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 itâ€™s generally fewer calculator button presses if you solve it as a recurrence relationship:

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

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 \(f_{ck} \leq 50\textrm{ MPa}\)