--------------- | ------------------------------- | ----------------------- | -------------------------------- | ------------------------- | | Buckling curve | Single curve (SSRC 2P) | Multiple alpha_b values | 5 curves (a0-d) | Single curve | | Resistance factor | phi_c = 0.90 | phi = 0.90 | gamma_M1 = 1.00 | phi = 0.90 | | Interaction equation | H1-1a/H1-1b (bilinear) | Section 8.4 (bilinear) | 6.3.3 (linear + N-M interaction) | 13.8 (similar to AISC H1) | | Second-order analysis | Direct analysis method (Ch. C) | Amplified moment method | EN 1993-1-1 Cl. 5.2.2 | Amplified first-order | | K-factor approach | Alignment chart or K=1 with DAM | Effective length ratios | Buckling length ratios | Similar to AISC | | Slenderness limit | KL/r <= 200 (recommended) | L_e/r <= 200 | lambda_bar practical limit | KL/r <= 200 |

Common mistakes to avoid

• Using K = 1.0 for columns in unbraced frames — sway-permitted K values are significantly larger (1.2 to 2.5+), dramatically reducing capacity. The direct analysis method (AISC Chapter C) permits K = 1.0 only when notional loads and stiffness reductions are applied to the analysis model.
• Neglecting weak-axis buckling — W-shapes have r_y much less than r_x (often 40-60%), so weak-axis KL/r usually governs unless the column is braced about the weak axis at shorter intervals.
• Ignoring P-delta amplification — second-order effects increase column moments by 10-30% in typical braced frames and much more in moment frames. The B1 amplifier (member P-delta) and B2 amplifier (story P-Delta) must both be applied.
• Using axial-only tables for beam-columns — AISC Table 4-1 gives pure axial capacity. Any bending reduces the available axial capacity via the H1 interaction equations. Even small moments from connection eccentricity should be checked.
• Not checking both strong and weak axis bending — for corner columns and columns at re-entrant corners, bending occurs about both axes simultaneously. Both M_rx and M_ry terms appear in the interaction equation.

Torsional and Flexural-Torsional Buckling (AISC 360 Section E4)

For singly-symmetric and asymmetric sections (angles, WT tees, channels), torsional or flexural-torsional buckling may govern over flexural buckling. This occurs when the shear centre does not coincide with the centroid, causing the member to twist as it buckles rather than bending about a principal axis. The elastic buckling stress F_e for these sections is the minimum root of the cubic equation incorporating flexural buckling about both axes and torsional buckling about the shear centre.

For single angles (a common bracing and truss member), the AISC Specification Section E5 provides simplified equations. For equal-leg angles loaded through one leg:

F_e = [pi^2 × E × C_w / (K_z × L_z)^2 + G × J] / (I_x + I_y)

where C_w is the warping constant (negligible for angles), J is the torsional constant, and I_x + I_y is the polar moment of inertia about the shear centre. The design procedure for single angles in compression is covered in AISC 360 Section E5, with modified slenderness limits accounting for the eccentricity inherent in single-angle connections.

WT (structural tee) sections, commonly used as truss chords and bracing members, are also susceptible to flexural-torsional buckling. The stem of the tee is connected, and the flange is free (or vice versa), creating an unsymmetric buckling mode. The buckling load for a WT section with the stem in compression is typically lower than the weak-axis flexural buckling load would suggest. Use AISC Table 4-7 for WT section axial capacities, which account for flexural-torsional buckling directly.

For the typical building column (doubly-symmetric W-shape with K_x × L_x/r_x and K_y × L_y/r_y as the primary checks), torsional buckling rarely governs. However, for columns with cruciform sections, very wide flanges, or asymmetric bracing, this check should not be overlooked.

AISC Chapter E design procedure — step by step

The following numbered procedure covers the full AISC 360-22 Chapter E (Section E3) design check for a doubly symmetric section under concentric axial compression. Every column design in a steel building follows these six steps.

Step 1 — Determine the factored axial load P_u. Apply LRFD load combinations from ASCE 7-22 / IBC 2024 to obtain the maximum required compressive strength. For gravity-only columns, the controlling combination is typically 1.2D + 1.6L. For columns in lateral-force-resisting systems, include wind and seismic combinations.

Step 2 — Select a trial section. Start with a W-shape from AISC Table 4-1 (available axial strength) based on the required capacity and KL. For beam-columns with significant bending, also check Table 6-1 (shapes for combined loading). The W14 family is the default starting point for building columns because of its favorable weak-axis radius of gyration and wide range of weights (W14x22 through W14x730).

Step 3 — Compute the slenderness ratio KL/r. Determine the effective length factor K (see the K-factor section below) and the unbraced length L for each axis. Calculate KL/r for both the strong axis (KL_x/r_x) and the weak axis (KL_y/r_y). The larger value governs. AISC recommends KL/r less than or equal to 200 for compression members.

(KL/r)_governing = max[ (K_x * L_x) / r_x ,  (K_y * L_y) / r_y ]

Step 4 — Determine the critical stress F_cr per AISC E3. First compute the elastic buckling stress:

F_e = pi^2 * E / (KL/r)^2

Then determine the limit slenderness:

lambda_r = 4.71 * sqrt(E / F_y)

For A992 steel (F_y = 50 ksi, E = 29,000 ksi), lambda_r = 4.71 * sqrt(29000/50) = 113.4.

• If KL/r <= lambda_r (inelastic buckling):

F_cr = [0.658^(F_y / F_e)] * F_y

• If KL/r > lambda_r (elastic / Euler buckling):

F_cr = 0.877 * F_e

Step 5 — Calculate the design compressive strength.

phi * P_n = phi_c * F_cr * A_g    (phi_c = 0.90)

Where A_g is the gross cross-sectional area from the AISC Steel Construction Manual Part 1.

Step 6 — Check the demand-to-capacity ratio.

P_u <= phi * P_n
D/C = P_u / (phi * P_n)  <=  1.0

If D/C exceeds 1.0, select a larger section and repeat from Step 3. If D/C is well below 1.0, consider a lighter section. Target a D/C between 0.80 and 0.95 for efficient designs with reasonable reserve.

Euler buckling vs inelastic buckling

Column buckling behavior divides into two distinct regimes based on the slenderness parameter KL/r. Understanding which regime governs is essential because the failure mechanism and the F_cr equation are fundamentally different.

Inelastic buckling (KL/r <= 4.71 * sqrt(E/F_y)). Short and intermediate columns buckle inelastically. At these slenderness levels, the column yield stress is reached on at least part of the cross section before elastic buckling can occur. Residual stresses from rolling and welding play a major role — they cause premature yielding at the tips of flanges, reducing the effective stiffness and triggering buckling at loads below the squash load F_y * A_g.

The AISC E3 equation for this regime is:

F_cr = [0.658^(F_y / F_e)] * F_y

This is a parabolic-type curve that transitions smoothly from the squash load (F_cr = F_y when KL/r = 0) toward the Euler curve at the transition point. The exponent (F_y/F_e) controls how rapidly the capacity drops with increasing slenderness. For stocky columns, the exponent approaches zero and F_cr approaches F_y. For columns near the transition slenderness, the exponent is close to 1.0 and F_cr drops significantly.

Elastic (Euler) buckling (KL/r > 4.71 * sqrt(E/F_y)). Slender columns buckle elastically — the material remains entirely elastic at the point of bifurcation, and failure is governed by the classic Euler hyperbola. The AISC equation applies a 0.877 reduction factor to account for initial out-of-straightness (taken as L/1000) and residual stresses:

F_cr = 0.877 * F_e = 0.877 * pi^2 * E / (KL/r)^2

The 0.877 factor comes from the Structural Stability Research Council (SSRC) Curve 2P. Without this factor, the Euler equation would be unconservative for real columns with geometric imperfections.

Quick threshold values for common steels:

For A992 columns, any member with KL/r below 113 is in the inelastic regime, which covers the vast majority of practical building columns. Elastic buckling only governs for unusually long, slender columns or unbraced members.

Verification at the boundary. At the transition slenderness (KL/r = 4.71 * sqrt(E/F_y)), both equations give the same result. For A992:

At KL/r = 113.4:
F_e = pi^2 * 29000 / 113.4^2 = 22.3 ksi
F_cr (inelastic) = 0.658^(50/22.3) * 50 = 0.658^2.242 * 50 = 0.392 * 50 = 19.6 ksi
F_cr (elastic)    = 0.877 * 22.3 = 19.6 ksi   (matches)

Effective length (K factor) quick reference

The K factor translates the actual end-restraint conditions of a column into an equivalent pin-ended length. The AISC Commentary (Table C-A-7.1) provides theoretical and recommended design K values for six idealized end conditions:

Why recommended values exceed theoretical. True fixed supports do not exist in practice — even stiff foundations rotate slightly, and beam-to-column connections have some flexibility. The recommended K values account for this partial fixity.

Alignment chart method. For columns in frames, the K factor is determined from the alignment chart (nomograph) using the G parameters at each end:

G = sum(I_col / L_col) / sum(I_beam / L_beam)

Where the summation includes all columns meeting at the joint (numerator) and all beams framing into the joint (denominator). Enter the alignment chart with G_A (bottom) and G_B (top) to read K. For sidesway-inhibited (braced) frames, use the braced frame chart. For sidesway-uninhibited (moment) frames, use the sway frame chart.

Direct analysis method simplification. AISC Chapter C permits the use of K = 1.0 for all columns when the direct analysis method is used with: (a) notional loads equal to 0.002 times the gravity load applied laterally at each level, and (b) reduced flexural stiffness (0.80 * EI) for all members contributing to the stability of the structure. Most modern designs use this approach because it eliminates the alignment chart and avoids the difficulty of estimating true K values in complex frames.

Worked example — W14x61 column (axial only)

Given: W14x61, A992 steel (F_y = 50 ksi, E = 29,000 ksi), braced frame, K = 1.0, L = 14 ft, P_u = 400 kips. Pure axial compression — no applied bending.

Section properties (AISC Manual Table 1-1):

Step 1 — Slenderness ratio. Weak axis governs for W-shapes unless braced differently:

KL/r_y = (1.0 * 14 * 12) / 2.45 = 168 / 2.45 = 68.6

Check against the recommended limit: 68.6 < 200. OK.

Step 2 — Check the transition slenderness.

lambda_r = 4.71 * sqrt(29000 / 50) = 4.71 * 24.08 = 113.4

Since KL/r = 68.6 < 113.4, the column is in the inelastic buckling regime.

Step 3 — Elastic buckling stress.

F_e = pi^2 * 29000 / 68.6^2 = 286,754 / 4706 = 60.9 ksi

Step 4 — Critical stress F_cr.

F_cr = 0.658^(50 / 60.9) * 50
     = 0.658^0.821 * 50
     = 0.712 * 50
     = 35.6 ksi

Step 5 — Design compressive strength.

phi * P_n = 0.90 * 35.6 * 17.9 = 573.8 kips

Step 6 — Demand-to-capacity ratio.

D/C = P_u / (phi * P_n) = 400 / 573.8 = 0.697

The W14x61 carries the 400-kip demand with approximately 30% reserve capacity. This is an efficient but not overly stressed column for this load. A W14x53 (phi*P_n ~ 495 kips at this KL) would be marginal at D/C = 0.81, and a W14x48 would be overstressed.

Worked example — HSS6x6x3/8 column (axial only)

Given: HSS6x6x3/8, A500 Gr. B (F_y = 46 ksi, E = 29,000 ksi), K = 1.0, L = 14 ft, P_u = 250 kips. This is a lighter column typical of mezzanine or low-rise construction.

Section properties (AISC Manual Table 1-12):

Step 1 — Slenderness ratio. Square HSS has equal radii of gyration about both axes:

KL/r = (1.0 * 14 * 12) / 2.27 = 168 / 2.27 = 74.0

Check: 74.0 < 200. OK.

Step 2 — Transition slenderness for A500 Gr. B.

lambda_r = 4.71 * sqrt(29000 / 46) = 4.71 * 25.12 = 118.2

Since 74.0 < 118.2, inelastic buckling governs.

Step 3 — Elastic buckling stress.

F_e = pi^2 * 29000 / 74.0^2 = 286,754 / 5476 = 52.4 ksi

Step 4 — Critical stress F_cr.

F_cr = 0.658^(46 / 52.4) * 46
     = 0.658^0.878 * 46
     = 0.697 * 46
     = 32.1 ksi

Step 5 — Design compressive strength.

phi * P_n = 0.90 * 32.1 * 7.58 = 218.9 kips

Step 6 — Demand-to-capacity ratio.

D/C = P_u / (phi * P_n) = 250 / 218.9 = 1.14

The HSS6x6x3/8 is overstressed at P_u = 250 kips with KL = 14 ft. The demand exceeds capacity by 14%. The designer must select a larger section such as HSS6x6x1/2 (phiP_n ~ 280 kips) or HSS8x8x3/8 (phiP_n ~ 310 kips), or reduce the unbraced length by adding bracing.

Comparison with W14x61 above: The W14x61 (phiP_n = 573.8 kips) has 2.6 times the axial capacity of the HSS6x6x3/8 (phiP_n = 218.9 kips), but it also weighs 61 lb/ft vs. 25.84 lb/ft — roughly 2.4 times heavier. The capacity-to-weight ratio is similar, but the W-shape provides much more absolute capacity. HSS sections are preferred for architectural exposure (clean appearance, equal properties about both axes) and torsional resistance, while W-shapes are preferred for maximum axial efficiency.

Column selection table — phi*P_n (kips) for common W-shapes

The following table provides the design compressive strength phi*P_n (kips, LRFD) for five common column sections at various effective lengths KL. All sections are A992 steel (F_y = 50 ksi). Use this table for quick trial selection before running a detailed calculation.

How to use this table:

1. Determine your P_u and KL.
2. Find the KL column that matches your condition (interpolate for intermediate values).
3. Scan down to find the lightest section with phi*P_n >= P_u.
4. Verify with a full calculation (local buckling checks, actual KL/r for your geometry).

Key observations:

• The W14x82 provides 30-35% more capacity than the W14x61 at every KL length, but at only 34% more weight (82 vs. 61 plf). This is because the extra material goes primarily into the flanges, increasing A_g without proportionally increasing r_y.
• The W12x65 outperforms the W14x61 at all KL lengths despite similar weight, because r_y = 3.02 in. vs. 2.45 in. — a 23% advantage in the governing slenderness parameter. For pure axial capacity, W12 shapes often beat W14 shapes at the same weight per foot.
• The W8x31 is only viable for shorter columns (KL < 15 ft) with moderate loads. Its small r_y = 2.02 in. causes rapid capacity loss with increasing KL.

Combined axial and bending — AISC Chapter H interaction equations

When a column carries bending in addition to axial compression (a beam-column), the AISC Chapter H interaction equations limit the combined demand. This is the general case for columns in moment frames, where lateral loads and frame continuity induce bending moments.

When does this apply? Any structural member with P_u > 0 and non-zero bending moments. Common scenarios include:

• Columns in moment frames (bending from lateral forces)
• Columns with eccentric connections (moment = P * eccentricity)
• Corner columns with beams framing in from two directions
• Columns supporting continuous beams (moment transfer through connections)

Equation H1-1a (when P_r / P_c >= 0.2):

P_r/P_c + (8/9) * [M_rx/M_cx + M_ry/M_cy]  <=  1.0

Equation H1-1b (when P_r / P_c < 0.2):

P_r/(2*P_c) + [M_rx/M_cx + M_ry/M_cy]  <=  1.0

Where:

Second-order effects are mandatory. The required moments M_rx and M_ry must include the amplification from P-delta effects. AISC Chapter C provides two approaches:

1. B1-B2 amplifier method — Amplify first-order moments using B1 (member P-delta, Eq. C2-2) and B2 (story P-Delta, Eq. C2-3). This is the traditional hand-calculation approach.

2. Direct second-order analysis — Run a geometric nonlinear (P-Delta) analysis in structural analysis software. The analysis model directly captures the amplification. This is the preferred method for complex frames.

Biaxial bending. When both M_rx and M_ry are non-zero (corner columns, columns at re-entrant corners), both terms appear in the interaction equation. This significantly reduces available capacity compared to uniaxial bending. A column that is adequate for P + M_x alone may fail the interaction check when a small M_y is added.

Practical design tip. For initial sizing of beam-columns, first size for axial alone (phi*P_n >= P_u with D/C around 0.70-0.80). This leaves 20-30% of the interaction equation for the bending terms. Then verify the full interaction check. If bending demands are high (M > 0.5 * M_c), start with a section selected for bending (using Z_x) and then verify axial capacity.

Frequently Asked Questions

What is the difference between AISC 360 column design and EN 1993-1-1 column design?

The fundamental approach is similar (slenderness parameter, buckling curve, reduction factor), but the details differ: AISC uses a single buckling curve (SSRC Curve 2P) with phi_c = 0.90, while EN 1993 uses five buckling curves (a0 through d) selected based on section type, fabrication method, and buckling axis with gamma_M1 = 1.00. EN 1993 also provides different buckling curves for strong-axis vs weak-axis buckling of the same section. For a typical W-shape column designed to both codes, the AISC capacity is typically 5-10% higher than the EN 1993 capacity due to the difference in imperfection factors.

When should I use a W12 column instead of a W14 column?

W12 columns are preferred when: (1) the column is in a moment frame (W12 sections have deeper flanges and thicker webs, providing better moment capacity for the same weight), (2) the platform width must be minimised (W12 sections are narrower), or (3) the available A_g values in the W14 family are insufficient for the axial demand. W14 columns are preferred for pure axial compression because their larger r_y gives lower KL/r and higher F_cr for the same weight. The W14 is the default column for braced frames; the W12 is the default for moment frames. AISC Table 4-1 and Table 6-1 provide pre-calculated capacities for both families at various KL values.

What is the maximum KL/r allowed for a steel column?

AISC 360 Section E2 recommends KL/r less than or equal to 200 for compression members. This is a practical recommendation, not an absolute limit — members with KL/r above 200 may be designed per the E3 provisions but will have very low capacity, are susceptible to vibration and accidental damage during handling and erection, and are generally uneconomical. AS 4100 and EN 1993-1-1 have similar recommendations (L_e/r less than or equal to 200 and lambda_bar practical limit, respectively). For tension-only bracing, KL/r less than or equal to 300 is permitted per AISC 360 Section D1.

How do I account for column splice in the design?

Column splices are typically located 1.2-1.5 m (4-5 ft) above the finished floor level for erection convenience. The splice must develop the column axial load plus any moment from frame action. For columns in braced frames (moment at splice is small), a partial-joint-penetration groove weld or bolted flange plates with a shear web plate are common. The splice plate design follows the same limit states as described in the Steel Plate Design reference: gross yielding, net section fracture, and block shear at the bolt groups. For heavy columns (W14x283 and above), the splice may require full-penetration groove welds to develop the flange capacity.

What is the direct analysis method and when should I use it?

The direct analysis method (DAM), described in AISC 360 Chapter C, replaces the traditional effective length method (alignment chart) with a second-order analysis that directly accounts for P-Delta effects, member out-of-straightness (via notional loads), and stiffness reduction due to residual stresses. DAM permits the use of K = 1.0 for all columns, eliminating the alignment chart entirely. DAM is required for structures where the ratio of second-order drift to first-order drift exceeds 1.5, for structures with nominal hinged bases, and for any structure where the engineer prefers not to compute K factors. Most modern structural analysis software (ETABS, SAP2000, RAM, Tekla Structural Designer) includes automated DAM workflows. For hand calculations and simple frames, the effective length method with alignment chart K factors remains acceptable.

Column Base Design — Brief Overview

Steel columns transfer axial load and moment to the foundation through a base plate that distributes the load to the concrete footing or pier. The column base is the interface between the steel superstructure and the concrete substructure, and its design involves both AISC 360 (steel plate and anchor rod design) and ACI 318 (concrete bearing and anchorage design). Key design steps:

1. Base plate area: Select plate dimensions so the bearing pressure on the concrete does not exceed phic * 0.85 _ f'c _ sqrt(A2/A1), limited to phic * 1.7 * f'c per ACI 318 Section 22.8 (phi_c = 0.65 for bearing)
2. Base plate thickness: Determine from bending of the plate cantilevering beyond the column footprint, using the yield line method (AISC DG1) or the simplified method: t = l _ sqrt(2 _ Pu / (0.9 * Fy * B * N)) where l is the greater of m (flange cantilever) and n (web cantilever)
3. Anchor rod design: Size rods for tension (uplift and moment) and shear. Anchor rod tension capacity per AISC 360 Table J3.2 (F_nt = 0.75 * F_u, phi = 0.75). Shear is typically resisted by anchor rods in bearing against the base plate (shear lug required for high shear)
4. Shear transfer: If the factored shear exceeds the friction capacity (mu * P_u where mu = 0.2 for steel on grout), a shear lug or anchor rod bearing must be provided
5. Grout pad: Non-shrink grout (typically 25-50 mm thick) provides full bearing and levels the base plate. The grout must have compressive strength at least equal to the concrete footing strength

For pinned bases (shear only, no moment), the base plate is compact and anchor rods are placed inside the column footprint. For fixed bases (moment-resisting), the base plate extends well beyond the column, anchor rods are placed outside the flanges, and the plate thickness increases significantly to resist bending from the anchor rod tension couple. Use the Column Base Design Calculator for automated sizing per AISC DG1.

Built-Up and Composite Columns

When rolled W-shapes lack the required capacity, built-up or composite columns may be specified. Built-up columns consist of two or more rolled sections laced, battened, or welded together to form a single compression member. Common configurations include: double channels toe-to-toe with battens (common for industrial bracing and struts), four angles laced to form a box column (historic construction, transmission towers), and cover-plated W-shapes (the most common modern built-up column, where plates are welded to the flanges to increase A_g and r_y).

Composite columns combine structural steel with reinforced concrete to create a member with higher axial capacity and inherent fire resistance than either material alone. Concrete-filled HSS columns (CFT) and encased W-shapes are the two primary types, designed per AISC 360 Chapter I (composite members). The concrete fill provides axial capacity, prevents inward local buckling of the steel tube, and provides inherent fire resistance (the concrete core absorbs heat and delays the temperature rise of the steel).

For typical building columns, built-up and composite solutions are reserved for: (1) axial demands exceeding 3,000 kips, where the largest rolled W14 section (W14x730, phi*P_n ~ 3,000 kips at KL=15 ft) is insufficient; (2) architectural requirements for exposed column appearance; and (3) blast or impact resistance requirements where the concrete fill provides additional robustness.

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Related references

• K-Factor Guide
• Column K-Factor
• How to Verify Calculations
• Column Buckling Reference
• Effective Length Factors
• Frame Analysis Methods
• Composite Column
• Steel Buckling

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This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.

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