## 4 Case Study: The Grottalunga Viaduct
### Non-Linear Time History Analysis (NLTHA)  
With the aim of evaluating the effectiveness of MDA and the pushover procedures described above, the results of non-linear time history analysis for the design value of PGA\({}_{\text{d}}\) = 0.33 g are shown. Three accelerograms compatible with the elastic response spectrum for the PGA\({}_{\text{d}}\) value are generated according to the procedure described in [40], and they were made to act separately in both transverse and longitudinal directions. In Figure 14, the transverse displacement along the longitudinal bridge axes obtained by single-run single-mode pushover and MPA are compared with the results of NLTHA. Preliminarily, it can be pointed out that the results of the maximum displacement at the midspan and the piers are slightly larger (roughly +6%) than the displacements caused by MDA, while since in MDA the abutments are restrained, the analysis method is not able to assess the actual response. Moreover, the figure shows that both the POA procedure predicts the same displacement of the control node (x = 101 m), equal to the maximum displacement obtained with the NLTHA, which must be taken as the design displacement in the case of using only three accelerograms. By contrast, on the piers (abscissa x = 51 m, and x = 171 m), and especially at the ends, MPA overestimates the displacement obtained by NLTHA, which is assessed with more accuracy by the simpler single-run analysis with a force profile proportional to the second mode only. With reference to the response in the longitudinal direction, in Figure 15, the time history of the displacement obtained for the three NLTHA is compared with the displacement limit that produces the bearing exit from the seat, the unseating of all the deck ends, and the bounding of the deck with the abutment. The comparison with the results provided by MDA proves that linear analysis provides an underestimation of 27% of the maximum expected displacement. Moreover, the displacement estimation provided by the combination of the displacement obtained in single-mode (SM) pushover for 1 st mode and 3 rd mode with the hypothesis of synchronous motion is reported together with the results obtained by using single-run pushover analyses with load profiles obtained by the combination of first mode and third mode (PO SINCR MM), again with the hypothesis of synchronous motion. The results show that the latter is the procedure able to assess the actual displacement since it can consider the interaction between the different parts of the bridge. Figure 14: Comparison of assessment of transverse displacement along the bridge axis. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **PH. SHIFT 0\({}^{\circ}\) + x** & **PH. SHIFT 0\({}^{\circ}\) – x** & **PH. SHIFT 180\({}^{\circ}\) APRR** & **PH. SHIFT 180\({}^{\circ}\) SEP** & **PH. SHIFT 180\({}^{\circ}\) SEP** & **PH. SHIFT 180\({}^{\circ}\)** & **SP.** \\ \hline
**SRSS** & **CQC** & **SRSS** & **CQC** & **SRSS** & **CQC** & **SRSS** & **CQC** \\ \hline LEFT ABUT. (mm) & 230 & 235 & 202 & 202 & 199 & 199 & 233 & 233 \\ \hline RIGHT ABUT. (mm) & 214 & 250 & 190 & 193 & 199 & 190 & 206 & 126 \\ \hline RELC-LEDIS (mm) & 243 & 171 & 241 & 216 & 268 & 293 & 213 & 237 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Assessment of seismic displacement by MPA. In Figure 16, the same comparison is shown for the relative distance of RELC-LEDIS due to the relative displacement, considering that the initial gap between the two ends after the effect of the gravitational load is 150 mm. The comparison with the results of MDA shown in Table 2, taking into account that the initial gap between the RELC-LEDIS is 150 mm; shows that only appropriate modal combination rules, e.g., DDC [28] and M&J [24], can provide a rough estimate of the actual response. Regarding POA methods, the results confirm that only PO SINCR MM can provide a reliable assessment of the response.