# Effect of Oblique SV Wave on the Seismic Response of Mountain Tunnel

Authors: Mingxing Cao; Songhong Yan; Jiaxuan Du; Weiyu Sun; Yuxiang Li; Junjie Zhao
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

In order to study the dynamic response of parallel mountain tunnels under the oblique incidence of seismic waves, based on the display finite element method and using viscoelastic artificial boundary, the oblique incidence of three-way seismic waves was realized by angular incident mode. The displacement and stress distribution characteristics of the tunnel lining under different propagation angles and vibration angles of SV waves were studied. The results show that the oblique incidence of SV wave has a certain effect on the displacement of the double tunnel, the forces in the tunnel are symmetrical and the axis displacement increases with the increase of incident angle, and the vertical displacement changes greatly. The stress of the tunnel lining under the oblique incidence of the SV wave is elliptical. The peak value of the maximum principal stress appears at the maximum span on both sides, and the maximum principal stress decreases with the increase of the vibration angle. The maximum principal stress of the right tunnel is flat. The minimum principal stress of the left and right holes decreases with the increase of vibration angle, and the minimum principal stress of the left hole is 90°∼270°. The distribution of the minimum principal stress in the range is large. Mises stress increases with the increase of the incidence angle of seismic waves.


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## Body

## 1. Introduction

The underground structure usually interacts with the surrounding rock, and the stress and deformation of the tunnel are closely related to the interaction between the surrounding rock. Because the tunnel structure gene really exists underground, the seismic performance is better than that of the above-ground structure, the tunnel has symmetry, and its stress has a great relationship with the symmetry, so there are few studies. The earthquake resistance of underground structures has received attention, especially for some traffic tunnels located in earthquake-prone areas. It is difficult to repair the large damage to the tunnels caused by earthquakes, such as the Italian earthquake [1], studies the influence of earthquake on the tunnel in Italy, analyzes the seismic damage phenomenon and seismic records, and quantitatively analyzes the seismic damage based on the theoretical mechanical mechanism of the interaction between surrounding rock and tunnel on the basis of the simplified model, which provides guidance for the seismic design of the tunnel. However, there is a certain gap between the equivalent static load and the actual seismic load. Wang et al. [2], by investigating the damage of the tunnel caused by Taiwan Jiji earthquake and studying the damage forms of the tunnel lining and the distribution of cracks, it is found that each tunnel has been damaged to different degrees. The failure law of the tunnel is summarized, and the analysis is carried out based on the geological conditions of the tunnel, design documents, and maintenance records. The influence factors of the earthquake on tunnel damage are mainly related to the geographical location of the tunnel, whether the tunnel passes through the fault fracture zone, the distance of the epicenter, the type of tunnel lining, and so on. However, only the tunnel failure phenomenon was recorded, and the factors influencing the tunnel earthquake failure were analyzed, but the mechanism of seismic wave action was not studied. Jiang et al. [3], by analyzing the various damage phenomena of the tunnel after the earthquake, the factors of the damage are analyzed, and the damage database of the tunnel is established by using the geographic information system to evaluate the damage degree of the tunnel that occurred in recent years and which caused great damage to the tunnel. Therefore, it is necessary to study the laws of tunnels under the action of earthquakes. In engineering earthquake resistance, there is no authoritative theoretical method for ground motion input. By Yan et al. [4], using finite element software, based on wave theory and obliquely incident P and SV waves, the seismic response of the tunnel under the interaction of soil structure is studied, and the comparison with the analytical solution is carried out to verify the accuracy, and the type of incident wave and the incident angle are, respectively, studied. Only the ground motion response of a circular tunnel is analyzed, and the seismic damage of a noncircular tunnel is not analyzed. However, most mountain railway and highway tunnels are horseshoe-shaped tunnels, so it is of greater use value to study the seismic response of noncircular tunnel burial depth and tunnel diameter on the seismic response of the tunnel. By Dana et al. [5], using the finite element method and boundary element method to calculate the near field and far field, the influence of incident angle, terrain, and geometric characteristics on the ground motion response of valley terrain is analyzed.This research can provide some reference for building design on this terrain. By Davis et al. [6], the Fourier series is used to deduce the free-form surface in half space, and the analytical solution of the unlined tunnel is calculated. The calculation results are compared with the previous research results. The approximate solution of the hoop stress of the tunnel when the incident wave is much larger than the diameter of the tunnel is analyzed, and the research results can provide a reference for the earthquake resistance of small-diameter tunnels, according to Zhang et al. [7]. The analytical solutions of the P wave and SV wave in the composite lining are studied by the wave function expansion method, and springs are set between the tunnel surrounding rock and the support. The influence of incident wave type and isolation layer on the dynamic stress concentration factor of the tunnel is analyzed, and it is obtained that the dynamic stress concentration factor of the tunnel under the action of SV wave is more significant according to Zhang et al. [8]. By analyzing the seismic responses of shafts and tunnels receiving SH waves, in the case of obliquely incident SH waves, the incident waves have a significant impact on the tunnel dynamics. In this paper, the stiffness matrix method is used to solve the analytical solution of the dynamic response of the shaft and the tunnel. The influence of the shaft on the tunnel decreases as the distance between the shaft and the tunnel increases according to Liao et al. [9]. The seismic response of seismic waves written into the radioelastic half-space tunnel is studied. The transition matrix is used to construct basis functions with the moving P wave, SH wave, and SV wave, and the scattered wave field and the refracted wave field are deduced according to Wolf et al. [10]. Based on the explicit finite element method combined with the viscoelastic artificial boundary, the dynamic response of the subway station under the oblique incidence of the SV wave is analyzed, and some beneficial conclusions are drawn. Under the oblique incidence of seismic waves, the shear force and axial force of the station pillars changed significantly, and they increased with the increase of the incident angle. At present, there are many researches on the two-dimensional incidence of seismic waves, there are few three-dimensional oblique incidence models, and there are no relatively mature results. Although the above scholars have conducted extensive researches on the seismic response of tunnel, most of them regard the seismic wave as a vertically incident on the underground structure, but, in fact, the seismic wave is an oblique incident to propagate when the earthquake occurs, and the study on the influence of the incidence angle of the seismic wave on the seismic response of the underground structure is far from sufficient.Based on the displayed finite element method, this paper deduces the equivalent nodal load of the oblique incidence of three-dimensional seismic waves, compiles the corresponding program, and conducts the seismic response of the double-hole tunnel in the mountains under the oblique incidence of three-dimensional seismic waves in ABAQUS.


## 2. Three-Dimensional Viscoelastic Artificial Boundary and Ground Motion Input

### 2.1. Viscoelastic Artificial Boundary

The viscoelastic artificial boundary can absorb the energy radiated from the computing area, which can well simulate the influence of infinite foundation on the computing area. In the three-dimensional viscoelastic artificial boundary, it is required to install springs and damping elements in both directions of each node on the boundary of the finite element model. The spring-damping elements are realized by the Springs/Dashpots element in the ABAQUS software. The schematic diagram is shown in Figure1. Relevant studies have shown that the accuracy of the viscoelastic boundary is relatively high, and the recovery force of the soil in the later stage is relatively good [11].Figure 1
3D viscoelastic boundary.The viscoelastic artificial boundary can absorb scattered waves and simulate the restoring force of semi-infinite foundations. The spring and damping parameters of the viscoelastic boundary are calculated by the following formulas [12, 13]:(1)Normal:kN=Al11+Aλ+2Gr,CN=AlBρcP.(2)Tangential:kT=Al11+AGr,CT=AlBρcs.In the formula,ρ is the density, G is the shear modulus, λ is the Lame constant, and cP, cs is the wave speed, according to the relevant literature A = 0.8, B = 1.1. Compile the corresponding program in Matlab, import the corresponding node information file and displacement velocity time history file of the model, and generate the amplitude curve file, load file, and spring damping file. Then, write these three files into the inp file, and perform calculations in Abaqus. The seismic wave oblique incidence model is shown in Figure 2. The left and right boundaries and the bottom boundary are set with viscoelastic artificial boundaries, and the upper surface is a free boundary.Figure 2
SV wave oblique incidence model.


### 2.2. Earthquake Input Method

The total wavefield motion equation of the finite element time domain method is shown as follows:(3)m1u¨li+∑k=1ne∑j=1nclikju˙kj+∑k=1ne∑j=1nklikjukj=Alfli.In the formula,ml is the mass of the node; klikj is the stiffness of the node k direction j to the node l direction; clikj is the damping coefficient; u˙kj is the displacement of the node k direction j ; u˙kj is the velocity; u˙lj is the acceleration; fli is the node l the force of the infinite far field at the direction i on the finite near field; Al is the influence area of the node. In three-dimensional problems, n = 3, i, j = 1, 2, 3. The total wave field of the artificial boundary is decomposed into an inner field and an outer field. The inner field is denoted by superscript R, and the outer field is denoted by superscript s. The total wave field displacement and force can be expressed as follows:(4)uli=uliR+ulis,(5)fli=fliR+flis.The motion relation of artificial boundary stress in layman’s field is(6)flis=−kliulis−cliu˙lis,where kli and cli are the viscoelastic boundary parameters. Taking formula (4) into (6), then into (6) into (5), and then into (5) into (3), the finite element equation of motion of the viscoelastic boundary is obtained:(7)m1u¨li+∑k=1ne∑j=1nclikj+δlkδijAlcliu˙kj+∑k=1ne∑j=1nklikj+δlkδijAlcliukj=AlkliuliR+cliu˙liR+fliR.


### 2.3. Viscoelastic Boundary Displacement Field

It can be known from the wave theory that the wave is delayed in the process of propagation. The SV waveu0t−∆t1 incident at angle α on the left, the reflected SV wave A3u0t−∆t2 at angle α, and the reflected P wave A4u0t−∆t3 at angle β are composed of the left inner field displacement:(8)ul1Rt=u0t−∆t1cosα−A3u0t−∆t2cosα+A4u0t−∆t3sinβ,ul2Rt=−u0t−∆t1sinα−A3u0t−∆t2sinα−A4u0t−∆t3cosβ.The front and back inner field displacements are as follows:(9)ul1Rt=u0t−∆t4cosα−A3u0t−∆t5cosα+A4u0t−∆t6sinβ,ul2Rt=−u0t−∆t4sinα−A3u0t−∆t5sinα−A4u0t−∆t6cosβ.The displacement of the row field in the bottom edge is as follows:(10)ul1Rt=u0t−∆t7cosα,ul2Rt=−u0t−∆t7sinα,where ∆t1 ∼ ∆t7 is the time for the incident wave to travel from the wavefront to the artificial boundary.


### 2.4. Viscoelastic Boundary Stress Field

The viscoelastic boundary stress is analyzed by references [14–16].Left side:(11)fl1R=Gcssin2αu˙0t−∆t1−A3u˙0t−∆t2+A4λ+2Gsin2βpcpu˙0t−∆t3,fl2R=Gcscos2αu˙0t−∆t1−A3u˙0t−∆t2−A4Gsin2βpcpu˙0t−∆t3,fl3R=0.Front side:(12)fl1R=fl2R=0,fl3R=A4λcpu˙0t−∆t6.Bottom:(13)fl1R=Gcscos2αu˙0t−∆t7,fl2R=−Gcscos2αu˙0t−∆t7,fl3R=0,where λ and G are Lame constants; cp is the P wave velocity; cs is the SV wave velocity.


## 2.1. Viscoelastic Artificial Boundary

The viscoelastic artificial boundary can absorb the energy radiated from the computing area, which can well simulate the influence of infinite foundation on the computing area. In the three-dimensional viscoelastic artificial boundary, it is required to install springs and damping elements in both directions of each node on the boundary of the finite element model. The spring-damping elements are realized by the Springs/Dashpots element in the ABAQUS software. The schematic diagram is shown in Figure1. Relevant studies have shown that the accuracy of the viscoelastic boundary is relatively high, and the recovery force of the soil in the later stage is relatively good [11].Figure 1
3D viscoelastic boundary.The viscoelastic artificial boundary can absorb scattered waves and simulate the restoring force of semi-infinite foundations. The spring and damping parameters of the viscoelastic boundary are calculated by the following formulas [12, 13]:(1)Normal:kN=Al11+Aλ+2Gr,CN=AlBρcP.(2)Tangential:kT=Al11+AGr,CT=AlBρcs.In the formula,ρ is the density, G is the shear modulus, λ is the Lame constant, and cP, cs is the wave speed, according to the relevant literature A = 0.8, B = 1.1. Compile the corresponding program in Matlab, import the corresponding node information file and displacement velocity time history file of the model, and generate the amplitude curve file, load file, and spring damping file. Then, write these three files into the inp file, and perform calculations in Abaqus. The seismic wave oblique incidence model is shown in Figure 2. The left and right boundaries and the bottom boundary are set with viscoelastic artificial boundaries, and the upper surface is a free boundary.Figure 2
SV wave oblique incidence model.


## 2.2. Earthquake Input Method

The total wavefield motion equation of the finite element time domain method is shown as follows:(3)m1u¨li+∑k=1ne∑j=1nclikju˙kj+∑k=1ne∑j=1nklikjukj=Alfli.In the formula,ml is the mass of the node; klikj is the stiffness of the node k direction j to the node l direction; clikj is the damping coefficient; u˙kj is the displacement of the node k direction j ; u˙kj is the velocity; u˙lj is the acceleration; fli is the node l the force of the infinite far field at the direction i on the finite near field; Al is the influence area of the node. In three-dimensional problems, n = 3, i, j = 1, 2, 3. The total wave field of the artificial boundary is decomposed into an inner field and an outer field. The inner field is denoted by superscript R, and the outer field is denoted by superscript s. The total wave field displacement and force can be expressed as follows:(4)uli=uliR+ulis,(5)fli=fliR+flis.The motion relation of artificial boundary stress in layman’s field is(6)flis=−kliulis−cliu˙lis,where kli and cli are the viscoelastic boundary parameters. Taking formula (4) into (6), then into (6) into (5), and then into (5) into (3), the finite element equation of motion of the viscoelastic boundary is obtained:(7)m1u¨li+∑k=1ne∑j=1nclikj+δlkδijAlcliu˙kj+∑k=1ne∑j=1nklikj+δlkδijAlcliukj=AlkliuliR+cliu˙liR+fliR.


## 2.3. Viscoelastic Boundary Displacement Field

It can be known from the wave theory that the wave is delayed in the process of propagation. The SV waveu0t−∆t1 incident at angle α on the left, the reflected SV wave A3u0t−∆t2 at angle α, and the reflected P wave A4u0t−∆t3 at angle β are composed of the left inner field displacement:(8)ul1Rt=u0t−∆t1cosα−A3u0t−∆t2cosα+A4u0t−∆t3sinβ,ul2Rt=−u0t−∆t1sinα−A3u0t−∆t2sinα−A4u0t−∆t3cosβ.The front and back inner field displacements are as follows:(9)ul1Rt=u0t−∆t4cosα−A3u0t−∆t5cosα+A4u0t−∆t6sinβ,ul2Rt=−u0t−∆t4sinα−A3u0t−∆t5sinα−A4u0t−∆t6cosβ.The displacement of the row field in the bottom edge is as follows:(10)ul1Rt=u0t−∆t7cosα,ul2Rt=−u0t−∆t7sinα,where ∆t1 ∼ ∆t7 is the time for the incident wave to travel from the wavefront to the artificial boundary.


## 2.4. Viscoelastic Boundary Stress Field

The viscoelastic boundary stress is analyzed by references [14–16].Left side:(11)fl1R=Gcssin2αu˙0t−∆t1−A3u˙0t−∆t2+A4λ+2Gsin2βpcpu˙0t−∆t3,fl2R=Gcscos2αu˙0t−∆t1−A3u˙0t−∆t2−A4Gsin2βpcpu˙0t−∆t3,fl3R=0.Front side:(12)fl1R=fl2R=0,fl3R=A4λcpu˙0t−∆t6.Bottom:(13)fl1R=Gcscos2αu˙0t−∆t7,fl2R=−Gcscos2αu˙0t−∆t7,fl3R=0,where λ and G are Lame constants; cp is the P wave velocity; cs is the SV wave velocity.


## 3. Earthquake Input Validation

A finite element model is established in ABAQUS to verify the seismic response when SV wave 15° is obliquely incident in a three-dimensionalhalf-space. The model size is 1000 m × 1000 m × 1000 m, the soil parameters are densityρ = 2000 kg/m3, elastic modulus E = 2 GPa, Poisson’s ratio is 0.3, and the displacement time-history curve of the incident wave is shown in Figure 3, and the center point A of the top surface of the model is taken as the analysis point.Figure 3
Incident wave displacement time-history curve.Build the finite element model to SV wave 15°. The incident half-space field, as can be seen from the cloud in Figure4, is the reflected wave that occurs after the seismic wave is incident. It can be seen from Figure 5 that the finite element model solution is consistent with the theoretical circle, which verifies the correctness of the model.Figure 4
SV wave 15° cloud map of half-space displacement field at oblique incidence.Figure 5
Displacement time history curve.In the figure, (1) is the displacement nephogram of SV wave 15° oblique incident at 0.7 s, (2) is the displacement nephogram of SV wave 15° oblique incident at 1.5 s, and (3) is the displacement nephogram of SV wave 15° oblique incident at 2.5 s.


## 4. Influence of Oblique Incidence of SV Wave on Seismic Response of Parallel Tunnel

### 4.1. Computational Model

A three-dimensional finite element model of a parallel tunnel is established in ABAQUS, as shown in Figure6. The longitudinal (Z) of the model is 100 m, the transverse (X) width is 200 m, the height (Y) is 100 m, the tunnel depth is 30 m, the distance between the two holes is 60 m, the soil adopts the Mohr–Coulomb constitutive model, the binding constraint is adopted between the soil and the primary lining, and the surface-to-surface contact between the primary lining and the secondary lining is adopted. The finite element model parameters are shown in Table 1.Figure 6
3D finite element model of the double-hole tunnel.Table 1
Material parameters.
Elastic modulus (GPa)Density (kg.m−3)Poisson’s ratioInternal friction angle (°)CohesionSoil321000.33300.1Initial lining3025000.2Second lining31.525000.2


### 4.2. Seismic Input

The American El-Centro wave is selected as the ground motion input, and the three-directional seismic wave is applied by means of the incident vector of the corner incident function. The acceleration time-history curve is shown in Figure7. The seismic wave propagation angle θ is the angle between the SV wave and the XY plane, which is taken as 0°, respectively, 15°, and 25°, for each propagation angle θv corresponding to a vibration angle 0°, 30°, and 60°.Figure 7
EI-Centro acceleration time-history curve.


### 4.3. Influence of Incident Angle on Axis Displacement

As shown in Figure6, the left and right double-hole tunnels are, respectively, determined at two points. The two points on the left tunnel are points A and B, and the two points on the right tunnel are points A′ and B′. The distance between points A and B is 80 m. The distance between “point” and B′ is also 80 m, and the SV wave is 15 m, respectively, 25°, and 90° oblique incidence double-hole tunnel. It can be seen from Figure 8 that the horizontal displacement of point A of the left hole increases with the increase of the incident angle of the seismic wave, but the increase is small; the relative vertical displacement of point A of the left hole increases with the increase of the incident angle. And the displacement in the case of normal incidence is significantly larger than the displacement in the other two incident angles and is 90°. The change curve of tunnel displacement peak when the incidence angles of seismic waves are 15° and 25°. The horizontal and vertical displacements of the right hole also increase with the increase of the incident angle, and the vertical displacement varies greatly, showing an obvious traveling wave effect. When the seismic wave is obliquely incident on the tunnel, there is an obvious traveling wave effect along the longitudinal direction of the tunnel. The nonuniform vibration effect of the seismic wave makes the movement of the tunnel along the longitudinal direction obviously different and accelerates the damage of the tunnel.Figure 8
Comparison of the displacement of the left and right tunnels: (a) horizontal displacement of point A in the left tunnel, (b) vertical displacement of point A in the left tunnel, (c) horizontal displacement of the right tunnel A′ point, and (d) vertical displacement of the right tunnel A′ point.
(a)(b)(c)(d)


### 4.4. Influence of Incidence Angle on Lining Stress Distribution

The finite element model was established, and the SV was 15°, respectively, and 25°. The propagation angles are obliquely incident on the tunnel, and each propagation angle corresponds to three vibration angles of 0°, 30°, and 60°. The distribution curves of the absolute value of the maximum principal stress and the minimum principal stress of the double-hole tunnel are shown in Figures9 and 10.Figure 9
Amplitude distribution of principal stress when SV wave is incident at 15: (a) maximum principal stress and (b) minimum principal stress.
(a)(b)Figure 10
Amplitude distribution of principal stress when SV wave is incident at 25: (a) maximum principal stress and (b) minimum principal stress.
(a)(b)SV wave 15° can be seen as the maximum principal stress of the left and right holes of the tunnel is elliptically distributed under the incident, and the distribution is basically symmetrical. The maximum principal stress of the left hole is smaller at the vault (−60°∼60°) and the inverted arch (150°∼240°), and the maximum span on the left (240°∼300°) and the right maximum span (90°∼150°) are larger, the maximum value appears at the left and right maximum span, and the maximum principal stress decreases with the increase of the vibration angle; the maximum principal stress of the right hole is flat, the left and right sides are large, and the upper and lower ends are smaller, the maximum principal stress occurs at the largest span on the right side (90°∼120°) and the largest span on the left (270°), the maximum principal stress at the vault (0°) and the bottom of the vault (180°) is the smallest, and the maximum principal stress in the right hole is the largest. The stress is less than the distribution of the maximum principal stress value of the left hole. The minimum principal stress of the left and right holes decreases with the increase of the vibration angle. The minimum principal stress of the left hole is larger in the range of 90°∼270° and is smaller in the range of (−60°∼60°), and the minimum principal stress is small. The stress appears at the largest span on the left and right. The distribution law of the minimum principal stress of the right hole is the same as that of the left hole, but the minimum principal stress of the vault of the right hole is greater than the minimum principal stress of the left hole.When the seismic wave is incident at 25°, the maximum principal stress is large on both sides and small at the upper and lower ends. The maximum value appears at the left and right maximum spans 90° and 270°, and the distribution range of the maximum principal stress in the left hole is larger than that in the right hole. The minimum principal stress is mainly distributed in the range of 90°∼270°, and the minimum principal stress of the right hole is axisymmetrically distributed, and the two largest spans on the left and right bear the greatest force.The Mises stress at the longitudinal vault of the tunnel is extracted. It can be seen from Figure11 that with the increase of the incident angle of the seismic wave, the Mises stress increases. Along the direction of the tunnel axis, the stress value changes due to the traveling wave effect, showing nonuniformity sex. The variation law of Mises stress in the right tunnel is the same as that in the left tunnel, the Mises stress in the right tunnel is significantly smaller than that in the left tunnel. It can be seen that under the action of seismic waves, the left tunnel receives a greater force.Figure 11
Distribution of peak Mises stress along the tunnel axis under different incident angles: (a) left tunnel Mises stress and (b) right tunnel Mises stress.
(a)(b)


## 4.1. Computational Model

A three-dimensional finite element model of a parallel tunnel is established in ABAQUS, as shown in Figure6. The longitudinal (Z) of the model is 100 m, the transverse (X) width is 200 m, the height (Y) is 100 m, the tunnel depth is 30 m, the distance between the two holes is 60 m, the soil adopts the Mohr–Coulomb constitutive model, the binding constraint is adopted between the soil and the primary lining, and the surface-to-surface contact between the primary lining and the secondary lining is adopted. The finite element model parameters are shown in Table 1.Figure 6
3D finite element model of the double-hole tunnel.Table 1
Material parameters.
Elastic modulus (GPa)Density (kg.m−3)Poisson’s ratioInternal friction angle (°)CohesionSoil321000.33300.1Initial lining3025000.2Second lining31.525000.2


## 4.2. Seismic Input

The American El-Centro wave is selected as the ground motion input, and the three-directional seismic wave is applied by means of the incident vector of the corner incident function. The acceleration time-history curve is shown in Figure7. The seismic wave propagation angle θ is the angle between the SV wave and the XY plane, which is taken as 0°, respectively, 15°, and 25°, for each propagation angle θv corresponding to a vibration angle 0°, 30°, and 60°.Figure 7
EI-Centro acceleration time-history curve.


## 4.3. Influence of Incident Angle on Axis Displacement

As shown in Figure6, the left and right double-hole tunnels are, respectively, determined at two points. The two points on the left tunnel are points A and B, and the two points on the right tunnel are points A′ and B′. The distance between points A and B is 80 m. The distance between “point” and B′ is also 80 m, and the SV wave is 15 m, respectively, 25°, and 90° oblique incidence double-hole tunnel. It can be seen from Figure 8 that the horizontal displacement of point A of the left hole increases with the increase of the incident angle of the seismic wave, but the increase is small; the relative vertical displacement of point A of the left hole increases with the increase of the incident angle. And the displacement in the case of normal incidence is significantly larger than the displacement in the other two incident angles and is 90°. The change curve of tunnel displacement peak when the incidence angles of seismic waves are 15° and 25°. The horizontal and vertical displacements of the right hole also increase with the increase of the incident angle, and the vertical displacement varies greatly, showing an obvious traveling wave effect. When the seismic wave is obliquely incident on the tunnel, there is an obvious traveling wave effect along the longitudinal direction of the tunnel. The nonuniform vibration effect of the seismic wave makes the movement of the tunnel along the longitudinal direction obviously different and accelerates the damage of the tunnel.Figure 8
Comparison of the displacement of the left and right tunnels: (a) horizontal displacement of point A in the left tunnel, (b) vertical displacement of point A in the left tunnel, (c) horizontal displacement of the right tunnel A′ point, and (d) vertical displacement of the right tunnel A′ point.
(a)(b)(c)(d)


## 4.4. Influence of Incidence Angle on Lining Stress Distribution

The finite element model was established, and the SV was 15°, respectively, and 25°. The propagation angles are obliquely incident on the tunnel, and each propagation angle corresponds to three vibration angles of 0°, 30°, and 60°. The distribution curves of the absolute value of the maximum principal stress and the minimum principal stress of the double-hole tunnel are shown in Figures9 and 10.Figure 9
Amplitude distribution of principal stress when SV wave is incident at 15: (a) maximum principal stress and (b) minimum principal stress.
(a)(b)Figure 10
Amplitude distribution of principal stress when SV wave is incident at 25: (a) maximum principal stress and (b) minimum principal stress.
(a)(b)SV wave 15° can be seen as the maximum principal stress of the left and right holes of the tunnel is elliptically distributed under the incident, and the distribution is basically symmetrical. The maximum principal stress of the left hole is smaller at the vault (−60°∼60°) and the inverted arch (150°∼240°), and the maximum span on the left (240°∼300°) and the right maximum span (90°∼150°) are larger, the maximum value appears at the left and right maximum span, and the maximum principal stress decreases with the increase of the vibration angle; the maximum principal stress of the right hole is flat, the left and right sides are large, and the upper and lower ends are smaller, the maximum principal stress occurs at the largest span on the right side (90°∼120°) and the largest span on the left (270°), the maximum principal stress at the vault (0°) and the bottom of the vault (180°) is the smallest, and the maximum principal stress in the right hole is the largest. The stress is less than the distribution of the maximum principal stress value of the left hole. The minimum principal stress of the left and right holes decreases with the increase of the vibration angle. The minimum principal stress of the left hole is larger in the range of 90°∼270° and is smaller in the range of (−60°∼60°), and the minimum principal stress is small. The stress appears at the largest span on the left and right. The distribution law of the minimum principal stress of the right hole is the same as that of the left hole, but the minimum principal stress of the vault of the right hole is greater than the minimum principal stress of the left hole.When the seismic wave is incident at 25°, the maximum principal stress is large on both sides and small at the upper and lower ends. The maximum value appears at the left and right maximum spans 90° and 270°, and the distribution range of the maximum principal stress in the left hole is larger than that in the right hole. The minimum principal stress is mainly distributed in the range of 90°∼270°, and the minimum principal stress of the right hole is axisymmetrically distributed, and the two largest spans on the left and right bear the greatest force.The Mises stress at the longitudinal vault of the tunnel is extracted. It can be seen from Figure11 that with the increase of the incident angle of the seismic wave, the Mises stress increases. Along the direction of the tunnel axis, the stress value changes due to the traveling wave effect, showing nonuniformity sex. The variation law of Mises stress in the right tunnel is the same as that in the left tunnel, the Mises stress in the right tunnel is significantly smaller than that in the left tunnel. It can be seen that under the action of seismic waves, the left tunnel receives a greater force.Figure 11
Distribution of peak Mises stress along the tunnel axis under different incident angles: (a) left tunnel Mises stress and (b) right tunnel Mises stress.
(a)(b)


## 5. Conclusions

(1)
When the SV wave is obliquely incident on the double-hole tunnel, the displacement of the left and right tunnels increases with the increase of the incident angle, and the vertical displacement is significantly larger than the horizontal displacement value, showing the traveling wave effect along the longitudinal direction of the tunnel.(2)
SV wave 15°: The maximum principal stress of the left and right holes of the tunnel is elliptically symmetrically distributed under the incident. The maximum principal stress of the left hole appears at the left and right maximum spans, and the maximum principal stress decreases with the increase of the vibration angle; the maximum principal stress of the right hole is flat with the right maximum span (90°∼120°). The maximum principal stress occurs at the maximum span (270°), and the maximum principal stress of the right hole is smaller than the distribution of the maximum principal stress value of the left hole. The minimum principal stress of the left and right holes decreases with the increase of the vibration angle, and the minimum principal stress of the left hole is larger in the range of 90°∼270°. When the SV wave is incident at 25°, the maximum principal stress is large on both sides and small at the upper and lower ends. The maximum value appears at the left and right maximum spans (90° and 270°), and the distribution range of the maximum principal stress in the left hole is larger than that in the right hole. The minimum principal stress is mainly distributed in the range of (90°∼270°), and the minimum principal stress of the right hole is axisymmetrically distributed, and the two largest spans on the left and right bear the greatest force.(3)
With the increase of the incident angle of the seismic wave, the Mises stress increases. Along the direction of the tunnel axis, the stress value changes due to the traveling wave effect, showing inconsistency. The Mises stress of the right tunnel is obviously smaller than that of the left tunnel, which shows that the left tunnel is subjected to a larger force under the action of seismic waves.(4)
In this paper, a three-dimensional finite element model is established to analyze the seismic response of the tunnel under the oblique incident SV wave. The three-dimensional model can consider the longitudinal seismic response of the tunnel, which is more similar to the actual project.



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*Source: 4368949-2023-03-09.xml*