Archive · 24 September 2026 · Thermal axial force to stress

Part VI · VerificationVI–C11

Part VI

Verification

C. Stress recovery · Thermal axial force to stress

A straight pipe 6.0 m long carries a thermal axial force of 240.0 N. The force is entered directly; it is not worked out from a temperature or a material. The pipe does not move. What axial force and axial stress does the solver report at end j, at midspan, and at stations along the length? No pressure is supplied.

Fig. VI–C11.—Thermal axial force 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	6.0 m
Elastic modulus	E	1200.0 Pa
Shear modulus	G	500.0 Pa
Pipe area	A	6.0 m²
Second moment about local y	I_y	2.0 m⁴
Second moment about local z	I_z	2.0 m⁴
Torsion constant	J	1.0 m⁴
Thermal axial force	F_t	240.0 N
Displacements	—	all zero
Station	x/L	0.5
Sweep stations	x/L	0.0, 0.5, 1.0
Stress area	A	6.0 m²
Section moduli	Z_y, Z_z	3.0 m³
Torsion radius	r	0.5 m

2.Method.

The thermal axial force enters as equivalent local loads of −F_t at the I end and +F_t at the J end. With no displacement, the recovered end forces are the negatives of those loads.

The station path starts from the I-end force. With no other loads, the axial force is the same at every station. Each force is divided by the area to give the axial stress.

F_x,i = 240.0 N, F_x,j = −240.0 N(1)

σ_end,j = F_x,j / A = −240.0 / 6.0 = −40.0 Pa(2)

N_station = 240.0 N, σ_station = N_station / A = 240.0 / 6.0 = 40.0 Pa(3)

3.Results.

Expected axial forces and stresses
Quantity	Expected
End-j axial force	−240.0 N
End-j axial stress	−40.0 Pa
Midspan axial force	240.0 N
Midspan axial stress	40.0 Pa
Sweep at 0.0, 0.5 and 1.0: axial stress	40.0 Pa at each station
Pressure stresses	None reported

The tests check that:

The end and midspan stress recoveries complete without being blocked, and the end result is for end j.
The end-j axial force matches −240.0 N and its stress matches −40.0 Pa.
The midspan axial force matches 240.0 N and its stress matches 40.0 Pa.
The sweep returns three stations at 0.0, 0.5 and 1.0, each with 240.0 N and 40.0 Pa, none blocked.
No hoop or longitudinal pressure stress is reported at the end or at midspan.

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 force and stress must match the hand calculation to within an absolute difference of 1.0 × 10⁻⁹ in its own units. Counts, labels and the absence of pressure stresses are checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

The end value and the station value have opposite signs, yet they describe one constant axial force. The end figure is the force on the element at end j in the direction of local x; the station figure is the internal force carried forward from the I end. Always read an end-force result with its sign convention in hand.

4.Run it yourself.

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

Hand calculation: validation/hand_calcs/stress/tp_phys_008_thermal_axial_effect_to_stress.md. Test record, with the recorded run of 2026-07-10: stress-tp-phys-008-thermal-axial-effect-to-stress.md.