Shear force

What this calculator checks

The calculator determines the plastic shear resistance of the section, \(V_{pl,Rd}\), according to EN 1993-1-1 §6.2.6, expression (6.18):

\(V_{pl,Rd} = \frac{A_v \cdot f_y / \sqrt{3}}{\gamma_{M0}}\)

and checks that \(V_{Ed} / V_{c,Rd} \le 1.0\).

The \(\sqrt{3}\) comes from the von Mises yield criterion: the pure shear yield limit of steel is \(f_y/\sqrt{3} \approx 0.577 \, f_y\), not \(f_y\).

The shear area \(A_v\)

The governing quantity is not the total area but the shear area — the part of the section that actually carries shear. In an I-profile loaded vertically that is essentially the web; the flanges contribute very little.

It is computed from EN 1993-1-1 Table 6.2, by profile family:

I and H, with \(\eta = 1.0\):

\(A_v = \max\left(A - 2 b t_f + (t_w + 2r)\,t_f \; ; \; \eta \, h_w t_w\right)\)

• Channels (UPN): \(A_v = A - 2 b t_f + (t_w + r)\,t_f\)
• RHS: \(A_v = A \cdot h / (b + h)\) — for shear parallel to the depth
• CHS: \(A_v = 2A / \pi\)

Shear buckling of the web

For I and H profiles the calculator additionally checks the condition of §6.2.6(6):

\(\frac{h_w}{t_w} \le \frac{72 \, \varepsilon}{\eta}, \qquad \varepsilon = \sqrt{235 / f_y}\)

If the web is too slender it buckles in shear before it yields, and expression (6.18) is no longer on the safe side. The check must then follow EN 1993-1-5 §5, accounting for the web contribution and possibly transverse stiffeners — not the formula used here.

Standard rolled profiles satisfy the condition with comfortable margin. It becomes relevant for welded plate girders, where the web is deliberately kept thin to save steel.

Input data

• \(V_{Ed}\) — the design shear force, in kN, at Ultimate Limit State.
• Profile and section — the dimensions (\(h\), \(b\), \(t_w\), \(t_f\), \(r\)) and area are taken from the catalogue.
• Steel grade — \(f_y\) per Table 3.1, as a function of the governing thickness.
• \(\gamma_{M0}\) — the recommended value is 1.0, but the National Annex takes precedence.

Assumptions and limitations

• Shear parallel to the web (along the z axis). Shear in the other direction is not covered.
• Plastic resistance: it assumes the section can yield in shear, which is exactly what the slenderness condition above verifies.
• Interaction with bending is not treated — when \(V_{Ed} > 0.5 \, V_{pl,Rd}\) the moment resistance is reduced per §6.2.8; see bending and shear.
• Local shear from concentrated loads (§6.2.6(2)) and torsion are not treated.

Related calculations

• Bending and shear — \(M\)–\(V\) interaction
• Bending moment — \(M_{c,Rd}\) by cross-section class
• Local plate buckling — slender plates under direct stress