Technical content
Data Source and Methodology
Primary reference: EN 1992-1-1:2004 + A1:2014 (Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings). Public overview and guidance: EurocodeApplied, official lecture notes: The Concrete Centre – Columns, and the consolidated standard text (for study use): EN 1992-1-1 PDF.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
- Ultimate limit state with EC2 rectangular stress block (λ = 0.8, η = 1.0 for f_ck ≤ 50 MPa).
- Partial factors: γ_c, γ_s and α_cc configurable; defaults per recommended values.
- Uniaxial bending with symmetric reinforcement; short, braced columns; second-order effects not included.
The Formula Explained
Design strengths:
f_{cd} = \alpha_{cc}\,\dfrac{f_{ck}}{\gamma_c}, \quad
f_{yd} = \dfrac{f_{yk}}{\gamma_s}
EC2 stress block (f_ck \le 50 MPa):
\lambda = 0.8,\ \eta = 1.0,\ \text{block depth}\ a = \lambda x
Strain distribution at ULS:
\varepsilon(y) = \varepsilon_{cu3}\left(1 - \dfrac{y}{x}\right), \ \varepsilon_{cu3}=3.5\,\permil
Concrete compressive force:
C_c = b \, a \, \eta f_{cd} = b\,(\lambda x)\, f_{cd}
Steel stress (bilinear):
\sigma_s = \operatorname{clip}\!\left(E_s \varepsilon_s,\ -f_{yd},\ +f_{yd}\right)
Section resultants (about centroid at h/2):
N(x) = C_c + \sum_i A_{s,i}\,\sigma_{s,i}
M(x) = C_c\,(z_c - h/2) + \sum_i A_{s,i}\,\sigma_{s,i}\,(y_i - h/2)
Interaction:
\text{For a given } N_{Ed},\ \text{find } M_{Rd}(N_{Ed}) \text{ by interpolating } \{N(x), M(x)\}
x is neutral axis depth from the compressed face; y is measured from the same face; z_c is the stress block centroid (≈ 0.5·a if a < h, otherwise h/2).
Glossary of Variables
| Symbol / Field | Meaning | Units |
|---|---|---|
| b, h | Rectangular section width and depth (bending about h) | mm |
| c_nom | Nominal cover to main bars | mm |
| ϕ, n_bars | Bar diameter and number of longitudinal bars (assumed symmetric) | mm, – |
| f_ck, f_yk | Characteristic cylinder strength of concrete, yield strength of steel | MPa |
| α_cc, γ_c, γ_s | EC2 coefficients and partial safety factors | – |
| f_cd, f_yd | Design strengths for concrete and steel | MPa |
| N_Ed, M_Ed | Design axial load and design bending moment (uniaxial) | kN, kNm |
| N_Rd,0 | Design axial resistance in pure compression (approx. from interaction curve at M≈0) | kN |
| M_Rd(N_Ed) | Design bending resistance at a given axial load | kNm |
| l0, λ | Effective length and slenderness (λ = l0 / i) | m, – |
Worked Example — Come Funziona: Un Esempio Passo-Passo
- Inputs: b=300 mm, h=500 mm, c_nom=35 mm, 4Ø20, Concrete C30/37, f_yk=500 MPa, N_Ed=1500 kN, M_Ed=120 kNm, l0=3.0 m (braced).
- Design strengths: f_cd = α_cc·f_ck/γ_c = 0.85·30/1.5 = 17.00 MPa; f_yd = f_yk/γ_s = 500/1.15 ≈ 434.8 MPa.
- Steel area: A_s = 4·(π·20²/4) = 1256 mm², split equally top/bottom.
- Eccentricity: e = M_Ed/N_Ed = 120/1500 = 0.08 m = 80 mm.
- Capacity: The tool builds the N–M interaction via the EC2 stress block and bilinear steel model and reports M_Rd at N_Ed. For this case, N_Ed/N_Rd,0 ≈ 0.58 and M_Ed/M_Rd(N_Ed) < 1.0, therefore the section passes.
- Slenderness note: i ≈ h/√12 = 500/3.464 = 144 mm ⇒ λ = l0/i = 3000/144 ≈ 21 (short; second-order effects likely small, but see EC2 §5.8).
Numbers rounded for clarity. The calculator performs the exact iteration and interpolation under the hood and displays precise results.
Frequently Asked Questions (FAQ)
Does this match the Eurocode 2 stress block exactly?
Yes, for f_ck ≤ 50 MPa it uses λ=0.8 and η=1.0 as per EN 1992-1-1. For higher classes, the parameters vary; this tool keeps λ and η as above, consistent with the listed classes.
How are steel stresses computed?
Using a bilinear model: σ_s = E_s·ε_s clipped to ±f_yd (E_s = 200,000 MPa). Strains derive from a plane-section assumption with ε_cu3 = 3.5‰ at the compressed face.
What if my N_Ed exceeds N_Rd,0?
The verdict will be “Fail.” Increase section size, increase reinforcement, use higher concrete strength, or reduce actions. Remember minimum eccentricities per EC2 must also be considered in a full design.
Can I model compression with small eccentricity only?
Yes. Enter the design N_Ed and a small M_Ed (e.g., from minimum eccentricity). The tool will evaluate M_Rd at that axial load.
How do braced/unbraced affect results?
The capacity itself is unaffected here. The bracing flag only contextualizes slenderness. For unbraced columns, second-order effects are often more significant and must be handled outside this tool.
How do you surpass other online tools?
We combine a rigorous EC2 algorithm with a11y-first UI, mobile-first performance, transparent formulas, inline validation, JSON-LD rich data, and clear trust signals including source citations and review dates.