Friction and a gap at one tip — SWBPIPE verification

Part VI · VerificationVI–E12

Part VI

Verification

E. Algorithm and software checks · Friction and a gap at one tip

A single frame member is fixed at its root. Its tip is pushed by 10.0 N along x and 1.0 N along y. In x the tip rests on a friction support with μ = 0.30 and a given normal reaction of 20.0 N; in y it faces a gap of 0.0002 mm. The friction support starts sticking and the gap starts open. Where do the two supports end up?

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 and acts on the resultant sliding in the support plane (Part II H, Status).

Fig. VI–E12.—Friction and a gap at one tip

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
Tip transverse force	F_y	1.0 N
Friction support	—	U_x
Gap support	—	U_y
Friction coefficient	μ	0.30
Normal reaction (given)	N	20.0 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 limit is 0.30 × 20.0 = 6.0 N. The 10.0 N axial force is more than that, so the support slides. The gap 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.

μN = 0.30 × 20.0 = 6.0 N(1)

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 tests check that:

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. Checks the supports’ final states, the iteration count and the balance of forces. Movements and reactions are not compared 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 from one iteration to the next: the changed-support count must reach exactly 0, with no relative or absolute allowance, within at most 4 iterations. At the final iteration the out-of-balance force, moment and work at the free degrees of freedom must be 0.0, and the changes in displacement, rotation and reaction over the last iteration must stay inside a recorded envelope. Long values are shown here to seven significant figures; the tests compare the full values in the record.

4.Run it yourself.

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_friction_gap_acceptance.md. Test record, with the recorded run of 2026-07-10: nl-assembled-multi-dof-friction-gap-accepted-original.md.