Evaluation of the Damping Layer between the Tunnel Lining and Surrounding Rock via a Shaking Table Test

Abstract

This paper primarily investigates the protective effect of the damping layer in tunnel lining structures under dynamic loads. A series of shaking table tests was conducted to investigate the seismic response mechanism of tunnel linings and the influence of surrounding rocks using the Wenchuan earthquake (magnitude 8.0) as a reference. The results show that the effect of the damping layer protection measures is accurate using the efficiency evaluation method for the damping layer under seismic excitation. The lower the excitation acceleration is, the better the effect will be. In addition, the damping coefficient is introduced to optimize the efficiency evaluation method for the damping layer. Among the factors influencing the seismic response of lining structures, the type of surrounding rock has a significant impact while the thickness of the damping layer has a relatively lesser influence. In seismic intensity areas of equal magnitude, an increase in the damping layer thickness leads to a more noticeable effect. In the different seismic intensity areas, the difference in the protection effect with the change in thickness is no longer obvious with the increase in seismic intensity. Moreover, the presence of a damping layer alters the intrinsic vibration characteristics of the tunnel lining structure, creating a space for deformation between the lining and the surrounding rock.

1. Introduction

Compared with other infrastructure, tunnels generally have good aseismic performance, but tunnel structures will still suffer damage or even destruction under strong earthquakes [1]. Generally, the parts that are vulnerable to earthquake damage are located in the fault fracture zone, the interface between soft and hard rock, adverse geological zones, and locations where there is an abrupt change in the tunnel cross-section shape. A large number of tunnel lining structures exhibited large-scale damage under strong earthquakes such as the Kobe earthquake in 1995 (magnitude 7.2), the Jiji earthquake in 1999 (magnitude 7.6), and the Wenchuan earthquake in 2008 (magnitude 8.0). In particular, the recent Wenchuan earthquake caused at least 56 highway tunnels in disaster areas to experience varying degrees of earthquake damage [2], and the lining structure of highway tunnels, which accounted for 24.72% of the length, was seriously damaged by the earthquake [3]. At the same time, it has been largely confirmed that the movement of tectonic plates entered the fifth active period [4]. Notably, China has experienced several major earthquakes in the past decade that posed significant threats to tunnel structures, including the Yushu earthquake in 2010 (magnitude 7.1), the Lushan earthquake in 2013 (magnitude 7.0), the Jiuzhaigou earthquake in 2017 (magnitude 7.0), and the Yibin earthquake in 2019 (magnitude 6.1) [5,6,7,8]. At present, the aseismic performance of tunnels has not received enough attention in all aspects of tunnel structure [9]. However, if a tunnel structure is severely damaged by a strong earthquake, it may disrupt transportation and become a focus of the public [10].

When a tunnel is located in an area with better geological conditions, it experiences less damage under seismic effects because the tunnel and its associated structures harmoniously integrate with the surrounding geological formations. Conversely, in tunnel construction sections with complex geological conditions and adverse geological environments, implementing certain aseismic measures can significantly reduce the seismic response of the tunnel structure. A damping layer serves as an economical control measure that reduces the seismic excitation impact of the surrounding rock on the tunnel [11]. By incorporating a damping layer, the impact of earthquakes can be significantly reduced, leading to decreased destruction and minimized economic losses [12,13,14]. The original rock-lining structure can be modified to include a rock-damping layer-lining structure, a rock-lining-damping layer structure, or even a rock-damping layer-rock-lining structure [15,16]. By setting a damping layer, the energy flowing from the surrounding rock into the lining is absorbed [16]. Moreover, the presence of the damping layer reduces the overall strength of the lining perimeter system, significantly decreasing the seismic forces acting on the tunnel lining [17]. Specifically, the damping layer can increase the displacement margin of the lining and absorb the relative displacement between the surrounding rock and the lining under dynamic loads [18]. However, if the damping layer is improperly designed or implemented, it can exacerbate the dynamic response of the lining, resulting in more severe damage.

Currently, there are two main types of damping layers used in engineering: the plate type and the grouting type [19]. The plate-type damping layer generally utilizes engineering rubber [20] while the grouting-type damping layer typically employs foam concrete [21]. Both types of damping layer materials are characterized by their ability to deform easily and resist shear failure. As an excellent anti-vibration measure, the damping layer has been applied in some practical projects [22]. Furthermore, modifying the dimensions of the damping layer allows for adjustments in its seismic performance. By using an appropriately sized damping layer, more effective protection can be achieved [4]. Therefore, additional research is required to develop evaluation methods and indicators that accurately assess the effectiveness of damping layers.

In this study, large-scale shaking table seismic simulation tests were conducted using the Wenchuan earthquake wave as the loading seismic wave. Through similar material design and calculations, a reduced-scale model of the tunnel–rock structure was created to simulate a real tunnel project. The seismic action mechanism of the isolation layer was analyzed. The experimental results were compared with the assessment results on the effectiveness of the isolation layer assessment method. This study aims to optimize the damping layer from both geometric and material perspectives and enhance its effectiveness. The research results can be applied to the design and construction of transportation tunnels in high-intensity seismic areas. They can also serve as a reliable theoretical reference and technical support for further research on efficiency evaluation models for damping layers and protective measures for tunnel lining structures under other dynamic loads.

2. Design of the Shaking Table Model Test Scheme

2.1. Shaking Table Test Device

This series of shaking table model tests used the three-directional and six-degree-of-freedom earthquake simulation shaking table device at Chengdu University of Technology, China.

Table 1. Main technical parameters of the shaking table.

Technical Indicators	Technical Parameters
Table size (m × m)	4.0 × 6.0
Frequency range (Hz)	0.1–60
Degree of vibration freedom	Three directions and six degrees of freedom
Maximum effective load on the table (t)	40
Maximum displacement of the table (mm)	X: ±300; Y: ±250; Z: ±150
Maximum velocity of the table (mm/s)	X: ±1500; Y: ±1200; Z: ±1000
Maximum acceleration under full load (g)	X: ±1.5; Y: ±1.2; Z: ±1.0
Maximum allowable overturning moment (t·m)	120

shows the related performance of the shaking table. The shaking table is shown in Figure 1.

2.2. Model Test Similarity Relation Design

Since the purpose of this series of tests is to investigate the effectiveness of the damping layer and the seismic dynamic response of the lining structures during seismic activity, it is essential to ensure that the elastic resistance, strain relationship, and inertia force of the test model are similar to those of the prototype. Unlike centrifugal fields, shaking table model tests are typically conducted in a 1.0 g gravity field, which will lead to gravitational distortion. To simulate the prototype in a 1 g gravity field and considering that it is difficult to increase the weight of the surrounding rock model, as well as considering the limitation of the model box on the size and burying depth of the lining structure, this model experiment adopted a gravity distortion model. The critical similarity ratio should meet the relationship shown in Equation (1) [21].

$$C_{\epsilon }\cdot C_E=C_L{C}_{\rho }\cdot C_a$$

(1)

In this equation, C_ε represents the similarity ratio of strain, C_E represents the similarity ratio of elastic modulus, C_L represents the similarity ratio of geometry, C_ρ represents the similarity ratio of density, and C_a represents the similarity ratio of acceleration.

Since the model and prototype are both in the same gravity field, the similarity ratio of acceleration C_a = 1. In addition, the similarity ratio of strain C_ε = 1 can easily restore the seismic response and failure mode of the prototype. The remaining three similarity indicators, including the similarity ratio of elastic modulus (represented by C_E) and density (represented by C_ρ), require orthogonal matching experiments to find suitable similar materials, while the similarity ratio of geometry (represented by C_L) determines the reliability and accuracy of the test results. To consider the limiting performance parameters of the shaking table test system and to maximize the performance of the shaking table, the size of the model box was determined by design and calculation to be 2.5 m (length) × 2.5 m (width) × 2 m (height) (as shown in Figure 2a). To reduce the influence of the scaled model boundary on the experimental results of the shaking table, a foam board with a thickness of 10 cm is set at the junction of the surrounding rock and the model box, and a mortar layer is poured at the bottom (Figure 2b). This method can ensure that the seismic wave propagation is not affected, and the similarity relationships for the other control indices are shown in

Table 2. Similarity relationships of shaking table model tests.

Physical Quantities	Dimension	Similarity Relationship	Similarity Ratio
Dimension	$L$	CL	1/30
Elastic modulus	FL−2	CE	1/45
Density	FTL−4	Cρ	1/1.5
Poisson’s ratio μ	-	Cμ	1
Stress σ	FL−2	Cσ	1/45
Strain ε	-	Cε	1
Acceleration a	LT−2	Ca	1
Time t	T	Ct	1/5.5

.

2.3. Shaking Table Test Scheme Design

The primary objective of this series of shaking table tests was to evaluate and optimize the material and geometric parameters of the damping layer, as well as propose an optimal selection method for the damping layer. Therefore, these tests were enhanced and optimized based on the underlying mechanism of the damping layer. Due to the higher stiffness of the tunnel lining structure compared to the surrounding rock, a relative displacement occurs between the lining and surrounding rock under the influence of seismic dynamics. The presence of the lining structure hinders the deformation of the surrounding rock and subjects it to significant inertia forces, ultimately leading to failure. The damping layer acts as a buffer zone to bear the relative displacement of the surrounding rock and reduce the pressure exerted by the surrounding rock on the lining [4]. When developing the shaking table test scheme, two surrounding rock environments of class IV and class V were set. The selected material properties for class IV surrounding rock were as follows: density = 1400 kg·m⁻³, cohesion = 3 kPa, and internal friction angle = 38°. For class V surrounding rock, the target properties were density = 1133 kg·m⁻³, cohesion = 1.1 kPa, and internal friction angle = 27°. To construct schemes with varying damping layer thickness (t_b), elastic modulus of the surrounding rock (E_S), and Poisson’s ratio of the surrounding rock (ν_S), three types of damping layers were combined: no damping layer, damping layer with a thickness of 1 cm, and damping layer with a thickness of 2 cm (refer to

Table 3. Shaking table test schemes and purposes.

Schemes	Surrounding Rock	Damping Layer	Test Purpose
Case 1	Class IV surrounding rock	Without damping layer	Control group
Case 2	Class IV surrounding rock	1 cm thickness damping layer	To study the effect of varying damping layer thickness and surrounding rock conditions on lining seismic response when compared with Condition 3 and 5
Case 3	Class IV surrounding rock	2 cm thickness damping layer	To study the effect of varying damping layer thickness and surrounding rock conditions on lining seismic response when compared with Condition 2 and 6
Case 4	Class V surrounding rock	Without damping layer	Control group
Case 5	Class V surrounding rock	1 cm thickness damping layer	To study the effect of varying surrounding rock conditions and damping layer thickness on lining seismic response when compared with Condition 2 and 6
Case 6	Class V surrounding rock	2 cm thickness damping layer	To study the effect of varying surrounding rock cases and damping layer thickness on lining seismic response when compared with Case 3 and 5

). In the case of earthquake damage in tunnel engineering, a deeply buried tunnel usually sustains less damage than a shallow tunnel. In this shaking table test, the buried depth of the tunnel was set as 47 cm to simulate the shallow buried condition of approximately 15 m in actual tunnel engineering. The lining structure was selected from the general design drawing of the lining structure of the Dujiangyan-to-Wenchuan-Highway tunnel, which was severely damaged in the Wenchuan earthquake. The inner diameter of the lining structure was 17 cm, which was scaled down according to the similarity relation.

2.3.1. Tunnel Lining: Material Design

According to the similarity calculation, the lining thickness of the generalized tunnel lining structure was 2 cm. An orthogonal experiment was designed, and a uniaxial compressive test was carried out to design similar materials for the lining structure. The model for the lining structure was determined to be barite:quartz sand:diatomite:gypsum:water = 0.4:0.2:0.6:0.6:1. The corresponding parameters are provided in

Table 4. Tunnel lining material parameters.

Project	Density/kg·m−3	Elasticity Modulus (GPa)	Intensity (MPa)
Prototype value	2400	30	20.1
Target value	1600	0.67	0.45
Measured value	1700	1.0	0.53

.

In the model experiment, the surface of the lining model was coated with a layer of varnish to simulate the waterproof layer on the surface of the lining structure. The lining model was reinforced with a square grid steel mesh with 0.8 mm diameter reinforcement and 10 mm spacing, as shown in Figure 3b.

2.3.2. Surrounding Rock: Simulated Material

The surrounding rock model was designed with river sand, oil, and fly ash as raw materials. According to the quality of the surrounding rock of the prototype, the generalized scheme of the surrounding rock model was designed through field investigation, proportioned design and formula calculation, and a large number of direct shear tests were carried out to verify that the mechanical property parameters of the generalized model met similar proportions. Finally, the mass ratio of river sand, oil, and fly ash in the Class IV surrounding rock was determined to be 40:10:50. The mass ratio of river sand, oil, and fly ash in the Class V surrounding rock was 40:15:45. The physical and mechanical parameters of similar materials in the surrounding rock model and their similarity degrees are presented in

Table 5. Surrounding rock material parameters.

Project		Density /kg·m−3	Elasticity Modulus/GPa	Cohesion/kPa	Internal Friction Angle/°
Class IV surrounding rock	Prototype value	2100	10	135	38
	Target value	1400	0.22	3	38
	Measured value	1300	0.06	1.3	33
Class V surrounding rock	Prototype value	1700	1.5	50	27
	Target value	1133	0.03	1.1	27
	Measured value	1100	0.02	0.6	25

. Surrounding rock materials were shown in Figure 4.

2.3.3. Damping Layer: Similar Material

In this series of tests, a B1-grade rubber–plastic sponge was selected as the damping layer’s simulated material for model testing (as shown in Figure 5). The density of the B1 rubber sponge was 60 kg/m³, the elastic modulus was 1.38 MPa, and the Poisson’s ratio was 0.3. This met the requirements of the similarity relationship and related requirements of the shaking table test. Three test cases were set up for shaking table tests, including no damping layer, a damping layer with a thickness of 1 cm, and a damping layer with a thickness of 2 cm. The length of each segment of the lining structure without a damping layer was 80 cm, and the length of each segment of the lining structure with a 1 cm or 2 cm thickness damping layer was 20 cm (as shown in Figure 6a).

2.3.4. Monitoring Scheme Design

The primary objective of installing a damping layer in the tunnel is to mitigate the seismic response in the cross-sectional direction of the lining structure [23,24]. To reduce the influence of the lining ends on the damping layer, the strain monitoring cross-sections of the lining structure should be set at the middle position, which is 20 cm away from both ends of the lining structure. Among them, the strain monitoring cross-section is No. 2 in the Class surrounding rock and No. 3 in the Class IV surrounding rock. To compare and analyze the effect of the damping layer, two strain monitoring cross-sections were also set at a distance of 60 cm from the boundary of the surrounding rock for the lining structure without a damping layer. The strain monitoring cross-section is No. 1 in the Class V surrounding rock and No. 4 in the Class IV surrounding rock. Each strain monitoring cross-section is equipped with strain gauges at eight positions, including the arch crown, left and right shoulders, sidewalls, left and right arch feet, and the inverted arch, with a total of 16 strain gauges per strain monitoring cross-section (as shown in Figure 6b). Above the arch crowns in the respective strain monitoring cross-sections, accelerometers are installed to investigate the transverse seismic response of the lining structure with or without damping layer protection. The accelerometers are represented in the form of A and numbers, with the numbers corresponding to the strain monitoring cross-section numbers. In addition, an A0 accelerometer is fixed on the vibration table to monitor the actual input seismic wave signal strength, and an A5 accelerometer is fixed above the model box to monitor the overall seismic response of the experimental model. Two accelerometers (A6 and A7) are arranged at the bottom of the lining.

2.3.5. Test Loading Scheme

In the Wenchuan earthquake, significant damage occurred to numerous highway tunnels. Therefore, for the dynamic loading experiments, the east–west component of the seismic waves recorded at the Maoxian Seismic Station in the Wenchuan region were used as the input motion at the shake table. Shaking was applied in the horizontal direction perpendicular to the tunnel axis. Considering that the regions with small original seismic responses were not valuable for the study, the waveform data of 20–165 s in the original seismic wave were selected as the test data, and the peak acceleration and duration were compressed according to the time similarity ratio (1:5.5); the compression duration was 30 s. Six loading cases were designed by controlling the peak acceleration of the input seismic waves. They gradually increased in magnitude from 0.05 g, 0.1 g, 0.15 g, 0.2 g, and 0.3 g to 0.4 g in ascending order of peak acceleration. The values correspond to the design’s basic seismic acceleration values for seismic fortification intensity levels 6 to 9, as specified in the “Seismic Design Code for Underground Structure” (GB/T 51336-2018) [25] (see

Table 6. The relationship between seismic fortification intensities and the values of designed seismic accelerations [25].

Loading Schemes	1	2	3	4	5	6
Design seismic accelerations (g)	0.05	0.10	0.15	0.20	0.30	0.40
Seismic fortification intensity	6	7		8		9

). It simulates the dynamic response process and final damage state of tunnel lining structures under different peak acceleration cases with the effect of damping layers. The acceleration time history of the seismic wave is shown in Figure 7a, and the peak ground acceleration (PGA) of the seismic wave is approximately 0.4 g, equivalent to a seismic intensity of IX. Most of the energy is released in the first 15 s of the waveform, and the frequency is mainly concentrated in the 0–15 Hz range (as shown in Figure 7b).

3. Evaluation Method for Damping Layer Performance

Konagai et al. [26,27,28] of the University of Tokyo in Japan studied the seismic interaction between tunnel structures and surrounding rock and evaluated the aseismic performance of tunnel lining structures. They discovered that under seismic forces, the cross-sectional area of the tunnel structure undergoes alternating cyclic tensile and compressive deformations along two diagonals oriented at a conjugate angle of 45°. Therefore, the key to the aseismic effect of tunnel structures is to solve the stiffness matching between the lining structure and the surrounding rock. To better achieve a gradual transition of stiffness between the lining structure and the surrounding rock, a damping layer is set between the lining structure and the surrounding rock, forming a surrounding rock-damping layer-lining structure system, which is then simplified into a system of three parts connected in series (as shown in Figure 8). By using a theoretical derivation method, it was found that the effectiveness of the damping layer was related to six basic parameters: (1) the ratio of the internal diameter of the lining structure to the burial depth r₀/H; (2) the relative stiffness of the surrounding rock ξ_S; (3) the ratio of shear stiffness between the surrounding rock and the damping layer $\overline{G}$; (4) the Poisson’s ratio of the surrounding rock ν_S; (5) the Poisson’s ratio of the damping layer ν_b; and (6) the thickness ratio of the damping layer to the internal diameter of the lining structure t₀/r₀.

Xin et al. [29,30] used a nonlinear large-scale seismic dynamic numerical calculation method and a multivariate nonlinear regression analysis method to evaluate the effectiveness of the damping layer (Equation (2)). This evaluation method fully considers the physical and mechanical parameters of the surrounding rock and the damping layer material. In the practical selection of the damping layer, the geometric and material parameters of the damping layer can be changed according to the geological and construction conditions of the tunnel site.

$$\begin{array}{l}{R}_b=0.713-0.187e^{-30.31r_0/H}-0.144e^{-15.87t_b/r_0}+0.136e^{-215.05E_b/E_s}-0.001e^{6.89{\nu }_b}\\ -0.004e^{8.42{\nu }_s}\end{array}$$

(2)

In the equation, R_b is the coefficient for evaluating the effectiveness of the damping layer; R_b = 1 − μ_l,b/μ_l,s, μ_l,b and μ_l,s are the maximum deformation values of the outer surface of the lining structure under the deformation effects of the damping layer and the surrounding rock; r₀/H is the ratio of the internal diameter of the lining structure to the burial depth; t₀/r₀ is the ratio of the thickness of the damping layer to the internal diameter of the lining structure; Eb/Es is the ratio of the elastic modulus between the damping layer and the surrounding rock; ν_b is the Poisson’s ratio of the damping layer; and ν_s is the Poisson’s ratio of the surrounding rock [31].

4. Test Data Analysis

4.1. Performance Evaluation of the Damping Layer in Class IV Surrounding Rock

The main difference between the calculated evaluation results of the tunnel damping layer efficiency in the Class IV surrounding rock and the test results of the model is shown in Figure 9. The red dashed line represents the evaluation value calculated based on Equation (2). Regardless of the thickness of the damping layer, as the peak acceleration of the excitation increases, the actual experimental values deviate from the evaluation value. This is mainly because there is an accumulation of damage to the lining structure after multiple rounds of excitation, which affects the accuracy of the evaluation. Therefore, the evaluation method for the damping layer efficiency should consider the influence of the peak acceleration increase on the structural damage of the lining structure and introduce a seismic damage coefficient for the lining structure to improve the evaluation method for the damping layer efficiency based on the original Equation (2). This extends the scope and accuracy of this evaluation method in high-intensity earthquake areas. By calculating the standard deviation of the experimental and evaluation values for the 1 cm and 2 cm damping layer tests in Figure 9a,b, it can be obtained that the standard deviation between the experimental and evaluation values for the 1 cm damping layer is σ_1cm = 0.027, while the standard deviation between the experimental and evaluation values for the 2 cm damping layer is σ_2cm = 0.023.

4.2. Performance Evaluation of the Damping Layer in Class V Surrounding Rock

The difference between the evaluation values and the experimental results under the effect of the damping layer in the Class V surrounding rock is similar to that in the Class IV surrounding rock. As the peak acceleration of the input excitation gradually increases, the experimental results deviate from the evaluation value calculated by Equation (2). By calculating the standard deviation of the experimental and evaluation values in Figure 10a,b, it can be obtained that the standard deviation between the experimental and evaluation values for the 1 cm thickness damping layer is σ_1cm = 0.030, while the standard deviation between the experimental and evaluation values for the 2 cm thickness damping layer is σ_2cm = 0.029.

4.3. Validation of Efficiency Evaluation Method of Damping Layer

According to the comparison between the evaluation results and the experimental results of the damping layer efficiency, as the acceleration increases, the $\Delta {\mathrm{R}}_b$ (evaluation value- experimental value) value gradually increases. It can be seen that with increasing acceleration, the effectiveness evaluation method for the damping layer gradually loses its accuracy. At the same time, the standard deviation is obtained. The evaluation value is more accurate under the action of a thicker damping layer. A thicker damping layer can provide better protection for the lining structure and avoid serious damage. Compared with the test results in the Class IV surrounding rock, the worse conditions of the surrounding rock are more likely to cause damage to the lining structure, and the test values deviate more. Therefore, according to the data analysis of this series of model tests, the level of surrounding rock is more decisive to the seismic response of the lining structure than the thickness of the damping layer. By analyzing the deviation between the experimental and evaluation values of the damping layer in the Class IV and Class V surrounding rock from Figure 9 and Figure 10, it can be observed that when the surrounding rock conditions and peak acceleration of seismic input are changed, the evaluation results will exhibit errors due to deteriorating rock conditions and increasing acceleration peak values. Based on the experimental results and evaluation results, the dynamic damage coefficient K, obtained through data fitting, ensures the reliability of the evaluation results for the effectiveness of the damping layer. Therefore, in actual engineering, the parameters of the damping layer can be chosen reasonably based on the physical and mechanical parameters of the surrounding rock in the tunnel site and the local seismic fortification intensity. The K value under each cases is shown in

Table 7. The seismic reduction coefficients corresponding to different operating cases.

Schemes	Surrounding Rock	Damping Layer	Seismic Damage Coefficient K
Case 1	Class IV	Without damping layer	/
Case 2	Class IV	1 cm thickness damping layer	−0.471a2 + 0.018a + 0.992
Case 3	Class IV	2 cm thickness damping layer	−0.033a2 − 0.079a + 0.991
Case 4	Class V	Without damping layer	/
Case 5	Class V	1 cm thickness damping layer	−0.487a2 + 0.047a + 0.981
Case 6	Class V	2 cm thickness damping layer	0.072a2 − 0.147a + 0.979

.

4.4. Strain Response Analysis of the Tunnel Lining Structure

The absolute values of the cumulative strain on various parts of the lining structure after incremental loading are shown in Figure 11. Combined with the damping layer efficiency evaluation method, the seismic response status of the cross-section on the lining structure under different damping layer effects is analyzed. As shown in Figure 11a, under the no-damping-layer condition in the Class IV surrounding rock, the arch shoulder and arch foot obtained relatively high seismic strain values since the deformation of the lining structure has strong conformity with the surrounding rock, and the radius of the arch foot is relatively small, which makes it susceptible to larger deformation under seismic loads. Protected by the 1 cm thick damping layer, the seismic strain values of the lining structure decrease significantly, the highest seismic strain value occurs at the arch shoulder, and the seismic strain values at other locations are all below 60 με. The highest value occurs at the arch shoulder, indicating that the 1 cm thick damping layer cannot change the deformation mode in which the lining structure’s cross-section alternatingly stretches and compresses along the conjugate 45° diagonal. Under the effect of a 2 cm thick damping layer, the seismic strain response of the entire cross-section of the lining structure decreases further, with the most significant reduction occurring at the arch foot and inverted arch locations. The increase in the thickness of the damping layer makes the arch crown, arch foot, and inverted arch, which are strongly constrained by the surrounding rock, more likely to undergo rigid deformation rather than relative deformation, leading to a decrease in seismic strain values. Since the excitation direction is vertical to the tunnel axis, the arch shoulder and sidewall locations that are most vulnerable to seismic damage are more significantly protected by the damping layer.

Consistent with the conclusions obtained by the evaluation method, the class of surrounding rock plays a more decisive role in the seismic response of the lining structure compared to the thickness of the damping layer. Figure 11b demonstrates a significant increase in the seismic strain response of the lining structure with a damping layer in the Class V surrounding rock. Under the no-damping-layer case, the lining structure has poor conformity with the surrounding rock, leading to significant deformation at the arch shoulder. The presence of a damping layer significantly reduces the seismic strain response of the lining structure, particularly at the arch shoulder. This suggests that the protective effect of the damping layer on the lining structure is more pronounced in cases where the surrounding rock quality is poor. It is noteworthy that the existence of the damping layer can cause certain rigid deformations in parts originally firmly constrained by the surrounding rock, such as the arch crown, arch foot, and inverted arch. Therefore, to achieve the optimal design and application of the damping layer, it is recommended to enhance its form, which should cover the entire cross-section of the lining structure, by optimizing the efficiency evaluation equation. This approach aims to maximize the effectiveness of the damping layer while minimizing engineering costs.

4.5. Analysis of the Acceleration Response of the Surrounding Rock

4.5.1. Time History Analysis of Surrounding Rock Acceleration

The peak acceleration of the surrounding rock under different cases is shown in

Table 8. Peak acceleration of the surrounding rock under different cases.

Schemes	Surrounding Rock	Damping Layer	Peak Seismic Acceleration (g)
Case 1	Class IV	Without damping layer	0.254
Case 2	Class IV	1 cm thickness damping layer	0.124
Case 3	Class IV	2 cm thickness damping layer	0.115
Case 4	Class V	Without damping layer	0.366
Case 5	Class V	1 cm thickness damping layer	0.217
Case 6	Class V	2 cm thickness damping layer	0.144

. The results from the measurements of the A2 and A3 accelerometers in Figure 12 show the effects of setting a damping layer with different thicknesses on the acceleration response characteristics of the different classes of surrounding rock sections. As indicated by Figure 12, the worse the class of surrounding rock is, the more obvious the acceleration response will be. With no damping layer, the lowest and highest accelerations of the Class IV surrounding rock were −0.22 g and 0.25 g, respectively, while the lowest and highest accelerations of the Class V surrounding rock were −0.38 g and 0.38 g, respectively. However, the amplitude of the acceleration response of the surrounding rock significantly decreases when a damping layer is installed. The maximum acceleration of the Class IV surrounding rock decreased from 0.25 g to 0.11 g due to the presence of the damping layer. Similarly, the maximum acceleration of the Class V surrounding rock decreased from 0.38 g to 0.14 g. These results clearly demonstrate the significant reduction in the dynamic response of the surrounding rock achieved by implementing a damping layer. Furthermore, increasing the thickness of the damping layer enhances its protective effect.

4.5.2. Spectrum Analysis of Surrounding Rock Acceleration

Based on the results obtained from the measurements of the A2 and A3 accelerometers in Figure 13, it can be observed that the amplitude of the spectral curve is significantly smaller when a damping layer is present compared to when it is absent. Furthermore, the amplitude decreases noticeably with an increase in the thickness of the damping layer. When a 1 cm damping layer is set, the dynamic response in the Class IV surrounding rock is lower, the seismic wave amplitude at various frequencies is significantly reduced compared to that without the damping layer, and the first primary frequency increases to 2.6 Hz. When a 2 cm damping layer is installed, compared to that setting a 1 cm damping layer, the first primary frequency is somewhat reduced, and diagrams (a) and (b) are more similar to diagrams (e) and (f). This phenomenon can be explained by the fact that the setting of the damping layer changes the superior frequency of the surrounding rock. Based on the data, it is evident that the installation and thickening of the damping layer significantly reduce the dynamic response of the surrounding rock, exhibiting a clear protective effect on the lining structure during earthquake events.

4.6. Analysis of the Acceleration Response of the Lining Structure

4.6.1. Time History Analysis of the Lining Structure Acceleration

The peak acceleration of tunnel lining under different cases is shown in

Table 9. Peak acceleration of tunnel lining under different cases.

Schemes	Surrounding Rock	Damping Layer	Peak Seismic Acceleration (g)
Case 1	Class IV	Without damping layer	0.192
Case 2	Class IV	1 cm thickness damping layer	0.114
Case 3	Class IV	2 cm thickness damping layer	0.173
Case 4	Class V	Without damping layer	0.287
Case 5	Class V	1 cm thickness damping layer	0.205
Case 6	Class V	2 cm thickness damping layer	0.142

. Based on a comparative analysis of the acceleration time history curves in Figure 14, three damping layer cases were considered, of the Class IV surrounding rock in Section 3 and Class V surrounding rock in Section 4.3, as examples. The measured data of accelerometers A6 and A7 were obtained for the cases of the three damping layer conditions. Within the Class IV rock section, under the condition of no damping layer, 1 cm damping layer and 2 cm damping layer, the peak accelerations were 0.19 g, 0.11 g and 0.15 g, respectively. Within the Class V rock section, the peak acceleration values on the inner side of the second lining crown were 0.25 g, 0.21 g, and 0.14 g for the three damping layer cases. It is indicated that the response of the lining structure can be significantly reduced by setting the damping layer, and a thicker damping layer has better aseismic performance.

4.6.2. Spectrum Analysis of the Lining Structure Acceleration

Figure 15 indicates that for the lining structure of Class IV surrounding rock, the linear shape of the curves shows obvious similarity under the three kinds of damping layer cases. Moreover, the peak value of the Class IV surrounding rock is lower than that of the Class V surrounding rock. When a damping layer of 1 cm thickness is added, the acceleration amplitude decreases significantly in different frequency ranges. In the range of 0–5 Hz, the lining structure has a more obvious acceleration response, and the main frequency does not change significantly due to the setting of the shock absorber layer. With the increase in the damping layer, the amplitude data further decrease in each frequency domain. Therefore, this observation demonstrates that a thicker damping layer can significantly reduce the acceleration amplitude of the lining structure.

4.7. Seismic Damage Status of Tunnel Lining Structure

The lining structure model was removed from the surrounding rock model, and the final seismic damage pattern of the lining structure was marked with a marker pen, as shown in Figure 16. The lining structure under the no-damping-layer condition suffered severe damage. The bottom inverted arch fractured into a flat shape, the arch foot on both sides uplifted, and longitudinal cracks appeared along the tunnel axis direction, while the arch foot was partially destroyed. The arch shoulder and sidewall near the end of the lining suffered extremely severe damage, with significant lining peeling and detachment, resulting in an exposed steel bar (as shown in Figure 16a). The overall effect of the 1 cm damping layer in the Class IV surrounding rock was inferior to that of the 2 cm damping layer (as shown in Figure 16b,c). Under the protection of the damping layer, the upper structure, consisting of parts such as the arch crown, arch shoulder, and sidewall, did not show obvious signs of damage, but the arch foot and inverted arch locations still suffered severe damage. This is consistent with the measured strain data along the tunnel section, which indicates that under the influence of seismic excitation, the extent of damage varies at different locations along the tunnel section.

The 1 cm thick damping layer in the Class IV surrounding rock cannot prevent longitudinal cracks at the arch foot or the uplift of the inverted arch. However, the 2 cm thick damping layer in the Class IV surrounding rock effectively reduces seismic damage to the lining structure, resulting in only slight damage concentrated at the arch foot and inverted arch and without thorough cracks. The 1 cm thick damping layer in the Class V surrounding rock is far less effective than the 2 cm thick layer (as shown in Figure 16d,e). Under the 1 cm thick damping layer in the Class V surrounding rock, the lining structure’s damage status is similar to that of the Class IV surrounding rock, with the inverted arch fractured and uplifted and longitudinal cracks at the arch foot, but the upper structure is relatively intact. The effect of the 2 cm thickness damping layer in Class V surrounding rock is also similar to that in Class IV, with only minor damage at the arch foot and inverted arch. Overall, the damping layer can effectively prevent seismic damage to the lining structure, and the protective effect is more significant with a thicker damping layer. Compared to the effect of the damping layer, the impact of rock level differences on the lining damage response is more apparent. The research results can provide a reference and guidance for seismic resistance in tunnel engineering. In the construction of tunnels in high-frequency seismic areas, special attention should be given to sections with poor surrounding rock cases. In these cases, the seismic response is more pronounced compared to sections with stable rock cases, and tunnels are subjected to more severe damage under seismic excitation. Furthermore, further research and the optimization of seismic isolation methods have been provided. For tunnel sections that utilize damping layers, it is important to select a reasonable thickness for the damping layer to achieve the dual purpose of protecting the tunnel while considering economic feasibility. Additionally, further studies are needed to verify the reliability of the evaluation methods for the effectiveness of the isolation layer under different seismic wave excitations [32,33].

5. Conclusions

The reliability and accuracy of the damping layer effectiveness evaluation method were verified by shaking table tests. The response state of damping layers with different thicknesses under different surrounding rock level cases was evaluated. The improvement direction and specific measures of the damping layer effectiveness evaluation method were proposed. Based on this, the interaction between the lining structure with the damping layer and surrounding rock during the excitation process was studied. The experimental results can guide the selection and construction of damping layers based on geological cases and construction cases. The main conclusions of the study are as follows:

(1)

The accuracy of the effectiveness evaluation method for the damping layer was verified through a large shaking table test with different damping layer thicknesses and surrounding rock cases. This method can accurately evaluate the effect of the damping layer on lining structures under seismic forces and guide the selection of optimal geometric and material parameters for the damping layer to achieve the best aseismic performance. However, with an increase in peak acceleration, the effectiveness evaluation method for the damping layer will produce slight deviations.

(2)

Corresponding seismic damage factors K were introduced for different cases based on the correspondence between seismic fortification intensity and basic seismic acceleration value, which can improve the accuracy of the effectiveness evaluation method for the damping layer under different seismic fortification intensities and guide the rational selection of the damping layer in actual engineering.

(3)

In cases with poorer rock, lining structures are more susceptible to damage under seismic forces. The direct contact of the lining with the rock leads to a very obvious deformation of the lining. Therefore, the surrounding rock class is more important for the seismic response of lining structures compared to the thickness of the damping layer.

(4)

The existence of a damping layer actually changes the inherent vibration characteristics of lining structures, allowing for a certain rigid displacement margin between the lining and the surrounding rock under seismic forces, thereby reducing the relative deformation.