-- | ----- | ----- | | d | 12.1 | in. | | bf | 12.0 | in. | | tf | 0.605 | in. | | tw | 0.390 | in. | | A | 19.1 | in.^2 | | Ix | 533 | in.^4 | | Iy | 174 | in.^4 | | rx | 5.28 | in. | | ry | 3.02 | in. | | J | 1.97 | in.^4 | | Cw | 2,180 | in.^6 |

Loads (factored, from analysis):

• Dead load tributary at level 5: P_D = 245 kips
• Live load tributary: P_L = 130 kips (reducible per ASCE 7-22 Section 4.7 — applies to column since tributary area > 400 ft^2)
• Roof live load (reduced): P_Lr = 28 kips

Factored axial load (LRFD, ASCE 7-22 Section 2.3.1, Load Combination 2 — 1.2D + 1.6L + 0.5Lr):

P_u = 1.2 x 245 + 1.6 x 130 + 0.5 x 28 = 294 + 208 + 14 = 516 kips

Step 1: Effective Length (Alignment Chart Method)

The column is in a braced frame (sidesway inhibited). Use AISC 360-22 Commentary Table C-A-7.1 alignment chart or the recommended K = 1.0 for braced frames unless a refined analysis is performed.

For the strong axis (x-x), beams frame into the column web at every floor. The beam-to-column stiffness ratio G is computed at each end:

G_top = sum(Ic/Lc) / sum(Ig/Lg) = (533/150) / (1,350/360 + 1,350/360) = 3.55 / 7.50 = 0.474
G_bot = (533/150) / (1,350/360 + 1,350/360) = 0.474

From the AISC alignment chart for braced frames (sidesway inhibited):

K_x ≈ 0.78 (read from alignment chart at G_top = G_bot = 0.474)

For the weak axis (y-y), beams frame into the column flanges with similar stiffness:

G_top_y = (174/150) / (29.1/96 + 29.1/96) = 1.16 / 0.606 = 1.91
G_bot_y = 1.91
K_y ≈ 0.88 (read from alignment chart)

Alternatively, for beam-column design in braced frames, the AISC Specification Commentary permits using K = 1.0 as a conservative default unless a more refined buckling analysis is performed. Using K = 1.0 for simplicity:

Effective length x-axis: L_cx = K_x x L = 1.0 x 12.5 ft = 12.5 ft (conservative)
Effective length y-axis: L_cy = K_y x L = 1.0 x 12.5 ft = 12.5 ft

Step 2: Slenderness Parameters

KL/r_x = 1.0 x 12.5 x 12 / 5.28 = 150 / 5.28 = 28.4 (strong axis)
KL/r_y = 1.0 x 12.5 x 12 / 3.02 = 150 / 3.02 = 49.7 (weak axis — governs)

Limiting slenderness: KL/r <= 200 (AISC E2, recommended). KL/r = 49.7 < 200. OK.

Step 3: Elastic Buckling Stress (AISC 360-22 Chapter E3)

For flexural buckling about the weak axis (y-y), which governs because KL/r_y > KL/r_x:

F_e = pi^2 x E / (KL/r)^2 = pi^2 x 29,000 / (49.7)^2 = 286,480 / 2,470 = 116.0 ksi

Step 4: Critical Stress (AISC E3-2 or E3-3)

Check the slenderness limit for inelastic vs. elastic buckling:

Fy / Fe = 50 / 116.0 = 0.431
Limit = 2.25 (AISC E3-2 vs E3-3 transition)
Fy / Fe = 0.431 < 2.25 → Inelastic buckling (AISC E3-2)

Critical buckling stress:

F_cr = 0.658^(Fy/Fe) x Fy = 0.658^(0.431) x 50 = 0.837 x 50 = 41.8 ksi

Design compressive strength:

phi_c x P_n = 0.90 x F_cr x A_g = 0.90 x 41.8 x 19.1 = 0.90 x 799 = 719 kips

Check: P_u = 516 kips < phi_c P_n = 719 kips. Axial compression OK. Utilization = 516/719 = 0.718.

Step 5: Torsional Buckling Check (AISC E4)

For doubly symmetric W-shapes, torsional buckling is not a governing limit state (the torsional buckling stress always exceeds the minor-axis flexural buckling stress for W-shapes). However, the check is performed for completeness:

F_e_torsion = (pi^2 x E x Cw / (K_z x L)^2 + G x J) / (Ix + Iy)
            = (pi^2 x 29,000 x 2,180 / (1.0 x 150)^2 + 11,200 x 1.97) / (533 + 174)
            = (6.23e8 / 22,500 + 22,064) / 707
            = (27,690 + 22,064) / 707 = 49,754 / 707 = 70.4 ksi

Since F_e_torsion = 70.4 ksi > F_e_flexural_y = 116.0 ksi (wait — actually F_e_torsion < F_e_flexural? Let me recheck: F_e_torsion = 70.4 ksi means torsional buckling would occur at a lower stress than flexural buckling about the weak axis at 116.0 ksi of elastic stress. However, the weak-axis critical stress F_cr = 41.8 ksi still governs because both are in the inelastic range with different Fy/Fe ratios.)

For torsional buckling: Fy/Fe_torsional = 50/70.4 = 0.710 < 2.25, so F_cr_torsional = 0.658^0.71 x 50 = 0.744 x 50 = 37.2 ksi. This is lower than 41.8 ksi, but AISC E4 specifies that for doubly symmetric members, F_cr is the minimum of flexural and torsional buckling. Since 37.2 < 41.8, torsional buckling actually governs for this W12x65 (though in practice, the additional restraint from beam connections and floor diaphragm is relied upon to suppress torsional buckling).

Actual design capacity considering torsional buckling:

phi_c P_n = 0.90 x 37.2 x 19.1 = 0.90 x 710 = 639 kips

P_u = 516 kips < 639 kips. Still OK (utilization 0.808).

This illustrates an important point: for short, stocky columns (low KL/r), torsional buckling can sometimes control. The W12x65 has a relatively short unbraced length, large flange width, moderate web depth, and the torsional buckling check should not be neglected.

Step 6: Alternative — Direct Analysis Method

The Direct Analysis Method (DAM, AISC 360-22 Chapter C) eliminates the need for effective length factors by directly modeling geometric imperfections and stiffness reductions. In the DAM:

• Notional loads of 0.002 x Y_i are applied at each level (where Y_i is the gravity load at that level)
• A flexural stiffness reduction of 0.8 (tau_b = 0.8) is applied when alpha Pr/Pns > 0.5
• K = 1.0 is used for all members
• Second-order analysis is required (P-Delta and P-delta effects)

For this column with Pr/Pns = 516/719 = 0.718 > 0.5, the stiffness reduction applies. The DAM would reduce EI to 0.8 x EI in the analysis model, which may increase the second-order moments.

Step 7: Summary

The W12x65 column satisfies all AISC 360-22 requirements. The dominant limit state is torsional buckling (utilization 0.808). For a more optimized design, consider W12x58 (A=17.0 in.^2, ry=2.51 in.) or W10x60 depending on connection geometry requirements, but the W12x65 is adequate and commonly available.

Column Design Principles per AISC 360-22

• Effective length K (Chapter C and Commentary): Theoretically between 0.5 and 1.0 for braced frames, and 1.0 to infinity for sway frames. DAM uses K=1.0 always. For preliminary design, K=1.0 is safe and the DAM removes the judgment required for alignment charts.
• Residual stress: Hot-rolled W-shapes have a residual stress pattern that causes inelastic buckling to govern for intermediate slenderness (KL/r between ~20 and ~120 for 50 ksi steel). The 0.658^N factor in AISC E3-2 accounts for this.
• Built-up vs. rolled sections: Built-up columns (welded plates) have higher residual stresses, reducing F_cr. AISC E5 provides separate provisions.
• Slender elements (Chapter E7): If any plate element (flange or web) is slender, the effective area must be reduced using Q factors (Q_s for stiffened elements — webs; Q_a for unstiffened elements — flanges).

Frequently Asked Questions

Why is the W12x65 a popular column section?

The W12x65 has nearly equal flange width and depth (12.0 in. flange, 12.1 in. depth), making it a "balanced" column section with rx approximately equal to ry (5.28 vs. 3.02 — not equal, but the ratio of 1.75 is better than deeper W-shapes which can be 3-4x). The equal (or near-equal) radii of gyration mean the column capacity is similar in both directions, simplifying design and reducing the penalty for the weak axis. W12 and W14 series are the most common column shapes.

What is the difference between AISC Chapter E3 and E4?

E3 covers flexural buckling (member bends without twisting). E4 covers torsional and flexural-torsional buckling (member twists, with or without bending). For doubly symmetric shapes like W12, pure torsional buckling is possible (E4-b). For singly symmetric shapes (Tee, channel), flexural-torsional buckling couples bending and twisting. The F_cr used for design is the minimum of all applicable buckling modes.

How does the Direct Analysis Method change column design?

DAM removes K factors entirely (use K=1.0 for all columns). Instead, it captures stability effects through: (1) notional loads applied directly in the analysis model, (2) reduced stiffness (0.8 x EI for members with alpha Pr/Pns > 0.5), (3) full second-order analysis (P-Delta + P-delta). The result is generally more accurate and eliminates the notoriously difficult K-factor determinations for complex frames. Most modern structural analysis software implements DAM by default.

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Related References

• Column Buckling Equations
• Column Curve Reference
• Steel Allowable Stress Design
• AISC Steel Construction Tables
• K Factor Reference
• How to Verify Calculations

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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.