AISC 360 Steel Column Design

Compression Member Basics

A compression member (column) fails by buckling, not by crushing. Unlike a tension member where capacity is simply yield stress times area, a column's capacity depends on its length, cross-section shape, and end restraint. A short, stocky column can reach the full yield strength of the steel (squash load = F_y × A_g), while a long, slender column buckles at a stress far below yield, governed by Euler's elastic buckling theory.

AISC 360 Chapter E defines the nominal compressive strength as: P_n = F_cr × A_g, where F_cr is the critical buckling stress and A_g is the gross cross-sectional area. For LRFD design, the design strength is φ_c × P_n with φ_c = 0.90. For ASD, the allowable strength is P_n / Ω_c with Ω_c = 1.67. The critical stress F_cr depends on whether the column buckles elastically or inelastically, which in turn depends on the slenderness ratio.

The column must also be checked for local buckling of the flanges and web. If the width-to-thickness ratios of the cross-section elements exceed the limits in Table B4.1a, the section is "slender" and the effective area must be reduced. For standard W-shapes in A992 steel (F_y = 50 ksi), most shapes commonly used as columns are non-slender, but this should be verified for every design, especially when using higher-strength steels where the slenderness limits are tighter.

Slenderness Ratio and Effective Length

The slenderness ratio KL/r is the single most important parameter in column design. K is the effective length factor accounting for end conditions, L is the unbraced length, and r is the radius of gyration of the cross-section about the buckling axis. A higher KL/r means a more slender column with lower buckling capacity. AISC 360 recommends that KL/r not exceed 200 for columns, though this is a practical guideline rather than a hard limit.

The effective length factor K depends on the rotational and translational restraint at each end of the column. For idealized conditions: K = 1.0 for pinned-pinned (both ends free to rotate but not translate), K = 0.65 for fixed-fixed (both ends fully restrained), K = 0.80 for fixed-pinned, and K = 2.0 for a cantilever (fixed base, free top). In real structures, the alignment charts in the AISC Commentary (or the direct analysis method of Chapter C) provide more realistic K values based on the relative stiffness of columns and beams framing into each joint.

For wide-flange columns, buckling typically occurs about the weak axis (y-y) because r_y is smaller than r_x. Therefore, KL/r_y is usually the controlling slenderness ratio. However, if the column is braced about the weak axis at intermediate points (for example, by girts in a braced frame), the unbraced length about the weak axis may be shorter than about the strong axis, and both axes must be checked. The controlling case is whichever axis gives the higher KL/r.

Elastic vs. Inelastic Buckling

Euler's elastic buckling stress is F_e = π²E / (KL/r)², where E is the modulus of elasticity (29,000 ksi for steel). For very slender columns (high KL/r), F_e is low and the column buckles elastically, well below the yield stress. For stocky columns (low KL/r), Euler's formula would predict a buckling stress above yield, which is physically impossible because the steel yields first. The transition between these regimes defines the two branches of the AISC column curve.

AISC 360 Section E3 defines the critical stress F_cr as follows. When KL/r ≤ 4.71√(E/F_y) (equivalently, F_e ≥ 0.44 F_y), the column is in the inelastic buckling range: F_cr = (0.658^(F_y/F_e)) × F_y. This is the Johnson parabola equation, which transitions smoothly from the squash load at KL/r = 0 down to the Euler curve at the transition point.

When KL/r > 4.71√(E/F_y) (F_e < 0.44 F_y), the column is in the elastic buckling range: F_cr = 0.877 × F_e. The 0.877 factor accounts for the effect of initial imperfections and residual stresses on elastic buckling. For A992 steel (F_y = 50 ksi), the transition occurs at KL/r = 113.4. Below this value, inelastic buckling governs. Above it, elastic buckling governs.

The Johnson Parabola

The inelastic buckling equation F_cr = 0.658^(F_y/F_e) × F_y is sometimes called the Johnson parabola (though its mathematical form is exponential, not parabolic). It represents the behavior of real steel columns that contain residual stresses from the rolling and cooling process. These residual stresses, typically 10-15 ksi in compression at the flange tips of hot-rolled W-shapes, cause early yielding at the most stressed fibers before the full cross-section reaches yield.

As the column load increases on a stocky member, the flange tips yield first (due to combined applied stress plus residual stress), reducing the effective stiffness of the cross-section. This progressive loss of stiffness causes the column to buckle at a load below the full squash load but above the Euler prediction. The 0.658^(F_y/F_e) expression captures this transition mathematically, fitting experimental column test data closely.

At KL/r = 0 (a hypothetical zero-length column), F_cr equals F_y (the squash load). As KL/r increases, F_cr drops below F_y, reaching about 0.66 F_y at KL/r around 60 for 50 ksi steel. At the transition point (KL/r = 113.4 for 50 ksi), the inelastic curve meets the elastic curve tangentially, ensuring a smooth transition. Beyond this point, the 0.877 × F_e elastic curve governs.

From a practical standpoint, most building columns operate in the KL/r range of 30 to 80, firmly in the inelastic buckling regime. Columns with KL/r above 120 are rare in building construction because they are very inefficient (F_cr drops to a small fraction of F_y, so most of the steel's strength is wasted). When a design pushes KL/r above 100, it is usually more economical to add intermediate bracing or choose a heavier section with a larger radius of gyration.

Section Selection Tips

The AISC Steel Construction Manual includes column load tables (Table 4-1a for W-shapes) that list the available axial strength (φP_n for LRFD, P_n/Ω for ASD) for each section at various effective lengths. These tables are the fastest way to select a column. Enter the table with your required strength and effective length, and pick the lightest section that meets or exceeds the demand. The tables are organized by nominal depth, so you can compare sections across different depth groups.

When selecting a column section, consider these practical factors beyond pure strength:

• Weak-axis r_y: A section with a larger r_y relative to its weight is more efficient as a column. W14 shapes are the traditional column sections because they have nearly square flange proportions, giving a favorable r_y/r_x ratio. W10 and W12 sections are also commonly used in lighter-load applications.
• Connection detailing: Wider flanges provide more room for beam-to-column connections. A W14x90 is easier to detail than a W10x100, even though the W10 may have slightly more axial capacity at the same effective length.
• Splices and continuity: In multi-story buildings, column sections are typically carried for 2-3 stories between splices. Select a section that works for the heaviest loaded story in the splice group, and check that it is adequate (if overdesigned) for the lighter stories above.
• Availability: Not all W-shapes listed in the manual are readily available. Check with your steel supplier for stock shapes. W14x columns from W14x43 through W14x730 are generally available, but some intermediate weights may have longer lead times.