How it’s calculated
A slender column under axial compression stays straight until the load reaches the Euler critical load, when it suddenly bows sideways. The load depends on stiffness (E and the weakest-axis I) and on the effective length, not on material strength.
| End conditions | K (theory) |
|---|---|
| Pinned – pinned | 1.0 |
| Fixed – pinned | 0.7 |
| Fixed – fixed | 0.5 |
| Fixed – free | 2.0 |
Example: a 50 mm solid steel bar (E = 200 GPa), 2 m long, pinned at both ends. I = π(0.05)⁴/64 = 3.068 × 10⁻⁷ m⁴, so Pcr = π² × 200 × 10⁹ × 3.068 × 10⁻⁷ ÷ 2² = 151.4 kN. σcr = 151,400 ÷ 1.963 × 10⁻³ m² = 77.1 MPa, well below the 250 MPa yield, and KL/r = 2000 ÷ 12.5 = 160, so the column is slender and Euler applies.
Pcr is the theoretical failure load of a perfectly straight column. Real columns have imperfections and residual stresses; design codes apply column curves and safety or resistance factors. This is not a design check.
Frequently asked questions
When is Euler’s formula valid?
For slender columns that buckle elastically, i.e. when the critical stress is well below the yield strength. Short, stocky columns fail by yielding or inelastic buckling at lower loads than Euler predicts.
Which moment of inertia do I use?
The smallest one. A column buckles about its weakest axis unless it is braced in that direction.
Is Pcr the safe load?
No. It is the theoretical buckling load. Allowable loads are much lower after applying code column curves and safety factors.
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Sources
- ME 354 Mechanics of Materials Lab, Chapter 10: Compression and Buckling — University of Washington
- University Physics Vol. 1, 12.3 Stress, Strain, and Elastic Modulus (Table 12.1) — OpenStax
Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.
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