Authoritative Data, Methodology, and Guidance
Authoritative Data Source and Methodology
Standard: EN 1993‑1‑1:2005 + A1:2014 — Eurocode 3: Design of steel structures — General rules and rules for buildings. Clauses 6.2.1 (General), 6.2.6 (Shear), and 6.2.8 (Bending and shear interaction).
Publisher: CEN (European Committee for Standardization), 2014. Official publisher pages are accessible via National Standards Bodies or the Eurocodes portal: EU JRC Eurocodes.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
Notes and limitations: The present calculator addresses cross‑section resistances only. It does not include global stability (e.g., lateral–torsional buckling), shear buckling verification for slender webs, or interaction with axial force.
The Formulas Explained
Plastic bending resistance (classes 1–2):
\\[ M_{pl,Rd} = \frac{W_{pl,y}\, f_y}{\gamma_{M0}} \\]
Elastic bending resistance (class 3):
\\[ M_{el,Rd} = \frac{W_{el,y}\, f_y}{\gamma_{M0}} \\]
Shear resistance (web in shear, no buckling):
\\[ V_{pl,Rd} = \frac{A_v\, f_y}{\sqrt{3}\, \gamma_{M0}} \\]
Governing cross‑section bending resistance:
\\[ M_{c,Rd} = \begin{cases} M_{pl,Rd} & \text{for Class 1–2}\\ M_{el,Rd} & \text{for Class 3} \end{cases} \\]
Bending–shear interaction reduction (Class 1–2 webs; typical formulation when \\(V_{Ed} > 0.5\,V_{pl,Rd}\\)):
\\[ \rho = \left(2\frac{V_{Ed}}{V_{pl,Rd}} - 1\right)^2,\quad M_{V,Rd} = (1-\rho)\, M_{c,Rd} \\]
Utilization ratios:
\\[ \mu = \frac{M_{Ed}}{M_{Rd}}, \quad \nu = \frac{V_{Ed}}{V_{pl,Rd}} \\]
Glossary of Variables
- fy (MPa): Yield strength of steel (e.g., S355 → 355 MPa).
- γM0, γM1: Partial safety factors from EN 1993‑1‑1 and National Annex.
- Wpl,y (mm³): Plastic section modulus about the major axis y.
- Wel,y (mm³): Elastic section modulus about the major axis y.
- Av (mm²): Shear area of the web for shear parallel to the web.
- MEd (kNm): Design bending moment about the major axis y.
- VEd (kN): Design shear force.
- Mpl,Rd (kNm): Plastic cross‑section bending resistance.
- Mel,Rd (kNm): Elastic cross‑section bending resistance.
- Mc,Rd (kNm): Governing cross‑section bending resistance by class.
- Vpl,Rd (kN): Shear resistance without buckling.
- ρ (–): Shear–bending interaction factor when VEd > 0.5·Vpl,Rd.
- MV,Rd (kNm): Reduced bending resistance due to shear interaction.
- μ, ν (–): Moment and shear utilization ratios.
How it Works: A Step-by-Step Example
Given: fy = 355 MPa, γM0 = 1.0, class = 1, Wpl,y = 1.10×10⁶ mm³, Wel,y = 1.00×10⁶ mm³, Av = 900 mm², MEd = 200 kNm, VEd = 60 kN.
- Plastic bending resistance: Mpl,Rd = Wpl,y·fy/γM0 = (1.10×10⁶ × 355)/10⁶ = 390.5 kNm.
- Elastic bending resistance: Mel,Rd = (1.00×10⁶ × 355)/10⁶ = 355.0 kNm.
- Governing Mc,Rd (Class 1): Mc,Rd = Mpl,Rd = 390.5 kNm.
- Shear resistance: Vpl,Rd = Av·fy/(√3·γM0) = 900×355/(1.732×1000) ≈ 184.4 kN.
- Shear interaction: VEd/Vpl,Rd ≈ 0.326 ≤ 0.5 → no reduction, so MV,Rd = Mc,Rd = 390.5 kNm.
- Utilizations: μ = 200/390.5 ≈ 0.51; ν = 60/184.4 ≈ 0.33 → both pass.
Use the “Fill worked example” button to auto-populate these values in the calculator.
Frequently Asked Questions (FAQ)
What checks are covered by this calculator?
Cross-section bending resistance (plastic or elastic by class) and shear resistance without shear buckling. For Class 1–2 webs, if VEd > 0.5·Vpl,Rd, the bending resistance is reduced per EN 1993‑1‑1 6.2.8.
Which units should I use?
fy in MPa, Wpl,y and Wel,y in mm³, Av in mm², MEd in kNm, VEd in kN. The tool outputs M in kNm and V in kN.
Can I use this for Class 4 sections?
No. Class 4 requires effective section properties (EN 1993‑1‑1 and EN 1993‑1‑5). This tool does not compute effective widths.
Does it include lateral–torsional buckling?
No. LTB is a member stability check per EN 1993‑1‑1 6.3 and must be verified separately.
How do I estimate Av?
For rolled I/H sections, Av ≈ tw × hw (clear web height). For precise values, consult section tables or your National Annex recommendations.
Which partial factor applies?
Cross‑section resistances commonly use γM0. Some National Annexes may adapt factors; always follow the NA applicable to your project.
Is shear buckling considered?
No. If the web is slender, check shear buckling per EN 1993‑1‑5, or provide stiffeners to prevent buckling.