Derived-normal friction with rotational lift-off

Part VI · VerificationVI–E10

Part VI

Verification

E. Algorithm and software checks · Derived-normal friction with rotational lift-off

A single frame member is fixed at its root. At the tip an axial force of 10.0 N acts along x against a friction support with μ = 0.06, whose normal force is the reaction at a restraint in y loaded by −100.0 N. A moment of 2.0 N·mm acts about z, where the rotation is held by a lift-off support. The friction support starts sticking and the lift-off support starts in contact. Where do they 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). The one-way support on a rotation is a numerical test of the iteration on a rotational degree of freedom. No such device exists in piping.

Fig. VI–E10.—Derived-normal friction with rotational lift-off

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 moment about z	M_z	2.0 N·mm
Friction support	—	U_x at the tip
Normal source	—	U_y restraint
Lift-off support	—	R_z at the tip
Friction coefficient	μ	0.06
Derived normal reaction	N	100.0 N
Friction limit	μN	6.0 N
Initial friction state	—	sticking
Initial lift-off state	—	active (in contact)

2.Method.

The friction support takes the absolute reaction at the y restraint, 100.0 N, as its normal force, so 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 rotational lift-off support releases. Both 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 lift-off state	inactive (released)
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, rotational lift-off support released. 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_derived_normal_rotational_acceptance.md. Test record, with the recorded run of 2026-07-10: nl-assembled-multi-dof-derived-normal-rotational-accepted-original.md.