## 5 Numeric Example
### Special Moment Resisting Frame
\[\theta_{\rm p.mec}\ =\ \frac{n_{\rm s}\xi_{y}}{H_{0}}\left[\frac{\Psi_{I}^{\prime}}{ \Psi_{2}^{\prime}}\Psi_{3}^{\prime}\Big{(}\frac{\alpha_{max}}{\alpha_{y}}\ -\ 1\Big{)}\frac{\Psi_{4}^{\prime}}{1}\frac{1\ -\ \Psi_{5}^{\prime}\gamma_{5}}{1\ -\ \Psi_{6}^{\prime}\gamma_{5}}\right]\ =\ 0.04587\ {\rm rad}\]  
The calculation of the corresponding capacity, in the case of the third storey columns, provides a value of the ultimate plastic rotation equal to 8\(\theta_{y}\) = 8 \(\times\) 0.004212 = 0.033699 rad. Therefore, the ultimate conditions are governed by the columns of the first storey. Considering the plastic rotation capacity, the ultimate displacement is given by:  
\[\delta_{u}\ =\ \delta_{C}\ +\ \big{(}\theta_{p,u}\ -\ \theta_{p,mecc}\big{)}H_{0} =\ 0.76\ +\ (0.033699\ -\ 0.04587)\ \times\ 14.0\ =\ 0.5901\ m\]  
Since the plastic rotation capacity of the third storey columns is lower than that necessary for the complete development of the kinematic mechanism, the points C and D corresponding to the limit states \"Life Safety\" and \"Near Collapse\" are coincident and correspond to the aforementioned last displacement (\(\delta_{u}\)). All the verification procedures considered the use of the transformation of the MDOF system by means of the participation coefficient of the main vibration mode \(\Gamma\). For this reason, it is necessary to define:  
The eigenvector \(\underline{\Phi}\ =\ \{\Phi_{1},\ \Phi_{2},\ \Phi_{3},\ \Phi_{4},\ \Phi_{5},\ \Phi_{6}\}\)that, assuming \(\Phi_{k}\ =\ \frac{F_{k}}{F_{n}}\)is:  
\[\hat{\Phi}_{1}\ =\ 0.134\ \hat{\Phi}_{2}\ =\ 0.267\ \hat{\Phi}_{3}\ =\ 0.401\]  
\[\hat{\Phi}_{4}\ =\ 0.535\ \hat{\Phi}_{5}\ =\ 0.669\ \hat{\Phi}_{6}\ =\ 0.802\ \hat{\Phi}_{7}\ =\ 1.000\]  
The modal participation factor \(\Gamma\):  
\[\Gamma\ =\ \frac{\sum_{k=1}^{n}m_{k}\hat{\Phi}_{k}}{\sum_{k=1}^{n}m_{k}\hat{ \Phi}_{k}^{2}}\ =\ 1.4381\]  
being:  
\[m_{1}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\ m_{2}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\ m_{3}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\]  
\[m_{4}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\ m_{5}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\ m_{5}\ =\ 57.98\ \times\ 10^{3}\ {\rm kg}\ m_{7}\ =\ 61.94\ \times\ 10^{3}\ {\rm kg}\]  
The dynamic parameters of the equivalent SDOF system are reported in \[\Psi\ =0.28488\ -\ 0.14042\ \xi\ =\ 0.2572\ \mathrm{with}\ \xi\ =\ \frac{\sum\frac{EI_{h}}{L_{p}}}{\sum\frac{EI_{c}}{L_{c}}}\ =\ 0.1971\]0  
Therefore, the characteristic points of the capacity curve are defined in the planes \(\alpha\ -\ \delta_{s},F_{b}\ -\ d_{c},F^{\star}\ -\ D^{\star},\ S_{a}\ -\ S_{D}\) assessing the capacity in terms of accelerations for the Nassar and Krawinkler approach and ADRS spectrum approach. In particular, in Table 6, the results based on the use of the ADRS spectrum, and in Table 7, the results based on the use of the Nassar and Krawinkler formulation, are reported. \begin{table}
\begin{tabular}{c c c c c} \hline  & **FO** & **O** & **LS** & **NC** \\ \hline \(F\) **[kN]** & 1284.84 & 2009.08 & 2009.08 & 2009.08 \\ \(F^{\star}\) **[kN]** & 893.41 & 1397.01 & 1397.01 & 1397.01 \\ \(d\) **[m]** & 0.1802 & 0.2824 & 0.5901 & 0.5901 \\ \(d^{\star}\) **[m]** & **0.1253** & **0.1964** & **0.4103** & **0.4103** \\ \(Sa^{\star}\) **[g]** & **0.404** & **0.634** & **1.324** & **1.324** \\ \hline \end{tabular}
\end{table} 
Table 6: ADRS spectrum approach. \begin{table}
\begin{tabular}{c c c c} \hline \(m^{\star}\) & \(k^{\star}\) & \(\omega^{\star}\) & \(T^{\star}\) \\ \hline \([{\rm kg}\ 10^{3}]\) & \([{\rm kN/m}]\) & \([{\rm rad/s}]\) & \([{\rm s}]\) \\ \hline
**224.76** & **7113.36** & **5.6257** & **1.117** \\ \hline \end{tabular}
\end{table} 
Table 5: Dynamic parameters of the equivalent SDOF system. Seismic performance verification requires that, for each limit state, the inequality \(S_{\text{a.ls}}(T^{\bullet})_{capacity}\geq S_{\text{a.ls}}(T^{\bullet})_{demand}\) is satisfied.