Pressure through a realized bend — SWBPIPE verification

Part VI · VerificationVI–A1

Part VI

Verification

A. System benchmarks · Pressure through a realized bend

An L line has an anchor at A and a free end at D. A realized circular bend joins two straight pipes.

Internal pressure is 5.0e6 Pa. Both closed terminal caps transfer their pressure loads to the pipe wall. Thermal change and self-weight are zero.

The independent reference uses the direct unit-load method. The current test calls the captured product API in both solver modes and document versions.

This case compares independent numerical values. It does not exercise desktop authoring, the native CLI or report export.

The reference matches small-strain linear elasticity with homogeneous isotropic material and a user-entered bend flexibility factor.

The formulation excludes flow momentum, transient loads, Bourdon bend opening, pressure stiffening and ovalization.

The frozen expectations predate this refresh. The date below records a new product comparison, not a new reference derivation.

The focused test passed on 2026-10-12: one test passed, none failed. Its assertions cover 316 reference, mode and document-version configurations.

Source 1. Source 2.

1.Inputs.

The table gives the declared reference inputs. Units are stated for each quantity.

Quantity	Symbol	Value
Node A coordinates	A	(0.0, 0.0, 0.0) m
Node B coordinates	B	(3.0, 0.0, 0.0) m
Node C coordinates	C	(3.25, 0.25, 0.0) m
Node D coordinates	D	(3.25, 4.25, 0.0) m
Bend radius	R	0.25 m
Bend flexibility factor	k	1
Outside diameter	OD	0.1683 m
Nominal wall	t	0.00711 m
Mill tolerance	m	0.000889 m
Elastic modulus	E	2.0e11 Pa
Poisson ratio	nu	0.3
Internal pressure	p	5.0e6 Pa
Terminal closure transfer	A and D	transfers_to_wall
Support	A	Anchor; all six freedoms restrained

2.Method.

The reference integrates axial, torsional and bending flexibility over the whole line. It does not use a curved-element stiffness matrix.

Pressure caps produce wall tension. The pressure eigenstrain includes Poisson contraction on each straight and arc.

Positive wall force means tension. The effective axial force is wall force minus pressure times bore area.

The reference uses meters and pascals. The product converts translations to millimeters and bending or torsion stresses to megapascals. Axial membrane stress stays in pascals.

t_eff = 0.00711 − 0.000889 = 0.006221 m(1)

N_w = p A_i; S = N_w − p A_i; sigma_m = N_w / A_s(2)

3.Results.

Selected independent reference values in SI units
Quantity	Expected
Effective wall	6.221e-3 m
Bore area	1.907867e-2 m²
Metal area	3.167647e-3 m²
Node B displacement X	1.806893e-4 m
Node C displacement X	1.957468e-4 m
Node C displacement Y	1.505744e-5 m
Node D displacement X	1.957468e-4 m
Node D displacement Y	2.559766e-4 m
Bend wall axial force at midspan	9.539335e+4 N
Bend effective axial force at midspan	0 N
Bend axial membrane stress at midspan	3.011489e+7 Pa
Anchor reaction; each force and moment component	0 N for force; 0 N·m for moment
Arc Lamé radial and hoop rows	Withheld

The case record describes these checks:

The test compares every valued reference case in both sparse_interactive and dense_scrutiny modes. Each document runs as 0.3.0 and 0.4.0.
The comparisons include node movement, support reactions, station forces and stresses, end actions and terminal cap transfers.
The test requires the pressure integrity verdict and checks required row absences on arcs.
The wall end action follows the end tangent. The arc elastic end-force rows use the chord frame. These quantities have different meanings.

What it shows. Compared with an independent reference calculation.

Path exercised. The captured product API, in both solver modes and both document versions. The comparison does not exercise desktop authoring, the native CLI or report export.

Agreement. The test requires |observed − expected| ≤ 1e-9 × max(|expected|, zero_scale(group)). The frozen reference defines each group floor. Exact unit conversion precedes comparison. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

The free L line grows under pressure while its anchor reaction remains zero. Wall tension remains nonzero; effective axial force is zero.

4.Recorded command.

This command ran against source a72f994d89d901b185116e6565bacb201b69e151 on 2026-10-12. The linked execution record states the result and its scope.

git checkout --detach a72f994d89d901b185116e6565bacb201b69e151
cd projects/chirality-piping
cargo test --offline --locked --manifest-path core/product_physics/Cargo.toml --test t4_u2_pressure_references i7_valued_cases_match_in_both_modes_and_document_versions -- --exact

Execution record: 2026-10-12.

Reference calculation: validation/references/t4_i7/u2_reference_cases.json. Test source for the execution of 2026-10-12: t4_u2_pressure_references.rs.