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

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

Column strength table — Fy = 50 ksi

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

Column strength table — Fy = 36 ksi

Column strength table — Fy = 65 ksi

High-strength steel (65 ksi) loses capacity faster relative to Fy because the transition to elastic buckling occurs at a lower slenderness ratio.

phiPn for common W-shapes — Fy = 50 ksi, K = 1.0

Values are phiPn in kips for weak-axis buckling. Strong-axis buckling (rx) typically gives higher capacity.

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 = (20*12)/2.48 = 96.8. Fe = pi^2*29000/96.8^2 = 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 = 0.90*25.2*24.0 = 544 kips. (AISC Manual Table 4-1a: 545 kips -- match.)

Worked example — W12x65 strong vs weak axis

Given: W12x65, Fy = 50 ksi, L = 18 ft, K = 1.0.

Properties: Ag = 19.1 in^2, rx = 5.28 in, ry = 3.02 in.

Weak axis (ry): KL/r = (18*12)/3.02 = 71.5. Fe = 55.9 ksi. 4.71*sqrt(29000/50) = 113.4. Inelastic. Fcr = 0.658^(50/55.9)*50 = 0.658^0.894*50 = 0.699*50 = 34.9 ksi. phiPn = 0.90*34.9*19.1 = 600 kips.

Strong axis (rx): KL/r = (18*12)/5.28 = 40.9. Fe = 171 ksi. Fcr = 0.658^(50/171)*50 = 0.658^0.292*50 = 0.878*50 = 43.9 ksi. phiPn = 0.90*43.9*19.1 = 755 kips.

Weak axis governs at 600 kips vs 755 kips. This is typical for W-shapes where ry < rx.

Multi-code comparison

AS 4100-2020: Five column curves (alpha_b based on section type). phiNc = phi*alpha_c*kf*An*fy. The alpha_c factor is determined from modified slenderness lambda_n and member capacity reduction factor alpha_b. Column curves range from alpha_b = -1.0 (cold-formed) to +1.0 (hot-rolled, heavily restrained).

EN 1993-1-1: Five buckling curves (a0, a, b, c, d). chi = 1/(Phi + sqrt(Phi^2 - lambda_bar^2)), where Phi = 0.5*(1 + alpha*(lambda_bar - 0.2) + lambda_bar^2).

CSA S16-19: Cr = phi*A*Fy*(1 + lambda^(2n))^(-1/n), n = 1.34 for hot-rolled. Uses a single column curve (no section-dependent curves like AS 4100 or EN 1993). phi = 0.90 for compression.

Cross-code comparison: phiPn for a W12x65 at Fy = 50 ksi (345 MPa), L = 18 ft

All codes produce similar results for typical sections. Differences become more significant for high slenderness ratios.

Flexural-torsional and torsional buckling (AISC E4)

Singly-symmetric sections (channels, tees, angles) and unsymmetric sections (structural tees, double angles) can fail in flexural-torsional buckling — combined lateral bending and twisting about the shear center. AISC E4 provides separate provisions:

For singly-symmetric sections (channels, tees, Z-shapes): The flexural-torsional buckling stress Fe is the smaller of the flexural Euler stress and:

Fe = (Fey + Fez) / (2 x H) x [1 - sqrt(1 - 4 x Fey x Fez x H / (Fey + Fez)^2)]

where Fey = pi^2 x E / (KyLy/ry)^2 (weak-axis Euler stress), Fez = (pi^2 x E x Cw / (KzLz)^2 + G x J) / (Ag x ro^2) (torsional buckling stress), H = 1 - (x_o^2 + y_o^2)/ro^2 (polar moment parameter), and ro = polar radius of gyration about the shear center.

For a typical C12x20.7 channel (KL = 10 ft, Fy = 36 ksi):

Fey = pi^2 x 29000 / (120/0.799)^2 = pi^2 x 29000 / 22584 = 12.7 ksi
Fez = (pi^2 x 29000 x 34.8 / 120^2 + 11200 x 0.194) / (6.09 x 2.86^2) = (578 + 2173) / 49.8 = 55.2 ksi
H = 1 - (0.698^2 + 0^2)/2.86^2 = 1 - 0.0596 = 0.940
Fe_FT = (12.7 + 55.2) / (2 x 0.940) x [1 - sqrt(1 - 4 x 12.7 x 55.2 x 0.940 / (12.7 + 55.2)^2)]
      = 36.1 x [1 - sqrt(1 - 2636 / 4607)] = 36.1 x [1 - 0.654] = 12.5 ksi

The flexural-torsional stress Fe_FT = 12.5 ksi is less than the weak-axis Euler stress Fey = 12.7 ksi — the torsional coupling reduces capacity by approximately 2%. For channels with short lips or tees with thin stems, the reduction can be 10-20%.

For double-angle sections: Flexural-torsional buckling almost always governs because the shear center offset creates strong torsional coupling. The E4-2 equation must be used. For a 2L6x4x3/8 (LLBB, long legs back-to-back) with KL = 8 ft: the flexural-torsional stress is approximately 50-60% of the weak-axis Euler stress, making the E4 check the governing limit state.

For HSS and round sections: These sections are doubly-symmetric and do not require flexural-torsional buckling checks (torsional buckling capacity for HSS is typically much higher than flexural buckling capacity). However, for thin-walled HSS (D/t > 0.11 x E/Fy), local buckling per AISC E7 may reduce capacity.

Built-up compression members (AISC E6)

Built-up columns (laced columns, battened columns, and pairs of channels or angles connected by stitch plates) require special provisions per AISC E6:

Modified slenderness: For built-up members with open sides (laced/battened pairs), the slenderness ratio of the individual component between connectors modifies the overall member slenderness. If the component slenderness (a/ri) exceeds 0.75 x (KL/r)_o, where (KL/r)_o is the overall slenderness:

(KL/r)_eff = sqrt((KL/r)_o^2 + (K_s x a/ri)^2)

where K_s = 0.50 for snug-tight bolted connections and 0.75 for welded connections. For a pair of C12x20.7 channels toe-to-toe with stitch plates at a = 40 in, ri = 0.796 in (individual channel weak axis), overall KL/r = 65: a/ri = 40/0.796 = 50.3, 0.75 x 65 = 48.8, since 50.3 > 48.8, the modified slenderness applies: (KL/r)_eff = sqrt(65^2 + (0.75 x 50.3)^2) = sqrt(4225 + 1423) = 75.1 — a 15% increase in effective slenderness.

Stitch spacing limit: Per AISC E6.2, the maximum spacing of stitch plates or lacing bars is a_max = 0.75 x ri x (KL/r)_o. For the C12 example: a_max = 0.75 x 0.796 x 65 = 38.8 in. The actual spacing of 40 in slightly exceeds this limit, so the modified slenderness check applies.

End connections: At the ends of built-up columns, weld or bolt lengths must develop 50% of the member force in each component per AISC E6.1. For compression, this force is 0.50 x Fy x Ag of the individual component. For tension, it is 0.50 x Fu x Ag.

Residual stress effects on column buckling

The AISC column curve (0.658^(Fy/Fe) x Fy) is calibrated to account for residual stresses from hot-rolling. During cooling, the flange tips and web center cool faster than the flange-web junction, creating compressive residual stresses at the tips (approximately 10-15 ksi for W-shapes) and tensile residual stresses at the junction.

How residual stresses affect buckling: When an axial load is applied, the portions of the section already in compressive residual stress yield earlier than the rest. This effectively reduces the elastic core of the section, lowering the buckling capacity. The AISC inelastic column equation (E3-2) captures this effect through the 0.658 factor. Without residual stresses, the column curve would follow a modified Johnson parabola with higher capacity in the inelastic range (KL/r = 40-80).

Imperfection sensitivity: The elastic curve multiplies Fe by 0.877 to account for initial out-of-straightness of L/1500 per AISC Commentary. Without this factor, the Euler load Fe is an upper bound that cannot be achieved in practice. The combined effect of residual stresses and initial crookedness reduces column capacity by approximately 20-35% at moderate slenderness (KL/r = 40-80) compared to an ideal column.

Effective length factor K — quick reference

See the Effective Length Factor K page for the full alignment chart method.

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. Using K = 1.0 for a sway frame overestimates capacity by 50-100%.

3. Forgetting the 0.877 factor. AISC elastic curve is 0.877*Fe, not Fe. Using Fe directly overestimates capacity by 14%.

4. Using wrong r for the axis being checked. rx for strong axis, ry for weak axis. Mixing these up is a critical error.

5. Neglecting slender element effects. HSS with thin walls need effective area reduction per AISC E7. The full Ag may not be available.

6. Using unsupported length instead of effective length. The effective length KL includes the K factor. KL/r uses KL, not just L.

7. Not checking KL/r <= 200. AISC recommends this limit. Above 200, the column is very sensitive to imperfections and capacity drops rapidly.

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. Some codes allow higher (AS 4100 permits up to 300 for secondary members).

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).

How does Fy affect column capacity? Higher Fy increases squash load (Fy*Ag) but also lowers the transition slenderness (4.71*sqrt(E/Fy)). This means high-strength steel transitions to elastic buckling sooner, reducing the advantage at moderate slenderness.

When do I need to use the effective length method vs the direct analysis method? AISC Chapter C permits the Direct Analysis Method (DAM) as the primary method, which uses K = 1.0 with reduced stiffness. The Effective Length Method (ELM) is allowed as an alternative for certain frames. DAM is simpler and preferred for most cases.

What is the difference between local and global buckling? Global (member) buckling involves the entire column bowing laterally over its length (governed by KL/r). Local buckling involves individual plate elements (flanges or web) buckling before the member reaches its overall capacity (governed by b/t or h/tw ratios). Both must be checked.

Can I interpolate the Fcr table? Yes. The AISC equations are continuous functions. For intermediate KL/r values not shown in tables, interpolate linearly or calculate directly using the equations. Direct calculation is more accurate.

What is flexural-torsional buckling and when does it apply?

Flexural-torsional buckling is a combined bending-and-twisting failure mode that occurs in singly-symmetric sections (channels, tees, double angles) and unsymmetric sections. Unlike doubly-symmetric W-shapes where the shear center coincides with the centroid, singly-symmetric sections have offset shear centers — axial load produces both bending and twisting. Per AISC E4, the flexural-torsional buckling stress Fe is calculated from the coupled Fey (weak-axis Euler) and Fez (torsional) terms. For a C12x20.7 channel at KL = 10 ft: Fey = 12.7 ksi, Fez = 55.2 ksi, and the coupled Fe_FT = 12.5 ksi — 2% less than Fey. For double angles, the reduction is 10-20% and flexural-torsional buckling almost always governs over pure flexural buckling. The E4 check is required for all singly-symmetric and unsymmetric compression members.

How does the Direct Strength Method apply to column buckling?

The Direct Strength Method (DSM) per AISI S100 Appendix 1 handles column buckling differently from AISC Chapter E by considering local, distortional, and global buckling separately. For CFS columns, DSM calculates: (1) elastic global buckling stress Fcre (AISC-like Euler), (2) elastic local buckling stress Fcrl from finite strip analysis, and (3) elastic distortional buckling stress Fcrd. The nominal capacity Pne (global) uses the same AISC equations: Pne = 0.658^(Fy/Fcre) x Fy x Ag for Fcre > 0.44 Fy, or Pne = 0.877 x Fcre x Ag for Fcre <= 0.44 Fy. Local buckling then reduces Pne to Pnl = Pne when lambda_l <= 0.776, or Pnl = (1 - 0.15 x (Fcrl/Fne)^0.4) x (Fcrl/Fne)^0.4 x Pne when lambda_l > 0.776. Distortional buckling separately gives Pnd. The controlling capacity is min(Pnl, Pnd, Pne). This is fundamentally different from AISC E7 (which uses effective area) because DSM uses the full gross section properties with elastic buckling stress ratios.

AISC E3 complete procedure with Fcr formula

AISC 360-22 Section E3 provides the compressive strength of members subject to flexural buckling. This is the most common limit state for steel columns.

Complete design procedure

1. Determine factored axial load Pu from LRFD load combinations.
2. Select trial section and obtain Ag, rx, ry from AISC Table 1-1.
3. Determine effective length KL for each axis (strong axis using rx, weak axis using ry).
4. Calculate slenderness ratio for each axis: KL/rx and KL/ry. The larger value governs.
5. Verify local buckling per Table B4.1a (compact/noncompact/slender elements).
6. Calculate elastic critical stress: Fe = pi^2 * E / (KL/r)^2.
7. Determine transition slenderness: 4.71 * sqrt(E/Fy).
8. Calculate Fcr:
   • If KL/r <= 4.71*sqrt(E/Fy): Fcr = 0.658^(Fy/Fe) * Fy (inelastic, Eq. E3-2)
   • If KL/r > 4.71*sqrt(E/Fy): Fcr = 0.877 * Fe (elastic, Eq. E3-3)
9. Calculate design strength: phi*Pn = 0.90 * Fcr * Ag.
10. Check phi*Pn >= Pu. If not, select a larger section.
11. Verify KL/r <= 200 (recommended maximum).

Euler vs inelastic buckling regions

The AISC column curve is divided into two regions based on the transition slenderness ratio:

For Fy = 50 ksi (A992), the transition occurs at KL/r = 113.4. Most practical building columns have KL/r in the range of 30-80 and fall in the inelastic region.

KL/r limitations and practical implications

Column selection by effective length table (Fy = 50 ksi)

This table shows the lightest W-section providing phiPn >= Pu for common effective lengths.

Values are approximate based on weak-axis buckling (ry) with K = 1.0. Always verify with AISC Manual Table 4-1.

Worked example — W14x82 column

Given: W14x82 (A992, Fy = 50 ksi), factored axial load Pu = 580 kips, K = 1.0, L = 14 ft (braced frame).

Section properties:

Step 1 — Slenderness:

KL/rx = 1.0 x 14 x 12 / 6.05 = 27.8
KL/ry = 1.0 x 14 x 12 / 4.26 = 39.4  (GOVERNS)

KL/ry = 39.4 < 200 (OK).

Step 2 — Elastic critical stress:

Fe = pi^2 x 29,000 / (39.4)^2 = 286,159 / 1,552 = 184.4 ksi

Step 3 — Transition check:

4.71 x sqrt(29,000/50) = 4.71 x 24.08 = 113.4
39.4 < 113.4 --> Inelastic buckling (E3-2)

Step 4 — Fcr calculation:

Fcr = 0.658^(Fy/Fe) x Fy = 0.658^(50/184.4) x 50
    = 0.658^0.271 x 50 = 0.880 x 50 = 44.0 ksi

Step 5 — Design strength:

phiPn = 0.90 x 44.0 x 24.1 = 954 kips > Pu = 580 kips (OK)
Utilization = 580 / 954 = 61%

Step 6 — Section comparison for same load:

The W14x82 is the most efficient choice because its large ry (wide flanges) keeps KL/ry low, resulting in high Fcr. The W14 shape is preferred for columns because the nearly equal flange width and depth provide similar rx and ry values.

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Related references

• Effective Length Factor K
• K-Factor Reference
• Beam Sizes
• Compact Section Limits
• Lateral-Torsional Buckling
• Column K Factor
• Column Base Plate
• Column Splice
• Structural Steel Weights
• Steel Grades Reference
• How to Verify Calculations

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.

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