{
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  "engine": "hollow-circular-section",
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    "boundary": {
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    "property": {
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  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
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  "reviewer_role": null,
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    "scope": [
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      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
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  "assumptions": [],
  "limitations": [
    "Hollow circle about centroidal diameters",
    "Ix = Iy = π(R⁴−r⁴)/4",
    "Sx = Sy = Ix/R",
    "kx = ky = √(R²+r²)/2",
    "Not a solid circle, a rectangle, bending stress, or a plastic modulus"
  ],
  "boundary_conditions": [
    "missing R or r → MISSING_REQUIRED_INPUT",
    "R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "r ≥ R → INVALID_INPUT"
  ],
  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia — circular area",
      "supports": "The centroidal moment of a hollow circle about a diameter is π(R⁴−r⁴)/4.",
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      "ok": true,
      "kind": "golden",
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        "r": 1
      },
      "expect": {
        "area": 9.42477796076938,
        "Ix": 11.780972450961723,
        "Iy": 11.780972450961723,
        "Sx": 5.890486225480862,
        "Sy": 5.890486225480862,
        "kx": 1.118033988749895,
        "ky": 1.118033988749895,
        "model": "hollow_circular_section"
      },
      "expectWarning": null,
      "actual": {
        "model": "hollow_circular_section",
        "R": 2,
        "r": 1,
        "area": 9.42477796076938,
        "Ix": 11.780972450961723,
        "Iy": 11.780972450961723,
        "kx": 1.118033988749895,
        "ky": 1.118033988749895,
        "Sx": 5.890486225480862,
        "Sy": 5.890486225480862,
        "solver": "area moment + section modulus",
        "convention": "Hollow circle about centroidal diameters. R is the outer radius and r is the inner radius. Ix = Iy = π(R⁴−r⁴)/4. Sx = Sy = Ix/R. kx = ky = √(R²+r²)/2. A solid circle is the circular-section page. A rectangle is the rectangular-section page."
      },
      "tol": 1e-12
    },
    {
      "id": "golden-four",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "R": 4,
        "r": 2
      },
      "expect": {
        "area": 37.69911184307752,
        "Ix": 188.49555921538757,
        "Iy": 188.49555921538757,
        "Sx": 47.12388980384689,
        "Sy": 47.12388980384689,
        "kx": 2.23606797749979,
        "ky": 2.23606797749979
      },
      "expectWarning": null,
      "actual": {
        "model": "hollow_circular_section",
        "R": 4,
        "r": 2,
        "area": 37.69911184307752,
        "Ix": 188.49555921538757,
        "Iy": 188.49555921538757,
        "kx": 2.23606797749979,
        "ky": 2.23606797749979,
        "Sx": 47.12388980384689,
        "Sy": 47.12388980384689,
        "solver": "area moment + section modulus",
        "convention": "Hollow circle about centroidal diameters. R is the outer radius and r is the inner radius. Ix = Iy = π(R⁴−r⁴)/4. Sx = Sy = Ix/R. kx = ky = √(R²+r²)/2. A solid circle is the circular-section page. A rectangle is the rectangular-section page."
      },
      "tol": 1e-12
    },
    {
      "id": "golden-alias",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "Ro": 2,
        "Ri": 1
      },
      "expect": {
        "area": 9.42477796076938,
        "Ix": 11.780972450961723,
        "Sx": 5.890486225480862,
        "kx": 1.118033988749895
      },
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      "actual": {
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        "r": 1,
        "area": 9.42477796076938,
        "Ix": 11.780972450961723,
        "Iy": 11.780972450961723,
        "kx": 1.118033988749895,
        "ky": 1.118033988749895,
        "Sx": 5.890486225480862,
        "Sy": 5.890486225480862,
        "solver": "area moment + section modulus",
        "convention": "Hollow circle about centroidal diameters. R is the outer radius and r is the inner radius. Ix = Iy = π(R⁴−r⁴)/4. Sx = Sy = Ix/R. kx = ky = √(R²+r²)/2. A solid circle is the circular-section page. A rectangle is the rectangular-section page."
      },
      "tol": 1e-12
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    {
      "id": "boundary-outer",
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      "inputs": {
        "R": 0,
        "r": 1
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
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        "r": 0
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        "r": 2
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      "id": "boundary-missing",
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      "kind": "boundary",
      "inputs": {
        "r": 1
      },
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "R is required."
    }
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