Buckling Calculator

Euler Column Buckling

Slender columns fail by buckling before the material reaches its compressive strength. Euler's formula gives the critical axial load at which a perfectly straight column becomes unstable.

Euler Critical Load

$$F_{cr} = \frac{\pi^2 E I}{(KL)^2}$$

$E$ is Young's modulus, $I$ is the second moment of area, $L$ is the unsupported length, and $K$ is the effective length factor that depends on end restraints.

End Condition Factors

Common $K$ values: fixed–fixed $K=0.5$, fixed–pinned $K=0.7$, pinned–pinned $K=1.0$, and fixed–free (cantilever) $K=2.0$. Smaller $K$ means shorter effective length and higher buckling capacity.

Critical Stress

$$\sigma_{cr} = \frac{F_{cr}}{A}$$

Dividing the critical load by cross-sectional area $A$ gives the buckling stress. Compare $\sigma_{cr}$ to yield strength to assess column design margin.

Frequently Asked Questions

What is the Euler buckling formula?

Euler's formula is $F_{cr} = \pi^2 E I / (KL)^2$. It predicts the maximum compressive load a slender column can carry before lateral buckling.

What does the K factor represent?

$K$ converts actual length to effective length $KL$, accounting for how end supports constrain lateral displacement and rotation. Fixed ends reduce $KL$; a cantilever increases it.

When does Euler buckling not apply?

Euler theory applies to long, slender columns where failure is elastic instability. Short, thick columns may fail by crushing or yielding instead.

How do I find the second moment of area I?

$I$ depends on cross-section shape. For a solid circle, $I = \pi d^4 / 64$. For a rectangle, $I = bh^3 / 12$ about the weak axis.

What units should I use?

This calculator uses GPa for $E$, mm for length, mm⁴ for $I$, and mm² for area. Results are in newtons and megapascals (N/mm²).

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Euler Column Buckling

Euler Critical Load

End Condition Factors

Critical Stress

Frequently Asked Questions

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