Part VI
Verification
B. Elements and loads · Fully restrained thermal growth
A straight prismatic member is held at both ends and warmed uniformly by 75.0 K. It wants to grow but cannot, so the restraints must push it back to its original length. What is the free thermal strain, and what axial force does full restraint produce?
No stiffness system is solved here: with both ends held there is nothing free to move. The case checks the product’s thermal-restraint calculation, which turns these inputs into the axial force the restraints must supply.
The case checks the equivalent thermal load, EAαΔT, that the solver is given for a pipe held at both ends.
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.
3.Results.
| Quantity | Expected |
|---|---|
| Free thermal strain | 0.0009 |
| Restrained axial force magnitude | 5.4 N |
The tests check that:
- 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 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 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.
The length of the member does not appear anywhere. A fully restrained bar develops the same force whether it is one metre long or a hundred.
4.Run it yourself.
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.