Source: https://guthlabs.ai/docs/instruments/johnson-parabolic-formula-intermediate-columns/

instrument / verified-clean

# Johnson Parabolic Formula (Intermediate Columns)

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/johnson-parabolic-formula-intermediate-columns/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/johnson-parabolic-formula-intermediate-columns/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/johnson-parabolic-formula-intermediate-columns/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/johnson-parabolic-formula-intermediate-columns/qualification/>)

## Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

**Area**: Mechanical, Civil & Structural Engineering — Statics/Dynamics, Strength of Materials, Machine Design, Civil, HVAC, Vibration

**Tranche**: tranche-4

**Evidence ID**: t4-118

**Ledger rows**: 1

No ledger note is attached.

## Review findings

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.

### Formula and scope findings

- 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.

### Unit assumptions

- 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.

## Boundary hardening

**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.

## Errata and corrections

No applied erratum is attached to this unique-name record.

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "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."
}
````

## Related workflows

[Beam & Column Structural Benchintermediate columns](<https://guthlabs.ai/docs/instrument-workflows/structural-bench/>)

## Sources and evidence

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

- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`

- `evidence-tranche4/tranche4-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`