Effective length
m
Product K·L.
Mechanics · Stability and buckling
Euler buckling
Estimate the elastic critical load of a slender compression member including its end conditions.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
MetricResults
Minimum radius of gyration
mm
r = √(Imin/A).
Slenderness ratio
—
Ratio KL/r.
Euler critical load
kN
Theoretical elastic bifurcation load.
Critical stress
MPa
Ratio Pcr/A.
View calculation detailsFormulas, SI conversion, numerical substitution and resultsOpen ↓
Calculation note
Full calculation details
Straight prismatic initially perfect member under axial load. Linear-elastic material. Global buckling about the weakest axis. Euler applies to sufficiently slender members and does not replace a design code.
01Formulas and symbols
Formulas used
Le
KLEffective lengthr
√(Imin/A)Minimum radius of gyrationλ
KL/rSlenderness ratioPcr
π²EImin/(KL)²Critical load02Assumptions and limits
Scope of validity
- Straight prismatic initially perfect member under axial load.
- Linear-elastic material.
- Global buckling about the weakest axis.
- Euler applies to sufficiently slender members and does not replace a design code.
03Validation example
Reference numerical case
- E = 210 GPa, Imin = 8,000,000 mm⁴, A = 2,500 mm², L = 3 m and K = 1.
- r = 56.5685 mm and λ = 53.033.
- Pcr = 1,842.33 kN.
- σcr = 736.93 MPa.
04Frequently asked questions
Questions about the calculation
Why use Imin?
A member generally buckles about the axis with the lowest flexural rigidity EI, so the relevant minimum second moment is required.
How should K be selected?
K idealizes end rotation and translation. Real frames may require a value from a design code or a global stability model.
Does a high Euler load guarantee safety?
No. Section strength, imperfections, second-order effects, buckling curves and the applicable design code must also be checked.
05References
Technical references
- Timoshenko & Gere, Theory of Elastic Stability.
- Gere & Goodno, Mechanics of Materials, column buckling.