# large_deflection

Family 8: a slender cantilever bent far beyond small deflection by a dead load at its tip.

Exact anchor: the inextensible elastica (Bisshopp and Drucker, 1945). With the load parameter a = F L^2 / (E I), the tip
travels delta/L = 0.3017 down and 0.0566 in at a = 1 (small-deflection theory says 1/3 and 0). One load curve gives every
level at once, so the answers are read at a = 1, 3 and 6. Elastic to any strain here (Green strain, second Piola stress):
a real metal yields first once the surface strain a t / (2 L) passes its yield strain.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `t_over_L` | 0.01 - 0.03 |
| `w_over_t` | 1.0 - 4.0 |
| `nu` | 0.25 - 0.35 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `delta_1` | 0.0095 % / 0.016 % / 0.016 % | max 2.8e-06 % | max - |
| `short_1` | 0.016 % / 0.034 % / 0.034 % | max 5.8e-06 % | max - |
| `delta_3` | 0.015 % / 0.018 % / 0.018 % | max 1.4e-06 % | max - |
| `short_3` | 0.022 % / 0.039 % / 0.039 % | max 3.3e-06 % | max - |
| `delta_6` | 0.02 % / 0.034 % / 0.034 % | max 7.2e-07 % | max - |
| `short_6` | 0.024 % / 0.045 % / 0.045 % | max 2e-06 % | max - |

## Checks

- points: 120 solved, 0 failed
- three-mesh studies: 24 (every 5th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: delta_1 vs t_over_L ok (rho 0.321), delta_6 vs t_over_L ok (rho 0.591)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `delta_1_over_elastica` | 1 | 1 | 1.001 |
| `short_1_over_elastica` | 1 | 1.001 | 1.001 |
| `delta_3_over_elastica` | 1 | 1.001 | 1.001 |
| `short_3_over_elastica` | 1 | 1.001 | 1.001 |
| `delta_6_over_elastica` | 1 | 1.001 | 1.001 |
| `short_6_over_elastica` | 1 | 1.001 | 1.002 |

## 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.