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      "sources",
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  "limitations": [
    "Sx = Ix/cy and Sy = Iy/cx about axes parallel to x and y",
    "cx and cy are distances from the composite centroid to the extreme fiber",
    "A blank distance omits that modulus",
    "Negative area is a hole",
    "Not bending stress, plastic modulus, principal-axis modulus, or beam deflection"
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  "boundary_conditions": [
    "no parts → MISSING_REQUIRED_INPUT",
    "neither cx nor cy → MISSING_REQUIRED_INPUT",
    "cx or cy ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "Ix < 0 when cy is set, or Iy < 0 when cx is set → INVALID_INPUT"
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    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia",
      "supports": "Elastic section modulus S = I/c, with c the distance from the centroid to the extreme fiber",
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        "cx": 1,
        "cy": 2
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        "Iy": 4,
        "Sx": 8,
        "Sy": 4,
        "area": 4,
        "x_bar": 0,
        "y_bar": 0,
        "model": "section_modulus"
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        "area": 4,
        "x_bar": 0,
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        "Ix": 16,
        "Iy": 4,
        "Sx": 8,
        "Sy": 4,
        "cx": 1,
        "cy": 2,
        "n": 1,
        "solver": "area moment",
        "convention": "Sx = Ix/cy and Sy = Iy/cx about the composite centroid. cx and cy are distances to the extreme fiber."
      },
      "tol": 1e-12
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      "id": "golden-sx",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "1,0,1,0,0; 1,0,1,0,2",
        "cy": 1
      },
      "expect": {
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        "Sx": 4,
        "Sy": null,
        "y_bar": 1
      },
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      "actual": {
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        "area": 2,
        "x_bar": 0,
        "y_bar": 1,
        "Ix": 4,
        "Iy": 0,
        "Sx": 4,
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        "cx": null,
        "cy": 1,
        "n": 2,
        "solver": "area moment",
        "convention": "Sx = Ix/cy and Sy = Iy/cx about the composite centroid. cx and cy are distances to the extreme fiber."
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      "tol": 1e-12
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        "parts": "8,8,4,0,0; -2,-2,-1,0,0",
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        "cy": 1
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        "Ix": 6,
        "Iy": 6,
        "Sx": 6,
        "Sy": 6
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        "cy": 1,
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        "cy": 0
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        "cy": 2
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      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Ix must be non-negative to form Sx."
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