## 6 Conclusions  
The simplified performance-based approach herein applied and validated has the aim of assessing the seismic vulnerability of existing steel Moment Resisting Frames (MRFs) subjected to seismic actions. The proposed methodology consists of a trilinear approximation of the structural-behaviour curve, whose first branch is obtained through elastic analysis, while rigid-plastic analysis, considering second-order effects, helps define the second and third branches. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{_Limit State_} & \multirow{2}{*}{\(\theta_{c}\)} & \multicolumn{4}{c}{\(S_{a}(T^{*})/g\)} \\ \cline{3-5}  & & **IDA** & **ADRS** & **Nassar and** \\  & & & **Spectrum** & **Krawinkler** \\ \hline Fully operational & - & 0.293 & 0.307 & 0.307 \\
[FO]—Point B & 0.0078 & 0.618 & 0.649 & 0.649 \\ \hline Life Safety [LS]—Point C & 0.047 & 2.180 & 1.983 & 2.040 \\ \hline Near Collapse [NC]—Point D & 0.063 & 3.010 & 3.071 & 3.151 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison between IDA results and Simplified Method in terms of spectral accelerations for case study 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{_Limit State_} & \multirow{2}{*}{\(\theta_{c}\)} & \multicolumn{4}{c}{\(S_{a}(T^{*})/g\)} \\ \cline{3-5}  & & **[rad]** & **IDA** & **ADRS** & **Nassar and** \\  & & & **spectrum** & **Krawinkler** \\ \hline Fully operational & - & 0.078 & 0.087 & 0.087 \\
[FO]—Point A & 0.009 & 0.0892 & 0.112 & 0.112 \\ \hline Life Safety [LS]—Point C & 0.053 & 0.201 & 0.171 & 0.182 \\ \hline Near Collapse [NC]—Point D & 0.070 & 0.230 & 0.224 & 0.236 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison between IDA results and Simplified Method in terms of spectral accelerations for case study 2. In this work, the validation of the method through pushover analysis and Incremental Dynamic Analysis (IDA), applied to a simulated design structure and a real building whose data are available in the literature, is carried out. The pushover analyses have been developed by the SAP2000 computer program. From the analysis of Figures 5 and 6, it is possible to notice the high levels of precision achieved, as evidenced by the mean value of the \(\delta_{SAP}/\delta_{th}\) ratios close to the unit and by the presence of points leaning against the trend line, which is very close to the bisector. The IDAs were executed using the OpenSees software, employing a highly accurate fibre model that accurately reflects the behaviour of the analysed structures. This allowed for evaluating the actual percentage error between the seismic capacity defined by the simplified methodology and that obtained through the IDAs for each considered limit state. The final part of the article is focused on the definition of the MIDR--Sa/g curves for each of the 7 earthquakes considered. The comparison in terms of spectral acceleration capacities between the simplified method and IDAs provided consistent results, with a maximum error of +2.00% for case study 1 and \(-\)2.50% for case study 2, both for the Near Collapse limit state. Future developments may involve extending the methodology to medium- and high-rise buildings or to other structural types [41] to ensure an even wider applicability of the developed method. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14031037/s1](https://www.mdpi.com/article/10.3390/app14031037/s1)]([https://www.mdpi.com/article/10.3390/app14031037/s1](https://www.mdpi.com/article/10.3390/app14031037/s1)). Conceptualization, V.P. and R.M. ; Methodology, P.T. and R.M. ; Software, P.T. ; Validation, E.N., A.P. and P.T. ; Formal analysis, P.T. ; Investigation, E.N. and P.T. ; Writing--original draft, A.P. and P.T. ; Writing--review & editing, A.P. and P.T. ; Supervision, V.P. and R.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study are available in Supplementary Materials.