forked from 1b-agent/fieldcraft-upstream
jyvikoandClaude Opus 5.5
9f6afe7a89
arc-shallow-arch: a sinusoidal arch of 40 × 2 × 1 hex20 under a sinusoidal dead load scaled to the one-mode closed form's upper limit load: the upper limit point at 0.992 (−0.8 %), the lower at 0.219 (0.8 % of the peak load) and exactly one negative eigenvalue between them. 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
46 lines
1.8 KiB
JSON
46 lines
1.8 KiB
JSON
{
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"title": "Snap-through of a shallow arch by arc length",
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"kind": "theory",
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"source": "One-mode closed form of a shallow sinusoidal arch y₀ = a sin(πx/L) on pins at a fixed span under q₀ sin(πx/L): q₀ = EI k⁴ [c + (a − c)(2ac − c²)/(4r²)], k = π/L, limit points at c = a ∓ r √((a²/r² − 4)/3) (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §7.6; Y. C. Fung, A. Kaplan, NACA TN 2840, 1952): L = 100, h = 1, unit width, E = 10⁴, ν = 0, a/r = 3.5; the load is scaled to the upper limit load, so the peak load factor is 1 and the valley 0.211035",
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"quantities": [
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{
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"id": "peak",
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"quantity": "upper limit load factor (snap-through)",
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"measure": {
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"from": "solve",
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"pointer": "/critical_points/0/load_factor"
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.02
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},
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"reason": "40 × 2 × 1 hex20 against shallow Euler–Bernoulli arch theory, the load piecewise constant per element, and the limit load picked from converged states (second order in the arc)"
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},
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{
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"id": "valley",
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"quantity": "lower limit load factor",
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"measure": {
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"from": "solve",
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"pointer": "/critical_points/1/load_factor"
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},
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"value": 0.21103504613969407,
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"tolerance": {
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"absolute": 0.02
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},
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"reason": "as the peak; the valley is a fifth of the peak load and a small difference of the bending and axial terms, so its tolerance is stated as 2 % of the peak load"
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},
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{
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"id": "negative",
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"quantity": "negative eigenvalues of the tangent past the upper limit point",
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"measure": {
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"from": "solve",
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"pointer": "/critical_points/0/negative_after"
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},
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"value": 1,
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"tolerance": {
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"exact": true
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},
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"reason": "the inertia of the LDLᵀ factor: exactly one unstable direction between the limit points"
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}
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]
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} |