Three-member branch assembly — SWBPIPE verification

Part VI · VerificationVI–B1

Part VI

Verification

B. Elements and loads · Three-member branch assembly

A straight header runs between two anchors. Its junction is two meters from the left anchor and three meters from the right. A branch rises four meters from the junction. Only movement along the branch is free at the junction and tip. A force of 90.0 N pulls along the branch at its tip. How far do the junction and tip move, and how do the anchors share the load?

The case tests assembly. Three members that meet at one node must form a single stiffness matrix.

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

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

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Only movement along the branch is free. Thus, the model reduces to the branch in series with two guided header segments. It checks assembly at a junction, not the behavior of a tee (Status).

Fig. VI–B1.—Three-member branch assembly

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	L_left	2.0 m
Right header length	L_right	3.0 m
Branch length	L_branch	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	I_y	3.0 m⁴
Second moment about local z	I_z	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. Rotations at the junction are restrained. Each header segment therefore acts as a guided beam, with lateral stiffness 12EI/L³. The header segments act in parallel. The branch acts in series with the header.

The restrained model has two unknowns: junction displacement and tip displacement. Junction displacement equals force divided by header stiffness. The tip moves farther by the branch extension. Each anchor reaction equals its segment stiffness times junction displacement, with the opposite sign. The reactions sum to the negative of the applied force.

k_branch = EA / L_branch = 1200.0 × 2.0 / 4.0 = 600.0 N/m(1)

k_left = 12EI_z / L_left³ = 7200.0 N/m, k_right = 12EI_z / L_right³ = 2133.3333333333335 N/m(2)

u_junction = F / (k_left + k_right) = 90.0 / 9333.333333333334 = 0.009642857142857142 m(3)

u_tip = u_junction + F / k_branch = 0.15964285714285714 m(4)

R_left = −k_left u_junction, R_right = −k_right u_junction(5)

3.Results.

Stiffnesses, displacements and anchor reactions
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 case record describes these checks:

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.

For the student

The header segments share the junction movement, so their stiffnesses add as springs in parallel. The branch carries the whole force and acts in series with the header. The shorter, stiffer left segment takes the larger reaction.

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 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.