# Interaction of Segmental Tunnel Liniings and Dip-Slip Faults--Tabriz Subway Tunnels
## 3 Materials and Methods
### Three-Dimensional Numerical Simulation  
Using FLAC\({}^{\text{3D}}\) finite difference software, the interaction between the segmental lining and fault motion was modeled. The numerical analysis was carried out in several stages. As a first step, the model reached initial equilibrium with in situ stresses. In the second step, the tunnel was excavated, and the segmental linings were installed. Lastly, dip-slip fault displacements were simulated. The hanging wall boundaries moved according to the type of dip-slip fault (normal or reverse) with a certain slope. At the same time, the football remained fixed. It was necessary to apply a specific velocity to the model hanging wall in order to achieve the desired fault displacements (Figure 3). The analysis results depend partly on mesh density and model dimensions. Therefore, after analyzing the sensitivity of the mesh, a series of parametric studies were conducted to evaluate the minimum dimensions of the model, resulting in Equations (2)-(4) [39]. Having a powerful computer enabled this study to construct a model with a minimum dimension that exceeded the values of these equations. The fault location was selected at the center of the model in order to minimize boundary effects. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameters** & **Symbol** & **Unit** & **Value** & **Value** & **Value** & **Value** & **Value** \\ \hline Soil type & L & - & Filling material (SM) & CL–ML & GM–GC–GW & CL & GM–GP & SM \\ Depth & D & m & 0–1.5 & 1.5–7 & 7–17 & 17–20 & 20–26 & 26–35 \\ Dry density & \(\rho\) & g/cm\({}^{3}\) & 1.65 & 1.775 & 1.775 & 1.775 & 1.850 & 1.825 \\ Cohesion & C & kPa & 0 & 11 & 6 & 25 & 5 & 5 \\ Angle of internal & \(\varphi\) & Degree & 27 & 24 & 32 & 23 & 33 & 32 \\ Triaxial loading secant & E\({}_{50}^{\text{ref}}\) & MPa & 10 & 25 & 47.5 & 35 & 65 & 55 \\ Young’s modulus & \(\text{F}_{\text{ur}}^{\text{ref}}\) & MPa & 30 & 75 & 150 & 105 & 195 & 165 \\ Oedometric loading & \(\text{F}_{\text{ood}}^{\text{ref}}\) & MPa & 12 & 30 & 60 & 42 & 78 & 66 \\ Young’s modulus & \(\text{F}_{\text{ood}}^{\text{ref}}\) & \(\text{M}\text{e}_{\text{ood}}\) & \(\text{e}_{\text{ood}}\) & 0.593 & 0.47 & 0.6 & 0.455 & 0.47 \\ Coefficient of earth & K\({}_{0}\) & - & 0.54 & 0.593 & 0.47 & 0.6 & 0.455 & 0.47 \\ Poisson’s ratio & \(\
u\) & - & 0.33 & 0.39 & 0.32 & 0.39 & 0.33 & 0.32 \\ Failure ratio & R\({}_{\text{f}}\) & - & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\ Janbu-type parameter & m & - & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\ Reference mean & \(\text{P}_{\text{ref}}\) & kPa & 100 & 100 & 100 & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geotechnical parameters of the critical section [36]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Unit** & **Value** \\ \hline \multicolumn{3}{c}{Properties of the segmental liner} \\ \hline Young’s modulus & E & GPa & 30 \\ Density & \(\rho\) & kg/m\({}^{3}\) & 2500 \\ External diameter & D & m & 9.5 \\ Thickness & t & cm & 35 \\ Poisson’s ratio & \(\
u\) & - & 0.2 \\ Width of the segment ring & W & m & 1.5 \\ \hline \multicolumn{3}{c}{Properties of the grouting layer (28-day)} \\ \hline Young’s modulus & E & MPa & 160 \\ Poisson’s ratio & \(\
u\) & - & 0.22 \\ Density & \(\rho\) & kg/m\({}^{3}\) & 2000 \\ Grouting layer thickness & t & cm & 15.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the segmental tunnel lining and grout of the Tabriz Subway Line 2 [38]. (H + 3D), for the model length (3) (3H), for the model width (4) where H is the tunnel axis depth, and D is the tunnel diameter. An appropriate constitutive model for the soil is crucial for the numerical modeling of a tunneling process and, consequently, for the correct prediction of lining loads. It is important to note that constant stiffness would not be an appropriate method of approximating surface settlements for all levels of shear strain [40,41].