## 5 Numeric Example
### Ordinary Moment Resisting Frame  
Ordinary moment resisting frames are designed without any seismic prescription. The beams and column sections are reported in Figure 4.  
The trilinear capacity curve is shown in Figure 5, which also shows the characteristic performance points of the model. Figure 4: Diagram of the frame constituting the numeric example with indication of beams, columns and seismic forces. Figure 3: Flowchart of the assessment procedure. Parameters obtained from the elastic analysis:  
* \(\delta_{1}(\alpha=1)=0.06305\) m
* \(k=15.8605\) m\({}^{-1}\)
* \(\delta_{A}=\delta_{y}=0.2602\) m
* \(\alpha_{A}=\alpha_{y}=k\delta_{y}=4.128\)  
Parameters obtained from the rigid-plastic analysis:  
* \(\alpha_{0}=5.219\)
* \(\gamma_{s}=3.729\)\(m^{-1}\)
* \(\alpha=\alpha_{0}-\gamma_{S}(\delta-\delta_{y})\rightarrow\alpha=5.219-3.729( \delta-0.2602)\)
* \(\alpha(\delta=0)=\alpha_{0}+\gamma_{S}\delta_{y}=6.1888\)
* \(H_{0}=3.5\)\(m\)(collapse mechanism Type 3, \(i_{m}=3\))  
Evaluation of the maximum multiplier using the calibrated Merchant\(-\)Rankine formula:  
* \(\alpha_{max}\ =\ \frac{\alpha_{0}}{1\ +\ \Psi\alpha_{0}\gamma_{s}\delta_{1}}\ =\ 4.2025\)  
where:  
\(\Psi\ =0.28488\ -\ 0.14042\ \xi\ =\ 0.1970\) with \(\xi\ =\ \frac{\sum\frac{Et_{b}}{L_{b}}}{\sum\frac{Et_{c}}{L_{c}}}\ =\ 0.6255\)  
and consequently \(\delta_{B}\ =\ \frac{\alpha_{max}}{k}\ =\ 0.265\) and \(\delta_{c}\ =\ \delta_{mecc}\ =\ \frac{\alpha_{0}-\alpha_{max}}{\gamma_{s}}\ +\ \delta_{y}=\frac{5.219-4.2025}{3.729}\ +\ 0.2602\ =\ 0.5326\ m\). Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):  
* \(\theta_{\rm p.mec}\ =\ \frac{\mathrm{n_{s}}\delta_{y}}{\mathrm{H_{0}}}\left[ \frac{\Psi_{1}}{\Psi_{2}}\Psi_{3}\left(\frac{\alpha_{\rm max}}{\alpha_{y}}\ -\ 1\right)\right]^{\Psi_{4}}\frac{1\ -\ \Psi_{5}\gamma_{s}}{1\ -\ \Psi_{6}\gamma_{s}}\right]\ =\ 0.06612\) rad  
The calculation of the corresponding capacity provides a final plastic rotation value of \(80_{y}=8\ \times\ 0.008257=0.06605\) rad. Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is partial type 3 \(i_{m}=3\), so the critical element is a third storey column):  
Figure 5: Trilinear model and characteristic points for the structure 754B_OMRF_6m. \[\theta_{\rm p.mec}\ =\ \frac{n_{\rm s}\delta_{y}}{H_{0}}\Bigg{[}\frac{\Psi_{1}^{ \prime}}{\Psi_{2}^{\prime}}\Psi_{3}^{\prime}\bigg{(}\frac{\alpha_{\rm max}}{ \alpha_{y}}\ -\ 1\bigg{)}^{\frac{\Psi_{4}^{\prime}}{4}}\frac{1\ -\ \Psi_{5}^{\prime}\gamma_{5}}{1\ -\ \Psi_{6}^{ \prime}\gamma_{5}}\bigg{]}\ =\ 0.07693\ {\rm rad}\]  
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 \(\times\) 0.005568 = 0.04454 rad. Therefore, the ultimate conditions are governed by the columns of the third storey. Considering the plastic rotation capacity, the ultimate displacement is given by:  
\[\delta_{u}\ =\ \delta_{\rm C}\ +\ \big{(}\delta_{p.u}\ -\ \delta_{p.mec}\big{)}H_{0} \ =\ 0.5326\ +\ (0.04454\ -\ 0.07693)\ \times\ 3.5\ =\ 0.4192\ m\]  
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 into an equivalent SDOF system using the participation coefficient of the main vibration mode \(\Gamma\).