Shaking Table Test and Dynamic Response Analysis of Saturated Soil–Submarine Tunnel

Abstract

With the increasing construction of undersea tunnels in seismic-prone areas, accurately assessing their response to seismic conditions is crucial. To grasp the dynamic response of undersea tunnel structures to seismic waves, the shaking table test of water–sea–sea submarine tunnel is designed and carried out based on the methods of orthogonal design and fuzzy method. A comprehensive time-domain model is developed to capture the nonlinear dynamic interaction of ocean engineering structures, taking into account seismic waves, seawater, and saturated soil. The research results show that as the burial depth at each measurement point of the submarine tunnel increases, the acceleration response decreases and the horizontal displacement relative to the seabed surface increases. Comparing test and finite element simulation results reveals that under seismic loading, the strain distribution pattern of the tunnel section is mainly in the arch shoulder, waist, and foot with larger strain peaks, whereas the strain peaks at the arch top and the superelevation arch are smaller. Simultaneously, doubling the water pressure induces a slight increase in the overall strain response peak of the tunnel, with an indistinct relative displacement change rule. When a vertically polarized shear wave (SV wave) is vertically incident, different dynamic response indices will have different trends with the change in water level. This study may provide a reference for shaking table tests for saturated soil–submarine tunnels at complex sites.

1. Introduction

Submarine tunnels have a significantly different geotechnical environment that encompasses the tunnel structure compared to ordinary mountain tunnels. The rock surrounding underwater tunnels is usually a fracture-saturated medium, and the overburden soil layer is often a saturated two-phase soil body, which results in a saturated soil structure coupled with a dynamic problem under seismic action. Due to the complexity and uncertainty of underwater conditions, this presents challenges in seismic studies of underwater tunnels.

For seismic research problems related to tunnels, shaking table model tests play a crucial role. It is the most direct method to investigate the seismic response and damage mechanism of a structure. It is an important means to study and evaluate the seismic performance of a structure [1]. While scholars by nationality have conducted numerous studies on the tunnel shaking table model test [2], the current object of the test research is mainly focused on mountain tunnels, with less emphasis on underwater tunnels. Fang et al. [3] performed a vibrating table modeling study of tunnels through faults in the background of a complex tunnel project in Tibet, China. Furthermore, Wang et al. [4] studied the impact of the cavity behind a circular tunnel on the seismic performance of the structure. They also analyzed the boundary treatment effect of the model box, which was constructed using gypsum as the material for the tunnel model. The acceleration response of the model foundation, the acceleration response of the tunnel structure, and the dynamic strain of the tunnel structure were investigated. The seismic response of the tunnel structure varied with the buried depth of the structure, and the results of the earthquake disaster investigation were verified, providing a basis for the seismic design of the tunnel. SeyedSaeid et al. [5] investigated the seismic response of tunnels and surrounding soil in the presence of eight-layer pile structures near the tunnel using shaking table tests. They analyzed four physical models: free field soil (S), tunnel-soil (TS), pile-soil structure (SP), and tunnel-pile-soil structure (TSP) on loose sandy soil subjected to three-frequency sinusoidal vibrations. Liang et al. utilized a plexiglass tube tunnel model to simulate soft soil containing silty clay. The use of fine gravel and angular gravel simulated hard soil, and the application of equivalent linearization accounted for soil nonlinearity in the frequency domain. The similarity ratio design method expands material selection possibilities for structural models and broadens the applicability of shakers with conventional loading properties [6]. By integrating numerical simulations of upper soft and lower hard fields, as well as pure soft and hard fields, the study unveils differences in tunnel seismic responses between upper soft and lower hard fields, and pure soft and hard fields [7]. Li et al. [8] divided the soil and rock fields into four areas with different conditions and studied the influence of tunnels on the surface acceleration response of soil-rock formation using a shaking table test. Lei et al. [9] developed a cross-tunnel-slope model to investigate the seismic failure mechanism and interaction of the cross-tunnel-slope. Shaking table tests were conducted to analyze the dynamic strain of the tunnel and the acceleration response of the slope. In recent times, an increasing number of researchers have begun to focus on the correlation between tunnels and adjacent sites, conducting shaking table tests.

Some scholars have conducted seismic model tests on underwater tunnels. Although the scaled-down model tests can more intuitively show the response law of underwater tunnel structures under seismic loading, the structural response results are affected by various factors, such as dimensional similarity ratios, boundary conditions, hydrogeological environments, and the materials used to develop the tunnel model. Yuan et al. [10] used a shield tunnel along the river in Shanghai as a prototype to investigate the longitudinal response pattern of a long shield tunnel under the influence of ground motion, taking into account the traveling wave effect. To simulate the traveling wave input along the tunnel, a segmental model box was designed and fabricated. Using the Shanghai artificial wave as the ground motion input, the dynamic response of the tunnel structure under consistent and traveling wave inputs was compared and analyzed. Peng et al. [11] assessed the risk of tunnel floating when a shield tunnel traverses a liquefied sand stratum and examined the response pattern of tunnel floating in such conditions. Their study did not delve into the investigation of the destructive properties of tunnel strength. Therefore, the design of similar models adheres only to the principle of the same equivalent density without performing ratio tests or analyses of model materials and foundation soil. The experiment was conducted to observe the development characteristics of excess pore pressure in sand with the same density under three different peak accelerations and to analyze the law of tunnel buoyancy. To investigate the seismic response of an immersed tube tunnel, Wang et al. [12] created a model with a similarity ratio of 1/100, using plexiglass as a similar material. The effect of water is considered in the experiment, and existing studies are compared and analyzed. The characteristics and peak values of seismic waves and water have a certain degree of influence on the seismic response of an immersed tube tunnel. The influence of water on the horizontal response of the immersed tube tunnel is greater than that on the longitudinal response. To study the longitudinal seismic response of shield tunnels through hard and soft strata, Zhang et al. [13] constructed a shield tunnel model with a geometric similarity ratio of 1/40 using a soft polyvinyl chloride material with a low elastic modulus. Furthermore, a steel wire skeleton with a high elastic modulus was embedded in the shield tunnel to strengthen the cross-sectional flexural stiffness of the model. The simulation involves a shield tunnel with a notable difference in stiffness between the transverse and longitudinal directions. Miao et al. [14] sorted out and summarized seismic research on submarine tunnels in China in detail. They also cataloged the seismic design, construction, and theoretical research of completed submarine tunnels in China. Their study has great reference value for further research on the seismic technology of submarine tunnels. In addition, Shi et al. [15] systematically organized and summarized the seismic analysis methods for various typical immersed tube tunnels globally. This contribution aims to enhance the design standards for immersed tube tunnels and other underwater tunnel projects in China. Abdolhosein et al. [16,17], using a series of shaking table tests, analyzed the uplift mechanism of the tunnel in the liquefiable sand layer. The excess pore water pressure dissipation in the soil overlying the uplifted tunnel was significant, which led to suction in the soil deposit. Helical piles can efficiently restrict the possibility of rapid uplift of the tunnel and shorten the duration of the primary uplift phase. Kyungtae et al. [18] present a series of shake table tests to investigate the seismic response of a scaled shallow tunnel with different backfill material properties and thicknesses of overburden soil. At the largest observed soil shear strains, tunnel racking was significantly lower than that of ground deformation due to the decrease in soil. Yasuo et al. [19] conducted a shaking table experiment under a centrifugal acceleration of 50× g in order to understand the effect of the ground-improvement pattern on the seismic behavior of a shallow overburden tunnel. For a Level 2 earthquake, the seismic behavior of a tunnel has different tendencies depending on the ground-improvement pattern because of the expansion of the plastic area in the improved ground and the difference in stiffness between the improved ground and the surrounding ground. Mohammad et al. [20] investigate seismically induced circumferential forces and the bending moment of tunnel linings using a series of large shaking table tests and numerical analyses. Low frequencies of input motions for the tunnel with a low flexibility ratio resulted in greater circumferential bending moment values of the tunnel lining. Xu et al. [21] tested the seismic response of prefabricated horseshoe segmental tunnels. With the increasing seismic intensity, the predominant frequencies of the soil-tube system were first decreasing and then tended to be rather steady, and the viscous damping ratio was first increasing and then rather steady.

The shaking table device was designed specifically to study the seismic response mechanisms of submarines and underwater tunnels. This device enables the simulation of saturated soil-tunnel lining interactions at various water levels under seismic effects. By adjusting the relative heights of the specially designed water tanks, different water depths can be replicated. Test data are collected from sensors placed on the tunnel lining, allowing for an investigation into the response patterns of submarine tunnels based on different water depths. To further analyze the seismic response of submarine tunnels, a two-dimensional comprehensive model incorporating structure, saturated soil, water bodies, and a uniform viscoelastic artificial boundary was constructed using OpenSees software (3.3.0 version).

2. Shaking Table Test Design

In Figure 1, a cross-section of a submarine tunnel is depicted with a height of 11.82 m, a width of 15.4 m, a soil thickness of 32 m, and a lining thickness of 0.6 m. The submarine tunnel utilizes composite lining, and C30 concrete is employed for both the initial support and the second lining. The tunnel site is situated at the seabed underwater shoal, gently tilted toward the southeast bay mouth. The average slope is less than 1 degree. The water depth ranges from 5 to 15 m, with a general depth of 15–20 m. The composition material is sand–chalk–clay or chalk mud and muddy silt. The seawater depth on the tunnel overburden is approximately 10–20 m. The soil body covering the tunnel is a water-saturated, two-phase sandy-soil medium.

2.1. Similarity Ratio and Material Proportioning Study

Using a mixed similarity model, the shaker is subjected to a maximum load of 22 t, determining the model geometric similarity ratio of 1:20. Using the similarity theory magnitude analysis method [13], the similarity ratio of each physical quantity can be deduced.

Table 1. Tunnel prototype material and model material parameters.

Material	Type	Elasticity Modulus	Density
C30	Archetype	30 GPa	2385 kg/m3
Gypsum + barite powder + water	Test model	0.983 GPa	1630 kg/m3

shows the comparison of elastic modulus and density between the tunnel prototype and the tunnel model.

Table 2. Similarity table.

Physical Quantity	Resemblance	Similarity Ratio
Geometric dimensions l	λl	1/20
Displacement u	λu = λlλε	1/20
Modulus of elasticity E	λE = λlλρλg/λε	1/30
Density ρ	λρ	2/3
Strain ε	λε	1
Stress σ	λσ = λlλρλg	1/4
Time t	λt = (λlλε/λg)0.5	1/4.472
Frequency ω	λω = (λlλε/λg)0.5	4.472
Acceleration a	λa = λg	1

shows the similarity of the model test table.

Gypsum, a common building material with mechanical properties similar to those of concrete, is currently a good modeling material [22]. An 800-mesh barite powder, predominantly consisting of barium sulfate, was added to facilitate density adjustment. Through the proportioning test, the final ratio of the tunnel model material was determined as gypsum:barite powder:water = 0.25:0.4:0.35. Using the equal area ratio method, a galvanized iron wire mesh with a mesh size of 1.0 cm × 1.0 cm and a diameter of 0.7 mm was used instead of steel bars [23]. Waterproofing of gypsum products is performed using acrylic paint, which can form a dense water-repellent membrane similar to plastic after drying [24].

The orthogonal matching test was used in the experiment, and three orthogonal design factors were selected: Factor A ([calcium carbonate + gypsum mass and]/fine sand mass), Factor B (water quality/[sum of fine sand + calcium carbonate + gypsum + water quality]), and Factor C (gypsum mass/[sum of calcium carbonate + gypsum mass]). Three levels were set for each factor. A total of 9 groups of soil samples were designed for the similarity ratio test, as shown in

Table 3. Orthogonal test table.

Number	Factor A/%	Factor B/%	Factor C/%
Soil sample 1	15	5	25
Soil sample 2	15	10	50
Soil sample 3	15	15	75
Soil sample 4	20	5	50
Soil sample 5	20	10	75
Soil sample 6	20	15	25
Soil sample 7	25	5	75
Soil sample 8	25	10	25
Soil sample 9	25	15	50

and

Table 4. Actual ratio table of material.

Number	Fine Sand	Gypsum	Calcium Carbonate	Water
Soil sample 1	0.8261	0.031	0.0929	0.0500
Soil sample 2	0.7826	0.0587	0.0587	0.1000
Soil sample 3	0.7391	0.0832	0.0277	0.1500
Soil sample 4	0.7916	0.0792	0.0792	0.0500
Soil sample 5	0.7500	0.1125	0.0375	0.1000
Soil sample 6	0.7083	0.0354	0.1063	0.1500
Soil sample 7	0.7600	0.1425	0.0475	0.0500
Soil sample 8	0.7200	0.045	0.1350	0.1000
Soil sample 9	0.6800	0.085	0.085	0.1500

.

To determine the compression modulus of each soil sample, the rapid consolidation compression test (Figure 2) is employed. This involves maintaining a consolidation time of 1 h at each pressure level, with the scale reading recorded after the final pressure level. The test utilizes four pressure levels: 50 kPa, 100 kPa, 200 kPa, and 400 kPa, as outlined in

Table 5. Soil compression test data processing table.

Number	Initial Void Ratio e0	Void Ratio e1	Void Ratio e2	Void Ratio e3	Void Ratio e4	Coefficient of Compression a1-2	Compression Modulus (MPa)
Soil sample 1	1.125	1.058	1.039	1.004	0.963	0.191	11.11
Soil sample 2	0.922	0.812	0.787	0.741	0.698	0.250	7.69
Soil sample 3	0.775	0.766	0.755	0.734	0.706	0.115	15.38
Soil sample 4	1.132	1.079	1.054	1.013	0.956	0.256	8.33
Soil sample 5	1.306	1.242	1.181	1.144	1.118	0.611	3.77
Soil sample 6	0.670	0.660	0.653	0.641	0.623	0.075	22.22
Soil sample 7	1.119	1.051	1.024	0.985	0.954	0.275	7.69
Soil sample 8	0.809	0.794	0.773	0.767	0.747	0.208	8.70
Soil sample 9	0.682	0.673	0.665	0.651	0.628	0.084	20.00

for data processing. In the direct shear test (Figure 3), the stable horizontal displacement of the pressure box under vertical pressures of 100 kPa, 200 kPa, 300 kPa, and 400 kPa is measured in four-stage pressure increments for each soil sample. The shear stress corresponding to each vertical pressure is determined using the stress conversion coefficient on the measuring force ring, as detailed in

Table 6. Direct shear test of soil horizontal shear force corresponding to each level of pressure (unit: kPa).

	100 kPa	200 kPa	300 kPa	400 kPa
Soil sample 1	57.42	104.09	142.19	193.60
Soil sample 2	58.70	102.09	146.75	191.42
Soil sample 3	37.37	75.65	120.32	140.19
Soil sample 4	63.81	1132.21	155.14	195.43
Soil sample 5	51.41	97.35	138.73	180.48
Soil sample 6	62.71	114.12	174.10	215.48
Soil sample 7	65.26	117.77	164.98	180.48
Soil sample 8	55.05	104.09	150.40	205.09
Soil sample 9	59.79	111.57	143.47	192.69

. The horizontal shear force under vertical pressure in Table 6 is obtained according to the test. The linear fitting curve between the two was obtained through data fitting so as to obtain the internal friction angle (arctanθ) and cohesion force (intercept with τ axis) of each soil sample. The calculation results are shown in

Table 7. The matching ratio and parameter table of each soil sample.

Number	Fine Sand	Gypsum	Calcium Carbonate	Density	Cohesive Force	Internal Friction Angle	Compression Modulus	z
	%	%	%	g/cm3	kPa	(°)	MPa
Target value	-	-	-	1.5	2.5	20	1.5
Soil sample 1	82.61	3.1	9.29	1.33	12.67	24.06	11.11	0.58
Soil sample 2	78.26	5.87	5.87	1.54	14.04	23.88	7.69	0.58
Soil sample 3	73.91	8.32	2.77	1.77	5.10	19.44	15.38	0.60
Soil sample 4	79.16	7.92	7.92	1.33	22.70	23.59	8.33	0.61
Soil sample 5	75	11.25	3.75	1.30	9.84	23.20	3.77	0.51
Soil sample 6	70.83	3.54	10.63	1.88	12.03	27.39	22.22	0.59
Soil sample 7	76	14.25	4.75	1.34	33.90	21.41	7.69	0.69
Soil sample 8	72	4.5	13.5	1.65	4.56	26.38	8.70	0.49
Soil sample 9	68	8.5	8.5	1.86	19.23	23.27	20.00	0.66

.

The matching test of similar soil includes three materials: fine sand, gypsum, and calcium carbonate, and four measurement indices: density, cohesion, internal friction angle, and compression modulus, as outlined in Table 7. Simultaneously, a set of undisturbed soil parameter target values is established as the expected parameters for the soil samples.

During the test, it is not possible for all parameters to reach or approach the expected value simultaneously. The comprehensive evaluation index z of each soil sample phase relative to the parameter value of the target soil sample is calculated based on the fuzzy comprehensive evaluation rule [25]. As shown in Table 7, the group with the maximum comprehensive evaluation index z is selected, and the specific scheme of the soil sample ratio selected is the ratio of soil sample 7 as follows:

$$m_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e} \mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}}:m_{\mathrm{g}\mathrm{y}\mathrm{p}\mathrm{s}\mathrm{u}\mathrm{m}}:m_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{i}\mathrm{u}\mathrm{m} \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}}=0.76:0.1425:0.0475$$

2.2. Water–Seabed Subsea Tunnel Shaking Table Test Setting

As shown in Figure 4, a shaking table test set that can be used for a water–seabed submarine tunnel is designed. A water tank is placed on a liftable platform and connected to a model box with a hose. The height of the water tank is changed by the lift platform to simulate different water pressures at varied depths. Figure 4a shows the size of the scaled test model and the composition of the test device. The parts of the device in the figure include: (1) lifting platform, (2) water injection tank, (3) hose connecting the water tank and cover, (4) model box cover, (5) model box, (6) model box bottom plate, and (7) shaker table. The lifting device includes a scale, allowing the water pressure in the soil box to be calculated based on the water level in the water tank and the relative height difference between the water levels in the model box. The design water depth is converted into water pressure as follows:

$$p={\rho }_{w}gh$$

(1)

where p is the converted water pressure, ρ_w is the density of the water body in the tank, and h is the tank scale (i.e., the design water depth).

As shown in Figure 4b–d, this test involves the consideration of a water-saturated soil multifield coupling submarine tunnel shaking table test. To prevent the vibration overflow of the overlying water during the shaking table test, a 10 mm thick steel cover plate with waterproof rubber is specially designed. The cover plate is bolted to the box, and holes are drilled in the cover plate to facilitate cable extraction. To minimize model soil slippage during testing, a 30 cm layer of gravel is affixed to the bottom of the model box. Additionally, the gravel serves to simulate bedrock beneath the model soil.

2.3. Seismic Inputs and Loading Schemes

As per the specifications outlined in the “Specification of Seismic Design for Highway Engineering” (JTG B02-2013) [26], two classical ground motions are chosen for amplitude modulation based on the correlation between site category IV and the prescribed basic seismic intensity of 7 degrees, along with the peak acceleration of horizontal basic ground motions. As shown in Figure 5, the typical seismic wave time curves of two different acceleration spectra were selected for the test. Small-amplitude white noise excitation is performed beforehand at each increase in the acceleration peak to scan the model seismic response. To examine the seismic response of submarine tunnel models with different seismic intensities under water depths of 10 m and 20 m, shaking table tests were conducted. The conditions included seismic wave peak accelerations of 0.05 g, 0.1 g, and 0.2 g, corresponding to water depths of 0.5 m and 1.0 m, respectively. The following seismic waves are processed according to the determined acceleration similarity ratio, and the seismic waves are transversally loaded along the tunnel section. The loading conditions are shown in

Table 8. Loading condition.

Group Number	Relative Water Level (m)	Condition Number	Wave	Condition Name	Lateral Peak Acceleration (g)	Seismic Fortification Intensity
1	0.5	1	White noise excitation	BZ-1	-	-
		2	Loma Prieta	LP-1	0.05	6
		3	Kobe	KO-1	0.05	6
		4	Sweep wave	SP-1	-	-
	1.0	5	white noise excitation	BZ-2	-	-
		6	Loma Prieta	LP-2	0.05	6
		7	Kobe	KO-2	0.05	6
		8	Sweep wave	SP-2	-	-
2	0.5	1	white noise excitation	BZ-3	-	-
		2	Loma Prieta	LP-3	0.1	7
		3	Kobe	KO-3	0.1	7
		4	Sweep wave	SP-3	-	-
	1.0	5	white noise excitation	BZ-4	-	-
		6	Loma Prieta	LP-4	0.1	7
		7	Kobe	KO-4	0.1	7
		8	Sweep wave	SP-4	-	-
3	0.5	1	white noise excitation	BZ-5	-	-
		2	Loma Prieta	LP-5	0.2	8
		3	Kobe	KO-5	0.2	8
		4	Sweep wave	SP-5	-	-
	1.0	5	white noise excitation	BZ-6	-	-
		6	Loma Prieta	LP-6	0.2	8
		7	Kobe	KO-6	0.2	8
		8	Sweep wave	SP-6	-	-

. The vibration direction of the shaking table test is along the transverse axis of the tunnel section, as shown in Figure 6a.

2.4. Layout of Measurement Points

The sensors used in this test mainly include piezoresistive accelerometers, piezoelectric accelerometers, and strain gauges. The sensor layout is shown in Figure 6. In Figure 6a, the layout of the piezoelectric accelerometer is shown. The acceleration sensors are arranged on the model soil surface (or seabed surface), the model box, and the shaking table surface, respectively. Figure 6b shows the arrangement of the strain gauges for the submarine tunnel model. The locations of the arch, two spinners, two waists, two arch feet, and an inverter were selected on both sides of the 200mm tunnel section at the tunnel port.

3. Analysis of the Shaking Table Test Results

Figure 7 shows the acceleration and strain time-course curves of the tunnel vault under the action of 0.05 g, 0.1 g, and 0.2 g peak accelerations of the Kobe wave at 0.5 m relative water depth. It is evident from Figure 7a that the acceleration of the vault increases with the growth of seismic wave intensity. Figure 7b shows a section of the strain time-course curve of the vault measurement point under different seismic wave intensities.

By comparing the time course of the acceleration of the vault under the influence of the Kobe wave with 0.05 g peak acceleration under different water depth conditions in Figure 8, the acceleration of the tunnel vault measurement point decreases when the water depth is 1.0 m. The structural vibration is consistent under the conditions of different water depths. The change in depth does not affect the propagation frequency of the fluctuation in this test, and no vibration delay phenomenon is observed.

To analyze the influence of different depths on the structural response, the ratio of the difference when the water depth is increased from 0.5 m to 1.0 m is compared with that of the original 0.5 m water depth and set as the rate of change. The rates of change in acceleration, relative displacement, and strain values are given in Figure 9. The relative displacements between eight key positions of the tunnel section (A1~A8) and soil surface A9 are calculated.

Bounded by the zero axis, the peak acceleration of the structure under the two seismic waves decreases slightly when the water depth is increased to 1.0 m. The strain increases slightly, while the relative displacement changes in different positions. The magnitude of the change varies at each measurement point. As the intensity of ground shaking increases, the curve of acceleration and relative displacement rate of change is close to the zero axis, which means that the influence of water depth change is weakened under stronger earthquake action. With the increase in water depth, the rate of change in acceleration and relative peak displacement decreases at most locations of the tunnel. It may be due to the effect of overlying water on the compaction of the tunnel in the saturated soil layer, which limits the vibration of the tunnel in space, while the strain presents a complex changing trend, showing different changing trends at different measuring points.

4. Numerical Analysis

4.1. Finite Element Modeling

OpenSees software was used to establish a water-saturated soil-tunnel coupling model with a size of 48 m × 32 m. The water layer is modeled by the AC3D8 acoustic fluid unit without considering the compressibility and viscosity of the fluid. The contact surfaces of the water and soil layers are modeled by the contact unit ASI3D8. The AV3D4 unit is simulated as an absorbing boundary on the side of the fluid domain, with no restrictions applied to the top surface of the fluid. The saturated soil unit is represented by the four-node quadrilateral plane strain unit (FourNodeQuadUP) proposed by Yang et al. [27], which has three degrees of freedom. The first and second degrees of freedom are the displacements in the two directions, and the third degree of freedom represents the pore pressure. The tunnel structure is represented by the nonlinear BeamColumn beam unit, which is connected to the contact nodes of the soil unit and the beam unit with equal DOF. The bottom, left, and right boundaries are set as nondrainage conditions. The model is shown in Figure 10. A set of normal and tangential damping and springs is set up on both sides and at the bottom of the model, forming a two-dimensional, consistent viscoelastic artificial boundary unit. Damping and spring coefficients on both sides and bottom of the model are shown in

Table 9. Table of normal and tangential spring coefficients at the boundary nodes of the model side.

Node Number	Normal Spring Coefficient KBN (kg·m−2·s−2)	Tangential Spring Coefficient KBT (kg·m−2·s−2)	Node Number	Normal Spring Coefficient KBN (kg·m−2·s−2)	Tangential Spring Coefficient KBT (kg·m−2·s−2)
26	1,940,285	970,142.5001	226	187,265.8355	93,632.91776
51	970,142.5001	485,071.2501	251	167,688.7233	83,844.36163
76	615,384.6154	307,692.3077	276	151,810.5827	75,905.29133
101	447,213.5955	223,606.7977	301	138,675.0491	69,337.52453
126	350,486.3635	175,243.1817	326	127,628.8842	63,814.44212
151	287,926.3003	143,963.1501	351	118,210.8249	59,105.41246
176	244,225.0043	122,112.5022	376	110,085.9522	55,042.9761
201	211,999.576	105,999.788	401	103,005.2405	51,502.62026

and

Table 10. Table of normal and tangential spring coefficients at the bottom boundary nodes of the model.

Node Number	Normal Spring Coefficient KBN (kg·m−2·s−2)	Tangential Spring Coefficient KBT (kg·m−2·s−2)	Node Number	Normal Spring Coefficient KBN (kg·m−2·s−2)	Tangential Spring Coefficient KBT (kg·m−2·s−2)
1	2,000,000	1,000,000	14	2,495,131.446	1,247,565.723
2	2,060,104.81	1,030,052.405	15	2,480,694.692	1,240,347.346
3	2,119,995.76	1,059,997.88	16	2,457,180.467	1,228,590.234
4	2,178,938.843	1,089,469.421	17	2,425,356.25	1,212,678.125
5	2,236,067.977	1,118,033.989	18	2,386,199.945	1,193,099.973
6	2,290,393.337	1,145,196.669	19	2,340,822.944	1,170,411.472
7	2,340,822.944	1,170,411.472	20	2,290,393.337	1,145,196.669
8	2,386,199.945	1,193,099.973	21	2,236,067.977	1,118,033.989
9	2,425,356.25	1,212,678.125	22	2,178,938.843	1,089,469.421
10	2,457,180.467	1,228,590.234	23	2,119,995.76	1,059,997.88
11	2,480,694.692	1,240,347.346	24	2,060,104.81	1,030,052.405
12	2,495,131.446	1,247,565.723	25	2,000,000	1,000,000
13	2,500,000	1,250,000

. Simultaneously, a viscoelastic artificial boundary unit is set up through a zero-length unit. The nodes of the artificial boundary unit and the soil unit are linked via binding constraints.

4.2. Comparison of Experimental and Numerical Models

Model similarity test results can be used to infer the quantitative outcomes of prototype structures under similar or identical load conditions. Using the similarity relationship, the test results are used to deduce the prototype outcomes. These are then compared with the prototype numerical model results, enabling a systematic analysis of the seismic response law of the submarine tunnel. However, it should be noted that the similarity of the comparison results here refers to the similarity of the trend of quantity change when moving from one water depth to another, rather than the absolute similarity. In order to meet the operability of the multifield coupled undersea tunnel shaking table test, the scale model is adopted in the test. The boundary and water field conditions are simplified in the test, while the numerical modeling still adopts the real engineering scenario conditions. That is, the modeling process is not to reproduce the test results but to build a more realistic tunnel model of a real engineering case study. The corresponding relationship between the numerical simulation record number and the test record number is shown in

Table 11. Table of correspondence between numerical simulation record number and test record number.

Numerical Simulation Record Number	Arch Roof	Right Spandrel	Right Hance	Right Arch Footing	Inverted Arch	Left Arch Footing	Left Hance	Left Spandrel
Test record number	A1	A2	A3	A4	A5	A6	A7	A8

below.

Figure 11 illustrates the peak acceleration response at key positions of the submarine tunnel under a ground motion intensity of 0.1 g, influenced by shear waves. It depicts the variation patterns of the peak response under two water depth conditions. On the whole, with increasing buried depth, the peak response value gradually decreases at the location of the tunnel, and the peak value of the vault reaches a maximum.

Figure 12 shows the peak relative displacement response of each key position of the submarine tunnel when the ground motion intensity is 0.1 g under the influence of shear waves, and the variation rules of the peak response under two water depth conditions are given. The figure shows that the test value and numerical solution rule are the same. As the buried depth of the tunnel location increases, its displacement relative to the ground gradually increases, with the maximum relative displacement observed at the invert. Furthermore, the change in water depth has no obvious effect on the response rule.

Figure 13 shows the peak value of the strain response at each key position of the submarine tunnel when the ground motion intensity is 0.1 g under the influence of shear waves, and the variation rules of peak response under two water depth conditions are given. From the figure, the experimental values and numerical solutions are the same, while the peak value of the strain response is larger at the arch waist and arch foot of the tunnel but smaller at the tunnel arch and invert. Considering the effect of water depth, the response value at 0.5 m water depth is smaller than that at 1.0 m water depth, but the overall effect of water depth is small.

4.3. Effect of Water Depth on SV Waves

Specifically, this study examines the dynamic response of Kobe and Loma Prieta waves at seawater depths of 10 m, 20 m, 30 m, and 40 m. The focus is on submarine tunnels at key locations, including the arch, arch shoulder, arch waist, arch foot, and elevation arch, with SV wave inputs.

Considering the influence of water depth, both the Kobe and Loma Prieta waves present a complicated situation when an SV wave occurs. As shown in Figure 14, the peak value of the horizontal acceleration response at the tunnel arch, left waist, and two spandrels first increases and then decreases with water depth. However, the phenomenon of initially decreasing and then increasing occurred at the invert and foot of the two arches. In addition, the peak value of the horizontal acceleration response at the right arch increases gradually with water depth. For the Loma Prieta wave, the peak value of the horizontal acceleration response at the tunnel arch, left arch foot, right arch waist, and spandrel initially increases and then decreases with water depth. However, at the invert and left arched waist, there was a phenomenon of initially decreasing and then increasing. Furthermore, at the right arch foot of the tunnel, the peak of the horizontal acceleration response appears to be a zigzag increase process. Based on the above laws, it is evident that under the influence of SV waves, the peak value of horizontal acceleration response at the vault and spinner of the submarine tunnel initially increases and then decreases with the increase in overlying sea depth—the response peak value decreases between 10–20 m and gradually increases after 20 m. The peak value of the horizontal acceleration response at the invert of the tunnel initially decreases and then increases with the increase in the overlying water depth—the peak value of the response increases at 10–20 m and gradually decreases after 20 m.

Considering the influence of water depth when an SV wave is incident, Figure 15 illustrates the variation curve of the peak relative displacement response of the subsea tunnel concerning water depth when the peak seismic intensity is 0.1 g. The peak relative displacement response of the subsea tunnel remains relatively constant with increasing water depth, experiencing a change of approximately 0.01 cm.

Figure 16 shows that the peak value of the soil pore water pressure response under SV waves varies with water depth when 0.1 g is used. This shows that the pore water pressure changes slightly with increased water depth. The corresponding curves of the Kobe and Loma Prieta waves are relatively gentle. Specifically, the alteration in peak water depth response to the Kobe wave remains relatively stable, while the Loma Prieta wave shows a gradual increasing trend with water depth, albeit within a limited overall growth range.

Figure 17 shows that the peak response value of the submarine tunnel shear force under the influence of the SV wave varies with water depth when it is 0.1 g. Overall, the peak shear response at each tunnel position tends to increase with changes in water depth. Specifically, the peak shear response at the invert, vault, and arch only increases with increased water depth, indicating no significant change. The peak value of the shear response at the spandrel and foot of the tunnel increases with an increase in water depth.

Figure 18 illustrates the variation in peak bending moment response of the submarine tunnel in response to changes in water depth under the influence of SV waves. The peak moment response at the arch and invert of the tunnel showed almost no change with increasing water depth, indicating a weak sensitivity to water depth. In addition, the peak moment response at the spinner, waist, and foot of the tunnel exhibits a gradual increasing trend with the increase in water depth.

5. Conclusions

A shaking table model test was conducted to simulate a submarine tunnel using a completed submarine tunnel as a background. Similar material proportions were proposed for both the tunnel model and the model soil, followed by a corresponding material proportioning test. The shaking table test device was designed and fabricated, and waterproofing measures for the tunnel model and transducer were studied. Subsequently, the shaking table test was performed using the device designed for the water–seabed submarine tunnel. The response of the submarine tunnel structure was analyzed under varying water depths and seismic wave intensities. The main conclusions drawn from this study are as follows:

(1)

A shaking table test device for a water–seabed submarine tunnel was designed, and shaking table tests were conducted using this setup. The results indicate that the test device effectively simulates the hydrogeological environment of the tunnel and provides pressure compensation to simulate changes in water levels.

(2)

The acceleration response of the submarine tunnel decreases as the burial depth of each measurement point increases, attributed to the more significant constraint effect of earth pressure on the tunnel with deeper soil layers. In addition, the acceleration response of the tunnel shows a slight decrease when subjected to double water pressure from two different water levels. It may be due to the effect of overlying water on the compaction of the tunnel in the saturated soil layer, which limits the vibration of the tunnel in space.

(3)

As the burial depth of each measurement point in the submarine tunnel increases, there is a corresponding increase in the constraint effect of earth pressure on the tunnel relative to the horizontal displacement of the seabed surface. Analyzing the rate of relative displacement change reveals that doubling water pressure does not lead to a significant change in the relative displacement law; the change rate remains minimal.

(4)

During seismic loading, the strain peaks at the tunnel arch shoulder, arch waist, and arch foot are larger, whereas those at the vault and superelevation arch are smaller. Furthermore, a slight increase in the peak strain response of the tunnel is observed after doubling the water pressure.

(5)

Compared with the test results, the seismic response of the tunnel is mainly affected by the intensity of the ground motion, and the water depth has little influence on the tunnel’s dynamic response. The dynamic response of an undersea tunnel under the influence of a shear wave exhibits regular changes along the direction of tunnel height. Different response indices show varied trends in response peaks, emphasizing the need for comprehensive consideration in the design of undersea tunnels.

(6)

A two-dimensional model of a saturated soil–submarine tunnel was created to investigate the effects of SV wave input at different water depths on acceleration, relative displacement, pore water pressure, shear force, and bending moment responses of the tunnel structure. The results indicate that as water depth increases under the influence of shear waves, the dynamic response indices of different structures display diverse trends. Furthermore, the same dynamic response index exhibits distinct changing patterns at different positions within the tunnel structure. Among these, the peak acceleration of the structure demonstrates the most complex and variable trend, while the relative displacement of the horizontal peak is the gentlest.