## 2 Materials and Methods
### The Structure and Analysis Models
The interaction between the FEM model of the structure and the free-field soil layers is accounted for through the interaction nodes, present on the surface of the near-field soil volume and highlighted in Figure 6. The strong nonlinear behavior of the near-field soil has a major influence on the behavior of the pier. In line with the typical SSI approach [10; 13; 53; 54; 55; 14], the site is modelled as a multitude of horizontal layers on top of a uniform viscoelastic half-space. The solution is computed in the complex frequency domain, so only linear or linearized behavior is possible. Thus, the nonlinear behavior of the soil is taken into account through the Seed-Idriss equivalent linear model [56], as follows: the increase in displacement generates a reduction in stiffness and an increase in damping; each type of soil has multiple behavior curves that vary by depth. Figure 8 contains the Seed-Idriss behavior curves for sand and gravel, as presented in ASCE 4-16 [50]. Figure 7: Finite element model in ACS SASSI: (**a**) structural model; (**b**) near-field soil volume; (**c**) complete model. The usual vertical-propagating wave field is used as follows: SV-waves for the longitudinal direction (X), SH-waves for the transverse direction (Y) and P-waves for the vertical direction (Z). The seismic waves propagate through the free-field soil layers until they reach the surface, which may produce significant changes in the ground motion that actually loads the structure. The spatial variation of ground motion during a seismic event is influenced by the following 4 main factors, which are also shown in Figure 9[57]:  
* Attenuation effect, due to varying distances from the seismic fault;
* Spatial incoherence effect, which is divided in the following 3 categories:  
* wave passage effect, which refers to the systematical time delays due to the horizontal component of wave incidence;
* ray-path incoherence effect, which refers to variations in the wave field due to reflections/refractions of the wave in discontinuities or complexities along the ray path;
* extended source effect, which refers to the mixing of seismic waves from different parts of the fault, for sites close to large earthquakes;  
The spatial variation of ground motion or the ground motion incoherency is defined by the one-dimensional coherence function from a homogenous and isotropic Gaussian stochastic field [58]. [PERSON] has generated empirical plane-wave incoherency models for varying soil conditions and foundation types, from seismological data recorded in multiple dense arrays around the world. Figure 9: Sources of spatial variation for seismic ground motion: (**a**) attenuation effect; (**b**) wave passage effect; (**c**) ray-path incoherence effect; (**d**) extended source effect [57]. This study uses the 2006 [PERSON] model [59], which was generated for soil and rock sites and for shallow and deep foundations. The coherence function is:  
\[\gamma_{pw}(f,\xi)=\left[1+\left(\frac{fTanh(a_{3}\xi)}{a_{1}f_{c}(\xi)}\right)^ {n1}\right]^{-1/2}\left[1+\left(\frac{fTanh(a_{3}\xi)}{a_{2}f_{c}(\xi)}\right)^{ n2(\xi)}\right]^{-1/2} \tag{1}\]  
where \(f\) is the frequency (Hz) and \(\xi\) is the separation distance (m). The other coefficients are detailed in Figure 2:0  
Equation (1) leads to the graph shown in Figure 2:1 The coherency for both horizontal and vertical components decreases with larger separation distances and larger frequencies. Additionally, at frequencies of 40 Hz or higher, the coherency has practically dropped almost to 0.