Pipe section properties to stress — SWBPIPE verification

Part VI · VerificationVI–C13

Part VI

Verification

C. Stress recovery · Pipe section properties to stress

An invented pipe has an outside diameter of 2.0 m and a wall of 0.25 m. At midspan it carries a shear of 4.0 N and a bending moment of 4.0 N·m, with no axial force and no torque. What are the area, section modulus and torsion constant of this section, and what stresses do the midspan resultants produce?

The case also checks software plumbing. The section properties are taken from a separate section-property calculation that carries its own source record, rather than being typed into the stress benchmark.

Fig. VI–C13.—Pipe section properties to 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
Outside diameter	D_o	2.0 m
Wall thickness	t	0.25 m
Axial force	N	0.0 N
Shear force	V_y	4.0 N
Shear force	V_z	0.0 N
Torsional moment	T	0.0 N·m
Bending moment about y	M_y	0.0 N·m
Bending moment about z	M_z	4.0 N·m

2.Method.

The inside diameter is the outside diameter less twice the wall. The area, second moment and torsion constant follow from the annulus formulas. The section is symmetric, so the section moduli about y and z are equal, each the second moment divided by the outer radius.

The midspan resultants are those of the upstream mechanics case and enter the stress recovery at those values; this test does not solve the model again. A separate machine-readable form of the section calculation evaluates the same formulas independently of the solver and is compared with the solver’s exported section values.

D_i = D_o − 2t = 1.5 m, r = D_o / 2 = 1.0 m(1)

A = (π/4)(D_o² − D_i²) = 1.3744467859455345 m²(2)

I = (π/64)(D_o⁴ − D_i⁴), Z_y = Z_z = I / r = 0.5368932757599744 m³(3)

J = (π/32)(D_o⁴ − D_i⁴) = 1.0737865515199487 m⁴(4)

σ_b,z = M_z / Z_z = 4.0 / 0.5368932757599744 = 7.450270250336039 Pa(5)

σ_axial = 0.0 / 1.3744467859455345 = 0.0, σ_b,y = 0.0 / 0.5368932757599744 = 0.0, τ = 0.0 × 1.0 / 1.0737865515199487 = 0.0(6)

3.Results.

Expected section properties and midspan stresses
Quantity	Expected
Area, A	1.374447 m²
Section modulus about y and z	0.5368933 m³
Torsion constant, J	1.073787 m⁴
Torsion radius, r	1.0 m
Axial normal stress	0.0 Pa
Bending stress about z	7.45027 Pa
Bending stress about y	0.0 Pa
Torsional shear stress	0.0 Pa

The tests check that:

The section properties carry their source record, with an outside diameter of 2.0 m and a wall of 0.25 m.
The section moduli about y and z are equal.
The midspan stress recovery completes without being blocked and is labelled as this case’s midspan station.
The midspan shear matches 4.0 N and the bending moment matches 4.0 N·m.
The axial stress, bending stress about y and torsional shear stress match 0.0 Pa; the bending stress about z matches 7.450270250336039 Pa.
No pressure stress is reported, and the result carries the standard review flags.

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 checked value must match the hand calculation to within an absolute difference of 1.0 × 10⁻⁹ in its own units. Labels, flags and the source record are checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

4.Run it yourself.

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

Hand calculation: validation/hand_calcs/stress/tp_phys_015_canonical_resultant_stress.md (with validation/hand_calcs/stress/generated/tp_phys_015_section_property_stress_witness.md). Test record, with the recorded run of 2026-07-10: stress-tp-phys-015-canonical-resultant-stress-recovery.md.