Eurocode 3 Combined Shear & Bending Check (EN 1993-1-1 §6.2.8)

Steel members are often subjected to both shear forces and bending moments simultaneously. While checked separately, high shear forces can reduce the member's capacity to resist bending. This guide explains the practical application of the shear and bending interaction rules according to Eurocode 3 (EN 1993-1-1 §6.2.8).

When does shear affect bending resistance? The 50% rule

Eurocode 3 provides a simple threshold to determine if the shear force (\(V_{Ed}\)) is high enough to warrant considering its effect on the bending moment resistance (\(M_{c,Rd}\)).

According to EN 1993-1-1 §6.2.8(2), if the design shear force is less than or equal to 50% of the design plastic shear resistance (see our Shear guide) of the cross-section:

\[ V_{Ed} \le 0.5 \times V_{pl,Rd} \]

Then, the effect of the shear force on the bending moment resistance may be neglected. In this common scenario, the design bending resistance \(M_{c,Rd}\) is calculated purely based on the rules in §6.2.5 (considering section class, using \(W_{pl}\), \(W_{el,min}\), or \(W_{eff,min}\), as explained in our Bending moment check guide).

Important note: This allowance assumes shear buckling does not govern. If shear buckling resistance according to EN 1993-1-5 is less than \(V_{pl,Rd}\), then interaction rules from EN 1993-1-5 must be applied, regardless of the 50% threshold.

Calculating reduced moment resistance under high shear (\(V_{Ed} > 0.5 V_{pl,Rd}\))

When the shear force exceeds the 50% threshold, the bending moment resistance must be reduced. The code conceptually justifies this by considering that the shear area (mainly the web) has its yield strength reduced by the high shear stress. However, calculating this literally is complex. Instead, practical methods based on a reduction factor \(\rho\) are used.

Step 1: Calculate the reduction factor (\(\rho\))

This factor relates the applied shear force to the plastic shear capacity:

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

Note that \(\rho\) ranges from 0 (when \(V_{Ed} = 0.5 V_{pl,Rd}\)) to 1 (when \(V_{Ed} = V_{pl,Rd}\)).

Step 2: Calculate the reduced moment resistance (\(M_{V,Rd}\))

The method depends on the cross-section type:

Method for I and H sections (equal flanges, major axis bending)

For this common case, EN 1993-1-1 §6.2.8(5) provides a specific formula that directly incorporates the reduction factor \(\rho\) and the web geometry:

\[ M_{y,V,Rd} = \frac{\left( W_{pl,y} - \frac{\rho A_w^2}{4 t_w} \right) f_y}{\gamma_{M0}} \]

But the result must not exceed the original bending resistance without shear interaction: \( M_{y,V,Rd} \le M_{y,c,Rd} \).

  • \(M_{y,c,Rd}\): Bending resistance from §6.2.5.
  • \(W_{pl,y}\): Plastic section modulus (major axis).
  • \(A_w\): Shear area of the web (\(\approx h_w t_w \)).
  • \(t_w\): Web thickness.
  • \(\rho\): Reduction factor calculated above.
  • \(f_y\), \(\gamma_{M0}\): Yield strength and partial safety factor.

Method for other cross-sections (conservative approach)

For sections or loading cases not covered by the specific formula above (e.g., RHS, CHS, minor axis bending of I-sections), the general principle of reducing the stress with \(\rho\) applies. Practical interpretation suggests calculating the reduced moment resistance by applying a factor based on \(\rho\) to the original bending resistance:

\[ M_{V,Rd} = (1 - \rho) M_{c,Rd} \]

Where \(M_{c,Rd}\) is the design bending resistance calculated according to §6.2.5 without considering the shear effect. This approach is generally considered practical and conservative. The precice method is to only apply the yield strength reduction to the shear affected areas.

Important considerations and further checks

  • Scope: This check (§6.2.8) applies to cross-section resistance. Overall member stability (like Lateral-Torsional Buckling, §6.3) must still be checked and may govern the design. LTB checks also include interaction factors for combined loading.
  • Torsion: If torsion is present, it interacts with shear, affecting the calculation of \(V_{pl,Rd}\) and thus the reduction factor \(\rho\). Refer to §6.2.7 for details.
  • Shear Buckling: If the web is slender (\(h_w / t_w > 72 \epsilon / \eta\)), shear buckling resistance governs. Then these rules do not apply directly. Interaction must be checked according to EN 1993-1-5, Section 7.
  • Class 4 sections: While the reduction methods apply conceptually, the initial \(M_{c,Rd}\) must already be based on effective properties (\(W_{eff,min}\)). Interaction within EN 1993-1-5 is therefore more appropriate.

Frequently asked questions (FAQ)

Do I always need to reduce moment capacity if shear is present?

No. Reduction is only required if the design shear force \(V_{Ed}\) is greater than 50% of the plastic shear resistance \(V_{pl,Rd}\), AND shear buckling according to EN 1993-1-5 does not govern.

What is plastic shear capacity \(V_{pl,Rd}\)?

It's the plastic shear resistance of the cross-section, calculated as \( A_v \times (f_y / \sqrt{3}) / \gamma_{M0} \), where \(A_v\) is the shear area. See our guide on Shear Check for details.

Which moment reduction method should I use for an RHS or CHS section?

Use the general reduction method: \( M_{V,Rd} = (1 - \rho) M_{c,Rd} \), where \(M_{c,Rd}\) is calculated based on the section's class according to §6.2.5.

What if shear buckling governs?

If \(h_w / t_w > 72 \epsilon / \eta\), you must calculate the shear buckling resistance \(V_{b,Rd}\) and the interaction with the bending moment according to EN 1993-1-5.

Does high bending reduce shear resistance?

EN 1993-1-1 §6.2.6 does not generally require reducing the shear resistance \(V_{pl,Rd}\) due to the presence of bending moment. However, EN 1993-1-5 (for plated structures/buckling) does have rules for interaction where bending stresses can affect shear buckling resistance.

What if I have bending about both axes AND high shear?

Eurocode 3 §6.2.9 and §6.2.10 do cover biaxial bending. High shear primarily affects the resistance related to the axis where shear is resisted (usually bending about the major axis for I-sections). You would typically use the reduced moment resistance \(M_{y,V,Rd}\) within the biaxial bending or combined axial force interaction equations if high shear (\(V_{z,Ed}\)) is present.

Conclusion

The interaction between high shear force (\(V_{Ed} > 0.5 V_{pl,Rd}\)) and bending moment must be considered in Eurocode 3 design. For standard I/H sections under major axis bending, a specific formula is provided to calculate the reduced moment resistance \(M_{y,V,Rd}\). For other cases, a practical approach is to apply the reduction factor \((1 - \rho)\) to the moment resistance calculated ignoring shear (\(M_{c,Rd}\)). Always remember to first check if shear buckling (EN 1993-1-5) or lateral-torsional buckling (EN 1993-1-1 §6.3) are the governing failure modes, as their interaction rules take precedence.