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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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# Euler buckling of a clamped–free column of solids
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A square column, length 10, side 1, E = 1000, ν = 0 (2 × 2 × 10 hex20), clamped at its base and
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pressed by a unit dead pressure on its free top: effective length 20. Reference: Euler with
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Engesser's shear correction.
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{
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"title": "Euler buckling of a clamped–free column of solids",
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"kind": "theory",
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"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, length 10 clamped–free, effective length 20, P_e = π² E I/20², so σ_cr = P/A = 2.046071",
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"quantities": [
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{
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"id": "lambda",
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"quantity": "first buckling stress (load factor of a unit end pressure)",
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"measure": {
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"from": "solve",
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"pointer": "/modes/0/load_factor"
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},
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"value": 2.046070629714299,
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"tolerance": {
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"relative": 0.005
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},
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"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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}
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]
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}
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{
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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.",
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"values": {
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"lambda": {
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"band": 2.0460706297142995e-9,
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"value": 2.042374584299829
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}
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}
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}
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[
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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"}}},
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Column"}]}},
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{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Column", "face": "x0"}]}},
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{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Column", "face": "x1"}]}},
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{"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 1000.0, "poisson": 0.0}}},
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{"command": "model.section.create", "params": {"name": "col", "set": "all", "material": "Mat", "kind": {"type": "solid"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "base", "type": "fixed", "target": {"set": "x0"}}},
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{"command": "model.load.create", "params": {"name": "press", "type": "pressure", "set": "x1", "magnitude": 1.0}},
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{"command": "model.step.create", "params": {"name": "Buckle", "procedure": {"type": "buckle", "modes": 2}, "boundary_conditions": ["base"], "loads": ["press"]}},
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{"command": "solve.buckle", "params": {"step": "Buckle"}, "id": "solve"}
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]
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