Euler Column Buckling

Euler buckling of a slender column

For elastic buckling of a slender column, the critical load is P_cr = π²·E·I / (K·L)². Here E is Young's modulus, I the smallest second moment of area, L the column length, and K the effective-length factor that reflects how the ends are restrained. The typical values run pinned-pinned K = 1, fixed-free (cantilever) K = 2, fixed-fixed K = 0.5, and fixed-pinned K ≈ 0.7. The slenderness ratio is λ = K·L/r with r = √(I/A). As a quick example, E = 200 GPa, I = 5000 cm⁴, L = 3 m, K = 1 gives P_cr ≈ 1097 kN.

Euler's formula only holds in the elastic range. Once a column is stocky (λ below the slenderness limit λ_1 in NBR 8800 §5.3), inelastic buckling takes over and you turn to curves like Johnson's or the NBR 8800 column curves. American practice (ASCE 7 / AISC 360) keeps the same K and r definitions but draws the transition slenderness a little differently.

Applications

Sizing slender steel columns and bracing under NBR 8800; checking long pipe runs and tubular supports against buckling; designing struts in trusses and shoring; verifying compression flanges in deep beams; running preliminary structural-code calculations per ASCE 7 / AISC 360.

FAQ

Why does K matter so much? Because P_cr scales with 1/K². Swap pinned-pinned (K = 1) for a cantilever (K = 2) at the same length and the critical load drops to a quarter.

Which I do I use? The smallest principal moment of area, since the column buckles about its weakest axis unless bracing stops it.

Is Euler always conservative? No, and that catches people out. For stocky columns (low λ) it is unconservative, because a real column yields before it ever reaches the elastic P_cr. Once λ < λ_1, switch to the inelastic-regime curves from NBR 8800 or AISC 360.