Seismic Response of Tunnel across Inactive Fault: Numerical Analysis

Earthquake damage investigations in recent years have revealed that fault zone is one of the most dangerous areas of tunnels in earthquakes, leading scholars to investigate the mechanisms of seismic response of fault-crossing tunnels and the aseismic measures. However, due to the complex fault-rock-tunnel interaction, the seismic response mechanism of tunnels crossing inactive faults is not yet complete and the suitability of different aseismic measure has not been evaluated. In this paper, a series of 3D numerical models of fault-crossing tunnels were established to study the seismic response of tunnels crossing inactive faults with different widths. On this basis, the suitability of grouting and flexible joints for fault-crossing tunnels were discussed. Due to the difference of physical property between the fault and the surrounding rock, the differential deformation will be produced at the interface, and resulting in a shear damage of the tunnel. The tunnel crossing wide fault is more vulnerable in earthquake than that crossing narrow fault. For the tunnel crossing inactive faults, especially those with wide widths, grouting reinforcement is a desirable aseismic measure. It can reduce the acceleration response of the tunnel within the fault and significantly reduce the tensile damage, while flexible joints have little effect on the acceleration response and the damage reduction of the tunnel within the fault. The present study can contribute to a better understanding of the seismic response of tunnels crossing inactive faults and provide some guidance to the seismic design.


Introduction
Driven by the growing demand for infrastructure in mountainous areas, the construction of tunnels in highway and railway networks have been accelerated. Challenges and complex geological conditions are met with in tunnel projects than in the past. Numerous cases of damages of mountain tunnels have been reported in earthquakes [1][2][3][4], which have led scholars and engineers into topics researching seismic response and seismic design of tunnels and other underground facilities.
Earthquake damage investigations on mountain tunnels reveal that fault or fracture zone is one of the most dangerous areas of tunnels in earthquake [5][6][7]. In order to study the seismic response and earthquake damage mechanism of fault-crossing tunnels, scholars have carried out numerous researches including scaled model tests [8][9][10] and numerical simulations [11][12][13]. Recent evidence suggests that the tunnel will be severely damaged by the violent shearing action when the fault dislocated [14,15]. Meanwhile, if the fault is not dislocated, the discontinuity of the strata will cause 2 the tunnel to deform unevenly in the longitudinal direction [16], and the fault interface will reflect and refract the incident seismic waves, causing increased tunnel deformation [17,18]. However, due to the complex fault-rock-tunnel interaction, the seismic response mechanism of tunnels crossing inactive faults is not yet complete, leading to a gap between research and engineering practice.
A number of aseismic measures have been proposed to reduce seismic damage in fault-crossing tunnels, including flexible joint [19], buffer layer [20], and fibre reinforcement concrete [21], etc. However, few studies have been carried out to compare the aseismic effects of different aseismic measures, and the suitability of each measure has not been evaluated.
In this paper, a series of 3D numerical models were established to investigate the seismic response and damage pattern of tunnels crossing inactive faults with different widths. Two aseismic measures, grouting reinforcement and flexible joints were proposed, and the aseismic effects of two measures on tunnels crossing different widths of faults were analysed and their suitability was evaluated. This study may provide some reference for the seismic design of fault-crossing tunnels.

Numerical modelling
In order to investigate the influence of different fault widths on the seismic response of tunnels and evaluate the seismic performance of different aseismic measures, a series of 3D numerical models of a mountain tunnel crossing faults were established.

Engineering background
The Xianglushan Tunnel is part of the Central Yunnan Water Diversion Project, located in the central part of China's Yunnan Province. The objective of the project is to deliver water from the Shigu River to the central part of Yunnan Province. The tunnel is a key project along the entire route, with a total length of 63.4km. The geological conditions along the tunnel route are complicated, crossing three major active fault zones, including the Longpan-Qiaohou fault, the Lijiang-Jianchuan fault and the Heqing-Eryuan fault. Among them, the Heqing-Eyuan fault was chosen as the prototype for numerical calculations due to its strong seismic activity. Figure1 shows the sketch of Heqing-Eryuan fault and the longitudinal profile of Xianglushan Tunnel. The width of the Heqing-Eryuan fault is 120m and the dip angle of the fault is 60°. The average burial depth of the tunnel is 1000m and the rock on both sides of the fault is mainly basalt and limestone, which can be classified as type IV according to the Chinese code. It should be noted that this paper focuses on the seismic response of the tunnel lining and the effect of the high geo-stress is ignored. The tunnel has a circular cross section with an inner radius of 4.2 m and outer radius of 4.75m. The lining has a thickness of 550 mm and is made of C30 concrete.

Finite element model
Nonlinear finite element models were established in the finite element program ABAQUS to simulate the seismic response of fault-crossing tunnel and the aseismic effect of different aseismic measures. Figure 2 shows the 3D models with a fault dip angle of 60°and with a fault width of 10 m and 100 m. The overall dimensions of the numerical model are 300 m (length) ×100 m (width) × 50 m (height). The diameter of the tunnel is 9.5 m and the burial depth is 20 m. For models setting flexible joint as aseismic measure, the lining within the fault is installed with flexible joints with 0.5m width at 6m intervals. For models setting grouting as aseismic measure, the strata within 0.5 times the tunnel diameter on the outside of the lining within the fault are set as the grouting zone. The details of the two aseismic measures are plotted in Figure 2. Eight-node reduced-integration brick elements (C3D8R) is used in the simulation. The tunnel lining is tied to the surrounding rock in the numerical simulation by assuming that there is no relative displacement between the lining and the rock. The interaction between the fault and the surrounding rock is simulated by applying "surface to surface contact" in ABAQUS to evaluate the potential slipping and the friction coefficient between them is set to 0.4 [22]. The Mohr-Coulomb model of elasto-plasticity is adopted for the constitutive model of surrounding rock, fault, grouting, and flexible joint [23], as listed in Table 1. The concrete damaged plasticity model is used to simulate the tunnel lining. The tension and compression stress-strain relationships of the C30 concrete are shown in Figure 3. The Rayleigh damping is used in numerical simulation and the damping ratio is set as 0.05. In order to simulate the shear deformation of the ground under shear wave action, the vertical boundaries introduce kinematic constrains are conducted [24], forcing the nodes at the same height to move simultaneously preventing any rotation. A synthetic motion is applied in the numerical simulation, which is provided in the " Evaluation of Seismic Safety " section of the geotechnical investigation report for the Xianglushan Tunnel, as shown in Figure 4 and the peak ground acceleration is 0.12 g, for a 63% probability of exceedance in 50 years. The seismic wave energy is mainly concentrated in the frequency band from 0 Hz to 20 Hz, and the first dominant frequency is 6.3 Hz.
(a) compression (b) tension Figure 3. Stress-strain, damage-strain and damage factor constitutive relationships for concrete. (a) accelerogram (b) Fourier spectrum Figure 4. Synthetic motion used in numerical simulation.

Analysis procedures
In order to investigate the aseismic effect of flexible joint and grouting, six numerical models were established and numbered from 1 to 6, as shown in Table 2. The models with 10 m fault and 100m fault are conducted to compare the effect of fault width on seismic response of fault-crossing tunnel. In addition, each model is set up with two aseismic measures, flexible joints and grouting within the faults. Furthermore, the seismic mitigation effect of different aseismic measures can be compared with different models

Effects of fault width
The  longitudinal direction, it can be seen that the acceleration in the hanging wall is larger than that in the footwall, which is consistent with the so-called "hanging wall effect" observed in seismic investigation [1].
Three crests and troughs are observed along the longitudinal direction of the lining in the 100 m fault. This might be a result of the superposition of seismic waves reflected within the fault. A similar phenomenon has been found in other study investigating the relationship between the burial depth and the seismic response of tunnel [25]. However, this phenomenon is not evident in the model with 10m fault because the fault width is less than the wavelength of the main frequency component of the seismic wave, which also makes the acceleration within the fault exceed that within the 100 m fault. Figure 6 presents the distribution of lining tensile damage of two models. It can be found that the lining damage is mainly concentrated near the interfaces between the fault and the surrounding rock. Due to the difference of physical property between the fault and the surrounding rock, the differential deformation will be produced at the interface, and resulting in a shear damage of the tunnel. The maximum tensile damage in the model with 100 m fault is 0.95, which is greater than that in the model with 10 m fault (0.23). It can be explained by the fact that, within a certain range, the wider the fault width, the greater the differential deformation and the more severe the tunnel damage. This is also in line with the earthquake damage investigation of the Wenchuan earthquake [26].

Comparison between different aseismic measures
The distributions of the maximum acceleration of the tunnel vault along the longitudinal direction for the models with different aseismic measures are plotted in in Figure 7. As can be seen in Figure 7(a), the installation of aseismic measures has little effect on the acceleration of the tunnel. The grouting reinforcement slightly reduces the acceleration of the tunnel within the fault, while the flexible joints do not change the overall acceleration response of the tunnel. As shown in Figure 7(b), for the model with 100 m fault, the grouting reinforcement effectively reduces the acceleration of the tunnel within the fault, especially in the area close to the fault interface, while the flexible joints only slightly reduce the acceleration in a region close to the footwall.  Figure 8 and Figure 9 presents the distribution of lining tensile damage of the models with different fault width setting different aseismic measures. As can be seen from Figure 8, the grouting reinforcement efficiently reduces the tensile damage of the tunnel in the fault with 10 m width. Although the flexible joints concentrate the tunnel damage in the vicinity of the joints, they give no reduction in the tensile damage of the tunnel. As shown in Figure 9, the grouting reinforcement significantly reduces the tensile damage of the tunnel in the fault with 100 m width from 0.95 to 0.34, while the flexible joints only marginally reduce the tensile damage of the tunnel from 0.95 to 0.93.

Conclusions
In this paper, a series of numerical models for the tunnel crossing fault were conducted. The influence of different fault widths on the seismic response of tunnel and the suitability of different aseismic measures were investigated.
The following conclusions could be drawn: (1) The acceleration response of the tunnel within the fault with 10 m width is greater than that of the tunnel within the fault with 100 m width; (2) There is a " hanging wall effect " in fault-crossing tunnels in earthquake, i.e. the acceleration response in the hanging wall is greater than that in the footwall, while the "hanging wall effect" of the fault with 10 m width is more remarkable than the fault with 100 m width; (3) Due to the difference of physical property between the fault and the surrounding rock, the differential deformation will be produced at the interface, and resulting in a shear damage of the tunnel. The tensile damage of the tunnel in the fault with 100 m width is much more severe than that in the fault with 10 m width, and the damage is mainly concentrated at the interface between the fault and the surrounding rock; (4) Grouting reinforcement can reduce the acceleration response of the tunnel within the fault and effectively reduce the tensile damage of the tunnel; (5) Flexible joints have little effect on the acceleration response of the tunnel within the fault. Although flexible joints concentrate the tensile damage of the tunnel in the vicinity of the joints, they have a slight effect on tensile damage reduction; (6) For the tunnel crossing inactive faults, especially those with wide widths, grouting reinforcement is a desirable aseismic measure. It can reduce the acceleration response of the tunnel within the fault and significantly reduce the tensile damage.