{
  "capability_id": "mechanical.statics.rectangular_plastic_modulus",
  "tool_id": "rectangular-plastic-modulus",
  "engine": "rectangular-plastic-modulus",
  "calculation_version": "1.0.0",
  "generated_at": "2026-09-23T21:48:01.335Z",
  "engine_build": "2026-09-23",
  "test_count": 4,
  "passed": 4,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 2,
      "passed": 2
    },
    "boundary": {
      "total": 2,
      "passed": 2
    },
    "property": {
      "total": 0,
      "passed": 0
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/physics/rectangular-plastic-modulus",
  "api": "https://www.calculatorx.com/api/v1/calc/rectangular-plastic-modulus",
  "schema_input": "https://www.calculatorx.com/schemas/mechanical.statics.rectangular_plastic_modulus.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/mechanical.statics.rectangular_plastic_modulus.output.json",
  "assumptions": [],
  "limitations": [
    "One solid rectangle",
    "Zx = b h²/4 and Zy = h b²/4",
    "Shape factor 3/2",
    "Not a circle, bending stress, or a plastic moment"
  ],
  "boundary_conditions": [
    "missing b or h → MISSING_REQUIRED_INPUT",
    "b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE"
  ],
  "sources": [
    {
      "title": "Hibbeler, Mechanics of Materials",
      "section": "Plastic deformation — rectangular section",
      "supports": "The plastic modulus of a rectangle about the axis through mid-depth is b h²/4, and the shape factor is 3/2.",
      "kind": "reference"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:5006153e3ecd5813500bc3b14b57d673e049a4572f73ca38399126de790ad543",
  "tests": [
    {
      "id": "golden-wide",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "b": 2,
        "h": 6
      },
      "expect": {
        "area": 12,
        "Zx": 18,
        "Zy": 6,
        "Sx": 12,
        "Sy": 4,
        "fx": 1.5,
        "fy": 1.5,
        "model": "rectangular_plastic_modulus"
      },
      "expectWarning": null,
      "actual": {
        "model": "rectangular_plastic_modulus",
        "b": 2,
        "h": 6,
        "area": 12,
        "Sx": 12,
        "Sy": 4,
        "Zx": 18,
        "Zy": 6,
        "fx": 1.5,
        "fy": 1.5,
        "solver": "rectangular section",
        "convention": "Zx = b h²/4 and Zy = h b²/4 about the centroidal axes. Sx and Sy come from the rectangular-section engine. The shape factor is 3/2. Bending stress is the bending-stress page. A circle is its own page."
      },
      "tol": 1e-12
    },
    {
      "id": "golden-square",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "b": 2,
        "h": 2
      },
      "expect": {
        "area": 4,
        "Zx": 2,
        "Zy": 2,
        "Sx": 1.3333333333333333,
        "Sy": 1.3333333333333333,
        "fx": 1.5,
        "fy": 1.5
      },
      "expectWarning": null,
      "actual": {
        "model": "rectangular_plastic_modulus",
        "b": 2,
        "h": 2,
        "area": 4,
        "Sx": 1.3333333333333333,
        "Sy": 1.3333333333333333,
        "Zx": 2,
        "Zy": 2,
        "fx": 1.5,
        "fy": 1.5,
        "solver": "rectangular section",
        "convention": "Zx = b h²/4 and Zy = h b²/4 about the centroidal axes. Sx and Sy come from the rectangular-section engine. The shape factor is 3/2. Bending stress is the bending-stress page. A circle is its own page."
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-width",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "b": 0,
        "h": 6
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "Width b must be positive."
    },
    {
      "id": "boundary-missing",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "h": 6
      },
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "b is required."
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 4,
      "total": 4,
      "golden": 2,
      "boundary": 2,
      "property": 0
    },
    "contract_checks": {
      "passed": 6,
      "total": 6
    }
  },
  "evidence_core_generated_at": "2026-09-23T14:32:49.903Z",
  "rendered_at": "2026-09-23T21:48:01.335Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.0.0",
    "artifacts": {
      "engine": "1.0.0",
      "page": "1.0.0",
      "capability": "1.0.0",
      "evidence": "1.0.0",
      "input_schema": "1.0.0",
      "output_schema": "1.0.0",
      "mcp_contract": "1.0.0"
    },
    "digests": {
      "input_schema_sha256": "e4c0bb46e9deb022b578471dbd1660c063ae82c046ee176cc5832053e641befb",
      "output_schema_sha256": "f7ecd0aa4870d51abafba438738360113dd27fd302514034b07662ded196ddb0",
      "evidence_sha256": "055fe0532c14d7529f5b271aa2c17988f8a293e87ee08a70b0c71cba79ef5da5",
      "evidence_core_sha256": "055fe0532c14d7529f5b271aa2c17988f8a293e87ee08a70b0c71cba79ef5da5",
      "capability_sha256": "5eb0ba19228852807593bbad41e0585e4952533bf83a4a61d3236cf0aad29c99",
      "engine_sha256": "9c01357c59289b486db75219c319b48412c2e4bcd3a700b0d2dd8145ce528650",
      "mcp_contract_sha256": "70bebadd7ed143df4a08cee8eb28a96620802497fda67dd843f4f57c33fd45d3"
    },
    "contract_checks": {
      "total": 6,
      "passed": 6,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.0.0"
        },
        {
          "id": "schema-output-required-declared",
          "status": "pass"
        },
        {
          "id": "schema-valid-input-executes",
          "status": "pass"
        },
        {
          "id": "engine-success-output-validates-schema",
          "status": "pass"
        },
        {
          "id": "schema-rejects-incomplete-input",
          "status": "pass"
        },
        {
          "id": "rest-success-payload-validates-schema",
          "status": "pass"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  }
}