Johnson Parabolic Formula (Intermediate Columns) mechanism

Model scope

The formula itself is the standard Johnson parabolic column formula, but the specification needs to define Cc=√(2π²E/σy), state the equality convention, and provide the Euler comparison/regime-selection rule before the instrument can be verified as complete.

• The stated Johnson formula is standard: σcr=σy−[σy²/(4π²E)](KL/r)², equivalently σcr=σy[1−σy(KL/r)²/(4π²E)].
• The critical slenderness ratio is not defined. The standard Johnson-to-Euler transition value is Cc=√(2π²E/σy).
• The output 'applicable range check vs Euler' is ambiguous because no Euler formula, comparison rule, or boundary convention is specified.
• The condition is written as KL/r < critical slenderness ratio, but the specification does not state whether equality at Cc is accepted; at equality, Johnson and Euler give the same stress.

Units and conventions

• σy and E must use the same stress units, such as both MPa or both ksi.
• KL/r is dimensionless; K is the effective-length factor and L and r must use the same length units before forming the ratio.
• The standard transition ratio is Cc=√(2π²E/σy), requiring positive, consistently dimensioned E and σy.
• The formula assumes the usual idealized column conditions for Johnson buckling: straight prismatic member, centrally applied axial load, specified effective-length factor, and material properties appropriate to the selected material.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "yield_strength": {
      "value": 250,
      "unit": "MPa"
    },
    "elastic_modulus": {
      "value": 200000,
      "unit": "MPa"
    },
    "slenderness_ratio_KL_over_r": 0
  },
  "expected_outputs": {
    "critical_stress": {
      "value": 250,
      "unit": "MPa"
    },
    "critical_slenderness_ratio": 125.663706,
    "applicable_to_strict_less_than_range": true
  },
  "working": "Cc=√(2π²E/σy)=√(2π²×200000/250)=√(1600π²)=40π=125.663706. σcr=σy−[σy²/(4π²E)](KL/r)²=250−[250²/(4π²×200000)]×0²=250−0=250 MPa. Since 0<125.663706, the Johnson range check is true."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The zero-modulus input makes the Johnson expression and critical slenderness ratio undefined through division by zero. The instrument should explicitly reject E = 0 rather than return a numeric critical stress.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json