{
  "version": 1,
  "topic": "GM11",
  "title": "How a stress sign convention changes an apparent failure result",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Consistent and inconsistent sign transformations",
    "premise": "Compression-positive effective normal stress varies. Cohesion 500 psi, friction coefficient 0.6. Tension-positive input is the negative of the same physical normal compression.",
    "formula": "strength+ = c + μσc; strength− = c − μσt, with σt = −σc",
    "variables": "The deliberately wrong column uses c + μσt without transforming the envelope.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Effective compression (psi) = 2000",
        "explanation": "Compression-positive effective normal stress varies. Cohesion 500 psi, friction coefficient 0.6. Tension-positive input is the negative of the same physical normal compression."
      },
      {
        "label": "Select the relation",
        "calculation": "strength+ = c + μσc; strength− = c − μσt, with σt = −σc",
        "explanation": "The deliberately wrong column uses c + μσt without transforming the envelope."
      },
      {
        "label": "Substitute the values",
        "calculation": "At σc = 2,000 and σt = −2,000: 500 + 0.6 × 2,000 = 500 − 0.6 × (−2,000) = 1,700 psi; wrong = −700 psi",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Correct compression-positive strength (psi) = 1700; Correct tension-positive strength (psi) = 1700; Wrong mixed-convention strength (psi) = -700",
        "explanation": "A convention change preserves the physical result only when the equation is transformed too."
      }
    ],
    "result": "Correct compression-positive strength (psi) = 1700; Correct tension-positive strength (psi) = 1700; Wrong mixed-convention strength (psi) = -700",
    "interpretation": "A convention change preserves the physical result only when the equation is transformed too.",
    "control": "Effective compression (psi)",
    "columns": [
      "Effective compression (psi)",
      "Correct compression-positive strength (psi)",
      "Correct tension-positive strength (psi)",
      "Wrong mixed-convention strength (psi)"
    ],
    "rows": [
      [
        0,
        500,
        500,
        500
      ],
      [
        1000,
        1100,
        1100,
        -100
      ],
      [
        2000,
        1700,
        1700,
        -700
      ],
      [
        3000,
        2300,
        2300,
        -1300
      ],
      [
        4000,
        2900,
        2900,
        -1900
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Effective compression (psi) = 0. Correct compression-positive strength (psi): 500; Correct tension-positive strength (psi): 500; Wrong mixed-convention strength (psi): 500.",
      "Effective compression (psi) = 1000. Correct compression-positive strength (psi): 1100; Correct tension-positive strength (psi): 1100; Wrong mixed-convention strength (psi): -100.",
      "Effective compression (psi) = 2000. Correct compression-positive strength (psi): 1700; Correct tension-positive strength (psi): 1700; Wrong mixed-convention strength (psi): -700.",
      "Effective compression (psi) = 3000. Correct compression-positive strength (psi): 2300; Correct tension-positive strength (psi): 2300; Wrong mixed-convention strength (psi): -1300.",
      "Effective compression (psi) = 4000. Correct compression-positive strength (psi): 2900; Correct tension-positive strength (psi): 2900; Wrong mixed-convention strength (psi): -1900."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Correct compression-positive strength (psi) without rounding intermediate values.",
        "answer": "At σc = 2,000 and σt = −2,000: 500 + 0.6 × 2,000 = 500 − 0.6 × (−2,000) = 1,700 psi; wrong = −700 psi. Correct compression-positive strength (psi) = 1700. A convention change preserves the physical result only when the equation is transformed too."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Effective compression (psi) changes from 0 to 4000. Other controls remain as stated in the premise. Correct compression-positive strength (psi) changes from 500 to 2900."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. The deliberately wrong column uses c + μσt without transforming the envelope. A convention change preserves the physical result only when the equation is transformed too."
      }
    ]
  },
  "recorded": null
}