Fully restrained thermal growth — SWBPIPE verification

Part VI · VerificationVI–B3

Part VI

Verification

B. Elements and loads · Fully restrained thermal growth

A straight prismatic member is held at both ends and heated uniformly by 75.0 K. The restraints prevent free thermal growth. What is the free thermal strain, and what axial force does full restraint produce?

No stiffness system is solved here. Both ends are held, so no degree of freedom remains free. The case tests the recorded thermal-restraint calculation for the force that the restraints must supply.

This page preserves a recorded benchmark and its expected values. The dated run below is not a new run of the current product.

This benchmark isolates the recorded axial-effect helper.

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

Read with

The case checks the equivalent thermal load, EAαΔT, that the solver is given for a pipe held at both ends.

Fig. VI–B3.—Fully restrained thermal growth

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
Elastic modulus	E	2000.0 Pa
Area	A	3.0 m²
Thermal expansion coefficient	α	0.000012 1/K
Temperature change	ΔT	75.0 K

2.Method.

The free thermal strain is the expansion coefficient times the temperature change. Full restraint must supply a mechanical strain equal and opposite to it, so the axial force is EA times that strain. This reference reports the restraint-force magnitude. Physical thermal restraint of the heated bar produces compression.

ε = α ΔT = 0.000012 × 75.0 = 0.0009(1)

F = EAα ΔT = 2000.0 × 3.0 × 0.000012 × 75.0 = 5.4 N(2)

3.Results.

Free strain and restrained force
Quantity	Expected
Free thermal strain	0.0009
Restrained axial force magnitude	5.4 N

The case record describes these checks:

The free thermal strain equals 0.0009, checked as reference arithmetic.
The product’s thermal-restraint calculation, given the same invented inputs, returns a restrained axial force of 5.4 N, matching the hand value.

What it shows. Checks the loads or boundary conditions prepared for the solver against a hand calculation. No system is solved.

Path exercised. The axial-effect helper prepares the thermal load. No global frame system is solved.

Agreement. Each computed value must match the reference within an absolute difference of 1.0 × 10⁻⁹ in the case’s own units. Counts and structural outcomes must match exactly. Long values are shown here to seven significant figures; the tests compare the full values in the record.

For the student

Member length does not enter this calculation. With these uniform properties and full restraint, a bar develops the same restraint-force magnitude at one meter or a hundred meters.

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/mechanics/Cargo.toml thermal_growth_fixture_records_open_axial_restraint_formula

Hand calculation: validation/hand_calcs/mechanics/fixed_fixed_thermal_axial.md. Test record, with the recorded run of 2026-07-10, with the strengthened check first run on 2026-09-22: mech-fixed-fixed-thermal-axial.md.