**5. Verification of the Method**

In order to verify the accuracy of the improved dynamic contact force method for simulating large sliding displacements, the movement response of a sliding block under external loads (as shown in Figure 5) was selected for analysis [25]. The block was a cube with an edge length of 1.0 m, and the sliding surface under the block was sufficiently large. The block and the bottom surface were made of the same linear elastic material, with an elastic modulus of 0.3 GPa, a Poisson ratio of 0.3, and a density of 300 kg/m3. The finite element mesh was composed of hexahedral elements with dimensions of 0.5 m × 0.5 m × 0.5 m. It was assumed that the gravity of the block, the cohesive force of the sliding surface, and the vibration deterioration effect of the sliding surface need not be considered. The static and dynamic friction coefficients of the sliding surface were approximately equal. In order to investigate the mutual transformation of different contact states, two calculation cases were designed, as shown in Table 1 (where *t* is the time).

**Figure 5.** Contact model of the elastic block.


**Table 1.** Design of calculation cases.

Taking the bottom center of the block as a monitoring point (as shown by the red dot in Figure 5), the velocity and displacement time histories of the block, calculated by the method developed in this paper and the analytical method, are plotted in Figures 6 and 7. In the two calculation cases, the numerical solutions of the velocity and displacement time histories of the block were basically consistent with the analytical solutions. The results show that under the above assumption, the improved dynamic contact force method is feasible, and is suitable for simulating different contact states of a contact system and the large sliding problem for contact interfaces.

**Figure 6.** Velocity time histories of the elastic block: (**a**) case 1; (**b**) case 2.

**Figure 7.** Displacement time histories of the elastic block: (**a**) case 1; (**b**) case 2.

#### **6. Engineering Case Study**

#### *6.1. Engineering Profiles and Calculation Model*

The water diversion project of Central Yunnan is a super-large inter-basin water transfer project, which takes water from the Shigu segment of the upper reaches of the Jinsha River to solve the water shortage problem in the Central Yunnan areas. The control

project for the main canal is the Xianglushan tunnel in Dali section I, with a total length of 63.43 km and a buried depth of more than 300 m. Many active fault zones exist along the line, most of which intersect with the line at medium or large angles. The basic earthquake intensity degree of the tunnel area is degree 8, and the peak acceleration of the horizontal ground motion with 10% exceedance probability in 50 years is 0.2–0.3 g. A tunnel section with a fault and a buried depth of 400 m was selected for the seismic calculation in this study. The rock mass is type IV, and the tunnel diameter is 9.8 m. The thickness of the lining support is 55 cm, with a C30 concrete structure.

Due to the great depth of the tunnel, the number of elements would be very large and the dynamic calculation would take a great deal of time if the 3D finite element model was built to the ground surface. Therefore, a depth of only 55 m was used at the top of the tunnel, to save calculation time. A normal fault with a thickness of 20 m and a dip angle of 60◦ was considered, with an inclination parallel to the tunnel axis. The finite element model was composed of 69,888 rock mass elements, 17,472 fault elements, and 8640 lining elements, all of which were eight-node hexahedrons, as shown in Figure 8. The maximum mesh size was within 5 m, which satisfies the accuracy requirement of the dynamic calculation. The initial contact node pairs were set between the hanging wall, the fault, and the footwall, respectively. It should be noted that the contact interface between the concrete lining and the fault or the rock mass was not considered.

**Figure 8.** Three-dimensional calculation model of the tunnel with a normal fault: (**a**) whole model; (**b**) lining.

Based on a comprehensive analysis of in situ stress test results in the tunnel area, the lateral pressure coefficients of the rock mass were taken as: *kX* = 1.2, *kY* = 0.74, *kZ* = 1.0. The values of the mechanical parameters of the tunnel materials are provided in Table 2. The cohesive force of the contact interface between the rock mass and the fault was taken as 0.35 MPa, and the static and dynamic friction coefficients were both taken as 0.53. The undetermined coefficients in Equation (4) were taken as: *α* = 0.883, *β* = 0.02, λ = 0.032, *P*<sup>0</sup> = 0.8, *ξ* = 5. As the research was limited by current test conditions, the values of these undetermined coefficients were mainly based on the test fitting curves in [19,23].


**Table 2.** Mechanical parameters of tunnel materials.
