# Interaction of Segmental Tunnel Liniings and Dip-Slip Faults--Tabriz Subway Tunnels
## 3 Materials and Methods
### Model Validation  
Using experimental results, numerical simulations can be validated for their ability to estimate the interaction effects between tunnels and faults. In this study, the numerical modeling was validated by the experimental results of [2; 12]. Using centrifuge tests, they investigated the behavior of segmental tunnels under normal faults. In their tests, considering a gravity acceleration of 50\(\times\)\(g\) (centrifuge scale coefficient of 50), the prototype model's dimensions were 20, 25, and 35 m (test dimensions: 0.4, 0.5, and 0.7 m). The diameter of the tunnel and the thickness of the segments were 5.9 m and 0.35 m, respectively. Six segments with a width of 1.15 m were present in each ring. The soil was composed of Firoozkuh sand No. 161 with a water content of 5% and a relative density of 50%. A more detailed description can be found in [2; 12]. Table 3 lists the soil and segmental lining properties used in the centrifuge test. Figure 4: \(K_{\rm axi}\), \(K_{\rm rad}\), \(K_{\rm rot}\) stiffness in the axial, radial, and rotational directions of a segmental joint. Six centrifuge tests were conducted considering fault angles of 60 and 75 relative to the horizon and overburden/diameter ratios (h/D) of 0.75, 1, and 1.2. In the centrifuge tests, the footwall was fixed, and the movable hanging wall was moved downwards using the jack hydraulic force for normal faulting. According to Figure 5, the validation model consists of 1,002,500 zones with a height of 33 m, a length of 60 m, and a width of 110 m. Figure 5 illustrates the mesh and groups of the numerical model. Table 4 provides stiffness values for the tunnel lining segments and rings based on the springs and stiffnesses discussed in Section 3.2. Figure 8: The cross-sectional deformation of the segmental tunnel lining under normal faulting: (**a**) the fault slope angle of \(60^{\circ}\); (**b**) the fault slope angle of \(75^{\circ}\). Figure 11: Soil surface settlement for different overburdens and fault angles in numerical validation models. Figure 12: PGD in the numerical validation and the centrifuge test at the fault angle of 75\({}^{\circ}\). Figure 10: Vertical displacement diagram of the segmental lining at 60\({}^{\circ}\) and 75\({}^{\circ}\) fault slope angles. Figure 9: Vertical displacement diagram of the segmental lining under normal faulting: (**a**) the fault slope angle of 60\({}^{\circ}\); (**b**) the fault slope angle of 75\({}^{\circ}\). As shown in Figure 6a,b, deformations of the segmental lining were caused by normal faulting at angles of 60\({}^{\circ}\) and 75\({}^{\circ}\) with overburden (h)/diameter (D) ratios of 0.75, 1, and 1.2, respectively. Figure 7a,b shows the value of Le (the length affected by the fault in the lining) for both centrifugal tests presented by [2; 12] and the numerical validation model. There is a relatively strong correlation between the Le value in the centrifuge test and the numerical model. It is observed that Le decreases with increasing depth. In other words, the fault's effect on the tunnel is concentrated in the fault plane area, and it affects a smaller length of the tunnel lining as a result of increasing the tunnel's depth. Figure 8a,b illustrates the cross-sectional deformations of segmental linings under normal faulting at angles of 60\({}^{\circ}\) and 75\({}^{\circ}\) with different overburden/diameter ratios from the side and inside the tunnel. According to these figures, the longitudinal joint separation increases with increasing tunnel depth and fault angle. This rise in the 1.2D overburden ratio is quite evident, and includes more rings. A greater amount of separation occurs at an angle of 75\({}^{\circ}\). As a result, an increase in fault angle leads to an increase in the vertical component of the faulting force and an increase in segmental ring ovalization. A diagram of the segmental lining vertical displacement following faulting at 60\({}^{\circ}\) and 75\({}^{\circ}\) slope angles with varying overburden ratios is shown in Figure 9a,b.