Required steel area of a concrete section

Figure 1: Stress distribution using the rectangular stress block method
Figure 1: Stress distribution using the rectangular stress block method

Symbols

Symbol Description
\(b\) breadth of section
\(h\) height of section
\(d\) effective depth of section
\(M_{Ed}\) design bending moment
\(f_{cd}\) design compressive strength of concrete
\(f_{yd}\) design tensile strength of reinforcement
\(x\) neutral axis depth
\(A_{s}\) steel area

Method

Begin by estimating the neutral axis at the midpoint:

\[ x_{0} = \frac{h}{2} \]

Then use the following recurrence relationship to iterate out:

\[ x_{i+1} = \frac{M_{Ed}}{0.8 f_{cd} b d} + 0.4\frac{x_{i}^{2}}{d} \]

Finally, calculate minimum required area of steel:

\[ A_{s} = \frac{0.8 f_{cd} x b}{f_{yd}} \]

Derivation

Balance axial forces in Figure 1.

\[\begin{align} A_{s} f_{yd} & = f_{cd} \times 0.8xb \\ A_{s} &= \frac{f_{cd}}{f_{yd}}0.8xb \label{a} \tag{Eq.1} \end{align}\]

Write equation for net moment in Figure 1.

\[\begin{align} M_{Ed} = A_{s} f_{yd} (d - x) + f_{cd} \times 0.8xb \times 0.6 x \end{align}\]

Substitute expression for \(A_{s}\) from \(\ref{a}\)

\[\begin{align} M_{Ed} & = \frac{f_{cd}}{f_{yd}}0.8 x b f_{yd}(d - x) + f_{cd} \times 0.8xb \times 0.6 x \\ \frac{M_{Ed}}{0.8f_{cd} bd} & = (d - x) \frac{x}{d} + 0.6 \frac{x^{2}}{d} \\ \frac{M_{Ed}}{0.8f_{cd} bd} & = x - \frac{x^{2}}{d} + 0.6 \frac{x^{2}}{d} \\ x & = \frac{M_{Ed}}{0.8f_{cd} bd} + 0.4\frac{x^{2}}{d} \end{align}\]

This can be solved for \(x\) using the quadratic equation, but it’s generally fewer calculator button presses if you solve it as a recurrence relationship:

\[ x_{i+1} = \frac{M_{Ed}}{0.8 f_{cd} b d} + 0.4\frac{x_{i}^{2}}{d} \]

Then obtain area of steel from \(\ref{a}\):

\[\begin{align} A_{s} &= \frac{f_{cd}}{f_{yd}}0.8xb \end{align}\]

References

This is derived using the assumptions of the rectangular stress block method that appears in Figure 3.5 of EN 1992-1-1:2004+A1:2014 where \(f_{ck} \leq 50\textrm{ MPa}\)

Similar rules are present in other design codes: