Study on the Seismic Response of a Water-Conveyance Tunnel Considering Non-Uniform Longitudinal Subsurface Geometry and Obliquely Incident SV-Waves

Abstract

The longitudinal seismic response characteristics of a shallow-buried water-conveyance tunnel under non-uniform longitudinal subsurface geometry and obliquely incident SV-waves was studied using the numerical method, where the effect of the non-uniform longitudinal subsurface geometry due to the existence of a local one-sided rock mountain on the seismic response of the tunnel was focused on. Correspondingly, a large-scale three-dimensional (3D) finite-element model was established, where different incidence angles and incidence directions of the SV-wave were taken into consideration. Also, the non-linearity of soil and rock and the damage plastic of the concrete lining were incorporated. In addition, the wave field of the site and the acceleration response as well as damage of the tunnel were observed. The results revealed the following: (i) a local inclined subsurface geometry may focus an obliquely incident wave due to refraction or total reflection; (ii) a tunnel in a site adjacent to a rock mountain may exhibit a higher acceleration response than a tunnel in a homogeneous plain site; and (iii) damage in the tunnel in the site adjacent to a rock mountain may appear in multiple positions, and the effect of the incidence angle on the mode of compressive deformation and damage of the lining is of significance.

1. Introduction

Several earthquake events have caused heavy human tolls and substantial property damages [1,2,3] in recent years, and, thus, the seismic response characteristics of tunnels have attracted extensive attention. Many researchers have studied the dynamic response of underground tunnels using different methods [4], and, due to the complexity of wave propagation and scattering in soil media, the dynamic analysis of underground structures is always carried out economically and under realistic conditions with the aid of the numerical method [5,6,7].

The angle of incidence is a critical factor that determines the seismic tunnel response [8,9,10,11,12,13], and the influence of the incidence direction and the angles of the seismic wave on the response of a tunnel was first noticed in the anti-seismic studies of underground tunnels. Through a series of studies and analyses, Huang et al. [7] stated that the incidence direction of earthquake waves has significance on the seismic responses of long lined tunnels, and it is important to include the incidence angles of earthquake waves when mapping the seismic risk of tunnels. Li and Song [14] developed a novel numerical method for analyzing the longitudinal seismic response of tunnels under an asynchronous earthquake wave input, based on which the influence of the incidence angle and incidence direction on the longitudinal seismic response of tunnels was discussed. Considering obliquely incident seismic waves, Zhu et al. [15,16] carried out a study on the dynamic response of tunnels using a 2.5D coupled finite-element—boundary-element model. Zhou et al. [17] conducted a 3D seismic response analysis of an immersed tunnel in a horizontally layered site subjected to obliquely incident waves based on the precise dynamic stiffness matrix of the soil layer and half-space via combined viscous–spring boundary and equivalent node stress methods. Zhu et al. [18] investigated the effects of pulse-like ground motions on tunnels in saturated poroelastic soil for obliquely incident seismic waves using a 2.5D modeling technique. Considering 3D obliquely incident P- and SV-waves, Yang et al. [19] calculated the 3D seismic response of a lined tunnel embedded in a half-space using the 2.5D finite/infinite-element method. Huang et al. [20] investigated the dynamic performance of prefabricated utility tunnels considering the oblique incidence of P-waves. Huang et al. [21] developed a viscous–slip interface model to reflect the contact between the surrounding rock and tunnels, and the dynamic stress concentrations and surface displacements of two shallowly buried lining tunnels were examined using IBIEM, during which the effects of the angle on the medium’s responses were considered. Zhang et al. [22] evaluated the seismic response of an immersed tunnel in an ocean space under obliquely incident earthquake waves.

In addition, the responses of a tunnel in the longitudinal direction should not be neglected [23], especially when considering the tunnel lining damage which has been caused by recent earthquakes, and the longitudinal seismic response is another important perspective in the field of anti-seismic analyses of tunnels. Considering the longitudinal variation in ground properties, Park et al. [24] developed a longitudinal displacement profile-based procedure for simulating the dynamic response of tunnels and performed a series of pseudo-static 3D finite-element analyses of tunnels under spatially varying ground motion. Based on the generalized response displacement method, Miao et al. [25,26] studied the longitudinal seismic response of a metro tunnel in a site with non-uniform soil properties and found that the inter-segment opening of a tunnel in soft soil is larger than that in hard rock. Liu et al. [27] proposed a new multi-scale numerical model and calculated the longitudinal seismic response of a shield tunnel through a typical site with a soft–hard strata junction, and it was found that the dynamic response of the shield tunnel rose dramatically near the soft–hard strata interface because of the non-uniform deformation of the soil in that area. Yuan et al. [28], Yu et al. [29], and Yan et al. [30] carried out a series of shaking-table tests to investigate the longitudinal seismic response of a tunnel under non-uniform ground motion.

As a matter of fact, many circular underground structures such as cavities and tunnels have been constructed near various canyons in seismic zones [31], and the local subsurface geometry may possess a remarkable effect on the seismic response of underground structures, and the longitudinal non-uniform subsurface geometry (i.e., subsurface geometry variation) may affect the longitudinal seismic response characteristics of tunnels. However, although studies on the seismic response analyses of tunnels are extensive, the longitudinal dynamic response and damage characteristics of tunnels under non-uniform longitudinal subsurface geometry are not well known and understood. Therefore, in the present paper, taking the incidence angle and direction of SV-waves into account, the authors carried out a numerical study of this problem. The rest of this paper is organized as follows. The definition of the problem discussed in this paper is presented in Section 2. In Section 3, the setup of the numerical models is presented in detail. Subsequently, Section 4 offers the results of the longitudinal dynamic responses of the tunnels. Finally, concluding remarks are given in Section 5.

2. Problem Definitions

The Dianzhong Water Diversion Project is the largest water diversion project under construction in China. It is located in the Yunnan province, China, and passes through Lijiang city, the Dali Prefecture, the Chuxiong Prefecture, Kunming city, Yuxi city, and the Honghe Prefecture. The total length of the project is about 664 km, more than 90% of which is tunnel engineering. What is noticeable is that multiple intervals of tunnel engineering in Kunming city are in soft soils, significantly near a rock mountain, as shown in Figure 1, and the burial depth of the tunnel is generally small (tens of meters). In other words, the tunnel might pass through a plain soft-soil site first, then a soft-soil site near a rock mountain, and, subsequently, a plain soft-soil site again, which leads to the fact that the longitudinal non-uniform subsurface geometry along the tunnel, induced by a local rock mountain, may possess a considerable influence on the seismic response of said tunnel. The existence of a local one-sided rock mountain near the tunnel may affect the propagation of incident seismic waves and, thus, brings about differences in the seismic responses between tunnels with and without adjacent rock mountains. The mechanism and regularity of this influence are studied in this paper, where different incidence directions and angles of SV-waves have been considered, as shown in Figure 2.

In summary, the influence of subsurface geometry variations along a tunnel’s path on the seismic response and damage characteristics of said tunnel are the main focus of this paper. It is hereby noted that the original and practical subsurface geometry was not completely reproduced in the numerical model in this paper, because the local elevation and site information was classified, and, thus, the authors could not attain or publish the corresponding detailed information at present. Therefore, simplified models were established instead.

3. Establishing the Numerical Model

3.1. General Model Information

Figure 3 and Figure 4 demonstrate the dimensions of the one-sided mountain–plain site tunnel model and the corresponding mesh, respectively. The one-sided mountain–plain site tunnel model was 210 m, 100 m, and 50 m in length, width, and depth, respectively, while the mountain was 90 m, 40 m, and 50 m in length, width, and height. A circular underground tunnel was set in the model, and the length, outer radius, lining thickness, and burial depth (the distance between the site surface and the tunnel center) of the tunnel were 210 m, 2.5 m, 0.3 m, and 20 m, respectively. In terms of mesh, the C3D8R element was used to discretize the model, and the maximum size of the mesh for the whole model should be set to 1/6~1/12 of the wavelength in soft soil [32]. Herein, eight elements for the wavelength were assumed and employed (i.e., l_max ≤ Vs/8f_max, where l_max is the maximum element length; Vs denotes the shear wave velocity; and f_max is the maximum accurate frequency, set to 25 Hz in this study). The element size was set to 1.5 m for the whole site model, which can assure the adequate transmission of a wave in soil and rock. Although this size is much too small for rock material and lowers the calculation efficiency, it can make sure that seismic waves can propagate in a rock medium more adequately and brings about no influence on the results. The element number of the one-sided mountain–plain site tunnel model was 371340. Moreover, the maximum time increment was set to 0.00005 s. As for the tunnel lining, the C3D8R element was also used, and the element size was set to 1.5 m as well so that the nodes of the soil overlapped those of the lining. The coupled Lagrangian–Eulerian method was employed to describe the water–tunnel lining interaction and the sloshing effect of water under seismic waves [33]. The Eulerian element was utilized to discrete the water domain. In this model, the normal direction contact passes via the contact compressive stress transfer interaction, and the element nodes on the surfaces satisfy Hook’s Law and the Harmonized Condition of Displacement. Tangential contact obeys Coulomb’s law of friction. To obtain the ultimate stress to satisfy Coulomb’s law, the following formula was used:

$${\tau }_{\mathrm{crit}}=\mu \cdot P,$$

(1)

where $\mu$ and P denote the coefficient of friction and the normal contact pressure, respectively. Herein, the soil–tunnel lining interface friction coefficient was set to 0.4 [33,34].

In this study, the SV-wave was considered, and two sets of numerical simulations were designed, each one consisting of four cases with different incidence angles, as shown in

Table 1. Sets of numerical simulation.

No.		Model	Incidence Direction	Wave Type	Incidence Angles
Set-1	1	One-sided mountain tunnel model	Left-side incidence	SV-wave	33.69°
	2				26.57°
	3				18.43°
	4				11.31°
Set-2	5		Right-side incidence		33.69°
	6				26.57°
	7				18.43°
	8				11.31°

. The incidence angles that were taken into consideration were 11.31°, 18.43°, 26.57°, and 33.69°, which corresponded to following vector directions: (x, y) = (1, 5), (1, 3), (1, 2), and (1, 1.5), respectively. Two different incidence directions were set, i.e., left-side incidence and right-side incidence. In the cases with left-side incidence, the wave passed through the rock mountain first and then the tunnel (see Figure 2a), while the wave passed through the tunnel first and then the rock mountain in the cases with right-side incidence (see Figure 2b).

3.2. Constitutive Model

The modified Davidenkov viscoelastic dynamic constitutive model was used to describe the dynamic characteristics of soil and rock [35], through the following expressions:

$$\tau -{\tau }_{\mathrm{c}}=G_{\max }\cdot \left(\gamma -{\gamma }_{\mathrm{c}}\right)\cdot \left[1-H\left(\frac{\left|\gamma -{\gamma }_{\mathrm{c}}\right|}{2n_a}\right)\right],$$

(2)

$${\left(2n_a{\gamma }_0\right)}^{2\mathrm{B}}={\left({\gamma }_{\mathrm{ex}}\pm {\gamma }_{\mathrm{c}}\right)}^{2\mathrm{B}}\cdot \left(\frac{1-R}{R}\right),$$

(3)

$$R={\left(1-\frac{{\tau }_{\mathrm{ex}}\pm {\tau }_{\mathrm{c}}}{G_{\max }\cdot ({\gamma }_{\mathrm{ex}}\pm {\gamma }_{\mathrm{c}})}\right)}^{\frac{1}{A}},$$

(4)

where τ and γ are the shear stress and the strain, respectively; G_max is the initial shear modulus; A, B, and γ₀ are the dimensionless fitting parameters; n_a is the coefficient controlling the scale of hysteresis loop; τ_c and γ_c denote the shear stress and the strain at the last stress reversal point, respectively; and the τ_ex and γ_ex are the shear stress and strain at the last extreme value point, respectively. The parameters of this model for soil and rock are shown in

Table 2. Values of the fitting parameters of the modified Davidenkov model.

Soil Types	Density (kg/m3)	Poisson’s Ratio	Shear Wave Velocity (m/s)	Compressive Wave Velocity (m/s)	Initial Shear Modulus (MPa)	Fitting Parameters
						A	B	γ0 (×10−4)
Medium-coarse sand	2000	0.25	280.18	485.28	157	1.20	0.37	7.4
Bed rock	2522	0.2	899.82	1469.40	2042	1.30	0.40	10.0

.

The tunnel lining’s concrete was assumed to be C30, and, to describe the compressive and tensile damage and degrading of the concrete, the concrete damage–plasticity constitutive model proposed by Jeeho and Fenves [36] was adopted. The corresponding material property parameters for the tunnel’s concrete lining are shown in

Table 3. Material property of the tunnel’s concrete lining.

Parameter	Description	Value
$\rho$	Density	2450 kg/m3
$E_0$	Elastic modulus	30,000 MPa
$\mu$	Poisson’s ratio	0.18
$\Psi$	Dilation angle	36.31°
${\epsilon }_0$	Flow potential eccentricity	0.1
${\sigma }_{b0}/{\sigma }_{c0}$	The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress	1.16
K	The ratio of the second stress invariant on the tensile meridian	0.6667
$V_c$	Viscosity coefficient	0.0005
${\omega }_c$	Compression stiffness recovery parameter	1.0
${\omega }_t$	Tensile stiffness recovery parameter	0.0

. The physico-mechanical parameters of the fluid in the present analysis are described in

Table 4. Physico-mechanical parameters of the water inner tunnel.

Material	Density (kg/m3)	Dynamic Viscosity Coefficient (Ns/m3)	Equation of State Parameter (m/s)
Water	1000	0.0013	1400

.

3.3. Input Ground Motion and Boundary Conditions

Since most ground motion records have a long duration, the artificial simulation of ground motion was used in this paper to generate ground motion with a short duration and reduce the time cost of the calculation. The classical spectral representation method [37] was adopted to simulate the input ground motion:

$$f_j(t)=A(t){\displaystyle \sum _{l=1}^N\sqrt{2S^0({\omega }_l)\Delta \omega }}\cos ({\omega }_{l}t+{\phi }_l)$$

(5)

with $S^0({\omega }_l)$ being the auto-power spectral density (PSD) and $\Delta \omega$ denoting the bandwidth and N the number of frequency intervals. ${\omega }_l$ is the lth value of frequency $\omega$ and can be calculated using ${\omega }_l=l\Delta \omega$, and Δω can be calculated using $\Delta \omega ={\omega }_u/N$, where ${\omega }_u$ represents an upper cut-off frequency beyond which the elements of auto-PSD may be assumed to be zero for any time instant. Herein, ${\omega }_u$ = 25 Hz, and N = 2048. ${\phi }_l$ is a random phase angle uniformly distributed in [0, 2π], and the relevant probability density function can be expressed as $f(x)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$(2\pi -0)$}\right.$. Moreover, the following intensity envelope function was employed to proceed with the simulation of temporal non-stationary ground motion:

$$A(t)=\left\{\begin{array}{c}{(t/t_0)}^2\\ 1\\ \exp (-0.155\cdot (t-t_n))\end{array}\right.\begin{array}{c}\\ \\ \end{array}\begin{array}{c}0\le t\le t_0\\ t_0<t\le t_n\\ t_n<t\le T\end{array}$$

(6)

with $t_0=2s$, and $t_n=4s$.

The Clough–Penzien model [38] was utilized to establish the PSD mentioned above, with the following expression:

$$S^0(\omega )=S_0\frac{\left(1+4{\zeta }_g^2{\left(\frac{\omega }{{\omega }_g}\right)}^2\right)}{{\left(1-{\left(\frac{\omega }{{\omega }_g}\right)}^2\right)}^2+4{\zeta }_g^2{\left(\frac{\omega }{{\omega }_g}\right)}^2}\times \frac{{\left(\frac{\omega }{{\omega }_f}\right)}^4}{{\left(1-{\left(\frac{\omega }{{\omega }_f}\right)}^2\right)}^2+4{\zeta }_f^2{\left(\frac{\omega }{{\omega }_f}\right)}^2}.$$

(7)

with ${\omega }_g$ and ${\zeta }_g$ denoting the central frequency and damping ratio of the Tajimi–Kanai PSD function, and ${\omega }_f$ and ${\zeta }_f$ being the central frequency and damping ratio of the high-pass filter. And, according to Ref. [39],

$$\begin{array}{c}{\omega }_g=10\pi ;{\zeta }_g=0.6;\\ {\omega }_f=0.5\pi ;{\zeta }_f={\zeta }_g;\end{array}$$

(8)

$$S_0=\frac{{\sigma }^2}{\pi {\omega }_g\left(2{\zeta }_g+\frac{1}{2{\zeta }_g}\right)};\sigma =1.5 \mathrm{m}/{\mathrm{s}}^2$$

(9)

The duration of the simulated ground motions was set to 20 s and the time interval to 0.01 s. The simulated ground motion is shown in Figure 5. It should be noted that the peak value of the simulated ground motion was about 2.58 m/s², and, clearly, this ground motion was a strong earthquake motion.

The viscous–elastic artificial boundary [7,33,40] composed of linear springs and dampers was used on the lateral and bottom sides of the site model. The velocity and displacement time-histories shown in Figure 5 were employed to derive the equivalent nodal force on the model boundary nodes.

4. Results and Analysis

4.1. Analysis of Wave Field

Figure 6 presents the diagrams of the wave field in the mountain–plain site for the SV-wave. Based on Snell’s law, the incidence angle, and the wave velocities in soil and rock, the angles of the transmitted SV- and P-waves at the soil–rock interface for the left-side-incident SV-wave are shown in Table 4, and the angles of the reflected waves for right-side incidence are in

Table 5. Angles of the transmitted SV- and P-waves at the soil–rock interface for a left-side-incident SV-wave.

Incidence Angle/°	Angle between Incident Wave and Normal Direction of Soil–Rock Interface/°	Transmitted Angle of SV-Wave/°	Transmitted Angle of Converted P-Wave/°
33.69	46.31	13.01	22.95
25.67	54.33	14.65	25.98
18.43	61.57	15.89	28.31
11.31	68.69	16.86	30.16

. Under the considered incidence angles, total reflection would occur on the left or right interface between soil and rock for left- or right-side-incident SV-waves. Since the right side of the mountain was inclined, total reflection was more likely to show up on the right side of the mountain under the right-side incidence of the wave. In the subsequent text, if lacking a special statement, the interface stands for the soil–rock interface on the right side of the mountain. Moreover, based on Snell’s law, the incidence angles considered herein, and the wave velocities in soil and rock, the critical angle that reflected P-wave can show up is 35.26° on the interface; however, the angle between the incident SV-wave and the normal direction of the interface was beyond the above-mentioned critical angle, and, thus, no reflected P-wave could be produced on the interface. Additionally, for the left-side-incident SV-wave, the tunnel was subjected to the incident wave, the transmitted converted waves coming from the rock mountain side, and the reflected converted waves from the site’s surface. For the right-side-incident SV-wave, the tunnel was mainly subjected to the incident wave, the reflected SV-wave on the soil–rock interface after total reflection, the reflected converted waves coming from the site’s surface, and the subsequent reflected waves. Due to the obstacle induced by total reflection, part of the waves could not go into the rock mountain from the soil, and, thus, multiple reflections would occur in the soil for right-side incidence. Taking all the above-mentioned information into consideration and based on Figure 6, it can be found that, for both left-side incidence and right-side incidence, the tunnel’s lining may be subjected to multiple waves pointing to the tunnel, and an inclined subsurface geometry focuses both left-side and right-side obliquely incident waves.

Figure 7 displays the displacement contour of the site for the SV-wave with an incidence angle = 33.69°, where the left, bottom region of Figure 7b is enlarged to show the wave’s front. Note that the observation time instant during which the incident wave just passes through the rock mountain has been chosen to observe the propagation direction of the transmitted wave. Firstly, it can be seen that the displacement contour is obliquely layered, and this demonstrates that the waves are, indeed, obliquely incident to the site. Secondly, as shown in the enlarged zone in Figure 7b, as the wave propagates, the inclined angle of the wave’s front, near the soil–rock interface on the left side of the mountain, increases gradually, and this might suggest that, when arriving at the interface between soil and rock, waves encounter some obstacles. The wave’s front tends to up-warp at the soil–rock interface in comparison to the initial incidence, which suggests that total reflection indeed occurs. Thirdly, the propagation direction in the rock mountain and that in the soil site behind the mountain are noticeably different due to refraction. Based on Snell’s law, the shear wave velocity in soil and rock, and the incidence angle, the transmission angle is equal to 13° for the SV-wave transmitting from the rock to the soil. Considering that the slope angle of the right side of the mountain is 80°, the angle between the transmitted wave coming from the rock mountain to the soil site and the horizon line should be 23°. Then, the propagation direction of the transmitted wave (solid line with a single arrow) derived from the theoretical calculation is given in Figure 7b and compared with the wave’s front in the displacement contour obtained by numerical simulation. It can be seen that the wave’s front of the transmitted wave in the numerical results is just perpendicular to the theoretical propagation direction. In the meantime, the predefined direction of the incident wave (solid line with double arrows) is also presented in Figure 7, and it can be found that the numerical results are in good agreement with the theoretical propagation direction. Taking all the above-mentioned information into account, the analysis of the wave field shown in Figure 6 can be verified, and the validity of this study can also be verified to some extent.

4.2. Acceleration Response of Tunnel Lining

The acceleration responses of the tunnel under SV-waves with different incidence directions and angles were calculated and analyzed. Note that, since the longitudinal seismic response of the tunnel is the object of concern, the longitudinal envelope of the acceleration response is presented and analyzed herein. As a matter of fact, the seismic responses at all the outer element nodes at the arch vault, arch bottom, left arch, and right arch along the tunnel model were extracted, and then the envelopes of the seismic response for the arch vault, arch bottom, left arch, and right arch were obtained and are displayed in Figure 8, Figure 9, Figure 10 and Figure 11, respectively. According to Table 5 and

Table 6. Angles of the reflected SV- and P-waves at the soil–rock interface for a right-side-incident SV-wave.

Incidence Angle/°	Angle between Incident Wave and Normal Direction of Soil–Rock Interface/°	Reflected Angle of SV-Wave/°	Reflected Angle of Converted P-Wave/°	Transmitted Angle of SV- or P-Wave/°
33.69	66.31	66.31	No reflected converted P-wave	No transmitted wave
25.67	74.33	74.33
18.43	81.57	81.57
11.31	88.69	88.69

, the propagation and vibration directions of the various waves in the site for incidence angles = 11.31° and 33.69° are drawn in Figure 12 and Figure 13, respectively, to facilitate a relevant analysis.

Figure 8, Figure 9, Figure 10 and Figure 11 show that the acceleration response at the arch vault, arch bottom, left arch, and right arch of the tunnel’s lining in the vertical direction are ultimately lower than those in the horizontal direction. This can be attributed to the fact that the vibration direction of the SV-wave is perpendicular to the propagation direction, and the incidence angles taken into consideration are lower than 45°, and, thus, the vertical component of motion is smaller. As the incidence angle increases, the horizontal component of the acceleration response of the lining decreases, while the vertical component increases gradually, because, as the incidence angle increases, the horizontal component of the incident SV-wave decreases, whereas the vertical component increases. The above-mentioned results are in agreement with theoretical knowledge, which verifies the validity of this study to some extent.

Under either left incidence or right incidence, the horizontal and vertical acceleration responses of the 60~150 m tunnel interval (i.e., the tunnel interval near the rock mountain) are noticeably higher than those of other intervals. In other words, the acceleration response of the tunnel in the plain–mountain site is greater than that in the plain site. As stated in Section 4.1, the wave field of the soil site located on the right side of the mountain for right incidence is remarkably complex due to the total reflection at the soil–rock interface and surface reflection, and this results in the fact that the tunnel’s lining may be subjected to multiple waves, which aggregate the vibration of the lining. Similarly, for left incidence, the transmitted SV- and P-waves produced by the left-side-incident SV-wave at the soil–rock interface possess different propagation and vibration directions from the incident wave that comes from the boundary beneath the tunnel (see Figure 12 and Figure 13), and, thus, the superposition of these waves may aggregate the vibration of the tunnel’s lining. However, in the plain-soil site, the site conditions are uniform, and no total reflection or refraction occur. Therefore, compared to the tunnel in the plain site, the tunnel interval adjacent to the rock mountain may be subjected to multiple waves possessing different directions of propagation and vibration due to obliquely incident wave and non-uniform subsurface geometry conditions, and, thus, the acceleration response of the tunnel in the 60~150 m range is higher.

Moreover, the horizontal and vertical acceleration responses of the arch vault, arch bottom, left arch, and right arch of the 60~150 m tunnel interval under left-side incidence are lower than those under right-side incidence. This is because multiple reflections occur at the site’s surface and soil–rock interface in the cases with right-side incidence, and, thus, multiple conversions of waves may occur to a great extent. Therefore, the complex superposition of waves induces the aggravation of the lining’s acceleration response. Regarding left-side incidence, the propagation direction of the transmitted and reflected waves generally points to the lateral model boundary, and, thus, it is easier for the waves to run outside of the site, meaning that the degree of multiple reflections is lower.

4.3. Damage Assessment of Tunnel Lining

The tensile damage contours in the longitudinal view of the tunnel lining under right-side-incident SV-waves for the different incidence angles considered are shown in Figure 14, where the selected time intervals are (i) the moment when the damage first appeared and (ii) the end time of the total duration of the input wave. As it can be seen in the longitudinal dimension, the position in which the damage first showed up is located in the tunnel interval of 60~150 m. This is because the existence of a local rock mountain brings about the reflection of waves (see Figure 12 and Figure 13), and the superposition of the reflected wave and the incident wave aggravates the deformation of the lining and then damages it, whereas, in a plain site, the above-mentioned reflection and superposition of waves are absent. In the circumferential dimension, the positions are different among the cases: as the incidence angle decreases, the position in which the damage first showed up tends to move from the vault towards the hance. Since the damage of the tunnel’s lining is highly related to deformation, the deformation of the lining under right-side-incident SV-waves with different incidence angles at the moment when the damage first appeared was observed, as shown in Figure 15, where the scaling factor is set to 300 so that the deformation of the lining can be investigated more clearly and two profiles are selected for each case therein: (i) the middle of the plain site with a longitudinal distance = 30 m; and (ii) the middle of the plain–mountain site with a longitudinal distance = 105 m. As it can be seen, thanks to the 300 scaling factor used, the ovaling deformation of the lining seems to be significant, and the compressive deformation is approximately perpendicular to the incidence direction of the SV-wave. To better observe the relation between the ovaling deformation of the tunnel lining and the incidence angles, the major axis (dash line) of the ovaling lining in the profile for the middle of the plain–mountain site with a longitudinal distance = 105 m is given for each case. Clearly, the inclined angle of the major axis tends to be greater with a decreasing incidence angle, meaning that the part of lining which is initially subjected to severe tension, i.e., the position where tensile damage first appears, gradually changes from the vault to the hance with a decreasing incidence angle.

In addition, for each case with right-side incidence, the final damage area reduces as the incidence angle decreases. As shown in Figure 13b, when the incidence angle = 33.69°, the tunnel lining is subjected to a directly incident SV-wave and reflected SV-waves from the soil–rock interface and the site’s surface, and the difference in the angle between the incident and reflected waves essentially induces the tunnel lining to experience compressive deformation in two directions. The fact that tensile damage shows up at the arch vault, arch bottom, left arch, and right arch of the 60~150 m tunnel interval can well verify the above-mentioned statement. By contrast, when the incidence angle = 11.31° (see Figure 12b), the difference in the angle between the incident and reflected waves is much smaller, and, thus, the tunnel lining is approximately subjected to a unidirectional deformation, and, thus, tensile damage only appears in a single, diagonal direction. In summary, as the incidence angle decreases, the mode of compressive deformation changes from multi-directional compression to unidirectional compression. To verify this statement, the deformation of tunnel lining in the profile for longitudinal distance = 105 m at time interval = 6.0 s under right-side-incident SV-wave was observed, as shown in Figure 16. It can be obviously found that at the same time interval the deformation modes of lining under right-side-incident SV-wave with predefined various incidence angles are significantly different, and when the incidence angle = 33.69°, the shape of deformed lining tends to be rectangular. This demonstrates that the lining may be subjected to multi-directional deformation.

The tensile damage contours in the longitudinal view of the tunnel’s lining under left-side-incident SV-waves for the different incidence angles considered and different time intervals are shown in Figure 17. The deformation of the lining under left-side-incident SV-waves with different incidence angles at the moment when damage first appears is shown in Figure 18. Similar to the cases with right-side incidence, the part of lining that is initially subjected to severe tension gradually changes from the vault to the hance with varying incidence angles in the cases with left-side incidence. However, what is different from the right-side cases is that, in all four cases of left-side incidence, the tunnel lining displays damage in multiple directions. As shown in Table 5, the transmitted angle of the SV- and P-waves at the soil–rock interface varies slightly with varying incidence angles, meaning that the propagation direction of the transmitted SV- and P-waves is always significantly different from that of the incident waves (see Figure 12 and Figure 13). Therefore, for all the incidence angles considered, the tunnel lining always experiences compression in multiple directions. Thus, the damage distribution of the lining for the cases with left incidence is always multi-directional.

5. Conclusions

Considering longitudinal non-uniform subsurface geometry conditions and different incidence directions and arbitrary angles of the SV-wave, the longitudinal seismic response and damage characteristics of a water-conveyance tunnel were investigated in this paper using the numerical method. The main conclusions can be summarized as follows: (i) an inclined subsurface geometry may focus both left-side and right-side obliquely incident waves due to refraction or total reflection; (ii) a tunnel in a site adjacent to a rock mountain may exhibit higher horizontal and vertical acceleration responses than a tunnel in a homogeneous plain site due to the refraction or reflection induced by the rock mountain; and (iii) a tunnel in a site adjacent to a rock mountain may be subjected to compression in multiple directions, thus damage may appear in multiple circumferential directions, and, as the incidence angle decreases, the mode of tensile damage for right-side incidence may change from multi-directional damage to unidirectional damage, while that of tensile damage for left-side incidence may always be multi-directional.

The chosen incidence angles in this study were all smaller than the critical incidence angle of the SV-wave, and, thus, the horizontal component of the input wave and the seismic response were higher than those in the vertical direction. The calculation and analysis results of cases in which the vertical component is higher may be needed in the future.