## 5 Incremental Dynamic Analysis (IDA)
### Results  
The maximum inter-storey drift, i.e., the ratio between the maximum relative inter-storey displacement and the inter-storey height, has been evaluated as a function of the peak ground acceleration (PGA) for each earthquake and intensity level. The rotation capacity has been set according to EN 1998-1-3 [31], and the results are reported in Table 10 with reference to each case study and in accordance with the specific limit state.  
The MIDR provides an estimate of the maximum rotation exhibited by the members (columns) of the structure. The corresponding \(S_{a}/g\) value was evaluated for each rotation as reported in Figure 14.  
For each earthquake, the PGA/g corresponding to the achievement of the four limit states considered was derived, and the average value was determined. Given the average PGA/g, the corresponding spectral acceleration was derived by constructing a specific response spectrum in terms of accelerations.  
Figure 13: Selected accelerograms.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Event ID** & **PGA** & **Length** & **N\({}_{pt}\)** & **ScaleFactor** \\  & & **[cm/s\({}^{2}\)]** & **[s]** & **[s]** & **[-]** & **[-]** \\ \hline S1 & GR-1995-0047 & 510.615 & 6.18 & 7958 & 1.50 \\ \hline S7 & IT-2009-0009 & 355.460 & 11.75 & 20000 & 1.28 \\ \hline S9 & IT-1976-0030 & 341.508 & 4.795 & 4919 & 0.50 \\ \hline S13 & IT-2009-0009 & 644.247 & 7.695 & 20001 & 0.75 \\ \hline S21 & EMSC-20161030 & 476.428 & 10.395 & 10000 & 0.63 \\ \hline S25 & IT-1980-0012 & 314.302 & 39.005 & 14152 & 0.80 \\ \hline S26 & IT-1980-0012 & 58.702 & 35.200 & 10602 & 1.50 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Selected set of earthquakes.  
The accelerations thus obtained were compared with those obtained through the application of the verification procedures described in the proposed simplified method, and the results are shown in Tables 11 and 12.  
\begin{table}
\begin{tabular}{c c c} \hline _Case Study_ & _Limit State_ & _Rotation Capacity_\(\theta_{e}\) \\ \hline \multirow{4}{*}{1} & Damage Limitation [DL] & 0.0078 \\ \cline{2-3}  & Significant Damage [SD] & 0.047 \\ \cline{2-3}  & Near Collapse [NC] & 0.063 \\ \hline \multirow{4}{*}{2} & Damage Limitation [DL] & 0.0088 \\ \cline{2-3}  & Significant Damage [SD] & 0.053 \\ \cline{2-3}  & Near Collapse [NC] & 0.070 \\ \hline \end{tabular}
\end{table}
Table 10: Rotation capacity according to EN 1998-1-3 [31].  
Figure 14: Sa/g value evaluated for each MIDR limit provided by EN1998-1-3 [31]: Case study 1 (a); Case study 2 (b).  
In particular, for case study 1, i.e., Table 11, there is a percentage error in the evaluation of capacity in terms of spectral acceleration between the simplified approach and IDA equal to +4.50% for the \"Fully Operational\" limit state, \(-3.30\%\) for the \"Operational\" limit state, \(-14.00\%\) for the \"Life Safety\" limit state, and +2.00% for the \"Near Collapse\" limit state. The simplified method, in this case, underestimates the actual capacity of the MRF for the \"Life Safety\" limit state while slightly overestimating the one at the \"Near Collapse\" limit state.  
In case study 2, i.e., Table 12, it can be observed that the percentage error in the evaluation of capacity in terms of spectral acceleration between the Simplified Method and IDA is equal to +10.30% for the \"Fully Operational\" limit state, \(+19.70\%\) for the \"Operational\" limit state, \(-9.40\%\) for the \"Life Safety\" limit state, and \(-2.50\%\) for the \"Near Collapse\" limit state. In other words, the simplified method underestimates the capacity in terms of spectral acceleration for both the \"Life Safety\" and \"Near Collapse\" limits states resulting on the safe side.