Engineering Reference

Column Buckling Calculator

Critical buckling load for a slender column by the Euler formula, with the effective length factors for the common end conditions.

Page last changed 2026-10-19 · based on n/a — standard engineering relationships, no single governing revision

Quick Answer

Euler buckling load is Pcr = π²·E·I ÷ (K·L)². Everything hinges on the effective length factor K: a column pinned at both ends (K = 1) carries four times the load of the same column fixed at one end only (K = 2).

Euler Column Buckling

The Formulas Used

Euler critical load: Pcr = π²·E·I ÷ (K·L)²
Critical stress: σcr = Pcr ÷ A   Slenderness ratio: K·L ÷ r
Effective length factors: pinned-pinned K = 1.0 · fixed-pinned K = 0.7 · fixed-fixed K = 0.5 · fixed-free K = 2.0

Using the Effective Length Factor

K accounts for how the ends are held, and it has a bigger effect than anything else in the formula because it is squared. The four ideal cases are: pinned-pinned K = 1.0, fixed-pinned K = 0.7, fixed-fixed K = 0.5, and fixed-free (a flagpole) K = 2.0.

Going from K = 1.0 to K = 2.0 quadruples the effective length and reduces the critical load to a quarter. Real connections are never perfectly pinned or perfectly fixed, so codes give K values that account for the actual rotational stiffness of the joints — and in frames, K depends on the stiffness of the members framing into the joint, which is why the alignment charts exist.

Euler's formula assumes the column is initially straight, homogeneous, centrally loaded, and that the stress at buckling is below the proportional limit of the material. For intermediate columns, where the column fails partly by yielding, Euler overestimates the capacity and empirical formulas such as Johnson's parabolic formula are used instead. Most design codes handle this with a single interaction curve rather than two separate formulas.

Frequently Asked Questions

What is Euler's buckling formula?
Pcr = π²·E·I/(K·L)², where E is the modulus of elasticity, I is the moment of inertia about the weak axis, L is the unbraced length and K is the effective length factor. It gives the axial load at which a slender column becomes unstable.
Why does buckling depend on E and not on yield strength?
Because buckling is a stiffness instability, not a strength failure. The column deflects sideways and the load creates additional moment; whether it recovers depends on how stiff the section is. Switching to a higher-strength steel raises E only marginally, so it does almost nothing for a slender column's capacity.
What is the effective length factor K?
A multiplier on the actual length that accounts for end conditions. K = 1.0 for pinned ends, 0.7 for fixed-pinned, 0.5 for fixed-fixed and 2.0 for fixed-free. Because it is squared in the formula, going from 1.0 to 2.0 cuts the capacity to a quarter.
Which axis governs a column?
The weak axis — the one with the smaller moment of inertia. A column buckles in the direction that is easiest to bend, so the governing I is the smaller value, and the unbraced length in that direction is often longer as well.
When does Euler's formula stop applying?
When the stress at buckling exceeds about half the yield strength. At that point the column is 'intermediate', part of the section has yielded, and Euler overestimates capacity — empirical formulas such as Johnson's parabolic formula or a code interaction curve are used instead.

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