# plate_hole

Family 2: a plate of width W and thickness t with a central through hole of diameter d, pulled along x.

The classic stress concentration: at the hole edge on the x = 0 section the axial stress peaks. Hand-checkable anchors:
Kirsch (infinite thin plate: Kt = 3 on the gross stress), Howland's finite-width values in Pilkey's fit for thin plates,
and the 3D thickness effect (the mid-plane value rises a few % above the plane-stress one as t/d and nu grow).

Only one eighth is solved: symmetry on x = 0, y = 0 and the mid-plane z = 0; the end face x = Lh is pulled with a
uniform traction. Lengths are scaled by W (W = 40 mm in the solves); in linear elasticity
Kt and the compliance E * t * u_end / (F * (Lh / W)) depend only on d/W, t/W and nu.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `d_over_W` | 0.1 - 0.6 |
| `t_over_W` | 0.05 - 0.5 |
| `nu` | 0.25 - 0.35 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `kt_net` | 0.75 % / 2.7 % / 2.7 % | max 0.64 % | max 0.013 % |
| `compliance_star` | 0.012 % / 0.025 % / 0.025 % | max 0.00078 % | max 2.4e-05 % |

## Checks

- points: 300 solved, 0 failed
- three-mesh studies: 30 (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): 0.15 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: kt_net vs d_over_W ok (rho -0.986), compliance_star vs d_over_W ok (rho 1.0), compliance_star vs t_over_W ok (rho -0.011)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `kt_over_pilkey` | 1.036 | 1.051 | 1.061 |
| `kt_peak_over_mid` | 1.001 | 1.004 | 1.009 |

## 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.