Seismic Response and Damage Analysis of Shield Tunnel with Lateral Karst Cavity under Oblique SV Waves

Abstract

The dynamic effects of the lateral karst cavity on the shield tunnels under different incident angles of seismic waves are investigated by numerical analysis in this paper, based on the Dalian Metro Line 5 project. The viscous-spring artificial boundary is applied and verified to guarantee the accuracy of seismic input. A simplified finite element model of shield tunnel is established based on the equivalent bending stiffness model. This paper compares the seismic response characteristics and damage states of the tunnel under different incident angles by analyzing the axial deformation, stress distribution, and damage severity, respectively. A new damage state classification criterion is proposed by introducing the relationship between cracks and tensile damage. The results show that the tunnel’s affected scope by the lateral karst cavity is twice the cavity diameter. As the incident angle increases, the tunnel’s displacement and stress increase and show the structural spatial difference, and the tunnel’s damage state is increasingly severe. The displacement and stress reach the max values when the incident angle is 30°. The cracks along the axial direction extend on the outer surface of the vault and bottom, and the crack width is greater than 0.2 mm, as that angle is 30°. The damage severity at the tunnel’s central zone is minimum along axial directions during seismic action, while the damage concentration occurs on the bottom at the end of seismic. The lateral karst cavity plays an energy dissipation and vibration reduction role to a certain extent, but it also aggravates the local damage. This paper can serve as a reference for the seismic design of tunnels in karst regions.

1. Introduction

Tunnel structure is an indispensable and important link in urban lifeline engineering systems, which is an essential element to guarantee the normal operation of urban life. In recent years, many surveys reported that the huge damage to tunnel structures under earthquakes gained great attention, such as the tunnels in the Chi-Chi (1999), Mid-Niigata (2004), Wenchuan (2008), and Kumamoto (2016) earthquakes [1,2,3,4]. Restricted by the urban route planning and site conditions, the tunnel may inevitably pass through adverse geology, such as karst regions and fault fracture zones. The karst region is a typical type among adverse geological conditions, covering about one-tenth of the Earth’s terrestrial areas and about one-third of the Chinese mainland [5,6]. In the past decades, numerous tunnels were constructed in the karst regions, such as the Albula Tunnel II in Switzerland [7], the Sol-an Tunnel in South Korea [8], the Gavarres Tunnel in Spain [9], and the Huaguoshan Tunnel in China [10]. The rock mass degradation caused by karstification will reduce the tunnel’s stability, and the external vibration will cause ground collapse and other problems. The tunnels in karst regions are vulnerable to excessive deformation, leading to huge disasters during earthquake events. Therefore, it is crucial to explore the seismic response of tunnels for ensuring structural safety in karst regions.

A tunnel’s safety and stability in karst regions are state of art issues. Tunnels in karst regions will face many great risks, such as water and mud inrush, partial karst ground collapse, and collapse of the karst cave roof. Many assessment models were proposed to evaluate the stability of tunnels in karst regions. Based on the prediction of karst characteristics and the mechanism of karst-induced collapse, some assessment models were successfully applied in tunneling projects [11,12,13], such as the Wuhan Metro Line 11 and Doupengshan tunnel. Wang et al. [14] put forward a risk assessment model of water inflow and inrush based on 13 indicators, and utilized it to evaluate the construction of the Qiyueshan Tunnel. Huang et al. [15] and Wang et al. [16] established a prediction model for the safe thickness between the tunnel and the karst cavity by theoretical derivation and model calculation, respectively. Labiouse et al. [17] and Blümling et al. [18] investigated the fracture mechanics characteristics of geotechnical materials based on the laboratory and in situ simulation tests of tunnel excavation. With recent advances in geophysical technologies (such as electromagnetic prospecting, ground penetrating radar, seismic prospecting, elastic wave testing, and tomography), it is achievable to assess the scale, filling degree, and boundary of karst regions with high technical accuracy. On this basis, a series of effective and timely measures are developed and adopted to the tunnel structure, such as in the Yichang–Wanzhou railway case [19]. To avoid ground collapse caused by karst caves, the full grouting filling method is applied in the shallow karst caves around the Wuhan Metro Line 6, i.e., a vertical isolated diaphragm wall by rows of curtain grouting holes [20]. To deal with the sink hole in Guangzhou Metro Line 9, Cui et al. [21] designed a treatment technique of slurry grouting by sleeve valve pipe grouting. Moreover, many measures are set to keep the structural safety and stability during construction. Among them, it is noteworthy that 3 m is set as the red line of horizontal distance between the cave and the tunnel [20,21], while there is no treatment for the cave when the distance is longer than 3 m. Generally, the operational life of a tunnel system is decades, even more than a hundred years. This distance setting can greatly reduce the occurrence of disasters during tunnel excavation, but the potential hazards caused by the karst cavity that threaten the long-term stability of the tunnels are still not to be ignored. If the tunnel is located in karst regions and earthquake prone areas at the same time, the structural damage may be much more serious. At present, there are only a few studies on this issue. Zang et al. [22] conducted the seismic analysis with a lateral karst cavity based on the elastic models; however, they ignored the axial deformation and damage characteristics of the tunnel. Therefore, it is necessary to study the dynamic characteristics of the tunnel with lateral karst cavity under earthquakes.

The seismic response characteristics of the structure are related to the types and the incident angles of the seismic wave. Compared with P waves, the tunnel’s damage caused by SV waves is more serious [23,24,25,26]. Savigamin et al. [27] derived two analytical solutions to explore the structural axial direction response of the circular tunnel under the shear wave. Lyu et al. [28] studied the tunnel collapse mechanism subjected to seismic force and seepage force, and deduced the critical height between the karst cave and the tunnel. Some studies show that the oblique SV waves can cause severe damage to tunnels, and the damage severity is positively correlated with the incident angle of SV waves [29,30]. The above research reveals that the obliquely incident angles of SV waves will affect the damage severity to the tunnel. Therefore, this paper focuses on the dynamic characteristics of tunnels with a lateral karst cavity under obliquely incident SV waves.

This paper investigates the seismic response characteristics and damage analysis of the shield tunnels in karst areas under the oblique SV waves. The seismic input is based on Python coding, and a three-dimensional model is used to verify the exactitude. A simplified finite element model of the shield tunnel is established based on the equivalent bending stiffness model. The displacement deformation and stress distribution characteristics of the tunnel are investigated under oblique waves. The affected tunnel’s scope and the most unfavorable incident angle affecting the tunnel is analyzed. By introducing the relationship between tensile damage and cracking, a new damage state classification criterion is proposed. The tunnel’s damage process during the seismic action at the most unfavorable incidence angle is analyzed. Additionally, the tunnel’s damage states at each incident angle are compared.

2. Artificial Boundary and Model Validation

2.1. 3D Viscous-Spring Artificial Boundary

The input method of ground motion should be reasonably addressed while simulating the structural seismic response. In recent years, many numerical methods were established to simulate artificial boundary, such as the viscous-spring artificial boundary (VSAB) [31], infinite element method [32,33], proportional boundary finite element method [34], and perfectly matched layer [35]. Among them, VSAB is widely used in numerical calculation, because it not only can simulate the elastic recovery and radiation damping effectively, but can also reduce the low-frequency drift error in the viscous boundary successfully.

Based on the Python coding, the VSAB is set to placing springs and dampers on the artificial cut-off boundary [36]. Specifically, the incident wave is transformed into an equivalent effect attached to an artificial boundary node. The 3D viscous-spring artificial boundary is as shown in Figure 1, and K, C represents the springs and dampers, respectively. By setting the springs and dampers, the reflection of scattered waves at the artificial boundary can be eliminated, and the propagation of waves from the finite domain to the infinite domain can be accurately simulated.

In this model, the coefficients of spring and damper are as follows:

In the normal direction:

$$K_n={\alpha }_{n}G\times A_l/R,C_n=\rho c_p\times A_l,$$

(1)

In the tangential direction:

$$K_{\tau }={\alpha }_{\tau }G\times A_l/R,C_{\tau }=\rho c_s\times A_l,$$

(2)

where the subscript n and τ represent the normal and tangential direction, respectively; α represent the modified coefficient, the suggested value of normal and tangential direction are 1.333 and 0.667, respectively [31]; R represents the distance between the boundary point and the load point; ρ is the mass density of the material; G is the shear modulus; c_s and c_p are the velocities of the shear and compressive wave in the material, respectively; and A_l is the truncated boundary area of node l.

To input the seismic wave accurately, the stresses and displacements of each node calculated on the artificial boundary should be equivalent to the free field [38,39]. Hence, stress fields at node $l\left(x_l,y_l,z_l\right)$ generated by the force are as follows:

$$\sigma \left(x_l,y_l,z_l,t\right)=F_l\left(t\right)-f_l\left(t\right)$$

(3)

where $F_l\left(t\right)$ represents the stress assumed to be exerted on the node l of the VSAB; $\sigma \left(x_l,y_l,z_l,t\right)$ represents the stress at node l of the artificial boundary, which is generated by the displacement $u\left(x_l,y_l,z_l,t\right)$, and $f_l\left(t\right)$ represents the stresses calculated by the springs and dampers. The equation of dampers and springs can be given as

$$f_l\left(t\right)=C\dot{u}\left(x_l,y_l,z_l,t\right)+Ku\left(x_l,y_l,z_l,t\right)$$

(4)

where C and K represent the coefficients of dampers and springs.

Combining Equation (3) and Equation (4), we can get the stress on the artificial boundary, as

$$F_l\left(t\right)=\sigma \left(x_l,y_l,z_l,t\right)+C\dot{u}\left(x_l,y_l,z_l,t\right)+Ku\left(x_l,y_l,z_l,t\right).$$

(5)

2.2. Model Validation

This section mainly verified the accuracy of the VSAB by a 3D finite element model. In finite element analysis, element size can affect the results’ stability and accuracy. According to the Liao’s research [40], the optimal element size is analyzed to ensure the calculation accuracy, and that suggested size $x_e$ is as follows:

$$x_e\le \left(1/10\sim 1/8\right){\lambda }_w$$

(6)

where ${\lambda }_w$ is the minimum wavelength required for seismic waves.

The 3D finite element model size is 1200 m (x) × 1000 m (y) × 1200 m (z), and the element size is 20 m, as shown in Figure 2. The free boundary is set at the top boundary of the model, and the other boundaries are VSAB. The model parameters are listed in

Table 1. Parameters of the validation model.

Elastic Modulus (GPa)	Density (kg/m3)	Poisson Ratio	P Wave Speed (m/s)	SV Wave Speed (m/s)
2	2200	0.3	1106	591

[41].

The SV wave of the validation model is input by the curve of the pulse function Equation (7), which is shown as follows:

y(t) = 0.5 [1 − cos(2πt)] (0 s ≤ t ≤ 8 s)

(7)

where y(t) represents the displacement, t is the time, and the time’s step is 0.01 s.

When the oblique wave is SV wave, there is a critical angle (${\theta }_{cr}$). When the incident angle is less than ${\theta }_{cr}$, this incident wave can be converted into the reflected SV wave and the reflected P wave. While the incident angle is over ${\theta }_{cr}$, this oblique wave is only reflected in the SV wave, and ${\theta }_{cr}$ is related to Poisson ratio $\mu$, which can be calculated by Equation (8) [42].

$${\theta }_{cr}=\arcsin \left(\sqrt{\frac{1-2\mu }{2\left(1-\mu \right)}}\right)$$

(8)

In this validation calculation, the incident angle of the oblique wave is set as 20° (the angle is between the incident wave and the y-axis), which is smaller than ${\theta }_{cr}$. Figure 3a shows the wave propagation in the model at different times. The reflected SV wave and P wave can be clearly seen in this figure, which is similar to the Huang’s research [43]. Figure 3b shows the displacement time history curves at point A of the theoretical and numerical solutions, and the accuracy can be proved according to the research [43]. In this model validation, we can find that this VSAB simulates well the structural seismic response. Therefore, the seismic response analysis of the tunnel structure is carried out in this paper based on the VSAB.

3. Underground Structure Model

3.1. Model Introduction

This paper studies the seismic response of a shield tunnel with a lateral karst cavity under oblique SV waves by taking the Dalian Metro Line 5 as an example. The tunnel passes through various complex geological conditions, mainly including dolomitic limestone, karst development regions, calcareous slate, etc. Therefore, the tunnel’s stability and safety are of high concern to the design, construction, and operation departments, as well as the public. Among them, the karst region has the most noteworthy impact on the metro, which may cause ground collapse, construction delay, and structural damage. Moreover, the metro also located the Tanlu seismic zone. If the metro is damaged under an earthquake, it will cause serious casualties and economic losses.

The shield tunnel is constructed by bolting the prefabricated segments. In seismic calculation, the tunnel is usually simplified based on the equivalent bending stiffness model [44]. The suggested parameter is usually 0.6–0.8. In this paper, the value is set to 0.8. Therefore, the lining ring is simplified as a constant stiffness ring, as shown in Figure 4. The outer diameter D of the tunnel is 11.8 m, the outer radius R is 5.9 m, the width is 2 m, and the thickness is 0.5 m. The material parameters of concrete lining are determined according to the equivalent bending stiffness model. The material parameters of the surrounding rock and tunnel structure are shown in

Table 2. Material parameters of surrounding rock and concrete lining.

Material	Constitutive Model	Input Parameter	Magnitude
Concrete lining	CDP	Mass density (kg/m3)	2500
		Elastic modulus (GPa)	31.5
		Poisson ratio	0.2
		Tensile yield stress (MPa)	2.2
		Compressive yield stress (MPa)	23.4
Surrounding rock	MC	Density (kg/m3)	2200
		Elastic modulus (GPa)	2
		Poisson ratio	0.3
		Friction angle (°)	40
		Cohesion (MPa)	4

[41,45]. The Mohr Coulomb (MC) constitutive model and the concrete damage plasticity (CDP) constitutive model are employed for the surrounding rock and the concrete lining. The damping ratios of the tunnel structure and the surrounding rock are set as 5%.

3.2. 3D Finite Element Models of Underground Structure

According to the above-mentioned structural details. A 3D finite element model of underground structure is established by the ABAQUS software, as shown in Figure 5a. That model size is 120 m (x) × 100 m (y) × 100 m (z). The structure is buried at a depth of 12 m. The karst cave is simplified as a sphere of diameter D, and the net distance between the cavity and the tunnel is 0.2 D. The VSAB is applied to all the surrounding and bottom of the structure. It is assumed that there is no relative displacement between the surrounding rock and the tunnel, since they cannot be detached from each other. Therefore, the interfaces of the tunnel and the rock use tie constraint in ABAQUS. Six typical sections in the axial direction are selected to observe structural deformation: Section 1-1 to Section 6-6, as shown in Figure 5b. Section 1-1 is located at the central position; Section 2-2 is at the distance R from the center line; and Section 3-3, 4-4, 5-5 is far from Section 2-2 at every 0.5 R. Section 6-6 is far form 2.0 R from Section 5-5.

3.3. Seismic Input

In this paper, the El-Centro earthquake wave is employed as the input SV wave, which is adapted to the site category of the metro. In Dalian, the acceleration of rare ground motion is 2 m/s², and the duration of the cut-off earthquake wave is 20 s. Figure 6a shows the acceleration time history curve. As can be seen from the acceleration response spectrum in Figure 6b, compared the seismic information between the original with cut-off wave, it is clear that the cut-off wave matches the original wave well. Hence, the cut-off wave can satisfy the numerical calculations in this paper.

4. Results and Discussion

In this section, the seismic responses of the tunnel are analyzed under oblique SV waves in terms of the structural deformation, stress distribution, and damage severity. The SV waves are input on the structural bottom along the X direction. The incident angle of the waves is set from 0° to 30°, since the critical angle is 32.3°, as calculated by Equation (8). Structural deformation and stress distribution are analyzed by the vertical displacement and stress envelope diagram. The structural damage severity is evaluated based on the relationship between tensile damage value and crack width.

4.1. Structural Deformation

In this sub-section, the vertical displacement (VD) is used to reflect the structural deformation. The vertical displacements of the vault and bottom of the tunnel’s inner surface are analyzed based on the characteristics of the tunnel shape. The maximum VD values during the earthquake are extracted to analyze the tunnel deformation. Figure 7 shows the displacement max value of each tunnel’s point during the seismic action. When the VD > 0, it indicates that the point on the tunnel is uplifted; when the VD < 0, it indicates settlement.

Figure 7a,b provide the displacement curves of the tunnel’s bottom and vault along the axial direction under the incident angle of 0° and 30°, respectively. Figure 7a shows the vault settlement and bottom uplift as the incident angle is 0°, respectively. In Figure 7a1, the curve shows vault settlement at each point of the tunnel. The settlement is obvious at 3-times the length of the cave diameter along the axial direction, with less difference in settlement at other points. The max value of settlement is located at Section 1-1, and the value is 0.655 mm. The max difference of the settlement is 0.138 mm. Figure 7a2 shows the curve of the bottom uplift. The uplift is evident over a length of twice the cave diameter, and less variation is in the amount of uplift at other locations. The max value of uplift is also located at Section 1-1, and the value is 0.434 mm. The max difference of the uplift is 0.108 mm. In Figure 7, the VD is highly influenced near the tunnel’s center affected by the lateral karst cavity. The max values of settlement and uplift are all located at Section 1-1, the shape of Section 1-1 changes from circle to ellipse, and the distance between the vault and bottom decreases 1.089 mm. The affected scope and displacement difference of the vault is noticeably larger than that of the bottom. The tunnel’s VD varies sharply within the range of 2–3 times of the cave diameter on axial direction affected by the lateral karst cavity.

Similarly, when the incident angle is 30°, the structure shows settlement as shown in Figure 7b. The max value of the vault and bottom settlement is −171.969 mm and −170.625 mm, respectively. The distance between the vault and bottom at Section 1-1 decreases by 1.344 mm. However, there are some differences compared with the Figure 7b, especially near the tunnel’s center. The VD trend of the bottom looks abnormal, settlement in the affected scope may be due to the tunnel’s damage, and that abnormal values near the end of the tunnel may be caused by a lack of restraint. The VD trend of the vault is such as that under an incident angle of 0°. On this basis, the five sections in the affected scope (Section 1-1 to 5-5) and one section (Section 6-6) out of scope are selected for further analysis below.

The max VD values are located at Section 1-1 by observing the deformation, and Section 6-6 is selected as an unaffected section to compare. The curves of max values of the vault and bottom are subjected to different incident angles as shown in Figure 8. In Figure 8a, as the incident angle of SV wave increases, the trend of the curve at the vault and bottom is approximately the same; their values first decrease sharply and then decrease slowly. The tunnel’s bottom and vault all show settlement, and values decrease from about −40 mm to −171 mm. The displacement difference between the bottom and vaults at the same angle is small, and the value increases from 0.873 mm at 5° to 1.344 mm at 30°. The curves in Figure 8b show a similar trend, which means the difference is small along the tunnel’s axial direction. It is clear that the change in the tunnel’s VD is due to the vertical component generated by the oblique SV wave. The greatest effect on structural deformation is observed at an incidence angle of 30°. Consequently, it is deserved to focus on the seismic response of the tunnel, as the incident angle is 30°.

4.2. Structural Stress

Based on the displacement curves, six sections are selected to reflect the stress distribution. In this section, the stress envelope of the tunnel’s section is calculated when the incident angle is 0° and 30°, respectively. Stress envelope is an effective way to observe stress distribution. Figure 9 and Figure 10 show the tensile and compressive stress of the tunnel’s inner surface, respectively.

Figure 9 shows the tensile stress envelopes and max value curves of each section. In Figure 9a, as the incident angle is 0°, the stress envelopes are symmetrical along the diagonal of the spandrel and foot. The stresses near the cavity (0°–180°) are visibly different from the other side (180°–360°). The stresses near the cavity increase with increasing distance from the center, and the opposite trend is observed on the other side. The stresses at Section 6-6 differ very little from those at Section 5-5, and the values of sections 4-4 and 5-5 are almost equal, which verifies the range of influence of the cavity on the tunnel in the above. The max stress of each section is located at the tunnel’s foot, and the max value is 0.91 MPa at Section 1-1. Similarly, Figure 9b shows the stress envelopes at an incident angle of 30°, which are shaped as hourglasses. The difference between the stresses in each section is small. The stresses at the vault and bottom are greater than those at other locations, and the values are greater than 2.2 MPa (the tensile yield stress), which means that the tunnel is damaged. Combining the Figure 9a,b, the change in the envelope shape shows that the change in stress distribution depends on the incident angle. To better observe the stress changes at different sections, the stress values are selected to draw the curves in Figure 9c, which are circled by dotted line at the foot and spandrel in Figure 9a and the vault and bottom in Figure 9b. When the incident angle is at 0°, the stress of tunnel’s foot decreases from 0.91 MPa (Section 1-1) to 0.83 MPa (Section 5-5), which decreases 8.79%. Oppositely, the stress of the spandrel increases from 0.36 MPa (Section 1-1) to 0.83 MPa (Section 5-5), which increases 88.89%. When the incident angle is at 30°, the stress of the vault decreases with the distance from the center. However, the stress of the bottom first decreases and then increases, which means that the affected scope is smaller than that scope under an incident angle of 0°. Affected by the lateral karst cavity, the cavity changes the stress distribution of the tunnel under the earthquake action.

Similar to Figure 9, Figure 10 shows the compressive stresses of each section. The envelopes are similar to the four-leaf clover in Figure 10a. The stresses near the cavity (0°–180°) are distinct, and decrease with increasing distance from the center. However, the stress difference on the other side is small. In Figure 10b, the envelopes look the same as the dumbbells. The stresses near the cavity (0°–180°) increase with increasing distance from the center. Here, the same points as in Figure 9c are selected for analysis in Figure 10c. When the incident angle is at 0°, the stress of the tunnel’s foot decreases from 4.73 MPa (Section 1-1) to 4.67 MPa (Section 5-5), which decreases 1.27%. The spandrel stress decreases from 4.11 MPa (Section 1-1) to 3.88 MPa at (Section 5-5), which decreases 5.60%. When the incident angle is at 30°, the stress of the vault and bottom increases with the distance from the center. In summary, uneven vertical settlement and axial stress distribution are shown under earthquakes affected by the lateral karst cavity.

This paper also explores the tensile and compressive stresses of sections 1-1 and 6-6 of the tunnel inner surface subjected to different incident angles, as shown in Figure 11 and Figure 12. Figure 11 shows the tensile envelopes and stress curves of the vault and bottom. In Figure 11a, the tensile stresses increase as the incident angle increases, and the stress distribution changes to an hourglass. The trend of increasing stress with the incident angle is similar to the vertical displacement above. This is mainly influenced by the vertical component of the oblique wave. When the incident angle is 20°, the stresses of the vault and bottom are both higher than 2.20 MPa, indicating that the tunnel is damaged. The increase in tensile stress is not obvious when it is greater than 20°, showing that the stress will not increase greatly once exceeding the tensile yield stress. The max stresses of the vault and bottom are 2.33 MPa and 2.30 MPa as the incident angle is 30°, respectively. The trend of the stress change in Figure 11b is similar to that in Figure 11a. The stress values in Figure 11b are larger than those in Figure 11a at the same incident angle, which means that the lateral cavities can reduce the stresses to a certain extent. Here, the stresses of the vault and bottom at Section 1-1 and 6-6 are drawn in Figure 11c. The black and red lines indicate the points of Section 1-1 and Section 6-6, separately. The stresses increase sharply with increasing incident angle as the incident angle is less than 20°, and then increase very slightly. When the incident angle increases from 5° to 30°, the stress of the vault and bottom at Section 1-1 grows from 0.42 MPa to 2.33 MPa and from 0.52 MPa to 2.30 MPa, respectively. It can be seen that the tunnel’s damage is most severe at an incident angle of 30° compared with other lower angles.

Figure 12 shows the compressive stress envelopes and stress curves of the vault and bottom. In Figure 12a, the stresses increase obviously with the increase in the incident angle, the stress distribution changes from a four-leaf clover to a dumbbell. The stresses of the vault and bottom are greater than other locations. When the incident angle is 30°, the stresses of the vault and bottom are both lower than 23.4 MPa (the compressive yield stress). This indicates that the tunnel will not be damaged for suffering compression. The stress values in Figure 12b are larger than those in Figure 12a at the same incident angle. The stresses increase sharply with increasing incident angle, as shown in Figure 12c. In the two sections, the difference of stress is not obvious. When the incident angle increases from 5° to 30°, the stress of the vault and bottom at Section 1-1 grows from 2.53 MPa to 13.90 MPa and from 2.77 MPa to 13.12 MPa, separately.

It is obvious that the incident angle is a key factor in stress distribution, and the maximum stress usually occurs at the tunnel’s bottom and vault because of the vertical component of seismic waves. Compared with Section 1-1 and Section 6-6, the lateral cavities can reduce the stresses to a certain extent. However, the effect of lateral cavity on the extent of tunnel damage needs further analysis.

4.3. Damage Analysis

In this sub-section, the tunnel’s damage is analyzed under different incident angles. According to the mechanical properties of concrete structures, the tunnel is easily damaged when the tensile stress is greater than the yield stress. In order to better quantify the damage severity, tensile damage value ($d_t$) is used to describe the structural damage severity, which can be obtained by the CDP constitution model. The relationship between the tensile damage value $d_t$ and the concrete crack width $w_t$, is shown in Equation (9) [46]:

$$d_t=\frac{w_t}{\left[w_t+\left({\sigma }_t{h}_c\right)/E_c\right]}$$

(9)

where h_c represents the length of the eigenvalue, which is equal to the element side length for an eight-node integral element; ${\sigma }_t$ and $E_c$ are the concrete tensile stress and elastic modulus, respectively; and $w_t$ denotes the tunnel’s crack width.

The Chinese code [45] specifies 0.2 as the threshold value of crack width ($w_t$) to evaluate the structural safety. Hence, the critical tensile damage value $d_{tc}$ can be calculated as 0.74. Based on Equation (9), the concrete damage severity can be divided into three levels, as shown in

Table 3. Material parameters of the surrounding rock and concrete lining.

Category	Damage State	Crack Width	Tensile Damage Value
I	No damage	$w_t$ = 0	$d_t$ = 0
II	Slight damage	$0.2 > w_t$ > 0	$d_{tc}$$> d_t$ > 0
III	Severe damage	$w_t$ ≥ 0.2	$d_t$$\ge d_{tc}$

. Compared with the Yu’s research [47], this classification criterion makes it easier to evaluate the damage state of the tunnel structure in a numerical simulation.

In the displacement and stress analysis, the tunnel is most affected at an incidence angle of 30°. Consequently, the tunnel’s damage at different periods subjected to an incident angle of 30° is analyzed here, as shown in Figure 13. Figure 13a–c shows the tunnel’s damage at 2 s, 3.14 s, and 12 s, respectively. As mentioned before, the stress of the vault and bottom shows that these locations are vulnerable to damage. Therefore, the damage details of the vault and bottom of the tunnel’s inner surface are shown in those figures.

Figure 13a shows the tunnel damage when time is at 2 s. In this figure, the acceleration is yet to reach its peak and the structural damage is slight. The vault and bottom of the tunnel’s inner surface are both damaged, while there is no damage on the outer surface. The damage value at the center of the vault is 0.03, while the bottom’s damage is 0. The damage at the tunnel’s center zone is lower than other locations in the axial direction, and this difference is caused by the lateral karst cavity. That phenomenon indicates that the karst cavity has the effect of energy dissipation and vibration reduction on seismic waves to a certain extent. The similar trend can be found in Figure 13b. At this moment, the acceleration exceeded the peak when the time was 3.14 s and the damage gradually accumulated, approaching the critical state of severe damage. The max value of d_t is 0.73, located on the inner surface of the vault. The vault and bottom of the inner and outer surfaces are all damaged. The damage severity of the inner surface is more serious than that of the outer surface. As can be seen from the damage details of bottom, the damage in the central zone (Z2) is lower than that of other locations in the axial direction. In zone Z2, the damage value in the tunnel’s center is 0.09. The max value of damage is 0.16 in zone Z1 and Z3. When the time reaches 12 s, the damage value of the tunnel reaches the maximum, as shown in Figure 13c. The damage values of the outer surface of the vault are all greater than 0.74, which indicates cracks on the outer surface with the crack width greater than 0.2 mm. The damage in the central zone (Z2) of the bottom is lower than other locations, and the d_t at the central location is 0.57. However, there is a concentration of damage in the adjacent zone (Z1, Z3) with a greatest value of 0.85. This may be caused by the lateral karst cavity boundary, where the material properties suddenly change at that boundary.

The damage states of the tunnel under different angles of oblique SV wave are also investigated in this paper. Figure 14a–c shows the tunnel’s damage under an incident angle of 10°, 20°, and 30°, respectively. The tunnel is not damaged when subjected the incident angle of 10°, as shown in Figure 14a. The tunnel’s damage values at all locations are 0. In Figure 14b, the tunnel’s damage state is severe damage when the incident angle is 20°. The damage zone is located at the vault and bottom. The central zone of the bottom differs obviously from other locations. The damage state at zone Z1 and Z3 is more serious than Z2, and the max value of those zones is 0.74. The damage area is shifted a little towards the circumferential direction due to the lateral karst cavity. As for the structural damage at an incident angle of 30° in Figure 14c, the damage state is similar to that at an incident angle of 20°. However, the damage is more severe. The damage states of the vault and bottom are consistent with the stress distribution above. The tunnel is vulnerable to damage along the axial direction under the oblique wave, which can be contributed to two key factors: one is the wave’s vertical component and the other is the free space between the bottom and vault.

Compared with the published studies [30,48], the results show that the incident angles of SV waves have a significant influence on displacement and stress, and the structural deformation and stress distribution have an obvious spatial difference. The damage state in the central zone of the tunnel reflects the energy dissipation and vibration reduction due to the scattering and reflection of the waves that are weakened by the lateral karst cavity, while the damage concentration occurs at the location corresponding to the boundary of karst cavity due to the sudden change in material properties. This further indicates that the tunnel’s damage cannot be ignored due to the lateral karst cavity under the oblique wave.

5. Conclusions

In this paper, the structural seismic characteristics are explored under different incident angles of oblique SV waves, which include the structural deformation, stress distribution, and damage state. Through the analysis of deformation and stress distribution, the dynamic effects of lateral karst cavity on the tunnel are studied thoroughly. A new damage state classification criterion is proposed to evaluate the tunnel’s damage by introducing the relationship equation between cracks and damage.

The oblique seismic wave increases the spatial difference of structural deformation. With the increase in the incident angle, the vertical displacement of the tunnel increases significantly. The affected scope of the tunnel is twice the cavity diameter. Affected by the free space between the vault and bottom, the vertical displacement at the vault is slightly larger than that at the bottom.

The stress distribution increases noticeably with the increase in the incident angle. As the incident angle increases from 0° to 30°, the location of max stress changes from the foot to the vault, which is mainly due to the vertical component of the seismic wave. When the incident angle is greater than 20°, the tunnel’s stress is higher than the tensile yield stress, which indicates that the tunnel is damaged. The tunnel’s stress reaches the max value when the incident angle is 30°. The vault and bottom are vulnerable and deserve more attention under the oblique waves.

As the incident angle increases, the tunnel’s damage state goes from no damage to severe damage. When the incident angle is higher than 20°, the tunnel is in a severe damage state. During the seismic action, the damage at the tunnel’s center zone is lower than other axial locations. This is due to the cavity weakening the reflection and the scattering of waves, which plays a key role in energy dissipation and vibration reduction. At the end of seismic action, the damage concentration of the bottom affected by the cavity boundary is heavier than that of the central zone, which is caused by the sudden change in material properties. The tunnel’s damage is the most severe as the incident angle is 30°. The damage values of the vault’s outer surface are all greater than 0.74, which indicates cracks on the outer surface of the vault and bottom, with the crack width greater than 0.2 mm.

In the karst regions, measures should be taken to ensure the tunnel’s stability under earthquakes as much as possible. More advanced technology should be used to detect the geological conditions. During construction, the stability of surrounding rock can be improved by grouting, and the reinforcement range should be more than twice the diameter of the karst cave. The reserved grouting holes can also be added to ensure the ground reinforcement conditions after the metro is put into operation.