# l_bracket_plastic

Family 1P: the L-bracket of family 1 in elastic-perfectly-plastic steel (J2, no hardening, small strain).

The plastic correction on top of the linear space. Loads are measured in units of the first-yield load F_y of the
same shape, taken from the certified linear answer (F_y = sy L^2 / sigma*, sigma* = von Mises peak of family 1):
  lam_limit  the collapse load / F_y  (the plastic reserve beyond first yield; the solver stops at collapse)
  amp_15     tip deflection at 1.5 F_y / the linear deflection at the same load (how much plasticity adds)
For an elastic-perfectly-plastic material at small strain both depend only on the shape and nu (not on sy or E), so
one row again holds for any steel grade, size and stiffness. Hand anchor: a rectangular section collapses at 1.5 x its
first-yield moment; with the fillet's concentration the bracket yields earlier, so lam_limit should exceed about 1.5
for slender brackets (beam reference = plastic hinge at the flange root).

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `h_over_L` | 0.4 - 1.0 |
| `t_over_L` | 0.04 - 0.15 |
| `w_over_L` | 0.25 - 0.75 |
| `r_over_t` | 0.25 - 1.5 |
| `nu` | 0.25 - 0.35 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `lam_limit` | 1.9 % / 4.3 % / 4.3 % | max 0.61 % | max - |
| `amp_15` | 1.4 % / 1.5 % / 1.5 % | max 0.037 % | max - |

## Checks

- points: 29 solved, 1 failed
- three-mesh studies: 8 (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 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: 

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `lam_over_hinge` | 1.002 | 1.026 | 1.036 |
| `hinge_lambda` | 2.12 | 2.656 | 2.763 |
| `steps` | 12 | 15.6 | 16 |

## 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.