## 4 Case Study: The Grottalunga Viaduct
### Preliminary Discussion for Developing Modal Dynamic Analysis and POA in Longitudinal Direction
#### 4.3.4 Single-Run Multi-Modal Pushover Analyses
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Dir x+ Phase Shift 0\({}^{\circ}\)**} & \multicolumn{2}{c}{**Dir x- Phase Shift 0\({}^{\circ}\)**} & \multicolumn{2}{c}{**Separ, Phase Shift 180\({}^{\circ}\)**} & \multicolumn{2}{c}{**Appr. Phase Shift 180\({}^{\circ}\)**} \\ \hline \multirow{2}{*}{**Event**} & **Displ** & **PGA** & **Event** & **Displ** & **PGA** & **Event** & **Displ** & **PGA** & **Event** & **Displ** & **PGA** \\  & **(mm)** & **(g)** & **Event** & **(mm)** & **(g)** & **Event** & **(mm)** & **(g)** & **Event** & **(mm)** & **(g)** \\ \hline PHA LPB & 112 & 0.111 & PHA RPB & \(111\) & 0.114 & ES DIS & 13 & 0.091 & PHA LPB & \(-88\) & 0.064 \\ \hline ES LA & 146 & 0.166 & ES LA & 131 & 0.146 & ES RA & 63 & 0.176 & POU DIS & \(-99\) & 0.079 \\ \hline ES RA & 149 & 0.171 & PHA LPB & 146 & 0.171 & PHA RPB & 75 & 0.196 & & & \\ \hline PHA RPB & 214 & 0.274 & ES RA & 148 & 0.174 & ES LA & 81 & 0.206 & & & \\ \hline PHF LPB & 214 & 0.275 & PHF LBP & 205 & 0.212 & UNS DIS & 109 & 0.254 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Events that characterize the single-run multi-modal PO curve, with corresponding values of the displacement of the control node and related PGA. Figure 13: Single-run multi-modal modal POA: (**a–d**) deformed shapes; (**e–h**) pushover curves: (**a–e**) dir x+ phase shift. 0\({}^{\circ}\); (**b–f**) dir x- phase shift. 0\({}^{\circ}\); (**c–g**) separation, phase shift. +180; (**d–h**) approaching, phases shift. +180. In Figure 13g, the pushover curve for the modal expansion of the load vector according to the combination (f1) - \(\alpha\)(f3) that produces a separation of the two parts of the deck is represented, reporting on the vertical axis of the base shear obtained as the difference of the absolute values of the two parts of the load vector. As above-mentioned, in order to link the level of PGA to the displacement of the structure, the period of the SDOF equivalent system pertaining to the first mode only and the displacement of the corresponding control node are considered. The behavior is characterized (cfr. Table 4) by the exit of the supports of the RELC-LEDIS from its seat for a displacement due to seismic action of 54 mm (corresponding to a total displacement of d\({}_{\rm(f1)\mathchar 45\alpha(f3),1}=12\) mm and a PGA\({}_{\rm(f1)\mathchar 45\alpha(f3),1}=0.091\) g). This is followed by the exit of the supports at the right abutment for a seismic displacement of 104 mm (total displacement d\({}_{\rm(f1)\mathchar 45\alpha(f3),2}=63\) mm and PGA\({}_{\rm(f1)\mathchar 45\alpha(f3),2}=0.176\) g). Afterward, the plastic hinge at the base of the right pier is activated for a seismic displacement of 117 mm and PGA\({}_{\rm(f1)\mathchar 45\alpha(f3),3}=0.196\) g, followed by the exit of the supports at the left abutment for a seismic displacement of 123 mm and PGA\({}_{\rm(f1)\mathchar 45\alpha(f3),3}=0.206\) g. The collapse is due to the unseating of the RELC-LEDIS that occurs for a seismic displacement of 151 mm and PGA\({}_{\rm(f1)\mathchar 45\alpha(f3),c}=0.254\) g. The analysis was continued until a plastic hinge was activated at the base of the left pier, and the failure of the plastic hinge at the base of the right pier occurred for a seismic displacement of 217 mm and PGA = 0.347 g.  
The pushover curve for the modal expansion of the load vector according to the combination (f1) + \(\alpha\) (f3) that produces an approach of the two parts of the deck is represented in Figure 13h. The curve between the base shear (obtained as the difference of the absolute values of the base shear for each mode) and the displacement of the right end of the drop in span (control node) is almost linear, thus the period of the SDOF equivalent system is taken as that of the 1 st mode.