*6.2. Calculation Conditions*

Based on the improved dynamic contact force method, a seismic response process simulation of the Xianglushan tunnel with a normal fault was performed, using the dynamic time history finite element analysis program for an underground cavern [26]. The same elastoplastic damage constitutive relation was applied to the rock mass and the concrete lining, and the Mohr–Coulomb yield function with a tensile cut-off limit was applied as the yield condition, expressed as

$$f^{\circ} = \sigma\_1 - \sigma\_3 N\_{\theta'} + 2(1 - D)c\sqrt{N\_{\theta'}}f^{\circ} = \sigma\_3 - \sigma^{\circ} \tag{23}$$

where *f* <sup>s</sup> = 0 and *f* <sup>t</sup> = 0 represent the shear and tensile yield conditions, respectively, and *σ*<sup>3</sup> and *σ*<sup>1</sup> are the maximum and minimum principal stresses, respectively. It was assumed that tensile stress is positive and compressive stress is negative. In addition, *c* is the cohesive force, *σ*<sup>t</sup> is the tensile strength and *N<sup>ϕ</sup>* is a parameter expressed as

$$N\_{\varphi} = \frac{1 + \sin \varphi}{1 - \sin \varphi} \tag{24}$$

where *ϕ* is the internal friction angle. *D* is the damage coefficient, which is expressed as

$$D = \sqrt{D\_1^2 + D\_2^2 + D\_3^2} \tag{25}$$

$$D\_i = 1 - \exp\left(-R\sqrt{\varepsilon\_i^p \varepsilon\_i^p}\right), \ i = 1, 2, 3\tag{26}$$

where *Di* is the damage coefficient of the *i*-th principal stress direction, *ε p <sup>i</sup>* is the cumulative plastic deviator strain of the *i*-th principal strain direction and *R* is a dimensionless constant.

A free field artificial boundary condition was applied at the four vertical boundaries, and a viscoelastic artificial boundary condition was applied at the top and bottom of the finite element model.

In this study, a representative Kobe NS wave was selected for input to the model in the form of acceleration time histories. A duration section of 15 s with high amplitude was adopted to shorten the calculation time. Then, the amplitude of the input wave was adjusted to 0.15 g to meet the requirements of seismic fortification intensity and the amplitude reduction of a deep tunnel. The processed acceleration time histories are shown in Figure 9, where only the x-direction seismic excitation is considered.

In addition, a contrasting calculation case was designed, with no contact node pairs between the rock mass and the fault. In other words, the rock mass and the fault were regarded as a continuous structure. Then, the seismic responses of the tunnel in the two calculation cases (considering or ignoring the dynamic interaction between the rock mass and the fault (with RFI or without RFI)) were analyzed comparatively. In order to monitor the displacement and stress time histories of the lining, three monitoring points were set at typical parts of the lining: the bottom arch, the haunch, and the top arch at the middle position of the fault. It should be noted that the tunnel excavation and the lining support calculations should be performed before the seismic action is applied.

**Figure 9.** Time history of x-direction acceleration of the input wave.

#### *6.3. Analysis of Calculation Results*

6.3.1. Relative Movement and Seismic Deterioration Analysis of the Contact Interface

In the seismic loading process of the system, the relative displacement and relative velocity between the rock mass and the fault leads to the seismic deterioration of the contact interface. A contact node pair close to the lining haunch was selected at the contact interface between the hanging wall and the fault, to monitor the relative movement characteristics of the rock mass and the fault. With the dynamic interaction included, the time histories for the relative x-direction movement of the contact interface between the hanging wall and the fault are plotted in Figure 10, and the time history of the vibration deterioration coefficient of the contact interface is plotted in Figure 11.

**Figure 10.** Time histories of the relative movement of the contact interface between the hanging wall and the fault: (**a**) displacement; (**b**) velocity.

**Figure 11.** Time history of the vibration deterioration coefficient of the contact interface.

We can see from Figure 10a that the relative displacement of the contact interface was very small at time 0–1.5 s, and it fluctuated violently and increased sharply at time 1.5–6.0 s. The maximum relative displacement was −5.14 cm, appearing at time 4.9 s. After time 6.0 s, the relative displacement basically remained at −4.40 cm. This indicates that the movements of the rock mass and the fault were not synchronous under a large seismic action, and an obvious dislocation displacement appeared between them. That is to say, the rock mass and the fault were in a sliding contact state after time 6.0 s. We can see from Figure 10b that the relative velocity of the contact interface presented similar fluctuating change laws, due to the fluctuation of the input wave: the relative velocity fluctuated violently at time 1.5–6.0 s and fluctuated gently after time 6.0 s.

We can see from Figure 11 that the vibration deterioration coefficient of the contact interface presented clear negative exponential attenuation laws over time: the coefficient decreased rapidly at time 0–1.5 s, then decreased to a certain extent at time 1.5–6.0 s, tending to be stable after time 6.0 s. This indicates that the seismic deterioration effect of the contact interface accumulated gradually over time, and its deterioration degree can be directly reflected in the value of the vibration deterioration coefficient.

#### 6.3.2. Displacement Analysis of the Lining

With the dynamic interaction between the rock mass and the fault considered or ignored, the time histories of the x-direction displacement of the lining monitoring points in the two calculation cases are plotted in Figure 12.

**Figure 12.** Time histories of the x-direction displacement of the monitoring points: (**a**) without RFI; (**b**) with RFI.

We can see from Figure 12 that the movement laws of the three typical parts were basically the same, and only the magnitudes at each point in time were somewhat different. On the whole, the displacement time histories of the bottom arch and the top arch were basically the same, while the displacement of the haunch along the x-direction was larger than that of the other parts. The main reason for this is that, influenced by the spatial structure of the tunnel, the seismic response of the haunch was more intense than that of the other parts under a transverse seismic wave. When the dynamic interaction between the rock mass and the fault was ignored, the maximum displacement of the haunch was −6.16 cm, appearing at time 3.0 s, and the maximum relative displacement between the haunch and the bottom arch was −0.33 cm. When the dynamic interaction was considered, the maximum displacement of the haunch was −6.31 cm, also appearing at time 3.0 s, and the maximum relative displacement between the haunch and the bottom arch was −0.64 cm. In the two calculation cases, the displacement responses of the lining are clearly different. The main reason for this is that an obvious transverse dislocation displacement appears between the rock mass and the fault under seismic action, which has an important influence on the seismic response of the lining monitoring section, especially the displacement of the haunch. This indicates that the transverse dislocation displacement between the rock mass and the fault leads to large relative deformations between different parts of the lining. Therefore, considering the dynamic interaction between the rock mass and the fault can more objectively reflect the displacement response characteristics of the tunnel structure.

According to the above analysis, the displacement of the lining haunch along the xdirection reached a maximum at time 3.0 s. With the dynamic interaction between the rock mass and the fault considered, the changing laws of the transverse maximum displacement of the haunch along the tunnel axis at time 3.0 s are plotted in Figure 13. Here, the minus sign indicates the direction.

**Figure 13.** Changing laws of the maximum displacement of the haunch along the tunnel axis.

We can see from Figure 13 that the maximum displacements of the haunch at different y coordinates of the footwall were basically the same. From the footwall to the fault, the displacement decreased sharply at the contact interface, and increased rapidly at the fault. From the fault to the hanging wall, the displacement appeared to change suddenly at the contact interface, decreasing slowly at the hanging wall. This indicates that the displacement of the lining was greatly affected by the fault dislocation, especially near the contact interface. Compared with the lining at the footwall, the fault was more likely to affect the lining at the hanging wall, which was characterized by a larger displacement.

#### 6.3.3. Stress and Damage Analysis of the Lining

In view of the fact that the tensile strength of concrete is much smaller than its compressive strength, only the changing laws of the tensile stress of the lining were analyzed in this study, which is reflected in the maximum principal stress. With the dynamic interaction between the rock mass and the fault considered or ignored, the time histories of the maximum principal stress of the lining monitoring points in the two calculation cases are plotted in Figure 14.

**Figure 14.** Time histories of the maximum principal stress of the monitoring points: (**a**) without RFI; (**b**) with RFI.

We can see from Figure 14 that the changing laws of the maximum principal stress of the three typical parts were basically the same: the stress increased slowly at time 0–1.5 s, then changed and increased rapidly at time 1.5–6.0 s, changing little after time 6.0 s. On the whole, the tensile stress of the haunch was larger than that of the other parts. The maximum tensile stress of the lining was 1.05 MPa when the dynamic interaction between the rock mass and the fault was ignored, while the value was 1.43 MPa when the dynamic interaction was considered, which is clearly larger than that in the former case. In addition, when the dynamic interaction was considered, the maximum tensile stress of the haunch reached the tensile strength of concrete, and the haunch may then suffer damage. This indicates that the stress of the lining is greatly affected by the dislocation displacement between the rock mass and the fault.

According to the above analysis, tensile damage appears on the lining when the dynamic interaction between the rock mass and the fault is considered. Based on the damage constitutive relation of concrete, the distribution of the damage coefficient of the lining structure after the earthquake can be obtained, as shown in Figure 15, and the changing laws of the damage coefficient of the lining haunch along the tunnel axis after the earthquake are plotted in Figure 16.

**Figure 15.** Distribution of the damage coefficient of the lining after the earthquake.

**Figure 16.** Changing laws of the damage coefficient of the lining haunch along the tunnel axis.

We can see from Figures 15 and 16 that the damage area of the lining was mainly distributed in a certain range near both sides of the fault and at the parts where the fault passes through, especially at the haunch. From the footwall to the fault, the damage coefficient of the haunch first increased and then decreased. From the fault to the hanging wall, the damage coefficient of the haunch also first increased and then decreased. Compared with the other parts, the damage to the lining at the contact interface between the rock mass and the fault was the most serious. Compared with the contact interface between the footwall and the fault, the damage to the lining at the contact interface between the hanging wall and the fault was more serious. The local damage coefficient of the haunch at this part was close to 1.0, and the haunch may then crack. Based on the above analysis, the influences of fault dislocations should be considered in the seismic design of a tunnel structure through a fault, from the perspective of engineering safety. In addition, according to the distribution length of the damage area of the lining, the longitudinal layout range of seismic fortification for the tunnel structure can be determined.

#### *6.4. Discussion*

The seismic response of the tunnel through fault is affected by many factors, including fault thickness and fault dip angle. In this section, when the influences of different fault thicknesses on the seismic response of the tunnel were considered, the fault dip angle was uniformly set at 60◦. When the influences of different fault dip angles on the seismic response of the tunnel were considered, the fault thickness was uniformly set at 20 m. In all the above calculation cases, the other calculation conditions remained consistent, including the fact that the dynamic interaction between the rock mass and the fault was considered.
