Column buckling: Euler's formula and effective length

A short, stocky column fails by crushing. A long, slender one fails much earlier by buckling: it suddenly bows sideways. Euler worked out the load at which this happens.

Euler's critical load

P_cr = π² E IL_e²

E is Young's modulus, I the least second moment of area (buckling happens about the weaker axis) and L_e the effective length.

Effective length for different end conditions

End conditions	Effective length Le
Both ends hinged	L
One end fixed, other free	2L
Both ends fixed	L/2
One end fixed, other hinged	L/√2 ≈ 0.7L

Slenderness ratio

Using the least radius of gyration, k = √(I/A):

λ = L_ek σ_cr = π² Eλ²

Euler's formula only holds when σ_cr is below the proportional limit. For mild steel that means λ above roughly 80 to 100. Shorter columns are designed with Rankine's or the code's formulas instead.

Worked example

A hinged–hinged steel strut, 3 m long, is a solid round bar of 50 mm diameter. E = 2 × 10⁵ N/mm².

I = π × 50⁴64 = 3.07 × 10⁵ mm⁴

P_cr = π² × 2 × 10⁵ × 3.07 × 10⁵3000² = 67.3 kN

Rankine's formula

1P_R = 1P_c + 1P_E

It blends crushing load P_c and Euler load P_E, so it works for short and long columns alike.