Column Design Guide — Engineering Reference

AISC 360 Chapter E steel column design: Fcr formula, effective length K, KL/r slenderness check, combined axial-bending interaction, and section selection.

Overview

Steel column design per AISC 360 Chapter E determines axial compression capacity based on the slenderness ratio KL/r, where K is the effective length factor, L is the unbraced length, and r is the governing radius of gyration. The designer must identify the critical buckling axis (usually weak-axis for W-shapes), classify the section per Table B4.1a, and compute the critical stress Fcr.

For beam-columns subjected to combined axial compression and bending, the interaction equations of Chapter H (H1-1a and H1-1b) govern the design. Selecting an efficient column requires balancing axial demand against bending from frame action, eccentricity, or lateral loads.

Axial compression capacity

AISC 360 Chapter E uses two equations depending on the slenderness parameter KL/r relative to 4.71*sqrt(E/Fy):

• Inelastic buckling (KL/r <= 4.71*sqrt(E/Fy)): Fcr = [0.658^(Fy/Fe)] * Fy, where Fe = pi^2 * E / (KL/r)^2 is the Euler elastic buckling stress.
• Elastic buckling (KL/r > 4.71*sqrt(E/Fy)): Fcr = 0.877 * Fe. This reduction accounts for initial imperfections and residual stresses.

The available axial strength is phi*c * Pn = 0.90 _ Fcr * Ag.

Effective length K-factors

The K-factor depends on end restraint conditions. Common values from the AISC Commentary Table C-A-7.1:

• K = 1.0 for pinned-pinned (sidesway prevented)
• K = 0.65 for fixed-fixed (sidesway prevented, theoretical 0.5)
• K = 1.2 for pinned-fixed (sidesway permitted, practical)
• K = 2.1 for fixed-free cantilever (sidesway permitted, practical)

For frames with sidesway, the alignment chart (nomograph) or direct analysis method of Chapter C should be used. The direct analysis method applies notional loads and stiffness reductions, allowing K = 1.0 in many cases.

Combined axial and bending (H1 interaction)

For P_r/P_c >= 0.2: P_r/P_c + (8/9) x [M_rx/M_cx + M_ry/M_cy] <= 1.0 (Eq. H1-1a)

For P_r/P_c < 0.2: P_r/(2 x P_c) + [M_rx/M_cx + M_ry/M_cy] <= 1.0 (Eq. H1-1b)

Where P_r is the required axial strength, P_c is the available axial strength, M_rx/M_ry are the required flexural strengths (including second-order effects), and M_cx/M_cy are the available flexural strengths. Second-order effects (P-delta member curvature and P-Delta story sway) must be included in the required strengths via B1-B2 amplification or direct second-order analysis per AISC Chapter C.

The interaction equations produce a linear interaction surface in P-Mx-My space. Equation H1-1a (used when the axial ratio exceeds 0.2) weights the bending terms by 8/9, slightly reducing the penalty for bending. Equation H1-1b (used when axial is light) removes most of the axial penalty and lets the bending terms control.

Worked example — W14x82 beam-column

Given: W14x82, A992 (F_y = 50 ksi), braced frame, K_x = K_y = 1.0, L = 14 ft, P_u = 400 kip, M_ux = 180 kip-ft (strong axis), M_uy = 0. Properties: A = 24.0 in^2, r_x = 6.05 in., r_y = 2.48 in., Z_x = 139 in^3, L_p = 8.76 ft, L_r = 28.5 ft.

1. Axial capacity: KL/r_y = (1.0 x 14 x 12)/2.48 = 67.7. F_e = pi^2 x 29000/67.7^2 = 62.5 ksi. F_cr = 0.658^(50/62.5) x 50 = 0.658^0.80 x 50 = 0.706 x 50 = 35.3 ksi. P_n = 35.3 x 24.0 = 847 kip. phi x P_n = 0.90 x 847 = 762 kip.
2. Flexural capacity: L_b = 14 ft. Since L_p (8.76) < L_b (14.0) < L_r (28.5), inelastic LTB. C_b = 1.0 (conservative for beam-column). M_n = M_p - (M_p - 0.7 x F_y x S_x) x (L_b - L_p)/(L_r - L_p) = 6950 - (6950 - 4079) x (14 - 8.76)/(28.5 - 8.76) = 6950 - 2871 x 0.265 = 6189 kip-in. phi x M_n = 0.90 x 6189/12 = 464 kip-ft.
3. Interaction: P_r/P_c = 400/762 = 0.525 > 0.2, so use H1-1a: 0.525 + (8/9) x (180/464 + 0) = 0.525 + 0.889 x 0.388 = 0.525 + 0.345 = 0.870 <= 1.0. OK.

The interaction ratio of 0.870 indicates the W14x82 has 13% reserve capacity for this load combination.

Column selection strategy

Efficient column selection depends on the axial-to-bending demand ratio:

• High axial, low bending (P_r/P_c > 0.5): Select based on A_g (cross-sectional area). W14 columns are preferred because they have large A_g values in a compact footprint and favorable r_y values.
• Moderate axial, moderate bending (0.2 < P_r/P_c < 0.5): Use AISC Table 6-1 (W-shapes selected for combined loading) which plots the interaction equation graphically.
• Low axial, high bending (P_r/P_c < 0.2): Select based on Z_x. Deep sections (W18, W21, W24) are more efficient in bending but may have lower r_y, increasing KL/r.

The W14 family is the workhorse for columns because: (1) r_y is relatively large (2.48 in. for W14x82 vs. 1.57 in. for W21x83), reducing KL/r; (2) many weights are available from W14x22 to W14x730; (3) column splices are simplified when all columns share the same nominal depth.

Code comparison — column design

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.

Run this calculation

• Column Capacity Calculator
• Section Properties Database

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

Disclaimer

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.