Euler Column Buckling Calculator

What is the Euler buckling load?

A slender column loses its capacity by bowing sideways long before the material crushes. This is buckling. The load at which an ideal column starts to buckle is given by Euler's formula: P_cr = π² × E × I / (K × L)². E is the elastic modulus, I the smallest second moment of area of the section, L the column length and K the effective length factor set by the end conditions.

The critical load depends on stiffness, not strength. A higher grade of steel does not raise the buckling load of a slender column because the elastic modulus stays the same. Halving the length, on the other hand, multiplies the load by four.

End conditions and effective length

The effective length K × L is the distance between the two points of inflection of the buckled shape.

• Pinned at both ends: K = 1
• Fixed at one end, pinned at the other: K = 0.7
• Fixed at both ends: K = 0.5
• Fixed at one end, free at the other (flagpole): K = 2

These are theoretical values. A real base plate or welded joint rotates a little, so design uses larger K values for fixed ends.

Slenderness and validity

Slenderness is λ = K × L / i with the radius of gyration i = √(I / A). The critical stress is σ_cr = π² × E / λ². Euler's formula is correct only while this stress stays below the proportional limit of the material. When the slenderness is less than λ_p = π × √(E / σ_p) the column starts to yield before it buckles elastically and the Euler result is higher than the real capacity. The calculator flags this on a separate line.

Example

A steel tube with 75 mm outer radius and 70 mm inner radius, 7.2 m long and pinned at both ends, has I = 5.993 × 10⁶ mm⁴. With E = 200 GPa, P_cr = π² × 200 000 × 5 993 079 / 7200² = 228.2 kN. The area is 2278 mm², so the critical stress is 100.2 MPa. That is below the 250 MPa yield strength, so Euler's formula applies.

Limits

The Euler load is an upper bound. Real columns are not perfectly straight, carry residual stresses from rolling and welding, and are never loaded exactly through the centroid. Steel design standards therefore give column resistance through buckling curves, and the result is always below the Euler load. Use this tool for preliminary sizing and for learning.