Closed gap opens again — SWBPIPE verification

Part VI · VerificationVI–D5

Part VI

Verification

D. Nonlinear supports · Closed gap opens again

A straight member 1.0 mm long, with an axial stiffness of 100.0 N/mm, is fixed at one end. The other end can move only along the member’s axis. A stop 0.01 mm beyond the free end resists positive movement, and the gap starts closed, with the end resting against the stop. The end is then pulled back by −2.0 N. A stop can push but cannot hold on. Does the gap open, and where does the end come to rest?

Fig. VI–D5.—Closed gap opens again

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
Node count	—	2
Member length	L	1.0 mm
Axial stiffness	k	100.0 N/mm
Applied axial force	F	−2.0 N
Gap clearance	c	0.01 mm
Gap closing direction	—	positive displacement
Maximum iterations	—	4
Active-set residual tolerance	—	0.0 (count)
Initial state	—	active (gap closed)

2.Method.

With the gap closed, the first solve holds the end at the clearance, +0.01 mm. The reaction there is 100.0 × 0.01 − (−2.0) = +3.0 N. A stop that resists positive movement can bear only with a negative reaction. A positive reaction means the stop would have to pull the end onto itself, so the gap opens.

On the second iteration the end is free and moves −2.0 / 100.0 = −0.02 mm. That does not reach the +0.01 mm clearance, so the gap stays open, nothing changes state, and the loop has converged.

R₁ = k c − F = 100.0 × 0.01 − (−2.0) = +3.0 N(1)

u₂ = F / k = −2.0 / 100.0 = −0.02 mm(2)

3.Results.

Expected values
Quantity	Expected
Closed-gap pulling reaction (iteration 1)	3.0 N
Released free displacement (iteration 2)	−0.02 mm
Final reaction	0.0 N
Iteration count	2
Final changed-support count	0.0
Final state	inactive (gap open)
Converged	true

The tests check that:

The solve converges in exactly 2 iterations, ending with the gap open and a changed-support count of 0.
On the first iteration the end is held at 0.01 mm and the reaction is 3.0 N, compared exactly.
The final displacement is −0.02 mm and the final reaction is 0.0 N, compared exactly.
At the final iteration the out-of-balance force, moment and work at the free degrees of freedom are all exactly 0.0.
The solve carries no notice that its tolerance is unset.
Any change in displacement or reaction over the last iteration stays inside the recorded envelope.

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. 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 change in displacement 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.

For the student

A closed gap sits exactly at its clearance, so its displacement alone can never show that it should open. Only the sign of the contact force can: a stop that would have to pull is a stop the pipe has already left.

4.Run it yourself.

cd projects/chirality-piping
cargo test --manifest-path validation/benchmarks/nonlinear/Cargo.toml assembled_global_loop_seed_cases_converge_under_governed_policy
cargo test --manifest-path validation/benchmarks/nonlinear/Cargo.toml assembled_convergence_observations_record_measured_fixture_values_only
cargo test --manifest-path validation/benchmarks/nonlinear/Cargo.toml assembled_force_displacement_residual_observations_use_free_dof_force_moment_thresholds
cargo test --manifest-path validation/benchmarks/nonlinear/Cargo.toml dec_067_transition_fixtures_witness_state_switched_and_bounded_outcomes

Hand calculation: validation/hand_calcs/nonlinear/assembled_gap_lift_off.md. Test record, with the recorded run of 2026-07-10: nl-assembled-gap-lift-off-original.md.