A straight pipe 4.0 m long is fixed at node 0 and free at node 1. It carries −2.0 N/m along its length and a point force of −4.0 N at midspan. The companion mechanics case gives free-end deflection and rotation. What internal forces act at midspan? What axial, bending and torsional stresses do these forces produce?

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.

Independent numerical comparison. The recorded test exercises a benchmark crate. It does not exercise the desktop product route.

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.

2.Method.

From known free-end movement, recovery gives fixed-end shear V_y,i = 12.0 and moment M_z,i = 24.0. Recovery then proceeds to x = 2.0 m and includes each load encountered. The point force is exactly at the station and is included.

Midspan axial force, torque and moment about y are zero. Stress formulas use force divided by area, moment divided by section modulus, and torque times radius divided by torsion constant. This case does not include transverse shear stress.

3.Results.

The case record describes these checks:

• The midspan stress recovery completes without being blocked and is labeled as the midspan station.
• The midspan bending moment matches 4.0 N·m.
• The bending stress about z matches 2.0 Pa.
• The axial normal stress is exactly zero.

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. The moment and bending stress must match the hand calculation to within an absolute difference of 1.0 × 10⁻⁹ in their own units. The zero axial stress and the station label are checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

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 load_to_resultant_station_stress_fixture_recovers_bending_component

Hand calculation: validation/hand_calcs/stress/tp_phys_004_load_to_resultant_stress.md (with validation/hand_calcs/mechanics/tp_phys_004_load_to_resultant.md). Test record, with the recorded run of 2026-07-10: stress-tp-phys-004-load-to-resultant.md.