verification: Euler columns and Bryan's plate in buckling steps

buckle-euler-pinned and buckle-euler-cantilever (hex20 columns, −0.18 % from Euler with
Engesser's shear correction) and buckle-plate-square (a simply supported square plate, b/t =
100, −0.39 % from k = 4; the support and thickness choices in its README). ROADMAP and parity
updated, docs/oracle.md regenerated.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed 2026-10-14 13:30:59 -04:00
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# Euler buckling of a pinned column of solids
A pinned–pinned square column, length 20, side 1, E = 1000, ν = 0, modelled as its half with a
symmetry plane at midspan (2 × 2 × 10 hex20). The pinned end is held laterally over its face,
which leaves its rotation free, and pressed by a unit pressure; the first load factor is the
buckling stress. Reference: Euler with Engesser's shear correction.
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{
"title": "Euler buckling of a pinned column of solids",
"kind": "theory",
"source": "Euler's column with Engesser's shear correction P = P_e/(1 + P_e/(κ G A)), κ = 5/6 (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §2.1, §2.17): square section 1 × 1, E = 1000, ν = 0, effective length 20, P_e = π² E I/20², so σ_cr = P/A = 2.046071",
"quantities": [
{
"id": "lambda",
"quantity": "first buckling stress (load factor of a unit end pressure)",
"measure": {
"from": "solve",
"pointer": "/modes/0/load_factor"
},
"value": 2.046070629714299,
"tolerance": {
"relative": 0.005
},
"reason": "hex20, 2 × 2 across and 10 along: the 3D solid's bending and shear against beam theory, and the preload concentrated at the held face"
}
]
}
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{
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
"values": {
"lambda": {
"band": 2.0460706297142995e-9,
"value": 2.042400056453741
}
}
}
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[
{"command": "mesh.block", "params": {"body": "Column", "kind": "hex20", "corners": [[0, 0, 0], [10.0, 0, 0], [10.0, 1.0, 0], [0, 1.0, 0], [0, 0, 1.0], [10.0, 0, 1.0], [10.0, 1.0, 1.0], [0, 1.0, 1.0]], "divisions": [10, 2, 2], "faces": {"6": "x0", "4": "x1"}}},
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Column"}]}},
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Column", "face": "x0"}]}},
{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Column", "face": "x1"}]}},
{"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 1000.0, "poisson": 0.0}}},
{"command": "model.section.create", "params": {"name": "col", "set": "all", "material": "Mat", "kind": {"type": "solid"}}},
{"command": "model.boundary_condition.create", "params": {"name": "pin", "type": "displacement", "target": {"set": "x0"}, "values": [null, 0.0, 0.0, null, null, null]}},
{"command": "model.boundary_condition.create", "params": {"name": "mid", "type": "symmetry", "target": {"set": "x1"}, "plane": "x"}},
{"command": "model.load.create", "params": {"name": "press", "type": "pressure", "set": "x0", "magnitude": 1.0}},
{"command": "model.step.create", "params": {"name": "Buckle", "procedure": {"type": "buckle", "modes": 2}, "boundary_conditions": ["pin", "mid"], "loads": ["press"]}},
{"command": "solve.buckle", "params": {"step": "Buckle"}, "id": "solve"}
]