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Simple bending

Design for normal simple bending according to NBR 6118, with the parabola-rectangle diagram and the equivalent rectangular stress block.

Materials

\[ f_{cd} = \frac{f_{ck}}{\gamma_c} \qquad\qquad f_{yd} = \frac{f_{yk}}{\gamma_s} \]

with \(\gamma_c = 1.4\), \(\gamma_s = 1.15\) and \(\gamma_f = 1.4\) by default — all editable.

Diagram parameters

\[ \lambda = 0.8 \qquad \alpha_c = 0.85 \qquad \varepsilon_{cu} = 3.5\text{‰} \]
\[ \xi_{lim} = 0.8\,\beta - 0.35 \]
\[ \lambda = 0.8 - \frac{f_{ck}-50}{400} \qquad \alpha_c = 0.85\left(1 - \frac{f_{ck}-50}{200}\right) \]
\[ \varepsilon_{cu} = 2.6 + 35\left(\frac{90-f_{ck}}{100}\right)^{4}\text{‰} \qquad \xi_{lim} = 0.8\,\beta - 0.45 \]

The β coefficient and redistribution

\(\beta\) is the moment redistribution ratio. With \(\beta = 1\) — no redistribution — \(\xi_{lim} = 0.45\) for concretes up to C50 and \(0.35\) above, which are the code's ductility limits.

Reducing \(\beta\) tightens \(\xi_{lim}\): the more moment is redistributed, the more rotation capacity the section needs, and the shallower the neutral axis must be.

Steel

Perfectly elastic-plastic diagram:

\[ \sigma_s = \begin{cases} E_s\,\varepsilon_s & \varepsilon_s < \varepsilon_{yd} \\[4pt] f_{yd} & \varepsilon_s \ge \varepsilon_{yd} \end{cases} \qquad \varepsilon_{yd} = \frac{f_{yd}}{E_s} \]

Rectangular section

The reduced moment is

\[ \mu = \frac{M_d}{b\,d^2\,\sigma_{cd}} \qquad\text{with}\qquad \sigma_{cd} = \alpha_c\,f_{cd} \quad\text{and}\quad M_d = \gamma_f M_k \]

and the limit between tension-only and compression reinforcement,

\[ \mu_{lim} = \lambda\,\xi_{lim}\left(1 - 0.5\,\lambda\,\xi_{lim}\right) \]

Tension reinforcement only — μ ≤ μlim

\[ \xi = \frac{1 - \sqrt{1 - 2\mu}}{\lambda} \qquad\qquad A_s = \frac{\lambda\,\xi\,b\,d\,\sigma_{cd}}{f_{yd}} \]

With compression reinforcement — μ > μlim

The section cannot resist with tension reinforcement alone; compression reinforcement \(A'_s\) is added. Its strain, with \(\delta = d'/d\):

\[ \varepsilon'_s = \frac{\varepsilon_{cu}\left(\xi_{lim} - \delta\right)}{\xi_{lim}} \]

and hence

\[ A'_s = \frac{(\mu - \mu_{lim})\,b\,d\,\sigma_{cd}}{(1-\delta)\,\sigma'_s} \]
\[ A_s = \left[\lambda\,\xi_{lim} + \frac{\mu - \mu_{lim}}{1-\delta}\right]\frac{b\,d\,\sigma_{cd}}{f_{yd}} \]

Two conditions that stop the calculation

Compression reinforcement in domain 2. If \(\xi_{lim} < \varepsilon_{cu}/(\varepsilon_{cu}+10)\), the section would be in domain 2 with compression reinforcement — the concrete would not even be used. The program refuses and asks for a larger section.

Compression reinforcement in tension. If \(\xi_{lim} \le \delta\), the neutral axis passes above \(A'_s\), and the "compression" reinforcement would be in tension. It refuses as well.

In both cases the practical message is the same: enlarge the section. These are situations in which the geometry is the problem, not the reinforcement.

T section

The procedure is that of the rectangular section, with the flange contributing in compression. The logic splits depending on whether the neutral axis falls within the flange — in which case the section behaves as a rectangle of width \(b_f\) — or below it, in which case compression is shared between the flange and the web.

The inputs are: flange width \(b_f\), flange thickness \(h_f\), web width \(b_w\) and the effective depth.

Minimum reinforcement

\[ \rho_{min} = \begin{cases} \dfrac{0.078\,f_{ck}^{2/3}}{f_{yd}} & f_{ck} \le 50\ \text{MPa} \\[10pt] \dfrac{0.5512\,\ln(1 + 0.11\,f_{ck})}{f_{yd}} & f_{ck} > 50\ \text{MPa} \end{cases} \]

with the absolute floor

\[ \rho_{min} \ge 0.0015 \qquad\qquad A_{s,min} = \rho_{min}\,b\,h \]

Note that the minimum ratio applies to the gross area \(b\,h\), not to \(b\,d\).

Limits of validity

Range of application

  • Normal simple bending, in plane. Biaxial bending and bending with axial force are not covered.
  • The design is of a section, not of a member: the program does not calculate internal forces from spans and loads. You enter \(M_k\), \(V_k\) and \(T_k\) already obtained from your structural analysis.
  • Serviceability limit states — crack width and deflection — are not checked. For slender beams, deflection usually governs, and it is not checked here.
  • There is no check of anchorage, laps or fatigue.
  • The minimum skin reinforcement only enters the SBX optimiser; in direct design it is a detailing choice.