Combined axial force and bending

Most columns also bend. This lesson combines lessons 2 and 3: the cross-section under N and M (6.2.9), the member checks of 6.3.3 with the interaction factors of Annex B, the moment shape through C_m, and braced and sway frames, by hand and on the Eurocode 3 route of the STATIX steel member calculator.

1. Learning outcomes

After this lesson you can:

check an I-section under axial force and bending with the reduced plastic moment of 6.2.9;
explain why a member check is needed as well, and what the k factors do;
work out k_yy, k_yz, k_zy and k_zz by Annex B and check equations (6.61) and (6.62);
choose C_m from the moment shape, for braced and sway frames;
check a beam-column in the STATIX steel member calculator.

2. Before you start

Lesson 2 (χ_y, χ_z) and lesson 3 (χ_LT).
The free STATIX steel member calculator, set to Eurocode 3 (EN 1993-1-1).
A pocket calculator and about 55 minutes.
The same lesson on the SA code: Steel Design 1, lesson 4 (SANS 10162-1).

3. Where beam-columns are

Where beam-columns are. Most real columns carry a moment as well as the axial force: a beam bearing on a bracket or a cap plate off the column centre, the leg of a portal frame, a chord of a truss with a load between its nodes. EN 1993-1-1 checks them twice: the cross-section (6.2) and the member against buckling (6.3.3).

4. The cross-section (6.2.9)

The cross-section under N and M (6.2.9). A class 1 or 2 I-section loses little moment resistance under a small axial force, because the web near the neutral axis takes the force. The reduced plastic moment M_N,Rd follows the curve; the straight line of 6.2.1(7) is always safe but conservative.

The 6.2.9.1 rules for rolled I and H sections of class 1 and 2. Class 3 sections use the elastic stresses (6.2.9.2) or the straight line. At a member end with a large moment, or in a short member, this cross-section check can govern.

5. The member (6.3.3)

Why the member check is more than the section check. A compressed member that bends also bows, and the axial force acting on the bow adds moment. The nearer N_Ed is to the buckling resistance, the larger the extra moment. The k factors of 6.3.3 carry this effect.

The two member checks of 6.3.3(4). Each adds the axial force over its buckling resistance about one axis to the moments over their resistances, the major-axis moment reduced for lateral-torsional buckling. Both must hold.

n_y = N_Ed / (χ_y N_Rk/γ_M1), n_z = N_Ed / (χ_z N_Rk/γ_M1) class 1, 2: k_yy = C_my [1 + (λ̅_y − 0.2) n_y] ≤ C_my (1 + 0.8 n_y) k_zz = C_mz [1 + (2λ̅_z − 0.6) n_z] ≤ C_mz (1 + 1.4 n_z) (I-sections) k_yz = 0.6 k_zz; k_zy = 0.6 k_yy (Table B.1) or Table B.2 when free to twist

The interaction factors k_yy, k_yz, k_zy and k_zz (Annex B, the method 2 of EN 1993-1-1). They grow with the axial force and the slenderness, shrink with a favourable moment shape, and take the plastic or elastic form by class. A member free to twist (not restrained against lateral-torsional buckling) uses Table B.2 for k_zy, otherwise Table B.1.

The equivalent uniform moment factor C_m (Table B.3). A uniform moment (single curvature, ψ = 1) is the worst; a moment at one end only, or double curvature, is much kinder. The calculator takes ψ for each axis and for the LTB segment, or a typed C_m; blank means 1.0. Table B.3 also has rows for loads between the ends: the page takes a uniform load in the segment (C_m 0.95 with no end moments); with end moments as well as span load, type C_m.

6. Braced and sway frames

Braced and sway frames. In a braced frame the column ends do not move sideways and C_m follows the moment shape. In a sway frame Table B.3 takes C_m = 0.9 for the sway mode, and the sway effects themselves must be in the moments (second-order analysis or amplified moments, 5.2). On the calculator, Frame “unbraced” sets C_my = C_mz = 0.9 unless you type C_m, with a note; you still enter the sway buckling lengths L_cr and the second-order moments yourself.

7. Tension with bending

Tension with bending. A tie that also bends does not buckle as a whole, so the cross-section check governs. The straight-line sum of 6.2.1(7) is always safe; the calculator uses the reduced plastic moments of 6.2.9 with the tension force, and for an I-section in tension that can still buckle laterally under the moment it also checks M_b,Rd.

What the steel member calculator’s Eurocode 3 route uses: 6.3.3 with the Annex B factors (method 2; there is no Annex A); Table B.2 for an open section with an LTB length (free to twist), Table B.1 when the flange is fully restrained; C_m from ψ by Table B.3 for end moments, from the load in the segment, typed, or 1.0 when blank; Frame “unbraced” sets C_my = C_mz = 0.9 unless C_m is typed; k_zz in the I-section form for I and H sections and the hollow-section form for tubes; γ_M1 = 1.0. With compression it reports the 6.2.9 cross-section check (the “Bending + axial” row with M_N,y,Rd and M_N,z,Rd), (6.61) and (6.62), and also the torsional buckling of the column.

8. In STATIX

The order of work: the cross-section first, then the three reduction factors from lessons 2 and 3, then the k factors, then the two member equations.

The beam-column route in the STATIX steel member calculator: Eurocode 3, the section, N (positive for compression) and the moments; the buckling and LTB lengths; the end-moment ratios ψ or typed C_m under EC3 only; and the frame type. The calc sheet shows every k factor and both equations.