## 4 Numerical Modeling
### PLAXIS3D Model  
Full dynamic numerical analyses were carried out with the PLAXIS3D FE software [44]. A 3D model--including the caisson, the pier, and the soil surrounding/underlying the caisson--was built. Figure 5 shows the geometry and mesh of the model. The considered soil volume is 100 m \(\times\) 100 m wide, following [PERSON] et al. [45] who suggest having a ratio between the horizontal dimension vs. the caisson diameter higher than 10, to have the lateral boundary sufficiently far from the structure and avoiding negative interferences due to reflection/refraction effects. To further reduce the effects produced by the presence of the boundaries, free field conditions were assumed at the lateral boundaries. They simulate the propagation of waves towards the far-field by minimizing the reflection at the lateral boundary. The bottom of the soil domain was fixed at 50 m and the hypothesis of a compliant base was assumed. This type of boundary condition simulates the propagation of seismic waves in the soil layers by minimizing their reflection at the bottom boundary. The entire soil domain was subdivided into 28,415 tetrahedral cluster elements with 10 nodes each. The pier was modeled as a Timoshenko beam element of 12 m height and rectangular section of 6 m \(\times\) 2 m. The caisson was modeled as a cylinder element of 8.3 m in diameter and 9 m in depth (Figure 6). Both these structural elements (pier and caisson) were modeled as linear visco-elastic elements, with a 5% damping ratio. The deck was modeled as a lumped mass of 4880 kN. Rigid interface elements between the foundation and soil were assumed. It should be noted that no interaction with surrounding bridge piers has been considered in the simulation to keep the model as simple as possible, yet realistic, to highlight the relevance of SSI effects. The goal of the simulation, indeed, is not that of assessing the safety conditions of the real structure, but to use a specific case study to explore the relevance of including the soil deformability in the dynamic analyses. Figure 5: Three-dimensional FE model of the soil volume surrounding and underlining the examined pier. The stratigraphy was defined based on the detected depth of the Marl formation (at about 9 m from the surface). The stratigraphical contact between the two layers was assumed horizontal in the model, for the sake of simplicity and due to the limited volume considered. As shown in Figure 5, the soil volume was subdivided into two layers with different mechanical characteristics. The Marl formation was modeled using the Mohr-Coulomb elastoplastic constitutive model, and the dissipative behavior was considered using a damping ratio of 5%, as conducted, for example, by [23]. For the upper clayey layer, the constitutive model, called the Hardening Soil model, with small-strain stiffness (_HS small_) was adopted [44]. In contrast to a more traditional Hardening Soil model, the _HS small_ shows hysteresis in cyclic loading. The amount of hysteresis depends on the magnitude of the corresponding strain amplitude. When applied in dynamics calculations, the hysteretic behavior of the _HS small_ leads to damping. The amount of hysteretic damping depends on the applied load amplitude and corresponding strain amplitudes. The maximum amount of hysteretic damping obtained with the _HS small_ depends on the ratio of \(G^{ref}_{0}\), and \(G^{ref}_{ur}=E^{ref}_{ur}/2(1+v^{\prime}{}_{ur})\)). A larger ratio leads to a larger maximum amount of hysteretic damping. Table 2 shows the adopted soil properties, where the soil unit weight, cohesion, and friction angle derive from the technical reports of the original project, while the static and dynamic stiffness values (\(E^{ref}_{50}\); \(E^{ref}_{od}\); \(E^{ref}_{ur}\), \(E^{\prime}\), \(G^{ref}_{0}\), \(\gamma_{0.7}\), \(v^{\prime}\)) were adopted from literature values, considering documented cases of similar formations [46].