## 5 Results  
To investigate the effects of the SSI on the pier behavior under seismic conditions, the results of the two FE models presented in Section 4.1 (where the pier was considered fixed to the base--and then the SSI was neglected) and Section 4.2 (where the pier was considered founded on a deformable soil--and the SSI was considered) have been compared. Model results are obtained in terms of:  
* Natural periods under free vibrations and system period under seismic conditions, the latter computed as the period corresponding to the maximum value of the ratio between the pseudo-acceleration response spectrum with 5% damping at the top of the pier and the one at the base of the pier [47];
* Relative horizontal displacement (U\({}_{\rm rel}\)) time-histories, evaluated as the difference between the absolute horizontal displacement at the top (U\({}_{\rm top}\)) and base (U\({}_{\rm bas}\)) of the pier;
* Rotation time-history of the pier;
* Moment and shear time-histories at the pier base. As for the natural periods of the pier, under free vibrations, T\({}_{\rm free\_vib}\), and system period under seismic conditions, T\({}_{\rm seismic}\), the presence of a deformable soil produces a value of the T\({}_{\rm free\_vib,SSI}=0.29\) s, which is about double the one obtained under the hypothesis of rigid soil (T\({}_{\rm free\_vib,FIX}=0.148\) s). This value agrees with the simplified formulations by [PERSON] et al. [24] and the results of the parametric studies by [PERSON] et al. [1]. Similar results have been obtained under seismic conditions: considering the SSI produces a value of the T\({}_{\rm seismic,SSI}=0.37\) s, which is about 2.5 times the one obtained neglecting the SSI (T\({}_{\rm seismic,FIX}=0.148\) s). This agrees with widely-known behaviors associated with the SSI effects, which elongate the natural period of the considered structure [10; 11; 12] and has an evident effect even for a stocky structure founded on a relatively rigid soil. Figure 7 shows the comparison between the relative horizontal displacement time-histories obtained by the two models, for the seismic input n. 1 (F\({}_{\rm a}=1.5\)), U\({}_{\rm rel,1.5}\). It is apparent that the presence of the SSI leads to a higher deformability of the system and, accordingly, the observed displacements are larger. Assuming the presence of a deformable pier-base, the maximum value of the relative horizontal displacement of the pier is U\({}_{\rm rel,1.5}=0.027\) m, about 3.2 times higher than the one computed considering the fixed-base pier (see Table 3). The residual horizontal displacement at the top of the pier is U\({}_{\rm res,1.5}=0.0024\) m considering the SSI, while it is null in the fixed-base case (Table 3). \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Symbol** & **Clayey F**. & **Marly F**. & **nc.** & **Unit** \\ \hline Constit. model & - & HS-small & Mohr-Coulomb & Linear Elastic & - \\ Dry unit weight & \(\gamma_{\rm s}\) & 17 & 20 & 25/22 & kN/m\({}^{3}\) \\ Sat unit weight & \(\gamma_{\rm sat}\) & 20 & 22 & & kN/m\({}^{3}\) \\ Mod. sec & \(E_{50}^{ref}\) & 20 & - & & MPa \\ Mod. tan & \(E_{\mathit{o}el}^{ref}\) & 20 & - & & MPa \\ Mod. unl/rel & \(E_{\mathit{u}r}^{ref}\) & 100 & - & & MPa \\ Young Modulus & \(E^{\prime}\) & - & 1350 & 30,000/25,000 & MPa \\ Cohesion & \(c^{\prime}\) & 20 & 50 & - & kN/m\({}^{3}\) \\ Friction angle & \(\phi^{\prime}\) & 25 & 28 & - & \(\circ\) \\ Small-strain stiff. & \(G^{\prime\prime\prime}\)\(\mathit{o}\) & 150 & - & - & MPa \\ Shear strain level & \(\gamma_{0,7}\) & 0.1 \(\times\) 10\({}^{-3}\) & - & - & - \\ Poisson coeff. & \(\
u^{\prime}\) & 0.25 & 0.35 & 0.2 & - \\ Damp. Rayleigh & \(\delta\) & 3\% & 5\% & 5\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the adopted soil properties.