Euler's Buckling Load Formula
Euler's column buckling formula calculates the critical load at which a slender column buckles.
Learn structural stability with examples.
The Formula
Euler's buckling formula predicts the maximum axial load a slender column can support before it suddenly bows sideways and collapses. This critical load depends on the column's material stiffness, cross-section geometry, length, and how its ends are supported.
Named after the Swiss mathematician Leonhard Euler, who derived it in 1757, this formula is fundamental to structural engineering. Buckling is dangerous because it can happen suddenly and without warning, even when the material stress is well below the yield strength.
The formula applies only to long, slender columns where buckling occurs before the material yields. Short, thick columns fail by crushing rather than buckling.
Variables
| Symbol | Meaning |
|---|---|
| Pcr | Critical buckling load (measured in newtons, N) |
| E | Modulus of elasticity (Young's modulus) of the material (measured in pascals, Pa) |
| I | Minimum moment of inertia of the cross-section (measured in m⁴) |
| K | Effective length factor (depends on end conditions) |
| L | Actual length of the column (measured in meters, m) |
Effective Length Factor (K)
- Both ends pinned: K = 1.0
- One end fixed, one end free (cantilever): K = 2.0
- Both ends fixed: K = 0.5
- One end fixed, one end pinned: K = 0.7
Example 1
A steel column is 4 m long with both ends pinned. The steel has E = 200 GPa and the column's moment of inertia is I = 5.0 × 10⁻⁶ m⁴. What is the critical buckling load?
K = 1.0 (both ends pinned)
Pcr = π²EI / (KL)² = π² × 200 × 10⁹ × 5.0 × 10⁻⁶ / (1.0 × 4)²
Pcr = 9.8696 × 200 × 10⁹ × 5.0 × 10⁻⁶ / 16
Pcr = 9.8696 × 1.0 × 10⁶ / 16
Pcr ≈ 616,850 N ≈ 617 kN
Example 2
An aluminum rod (E = 69 GPa) is used as a cantilever column (one end fixed, one end free). It is 2.5 m long with a circular cross-section of diameter 50 mm. What is the critical buckling load?
K = 2.0 (cantilever)
I = π d⁴ / 64 = π × (0.05)⁴ / 64 = π × 6.25 × 10⁻⁶ / 64 = 3.068 × 10⁻⁷ m⁴
Pcr = π² × 69 × 10⁹ × 3.068 × 10⁻⁷ / (2.0 × 2.5)²
Pcr = 9.8696 × 69 × 10⁹ × 3.068 × 10⁻⁷ / 25
Pcr = 9.8696 × 21,169 / 25 = 208,932 / 25
Pcr ≈ 8,357 N ≈ 8.36 kN
When to Use It
Use Euler's buckling formula for slender column design and analysis.
- Designing structural columns in buildings and bridges
- Checking whether support struts in machines will buckle under load
- Sizing compression members in truss structures
- Analyzing the stability of scaffolding and temporary supports
- Determining safe working loads for slender rods and tubes