Beam-Column Under Combined Axial and Bending Load
A 254 x 254 x 73 UC in S275 steel forms a 4.5 m tall column in a braced frame. It carries a factored axial load of 1100 kN and a factored major-axis moment of 95 kNm at the top, applied by a cantilevered crane bracket. The column is pinned at both ends in the plane of bending and restrained against out-of-plane buckling at mid-height. Section properties: A = 93.1 cm2, Wpl,y = 992 cm3, ry = 6.48 cm, and the buckling curve is curve b. The designer has checked only the cross-section resistance (NEd/Npl,Rd + My,Ed/Mc,Rd ≤ 1) and declared the member adequate.
1. Identify the flaw in the designer's check and describe the correct verification approach for this member.
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The designer has checked only cross-section resistance at the point of maximum moment. For a member in compression plus bending, cross-section resistance rarely governs; member buckling does.
Because the column is pinned in-plane with a moment applied at the top, the moment is not uniform along the length. The member must be verified for:
- In-plane buckling about the major axis, using the member buckling resistance Nb,y,Rd and the interaction with My,Ed. The equivalent uniform moment factor Cmy must reflect the actual moment diagram (a triangular diagram gives a lower Cmy, reducing the interaction penalty).
- Out-of-plane buckling about the minor axis, combining NEd with the major-axis moment through the lateral-torsional interaction term. Even though the moment is about the strong axis, it drives weak-axis buckling and must be included with the appropriate CmLT.
- Lateral-torsional buckling of the column segment, since the top moment induces compression in one flange over an unrestrained length.
The interaction equations (e.g. the two-equation approach in EN 1993-1-1 clause 6.3.3) must both be satisfied. Given ry = 6.48 cm and a 4.5 m height with mid-height restraint, the minor-axis slenderness is significant and the out-of-plane check is likely to be the governing one.
2. Estimate whether the member is likely to be adequate, showing the governing slenderness and the reasoning.
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Minor-axis slenderness: with mid-height restraint, the out-of-plane buckling length is 2.25 m. λ = Lcr/ry = 2250/64.8 ≈ 34.7. Non-dimensional slenderness λ̄ = (λ/93.9)·√(235/275) ≈ (34.7/93.9)·0.924 ≈ 0.34. For curve b at λ̄ = 0.34, the reduction factor χ is high, around 0.94.
Major-axis slenderness: in-plane length is 4.5 m, ry replaced by ry for major axis (roughly 11 cm for this section), so λ ≈ 4500/110 ≈ 41, λ̄ ≈ 0.40, χ ≈ 0.90.
Axial resistance: Npl,Rd ≈ 9310 mm2 × 275/1.0 ≈ 2560 kN. Buckling resistance ≈ 0.90 × 2560 ≈ 2300 kN, comfortably above 1100 kN.
Moment resistance: Wpl,y × fy = 992000 mm3 × 275 ≈ 273 kNm. Applied 95 kNm is about 35% of capacity.
Interaction: axial utilisation ≈ 0.48, moment utilisation ≈ 0.35. The interaction sum with Cmy less than 1.0 is likely below unity, so the member is probably adequate - but this must be confirmed with the formal interaction equations, not asserted. The key point is that adequacy depends on the buckling interaction, which the original check omitted entirely.