Real columns carry moments and real beams can carry axial force. This lesson shows how SANS 10162-1 combines the two, why the moment is amplified, and how to check a beam-column by hand and in the STATIX steel member calculator.
1. Learning outcomes
After this lesson you can:
explain the interaction of axial force and bending and the second-order amplification U1;
choose ω1 from the end moments and find Ce about each axis;
carry out the 13.8.2 checks (a), (b), (c) and the extra biaxial check, and say which governs;
tell braced from unbraced members and use 13.9 for tension with bending;
check a beam-column in the STATIX steel member calculator.
A beam-column carries axial force and bending at the same time: a column with beams framing in off-centre, a portal rafter, a truss chord with loads between nodes. Each action alone is easy; together they use up the same steel.The interaction idea. Add the share of the axial resistance used to the share of the bending resistance used; the total must not pass 1. SANS 10162-1 (13.8) refines this straight line with factors for the moment shape, the amplification and the section class.
4. Amplification
Second-order amplification (SANS 10162-1, 13.8.4). The axial load acting on the bowed member adds moment, which bows it more. U1 estimates the result: it grows as Cf approaches the Euler load Ce of the axis in question.The Euler load Ce of each axis, with that axis's effective length, sets its amplification. STATIX uses the same per-axis lengths as for Cr.ω1 (13.8.4) turns an uneven moment into an equivalent uniform one. Equal end moments bending the member one way are the worst case (1.0); a member bent into an S needs much less. The STATIX page takes ω1x and ω1y typed, 0.4 to 1.0, default 1.0 (safe).
Ce = π² E I / (KL)² for each axis
U1 = ω1 / (1 − Cf/Ce) (13.8.4)
ω1 = 0.6 − 0.4 κ ≥ 0.4 (κ = Msmall/Mlarge, positive for double curvature)
U1x is not taken less than 1.0
5. The checks (SANS 10162-1, 13.8.2 and 13.8.3)
SANS 10162-1, 13.8.2 for class 1 and 2 I-shaped sections: three versions of the same sum, each with a different Cr and Mrx, plus a check of the moments alone. The largest result governs. For all other sections (13.8.3) the 0.85 and β become 1.0.Which clause. The extra factors of 13.8.2 are earned by the plastic reserve of stocky I-sections. Class 3 sections, channels, tubes and angles use 13.8.3. STATIX picks the clause from the class.The weak axis. An H-section is much weaker and more flexible about its minor axis, so even a small My (a beam bearing on one flange, a wind load on the side) can use a large share of the resistance. β scales that term with the weak-axis slenderness.
6. Braced and unbraced; tension with bending
Braced or unbraced. In a braced frame the member's own bow is amplified by U1. In an unbraced frame U1 = 1.0 and the sway effects must already be in the moments, from a second-order analysis or amplified first-order moments; the STATIX page has no separate sway factor. If you do not choose, the page uses the larger of the two results.A member in tension with bending, such as a truss bottom chord with a load between nodes, uses the simple sum of 13.9. Tension straightens the member, so there is no amplification. The STATIX page reports pure bending (N = 0) under this same row.
7. In STATIX
The beam-column route in the STATIX steel member calculator: N (positive for compression), Mx, My; the effective length, LTB length and ω2; under More inputs ω1x, ω1y and the frame type. The calc sheet lists every check (a), (b), (c) and which one governs.
8. Example A: a braced UC column with end moments
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9. Example B: add a small weak-axis moment
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10. Do it in STATIX
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11. Common mistakes
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12. Exercises
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13. Self-check quiz
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14. Lesson 4 on one page
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Next: Lesson 5, bolted connectionsFrom members to joints: bolts in shear and bearing, bolt groups and fin plates. Go to lesson 5
A worked example for learning, not a design. Check every value against the current code and your own project, and have the design checked and signed by a registered professional engineer.
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