## 5 Numeric Example
### Global Moment Resisting Frame  
Global moment resisting frames are designed according to the TPMC. The design results are reported in Figure 8.  
The trilinear capacity curve is shown in Figure 9, which also shows the characteristic performance points of the model. Parameters obtained from the elastic analysis:  
* \(\delta_{1}(\alpha=1)=0.02684\;m\)
* \(k=37.2606\;m^{-1}\)
* \(\delta_{A}=\delta_{y}=0.1602\;m\)
* \(\alpha_{A}=\alpha_{y}=k\delta_{y}=5.999\)  
Parameters obtained from the rigid-plastic analysis:  
* \(\alpha_{0}=10.149\)
* \(\gamma_{s}=0.53\;m^{-1}\)
* \(\alpha=\alpha_{0}-\gamma_{s}(\delta-\delta_{y})\rightarrow\alpha=10.149-0.53( \delta-0.1602)\)
* \(\alpha(\delta=0)=\alpha_{0}+\gamma_{s}\delta_{y}=10.234\)
* \(H_{0}=24.5\;m\) (Global collapse mechanism) \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **FO** & **O** & **LS** & **NC** \\ \hline \(F\) **[kN]** & 1284.84 & 2009.08 & 2009.08 & 2009.08 \\ \(F^{\bullet}\) **[kN]** & 893.41 & 1397.01 & 1397.01 & 1397.01 \\ \(d\) **[m]** & 0.1802 & 0.2824 & 0.5901 & 0.5901 \\ \(d^{\bullet}\) **[m]** & 0.1253 & 0.1964 & 0.4103 & 0.4103 \\ \(\mu\) **[m]** & – & – & 2.089 & 2.089 \\ \(Sa^{\bullet}\) **[g]** & 0.405 & 0.634 & 1.353 & 1.353 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: [PERSON] and [PERSON] approach. Figure 8: Diagram of the frame constituting the numeric example with the indication of beams, columns and seismic forces. Evaluation of the maximum multiplier using the calibrated Merchant\(-\)Rankine formula:  
\(\bullet\)  
\[\alpha_{max}\ =\ \frac{\alpha_{0}}{1\ +\ \Psi_{\alpha 0}\gamma_{s}\delta_{1}}\ =\ 9.7594\]  
where:  
\[\Psi\ =0.28488\ -\ 0.14042\ \zeta\ =\ 0.2763\ \text{with}\ \zeta\ =\ \frac{\sum\frac{E_{1}}{L_{b}}}{\sum\frac{E_{1}}{L_{c}}}\ =\ 0.06129\]  
and consequently \(\delta_{B}\ =\ \frac{\alpha_{max}}{k}\ =\ 0.2619\ \text{and}\ \delta_{C}\ =\ \delta_{mecec}\ =\ \frac{ \alpha_{0}-\alpha_{max}}{\gamma_{s}}\ +\ \delta_{y}=\frac{10.149\ -\ 9.7594}{0.53}\ +\ 0.1602\ =\ 0.89 46\ m\). Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):  
\[\theta_{\text{p.mec}}\ =\ \frac{\text{n}_{\text{s}}\delta_{y}}{\text{H}_{0}} \bigg{[}\frac{\Psi_{1}}{\Psi_{2}}\Psi_{3}\bigg{(}\frac{\alpha_{\text{max}}}{ \alpha_{y}}\ -\ 1\bigg{)}^{\frac{\Psi_{4}}{2}}\frac{1\ -\ \Psi_{5}\gamma_{s}}{1\ -\ \Psi_{6}\gamma_{s}}\bigg{]}\ =\ 0.01886\ \text{rad}\]  
The calculation of the corresponding capacity provides a final plastic rotation value of \(8\theta_{y}\) = 8 \(\times\) 0.008257 = 0.06605 rad. Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is global, so the critical element is one of the first storey columns):  
\(\bullet\)  
\[\theta_{\text{p.mec}}\ =\ \frac{\text{n}_{\text{s}}\delta_{y}}{\text{H}_{0}} \bigg{[}\frac{\Psi_{1}^{\prime}}{\Psi_{2}^{\prime}}\Psi_{3}^{\prime}\bigg{(} \frac{\alpha_{max}}{\alpha_{y}}\ -\ 1\bigg{)}^{\frac{\Psi_{4}^{\prime}}{2}}\frac{1\ -\ \Psi_{5}^{\prime}\gamma_{s}}{1\ -\ \Psi_{6}^{\prime}\gamma_{s}}\bigg{]}\ =\ 0.01774\ \text{rad}\]  
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 \(\times\) 0.003714 = 0.02971 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:  
\[\delta_{u}\ =\ \delta_{C}\ +\ \big{(}\theta_{p.u}\ -\ \theta_{p.mec}\big{)}H_{0} \ =\ 0.8946\ +\ (0.02971\ -\ 0.01774)\ \times\ 24.5\ =\ 1.188\ m\]  
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:  
Figure 9: Trilinear model and characteristic points for the structure 754B_GMRF_6m.