## 5 Numeric Example
### Special Moment Resisting Frame  
Special moment resisting frames are designed to fulfil the Eurocode 8 seismic provisions. The selected case study with the definition of the beam and column dimension is reported in Figure 6.  
The trilinear capacity curve is shown in Figure 7, which also shows the characteristic performance points of the model. \begin{table}
\begin{tabular}{c c c c c} \hline  & **FO** & **O** & **LS** & **NC** \\ \hline \(F\) [**kN**] & 1119.90 & 1140.12 & 1140.12 & 1140.12 \\ \(F^{\bullet}\) [**kN**] & 778.72 & 792.78 & 792.78 & 792.78 \\ \(d\) [**m**] & 0.2602 & 0.265 & 0.4192 & 0.4192 \\ \(d^{\bullet}\) [**m**] & **0.181** & **0.184** & **0.291** & **0.291** \\ \(\mu\) [**m**] & \(-\) & \(-\) & 1.582 & 1.582 \\ \(Sa^{\bullet}\) [**g**] & **0.353** & **0.359** & **0.575** & **0.575** \\ \hline \end{tabular}
\end{table} 
Table 4: Nassar and Krawinkler approach. Figure 6: Diagram of the frame constituting the numeric example with the indication of beams, columns and seismic forces. \begin{table}
\begin{tabular}{c c c c c} \hline  & **FO** & **O** & **LS** & **NC** \\ \hline \(F\) [**kN**] & 1119.90 & 1140.12 & 1140.12 & 1140.12 \\ \(F^{\bullet}\) [**kN**] & 778.72 & 792.78 & 792.78 & 792.78 \\ \(d\) [**m**] & 0.2602 & 0.265 & 0.4192 & 0.4192 \\ \(d^{\bullet}\) [**m**] & **0.181** & **0.184** & **0.291** & **0.291** \\ \(d^{\bullet}\) [**m**] & **0.181** & **0.184** & **0.291** & **0.291** \\ \(\mu\) [**m**] & \(-\) & \(-\) & 1.582 & 1.582 \\ \(Sa^{\bullet}\) [**g**] & **0.353** & **0.359** & **0.575** & **0.575** \\ \hline \end{tabular}
\end{table} 
Table 3: ADRS spectrum approach. Parameters obtained from the elastic analysis:  
* \(\delta_{1}(\alpha=1)=0.03814\ m\)
* \(k=15.8605\ m^{-1}\)
* \(\delta_{A}=\delta_{y}=0.1802\ m\)
* \(\alpha_{A}=\alpha_{y}=k\delta_{y}=4.736\)  
Parameters obtained from the rigid-plastic analysis:  
* \(\alpha_{0}=7.989\)
* \(\gamma_{s}=1.006\ m^{-1}\)
* \(\alpha=\alpha_{0}-\gamma_{s}(\delta-\delta_{y})\rightarrow\alpha=7.989-1.006( \delta-0.1802)\)
* \(\alpha(\delta=0)=\alpha_{0}+\gamma_{s}\delta_{y}=8.1704\)
* \(H_{0}=14.0\ m\) (collapse mechanism Type 1, \(i_{m}=4\))  
Evaluation of the maximum multiplier using the calibrated Merchant\(-\)Rankine formula:  
* \(\alpha_{max}\ =\ \frac{\alpha_{0}}{1\ +\ \Psi\alpha_{0}\gamma_{s}\delta_{1}}\ =\ 7.4056\)  
where:  
\[\Psi\ =0.28488\ -\ 0.14042\ \xi\ =\ 0.2572\ \mathrm{with}\ \xi\ =\ \frac{\sum\frac{EI_{h}}{L_{p}}}{\sum\frac{EI_{c}}{L_{c}}}\ =\ 0.1971\]  
and consequently \(\delta_{B}\ =\ \frac{\alpha_{max}}{k}\ =\ 0.2824\ \mathrm{and}\ \delta_{C}\ =\ \delta_{mec}\ =\ \frac{\alpha_{0}-\alpha_{max}}{\gamma_{s}}\ +\ \delta_{y}=\frac{7.989-7.4056}{1.006}\ +\ 0.1802\ =\ 0.7605\ m\). Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):  
* \(\theta_{\mathrm{p.mec}}\ =\ \frac{\mathrm{n_{s}}\delta_{y}}{\mathrm{H_{0}}} \Bigg{[}\frac{\Psi_{1}}{\Psi_{2}}\Psi_{3}\bigg{(}\frac{\alpha_{\mathrm{max}}} {\alpha_{y}}\ -\ 1\bigg{)}^{\Psi_{4}}\frac{1}{1\ -\ \Psi_{5}\gamma_{s}}\Bigg{]}\ =\ 0.03043\ \mathrm{rad}\)  
The calculation of the corresponding capacity provides a final plastic rotation value of \(8\theta_{y}=8\times 0.008257=0.06605\ \mathrm{rad}\). Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is partial type 1 \(i_{m}=4\), so the critical element is a first storey column):  
Figure 7: Trilinear model and characteristic points for the structure 7S4B_SMRF_6m.