Column Buckling: Euler & Johnson Formulas
1. Slenderness Ratio
The key parameter governing column behavior is the slenderness ratio:
λ = L / k, where k = √(I / A). Small λ indicates a short (stocky) column dominated by yielding; large λ indicates a
slender column dominated by elastic buckling.
2. Euler Buckling Formula (Elastic Buckling)
Euler’s formula applies to long, slender columns that buckle elastically before yielding. The critical buckling load and
stress are:
Pcr = π²EI / L²
σcr = π²E / (L / k)²
Assumptions: linear elasticity, perfect geometry, concentric loading, and no yielding.
3. Johnson (J.B. Johnson) Formula (Inelastic Buckling)
Johnson’s formula is used for short to intermediate columns where yielding and buckling occur together. The critical
stress is:
σcr = σy[1 − (σy / (4π²E))(L / k)²]
The critical load is Pcr = σcrA.
4. Validity Criterion (Formula Selection)
Before choosing a formula, the limiting slenderness ratio must be checked:
(L / k)lim = √(2π²E / σy)
If L / k < (L / k)lim, Johnson formula is valid (inelastic buckling).
If L / k > (L / k)lim, Euler formula must be used (elastic buckling).
5. Problem-Solving Procedure
1) Determine effective length L from end conditions.
2) Compute radius of gyration k = √(I / A) using the weak axis.
3) Calculate slenderness ratio λ = L / k.
4) Compare λ with √(2π²E / σy) to select Euler or Johnson.
5) Compute critical stress and then critical load Pcr = σcrA.
Key Takeaway
Short columns fail by yielding → Johnson formula. Long slender columns fail by elastic buckling → Euler formula. The
slenderness ratio check is essential to ensure physical validity.