Column Buckling Equations — Euler, AISC Chapter E, and Fcr Reference

Column buckling is the governing limit state for most steel compression members. The available compressive strength depends on the slenderness ratio KL/r, which determines whether failure is by inelastic buckling (yielding + instability) or elastic buckling (Euler). AISC 360-22 Chapter E provides the equations.

Euler elastic buckling

The theoretical foundation: Fe = pi^2*E/(KL/r)^2, where E = 29,000 ksi, K = effective length factor, L = unbraced length, r = radius of gyration.

Euler stress for common slenderness ratios

KL/r Fe (ksi) Notes
20 716 Very stocky, yielding governs
40 179 Inelastic buckling
60 79.5 Moderate slenderness
80 44.7 Transition zone
100 28.6 Approaching elastic
120 19.9 Slender column
160 11.2 Near practical maximum
200 7.15 AISC recommended max KL/r

AISC 360-22 Chapter E — column strength

AISC uses a two-equation curve transitioning at KL/r = 4.71*sqrt(E/Fy):

Inelastic buckling (KL/r <= 4.71*sqrt(E/Fy)):

Fcr = 0.658^(Fy/Fe) * Fy     [Eq. E3-2]

Elastic buckling (KL/r > 4.71*sqrt(E/Fy)):

Fcr = 0.877 * Fe              [Eq. E3-3]

The 0.877 factor accounts for initial out-of-straightness (L/1500). Design strength: phiPn = 0.90 _ Fcr _ Ag.

Transition slenderness

Fy (ksi) Transition KL/r
36 134
50 113
65 99.5

Column strength table (Fy = 50 ksi)

KL/r Fe (ksi) Fcr (ksi) phiFcr (ksi) % of Fy
0 -- 50.0 45.0 100%
20 716 49.1 44.2 98%
40 179 45.1 40.6 90%
60 79.5 38.3 34.5 77%
80 44.7 29.8 26.8 60%
100 28.6 22.0 19.8 44%
120 19.9 17.5 15.7 35%
160 11.2 9.8 8.8 20%
200 7.15 6.3 5.6 13%

A column at KL/r = 100 retains only 44% of its squash load.

Worked example — W14x82, Fy = 50 ksi

Given: W14x82, KL = 20 ft (braced frame, K = 1.0). Ag = 24.0 in^2, ry = 2.48 in.

KL/r = (2012)/2.48 = 96.8. Fe = pi^229000/96.8^2 = 30.5 ksi. Transition: 4.71*sqrt(29000/50) = 113.4. Since 96.8 < 113.4, use inelastic equation.

Fcr = 0.658^(50/30.5)_50 = 0.658^1.639 _ 50 = 0.503*50 = 25.2 ksi.

phiPn = 0.9025.224.0 = 544 kips. (AISC Manual Table 4-1a: 545 kips -- match.)

Multi-code comparison

AS 4100: Five column curves (alpha_b based on section type). phiNc = phialpha_ckfAnfy.

EN 1993-1-1: Five buckling curves (a0, a, b, c, d). chi = 1/(Phi + sqrt(Phi^2 - lambda_bar^2)). Imperfection factors: a=0.21, b=0.34, c=0.49, d=0.76.

CSA S16: Cr = phiAFy*(1 + lambda^(2n))^(-1/n), n = 1.34 for hot-rolled.

Common mistakes

  1. Checking only one axis. The axis with higher KL/r (lower capacity) governs. For W-shapes, weak axis (ry) typically governs.
  2. Using K = 1.0 for unbraced frames. K > 1.0 for sway frames.
  3. Forgetting the 0.877 factor. AISC elastic curve is 0.877*Fe, not Fe.
  4. Using wrong r for the axis being checked. rx for strong axis, ry for weak axis.
  5. Neglecting slender element effects. HSS with thin walls need effective area reduction.

Frequently asked questions

What is the maximum slenderness ratio? AISC recommends KL/r <= 200 for compression members. At KL/r = 200, only 13% of squash load capacity remains.

Does Euler buckling apply to real columns? Fe is the theoretical upper bound. Real columns fail at lower loads due to residual stresses and imperfections. AISC accounts for this with the inelastic curve and 0.877 factor.

Which axis governs? The axis with the largest KL/r. For unbraced columns with equal lengths in both directions, always the weak axis (ry < rx).

Run this calculation

Related references

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Chapter E and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.