Calculator · Structural Mechanics
Euler Buckling Load Calculator
The axial load at which an ideal slender column stops being straight and bows sideways instead. This is an elastic bifurcation load: it depends on stiffness and geometry, not on the strength of the material, and it carries no factor of safety.
Inputs
About the axis the column can buckle about.
Only needed for the critical stress and the slenderness.
Effective-length factor for these end conditions: K = 1.
Results
| Effective-length factor K | 1 |
|---|---|
| Effective length KL | 3 m |
| Critical load Pcr = π²EI/(KL)² | 548310 N |
| Critical stress σcr = Pcr/A | 137.08 MPa |
| Radius of gyration r | 25 mm |
| Slenderness ratio λ = KL/r | 120 |
Pcr is the load at which the ideal column becomes unstable, not a load it may safely carry. A real column with initial crookedness or an eccentric load starts bending immediately and fails below this value.
Assumptions
- · Slender, initially straight, prismatic column.
- · Linear-elastic material; buckling occurs below the proportional limit.
- · Concentric axial load and idealised effective-length end conditions.
- · Elastic bifurcation load, not a design capacity: imperfections, eccentricity and inelastic behaviour are not included.
- · Pcr is the elastic critical load at which an ideal column bifurcates. It is not a safe load, not an allowable load and not a design resistance, and it contains no factor of safety.
- · Initial crookedness, load eccentricity, residual stress, material yielding, local buckling and lateral-torsional buckling are all excluded.
- · Euler theory applies to sufficiently slender columns that remain elastic up to instability; no cutoff slenderness is asserted here, and no inelastic column formula is substituted automatically.
- · Buckling occurs about the axis with the smallest EI available to it. With a uniform E, the smaller I is generally the weaker axis, but the governing axis depends on how the column is braced.
Technical reference
Pcr = π² E I / (K L)²
σcr = Pcr / A = π² E / λ²
λ = K L / r
r = √(I / A)
- E
- Young's modulus[Pa]
- I
- second moment of area about the buckling axis[m⁴]
- L
- unbraced length[m]
- K
- effective-length factor[—]
- Pcr
- elastic critical load, positive in compression[N]
- σcr
- critical stress at the bifurcation load[Pa]
Pcr is an elastic bifurcation load for a perfectly straight, concentrically loaded, prismatic column. It is not a design resistance, not an allowable load and not a safe load, and it contains no factor of safety.
Initial crookedness, load eccentricity, residual stress, material yielding, local buckling and lateral-torsional buckling are all excluded. Every one of them reduces the load the real column can carry.
The relation is only meaningful while the column stays elastic up to instability. For a stocky column σcr exceeds the yield strength and the column squashes instead; no inelastic column formula is substituted here, and no cutoff slenderness is asserted.
Buckling happens about the axis with the least available EI. With a uniform E that is usually the axis with the smaller I, but bracing can change which axis governs, so I and K must be taken about the same axis.
Doubling the length quarters Pcr: slenderness, not strength, is what governs a long column.