Library / Civil / CIV-05
Civil
Euler buckling load
Compressive load at which a slender column bows sideways.
- Ref
- CIV-05
- Rev
- v1
- Field
- Civil
- Result
- Pcr [kN]
Pcr in newtons (shown in kN), E Young’s modulus, Imin the smaller second moment of area of the section, L the length and K the effective-length factor: 1 for pin–pin, 0.5 for fixed–fixed, 0.7 for fixed–pinned and 2 for fixed–free.
A long thin column loaded in compression suddenly bows sideways at a critical load, often long before the material would crush. Longer, more slender, or more weakly restrained columns buckle at lower loads.
Calculator
- Effective length KL3 m
- Critical load as a mass178.9 tonne
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Length enters squared (and with K, so does end restraint). Restraining the middle of a column halves its effective length and quadruples the capacity.
- Buckling is governed by stiffness EI, not strength. A high-strength steel buckles at the same load as a mild steel of the same size.
- Always use the weaker axis. A tall, narrow column buckles about its thin direction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Column length | 0.5 to 10 m | 157.9 | 63,170 | 63,010 | +10% | −17.4 % |
| Imin Second moment of area | 10 to 20,000 cm⁴ | 21.93 | 43,860 | 43,840 | +10% | +10 % |
| K Effective-length factor | 0.5 to 2 | 438.6 | 7,018 | 6,580 | +undefined% | — |
| E Young’s modulus | 10 to 210 GPa | 87.73 | 1,842 | 1,755 | +10% | +10 % |
Sweep of Column length
| Column length (m) | Critical buckling load (kN) |
|---|---|
| 0.5 | 63,170 |
| 0.7271 | 29,870 |
| 1.057 | 14,120 |
| 1.538 | 6,679 |
| 2.236 | 3,158 |
| 3.252 | 1,493 |
| 4.729 | 706.2 |
| 6.877 | 333.9 |
| 10 | 157.9 |
Sweep one variable
Variables and how each one contributes
Critical buckling load
What the equation gives you. Everything else feeds it.
Young’s modulus
Material stiffness. Directly proportional.
Second moment of area
About the weaker axis of the cross-section. Directly proportional.
Column length
Unsupported length. The dominant driver: inversely proportional to L², so +10% length cuts the load by about 17%.
Effective-length factor
Set by end restraint: 0.5 fixed–fixed, 0.7 fixed–pinned, 1 pin–pin, 2 fixed–free. Inversely proportional to K²: a fixed–free column (2) carries one quarter of a pin–pin column (1).
Worked example with the values above
- EI = 2×1011 Pa × 8×10−6 m⁴ = 1,600,000 N·m². Effective length KL = 3 m.
- Pcr = π²EI/(KL)² = 9.8696 × 1,600,000 / 9.
- Pcr = 1,755,000 N = 1,755 kN.
Assumptions and limits
- Ideal straight, slender, elastic column under concentric load.
- Pure elastic buckling (slenderness high enough that stress stays below yield).
- Constant section.
Common mistakes
- Using the strong axis I instead of Imin.
- Assuming Euler applies to short, stocky columns; those fail by crushing.
- Using the wrong K for the support conditions.
Statistics and uncertainty
Real columns have initial crookedness, load eccentricity and residual stress, so test loads fall below Euler by 10–30%. Design codes apply a safety factor or reduction curve. The squared effective length amplifies errors: a 5% error in K or L becomes 10% in Pcr.
Where it is used
- Struts, masts, tripod legs and ramp supports.
- Frame bracing and trusses.
- Pushrods and actuator rods.
Maths you need
- Second moment of area: I measures section stiffness
- Support conditions: pinned or fixed ends set K