# Seismic Performance of Modal Transfer Stations on Soft Clays
## 4 Numerical Model  
The seismic tunnel-building-bridge interaction was established through numerical models developed with the software FLAC3D v.6.0. FLAC3D is designed to simulate the mechanical behavior of three-dimensional continua under static or dynamic loading conditions, employing an explicit finite-difference scheme. The software is driven by strains, laws of motion, and constitutive equations, resulting in partial differential equations for specific geometries and properties. These equations, which are solved by the program, relate mechanical (stress) and kinematic (strain rate velocity) variables, allowing them to simulate complex geotechnical systems under specific boundary and initial conditions. The main advantage of FLAC3D regarding handling dynamic problems, such as seismic wave propagation, is that it incorporates inertial effects, which makes it well suited to simulate soil-structure interactions. The model has 166,263 solid elements and 239,548 nodes, representing the building and the soil. The minimum thickness of the elements was 2 m based on the recommendation provided by [PERSON] and [PERSON] [44] to adequately simulate wave transmission through the mesh. Similar element sizes have been employed in previous research, and good agreement has been found between FLAC3D and SHAKE models [8,22]. Wave reflection at the model base was avoided by using quiet boundary conditions, with the [PERSON] and [PERSON] [44] formulation. For the horizontal faces of the model, free-field boundaries available in FLAC3D were used to avoid energy reflection. Thus, the main grid's lateral boundaries are coupled to a free-field grid with the same properties as the lateral faces using viscous dashpots to simulate a quiet boundary, and the unbalanced forces from the free-field grid are applied to the main grid's boundary in such a way that plane waves traveling upward suffer no distortion at the boundary because the free-field grid provides similar forces to those in an infinite model [43]. The Mohr-Coulomb failure criterion was used to represent the stress-strain soil behavior, and the structure elements were simulated with an elastic relationship. The bridge superstructure was simulated with beam elements, and equivalent density and stiffness to account for inertial effects in the full-scale bridge, properly accounting for the rocking generated by the mass distribution [33]. The walls and slabs of the box-type foundation were simulated with shell elements and the concrete piles with pile elements, which are basically beam elements coupled with normal and tangential springs that are defined as a function of the surrounding soil parameters. The pile element parameter was defined following the procedure described by [PERSON] et al. [45], which allows the pile-soil to slip during ground shaking. Therefore, when a pile is loaded vertically or horizontally, a relative displacement between the pile and the host medium can occur. The lining of the tunnel and the box foundation of the bridge were simulated by SHELL elements. Figure 9a shows the numerical model considered for the seismic tunnel-building-bridge interaction analyses, with the assumed boundary conditions, and Figure 9b shows the corresponding structural elements. During the dynamic simulation, the soil elements develop hysteretic closed loops of load-unload cycles; nevertheless, the Mohr-Coulomb model is not capable of reproducing that behavior. Therefore, to simulate the soil nonlinearities, the hysteretic model Sig3, available in FLAC3D, was implemented. This model allows the simulation of stiffness degradation and generation of damping during cyclic stress-strain loops. Although it is not a constitutive model, it can be used to provide proper shear stiffness and damping according to the equivalent linear approach. It should be stated that, with this model, stress depends only on the deformation and not on the number of cycles, and the corresponding damping is given directly by the hysteresis loop during cyclic loading [43].