How to Size a Steel Column: AISC 360-22 Step-by-Step Guide
Selecting the right steel column section requires understanding effective length, slenderness, and the critical buckling stress Fcr. This guide walks through the AISC 360-22 Chapter E compression member design procedure — from boundary conditions to section selection — with a complete worked example.
1. Column Failure Modes
A steel column can fail in three distinct modes:
| Mode | Governing Parameter | Design Section |
|---|---|---|
| Flexural buckling | KL/r — slenderness about the weak axis | AISC 360-22 §E3 |
| Torsional buckling | Cross-section geometry (cruciform, built-ups) | AISC 360-22 §E4 |
| Flexural-torsional buckling | Singly symmetric or asymmetric sections (angles, tees) | AISC 360-22 §E4 |
For standard W-shapes (doubly symmetric I-sections), flexural buckling about the weak axis (y-y) almost always governs. Torsional and flexural-torsional buckling need checking only for non-standard cross-sections.
2. Step-by-Step Design Procedure (AISC 360-22 Chapter E)
The AISC 360-22 Chapter E compression member design follows a logical sequence. Each step feeds directly into the next:
| Step | Action | Code Reference |
|---|---|---|
| 1 | Determine the factored axial demand Pu from load combinations (ASCE 7-22 §2.3) | ASCE 7-22 Eq. 2.3-1 to 2.3-7 |
| 2 | Establish boundary conditions and compute effective length KL for each axis | AISC 360-22 Table C-E1.1 |
| 3 | Select a trial section — W14 shapes are most common for columns | AISC Steel Construction Manual |
| 4 | Compute governing slenderness ratio: KL/r = max(KLx/rx, KLy/ry) | AISC 360-22 §E3 |
| 5 | Check slenderness limit: KL/r ≤ 200 (recommended maximum) | AISC 360-22 §E2 |
| 6 | Compute elastic buckling stress Fe = π²E / (KL/r)² | AISC 360-22 Eq. E3-4 |
| 7 | Determine Fcr: inelastic (KL/r ≤ 4.71√(E/Fy)) or elastic zone | AISC 360-22 Eq. E3-2 or E3-3 |
| 8 | Compute design strength: φcPn = 0.90 × Fcr × Ag | AISC 360-22 §E3 |
| 9 | Check compactness: verify flange and web width-to-thickness ratios meet §E7 limits | AISC 360-22 Table E7.1 |
| 10 | DCR = Pu / φcPn ≤ 1.0. If > 1.0, select heavier section and repeat from Step 4. | — |
Key Formulas
The inelastic/elastic boundary occurs at KL/r = 4.71√(E/Fy). For Fy=345 MPa (A992) and E=200,000 MPa: boundary KL/r = 4.71√(200,000/345) = 113.4. Columns with KL/r above 113 are in the elastic Euler range.
3. Effective Length K Factors
The effective length factor K accounts for rotational and translational end restraint. AISC Commentary Table C-E1.1 provides theoretical and recommended design values:
| Boundary Condition | Theoretical K | Recommended K | Buckled Shape |
|---|---|---|---|
| Both ends pinned (braced frame, no rotation restraint) | 1.0 | 1.0 | Half sine wave |
| Fixed base, pinned top (one end pinned) | 0.70 | 0.80 | Quarter sine wave |
| Fixed base, fixed top (both ends rotationally fixed) | 0.50 | 0.65 | Double curvature |
| Fixed base, free top (cantilever column) | 2.00 | 2.10 | Quarter sine wave |
| Pinned base, fixed top (sway permitted) | 2.00 | 2.00 | Sway mode |
| Fixed base, fixed top, sway permitted | 1.00 | 1.20 | S-shape with sway |
Braced vs. Unbraced Frames
Braced frames (sidesway inhibited): Lateral bracing prevents sidesway — shear walls, braced bays, or rigid cores carry horizontal loads. Columns in braced frames use K ≤ 1.0. The governing axis is almost always the weak axis (y-y) where bracing is typically at floor levels only.
Moment frames (sidesway permitted): Columns must resist lateral loads through bending. K values exceed 1.0, and effective lengths can be significantly longer than the physical story height. For moment frames, K must be computed from the G-factor alignment charts (AISC Commentary §C-E1) based on the ratio of column stiffness to beam stiffness at each joint.
4. Critical Buckling Stress Fcr
Fcr vs KL/r for Fy=345 MPa (A992 steel)
| KL/r | Fe (MPa) | Zone | Fcr (MPa) | Fcr/Fy |
|---|---|---|---|---|
| 20 | 4,934 | Inelastic | 340 | 0.985 |
| 40 | 1,234 | Inelastic | 326 | 0.945 |
| 60 | 548 | Inelastic | 299 | 0.867 |
| 80 | 308 | Inelastic | 261 | 0.757 |
| 100 | 197 | Inelastic | 214 | 0.620 |
| 113 | 154 | Boundary | 183 | 0.530 |
| 140 | 100 | Elastic | 88 | 0.255 |
| 200 | 49 | Elastic | 43 | 0.125 |
At KL/r=100, the column carries only 62% of its yield load. At KL/r=200, only 12.5%. This is why very slender columns are avoided — you pay for steel you cannot use.
5. Local Buckling & Width-to-Thickness Limits (AISC 360-22 §E7)
A column's cross-sectional elements (flanges, web) must be compact enough not to buckle locally before the overall column buckles. AISC 360-22 §E7 defines slenderness limits for compression members:
| Element | λ (slenderness) | λr (limit for non-slender) | Comment |
|---|---|---|---|
| W-shape flange (uniform compression) | bf / 2tf | 0.56√(E/Fy) | = 13.5 for Fy=345 MPa |
| W-shape web (uniform compression) | h / tw | 1.49√(E/Fy) | = 35.9 for Fy=345 MPa |
| HSS rectangular wall | b / t | 1.40√(E/Fy) | = 33.7 for Fy=345 MPa |
| HSS circular (round) | D / t | 0.15E/Fy | = 87.0 for Fy=345 MPa |
Most standard AISC W-shapes have non-slender elements for Fy≤345 MPa. However, high-strength steel (Fy≥415 MPa) or built-up sections can have slender elements. When λ > λr, a reduced effective area Q must be computed (AISC §E7.2, §E7.3) and applied as:
Practical Check for W-Shapes
The AISC Steel Construction Manual tabulates λf and λw for every W-shape. The φcPn column load tables already account for local buckling — when using these tables you do not need a separate §E7 check. For custom built-up sections, the §E7 check is essential.
A quick rule: all standard W14 column sections from W14×48 through W14×730 are non-slender for Fy≤345 MPa. For high-strength steel (A913 Grade 450 MPa), check the W14 flange λf values individually — most heavier W14 shapes (≥W14×61) remain non-slender at 450 MPa, but lighter shapes may not.
6. Section Selection Strategy
Target DCR (Demand/Capacity Ratio): For economy, aim for DCR=0.80–0.95. A DCR below 0.60 suggests the section is oversized. Above 1.0 means it fails.
Why W14 Sections Dominate Column Design
The W14 family (nominal 14-inch depth) was developed specifically for column applications. Key reasons:
- Near-square flanges: Wide flanges relative to depth → large ry → lower KL/ry → more efficient use of steel area
- Wide weight range: W14×48 through W14×730 — covers 10× variation in axial capacity within one nominal depth, allowing consistent column dimensions from floor to floor
- Splice compatibility: Columns spliced every 2–3 floors need similar external geometry to simplify connection plates
Quick Selection Table — W14, Fy=345 MPa, KL=4.0 m
| Section | Ag (cm²) | ry (mm) | KL/ry | Fcr (MPa) | φcPn (kN) |
|---|---|---|---|---|---|
| W14×48 | 90.3 | 58.4 | 68.5 | 290 | 2,351 |
| W14×61 | 116 | 59.7 | 67.0 | 292 | 3,050 |
| W14×82 | 155 | 60.5 | 66.1 | 293 | 4,093 |
| W14×109 | 207 | 66.5 | 60.2 | 302 | 5,632 |
| W14×132 | 250 | 66.5 | 60.2 | 302 | 6,795 |
7. Worked Example — Interior Column, 3-Storey Building
Given: Interior column of a braced 3-storey office building. Factored axial load Pu=2,200 kN. Story height=4.5 m. Both ends pinned in the weak direction (K=1.0), fixed at base and pinned at top in strong direction (K=0.80). Fy=345 MPa.
Effective lengths: KLx=0.80×4.5=3.6 m | KLy=1.0×4.5=4.5 m
Trial section: W14×82 — Ag=155 cm², rx=152 mm, ry=60.5 mm
Slenderness:
KLx/rx=3,600/152=23.7 | KLy/ry=4,500/60.5=74.4 governs
Fe: π²×200,000/(74.4²) = 1,973,920/5,535 = 356.6 MPa
Fcr: KL/r=74.4 ≤ 113 → inelastic zone
Fcr = (0.658^(345/356.6)) × 345 = (0.658^0.968) × 345 = 0.665 × 345 = 229.4 MPa
φcPn: 0.90 × 229.4 × 15,500 = 0.90 × 229.4 × 15,500/1,000 = 3,200 kN
Check: φcPn=3,200 kN > Pu=2,200 kN ✓ | DCR=2,200/3,200=0.69
Try W14×61 (lighter): Ag=116 cm², ry=59.7 mm → KLy/ry=4,500/59.7=75.4 → Fcr≈228 MPa → φcPn=0.90×228×11,600/1,000=2,380 kN
DCR=2,200/2,380=0.924 ✓ — W14×61 is more economical.