Station resultants to stress — SWBPIPE verification

Part VI · VerificationVI–C7

Part VI

Verification

C. Stress recovery · Station resultants to stress

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

Fig. VI–C7.—Station resultants 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
Length	L	4.0 m
Elastic modulus	E	1000.0 Pa
Second moment about local z	I_z	2.0 m⁴
Uniform local y load	q	−2.0 N/m
Point local y force	P	−4.0 N
Point force station	a/L	0.5
Free-end deflection	v(L)	−0.04533333 m
Free-end rotation	θ(L)	−0.01466667 rad
Stress area	A	3.0 m²
Section modulus about y	Z_y	2.5 m³
Section modulus about z	Z_z	2.0 m³
Torsion constant	J	1.0 m⁴
Torsion radius	r	0.5 m

2.Method.

The solver recovers the loaded end forces at the fixed end from the free-end movement: a shear V_y,i = 12.0 and a moment M_z,i = 24.0. It then walks along the pipe to the station at x = 2.0 m, adding the effect of the loads it passes. The point force sits exactly at the station and is counted in.

The midspan axial force, torque and moment about y are zero. The stresses follow from force over area, moment over section modulus, and torque times radius over torsion constant. Transverse shear stress is not part of this case.

V_y(x) = V_y,i + q x + P = 12.0 + (−2.0 × 2.0) + (−4.0) = 4.0 N(1)

M_z(x) = M_z,i − V_y,i x − q x²/2 − P(x − a) = 24.0 − 12.0 × 2.0 − (−2.0 × 2.0²/2) − (−4.0 × 0.0) = 4.0 N·m(2)

σ_b,z = M_z / Z_z = 4.0 / 2.0 = 2.0 Pa(3)

σ_axial = N / A = 0.0 / 3.0 = 0.0 Pa(4)

3.Results.

Expected midspan values (the test checks the moment, the bending stress about z and the axial stress)
Quantity	Expected
Axial force, N	0.0 N
Shear force, V_y	4.0 N
Bending moment, M_z	4.0 N·m
Axial normal stress	0.0 Pa
Bending stress about z	2.0 Pa
Bending stress about y	0.0 Pa
Torsional shear stress	0.0 Pa

The tests check that:

The midspan stress recovery completes without being blocked and is labelled 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.Run it yourself.

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.