# stepped_shaft

Family 4: a round shaft stepping from diameter D down to d with a shoulder fillet of radius r, in tension.

The textbook case of machine design. Reference: Peterson's charts as fitted in Pilkey (stepped bar of circular
cross section, tension), a trend check only, not a certificate: Peterson's shoulder-fillet curves were estimated from
related geometries and are known to run low, by up to 24 % (Tipton, Sorem and Schmidt, Univ. of Tulsa, FEA) and up to
25 % in tension at small radii (Oda, Takahata, Yamashita, Noda, J. Japan Soc. Design Eng. 61(3), 2025, body force
method). The certificate here is the mesh study, CalculiX and the axisymmetry check.
A quarter is solved (symmetry on x = 0 and y = 0); the end of the small shaft is a symmetry plane (z = 0, as if the
bar continued mirror-wise), the end of the large one is pulled. Lengths are scaled by d (d = 20 mm in the solves).

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `D_over_d` | 1.2 - 2.0 |
| `r_over_h` | 0.2 - 0.9 |
| `nu` | 0.25 - 0.35 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `kt` | 1.2 % / 2.8 % / 2.8 % | max 0.79 % | max 0.79 % |

## Checks

- points: 200 solved, 0 failed
- three-mesh studies: 20 (every 10th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0.84 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: kt vs r_over_h ok (rho -0.783), kt vs nu ok (rho -0.003)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `kt_over_peterson` | 1.107 | 1.163 | 1.199 |
| `peak_angle_deg` | 15 | 19.5 | 21 |

## Columns of points.csv

parameters, the answers, `<answer>_band` (relative), the hand-theory ratios, `mirror_err`, `equilibrium_err` (where a single reaction exists), `<answer>_loo_err` / `_neighbour_flag` (leave-one-out prediction from all other points), `<answer>_recheck_diff` / `_flag_resolved` (finer re-solve of a flagged point), `has_gci` / `has_similarity` / `has_calculix`, `nodes`, `solve_s`, `studio` (version).

Solved with NovaSolver Studio through the same commands an AI client uses over MCP; the first section says which solver and mesh. The claim is verification (the equations are solved within the band), not validation against tests.

## License

This data is licensed under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/
Credit: NovaSolver, https://novasolver.jp/studio/verified/
Provided as is, without warranty. Design decisions remain the user's responsibility.