Column Buckling — Euler Critical Load mechanism

Model scope

The core Euler formulas are standard and numerically verified, but the specification needs explicit unit contracts, complete input requirements for stress and slenderness, domain validation, and a more accurate description of Euler versus Johnson and crushing/yield behavior.

• The algebraic formulas are correct: Pcr = π²EI/(KL)² and σcr = π²E/(KL/r)², provided r = √(I/A) and all quantities use a consistent unit system.
• Stress and slenderness ratio cannot be computed from E, I, L, and K alone; r or both I and A are required.
• The honest-label statement that short columns need the Johnson formula is imprecise: Johnson's parabolic formula generally applies to an intermediate inelastic range, while short columns are typically checked for yielding or crushing; applicability depends on material and slenderness.

Units and conventions

• E, stress, and σcr must use the same stress unit; I and A must use mutually consistent length powers.
• For SI inputs, use E in Pa, I in m^4, A in m^2, L and r in m; Pcr is then in N and σcr in Pa.
• For US customary inputs, using E in ksi, I in in^4, A in in^2, and L and r in inches produces Pcr in kip and σcr in ksi.
• K is dimensionless and is the effective-length factor: approximately 0.5 fixed-fixed, 0.7 fixed-pinned, 1.0 pinned-pinned, and 2.0 fixed-free.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "E_Pa": 200000000000,
    "I_m4": 0.000008,
    "A_m2": 0.004,
    "L_m": 3,
    "K": 1
  },
  "expected_outputs": {
    "Pcr_N": 1754596.34,
    "sigma_cr_Pa": 438649084.45,
    "slenderness_ratio": 67.08204
  },
  "working": "r = √(I/A) = √(0.000008/0.004) = √0.002 = 0.0447214 m. KL/r = (1×3)/0.0447214 = 67.08204. Pcr = π²EI/(KL)² = 9.8696044×200000000000×0.000008/(3)² = 9.8696044×1600000/9 = 1754596.34 N. σcr = Pcr/A = 1754596.34/0.004 = 438649084.45 Pa."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification has no numerical slender-column validity threshold or Johnson-formula branch, so it cannot reliably classify the formula's applicability near the slender/short-column transition. It also produces undefined results for zero length unless the input is rejected, and negative K has an unresolved convention that can cause either silent normalization or invalid lookup behavior.

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