## 4 Case Study: The Grottalunga Viaduct
### Preliminary Discussion for Developing Modal Dynamic Analysis and POA in Longitudinal Direction
#### 4.3.3 Single-Mode Pushover Analysis  
Figure 12a-d shows the deformed shape obtained with a load pattern consistent with a single-mode shape, while Figure 12e-h shows the corresponding pushover curves and the bilinear idealized curves. Table 3 gives, for each POA, the list of events with the corresponding values of the control node displacement and the corresponding level of PGA. Assuming a load pattern according to the 1 st mode in the +x direction (f1\({}_{+}\)), only the right pier is significantly loaded (see Figure 12a), and thus the control node is taken at the left end of the deck; in Figure 12e, the pushover curve is depicted with the bi-linear equivalent curve corresponding to an equivalent SDOF with period T1\({}_{+}\)* = 1.71 s. A linear behavior is shown until the supports on the left end of the drop-in beam exit the bearing seat, and the friction starts to transfer part of the seismic load to the left pier. This phenomenon occurs for a seismic displacement of 101 mm, corresponding to a total displacement of d\({}_{+1,1}\) = 59 mm (Figure 12a and Table 3) and an associated PGA\({}_{+1,1}\) = 0.169 g. This event is followed by the same at the right abutment, which exits the seat for a seismic displacement of 105 mm (total displacement d\({}_{+1,2}\) = 63 mm and PGA\({}_{+1,2}\) = 0.176 g). The plastic hinge at the pier base is activated for a seismic displacement of 115 mm (d\({}_{+1,3}\) = 74 mm, PGA\({}_{+1,3}\) = 0.194 g). Unseating (downfall) at the left end of the drop-in occurs when the seismic displacement reaches 150 mm (d\({}_{+1,c}\) = 108 mm, PGA\({}_{+1,c}\) = 0.251 g), while the failure of the plastic hinge at the pier base occurs when the seismic displacement is 216 mm for a total displacement of d\({}_{+1,\text{FPH}}\) = 164 mm and PGA\({}_{+1,\text{FPH}}\) = 0.362 g.  
Assuming a load pattern according to the 1 st mode in the opposite direction (\(-\)x) that is (f1\({}_{-}\)), the pushover curve, depicted in Figure 12b, is characterized by a trend with sections characterized by alternating increases and decreases in stiffness and a period of the equivalent SDOF of T1\(-\)* = 1.91 s. The first event is the activation of the plastic hinge at the base of the right pier for a seismic displacement of 69 mm (corresponding to a total displacement of d\({}_{1-,1}\) = \(-\)110 mm and PGA\({}_{1-,1}\) = 0.09 g); afterward, the deck at the right abutment exits, and the seat friction starts for a seismic displacement of \(-\)104 mm (d\({}_{1-,2}\) = \(-\)146 mm and PGA\({}_{1-,2}\) = 0.143 g. The closure of the gap between the deck and the left end of the drop-in span (rounding) occurs for a seismic displacement of 152 mm (d\({}_{1-,\text{FOU}}\) = \(-\)193 mm and PGA\({}_{1-,\text{FOU}}\) = 0.208 g). Lastly, the plastic hinge at the base of the right pier attains the ultimate rotation for a seismic displacement of \(-\)173 mm, \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Mode** & **1+** & \multicolumn{4}{c}{**1**–} & \multicolumn{4}{c}{**3+**} & \multicolumn{4}{c}{**3**–} \\ \hline
**Event** & \begin{tabular}{c} **Displ** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **PGA** \\ **(g)** \\ \end{tabular} & **Event** & \begin{tabular}{c} **Displ** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **PGA** \\ **(g)** \\ \end{tabular} & **Event** & \begin{tabular}{c} **Displ** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **PGA** \\ **(g)** \\ \end{tabular} & **Event** & \
\begin{tabular}{c} **Displ** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **PGA** \\ **(g)** \\ \end{tabular} \\ \hline ES DIS & 59 & 0.169 & PHA RPB & \(-111\) & 0.144 & PHA LPB & 110 & 0.097 & ES DIS & \(-59\) & 0.170 \\ \hline ES RA & 63 & 0.176 & ES LA & \(-146\) & 0.218 & ES LA & 148 & 0.152 & ES LA & \(-63\) & 0.176 \\ \hline PHA RPB & 74 & 0.194 & POU DIS & \(-193\) & 0.316 & POU DIS & 196 & 0.221 & PHA LPB & \(-76\) & 0.199 \\ \hline UNS DIS & 108 & 0.252 & PHF RPB & \(-214\) & 0.360 & PHF LPB & 217 & 0.251 & UNS DIS & \(-108\) & 0.251 \\ \hline \hline \end{tabular}
\end{table}