Steel column flexural buckling: an AISC 360-22 §E3 worked example

Axial compression strength of a steel column is a buckling problem long before it is a yielding problem. AISC 360-22 Chapter E covers it. For the common case, a rolled W-shape whose flanges and web are not slender, the governing limit state is usually flexural buckling under §E3, and the whole check fits in five equations. The equations are short; the mistakes happen in the setup: the wrong axis, the wrong effective length, or the wrong side of the inelastic/elastic boundary. This article walks through the procedure and then a complete example.

Step 1: confirm the section has no slender elements

§E3 applies to members without slender elements. Element slenderness for axial compression is classified under §B4.1 using Table B4.1a. For a rolled W-shape the two cases are:

• Flanges (Table B4.1a, Case 1): λ = b/t = b_f/2t_f, limit λ_r = 0.56√(E/F_y).
• Web (Table B4.1a, Case 5): λ = h/t_w, limit λ_r = 1.49√(E/F_y).

If λ ≤ λ_r for every element, the section is nonslender for compression and §E3 (with §E4 where it applies) gives the strength. If any element exceeds λ_r, the member goes to §E7, which reduces the area through effective widths. The AISC Manual flags W-shapes that are slender for compression at F_y = 50 ksi with a footnote in Table 1-1.

Step 2: effective length and slenderness

Since the 2016 edition, and continuing in AISC 360-22, the Specification writes the effective length as L_c rather than KL:

K comes from the stability method in use. Under the direct analysis method of Chapter C, the effective length for flexural buckling is taken as the unbraced length (K = 1.0) unless a smaller value is justified by rational analysis. Under the effective length method of Appendix 7, K comes from the frame analysis or the alignment charts. A column pinned at both ends in a braced frame has K = 1.0 either way.

Compute L_c/r about both principal axes, each with its own unbraced length, and use the larger. For a W-shape braced equally about both axes, the weak axis (r_y) governs. A user note in §E2 recommends L_c/r not exceed 200.

Step 3: elastic buckling stress

This is the Euler stress for the governing axis. For doubly symmetric members, §E4 also defines an elastic torsional buckling stress. For rolled W-shapes with the torsional unbraced length equal to the weak-axis length, flexural buckling about the weak axis governs; the example below checks this.

Step 4: critical stress, inelastic or elastic

The two forms of the limit are the same boundary: substituting L_c/r = 4.71√(E/F_y) into Eq. E3-4 gives F_e = 0.445 F_y, so F_y/F_e = 2.25. Eq. E3-2 is the inelastic range, where residual stresses cause partial yielding before buckling. Eq. E3-3 is the elastic range; the 0.877 factor reduces the Euler stress for initial out-of-straightness. For A992 steel (F_y = 50 ksi), the boundary is L_c/r = 113.4.

Step 5: nominal and available strength

Worked example: W10x33, A992, L_c = 14 ft

A W10x33 column in a braced frame, pinned at both ends, 14 ft between supports, with no intermediate bracing about either axis. A992 steel: F_y = 50 ksi, E = 29,000 ksi. K = 1.0 about both axes, so L_c = 14 × 12 = 168 in.

Element slenderness (Table B4.1a)

Governing slenderness

Elastic buckling stress, Eq. E3-4

Which F_cr equation

Critical stress, Eq. E3-2

Nominal and available strength

Torsional buckling cross-check (§E4)

With the torsional unbraced length also 168 in and G = 11,200 ksi, the §E4 elastic torsional buckling stress for a doubly symmetric member is (π²EC_w/L_cz² + GJ) / (I_x + I_y) = 70.1 ksi. That is well above the 38.17 ksi flexural value, so weak-axis flexural buckling governs, as expected for a W-shape.

AISC Manual Part 4 (Table 4-1a) tabulates φ_cP_n and P_n/Ω_c for W-shapes against L_c about the y-axis, which is a quick independent check of a hand or software result.

Practical notes

• Unequal bracing. If the weak axis is braced at mid-height but the strong axis is not, L_cy = 7 ft and L_cx = 14 ft. Compare L_cx/r_x = 40.1 with L_cy/r_y = 84/1.94 = 43.3; the weak axis still governs here, but only just. The Manual's r_x/r_y ratio (2.16 for this shape) converts a strong-axis length into an equivalent weak-axis length for use with the Part 4 tables.
• K is not always 1.0. Columns in moment frames designed by the effective length method can have K well above 1.0. Under the direct analysis method, K = 1.0 is permitted because the analysis itself carries the second-order and imperfection effects.
• Combined loading. A column with moment as well as axial load also needs the Chapter F flexural strengths and the Chapter H interaction check. §E3 supplies only the P_c term.

How StructuralClaw handles it

The StructuralClaw steel column module checks W-shape columns for flexural buckling about both axes, with an effective-length case picker, and carries the result into the Chapter H interaction check. Its calculation trace lists each step with substituted values, so a reviewer can see which F_cr equation was used and why. The module currently cites AISC 360-16 Chapter E; the §E3 equations shown here are unchanged in AISC 360-22. It is part of the StructuralClaw desktop program, which is coming soon. For beams, the free steel beam calculator runs a W-shape flexure and shear check in the browser today.

• AISC 360-22, Specification for Structural Steel Buildings, §B4.1 and Table B4.1a (element slenderness), Chapter C and Appendix 7 (stability and effective length), Chapter E: §E1, §E2, §E3 (Eqs. E3-1 through E3-4), §E4, §E7.
• AISC Steel Construction Manual, Table 1-1 (W-shape dimensions and properties) and Part 4 (design of compression members).