Cantilever with a tip force — SWBPIPE verification

Part VI · VerificationVI–B2

Part VI

Verification

B. Elements and loads · Cantilever with a tip force

A single straight member ten metres long is fixed at one end and free at the other. A lateral force of 6.0 N acts at the free end. How far does the tip deflect, and what moment does the fixed end carry? It is the simplest bending case in the suite: one element, one load, one answer.

Fig. VI–B2.—Cantilever with a tip force

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
Length	L	10.0 m
Elastic modulus	E	1200.0 Pa
Second moment of area	I_z	4.0 m⁴
Tip force	P_y	6.0 N

2.Method.

The reference uses the elementary cantilever results. The tip deflection is PL³/(3EI), and the moment at the fixed end is the force times the length. The solver’s fixed-end moment is recovered from the solved displacements as the reaction the root restraint must supply, so it carries the opposite sign to the applied moment: −60.0 N·m, magnitude 60.0.

δ_y = P_yL³ / (3EI_z) = 6.0 × 10.0³ / (3 × 1200.0 × 4.0) = 0.4166666666666667 m(1)

M_z = P_yL = 60.0 N·m(2)

3.Results.

Tip deflection and fixed-end moment
Quantity	Expected
Tip deflection in Y	0.4166667 m
Fixed-end moment about Z	60.0 N·m

The tests check that:

The solver builds one frame element, fixes all six freedoms at the root and solves for the tip.
The solved tip deflection in Y matches the hand value.
The fixed-end reaction moment recovered from the solution is −60.0 N·m, the hand magnitude with the reaction’s sign.

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.

4.Run it yourself.

cd projects/chirality-piping
cargo test --manifest-path validation/benchmarks/mechanics/Cargo.toml cantilever_tip_force_matches_open_mechanics_formula

Hand calculation: validation/hand_calcs/mechanics/cantilever_tip_force.md. Test record, with the recorded run of 2026-07-10, with the strengthened check first run on 2026-09-22: mech-cantilever-tip-force.md.