Seismic Responses in Shaking Table Tests of Spatial Crossing Tunnels

Abstract

To study the complex dynamic response characteristics of spatial crossing tunnels under seismic loads, shaking table model tests were carried out for typical spatial parallel, orthogonal, and oblique crossing tunnels. The propagation and energy distribution characteristics of seismic waves were quantitatively analyzed according to the fundamental frequency, acceleration, and strain response of the system. The results show the following: the addition of a tunnel structure significantly reduces the natural frequency of the system. In spatial crossing tunnel engineering, the axial acceleration responses of the arch top and arch bottom of the tunnel both exhibit the characteristic of a linear distribution, presenting a ‘linear’ shape. For spatial parallel-type and spatial orthogonal-type tunnels, the peak acceleration at the same measurement point of the overcrossing tunnel under the same working condition is generally greater than that of the undercrossing tunnel. However, for the spatial oblique intersection-type structure, the result is just the opposite, that is, the peak acceleration of the overcrossing tunnel is generally less than that of the undercrossing tunnel. For spatial crossing tunnels, unlike the amplification effect of acceleration in a single tunnel, due to the reflection and refraction of seismic waves between the two tunnels, a ‘superposition effect’ of acceleration is generated in space, resulting in an abnormal increase in the acceleration response within the crossing section, which is prone to becoming a weak link in the seismic resistance of the tunnel structure. The strain response of both spatially parallel and orthogonal overcrossing tunnels is greater at the central section than that of undercrossing tunnels and less on both sides. The strain response of the spatial oblique intersection-type overcrossing tunnel is generally greater than that of the undercrossing tunnel.

1. Introduction

With the rapid development of China’s transportation infrastructure, aboveground space resources have become saturated, which has spawned a large number of adjacent crossings and underground projects. Li et al. [1] categorized crossing tunnels into two types according to relative structural and spatial relationships: structural crossings (structural bifurcation, structural air shaft, and structural liaison channel) and spatial crossings (spatial orthogonal, spatial diagonal, and spatial parallel). Structural crossing tunnels are connected to each other, while spatial crossing tunnels are structurally unconnected to each other and only show a spatial crossing relationship.

China has more than 30 shaker simulation laboratories, which provide a solid research foundation for many seismological studies [2]. In order to study the influence of site liquefaction on the seismic response of underground structures, Zhang et al. [3] carried out a centrifuge shaking table test under the condition of a liquefiable site. The geometric similarity ratio was selected as 1/55, and the seismic response law of underground structures in a liquefiable site was obtained. Zhang et al. [4] performed shaking table tests on four shield tunnels crossing a soft–hard soil interface. At an interface dip angle of 20°, the results indicate that the seismic response of the underground structure increases markedly when the dominant frequency of the input seismic wave approaches the natural frequency of the hard stratum. Pai et al. [5] investigated a stereoscopic crossing between the Strawberry Ditch No. 1 Tunnel (overcrossing) and the Pandaoling Tunnel (undercrossing). The undercrossing tunnel has a circular cross-section, and shaking table tests were performed on this configuration. They concluded that the acceleration time history and frequency spectrum of the arch top of the tunnel under the interchange tunnel are larger than the overall response of the inverted arch, and the peak acceleration response of the vault has a superposition effect. Wu et al. [6] performed shaking table tests on stereoscopic crossing tunnels with an ultra-small clear distance and considered three excitation scenarios: frequent, design-level (basic), and rare earthquakes. The results show that the dominant frequency bands with significant influence on the tunnel structure are concentrated within 2–8 Hz and 12–20 Hz. Lei et al. [7] investigated the dynamic response of obliquely intersecting crossing tunnels using large-scale shaking table tests with an approximately 6 m table and a geometric similarity ratio of 1:8. The results indicate that the tunnel crown is a critical weak point in the seismic performance of crossing tunnels. Yu et al. [8] investigated the seismic performance of a cross-shaped subway station structure using shaking table tests. They revealed the propagation characteristics of seismic motion within the site and its impact on the seismic response characteristics of the station structure. The research shows that seismic wave propagation is significantly affected by site characteristics, and the cross-transfer connection section of the station has an important influence on the overall seismic performance of the station. Based on a 1 g shaking table test of a free field system, a single underground structure system, and a tunnel parallel undercrossing station structure system, Xiang et al. [9] discussed the interaction between the seismic response of an underground structure to soil and the close underground structure, and determined a seismic response law for underground structures and an interaction law between structures.

In summary, the research methods of seismic dynamic responses of cross tunnels should also be based on theoretical analysis and numerical simulation, and fewer shaking table tests have been carried out for spatial cross tunnels. In view of this, this paper takes the left line tunnel of Zhangjiacun Station to Chenzhai Station of Zhengzhou Metro Line 7 over the right line tunnel of Wenhua Road Station to Xisha Road Station of Metro Line 4 as the engineering background. The shaking table test was designed according to a similarity ratio to explore the acceleration and internal force response characteristics of spatial crossing tunnels under different amplitudes and different types of seismic loads.

2. Project Overview

This paper takes the overcrossing section of Line 7 and Line 4 of the Zhengzhou Metro in China as the engineering background. The buried depth of the right line tunnel of the overcrossing section of Line 4 is about 16.23–18.10 m (the soil layers are ② 33 clayey silt, ② 23 silty clay, ② 42 silt, and ② 51 fine sand), and the buried depth of the interval tunnel of Line 7 is 7.24–8.13 m (the soil layers are ② 31 clayey silt, ② 32 clayey silt, and ② 22 powdery clay). During the site investigation, no adverse geological effects, such as subsidence, karst, landslide, ground collapse, or ground fissure, were found. In addition, no adverse building deposits affecting the stability of the foundation were found, and no liquefaction points were observed under the structural footing of the zone under 7-degree seismic conditions. The main purpose of this experiment was to explore regularity. In order to facilitate the subsequent comparative analysis of the test results, the tunnel structure was simplified as a continuous circular tunnel, without considering the joint characteristics between shield tunnel segments.

The relative position of the engineering structure and the distribution of the soil layer are shown in Figure 1. The minimum vertical distance between the tunnel of Line 7 and the right tunnel of Line 4 is 2.788–3.986 m, the vertical net distance between the left tunnel of Line 7 and the right tunnel of Line 4 is 3.906 m, and the vertical net distance between the right tunnel of Line 7 and the right tunnel of Line 4 is 3.373 m. According to the design drawings and construction drawings, it can be seen that the underground side-crossing rail transit of Line 7 is a shield tunnel with an outer diameter of 6.20 m and an inner diameter of 5.50 m. The lining thickness is 0.35 m, and the buried depth of the vault is 12.77 m.

3. Shaking Table Test Design

The test was carried out on the seismic simulation shaking table of the Civil Engineering Training Center of Liaoning Technical University. The size of the model box matched with the shaking table was 2 m × 2 m × 1.5 m (length × width × height). The boundary of the model box is made of angle steel to limit its deformation. Hole grooves are reserved at the top and bottom of the model box, which were firmly installed within the steel platform of the shaking table using bolts, reinforced by steel brackets. Equipment indexes are shown in

Table 1. Basic performance index of the shaking table.

Property	Rated Load	Table Measurement	Excitation Directions	Test Frequency	Peak Acceleration	Peak Displacement	Peak Velocity	Overturning Moment
Index	10 t	3 m × 3 m	Two-directional horizontal	0–50 Hz	1.5 g	±15 cm	100 cm/s	300 kN·m

. The shaking table system and model box are shown in Figure 2.

Boundary effect treatment is an important part of shaking table model tests. Through calculation, analysis, and testing, Wang et al. [10] and Zhang et al. [11] have shown that flexible containers can well represent the response of actual engineering, are lightweight, meet the requirements of a multi-directional vibration test, and weaken the boundary effect. Since a rigid model box was used in the test, in order to reduce its boundary effect, the boundary of the model box was treated accordingly: (1) A layer (4 cm thick) of fine stone was paved at the bottom to form a friction boundary to prevent bottom slip; (2) a 200 mm-thick polystyrene foam board was pasted on the inner wall of the model box to form a flexible boundary to weaken the reflection of the seismic wave via the box’s steel plate; (3) after the inner wall of the model box was processed into a flexible boundary, a polyvinyl chloride plastic film was pasted onto the foam plate to form a sliding boundary to reduce the friction between the model and the boundary.

3.1. Similarity Ratio Design

In this test, the size of the structural model was first determined using the test hardware equipment, including the limitation of the table size and the allowable error limits of the model box boundary. Research has shown that the relative stiffness between an underground structure and the surrounding soil layer is one of the most important factors affecting the seismic response characteristics of an underground structure [12]. Therefore, in order to make the model test reflect the results in line with the objective facts, the similarity ratio of the relative stiffness between the underground structure and the surrounding soil layer in the model test and the prototype should be ensured as much as possible. The similarity ratio of the relative stiffness between the two soils and the structure is calculated as follows:

$$S_F={\displaystyle \frac{G_m}{Gp}}\times {\displaystyle \frac{{\Delta }_m}{\Delta p}}$$

(1)

In the formula, $G_m$ is the dynamic shear modulus of the model soil; $G_p$ is the dynamic shear modulus of the prototype soil; ${\Delta }_m$ is the deformation displacement of the structural model; and ${\Delta }_p$ is the deformation displacement of the prototype structure. The length, elastic modulus, and acceleration were chosen as the basic physical quantities of the structural model, and the shear wave velocity, density, and acceleration were chosen as the basic physical quantities of the model soil. The similar relationships of the rest of the physical quantities were deduced, as shown in

Table 2. Model similarity relationship.

Physical Property		Physical Quantity	Similarity Relationship Formula	Similarity Ratio
Geometric property		Length	$S_l$	1/30
		Displacement	$S_u=S_l$	1/30
Material property		Elastic modulus	$S_E$	0.087
		Stress	$S_{\sigma }=S_E$	0.087
		Strain	$S_{\rho }/S_E$	1
Material property	Density		$S_{\rho }=S_E{S_a}^{-1}{S_l}^{-1}$	2.61
Dynamic property	Time		$S_t={S_l}^{0.5}{S_a}^{-0.5}$	0.183
	Frequency		$S_{\omega }={S_l}^{-0.5}{S_a}^{0.5}$	5.477
	Velocity		$S_v={S_l}^{0.5}$	0.183
	Acceleration		$S_a$	1

.

At the same time, in order to meet the requirements of the quality similarity ratio, an artificial mass model was adopted, that is, an artificial mass block was added to the tunnel. The additional artificial mass was calculated according to the following formula:

$$m_a=S_E{S_l}^2{m}_p-m_m$$

(2)

In the formula, $m_a$ is the additional artificial mass; $S_E$ is the similarity ratio of the elastic modulus; $S_l$ is the geometric similarity ratio; $m_p$ is the mass of the prototype; and $m_m$ is the mass of the model itself. The total counterweight of the additional mass tunnel model was calculated to be 60 kg, and the counterweight was placed on the tunnel and fixed by using adhesive tape to prevent it from moving during vibration and to minimize the test error.

3.2. Test Soil and Model Structure

In this experiment, considering the engineering background and laboratory conditions, the soil model was simplified into a uniform field, so sand was selected as the medium to simulate the propagation of seismic waves. In the preparation stage of the experiment, the sand was first dried, and then the impure material was removed via screening to ensure that the relative density of the sand reached a predetermined value of 80%. By filling and rolling the sand layer by layer into the model box, the consistency of the relative density of the soil layer was guaranteed. The basic parameters of the sand are shown in

Table 3. Basic parameters of sandy soil.

Model Soil	Density (kg/m3)	Poisson Ratio	Elastic Modulus (MPa)	Angle of Internal Friction (°)
Sand	1614	0.3	12.69	30

.

In this paper, the tunnel model in the scale test was made of plexiglas. Plexiglass is a suitable material for machining and easy thermoforming. The thickness tolerance of the plate can be controlled within ±0.2 mm. There is a special glue used in the model assembly, which can effectively ensure the model meets the size, material strength, and other requirements. Plexiglas has the characteristics of good uniformity, high strength, transparency, and easy processing. It is suitable for simulating actual structural materials and conducting linear elastic research. Chi et al. [13] have used organic glass models in structural dynamic tests for verification. Combined with the actual engineering structure and similarity relationship, the structural dimensions and counterweights of each model can be converted. The outer diameter of the tunnel structure model is 0.21 m, the inner diameter is 0.19 m, and the length is 0.70 m. The tunnel structure model is shown in Figure 3, and the counterweight is shown in Figure 4.

Test Point Arrangement and Collection

The purpose of this test was to study the seismic response law of a small net spatial crossing tunnel system, and the following four groups of working conditions were designed for the shaking table model test: free field, soil–space parallel tunnel, soil–space orthogonal tunnel, and soil–space oblique tunnel.

The focus of the test was to measure the acceleration and strain values at different positions within the model soil and the tunnel structure. These data points are crucial for understanding the dynamic behavior of a small net space cross tunnel system. In order to accurately capture the dynamic response of the tunnel structure, especially in shaking table model tests, the sensors were strategically installed at the central cross-section location where the dynamic response of the tunnel structure is large [14]. According to the symmetrical characteristics of the model, 25 accelerometers and 16 strain gauges were deployed, labeled as A and S, respectively, and the dynamic data were recorded in detail. For example, ‘A01’ represents the first accelerometer, and ‘S01’ represents the first strain gauge. This naming rule helped to systematically record and analyze data. The sensor arrangement for the working condition is shown in Figure 5.

In this experiment, strain data acquisition was mainly completed by using a Donghua DH3817K acquisition instrument with 32 channels, and acceleration data acquisition was mainly completed by using a Donghua DH5925D acquisition instrument. This equipment was purchased from Jiangsu Donghua Testing Technology Co., Ltd., which is located in Taizhou, China. Detailed configurations of the relevant acquisition equipment are shown in Figure 6.

3.3. Ground Motion Selection and Loading

A total of two seismic waves, a Chi-Chi wave and an El-Centro wave, were selected for this experiment. Taking the peak acceleration of 0.1 g as an example, the acceleration time history curves and frequency spectrum curves of the two seismic waves are shown in Figure 7.

By observing the Fourier spectrum of different input waves in Figure 7, it can be found that the spectral characteristics of the two input waves with a peak acceleration of 0.1 g are obviously different. The predominant frequencies of the Chi-Chi wave and the El-Centro wave are 8.74 Hz and 7.81 Hz, respectively. In addition, the energy frequency bands of different seismic waves are also distributed differently.

In addition, horizontal earthquakes represent an important factor leading to the seismic damage of a structure, so the seismic waves were loaded horizontally (X direction) in the test. A micro-seismic sine sweep frequency (0.1 g) process was performed before and after the start of the test, as well as when the peak value of the input wave was changed to test the dynamic characteristics of the model [15]. The prototype tunnel site area of the project belongs to a 7-degree seismic basic intensity area [16]. The seismic fortification intensity of China is 7 degrees and 8 degrees, corresponding to the basic peak acceleration of 0.1 and 0.2 (0.3) g, respectively. Therefore, a total of 6 working conditions (excluding sine sweep) were loaded in the test. During the test loading process, the acceleration amplitude was input in a step-by-step increasing manner, and the amplitude range was 0.1–0.3 g.

Table 4. Test loading cases.

Input Wave Type	Condition Code	Input Amplitude/g	Peak Acceleration at Measuring Point A01/g
White noise	WN1	0.1	0.099
Chi-Chi wave	CC1	0.1	0.107
El-Centro wave	EL1		0.106
White noise	WN2	0.1	0.098
Chi-Chi wave	CC2	0.2	0.198
El-Centro wave	EL2		0.203
White noise	WN3	0.1	0.110
Chi-Chi wave	CC3	0.3	0.310
El-Centro wave	EL3		0.304

presents the seismic loading protocols for each test condition, and the rest of the working condition loading forms were the same as this. In the table, CC and EL represent the Chi-Chi wave and the El-Centro wave inputs, respectively, and CC1 represents the first loading of the Chi-Chi wave. It should be noted that due to the non-repeatable nature of shaking table tests, the peak acceleration data in this study were single typical test results. The previous preparatory tests show that the variation of the peak acceleration at the key measurement points under the same working conditions is within ±10%. All test conditions employed strictly identical sample preparation and loading procedures to ensure result comparability.

4. Analysis of Shaking Table Test Results

4.1. Boundary Effect Verification

In order to verify the effectiveness of the method of dealing with the boundary effect in this experiment, this section takes the free field condition (FF) as an example, with the Chi-Chi wave loaded with PGA = 0.1 g. Due to the symmetrical arrangement of the sensors, three monitoring points (A21, A22, and A24) in the left half were selected. The collected acceleration time history and Fourier spectrum curves are shown in Figure 8.

It can be seen from Figure 8 that the acceleration time history curves of different monitoring points on the land surface under the same ground motion are extremely consistent, and the time corresponding to the peak acceleration is also roughly the same. Comparing the acceleration time history curves, it can be seen that the peak acceleration of each monitoring point on the soil surface is basically 0.30–0.34 g, and the main shock time is about 3.0–9.5 s, with a duration of 6.5 s. In addition, the Fourier spectrum amplitudes corresponding to different monitoring points are similar, and the predominant frequency is 8.789 Hz in all cases. This can be explained by the fact that the method of laying polystyrene foam plates around the inner wall of the rigid model box in this experiment can better reduce the diffraction and reflection of seismic waves at the boundary, so that the seismic waves have good fidelity and achieve the result of reducing the boundary effect. Therefore, based on the above analysis, the boundary effect of the model box in this test is well treated, and the test results are reliable.

4.2. The Natural Frequency of the System

In the tests, in order to obtain the dynamic characteristics of the system, frequency sweeps using white noise with a PGA of 0.1 g were used at the beginning and at the end of the tests for all working conditions. An analysis of the natural frequency of the model system of each working condition was obtained by using the transfer function, which is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input [17]. The FRF curves for the four operating conditions are shown in Figure 9, and their natural frequencies are shown in

Table 5. Natural frequency of the system under different working conditions.

Working Condition	FF	SPT	SORT	SOBT
Natural frequency/Hz	14.34	13.53	13.18	12.98

.

Figure 9 shows the transfer function results of the four systems under the WN3 condition. It can be observed that the natural frequency of the free field is significantly higher than that of other working conditions. It is pointed out that the addition of the tunnel structure significantly reduces the natural frequency of the system. This phenomenon is attributed to the fact that the tunnel structure reduces the quality of the site compared to the free field, resulting in a decrease in the natural frequency of the system.

4.3. Spatial Parallel Crossing Tunnels

4.3.1. Acceleration Response of Spatial Parallel Crossing Tunnels

When the Chi-Chi waves and El-Centro waves were input under different amplitudes, the peak accelerations of the vault and arch bottom in the axial direction of the tunnel were comparatively analyzed, respectively, as shown in Figure 10.

In order to further analyze the seismic response law of the tunnel structure, the circumferential acceleration of the central section B (B′) of the tunnel was comparatively analyzed when the Chi-Chi wave and El-Centro wave were input at different amplitudes, as shown in Figure 11.

From Figure 10 and Figure 11, it can be seen that the loading of the Chi-Chi wave and the El-Centro wave shows basically the same law. With the increase in the amplitude of the input seismic wave, the peak acceleration of each measuring point increases. When the amplitude of the input seismic wave is 0.1–0.3 g, the peak acceleration of the tunnel vault and arch bottom is basically flat, in a ‘straight line‘ distribution at the monitoring points of the central section of the tunnel and the sections on both sides of the affected section. At the same time, the peak acceleration at the vault of the overcrossing tunnel and the undercrossing tunnel is significantly greater than that at the arch bottom under the same working condition, and the peak acceleration at the vault and arch bottom of the overcrossing tunnel is greater than that of the undercrossing tunnel under the same working condition. Taking the loading condition of a 0.1–0.3 g Chi-Chi wave as an example, the peak accelerations at the vault and arch bottom of section B are 0.300, 0.424, and 0.601 g and 0.287, 0.405, and 0.577 g, respectively, and the arch bottom is 0.957, 0.955, and 0.960 times of the vault, respectively. This is due to the existence of the undercrossing tunnel, so that the displacement at the arch bottom of the cross-section of the overcrossing tunnel is limited by the surrounding soil and the undercrossing tunnel, and part of the seismic wave is absorbed, so its response is significantly weaker than that of the arch top. Due to the influence of the overcrossing tunnel and the deeper burial of the undercrossing tunnel, its effect on seismic wave absorption is more obvious. This phenomenon can be explained by wave interference theory: when the seismic wave propagates between parallel tunnels, the incident wave and the reflected wave through the tunnel interface are superimposed on each other. When the two waves have the same phase, phase-length interference is generated, resulting in an increase in the acceleration response. The acceleration response of the overcrossing tunnel observed in the test is greater than that of the undercrossing tunnel because the position of the overcrossing tunnel is more susceptible to this interference effect.

4.3.2. Strain Response of Spatial Parallel Crossing Tunnels

The axial strain peaks of the vault and arch bottom of the overcrossing and undercrossing tunnels were analyzed. The strain peak distribution of each measuring point under different loading conditions of the Chi-Chi wave and the El-Centro wave is shown in Figure 12. Due to the damage of some strain gauges under the 0.1 g Chi-Chi wave condition, no effective data were collected, so no analysis was carried out.

For the same tunnel vault and arch bottom, the central section B and B′ are larger than the sections A and C (A′ and C′) on both sides of the affected area. This shows that the response of the A and C (A and C′) sections of the spatial parallel crossing tunnel to the seismic load is relatively stable, while the B (B′) section in the center of the cross zone is more intense under the seismic load.

Figure 13 shows the peak circumferential strain of section B and B′ under various tunnel working conditions when loading the Chi-Chi wave and the El-Centro wave with three different amplitudes of 0.1 g, 0.2 g, and 0.3 g in the X direction.

It can be seen from Figure 13 that when loading different types of seismic waves, the strain peaks in the 0.1~0.3 g conditions show basically the same pattern. When inputting seismic waves, the strain peak increases with the increase in loading conditions. The strain distribution of each part is as follows: left wall > right wall > vault > arch bottom. This is due to the influence of the undercrossing tunnel, where the seismic wave at the arch bottom of the cross-section of the overcrossing tunnel is absorbed by the surrounding rock. In addition, the left wall and the right wall are on the side of the free surface, so the strain peak is significantly larger than at the other monitoring points. The increase in the strain peak is greater than that of the El-Centro wave when the Chi-Chi wave is loaded.

4.4. Spatial Orthogonal Crossing Tunnels

4.4.1. Acceleration Response of Spatial Orthogonal Crossing Tunnels

The peak accelerations of the tunnel vault and arch bottom were comparatively analyzed when the Chi-Chi wave and El-Centro wave were input at different amplitudes, as shown in Figure 14.

In order to further analyze the seismic response law of the tunnel structure, the circumferential accelerations of the central section of the tunnel under different Chi-Chi wave and El-Centro wave amplitudes were comparatively analyzed, as shown in Figure 15.

From Figure 14 and Figure 15, it can be seen that the loading of the Chi-Chi wave and the El-Centro wave shows basically the same law. When loading the El-Centro wave, the peak acceleration of each measuring point is basically slightly larger than that of loading the Chi-Chi wave. With the increase in input conditions, the peak acceleration of each measuring point increases. Based on the A16 measuring point at the cross-center section II vault, when loading a 0.1 g El-Centro wave, the peak accelerations of 0.2 g and 0.3 g are 1.45 and 2.18 times that of 0.1 g, respectively. The special geometric shape of the orthogonal intersection leads to a complex wave–field interference phenomenon of seismic waves at the intersection node. According to the test data, when the amplitude of the input seismic wave is 0.1–0.3 g, the peak acceleration of the vault and arch bottom of the overcrossing tunnel and the undercrossing tunnel is basically flat at each measuring point in the central section of the tunnel and on both sides of the affected section. The distribution is a ‘straight line’, and the overall peak acceleration of the overcrossing tunnel is greater than that of the undercrossing tunnel. This phenomenon can be explained in terms of wave propagation characteristics: the orthogonal crossing structure causes multiple reflections of seismic waves in the crossing region, and phase-length interferences between the incident and reflected waves are formed at certain locations, resulting in an enhanced acceleration response.

Through analysis, it is known that the peak acceleration at the vault of the overcrossing tunnel is slightly larger than that at the arch bottom under the same working condition, while the peak acceleration at the vault of the undercrossing tunnel is significantly larger than that at the arch bottom under the same working condition. Taking the 0.3 g El-Centro wave loading condition as an example, the peak accelerations at the vault and arch bottom of sections II and II′ are 0.664 g and 0.580 g and 0.658 g and 0.526 g, respectively, and the arch bottom is 0.989 times and 0.907 times that of the vault. This is due to the existence of the undercrossing tunnel: the displacement at the arch bottom of the cross-section of the overcrossing tunnel is limited by the surrounding soil and the undercrossing tunnel, and some seismic waves are absorbed, so the response of the arch bottom of the overcrossing tunnel is significantly weaker than that of the vault. Due to the influence of the overcrossing tunnel and the deep burial of the lower tunnel, the absorption effect of the undercrossing tunnel on the seismic wave is more obvious. This difference in response reflects the spatial distribution characteristics of the wave propagation path and the interference effect in orthogonal crossing tunnels.

4.4.2. Strain Response of Spatial Orthogonal Crossing Tunnels

The axial strain peak values of the vault and arch bottom of the overcrossing and undercrossing tunnels were analyzed. The strain peak distribution of each measuring point under different Chi-Chi wave and El-Centro wave loading conditions is shown in Figure 16.

Through the comparative analysis of the axial strain peak values of the above two crossing tunnels, it is found that under the same working conditions, the strain peak value of section II of the overcrossing tunnel is smaller than that of section II′ of the undercrossing tunnel, while section I and III are larger than the section I′ and III′; the peak strain at the vault of the overcrossing tunnel is smaller than that at the arch bottom, and the peak strain at the vault of the undercrossing tunnel is larger than that at the arch bottom. At the same time, it can also be seen that the peak strain at the vault and arch bottom of the overcrossing tunnel is smaller than that on both sides of the affected zone at the central section of the cross zone. The undercrossing tunnel is larger than the cross-section on both sides of the affected area at the central section of the cross area at the vault and the arch bottom. This shows that the response of sections I and III of the overcrossing tunnel to the seismic load is relatively strong, and section II in the center of the cross area is relatively stable under the action of seismic load. Due to the deep buried depth, the undercrossing tunnel is relatively stable as a whole.

Since the crossing section is the key research object, this study only investigated the peak dynamic strains in the circumferential direction and axial direction of the tunnel at the intersection center (sections II and II′). When loading the Chi-Chi wave and the El-Centro wave with three different amplitudes of 0.1 g, 0.2 g, and 0.3 g in the X direction, the peak circumferential strain of section II and II′ under each working condition can be seen in Figure 17.

From Figure 17, when loading different types of seismic waves, the strain peaks at the 0.1–0.3 g conditions show basically the same pattern. When inputting seismic waves, the strain peak increases with the increase in loading conditions. For the overcrossing tunnel, the strain distribution of each part is as follows: vault > arch bottom > left wall > right wall. For the undercrossing tunnel, the strain distribution of each part is left wall > right wall > vault > arch bottom. This is due to the influence of the undercrossing tunnel, meaning that the monitoring points of the vault and arch bottom at the cross-section of the overcrossing tunnel are obviously larger. In addition, the left wall and the right wall are partially on the side of the free surface, so the strain peak is smaller than at the other monitoring points. The undercrossing tunnel is buried relatively deep, and the surrounding soil has an obvious absorption effect on the seismic wave. At the same time, the overcrossing tunnel also has a blocking effect on the propagation of the seismic wave, and the side wall part has no restriction on the overcrossing tunnel, resulting in a stronger response than the vault and the inverted arch. In addition, the strain peak values of the monitoring points at the circumferential vault and arch bottom of the overcrossing tunnel are basically larger than those of the undercrossing tunnel, while the strain peak values of the monitoring points on both sides of the overcrossing tunnel are much smaller than those of the undercrossing tunnel. At the same time, the growth rate of the strain peak value of the undercrossing tunnel is faster than that of the overcrossing tunnel.

4.5. Spatial Oblique Crossing Tunnels

4.5.1. Acceleration Response of Spatial Oblique Crossing Tunnels

A comparative analysis of peak acceleration at the tunnel vault and arch bottom for different amplitudes of the Chi-Chi and El-Centro input waves, respectively, is shown in Figure 18.

In order to further analyze the seismic response law of the tunnel structure, a comparative analysis of the circumferential acceleration of the central section of the tunnel at different amplitudes of the input Chi-Chi and El-Centro waves is carried out, as shown in Figure 19.

It can be seen that the loading of the Chi-Chi wave and the El-Centro wave shows basically the same law. The peak acceleration of each measuring point of the El-Centro wave is slightly larger than that of the Chi-Chi wave. With the increase in input conditions, the peak acceleration of each measuring point increases. Taking the peak acceleration of the A16 measuring point at the vault of the intersection center (cross-section II) under the 0.1 g El-Centro wave loading as the reference, the peak accelerations under the 0.2 g and 0.3 g loading are 1.45 and 2.80 times that under the 0.1 g loading, respectively. The overall peak acceleration of the overcrossing tunnel is greater than that of the undercrossing tunnel. Moreover, the peak acceleration of the vault and the arch bottom of the overcrossing tunnel and the undercrossing tunnel are basically flat at each measuring point in the center section of the tunnel and in the sections on both sides of the affected area. The distribution is a ‘straight line’, and the peak acceleration of the arch bottom of the overcrossing tunnel and the undercrossing tunnel is significantly greater than that of the vault. Taking the 0.3 g El-Centro wave loading condition as an example, the peak accelerations at the vault and the arch bottom of section II and II′ are 1.130 g and 1.001 g and 1.884 g and 1.598 g, respectively, and the arch bottom is 1.667 times and 1.596 times that of the vault. The oblique intersection angle alters the wave propagation paths, causing deflection of seismic wave reflection directions at the undercrossing tunnel surface. This geometric characteristic modifies wave energy distribution patterns, resulting in distinct interference mechanisms compared to parallel and orthogonal crossing conditions. Consequently, this leads to the unique phenomenon that the overcrossing tunnel exhibits lower acceleration responses than the undercrossing tunnel.

4.5.2. Strain Response of Spatial Oblique Crossing Tunnels

The strain peak values in the axial direction at the vault and arch bottom of the arches of the overcrossing and the undercrossing tunnel were analyzed. The strain peak distribution of each monitoring point under Chi-Chi wave and E-Centro wave loading conditions is shown in Figure 20.

Through the comparative analysis of the axial strain peak values of the above two intersecting tunnels, it is found that under the same working condition, the axial strain peak value of the overcrossing tunnel is larger than that of the undercrossing tunnel; the strain peak at the vault of the tunnel is greater than that at the arch bottom. At the same time, it can also be seen that the strain peaks at the vault and arch bottom of the overcrossing tunnel and the undercrossing tunnel are smaller than the cross-sections on both sides of the affected area at the central section of the cross-section. This indicates that cross-sections I and III (I′ and III′) of the affected section of the three-dimensional intersecting tunnel have a more intense response to seismic loads. Meanwhile, the central cross-section II (II′) of the intersection area is relatively stable under seismic loads. The reason is that the existence of the cross area of the two tunnels has an impact on the surrounding rock of the cross-center section. The surrounding rock around the tunnel produces a ‘compaction effect’, which enhances the absorption of seismic waves and also weakens the impact of seismic loads on the tunnel. In contrast, the two sides of the affected section are in a free state (relatively) during the earthquake. The compression of the soil on both sides makes it easier to deform, resulting in a more intense dynamic response.

Since the crossing section is the key research object, this study only investigated the peak dynamic strains in the circumferential direction and axial direction of the tunnel at the intersection center (sections II and II′). Figure 21 shows the circumferential strain peak of section II under various working conditions of the overcrossing tunnel when loading the Chi-Ci wave and the El-Centro wave with three different amplitudes of 0.1 g, 0.2 g, and 0.3 g in the X direction.

From Figure 21, it can be seen that when loading the Chi-Chi wave and El-Centro wave, the dynamic strain of the space oblique tunnel shows a similar law, and the dynamic strain response of the tunnel when loading the El-Centro wave is significantly greater than that of loading the Chi-Chi wave.

In addition, it can also be seen that the peak strain increases with the increase in the peak acceleration of the seismic wave, and its variation law shows consistency. It basically maintains the law of vault side wall > vault > arch bottom and diffuses in a ‘flat’ shape along the circumferential section of the tunnel.

5. Conclusions

In this paper, the acceleration and dynamic strain responses of space parallel, space orthogonal, and space oblique crossing tunnels are analyzed, and the dynamic response characteristics of spatial crossing tunnels under seismic load are revealed. The following research results were obtained:

(1)

The natural frequency of the free field is significantly higher than that of the spatial crossing tunnel. It is pointed out that the addition of the tunnel structure significantly reduces the self-vibration frequency of the system. This phenomenon is attributed to the fact that the tunnel structure reduces the quality of the site compared to the free field, resulting in a decrease in the self-vibration frequency of the system.

(2)

In the spatial crossing tunnel system, the axial acceleration response of the tunnel vault and the arch bottom shows the characteristics of a ‘linear‘ distribution. For spatial parallel and spatial orthogonal tunnels, for the same working condition, the peak acceleration at the same measuring point of the overcrossing tunnel is generally greater than that of the undercrossing tunnel. However, for spatial oblique crossing tunnels, the result is just the opposite, that is, the peak acceleration of the overcrossing tunnel is smaller than that of the undercrossing tunnel, indicating that the crossing type and angle have an impact on the seismic response of the tunnel.

(3)

For spatial crossing tunnels, in contrast to the amplification effect of acceleration in a single tunnel, because the seismic wave will be reflected and refracted between the two tunnels, the ‘superposition effect‘ of acceleration is generated in the space, resulting in the abnormal increase in acceleration response in the cross-section, which easily becomes the weak link regarding tunnel structure seismic resistance. Therefore, in the seismic design of the crossing tunnels, the crossing section should be the focus of attention.

(4)

For the axial strain response of the tunnels, the strain response of the space parallel and space orthogonal overcrossing tunnels is larger than that of the undercrossing tunnels in the center section and smaller than that of the undercrossing tunnels on both sides. The strain response of the space oblique crossing tunnels is larger than that of the undercrossing tunnels. This shows that the crossing type and angle have a certain influence on the seismic response of the spatial tunnels, and the smaller the net distance between the two tunnels and the smaller the intersection angle, the greater the deformation and damage that may occur in the upper tunnel.