## 6 Time-History Analysis  
Seven nonlinear time history analyses were carried out using the seven selected accelerograms, averaging--for the calculation of the stresses--the unfavourable results on the conducted time-histories as required by the Italian NTC 2018. In addition to the structural safety check, it is of interest to this study to understand which piles were most affected by the dynamics of the seismic event.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**q**} & \multirow{2}{*}{**M\({}_{\text{ed}}\)**} & \multirow{2}{*}{**M\({}_{\text{ultimate,d}}\)**} & **M\({}_{\text{Rd}}\)** & **ULS Safety Check** & **M\({}_{\text{ed}}\)/M\({}_{\text{rd}}\) (Sectional** \\  & & & **(0.75 \(\times\) M\({}_{\text{Ultimate,d}}\))** & **(Mrd \(>\) Med)** & **Safety Factors)** \\ \hline
5 & 1655 & 2445 & 1834 & OK & 0.90 \\
4.5 & 1815 & 2445 & 1834 & OK & 0.99 \\
4 & 1990 & 2445 & 1834 & **NOT VERIFIED** & **1.08** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Trial and error iterative safety check. Limit-seismic input for response spectrum scaled for q = 4.5.  
Figure 15: Example of displacement of northern piles 13 (**left**) and 14 (**right**) at step 54 for pushover curve n. 1 (main period).  
To properly understand this aspect, the damage to the individual piers of the viaduct at the end of each time history in the two vertical loading scenarios (A without traffic load, B with traffic load included) was analyzed according to the FEMA plastic hinges criterion. Regarding the ultimate limit state (ULS) check, the bending moments were extracted from the FEM model calculation, averaged from the time histories and compared with the resistant moments according to FEMA criterion.  
There was substantial compliance with the life safety limit state in the NSS approach (Table 9). By contrast, due to the much more amplified nature of some accelerograms based on a site-specific (SS) spectral matching, the damage reported for the viaduct was more significant (Table 9). Concerning the force verification, however, it is noted that in the direction of the weak axis (X direction), the verifications were satisfied for both load scenarios and both NSS and SS approaches. On the other hand, around the strong axis, some verifications were deemed non-compliant, with many elements remaining unverified and the slightly worse scenario B.  
Regarding the site-specific (SS) and spectrum-compatible accelerogram experimental approach (scenario B), due to the nature of the amplified accelerograms based on a site-specific spectral matching as obtained above, the damage reported for the viaduct is highly significant. Across all performed time histories, in fact, the plastic hinges almost systematically indicate serious damage only in the Y direction, with slightly better behaviour in the scenario A load-condition.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{4}{c}{**NLDA \(\rightarrow\) NSS**} & \multicolumn{4}{c}{**NLDA \(\rightarrow\) SS**} \\ \hline  & \multicolumn{2}{c}{**X-DIR**} & \multicolumn{2}{c}{**Y-DIR**} & \multicolumn{2}{c}{**X-DIR**} & \multicolumn{2}{c}{**Y-DIR**} \\ \hline
**Pile n\({}^{\circ}\)** & **Scenario A** & **Scenario B** & **Scenario A** & **Scenario B** & **Scenario A** & **Scenario B** & **Scenario A** & **Scenario B** \\ \hline
1 & 20.54 & 1.89 & 215.80 & 5.79 & 3.62 & 3.84 & 66.98 & 66.98 \\ \hline
2 & 2.13 & 3.46 & 4.45 & 9.25 & 1.00 & 1.15 & 1.41 & 1.13 \\ \hline
3 &