Study on the Seismic Response and Aseismic Measure of Fault-Crossing Tunnels under Combined Action of Fault Dislocation and Seismic Motions

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Introduction
With the rapid growth of infrastructure demand, the rate of highway and railroad tunnel construction has increased by leaps and bounds, which has led to more complex geological conditions that tunnel construction can meet with.Tunnel damages in recent years have challenged the traditional idea that mountain tunnels have good seismic performance [1][2][3][4], which has prompted many scholars and engineers to study the seismic response and improve the seismic design of tunnels and other underground structures.
Earthquake damage investigations have shown that a fault fracture zone is one of the most dangerous areas for mountain tunnels [5][6][7].Terefore, several studies on the seismic response of fault-crossing tunnels using model tests [8][9][10] and numerical simulations [11][12][13] have been carried out.When a fault dislocates, the tunnel will subject to severe damage due to the intense shear action [10,14].If the fault is not dislocated, the diference in the longitudinal mechanical properties of the strata will also cause uneven deformation in the longitudinal direction of the tunnel [7,15].Te refection and refraction of seismic waves on the fault interface also have negative efects on tunnel deformation [16,17].In the case of tunnels crossing secondary faults, faults may be permanently dislocated by causative faults during earthquakes and tunnels will sufer from the combined action of fault dislocation and seismic motions.Due to the complex interaction of the fault-rock-tunnel system, the understanding of the seismic response mechanism of faultcrossing tunnels is not yet complete, and there is no consensus on the best aseismic measure, which leads to engineering practice lagging behind research.
In order to mitigate the damage of fault-crossing tunnels, several aseismic measures have been proposed: fexible joints can reduce the deformation of the tunnel by concentrating the permanent deformation of the fault to the joints through their own deformation [18,19].Bufer layers can reduce the transfer of earthquake energy to the tunnel by absorbing the deformation of the ground and thus reduce the damage of the tunnel [20,21].Fiber concrete can reduce the tunnel damage by enhancing the mechanical properties of the tunnel [21,22].Grouting reinforcement can reduce the tunnel damage by enhancing the mechanical properties of the strata in a certain range outside the tunnel and reducing the deformation of the strata within the fault [23].However, few studies have been conducted to compare the seismic mitigation efects of diferent aseismic measures and evaluate the suitability of diferent aseismic measures.At the same time, few studies have been carried out to investigate the seismic response of tunnels crossing secondary faults, which may sufer from the combined action of seismic motions and fault dislocation during earthquakes.
In this paper, three-dimensional numerical models were developed for tunnels crossing secondary faults and the seismic response of tunnels crossing diferent widths of faults is investigated.Two aseismic measures, grouting reinforcement and fexible joints, were compared in terms of their aseismic efects in tunnels crossing diferent widths of faults.Te present study can provide references and suggestions for the seismic design of tunnels across faults.

Numerical Model
2.1.Model Setup.Nonlinear fnite element numerical models were established in ABAQUS to simulate the seismic response of the fault-crossing tunnel and verify the aseismic efect of diferent aseismic measures.Figure 1 shows the 3D model with a fault dip angle of 60 °and fault widths of 10 m and 100 m, respectively.Te diameter of the tunnel is 9.5 m, and the burial depth is 20 m.For the model with fexible joints, fexible joints with a width of 0.5 m are set at 6 m intervals in the lining of the fault.For the model with grouting zones, the strata within 0.5 times the diameter of the tunnel on the outside of the inner lining of the fault were set as grouting zones.Te details of the two damping measures are shown in Figure 1.
In the numerical simulation, the tunnel lining is tied with the surrounding rock, and it is assumed that there is no relative slipping between lining and surrounding rock.
Frictional contact between the fault and surrounding rock is used, and the friction coefcient is set to be 0.4.Te Mohr-Coulomb criterion is used to simulate the elastoplastic behavior of the surrounding rock, fault, and grouting zone, and tunnel lining and fexible joints are assumed to be linear elastic.Te model material parameters are shown in Table 1.Rayleigh damping is adopted in the simulation, and the damping ratio is set to be 0.05.Te frst two modes of the numerical model were selected to construct the damping matrix.To simulate the shear deformation of the ground under the action of shear waves, the equal displacement boundary is set on the lateral side of the model, so that the nodes at the same height move simultaneously.Te synthetic wave, Wenchuan wave, and Kobe wave were used in the simulation as input seismic motions, as shown in Figure 2, and the peak ground acceleration is 0.3 g, which corresponds to a peak acceleration of 10% of the exceedance probability in 50 years.

Ground Motion Input.
Fault-crossing tunnels may sufer from the combined action of fault displacement and ground motions under strong earthquakes.While the causes of permanent displacement of secondary faults are complex, some scholars believe that rupture occurs under the perturbation of earthquakes due to the initial stress level within the fault close to the material strength [24].It is difcult to realistically reproduce the complex stress conditions and rupture processes within the fault in numerical simulations, so this paper implements the simulation of the permanent displacement of the fault by applying ground motion considering the permanent displacement on both sides of the fault.
Te permanent displacement of a fault should be generated and ended at a certain moment of earthquake occurrence.Chao et al. [25] proposed the energy distribution ratio to determine the beginning and the end of the fault displacement and concluded that the fault movement starts when the energy distribution ratio reaches 25% and ends when the energy distribution ratio reaches 65%.
Te detailed description of the seismic wave construction process is as follows: the synthetic wave energy time history is plotted in Figure 3, and it can be found that the seismic wave energy reaches 25% at 3.5 s and 65% at 6.5 s, so the start and end moments of the permanent displacement of the fault are set to be 3.5 s and 6.5 s.Te displacement time history is obtained by integrating the synthetic wave acceleration and increasing the displacement linearly by a total of 0.1 m between 3.5 s and 6.5 s.Te displacement time history after adding the permanent displacement is obtained, as shown in Figure 4. Te acceleration time history considering the permanent displacement can be obtained by deriving the displacement time curve, as shown in Figure 5.A slip fracture surface is set in the center of the fault, and the original synthetic wave and the synthetic wave considering the permanent displacement are applied on both sides of the slip fracture surface to achieve the combined action of fault displacement and ground motions.

2
Shock and Vibration

Analysis Procedures.
In order to analyze the seismic response of tunnels crossing diferent widths of faults and explore the suitability of diferent seismic mitigation measures, eight numerical models were established, as shown in Table 2. Te models with fault widths of 10 m and 100 m are used to compare the efect of the fault width on tunnel response under the combined action of fault dislocation and ground motions.In addition, three types of aseismic measures, namely, grouting, fexible joint, and grouting reinforcement + fexible joint, are installed in the models with diferent fault widths to verify the suitability of various damping measures; the details of these aseismic measures are plotted in Figure 6.

Tunnel Responses with Diferent Fault Widths.
Te fnal deformation of the tunnel in the calculated conditions for the fault widths of 10 m and 100 m is plotted in Figure 7.It can be found that the fnal deformation of the tunnel is consistent with stratigraphic deformation, with a permanent deformation of 0.1 m, between the tunnels on both sides of the fault.Although a slip fracture surface is set in the center of the fault, tunnel deformation is not concentrated near the slip fracture surface but is evenly distributed over the fault width.Te peak acceleration of the tunnel vault along the longitudinal direction of the two models without aseismic measures is plotted in Figure 8, which indicates that the acceleration of the tunnel inside the fault is signifcantly greater than that of the tunnel in the surrounding rock on both sides of the fault.In addition, for tunnels located within a certain range on both sides of the fault, the acceleration of tunnels located at the hanging wall is greater than that located at the footwall, and this phenomenon is particularly signifcant when the fault width is small, which is also consistent with the upper plate efect observed in the seismic investigation [6].For a fault with a width of 100 m, peak tunnel acceleration appears along the longitudinal direction with two peaks located near the two interfaces of the fault.Tis is because seismic waves incident vertically from the bottom are refected through the fault intersections and superimposed with the incident waves at this location, which increases the acceleration there.For the fault with a width of 10 m, the two peaks overlap due to the small width of the fault, making the peak acceleration of the tunnel within the fault exceed the peak acceleration of the tunnel within the fault with a width of 100 m.
Te maximum and minimum principal stresses in the tunnel for the two cases without aseismic measures are plotted in Figure 9.For the model with a 10 m fault, the maximum principal stress of the tunnel is 111.1 MPa and the minimum principal stress is 70.1 MPa in the synthetic wave case; the maximum principal stress of the tunnel is 107.8MPa and the minimum principal stress is 94.0 MPa in the Wenchuan wave case; the maximum principal stress of the tunnel is 106.8MPa and the minimum principal stress is 88.3 MPa in the Kobe wave case.For the model with a 100 m fault, the maximum principal stress of the tunnel is 46.5 MPa and the minimum principal stress is 42.7 MPa in the synthetic wave case; the maximum principal stress of the tunnel is 32.8 MPa and the minimum principal stress is 40.0 MPa in  Shock and Vibration the Wenchuan wave case; the maximum principal stress of the tunnel is 33.1 MPa and the minimum principal stress is 28.9 MPa in the Kobe wave case.Diferent from the seismic response of fault-crossing tunnels only considering the seismic motions [26], the stress of the tunnel in narrow faults is greater than that in wide faults.Tis is because that the   Shock and Vibration same permanent displacement in narrow faults will produce greater relative deformation in the tunnel, as shown in Figure 7, which also indicates that the response of the faultcrossing tunnel is to some extent dominant by the permanent displacement of the fault.

Aseismic Efect of Diferent Aseismic Measures.
Te distribution of the peak acceleration of the tunnel in a 10 m fault is plotted in Figure 10.It can be seen that the installation of diferent aseismic measures has little efect on tunnel acceleration and does not change the distribution of tunnel acceleration.In the synthetic wave case, the peak tunnel acceleration in the fault is reduced from 8.88 m/s 2 to 8.52 m/s 2 by both grouting and grouting + fexible joints, while peak tunnel acceleration is slightly increased to 8.97 m/ s 2 by fexible joints.In the Wenchuan wave case, peak tunnel acceleration in the fault is reduced from 8.83 m/s 2 to 8.37 m/ s 2 by both grouting and grouting + fexible joints, while peak tunnel acceleration is slightly increased to 9.00 m/s 2 by fexible joints.In the Kobe wave case, peak tunnel acceleration in the fault is reduced from 8.44 m/s 2 to 8.06 m/s 2 by both grouting and grouting + fexible joints, while peak tunnel acceleration is slightly increased to 8.58 m/s 2 by fexible joints.Te distribution of the peak acceleration of the tunnel in a 100 m fault is plotted in Figure 11.It can be seen that the installation of grouting signifcantly changes the distribution of tunnel acceleration within the fault.For three cases, the maximum acceleration within the fault is reduced from 8.31 m/s 2 to 7.91 m/s 2 , 7.50 m/s 2 to 6.67 m/s 2 , and 7.61 m/s 2 to 7.11 m/s 2 , respectively.Te installation of fexible joints does not change the acceleration distribution pattern of the tunnel but increases the maximum acceleration of the tunnel to 8.67 m/s 2 , 8.08 m/s 2 , and 8.34 m/s 2 , respectively.
Te maximum and minimum principal stresses in the model with a fault width of 10 m are plotted in Figure 12.It can be found that the installation of diferent aseismic measures does not change the distribution pattern of the principal stresses of the tunnel: the peak principal stresses are mainly concentrated at the interface between the fault and footwall.In the synthetic wave case, grouting reduces the maximum principal stress to 69.8 MPa and the minimum principal stress to 48.8 MPa; fexible joints reduce the maximum principal stress to 76.9 MPa and the minimum principal stress to 65.5 MPa; for the model with both aseismic measures installed, the maximum principal stress is reduced to 48.0 MPa and the minimum principal stress is reduced to 42.6 MPa.In the Wenchuan wave case, grouting reduces the maximum principal stress to 67.3 MPa and the minimum principal stress to 67.3 MPa; fexible joints increase the maximum principal stress to 122.4 MPa and the minimum principal stress to 103.2 MPa; for the model with both aseismic measures installed, the maximum principal stress is reduced to 86.9 MPa and the minimum principal stress is reduced to 73.1 MPa.In the Kobe wave case, grouting reduces the maximum principal stress to 66.3 MPa and the minimum principal stress to 66.4 MPa; fexible joints increase the maximum principal stress to 121.1 MPa and the minimum principal stress to 102.4 MPa; for the model with both aseismic measures installed, the maximum principal stress is reduced to 86.3 MPa and the minimum principal stress is reduced to 72.2 MPa.
Te maximum and minimum principal stresses in the model with a fault width of 100 m are plotted in Figure 13.It can be found that the installation of grouting does not change the distribution pattern of principal stresses in the tunnel, and the peak maximum principal stresses are mainly concentrated at the interface between the fault and footwall as well as the center of the fault, while the peak minimum principal stresses are mainly concentrated at the vault of the tunnel at the fault interfaces.Te maximum and minimum principal stresses in the tunnel are concentrated at the       Te peak principal stresses of the tunnel-crossing different fault widths are summarized in Table 3.It can be found that grouting can signifcantly reduce the tunnel response and has the best aseismic efect.Flexible joints reduce the longitudinal stifness of the tunnel, which will increase the seismic response in earthquakes.Considering the feasibility of postearthquake restoration, the installation of both two aseismic measures can achieve both damping efects and economic benefts.

Conclusions
In this paper, three-dimensional numerical models of tunnels crossing secondary faults are established and the seismic response of the fault-crossing tunnel under the combined action of ground motions and permanent fault dislocation of the fault is investigated.Te aseismic efects of diferent aseismic measures are discussed.Te following conclusions are obtained: (1) For the model with a fault width of 100 m, peak tunnel acceleration has two peaks near the interfaces of the fault due to the refection of seismic waves at the fault interfaces.While for the model with a fault width of 10 m, the two acceleration peaks overlap, making the peak tunnel acceleration within the fault exceed the peak tunnel acceleration within the fault with a width of 100 m. (2) Since the permanent displacement of the fault will cause greater relative deformation of the tunnel within the fault when the fault width is narrow, the peak stress of the tunnel within the fault with a width of 10 m is much larger than that of the tunnel within the fault with a width of 100 m. (3) For the model with a fault width of 10 m, setting diferent aseismic measures will not change the acceleration distribution pattern of the tunnel during earthquakes.For the model with a 100 m fault, the installation of grouting will reduce the larger acceleration response near the fault interfaces and small acceleration response in the center of the fault.Regardless of the fault width, the installation of fexible joints will slightly increase the acceleration response of the tunnel within the fault.(4) For fault-crossing tunnels, grouting can signifcantly reduce the seismic response of tunnels.In some cases, the installation of fexible joints might increase the seismic response of tunnels.Tis is because installing fexible joints will reduce the longitudinal stifness and integrity of the tunnel, which will increase the deformation of the tunnel during earthquakes and thus increase the stresses of the tunnel.(5) Considering the feasibility of postearthquake restoration, the installation of both two aseismic measures can achieve both damping efects and economic benefts.

Figure 1 :
Figure 1: Diagram of models: (a) model with a 10 m fault and (b) model with a 100 m fault.

Figure 3 :
Figure 3: Energy time history of synthetic waves.
junctions of the tunnel and fexible joints when the fexible joints are installed.In the synthetic wave case, grouting reduces the maximum principal stress to 35.2 MPa and the minimum principal stress to 25.8 MPa; the fexible joint increases the maximum principal stress to 56.6 MPa and the minimum principal stress to 54.6 MPa.For the model with both aseismic measures installed, the maximum principal stress in the tunnel is reduced to 42.3 MPa and the minimum principal stress is reduced to 31.9 MPa.In the Wenchuan wave case, grouting reduces the maximum principal stress to 26.1 MPa and the minimum principal stress to 28.4 MPa; fexible joints increase the maximum principal stress to 51.3 MPa and the minimum principal stress to 55.4 MPa; for the model with both aseismic measures installed, the maximum principal stress is reduced to 32.7 MPa and the

FlexibleFigure 12 :
Figure 12: Principal stress of tunnels with diferent aseismic measures in a 10 m fault: (a) maximum principal stress in the synthetic wave case; (b) minimum principal stress in the synthetic wave case; (c) maximum principal stress in the Wenchuan wave case; (d) minimum principal stress in the Wenchuan wave case; (e) maximum principal stress in the Kobe wave case; (f ) minimum principal stress in the Kobe wave case.

FlexibleFigure 13 :
Figure 13: Principal stress of tunnels with diferent aseismic measures in a 100 m fault: (a) maximum principal stress in the synthetic wave case; (b) minimum principal stress in the synthetic wave case; (c) maximum principal stress in the Wenchuan wave case; (d) minimum principal stress in the Wenchuan wave case; (e) maximum principal stress in the Kobe wave case; (f ) minimum principal stress in the Kobe wave case.