## 2 Numerical Models
### Numerical Model for the Circular Tunnel  
Firstly, a numerical model for circular tunnels was developed using a finite difference program (FLAC\({}^{\text{3D}}\)) [24]. The purpose was to investigate the behavior of circular tunnel linings under quasi-static loading and make a comparison with those obtained by an analytical solution.  
A 2D plane strain model was created (Figure 3). The soil mass is discretized into hexahedral zones. The tunnel lining is modeled using embedded liner elements, which are attached to the zone faces along the tunnel boundary with interfaces. Two interface conditions were considered: no slip between the soil and lining and full slip. The liner-zone interface stiffness (normal stiffness \(k_{n}\) and tangential stiffness \(k_{s}\)) is determined based on a given rule of thumb [24]. \(k_{n}\) and \(k_{s}\) are set equal to 100 times the equivalent stiffness of the stiffest neighboring zone [24] for the no-slip condition case. When considering the full-slip condition, \(k_{s}\) is assigned to be equal to zero.  
The mesh is composed of a single layer of zones in the y-direction, and the element's dimension increases as one moves away from the tunnel. It reaches the maximum size of 2.6 \(\times\) 1 m at the model boundary (Figure 3). The boundary conditions of the numerical model are 120 m in the x-direction and 40 m in the z-direction. It consists of approximately  
Figure 3: Geometry and quasi-static loading conditions for the circular tunnel model [25].  
Figure 2: Circular tunnel with the same utilization space area, distances in millimeters.  
4800 zones and 9802 nodes. The bottom of the model was blocked in all directions, and the vertical sides were fixed in the horizontal one.  
In this study, similar to the research work of [PERSON] et al. [15], [PERSON] et al. [25], and [PERSON] [26], ovaling deformations due to the seismic loading are imposed as inverted triangular displacements, along with the model lateral boundaries. Uniform horizontal displacements are applied along the top boundary (Figure 3). The magnitude of the prescribed displacements assigned at the top of the model is dependent on the maximum shear strain \(\gamma_{max}\), estimated based on the horizontal acceleration \(a_{H}\). The bottom of the model is restrained in all directions.  
Before applying the ovaling deformation due to seismic loading, a steady state of the excavated tunnel under static conditions was established. When the tunneling process is performed in a 2D plane strain model, it is assumed that ground displacements surrounding the tunnel boundary prior to the lining installation are simulated by using the convergence confinement method with a relaxation factor, \(\lambda_{d}\), equal to 0.3 [25,27]. The numerical simulation of the tunnel ovaling was therefore performed through the following steps:  
* Step 1: In situ state of stresses before tunnel construction.
* Step 2: Excavation of the tunnel and use of the convergence confinement process with a relaxation factor, \(\gamma_{d}\), of 0.3. The concrete lining is then activated on the tunnel's periphery.
* Step 3: Assigning ovaling deformations caused by the seismic loading on the model boundaries using the prescribed displacements previously mentioned.  
It should be noted that only incremental internal forces are presented in this study. They are determined by subtracting the total lining forces computed at the end of the static loading (step 2) from those determined at the end of the ovaling step (step 3).