1 Introduction

Underground structures play a crucial role as essential infrastructure for transportation, utilities, and storage. However, when these structures are located in saturated sand deposits, they can be prone to lateral ground spreading, structural damage, and uplift during earthquake-induced liquefaction. Some examples include the subway system in Kobe, which was damaged in the 1995 Kobe earthquake (Iida et al. 1996), the uplifting of underground structures such as manholes and underground tanks in Chile during the 2010 Maule earthquake, and an underground tank in San Pedro del Valle that was uplifted by approximately 1.2 m (Yasuda et al. 2010). During liquefaction, a significant decrease in the shear strength and stiffness of the soil surrounding the underground structure happens. The excess pore water pressure in the soil increases because of the buildup of water pressure from the upward movement of groundwater. The increased pore water pressure and loss of soil strength can create a buoyant force on the underground structure. The buoyancy force acts in an upward direction, causing the structure to uplift. Numerous reports have highlighted the destructive effects of uplift pressure on underground structures (e.g., Khoshnoudian and Shahrour 2002; Aydingun and Adalier 2003). During the last few decades, various experimental and numerical studies have been conducted to investigate the seismic behavior and liquefaction-induced uplift of underground structures, as well as mitigation techniques to tackle the deteriorating effect of liquefaction around tunnels (e.g., Rashiddel et al. 2024; Fatima et al. 2023). Yang et al. (2004) performed numerical analyses to investigate the uplift of George Massey and the BART tunnels. The results demonstrated significant sand movement toward the bottom of the uplifted tunnels.

Liu and Song (2005) assessed the effects of soil liquefaction on underground structures. They showed that upward displacement of the structures occurred because of the liquefaction-induced pressure acting beneath the underground structures. Liu and Song (2006) used the fully coupled dynamic finite element code to assess the liquefaction-induced uplift behavior of a subway tunnel in saturated sand. They showed that the small effective unit weight of underground structures, excess pore pressure buildup, and flow of liquefied soils played key roles in applying upward pressure to underground structures during earthquakes. Azadi and Hosseini (2010) conducted numerical studies on the uplift and seismic behavior of shallow tunnels in liquefied ground. These studies showed that as the loading amplitude increased, the tunnel uplift and bending moments in the tunnel segments increased as well.

Tobita et al. (2011) did centrifuge tests and found that the input motion intensity and the creation of excess pore water pressure had a big effect on how the underground structure floated. Saeedzadeh and Hataf (2011) and Sharafi and Parsafar (2016) studied the effects of different parameters, such as dilatancy angle and density ratio of soil, burial depth and diameter of pipe, underground water table, and thickness of the saturated soil layer on pipeline floatation. The results showed that the burial depth of the pipe and the generation of excess pore water pressure around the pipe had the most prominent effect on the pipe response. Similarly, the importance of excess pore pressure at the bottom of the structure as a major cause of the uplift response of shallow structures was confirmed by Zhou et al. (2014). A numerical study on the liquefaction potential of soils surrounding circular tunnels showed that the most effective parameter in the liquefaction potential is tunnel depth (Unutmaz 2014).

Zhou et al. (2015) applied strong simulated earthquake loading to a centrifuge shaking table and studied the micro-behavior of saturated sand around a buried structure. They compared the seismic behavior of two underground structures with different burial depths. They demonstrated that, in cases where the burial depth of the structure was relatively shallow, the uplift force exerted by the excess pore pressure beneath the base of the structure surpassed the pressure exerted by the soil overburden. Hence, a large uplift force was applied to the structure. This resulted in considerable upward displacement of the structure. By contrast, when the structure was deeply buried, the sand above the structure generated a large overburden pressure. This overburden pressure opposed the uplift force beneath the bottom of the structure, thus preventing its uplift. Huang et al. (2015) conducted centrifuge tests to examine the correlation between the maximum uplift displacement of a simulated pipe and the maximum excess pore water pressure (EPWP) ratio. They discovered that the pipe experienced uplift when the maximum EPWP ratio, which represents the excess pore water pressure to the initial effective stress of the soil, surpassed a specific threshold value associated with the burial depth of the pipe.

Zheng et al. (2019) used a multivariate adaptive regression spline model based on numerically artificial data to analyze the main uplift-related parameters. These parameters included the diameter and burial depth of the tunnel, the friction angle and relative density of the soil, and the amplitude, duration, and frequency of the input motion. Zhao et al. (2019) considered soil-structure interaction and proposed a seismic analysis method for underground structures based on the response spectrum method. Zheng et al. (2021) employed a support vector machine and artificial neural network models to propose a model for the liquefaction-induced uplift of underground structures. Jitchaijaroen et al. (2024) developed soft-computing models by studying the different parameters like anisotropic strength ratios of clayey soils, overburden factor, and width-to-depth ratios to investigate the stability of rectangular tunnels.

Valizadeh and Ecemis (2022) studied liquefaction-induced uplift of buried pipes and mitigation techniques against liquefaction-induced uplift through shaking table tests and numerical models. Nokande et al. (2023) conducted a series of shaking table tests to investigate the uplift of circular tunnels in a liquefiable sand layer. Their research focused on the tunnel diameter, burial depth of the tunnel, tunnel weight, liquefaction extent, uplift mechanism, and safety factors against liquefaction-induced uplift. They showed that shallow burial depth, larger diameter, and lower weight can intensify tunnel uplift.

Ko et al. (2023) constructed a full-scale physical model of pipelines buried in liquefiable ground and conducted shaking table tests to investigate the liquefaction-induced uplift of buried pipelines. They observed that owing to the pipeline–soil interaction, the excess pore pressure buildup around the uplifted pipelines was lower than that in the free field and on the top of the pipeline.

Previous studies investigating liquefaction-induced uplift of underground structures have primarily focused on the possibility and understanding of the mechanisms behind the uplift process. The study conducted by Koseki et al. (1997) examined uplift mechanisms related to different underground structures. Researchers explored the correlation between uplift displacement and the factor of safety against uplift (FS), ultimately determining that uplift occurred when the FS approached a value of 1.0. Figure 1 shows the force distribution acting on a tunnel in saturated sand. The modified expression for the safety factor of the underground structure under liquefaction conditions is calculated using Eq. (1) as.

$${\text{FS}}=\frac{{F}_{{\text{WS}}}+{F}_{{\text{SP}}}}{{F}_{B}-{F}_{T}+{F}_{{\text{EPP}}}}$$
(1)

where FWS, FSP, FB, FT, and FEPP are the topsoil weight, frictional resistance force at the boundary between the sidewall of the structure and the surrounding subsoil, buoyant force due to hydrostatic pressure, weight of the structure, and uplift force due to excess pore water pressure caused by soil liquefaction, respectively. However, the factor of safety procedure only provides the triggering condition of uplift and cannot accurately predict the ultimate uplift displacement of underground structures.

Fig. 1
figure 1

Force distribution acting on a tunnel in saturated sand

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Most studies in the literature assume instantaneous liquefaction and neglect the variation in the liquefaction response over time. However, liquefaction-induced ground movements can develop over time, and the uplift behavior of underground structures is influenced by the rate and duration of the liquefaction process. Neglecting these time effects can limit the prediction accuracy.

In this study, a parametric analysis was conducted to investigate the effects of various parameters on liquefaction-induced uplift of tunnels. The examined parameters included the burial depth of the tunnel (H), the diameter of the tunnel (D), and the amplitude of the input acceleration (amax).

The results revealed significant influences of each parameter on the tunnel uplift. These findings provide valuable insights into the design and safety considerations associated with tunnel construction in seismically active regions affected by liquefaction.

2 Numerical Study

The simulation process consists of two sequential steps. The first step involves defining the primary state, which comprises two phases. The initial phase establishes a static steady state in a drained position, whereas the second phase involves modeling the tunnel by defining the tunnel lining and removing the corresponding soil element. The second step involves dynamic analysis, where the model is subjected to seismic loading in the form of harmonic acceleration applied at the bottom. During this step, an undrained analysis was conducted to simulate liquefaction. In this study, the liquefaction-induced uplift of buried tunnels was appraised using the PLAXIS2D software and the PM4SAND soil constitutive model under cyclic loading.

2.1 Model Geometry and Boundary Conditions

To verify the numerical model, a centrifuge experiment previously performed by Chian et al. (2014) was used. A circular tunnel with a diameter (D) of 5 m and buried at a depth (H) of 7.5 m from the ground surface to the center of the tunnel was considered in the analysis, as shown in Fig. 2. The model consisted of a 24-m-thick liquefiable sand layer with a relative density (Dr) of 45%. The water table was located at the top layer of the surface. The tunnel was simulated using a linear elastic material.

Fig. 2
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Illustrations of model dimensions and boundary conditions

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In the static analysis phase, the lateral boundaries of the model were set to an equal degree of freedom (equal DOF) condition. In the dynamic analysis phase, free-field boundaries (as drained material) were used to simulate the far field (as recommended in the PLAXIS2D manual). The base of the model was fixed in both horizontal and vertical directions, whereas the lateral boundaries were fixed only in the horizontal direction. A sinusoidal acceleration motion with a frequency of 0.75 Hz, amplitude of 0.22 g, and total duration of 28 s was applied at the rigid base of the model (Fig. 3). The generated mesh includes 553 elements and 5146 nodes, and the size of the elements is 0.12 m.

Fig. 3
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Input base acceleration in PLAXIS2D numerical model (amax = 0.22 g, f = 0.75 Hz)

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2.2 PM4SAND Soil Constitutive Model Calibration

A saturated sand layer was modeled using the PM4SAND constitutive model (Boulanger and Ziotopoulou 2017) implemented in PLAXIS2D software. The sand parameters used in the numerical model were defined according to Dr and the normalized SPT value (N1)60, which are correlated as \({D}_{r}=\sqrt{\frac{{({N}_{1})}_{60}}{60}}\) (Skempton 1986). The input values for the soil layer are listed in Table 1.

Table 1 Input parameters of saturated sand in the numerical analysis
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Within the PM4SAND model, there are three main input parameters: relative density, shear modulus coefficient (G0), and contraction rate parameter (hpo), which are employed to calibrate the undrained shear strength. In addition, there are 19 parameters that could be adjusted through calibration using laboratory test data. G0 is correlated to (N1)60 according to the relationship proposed by Boulanger and Ziotopoulou (2017) as

$${G}_{0}=167\sqrt{{\left({N}_{1}\right)}_{60}+2.5}$$
(2)

The contraction rate parameter is employed to adjust the model’s contractiveness and calibrate it to specific values of the cyclic resistance ratio (CSR). After gathering all the necessary input parameters, such as Dr, degree of saturation (S), dry density (ρd), maximum void ratio (emax), minimum void ratio (emin), water bulk modulus (Kw), friction angle (φ), and permeability (k) from the reference centrifuge tests, they were assigned to the model. The calibration of hpo was then performed by conducting a uniform cyclic direct simple shear test using the PLAXIS2D soil test facility. To achieve the desired CSR, a methodology was proposed to calibrate the hpo based on the liquefaction-triggering strength (Boulanger and Ziotopoulou 2017).

Specifically, a CSR curve available in the literature (Wu et al. 2003) was selected for the fully saturated sand, which matched the properties of the sand used in the reference centrifuge test. Figure 4 shows a visual representation of the selected CSR curve for calibration of the model. In the initial calibration process, hpo was adjusted to match the target CSR, aiming for a double-amplitude shear strain (DAγ) of 6% after 15 loading cycles, based on PM4SAND element tests with CSR curves at 80 kPa consolidation stress. The calibration results, as shown in Fig. 4, indicate the successful adjustment of hpo based on literature data. Figure 5 illustrates the stress–strain relationship obtained from the PM4SAND calibration simulations on fully saturated sand with a DAγ of 6%.

Fig. 4
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Cyclic stress ratio versus the number of cycles to reach DAγ = 6% in liquefiable sand

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Fig. 5
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Stress–strain responses from PM4SAND calibration for saturated sand

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2.3 Model Validation

To investigate the failure of tunnels to float, a combination of a centrifuge experiment and numerical simulation was conducted (Chian et al. 2014). The results of excess pore water pressure at the crown, springing, and invert of the tunnel, as well as the uplift time history of the tunnel at its springing level, were obtained. This study aims to simulate the centrifuge test using the PLAXIS2D software and validate the results of the numerical model by comparing them with both the experimental and numerical results from the original centrifuge test.

Figure 6 illustrates a comparison of the far-field acceleration time histories between the validation model and reference results. In addition, Fig. 7 shows a comparison of the uplift time histories at the springing level of the tunnel, comparing the reference numerical and experimental centrifuge tests with the results of the PLAXIS2D validation model. An imbalanced vertical pressure was applied to the tunnel by increasing the pore water pressure. Consequently, the tunnel started to move up, and the backfill soil started flowing from beside the tunnel to beneath the tunnel. This leads to tunnel uplift and the settlement of the surrounding soil.

Fig. 6
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Comparison of far-field acceleration time histories in numerical and experimental studies

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Fig. 7
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Comparison of the uplifts between the experimental study and the validation model in PLAXIS2D

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Figure 8 depicts the excess pore pressure time histories at various locations within the model, which were obtained and compared with the results of the centrifuge test. The comparison reveals favorable agreement between the numerical model developed in PLAXIS2D and the findings of the reference study. It is evident from Fig. 8 that the excess pore water pressure around the tunnel increased over time owing to the dynamic loading induced by shaking. During seismic shaking, the soil around the tunnel experiences cyclic stresses, causing the pore water pressure to increase for some initial cycles (12 cycles, equivalent to approximately 15 s). This excess pore pressure was initially high but gradually dissipated through the soil over time. However, the dissipation process around the tunnel is slower because of factors such as longer drainage paths and lower permeability. As a result, the excess pore pressure at these locations continued to increase over time after strong shaking, eventually reaching a peak before gradually dissipating.

Fig. 8
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Comparison of the excess pore water pressure time histories between the experimental centrifuge test and the validation model in PLAXIS2D

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Based on these results, it can be concluded that the model constructed in the present paper effectively predicts the liquefaction response of the tunnel in liquefiable sand. Therefore, by altering the parameters of this validated model, like burial depth, diameter of the tunnel, and amplitude of the input acceleration, a parametric study was done. Results obtained from the parametric study were used to propose an empirical function to predict the liquefaction-induced uplift of circular tunnels.

2.4 Parametric Numerical Investigation

2.4.1 Effect of Tunnel Diameter on Uplift

The diameter of a tunnel can affect its liquefaction-induced uplift. Because a larger tunnel diameter typically corresponds to a larger surface area exposed to the surrounding soil, it can lead to a higher potential for interaction between the tunnel and liquefiable soil during an earthquake-induced liquefaction event. Additionally, the size of the tunnel can influence the stress redistribution in the surrounding soil during liquefaction.

A larger tunnel diameter can result in a larger zone of influence, where excess pore water pressure is generated, potentially resulting in a more significant uplift response. Figure 9 compares the uplift response of tunnels with different diameters buried at a depth of 5 m. This indicates that at the same burial depth, enlarging the diameter of the tunnel resulted in a higher uplift value.

Fig. 9
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Comparison of uplifts for tunnels with different diameters (H = 5 m, amax = 0.22 g)

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2.4.2 Effect of Burial Depth of Tunnel on Uplift

The burial depth of the tunnel affects the magnitude of the overburden pressure acting on it. When the burial depth of the tunnel increased, a greater overburden pressure was applied to the tunnel. Consequently, it can help the tunnel resist the liquefaction-induced uplift forces. In addition, greater burial depths enhance the confinement provided by the surrounding soil. This can oppose the uplift forces. However, higher excess pore water pressures were observed in the deeper soil layers. Therefore, greater burial depths may have led to increased uplift forces. It is important to consider these issues and conduct comprehensive numerical and experimental analyses to evaluate the effects of burial depth on the liquefaction-induced uplift of tunnels.

Figure 10 compares the uplift responses of tunnels buried at different depths. This shows that increasing the burial depth provides a higher overburden pressure on the tunnel and reduces liquefaction-induced uplift of the tunnel.

Fig. 10
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Comparison of uplifts for tunnels buried in different depths (D = 5 m, amax = 0.22 g)

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2.4.3 Effect of Input Acceleration Amplitude on Uplift

The amplitude of the input acceleration has a significant impact on the uplift of tunnels caused by liquefaction. Increasing the input acceleration amplitude intensifies ground shaking and amplifies the generation of excess pore water pressure in the liquefiable soil. Consequently, a larger liquefaction zone forms around the tunnel, exerting higher uplift forces on the structure (see Fig. 11).

Fig. 11
figure 11

Illustration of the effect of input acceleration amplitude on ru,max around the tunnels at the end of shaking (D = 6.5 m, H = 10 m); a amax = 0.1 g, b amax = 0.22 g, c amax = 0.3 g

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The interaction between the tunnel and the surrounding soil during seismic events is greatly influenced by the amplitude of shaking. The dynamic response of both the tunnel and the surrounding soil is significantly impacted by the intensity of shaking, ultimately affecting the magnitude of uplift forces experienced by the tunnel. Figure 12 demonstrates the effect of input acceleration amplitude on the uplift of a tunnel with a diameter of 6.5 m buried in 10 m of depth. According to this figure, a higher amplitude of input acceleration increases the liquefaction-induced uplift response.

Fig. 12
figure 12

Effect of input acceleration amplitude on the uplift of the tunnel (D = 6.5 m, H = 10 m)

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Table 2 summarizes the parameters used in the numerical models, as well as the internal forces on the lining of tunnels obtained from numerical analyses. Totally, 72 numerical models with D of 3, 3.5, 4, 4.5, 5, 5.5, 6, and 6.5 m; H of 5, 7.5, and 10 m; and amax of 0.1 g, 0.22 g, and 0.3 g were simulated.

Table 2 Numerical study parameters and the internal forces of tunnel lining (at the end of shaking, t = 28 s)
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2.4.4 Effect of the Tunnel Burial Depth on Internal Forces in Lining

Figure 13 presents examples of recorded internal forces acting on tunnel linings with a diameter of 3.5 m, buried at depths of 5, 7.5, and 10 m, and subjected to an input acceleration of 0.22 g. As previously discussed, the depth of a tunnel plays a significant role in determining the internal forces experienced by its lining. As the depth increases, the pressure exerted by the overlying soil becomes more substantial, leading to greater forces on the tunnel lining. Deeper tunnels are also more susceptible to differential ground movements, such as settlements or deformations caused by variations in soil properties. These differential movements result in higher bending moments in the tunnel lining. However, it is worth noting that the impact of depth on shear forces is generally less pronounced compared to axial forces, particularly in shallow tunnels where the soil above the tunnel provides less confinement.

Fig. 13
figure 13

Illustration of internal forces recorded on the lining of tunnels (D = 3.5 m, amax = 0.22 g, t = 28 s) (not in scale)

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Figure 14 exhibits the effect of variation in the tunnel diameter and its burial depth on the internal forces recorded on the lining. It is observed from the figure that the most noticeable effect of the tunnel’s burial depth is on the axial force produced in the lining. The shear force and bending moment are less affected by the burial depth. Additionally, at the same burial depth, increasing the diameter of the tunnel resulted in a rise in the internal forces experienced by the tunnel.

Fig. 14
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Effect of the burial depth and diameter of the tunnels on the internal forces recorded on the lining (t = 28 s)

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3 Proposed Empirical Function

In the literature, a limited number of studies have developed functions to predict the liquefaction-induced uplift of underground structures. Sudevan et al. (2020) carried out some finite difference analyses of circular tunnels buried in liquefiable sands. They introduced a function to predict the ultimate uplift response of underground structures, as given in Eq. (3):

$$\frac{\mathrm{Ultimate\ uplift} \ (m)}{D}=0.5-0.005{D}_{r}-0.131\frac{h}{D}$$
(3)

where D, Dr, and h are the diameter of the tunnel, the relative density of the soil, and the embedment depth of the tunnel. Although it was shown that Eq. (3) can predict the ultimate uplift of the tunnels, it does not consider the effect of the applied shaking amplitude. It was shown in the literature (e.g., Chian et al. 2014) and Figs. 11 and 12 of the present study that the amplitude of the input shaking has a significant effect on the ultimate uplift response. So ignoring amax in the prediction model would be a questionable issue and could drop the accuracy of the model.

Another numerical study that has suggested a function to predict the uplift of underground structures is the one presented by Valizadeh and Ecemis (2022). As discussed before, they carried out numerical analyses to predict the liquefaction-induced uplift of the buried pipelines. However, they have not evaluated the applicability of their function for higher values of D and H that are attributed to tunnels buried in deep layers.

The aforementioned limitations impelled the author to perform a comprehensive numerical study to find a function to predict the liquefaction-induced uplift of tunnels buried in shallow and deep layers. The novelty of the function would be its focus on the dependency of uplift on amax and higher values of H and D. Also, this function would be able to predict not only the ultimate response of the uplift but also its time-dependent variations. Regarding the uplift values of models with different H/D ratios and drawing an exponential trend line (see Fig. 15) of numerical results for amax of 0.1 g, 0.22 g, and 0.3 g, an empirical function was proposed to predict the liquefaction-induced uplift of buried circular tunnels. The proposed function estimates the time-dependent uplift response of circular tunnels under sinusoidal excitation with a constant frequency of 0.75 Hz. Considering the uplift results presented in Table 2, it was observed that the liquefaction-induced uplift of tunnels is significantly affected by H/D ratio, amax, and time. So, after some trial and error, Eq. (4) was established:

$${\text{Uplift}} \left(m\right)=0.08t\times {e}^{\frac{H}{D}\left(4.61\frac{{a}_{{\text{max}}}}{g}-1.96\right)}$$
(4)

where t is the time parameter in the unit of seconds.

Fig. 15
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Uplift values obtained from numerical analyses for different burial depth ratios \(\left(\frac{H}{D}\right)\) under different input acceleration amplitudes

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Since the uplift response of tunnels is significantly dependent on the soil properties, shaking amplitude, and parameters related to the tunnels, the author suggests that the proposed function is acceptable and conservative for circular tunnels buried in loose soils with 0.5 ≤ H/D ≤ 3.5 and amax ≤ 0.3 g.

It should be noted that 2D numerical models with plane-strain conditions were used in this study, which may not fully capture the complex three-dimensional stress and strain distributions around the tunnel or the phenomenon of soil arching around the tunnel. 2D numerical models also may not accurately represent the boundary conditions. Using 3D numerical models would provide a better understanding of how tunnels interact with the surrounding soil during liquefaction events.

After the proposal of the empirical function, its accuracy in predicting the ultimate liquefaction-induced uplift was compared with the numerical analysis results in Fig. 16. According to the R2 values calculated for each set of data and the corresponding predicted function, it was concluded that the proposed function has an acceptable accuracy. However, it should be modified by more parametric or experimental studies to be able to predict the uplift response of tunnels buried in dense soils or subjected to higher amplitudes of input shaking.

Fig. 16
figure 16

Comparing uplift values obtained from numerical results and proposed function

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4 Validation of the Proposed Empirical Function with Experimental Studies

The last section of this study is allocated to the validation of the proposed empirical function with experimental studies available in the literature. Two different experiments were selected as reference tests to validate the proposed function.

4.1 Validation with Centrifuge Test Results

As previously mentioned, Chian et al. (2014) conducted a series of centrifuge tests to look into the liquefaction response of circular tunnels. Uplift time histories of two tests were selected to compare with the values predicted by the proposed function in the present study. Figure 17 illustrates the comparison of uplift time histories between centrifuge tests and those predicted by the function. For the soil-tunnel model with a burial depth ratio of H/D = 1.1, subjected to the base shaking of amplitude amax = 0.22 g, a good agreement was detected. However, for the case of H/D = 1.5 and amax = 0.1 g, a small dissimilarity was observed between the experimental result and the prediction model. The uplift predicted by the proposed function has been increased gradually, while the experimental result shows an abrupt increase in the rate of uplift. The probable reason for this difference is that a significant fluctuation of forces has acted on the tunnel, which was not properly simulated in the numerical analysis.

Fig. 17
figure 17

Comparing uplift time history predicted by the proposed function with the centrifuge test

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4.2 Validation with Shaking Table Test Results

Nokande et al. (2023) conducted a series of shaking table tests to investigate the uplift behavior of circular tunnels in a liquefiable sand layer. One of those experiments was selected here, and the relevant uplift was compared with the predicted uplift by the proposed function. Figure 18 presents the comparison of uplift responses for a tunnel with a diameter of D = 8 m, buried in H = 12.48 m, and subjected to sinusoidal shaking with an amplitude of amax = 0.14 g.

Fig. 18
figure 18

Comparing uplift time history predicted by the proposed function with the shaking table test

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According to this figure, not only is the trend of the uplift increment similar to the experimental one, but the ultimate uplift response predicted by the proposed function is very close to that of the shaking table test. It confirms the acceptable applicability of the proposed function in predicting the liquefaction-induced uplift of circular tunnels.

5 Conclusion

The dynamic behavior of buried circular tunnels in liquefiable loose sands was investigated using the PM4SAND model and the PLAXIS2D finite element program. The constitutive model and the liquefaction parameters were validated using centrifuge test data. Simulated tunnel uplift and excessive pore water pressure were compared to the outcomes of the experiments. Then, an extensive parametric study was conducted to determine how each parameter affected the tunnels’ uplift due to liquefaction by altering the tunnel diameter and/or burial depth ratio in each model. An empirical function was suggested to calculate the tunnel liquefaction-induced uplift using the numerical data. Finally, the suggested function’s accuracy was assessed by comparing its predicted results with those reported in the literature. The numerical parametric analysis revealed the following findings:

  • During the liquefaction, the soil experiences the development of excess pore water pressure, which results in the emergence of a buoyant force acting on the tunnel, leading to its uplift. The presence of upward pressure can pose significant challenges, especially in soils that are saturated and loose.

  • The burial depth (H) of a tunnel can have a significant impact on its susceptibility to liquefaction-induced uplift. In general, it can be observed that tunnels buried at greater depths exhibit a reduced vulnerability to uplift resulting from liquefaction in comparison to those located at shallower depths. Tunnels at greater depths are subjected to increased confining stresses exerted by the soil layers above. The heightened levels of stress contribute to the stabilization of the soil, thereby diminishing the likelihood of liquefaction.

  • Tunnels with greater depth exhibit an increased vertical separation between the ground surface and the crown of the tunnel, resulting in a larger volume of soil above the tunnel. Consequently, this additional soil mass serves to disperse the pore water pressure that arises during seismic events. This measure has the potential to mitigate the uplift forces exerted on the tunnel.

  • The diameter of the tunnel (D) can also impact its vulnerability to uplift caused by liquefaction, although the specific consequences may not be as significant as those associated with the depth of burial.

  • The peak ground acceleration, which represents the amplitude of input shaking (amax), has a notable impact on the phenomenon of liquefaction-induced uplift of the tunnels. An elevated amax has a direct correlation with an increased likelihood of liquefaction, thereby resulting in heightened pore water pressure and uplift forces exerted on tunnels.

  • The ratio of H/D has a significant impact on the bending moment, shear force, and axial force experienced by the tunnel lining. Increased H/D ratios result in enhanced overburden pressure, thereby inducing amplified bending moments. The increase in H/D ratios leads to a rise in the magnitude of shear forces, which is primarily attributed to the complemented lateral pressures exerted by the adjacent soil. Furthermore, it can be observed that the magnitude of the axial forces experienced by the tunnel is directly proportional to the H/D ratio. Consequently, this leads to the tunnel being subjected to greater vertical stresses exerted by the soil above it.

  • An empirical function was developed here to estimate the uplift caused by liquefaction in circular tunnels buried in loose sands. This estimation is based on variations in the acceleration ratio (amax/g) and the tunnel burial depth ratio (H/D). The proposed function provides a prediction for the time-varying uplift of buried tunnels in loose sand when subjected to sinusoidal excitation at a fixed frequency of 0.75 Hz. The function exhibits potential applicability in practical engineering for the cases in which 0.5 ≤ H/D ≤ 3.5 and amax ≤ 0.3 g.

  • Engineers can optimize tunnel designs to withstand liquefaction-induced uplifts, ensure regulatory compliance, and assess performance after construction by using the empirical function introduced in this study. With the help of this feature, engineers can design tunnels that conform to safety regulations, prevent excessive displacement, and evaluate the accuracy of their designs by comparing them to international standards. This results in tunnel infrastructure that is more resilient and economical in seismic regions.

6 Limitations and Future Work

This study focuses on tunnels designed as hollow circular sections. The effect of lining thickness was not considered here. When designing such tunnels, it is important to carefully assess their structural requirements and capacities. The thickness of the tunnel lining is particularly crucial for determining its ability to withstand bending or shearing forces. Furthermore, more experimental and numerical studies are required to modify the proposed function for higher amplitudes of the input shakings as well as tunnels that are buried in soils with higher relative densities.

Additionally, tunnels buried in soils other than liquefiable sandy soils can also face various hazards that may affect their stability. Some of the possible hazards include:

  • Ground settlement: Clay and sandy-clay soils can undergo settlement over time, especially if they are poorly compacted or if there are changes in the groundwater level. Settlement can cause damage to the tunnel structure and affect its stability.

  • Groundwater seepage: Clay soils tend to hold water, which can lead to increased groundwater pressure around the tunnel. This pressure can cause seepage into the tunnel, leading to potential flooding and erosion of the tunnel lining.

  • Swelling and shrinkage: Clay soils are prone to swelling when they absorb water and shrinking when they dry out. These volume changes can exert pressure on the tunnel walls, leading to deformation and potential structural damage.

Future research can focus on seismic performance and stability analysis of tunnels buried in different types of soils to further advance our understanding of underground structures.