What is Euler buckling?

Euler buckling is the sudden lateral deflection of a slender column under axial compression. The critical load Pcr = π²EI/(KL)² is the maximum axial load before buckling occurs. Below Pcr, the column remains straight; at Pcr, it becomes unstable.

What is the effective length factor K?

K accounts for end conditions: K=0.5 (both ends fixed), K=0.7 (one fixed, one pinned), K=1.0 (both pinned), K=2.0 (one fixed, one free/cantilever). Lower K means higher critical load.

What is the slenderness ratio?

Slenderness ratio λ = KL/r, where r is the radius of gyration. High λ (>100-120) means the column is slender and Euler's formula applies. For short columns (low λ), inelastic buckling or crushing governs.

When does Euler's formula not apply?

Euler's formula assumes elastic behavior and applies only when the critical stress σcr = Pcr/A is below the yield strength. For short/intermediate columns where σcr > σy, use Johnson's parabola or other inelastic buckling criteria.

How do I increase a column's buckling capacity?

Increase I (use a shape with more material far from the centroid), increase E (use stiffer material), reduce L (add intermediate bracing), or improve end fixity (lower K). The most effective approach depends on practical constraints.

Free Engineering Tool

Euler Buckling Calculator

Calculate the critical buckling load, critical stress, and slenderness ratio for columns using Euler’s formula P_cr = π²EI/(KL)².

Pcr = π²EI/(KL)²K = 0.5–2.0#168

Elastic Modulus E (GPa)

Moment of Inertia I (cm⁴)

Cross-Section Area A (cm²)

Column Length L (m)

End Condition (K factor)

Quick presets

Results

Critical Buckling Load Pcr

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Critical Stress σcr

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Slenderness Ratio λ

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Effective Length KL

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Radius of Gyration r

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Column Buckling Visualization

Euler’s Buckling Formula

Effective Length Factors

End Condition	K	KL (effective)
Both ends fixed	0.5	0.5L
Fixed–Pinned	0.7	0.7L
Both ends pinned	1.0	L
Fixed–Free (cantilever)	2.0	2L

⚠️ Note: Euler’s formula applies only for elastic buckling (σ_cr < σ_y). For intermediate columns, use Johnson's parabola or Eurocode/AISC column curves.