Hot-state properties and stress range

Part VI · VerificationVI–C15

Part VI

Verification

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

A pipe with metal area 0.004 m² is fully restrained and heated uniformly by 10.0 K. Hot-point elastic modulus is 1.8 × 10¹¹ Pa, and expansion coefficient is 1.3 × 10⁻⁵ 1/K. What axial force and stress arise? What is the range to the cold, unloaded state? Does the range record the property set for each state?

With properties entered at 300.0 K and 500.0 K, what modulus and expansion coefficient follow at 400.0 K from linear interpolation?

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 stress components, not code stress. The current ordinary preview reports supported straight circular normal-stress extrema. Torsional shear remains separate, and arc maxima are withheld. Component recovery in this benchmark does not establish the current maximum or its coverage.

This benchmark forms absolute differences of stress components. Current ordinary result-state subtraction is a labeled signed difference, not an equilibrium state. Current mechanics combinations are withheld for nonlinear supports, mixed moduli or constant effort whose factors do not sum to one. This recorded range calculation does not establish a current combination or maximum.

This benchmark tests stress recovery and property labels from entered forces. Its interpolation arithmetic does not establish current selection, thermal resolution or mixed-modulus combination behavior.

Read with

This case forms a range with the properties of the hot state. For the code range the calculation uses the reference (installed) modulus. Hot reactions are scaled from that basis (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.

Full restraint prevents thermal lengthening. Mechanical strain cancels free thermal strain αΔT. The reference restraint-force magnitude is EAαΔT, using hot-point properties. The benchmark enters that magnitude as positive axial force under its own convention. Physical thermal restraint of this heated bar produces compression. Dividing entered force by area gives positive benchmark stress, and the unloaded cold state gives the same range.

The range record names each property set exactly as entered. Recovery does not infer the sets. Between two entered temperatures, each property is interpolated linearly. The reference does not extend beyond those points.

The benchmark calculates hot axial force from the closed form, then passes it to stress recovery and range calculation. Interpolated values are tested as benchmark arithmetic. This test does not run 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 case record describes these checks:

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, thermal stress magnitude EαΔT is independent of area. Force depends on area. With the same properties and temperature change, a heavier wall raises restraint load but leaves stress unchanged. Stress depends on the selected modulus and expansion coefficient, so the range records those properties.

4.Recorded command.

This command belongs to the linked case record and its recorded execution path. This website update did not rerun it. A historical command or result does not establish current product support.

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.