ratio is 0.5/3.0 = 1/6. The thin-wall formulas here provide an idealized membrane approximation.

Read with

The formulas use the mean radius of a thin wall; code forms such as pD_o/4t differ slightly. This historical component comparison does not establish a pressure solve through the current product path. See the Status for current pressure paths and their conditions.

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 case record describes these checks:

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 longitudinal stress exactly half the hoop stress? Cut the cylinder along its axis. Per unit length, pressure acts on projected width 2r, resisted by two wall sections of thickness t. Cut the cylinder across its axis. Pressure acts on end area πr², resisted by ring area 2πrt. The first balance gives pr/t, and the second gives pr/2t.

4.Recorded command.

This command belongs to the archived source at 462f66975d36cd10d5b8455193ada9517d28d353. Use a checkout of that exact revision. This edition did not run the command. The recorded result does not establish current product support.

git checkout --detach 462f66975d36cd10d5b8455193ada9517d28d353
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.