Bending with axial force

What this calculator checks

The calculator verifies a steel section subjected simultaneously to bending moment and axial force (M+N), according to EN 1993-1-1 §6.2.9. It is the cross-section check for a compressed (or tensioned) and bent member, at the resistance level — without including loss of stability.

The M–N interaction depends on the class

How moment and axial force combine depends on the cross-section class, because the resistance method differs:

Classes 1 and 2 — plastic interaction (§6.2.9.1). The section can fully yield, so a reduced plastic moment \(M_{N,Rd}\) is used, smaller than \(M_{pl,Rd}\) the larger the axial force. For I and H profiles, the biaxial criterion is:

\(\left(\frac{M_{y,Ed}}{M_{N,y,Rd}}\right)^\alpha + \left(\frac{M_{z,Ed}}{M_{N,z,Rd}}\right)^\beta \le 1\)

with \(\alpha = 2\) and \(\beta = \max(1; 5n)\), where \(n = N_{Ed}/N_{pl,Rd}\). The reduction is not linear: at small axial forces (typically \(n < 0.25\)), the moment capacity is not reduced at all, because the axial force is carried by the central web zone, which contributes little to the moment anyway.

Class 3 — elastic criterion (§6.2.9.2). The stresses are summed: the first fibre to yield sets the limit.

\(\frac{N_{Ed}}{N_{pl,Rd}} + \frac{M_{y,Ed}}{M_{el,y,Rd}} + \frac{M_{z,Ed}}{M_{el,z,Rd}} \le 1\)

Class 4 — elastic criterion with effective properties (§6.2.9.3), accounting for local buckling through effective areas and moduli.

Formulas by profile family

For hollow sections, the plastic interaction has its own forms: \(M_{N,Rd} = M_{pl,Rd}(1-n^2)\) for RHS, \(M_{pl,Rd}(1-n^{1.7})\) for CHS (when \(n > 0.25\)). Channel profiles are treated as I/H.

Input data

• \(N_{Ed}\), \(M_{y,Ed}\), \(M_{z,Ed}\) — the axial force (kN) and the moments about the two axes (kNm).
• Profile and section — the areas and plastic/elastic moduli from the catalogue.
• Cross-section class (1–4) — decides the criterion applied.
• Steel grade and \(\gamma_{M0}\) (recommended 1.0; the National Annex takes precedence).

Assumptions and limitations

• A cross-section level check: it does not include loss of stability. For the real member, with buckling, see bending and compression.
• Channel profiles are treated as I/H (a single web).
• It does not treat interaction with large shear force — see bending and shear.

Related calculations

• Bending and compression — the same action, but with instability (real column)
• Bending moment — the case with \(N = 0\)
• Compression — the case with \(M = 0\)
• Cross-section class — decides the interaction criterion