Part VI
Verification
B. Elements and loads · Three-member branch assembly
A straight header runs between two anchors, with a junction two metres from the left anchor and three metres from the right. A branch rises four metres from the junction. Only movement along the branch is free at the junction and the tip. A force of 90.0 N pulls on the branch tip along its length. How far do the junction and the tip move, and how do the two anchors share the load?
The case tests assembly: three members meeting at one node must form a single stiffness matrix.
Only movement along the branch is free, so the model reduces to the branch in series with two guided header segments. It checks assembly at a junction, not the behaviour of a tee (Status).
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 |
|---|---|---|
| Left header length | Lleft | 2.0 m |
| Right header length | Lright | 3.0 m |
| Branch length | Lbranch | 4.0 m |
| Elastic modulus | E | 1200.0 Pa |
| Shear modulus | G | 500.0 Pa |
| Area | A | 2.0 m2 |
| Second moment about local y | Iy | 3.0 m4 |
| Second moment about local z | Iz | 4.0 m4 |
| Torsion constant | J | 1.5 m4 |
| Tip force along the branch | F | 90.0 N |
2.Method.
The branch carries the tip force axially, so its stiffness is EA/L. With rotations held at the junction, each header segment resists the junction movement as a guided beam, with lateral stiffness 12EI/L3. The two header segments act in parallel; the branch sits in series with them.
The restrained model reduces to two unknowns, the junction and tip displacements. The junction moves by the force over the header stiffness, and the tip moves further by the branch extension. Each anchor reaction is its segment stiffness times the junction displacement, and the two reactions sum to the applied force.
3.Results.
| Quantity | Expected |
|---|---|
| Branch axial stiffness | 600.0 N/m |
| Header lateral stiffness | 9333.333 N/m |
| Junction displacement | 0.009642857 m |
| Branch tip displacement | 0.1596429 m |
| Branch axial extension | 0.15 m |
| Left anchor reaction | −69.42857 N |
| Right anchor reaction | −20.57143 N |
| Sum of anchor reactions | −90.0 N |
The tests check that:
- The solver assembles the three members and solves the restrained system.
- The branch and header stiffnesses match the hand values.
- The junction and tip displacements and the branch extension match the hand values.
- Both anchor reactions, and their sum, match the hand values.
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 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 header segments share the junction movement, so they add like springs in parallel; the branch passes the whole force through itself, so it adds like a spring in series. The stiffer, shorter left segment takes the larger share of the reaction.
4.Run it yourself.
cd projects/chirality-piping cargo test --manifest-path validation/benchmarks/mechanics/Cargo.toml branch_assembly_fixture_matches_open_stiffness_network
Hand calculation:
validation/hand_calcs/mechanics/branch_assembly.md.
Test record, with the recorded run of 2026-07-10:
mech-branch-assembly-three-member.md.