Euler Buckling Reference
Euler buckling load, critical stress, effective length factors and the Johnson-Euler transition for intermediate slenderness.
Definition
Euler buckling is the elastic instability of a slender column under axial compression. The critical load is the load at which a perfectly straight column bifurcates into a bent configuration.
Equations
Euler buckling load:
π² · E · I
P_cr = ───────────
(K · L)²
Euler buckling stress:
π² · E
σ_cr = ───────────
(L'/r)²
where L' = K · L (effective length)
r = √(I/A) (radius of gyration)
Johnson-Euler transition (inelastic):
σ_allow = σ_y − σ_y² · (L'/r)² / (4 π² E)
Transition slenderness:
(L'/r)_trans = √(2 π² E / σ_y)Effective Length Factors
| End Condition | K (theoretical) | K (practical) |
|---|---|---|
| Pinned-pinned | 1.0 | 1.0 |
| Fixed-fixed | 0.5 | 0.65 |
| Fixed-pinned | 0.7 | 0.8 |
| Fixed-free (cantilever) | 2.0 | 2.1 |
| Fixed-guided | 1.0 | 1.2 |
Variables
| Symbol | Definition | Units |
|---|---|---|
| P_cr | Critical (Euler) buckling load | N |
| σ_cr | Critical buckling stress | MPa |
| E | Young's modulus | GPa |
| I | Minimum second moment of area | mm⁴ |
| A | Cross-sectional area | mm² |
| L | Actual column length | mm |
| K | Effective length factor | — |
| L' | Effective length (K·L) | mm |
| r | Radius of gyration | mm |
| L'/r | Slenderness ratio | — |
| σ_y | Yield strength | MPa |
Notes and Limitations
- Euler buckling assumes perfect geometry, linear elasticity and concentric loading
- Real columns have imperfections — apply a knockdown factor (typically 0.8-0.9)
- Buckling occurs about the weak axis (minimum I) unless bracing prevents it
- For thin-walled sections, local buckling (flange/web) may precede column buckling
- The practical K values account for less-than-perfect end fixity
Related Knowledge
See the Structural Analysis Knowledge category for buckling theory and post-buckling behaviour. See How to Assess Column Buckling for the practical workflow.