## 5 Incremental Dynamic Analysis (IDA)
### Evaluation of the Capacity in Terms of Spectral Acceleration by Simplified Method  
After defining the trilinear model and the performance points, the transformation procedure of the multiple degrees of freedom (MDOF) system into an equivalent single degree of freedom (SDOF) system was applied through the modal participation coefficient \(\Gamma\).  
Subsequently, the dynamic properties of the SDOF system, the simplified curve, and the performance points were represented in the ADRS plan. In this way, the capacity in terms of spectral displacements and accelerations was defined according to the \"ADRS spectrum\" and _Nassar and Krawinkler_ models [34; 35; 36].  
It was, therefore, necessary to define the nondimensional eigenvectors:  
\[\begin{array}{ll}\underline{\phi}=\{\phi_{1},\phi_{2},\phi_{3},\phi_{4},\phi _{5}\}&\underline{\phi}=\{\phi_{1},\phi_{2},\phi_{3}\}\\ \quad\quad\quad(\text{Case Study1})&(\text{Case Study2})\end{array} \tag{13}\]  
where:  
\[\phi_{k}=F_{k}/F_{n_{s}} \tag{14}\]  
where \(F_{k}\) is equal to seismic force at \(k-th\) storey and \(F_{n_{s}}\) is the seismic force at top storey according to EN 1998-1-1 [32]. The \(\phi_{i}\) values are reported in Table 4 for each case study.  
The modal participation factor \(\Gamma\) is given by:  
\[\Gamma=\frac{\sum_{k=1}^{n_{s}}m_{k}\phi_{k}}{\sum_{k=1}^{n_{s}}m_{k}\phi_{k}^ {2}} \tag{15}\]  
where the \(\Gamma\) and \(m_{k}\) values are reported in Table 5. While the dynamic parameters of the equivalent SDOF system are shown in Table 6 according to EN 1998-1-1 [32].  
Therefore, the characteristic points of the capacity curve are defined in the planes \(\alpha-\delta\), \(F_{b}-d_{c}\), \(F^{*}-D^{*}\), \(S_{a}-S_{D}\) assessing the capacity in terms of accelerations for _Nassar and Krawinkler_ approach and ADRS spectrum approach. The results based on the use of the ADRS spectrum and the _Nassar and Krawinkler_ formulations are reported in Tables 7 and 8 for different structural levels [33; 34; 35]. In particular, the following results are indicated: \(\alpha_{LS}\) is the multiplier of seismic forces for a specific limit state (LS); \(F_{LS}\) is base shear force corresponding to the specific LS; \(\delta_{LS}\) is the maximum displacement of the top storey  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline _Case_ & \(m^{*}\) & \(k^{*}\) & \(\omega^{*}\) & \(T^{*}\) \\ _Study_ & [**[kg\(\cdot\)10\({}^{3}\)]** & [**kh/m**] & [**rad/s**] & [**s**] \\ \hline
1 & 181.04 & 8668.21 & 6.920 & 0.91 \\
2 & 96.63 & 1152.32 & 3.443 & 1.82 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Dynamic parameters of the equivalent SDOF systems.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline _Case_ & \(\Phi_{1}\) & \(\Phi_{2}\) & \(\Phi_{3}\) & \(\Phi_{4}\) & \(\Phi_{5}\) \\ _Study_ & [**-**] & [**-**] & [**-**] & [**-**] & [**-**] \\ \hline
1 & 0.206 & 0.402 & 0.608 & 0.788 & 1.00 \\
2 & 0.262 & 0.519 & 1.00 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Values of \(\phi_{i}\) for each case study.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _Case_ & \(\Gamma\) & \(m_{1}\) & \(m_{2}\) & \(m_{3}\) & \(m_{4}\) & \(m_{5}\) \\ _Study_ & [**-**] & [**kg\(\cdot\)10\({}^{3}\)]** & [**kg\(\cdot\)10\({}^{3}\)]** & [**kg\(\cdot\)10\({}^{3}\)]** & [**kg\(\cdot\)10\({}^{3}\)]** & [**kg\(\cdot\)10\({}^{3}\)]** \\ \hline
1 & 1.364 & 60.346 & 60.346 & 60.346 & 60.346 & 60.346 \\
2 & 1.331 & 54.234 & 54.234 & 54.234 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 5: Values of \(\Gamma\) and \(m_{k}\) for each case study.  
corresponding to \(F_{LS}\); \(F^{*}\) is the base shear force of the equivalent SDOF system; \(\delta^{*}\) is the displacement of the equivalent SDOF corresponding to \(F^{*}\); \(S_{a}(T^{*})\) is the spectral acceleration obtained by using the ADRS spectrum and for _Nassar and Krawinkler_ approach.