What is the Euler buckling formula?

The critical load is P_cr = π²EI / (KL)², where E is the modulus of elasticity, I is the moment of inertia of the cross-section, L is the physical column length, and K is the effective-length factor that accounts for end conditions. The column buckles when the applied axial load reaches P_cr.

What is the effective-length factor K?

K converts the actual column length into the equivalent pinned-pinned length that would buckle at the same load. Standard values: K = 1.0 for pinned-pinned (both ends free to rotate), K = 2.0 for fixed-free (cantilever column), K = 0.7 for fixed-pinned (one end fixed, one pinned), and K = 0.5 for fixed-fixed (both ends fully restrained against rotation).

Which moment of inertia should I use?

Use the minimum (weak-axis) moment of inertia of the cross-section, because buckling will initiate about the axis of least resistance. For a wide-flange steel section this is typically I_y; for a rectangle it is bh³/12 about the narrower dimension.

Does Euler's formula apply to all columns?

Euler's formula is accurate only for slender columns where elastic buckling governs before material yielding. Most structural codes require a slenderness ratio KL/r above roughly 100–120 (steel) or 50 (timber) for Euler to apply. Short or intermediate columns fail by inelastic buckling or crushing and require Johnson, Rankine, or code-specific interaction equations instead.

What is the slenderness ratio and why does it matter?

The slenderness ratio KL/r (effective length divided by radius of gyration) is a dimensionless measure of how slender a column is. Higher values mean the column is more susceptible to elastic buckling. Most steel building codes limit KL/r to 200 for primary members and 300 for bracing elements.

What safety factor should I apply?

The Euler critical load is a theoretical elastic buckling load. In practice, apply a factor of safety of at least 3.0 for timber and 2.0–2.5 for steel when using this formula directly, or follow your governing design code (AISC 360, NDS, EN 1993, etc.) which incorporates imperfection factors and resistance factors automatically.

Column Buckling Calculator

Name: Column Buckling Calculator
Availability: InStock
Author: Nham Vu

Calculate Euler's critical buckling load for a structural column given modulus of elasticity, moment of inertia, effective length, and end-condition factor K.

Column Parameters

End Condition

Effective-Length Factor K

Filled automatically from end condition above, or override manually.

Column Length L (in)

Modulus of Elasticity E (ksi)

Moment of Inertia I (in⁴)

Radius of Gyration r (in) — optional

Applied Axial Load P (kips) — optional

Common Material Values

Material	E (ksi)	E (GPa)
Structural Steel (A992)	29,000	200
Aluminum 6061-T6	10,000	69
Douglas Fir (No. 2)	1,600	11
Southern Pine (No. 2)	1,400	9.7
Concrete (normal weight)	3,600	25–30

Summary