Buckling Calculator — Euler Critical Load for Columns
Compute the Euler critical load at which a slender column buckles under axial compression. Enter the material modulus, cross-section second moment of area, unsupported length, and end conditions to get Pcr and the slenderness ratio.
GPa
cm4
m
End conditions
cm2
Pcr = π² × E × I / (K × L)² — Euler column buckling
- 1
Effective length KL
1 × 3 m = 3End-condition factor K scales the physical length to an equivalent pin–pin length. - 2
EI product (kN·m²)
200 GPa × 100 cm⁴ × 0.01 = 200 - 3
π² × EI
π² × 200 = 1,973.92 - 4
Critical buckling load Pcr
1,973.92 ÷ 3² = 219.3
How does this calculator work?
Euler critical buckling load: Pcr = π²EI / (KL)², where E is Young’s modulus, I is second moment of area, L is column length, and K is the end-condition factor (0.5 fixed–fixed to 2.0 cantilever). Result is an elastic upper bound; real design codes apply additional imperfection reduction factors.
Formula
How this is calculated
When a long, slender column is loaded axially, it can suddenly bend sideways — buckle — at a load far below the material’s compressive strength. Euler’s formula Pcr = π²EI / (KL)² gives the theoretical load at which this instability occurs. E is Young’s modulus (material stiffness), I is the second moment of area about the weak axis, L is the unsupported length, and K is the effective-length factor: K = 1.0 for pin–pin, 0.5 for fixed–fixed, 0.7 for fixed–pin, and 2.0 for a cantilever.
The slenderness ratio KL/r (where r = √(I/A) is the radius of gyration) characterises buckling proneness. High slenderness (> 120 for steel) means Euler’s elastic formula is accurate; for stocky columns inelastic or plastic buckling governs and Euler overestimates capacity.
This calculator uses the ideal Euler formula, valid for perfectly straight, centrally loaded, elastic columns. Real columns have geometric imperfections, residual stresses, and load eccentricities that reduce capacity below Pcr. Design codes (Eurocode 3, AISC) apply additional reduction factors; use this result as an upper bound for preliminary sizing only.
Frequently asked questions
K converts the actual column length to an equivalent pin–pin length. K = 1.0 for pin–pin (both ends rotate freely), K = 0.5 for fixed–fixed (highest restraint and highest Pcr), K = 0.7 for fixed–pin, and K = 2.0 for a fixed–free cantilever, which has the lowest buckling resistance.
Euler's formula applies in the elastic range, where the critical stress Pcr/A is below the proportional limit. For steel (E ≈ 200 GPa, yield ≈ 250 MPa) the formula is reliable for slenderness KL/r above about 90. For stockier columns use the Johnson parabola or the relevant code buckling curve.
For a solid rectangle (b × d), I = b × d³ / 12 about the weak axis. For standard rolled sections look up I in the manufacturer’s section tables. Use the smaller (weak-axis) value when the column is unrestrained in both directions.
Also known as
TG we-Calculate Editorial Team. (2026). Buckling Calculator — Euler Critical Load for Columns [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/buckling-calculator
TG we-Calculate Editorial Team. "Buckling Calculator — Euler Critical Load for Columns." TG we-Calculate. 2026. https://we-calculate.com/calculator/buckling-calculator.
TG we-Calculate Editorial Team, "Buckling Calculator — Euler Critical Load for Columns," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/buckling-calculator
@misc{wecalculate_buckling_calculator, title = {Buckling Calculator — Euler Critical Load for Columns}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/buckling-calculator}}, year = {2026}, note = {TG we-Calculate} }
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