Column Buckling — Euler Critical Load

Material

End Condition

Geometry

Column length L 3.0 m

Square section side b 60 mm

Applied axial load P / P_cr 0%

Stable — no load applied

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Critical load P_cr

— kN

Applied load P

0 kN

Slenderness ratio λ

—

Effective length KL

— m

How it works

A slender column loaded in pure compression does not crush — it buckles sideways, sharply and often suddenly, well below the material's crushing strength. The load at which this instability begins is Euler's critical load:

P_cr = π² E I / (K L)²
λ = K L / r         (slenderness ratio, r = √(I / A))
δ = δ₀ / (1 − P / P_cr)   (Southwell amplification of a small initial crookedness δ₀)

E is the material's stiffness, I the cross-section's second moment of area, L the column's physical length, and K the effective-length factor set by how the ends are held: pinned–pinned (K=1), fixed–free / cantilever (K=2), fixed–fixed (K=0.5), fixed–pinned (K≈0.699). A stockier, shorter, more rigidly-held column buckles at a far higher load than a slender, long, weakly-held one of the same material.

Material / section / length — set E, I = b⁴/12 and A = b² for a solid square section, which fixes P_cr.
End condition — changes K and therefore the effective length KL and the buckled mode shape drawn on the column.
Load slider — sets P as a fraction of the current P_cr. Every real column has a tiny initial crookedness δ₀ ≈ L/500; the Southwell formula shows why deflection grows gently at low load and then explodes as P approaches P_cr — the column is stable everywhere except in the limit P → P_cr, where any small imperfection is amplified without bound.

Real-world relevance: this exact formula sizes every steel column, bracing strut and scaffold pole in a building — a member is chosen not just strong enough to avoid crushing, but stiff and short enough that P_cr sits safely above the load it will ever see (typically with a safety factor of 2–3).

Stable Near critical Buckled

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Column Buckling — Euler Critical Load Simulator