Finite-element analysis your agent can run

The loop's answer is the 3 mm pocket: 267.1 g against 292.7 g for the solid arm, 25.6 g (9 %) off with the peak at 173.5 MPa on the pocket end and SF 1.59 against S_y 276 MPa. At 4 mm (a 12 mm web) the SF drops to 1.27 and the loop stops banking mass. Every row is a solved case with its case directory beside it; the “where” column is the solver's peak coordinate placed on the geometry the loop itself built, so “pocket end” is the inboard face at the pocket's end arc, “pocket end wall” is that wall itself and “pocket floor” the floor or its fillet — not a guess from a picture.

Two peaks are reported on purpose. The arm peak is the judged one: the element-averaged integration-point von Mises over every element of the arm, read from the solver's own .dat with the modelled pedal left out — the spindle is a steel part in reality and is drawn here in the arm's aluminium, monolithic with it, only to carry the load 65 mm out, so its own bending stress and the sharp corner where it meets the crank face are artefacts of the stand-in. The body peak beside it is what cad_fea_run reported for the whole body, pedal included, so nothing is hidden: it is always on the pedal stub, and it is why the loop judges by zone rather than by the global maximum. Every solve here was cut with meshQuality: true (the boundary re-cut and the tet-quality pass, public on cad_fea_run since this page): the first two runs of this loop, on the plain mesher's 50–60 % slivers, put the arm's peak in a tet of shape quality 4×10⁻⁵ — a number that belonged to the mesh, not the part.

The first cut of this loop — a 26 mm arm on a Ø36 boss, the load applied in the crank plane at the pedal eye — failed at SF 1.19 before any pocket was cut, with the peak in the 34° arm/boss notch. The readout said where; the geometry was redrawn without the notch and the load was moved to where the standard puts it. That is the loop doing its job.

A 120×20×4 mm AISI 1018 steel bar, one end face fixed (#127 from cad_body_topology), cad_modal for six modes. Three calls; the deck was written in a sandbox with no solver (deckOnly: true) and solved on a workstation with CalculiX 2.21 — 29,022 nodes, 17,697 C3D10 tets on the quality mesh (< 0.01 % slivers, min dihedral 9.6°), 33.5 s for mesh and solve together.

Three references, each computed by the page generator from the case's own E = 205 GPa, ν = 0.29, ρ = 7870 kg/m³ and the deck's own bounding box: Euler–Bernoulli (the slender beam), Timoshenko (adds shear deformation and rotary inertia, κ = 10(1+ν)/(12+11ν) = 0.849, the clamped-free characteristic equation solved for each mode, no series truncation) and, for torsion, Saint-Venant with the exact series J and then Vlasov warping restraint at the clamp. A Δ is bold only where that formula's own slenderness assumption holds: L/h ≥ 20 in the bending direction for the slender formula (flap L/t = 30, yes; in-plane L/b = 6, no), L/h ≥ 5 for Timoshenko and Vlasov. Against those, all 6 modes land within 0.9 % on the first mesh, with no tuning. Which mode is which is read from the solver's own mass-participation vector, not assigned by hand.

Where the gaps go. In-plane, the bar is short (L/b = 6): shear and rotary inertia lower the frequency by 2.1 % and 12.3 %, which the slender formula cannot see — the solver's in-plane gaps of −1.8 % and −12.0 % against it close to +0.3 % and +0.4 % against Timoshenko. In torsion, Saint-Venant lets the section warp freely; the clamp holds it flat, and the bar is stiffened in a boundary layer about 1/k = 5.0 mm long at the root (kL = 24.2). Spreading that over the whole length as a one-term sine adds only +0.2 %; the exact warping-restrained eigenvalue adds +4.5 %, and the solver's +3.9 % becomes −0.6 %. In flap, the slender formula's closer match on the higher modes is partly a cancellation: shear and rotary inertia lower them by 0.1 %, 0.6 %, 1.4 %, while a first quadratic-tet mesh is stiff. What is left in bending (+0.3 % to +0.9 %) has one sign on every bending mode, which is what an unconverged displacement mesh does — it is an upper bound. Torsion's −0.6 % is the other way: Vlasov ignores the shear strain of the warping stresses, which overstates the clamp's stiffening for a section this thick (b/t = 5), so the warping reference is likely itself a little high there.

Convergence rung pending. The same deck at three mesh densities, and an independent second solver on the finest mesh, are queued for the workstation; until they report, this page quotes no converged value and does not split the residual between the formula and the mesh.

Slender: f_n = β_n²/(2π)·√(EI/ρAL⁴), cosβcoshβ + 1 = 0. Timoshenko: the 4×4 clamped-free determinant in b² = ρAL⁴ω²/EI with r² = I/AL², s² = EI/κAGL² (Han, Benaroya & Wei 1999). Torsion: J = βbt³, β = ⅓[1 − (192/π⁵)(t/b)Σ_{n odd} tanh(nπb/2t)/n⁵] = 0.2913; Saint-Venant f = (1/4L)·√(GJ/ρI_p); Vlasov: EC_wφ⁽⁴⁾ − GJφ″ = ρI_pω²φ, φ = φ′ = 0 at the clamp, C_w ≈ b³t³/144 (the one-term estimate ω² = [GJ(π/2L)² + EC_w(π/2L)⁴]/ρI_p gives 2432.4 Hz).

Public, in the tarball and the .mcpb

• cad_fea_case — body + boundary conditions (fixed / sliding / force / pressure / bearing on face ids), material preset or SI values, body loads, static or modal solve, tessellation and provenance blocks → a case directory.
• cad_fea_run — C3D10 deck, ccx, .frd read back: peak von Mises and its location and face, nodal peak, displacement, safety factor, reaction, warnings, one-line summary. solver: "auto" | "calculix" | <a registered adapter>; an unknown name is refused by name, never substituted.
• cad_modal — eigenpairs, restrained or free-free, with the rotor block for centrifugal prestress and Campbell sweeps; deckOnly hands the deck to a workstation or a queue; MYSTRAN as a second modal path.
• The design-loop driver: target → build → verdict → remedy → iterate → stop, with provenance per iteration, for this verdict and for cad_dfm_check, cad_mass_properties, cad_tolerance_stack and the rest.

Honest about the mesh

Every results.json carries the mesh it was solved on — node and tet counts, target size, the dihedral histogram and the sliver fraction — and a warning when the boundary tessellation is needle-rich. The cantilever above was cut with the boundary and tet quality passes (--improve-surface --improve-quality): < 0.01 % slivers. The first cut, without them, was 43.6 % slivers by count — and moved the three flap frequencies by under 0.2 %. That is the point of publishing the mesh block: you can see what the mesh did to the number instead of wondering. The same readout is what makes the bracket's two peaks (integration-point and nodal) useful — when they disagree by 2× the mesh is telling you to refine. The blade-qualification chain — Campbell, forced response, Goodman — runs on the same case directory and stays a private edition for now.