Effective Length Buckling (Euler) Calculator
Euler's buckling formula gives the theoretical elastic critical load for a column: Pcr = pi^2 * E * I / (KL)^2. This is the load at which a perfectly straight, perfectly elastic column becomes unstable and buckles. Real columns buckle at lower loads due to initial imperfections, residual stresses, and inelastic material behaviour; AISC 360 Chapter E accounts for these through design curves. This calculator is useful for understanding buckling behaviour and checking whether elastic (Euler) buckling governs versus inelastic buckling by comparing Pcr/A to the yield stress Fy.
Euler buckling formula
Pcr = pi² × E × I / (K × L)²
Fe = pi² × E / (KL/r)²
Fe is the elastic critical stress (Pcr/A = Fe if r^2 = I/A). If Fe is greater than 0.44 * Fy, elastic buckling does not govern and AISC 360 Section E3 (inelastic buckling) applies.
When does Euler buckling govern?
- Euler buckling governs when KL/r is greater than 4.71 * sqrt(E/Fy) (for A992 steel: 113). Below this, inelastic buckling governs and AISC uses a reduced Fcr formula.
- The critical slenderness KL/r = pi * sqrt(E/Fy) at which Fcr equals 0.50 * Fy (half the yield stress). Columns with lower KL/r are relatively stocky and governed by material yielding.
- Always check both principal axes; buckling about the weak axis (minimum I) typically controls for standard I-shaped columns.
- For braced frames, K = 1.0 is conservative and commonly used; for unbraced (sway) frames, K greater than 1.0 must be used.
Frequently asked questions
What is the Euler critical buckling load?
Pcr = pi^2 * E * I / (KL)^2 is the theoretical elastic buckling load for a pin-ended column, first derived by Leonhard Euler in 1744. For columns with other end conditions, the effective length KL replaces the physical length L to produce an equivalent pin-pin column.
Does Euler buckling apply to all columns?
Euler buckling (elastic buckling) applies when the critical stress Pcr/A is below the proportional limit of the material. For steel, this means KL/r must exceed the transition slenderness 4.71*sqrt(E/Fy). For shorter columns, inelastic buckling governs and AISC 360 Equation E3-2 applies.
What is the effective length KL?
KL is the distance between inflection points (points of zero moment) on the buckled shape. K is the effective length factor: pin-pin K=1.0 (inflection points at ends), fixed-fixed K=0.5, fixed-free K=2.0, fixed-pin K=0.7. The physical length is L.
How does the moment of inertia affect buckling?
Buckling occurs about the axis of minimum moment of inertia (the weak axis). For I-shaped columns, Ix is much larger than Iy, so buckling almost always controls about the y-axis unless bracing prevents this. Always check both axes.
What safety factor applies to Pcr?
Euler's Pcr is a theoretical elastic limit with no factor of safety. In AISC 360 LRFD, the design compressive strength is phi*Pn = 0.90*Pn, where Pn accounts for initial imperfections and residual stresses. Pcr from this calculator must be compared to factored loads after applying appropriate reduction factors per AISC 360 Chapter E.
Official sources
- American Institute of Steel Construction: AISC 360-22 Specification Chapter E Design of Members Subject to Compression.
- American Society of Civil Engineers: ASCE 7 Minimum Design Loads for structural frames.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.