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  "limitations": [
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    "Z = (4/3)(R³−r³)",
    "Shape factor Z/S",
    "Not a solid circle, a rectangle, or a plastic moment"
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    "R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE",
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      "title": "Hibbeler, Mechanics of Materials",
      "section": "Plastic deformation — tubular section",
      "supports": "The plastic modulus of a hollow circle about a diameter is (4/3)(R³−r³).",
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        "r": 1
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        "S": 5.890486225480862,
        "f": 1.584475877892647,
        "model": "hollow_circular_plastic_modulus"
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        "area": 9.42477796076938,
        "S": 5.890486225480862,
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        "r": 2
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        "S": 47.12388980384689,
        "f": 1.584475877892647
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