# Optimal Design of Earthquake-Resistant Buildings Based on Neural Network Inversion
## 2 Case Study to Test the Procedure
### Design Parameters and Performance Variables  
To obtain a database of cases for the training set, three parameters are taken as design parameters and referred to as \(x_{1}\), \(x_{2}\), and \(x_{3}\). The first one is the base of beams and columns, namely \(b_{b}=b_{c}=x_{1}\). The second one is the height of the beams \(h_{b}=x_{2}\). Finally, the third variable is the ratio between the area of the tension longitudinal steel (typically placed at the bottom of the beam section) and the beam section area, namely \(\rho_{b}=x_{3}\). The compression steel ratio was assumed to be \(\rho_{b,comp}=0.6x_{3}\). The ranges of variation of the design parameters are given in Table 3. A Phyton script [18] helped to vary the design parameters to obtain a dataset of examples. Eventually, 1011 cases were found, and a 3 \(\times\) 1011 input matrix was obtained to be used for the training set.  
Selected among those stated by the Italian Building Code [1], seven key structural checks were considered to build the performance target matrix. Based on the capacity design and meeting some strength hierarchy rules, these structural checks are expressed as difference between capacity and demand values and assigned to seven parameters \(y_{i}\) (\(i=1,\ 7\)). They are provided in Table 4. The building was assumed to be designed in a high ductility class (DCH) at the life protection (LP) limit state. The values of parameters \(y_{i}\) calculated at the most demanded sections for each building's configuration will be stored in the \(7\times 1011\) target matrix.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Beams**} & \multicolumn{5}{c}{**Columns**} \\ \hline beam length & \(l_{b}=\) & 5 & m & column length & \(l_{c}=\) & 3 & m \\ section height & \(h_{b}=\) & \(x_{1}\) & cm & section height & \(h_{c}=\) & \(x_{2}\) & cm \\ section base & \(b_{b}=\) & \(x_{2}\) & cm & section base & \(b_{c}=\) & \(x_{2}\) & cm \\ concrete cover & \(d=\) & 3 & cm & concrete cover & \(d=\) & 3 & cm \\ top rebars [\(\phi\)] & \(\phi_{b,t}=\) & f(\(x_{3}\)) & mm & rebars [\(\phi\)] & \(\phi_{c}=\) & f(\(x_{2}\)) & mm \\ top rebars [\(n^{\circ}\)] & \(n_{b,t}=\) & f(\(x_{3}\)) & - & rebars [\(n^{\circ}\)] & \(n_{c}=\) & f(\(x_{2}\)) & - \\ top rebars [\(\%\)] & \(\rho_{b,t}=\) & 0.6-\(x_{3}\) & - & rebars [\(\%\)] & \(\rho_{c}=\) & 0.025 & - \\ bottom rebars [\(\phi\)] & \(\phi_{b,bott}=\) & f(\(x_{3}\)) & mm & & & & \\ bottom rebars [\(n^{\circ}\)] & \(n_{b,bott}=\) & f(\(x_{3}\)) & - & & & & \\ bottom rebars [\(\%\)] & \(\rho_{b,b}=\) & \(x_{3}\) & - & & & & \\ stirrup \(\phi\) & \(\phi_{b,b,t}=\) & 10 & mm & stirrup \(\phi\) & \(\phi_{p,sl}=\) & 12 & mm \\ stirrup wheelase & \(s_{b}=\) & 60 & mm & stirrup wheelase & \(s_{p}=\) & 80 & mm \\ \hline \multicolumn{3}{c}{**Materials**} \\ \hline \multicolumn{3}{c}{**Characteristic Strength**} & \multicolumn{3}{c}{**Young Modulus**} \\ \hline concrete & f\({}_{\text{ck}}=\) 35 N/mm\({}^{2}\) & E\({}_{\text{cm}}=\) 40,600 N/mm\({}^{2}\) & & \\ steel & f\({}_{\text{yk}}=\) 450 N/mm\({}^{2}\) & E\({}_{\text{s}}=\) 200,000 N/mm\({}^{2}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main characteristics of the R/C structure.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Range of Variation** & **Increment Step** \\ \hline \(x_{1}=b_{b}=b_{c}\) & 42 \(\div\) 78 cm & +2 cm \\ \(x_{2}=h_{b}\) & 42 \(\div\) 90 cm & +2 cm \\ \(x_{3}=\rho_{b}\) & 0.0058 \(\div\) 0.019 & 0.0013 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ranges of variation of the design parameters.