Hot-state properties and stress range

Part VI · VerificationVI–C15

Part VI

Verification

C. Stress recovery · Hot-state properties and stress range

A pipe of metal area 0.004 m² is fully restrained at both ends and warmed uniformly by 10.0 K. Its elastic modulus and expansion coefficient are entered at a hot temperature point: 1.8 × 10¹¹ Pa and 1.3 × 10⁻⁵ 1/K. What axial force and stress arise? What is the stress range to the cold, unloaded state, and does the range record which set of properties each state used?

A second question concerns values between entered points. With properties entered at 300.0 K and 500.0 K, what modulus and expansion coefficient apply at 400.0 K by straight-line interpolation?

Read with

This case forms a range with the properties of the hot state. For the code range the reference (installed) modulus is used, and hot reactions are scaled from it (Part II D, Status).

Fig. VI–C15.—Hot-state properties and stress range

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
Hot-point elastic modulus	E_hot	1.8 × 10¹¹ Pa
Hot-point expansion coefficient	α_hot	1.3 × 10⁻⁵ 1/K
Temperature change	ΔT	10.0 K
Pipe metal area	A	0.004 m²
Hot-state property label	—	temperature_point:hot
Cold-state property label	—	material_base_values
Lower and upper entered temperatures	T₁, T₂	300.0 K, 500.0 K
Solve temperature	T_solve	400.0 K
Elastic modulus at T₁ and T₂	E₁, E₂	2.0 × 10¹¹ Pa, 1.8 × 10¹¹ Pa
Expansion coefficient at T₁ and T₂	α₁, α₂	1.2 × 10⁻⁵ 1/K, 1.4 × 10⁻⁵ 1/K

2.Method.

A bar fixed at both ends and warmed uniformly cannot lengthen, so its free thermal strain αΔT is cancelled by an equal and opposite mechanical strain. The axial force is EAαΔT, using the hot-point values, and the stress is that force over the area. The cold state carries no load, so the range equals the hot stress.

The range is formed together with a record that names the property set of each state exactly as entered; the solver does not infer it. Between two entered temperatures, each property is interpolated along a straight line. The reference does not extend beyond the two entered points.

In the test, the hot axial force is formed from the closed form in the benchmark and passed to the solver’s stress recovery and range. The interpolated values are checked as the benchmark’s own arithmetic, not through a solver interpolation routine.

F_hot = E_hot A α_hot ΔT = 1.8 × 10¹¹ × 0.004 × 1.3 × 10⁻⁵ × 10.0 = 93600.0 N(1)

σ_axial,hot = F_hot / A = E_hot α_hot ΔT = 1.8 × 10¹¹ × 1.3 × 10⁻⁵ × 10.0 = 2.34 × 10⁷ Pa(2)

Δσ_axial = σ_axial,hot − σ_axial,cold = 2.34 × 10⁷ Pa, with σ_axial,cold = 0.0 Pa(3)

r = (T_solve − T₁) / (T₂ − T₁) = (400 − 300) / (500 − 300) = 0.5(4)

E(400 K) = 2.0 × 10¹¹ + 0.5 × (1.8 × 10¹¹ − 2.0 × 10¹¹) = 1.9 × 10¹¹ Pa(5)

α(400 K) = 1.2 × 10⁻⁵ + 0.5 × (1.4 × 10⁻⁵ − 1.2 × 10⁻⁵) = 1.3 × 10⁻⁵ 1/K(6)

3.Results.

Expected hot-state values, range, property record and interpolated values
Quantity	Expected
Hot-state axial force, F_hot	93600.0 N
Hot-state axial stress	2.34 × 10⁷ Pa
Cold-state axial stress	0.0 Pa
Axial stress range	2.34 × 10⁷ Pa
Property record, first (hot) state	temperature_point:hot
Property record, second (cold) state	material_base_values
Interpolated elastic modulus at 400 K	1.9 × 10¹¹ Pa
Interpolated expansion coefficient at 400 K	1.3 × 10⁻⁵ 1/K

The tests check that:

The hot-state stress recovery completes without being blocked, and its axial stress matches 2.34 × 10⁷ Pa.
The range between the hot and cold states completes without being blocked, and its axial range matches 2.34 × 10⁷ Pa.
The range record names the hot state as temperature_point:hot and the cold state as material_base_values, exactly as entered.
A range formed without a record carries no property record and gives the same ranges.
The interpolated modulus and expansion coefficient match 1.9 × 10¹¹ Pa and 1.3 × 10⁻⁵ 1/K in the benchmark’s own arithmetic.

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. The property labels and the blocking conditions are checked exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

For a fully restrained bar, the thermal stress EαΔT does not depend on the area; only the force does. A heavier wall raises the load on the restraints but leaves the stress unchanged. The stress does depend on which modulus and expansion coefficient are used, which is why the range records them.

4.Run it yourself.

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

Hand calculation: validation/hand_calcs/stress/tp_pmm_p3_modulusbasis_range_stress.md. Test record, with the recorded run of 2026-07-10: stress-tp-pmm-p3-modulusbasis-range-stress.md.