Archive · 7 October 2026 · Derived-normal friction with a vertical gap

Part VI · VerificationVI–E9

Part VI

Verification

E. Algorithm and software checks · Derived-normal friction with a vertical gap

A frame member is fixed at its root. Tip force 10.0 N acts in x against friction with μ = 0.06. Normal force comes from a y restraint reaction under load −100.0 N. Force 6.0 N acts toward a gap of 0.0002 mm in z. Friction starts sticking, and the gap starts open. Where do the supports settle?

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

The current product limits active-set solves to four iterations. Current mechanics combinations with nonlinear supports are withheld. These recorded transitions and free-degree residuals do not establish general equilibrium or convergence for other models.

Read with

Friction here acts along one axis, with its normal force given or taken from another support. In a real line, friction belongs to the rest itself. It acts against the resultant sliding in the support plane (Part II H, Status).

Fig. VI–E9.—Derived-normal friction with a vertical gap

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
Tip axial force	F_x	10.0 N
Normal-source force	F_y	−100.0 N
Tip vertical force	F_z	6.0 N
Friction support	—	U_x at the tip
Normal source	—	U_y restraint
Gap support	—	U_z at the tip
Friction coefficient	μ	0.06
Derived normal reaction	N	100.0 N
Friction limit	μN	6.0 N
Gap clearance	c	0.0002 mm
Initial friction state	—	sticking
Initial gap state	—	inactive (open)

2.Method.

The friction support uses the absolute y restraint reaction, 100.0 N, as normal force. Its limit is 0.06 × 100.0 = 6.0 N. The 10.0 N axial force is more than that, so the support slides. The gap in z closes. Both supports change on the first iteration, and neither changes on the second.

The reference for this group records the expected support states, changed-support counts and final residuals. It does not work out displacements or reactions, so none are tabulated here.

N = |R_y| = 100.0 N(1)

μN = 0.06 × 100.0 = 6.0 N(2)

3.Results.

Expected outcome (the reference gives states, counts and residuals, not displacements or reactions)
Quantity	Expected
Changed supports, iteration 1	2
Final changed-support count	0
Iteration count	2
Free-DOF force residual	0.0 N
Free-DOF moment residual	0.0 N·mm
Final free-DOF work residual	0.0 N·mm
Final friction state	sliding
Final gap state	active (closed)
Converged	true
Diagnostic	none

The case record describes these checks:

The solve converges in exactly 2 iterations: 2 supports change state on the first iteration and none on the last.
Final states match the reference: friction sliding, gap closed. No diagnostics are raised.
At the final iteration the out-of-balance force, moment and work at the free degrees of freedom are all exactly 0.0.
The changes in displacement, rotation and reaction over the last iteration stay inside the recorded envelope for this group.

What it shows. Compares final support states, the iteration count, and force balance. It does not compare movements or reactions with a 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. The loop counts as converged only when no support changes state between iterations. The changed-support count must reach exactly 0 within at most 4 iterations, with no relative or absolute allowance. At the final iteration, out-of-balance force, moment and work at free degrees of freedom must be 0.0. Changes in displacement, rotation and reaction over the last iteration must stay inside the recorded envelope. Long values are shown here to seven significant figures; the tests compare the full values in the record.

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/nonlinear/Cargo.toml multisupport_acceptance_inventory_uses_narrow_dec_046_policy

Hand calculation: validation/hand_calcs/nonlinear/assembled_multi_support_derived_normal_gap_acceptance.md. Test record, with the recorded run of 2026-07-10: nl-assembled-multi-dof-derived-normal-gap-accepted-original.md.