Bending and shear

What this calculator checks

The calculator verifies a steel section subjected simultaneously to bending moment and shear force, according to EN 1993-1-1 §6.2.8. It does two things: it checks the shear (§6.2.6) and, if the shear is large, reduces the moment resistance to account for the interaction.

When shear reduces the moment

The central rule of the interaction is a threshold:

• if \(V_{Ed} \le 0.5 \, V_{c,Rd}\) — the shear is small, its effect on the moment is neglected, and \(M_{c,Rd}\) stays as in pure bending
• if \(V_{Ed} > 0.5 \, V_{c,Rd}\) — the shear is large, and the moment resistance is reduced

The physical reason: shear force is carried mainly by the web, where it produces shear stresses. When these become significant, the web has less reserve for the normal stresses of bending. Below half the shear capacity, the reserve is sufficient and the interaction can be ignored — hence the 0.5 threshold.

The moment reduction

Above the threshold, a reduction factor is applied to the web area:

\(\rho = \left(\frac{2 V_{Ed}}{V_{c,Rd}} - 1\right)^2\)

For I, H and channel profiles, the reduced moment lowers the shear-affected part of the web; for hollow sections the simplified form \(M_{y,V,Rd} = M_{c,Rd}(1-\rho)\) of §6.2.8(2) applies. The moment to check is the lesser of the pure-bending capacity and the reduced one.

Practical note: at \(V_{Ed} = V_{c,Rd}\), \(\rho = 1\) and the whole web contribution to the moment is lost — which is why a section loaded near its shear capacity has very little moment reserve.

Input data

• \(V_{Ed}\), \(M_{Ed}\) — the design shear force (kN) and moment (kNm), on the same section.
• Profile and section — the shear area \(A_v\) and the modulus \(W\) are taken from the catalogue.
• Cross-section class (1–3) — decides whether \(M_{c,Rd}\) uses \(W_{pl}\) or \(W_{el}\).
• Steel grade and \(\gamma_{M0}\) (recommended 1.0; the National Annex takes precedence).

Assumptions and limitations

• The moment check is performed only if the shear passes; otherwise the section is inadequate in shear, regardless of the moment.
• Axial force is not included — for M+N see bending and axial force.
• Shear buckling of the web (slender webs, EN 1993-1-5) is not treated — see shear force.
• Loss of stability (lateral-torsional buckling) is not included — see lateral-torsional buckling.

Related calculations

• Bending moment — the pure-bending capacity \(M_{c,Rd}\)
• Shear force — \(V_{pl,Rd}\) and the shear area
• Bending and axial force — the \(M\)–\(N\) interaction
• Cross-section class — it decides \(M_{c,Rd}\)