engine: crank-arm sample, a bicycle crank swept all-hex and solved

file.sample crank-arm: a 170 mm 6061-T6 crank built in SolidCraft
(crates/engine/samples/crank-arm.json), swept along z by mesh.hex.generate
as 7274 hex20 elements of 3 mm (scaled Jacobian min 0.17). The hex variant
drops the pocket floor fillet and boss chamfers, and its root arcs meet the
bosses 20 degrees off tangent so the arm's top face has no zero-angle cusp.
The square bore is fixed and 1500 N acts on the pedal hole; the faces are
picked by their surfaces, not their names. Its heavy test matches the root
bending stress to M/Z = 62.6 MPa within 3 % (0.3 % in practice).

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

docs/examples.md

@@ -111,3 +111,28 @@ peaks at 295 MPa at the upright's hole.
 **Check:** the tip of the foot moves 0.509 mm down, within 1 % of a 48 016-element tet10
 solution of the unpartitioned part (0.5093 mm, `crates/testkit/tests/bracket_blocks.rs`), and the
 clamp carries the whole 1 kN.
+
+## Bicycle crank arm (`crank-arm`)
+
+![Von Mises stress on the all-hex crank arm, the arm root and the square bore carrying the most](media/examples/crank-arm.png)
+
+A 170 mm aluminium crank built in SolidCraft (`crates/engine/samples/crank-arm.json`): a 16 mm
+arm tapering from 38 to 24 mm, root arcs into a Ø46 × 22 spindle boss with a 12.7 mm square bore
+(R1.5 corners) and a Ø30 × 20 pedal boss with a Ø14 hole, and an 82 × 14 × 8 pocket in its back
+(R5 corners). It is a stack of extrusions along z, so `mesh.hex.generate` sweeps it all-hex:
+7 274 hex20 elements of 3 mm, all valid, scaled Jacobian min 0.17, mean 0.83. To keep it
+sweepable the hex variant leaves out the pocket's floor fillet and the bosses' 1.5 mm chamfers,
+and its root arcs meet the bosses about 20° off tangent (a tangent meeting leaves a zero-angle
+cusp on the arm's top face, where any hex is degenerate). 6061-T6 (E = 68 900 MPa, ν = 0.33); the
+square bore's faces fixed, 1500 N on the pedal hole in the crank's plane, across the arm (−y).
+Linear static: about 4 s to open and 7 s to solve.
+
+**Look at:** the swept mesh (Prepare ▸ Mesh), eight layers through the arm, and the von Mises
+contours: the bending stress grows along the arm toward the root; the peak, 129 MPa, sits at a
+square-bore corner where the rigid clamp meets the free face, a support singularity.
+
+**Check:** σxx on the arm's sides at x = 29 mm is ±62.4 MPa against the beam formula
+M/Z = 1500 · 141 / (16 · 35.6² / 6) = 62.6 MPa (0.3 %); a 41 815-element tet10 solution of the
+full part gives 62.3 MPa. The pedal bore moves 0.75 mm down (tet10: 0.70–0.75 mm; Euler–Bernoulli
+with a rigid clamp: 0.58 mm, without shear or the boss's compliance), and the spindle carries the
+whole 1500 N and 255 N·m.