## 3 Seismic Input, Load Scenarios and Analyses
### Nonlinear-Static Analysis (NLSA--Multimodal Pushover Analysis or MMPO)
As a first step, to better understand the evolution of the amplification phenomenon, the fundamental frequency of the stratigraphy can be estimated using the following formula (in agreement with Table 3):  
\[\frac{\sum_{i}^{n}h_{i}\cdot\beta_{i}}{4\cdot H^{2}}=7.79\ \mathrm{Hz}\]  
where \(h_{i}\) is the thickness of the \(i\)-layer, \(\beta_{i}\) is the shear velocity of the \(i\)-layer and \(H\) is the total thickness of the stratigraphic profile over the bedrock (173.4 m). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Input Parameters**} \\ \hline a\({}_{\mathrm{g}}\) & F\({}_{0}\) & T\({}_{\mathrm{c}}\)\({}^{*}\) & S\({}_{\mathrm{s}}\) & C\({}_{\mathrm{c}}\) & S\({}_{\mathrm{t}}\) & q \\ \hline
0.243 & 2.560 & 0.307 & 1.327 & 1.550 & 1.100 & 1.000 \\ \hline \multicolumn{6}{c}{**Output Parameters**} \\ \hline S & \(\eta\) & T\({}_{\mathrm{b}}\) & T\({}_{\mathrm{c}}\) & T\({}_{\mathrm{d}}\) & & \\ \hline
1.459 & 1.000 & 0.159 & 0.476 & 2.573 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Implemented parameters for the national spectrum in the design code. Benchmarks of the Terenzo site (ULS, Vr = 200 years, Soil type C, topography \(T_{2}\)). Figure 3: Illustration of the geotechnical stratigraphic profile near the Vizzana-Zampogna viaduct highlighted in [42]. To understand the post-elastic behaviour of soils, an equivalent linear analytical approach with solutions in the frequency domain was adopted. As clearly known, under the effect of dynamic stresses, soil tends to soften. This effect is expressed according to the hypothesis of a linear-equivalent soil model in terms of the maximum elastic shear stiffness, \(G_{max}\). As the ratio \(G/G_{max}\) decreases with the increase in cyclic seismic shear strain, an increase in the soil damping ratio is observed. Consequently, for the calibration of the decay of stiffnesses \(G/G_{max}\) and the progressive increase in damping (%) as a function of the displacements of the ground, reference will be made to the seismic micro-zonation study that frames the lithostrattigraphic column (mainly gravel-type) of Selva Bocchetto (see Figure 4, where the mean curves of [PERSON] and Idriss [44] are also built and depicted for each soil layer). The procedure for the derivation of a site-specific spectrum can be summarized as follows:  
1. calculation of the response spectra of the input selected accelerograms (seven in total);  
Figure 4: \(G/G\)max, damping ratio (%) and shear strength (kPa) curves according to Seed and Idriss (1970) (**left**); and the stratigraphic profile model implemented in DeepSoil software and relative shear wave velocity (m/s) (**right**). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Depth [m]** & \multicolumn{2}{c}{**Shear Wave Velocity**} & **Thickness of Layer** \\ \hline
**From** & **To** & **(\(\beta\)) [m/s]** & **(\(H\)) [m]** \\ \hline
0 & 0.5 & 78 & 0.5 \\ \hline
0.5 & 1 & 138 & 0.5 \\ \hline
1 & 4.2 & 224 & 3.2 \\ \hline
4.2 & 6.4 & 296 & 2.2 \\ \hline
6.4 & 8.4 & 362 & 2 \\ \hline
8.4 & 173.4 & 810 & 165 \\ \hline
173.4 & infinite & 1645 & infinite \\ \hline \hline \end{tabular}
\end{table} 
Table 3: HVSR results at Selva Bocchetto, near the case study viaduct, in terms of shear wave velocities and depths. application of the local seismic response analysis (soil layer response) through the modelling carried out (Figure 4);
* calculation of the response spectra of the output accelerograms related to the surface soil layer response;
* calculation of the spectral amplification factors for discrete points (as the period \(T\) varies) relative to the same accelerogram, \(\frac{\delta_{S_{Surface}}}{\delta_{B_{elastic}}}\);
* calculation of the mean of the amplification factors (for each single period T) relative to the seven accelerograms;
* calculation of the code response spectrum on rock (cat.