The compression-rebound test of soil is carried out with a standard consolidation instrument (as shown in Figure 1). The size of the soil sample is φ61.8 mm in diameter and 20 mm in height. The test condition is to limit the deformation of soil laterally, and then apply the vertical load on the top of the sample. After the vertical load is loaded to a certain stress level, it is unloaded in stages. Each loading or unloading step needs to wait until the soil sample is stabilized in deformation, and the height of the soil sample is recorded to calculate the change of pore ratio and compression or rebound modulus of the soil sample. Specific test conditions are shown in Table 1.

Both elastic deformation and plastic deformation will occur during the compression process, and the deformation curve of soil appeared to have hysteretic characteristics during the rebound and re-compression process. This feature needs to be taken into account in the calculation to make it more realistic. Figure 2 shows typical compression-rebound test curves of different soils.

As can be seen from Figure 2, no matter whether it is soft soil or hard soil, the soil pore ratio could not return to the initial pore ratio when unloading under different overlaying loads, which indicates that both elastic deformation and plastic deformation occurred in the soil during compression, and the plastic deformation is relatively large.

It can also be seen that in the process of soil rebound, the load-pore ratio curve of soil is not a straight line (the rebound curve appears to be a straight line under the condition of low load unloading, it is a curve, but the bending is not obvious, which will be reflected in the fitting model later). This indicates that during the process of soil rebound, the stress-strain relationship of soil is elastic-plastic, and the deformation modulus of soil changes can not be used by the unified elastic modulus.

In addition, the hysteretic characteristics of soil rebound-recompression deformation are different under high pressure and low pressure. When soil is unloaded and reloaded under a low compression load (less than 100 kPa), the load-pore ratio curve of soil overlaps and there are no hysteretic characteristics. When the soil is unloaded and reloaded under a high compression load, the load-pore ratio curve shows obvious hysteretic characteristics. At the initial stage of unloading under a high compression load, the load-pore ratio curve of soil is relatively smooth and the rebound deformation is small, which indicates that the rebound modulus is large at the initial stage of unloading. When unloading to a certain extent, the load-pore ratio curve begins to bend and the rebound deformation occurs rapidly. The hysteresis characteristics of soil under high compression load are consistent with the existing results (Li and Teng, 2018; Wang et al., 2020), but that under low compression load is rarely reported. What should be emphasized is that the excavation on the ground will usually cause the deformation of the shallow soil layer. Therefore, the unloading rebound characteristics of soil under low compression load are more significant for shallowly buried tunnel engineering.

For numerical calculation, the most commonly used and most convenient constitutive model is the Mole Coulomb model. The parameters used in the mole Coulomb model are elastic modulus E ₀ , Poisson’s ratio ν, cohesion c, and friction Angle φ, among which, the impact of the elastic modulus on deformation is the largest. But for engineering investigation, the compression test is used to determine the modulus of soil mass. To avoid confusion, here we use E _s and E _sc to represent the compression modulus and rebound modulus determined by the compression-rebound test and use E ₀ and E _c0 to represent the elastic modulus and rebound modulus before and after the excavation construction in the numerical simulation. The problem of shallow buried tunnel deformation caused by soil excavation does not involve soil re-compression, therefore, the estimation of soil re-compression modulus is not included in this research.

The compression curve of soil is shown in Figure 3(1) , and the rebound curve of soil is shown in Figure 3(2). In terms of morphology, the compression curve and the rebound curve have similar morphology, so they can be fitted by one equation. After a comprehensive comparison, we choose the power function of three parameters as the fitting function of E _s and E _sc . The specific equation is as follows: e = a ( 1 + p / b ) c (1) Where: a,b,c are model parameters; e is porosity ratio; p is the current pressure value/kPa.

1) Comparison of test data and formula fitting data in the soil compression test

As shown in Figure 3, the fitting effect of the soil compression curve and rebound curve is good, and the fitting model can accurately reflect the compression and rebound characteristics of different soils. For the convenience of application, the fitting parameters of the compression curve are listed in Table 2. For rebound conditions, there are different fitting parameters for the rebound curve under different pressure values. The function relationship between the three fitting parameters and the pressure values can be further established to facilitate the application of numerical calculation in the later period. The fitting functions of different soil rebound parameters are also listed in Table 2 below.

For excavation construction, the soil should be unloaded at different load levels, so the rebound modulus E _sc of soil should be calculated according to the excavation depth and the unloading level. It is worth noting that the above model and calculation parameters are used to calculate the current pore ratio of soil, which need to be converted to modulus if the modulus is wanted.

Some scholars have discussed the relationship between soil compression modulus E _s and elastic modulus E ₀ under compression, and the conclusion is that there is a certain proportional relationship between E _s and E ₀ (Jia et al., 2008; Yu, 2014). However, there is no literature report on the relationship between E _sc and E _c0 , used in numerical calculation. Therefore, it is necessary to determine the relationship between them to provide reliable parameters for the subsequent calculation of shallow buried tunnel deformation.

Figure 4(1) shows the comparison between the numerical simulation and the test conditions of different soil compression-rebound tests. It can be seen that the conditions of the compression rebound test and the numerical simulation are the same: the numerical model has a diameter of 6.18 cm and a height of 2 cm, the horizontal deformation is constrained around the soil sample, displacement is completely constrained at the bottom of the soil sample, and only axial load is applied or reduced at the upper part of the numerical model. The loading step is also the same as the real test which is that after the consolidation under a certain load level is completed, the next load will be added to the surface of the model.

Figure 4(2) shows the datum comparison of the compression-rebound test and the numerical simulation of different soil samples under different rebound conditions. Here, the rebound modulus E _sc obtained by the compression-rebound test is constantly adjusted, and the multiple relationships between E _c0 used in numerical simulation and E _sc obtained by the test are determined, that is, E _c0 = 4.0E _sc . It can be seen from the figure that the soil rebound curve obtained by numerical simulation is in good agreement with the test curve, which can well reflect the deformation of soil under the rebound conditions. Therefore, in the subsequent deformation calculation of shallow buried tunnel, E _c0 will be set whose value is 4 times E _sc according to the different compression state and unloading level.

A previous study by Wang et al. (2006) reported on the influence of the Shanghai Daning Commercial Center foundation excavation on its nearby metro tunnel. Another study (Liang et al., 2017) reported a simplified analytical method to evaluate the effects of the excavation of the tunnel. In this literature, the deformation datum by the analytical calculation and in-site measuring are compared. This datum is used as the base datum to verify the calculated metro tunnel deformation obtained by the method proposed in this paper.

The layout of the foundation pit and its position in relation to the metro tunnel are shown in Figure 6. The length and width of the foundation pit are 240 m × 230 m, and the distance between the supported pile and the tunnel is only 5.4 m. In the adjacent area, there are four layers of soil, which respectively are: 1 fill; 2 silty clay; 4 muddy clay; 5 ₁clay. The external diameter of the metro tunnel is 6.2 and its liner thickness is 0.3 m. The numerical model is modeled according to the project and is shown in Figure 6.

The basic parameters of the soil and supported structure are the same as that used in other literature (Wang et al., 2006), which are 1 fill: Poisson ratio 0.32, cohesion 1 kPa, friction 22°; 2 silty clay: Poisson ratio 0.35, cohesion 18 kPa, friction 18°; 4 muddy clay: Poisson ratio 0.4, cohesion 12 kPa, friction 10°; 5 ₁clay: Poisson ratio 0.35, cohesion 20 kPa, friction 16°. In a contrasting situation considering the type of the surrounding soil, the resilience parameters are determined referred to the test results in this paper.

As the downline tunnel is closer to the foundation pit than the upline tunnel, the deformation of this tunnel is considered and contrasted to verify the rationality of the method proposed in this paper. The calculated deformation datum obtained from wider literature (Wang et al., 2006; Liang et al., 2017) and the proposed method are compared with the measured results. As shown in Figure 7, although the calculated datum obtained by the three methods all reflect the deformation of the metro tunnel, the datum calculated by the method in this paper is closer to the measured datum and better reflects the tunnel deformation trend.

To better verify the rationality of the proposed method, another project case is introduced below. In this case, the Bund Underground Passage in Shanghai was constructed on the metro tunnel. The layout of the underground passage and its position in relation to the metro tunnel is shown below in Figure 8(1).

The underground passage was constructed using the open-cut method. The foundation pit was supported by several retaining walls, as shown in Figure 8. In this case, four conditions with different Dz (Plan I: Dz = 11 m; Plan II: Dz = 16.5; Plan III: Dz = 22 m; Plan IV: Dz = 27.5) were calculated, and the results are compared with those in the literature (Huang et al., 2013).

The numerical model is shown in Figure 8(2). The engineering geological condition is that there are three soil layers: 1. Clay layer; 2. Silty clay layer; 3. Sandy silty clay layer. The mechanical parameters of the soil are the same as that used in the literature (Huang et al., 2013)which are ①clay: Poisson ratio 0.32, cohesion 8 kPa, friction 23°; ②silty clay: Poisson ratio 0.31, cohesion 13 kPa, friction 27°; ③sandy silty clay: Poisson ratio 0.3, cohesion 3 kPa, friction 30°. As with Case I, in this case, the resilience parameters are also referred to in the test results in this paper.

Figure 9 compares the results of this study and the wider literature (Huang et al., 2013). It can be seen that the tunnel deformation results calculated by the two methods can all reflect the law that with the increase of the burial depth of the tunnel, the deformation of the tunnel decreases, and the maximum deformations obtained by the two methods are almost the same. However, from the axial deformation variation law, the results of the two methods have certain differences. This could be caused by setting the different parameters of the supported structure. According to experience, we believe that under the supported structure with enough strength, the influence range should be limited in the cutting construction area. Because the supporting structure can effectively block the deformation surrounding the foundation pit wall. This understanding is also confirmed by Case I, in which the deformed range of the metro tunnel is consistent with that of the cutting construction (see Figure 8).

A real estate project was constructed above the Wuhan metro line 4, and before its construction, an influence assessment was done to ensure the engineering can be conducted.

The range of the soil unloading and its position in relation to the Wuhan metro line four is shown in Figure 10(1). As seen in the figure, the unloading soil lay directly on the metro tunnel. The whole unloading area is divided into two parts: 1) part I: the north of the red line and 2) part II: the south of the red line. The elevation of part I after soil unloading is 23.5 m, the unloading thickness is 8–12 m and the tunnel buried depth is 10.9–13.8 m. The elevation of part II after soil unloading is 22 m, the unloading thickness is 4.2–7.8 m and the tunnel buried depth is 13.8–19.6 m. Based on engineering experience, the influence of unloading construction in part I should be greater than that in part II due to the larger buried depth of the tunnel and unloading thickness. Therefore, the tunnel deformation calculation in part I was under heavy consideration in the assessment.

For establishing the numerical model, we selected two typical sections of part I, which can be seen in Figure 10(2). The distance between the two sections is about 50 m and the 3D numerical model is obtained by extrusion operation from sections A₁A₂ to A₃A₄. The 3D numerical model can be seen in Figure 10(3). The length and height of the model are respectively 220 m and 50 m. The diameter of the metro tunnel is 5.5 m and the thickness of the concrete supporting structure is 0.35 m.

The engineering geological condition is also shown in Figure 10(3). It can be seen that the soil layers under the unloading soil are respectively 1) fill; 2) clay; 3) silty clay; 4) clay; 5) highly weathered mud sandstone; and 6) medium weathered mud sandstone.

The initial parameters of the model are listed in Table 3 and the parameter adjustment in the calculation process is achieved by the method in Section 2.3. What needs illustration is that the initial parameters are necessary before the calculation and along with the calculation, the parameters would be periodically and continually adjusted according to the indoor test results.

As mentioned in Section 3, the whole calculation procedure is divided into three stages: I. initial stress balance stage; II. tunnel construction stage; and III soil unloading on the ground stage. In stage I, the whole model consists of soil, while in stage II, the tunnel construction is simulated. In this stage, the deformation of the tunnel is resisted by the concrete supporting structure. Finally, in stage III, all the deformation in stage II is eliminated, and the deformation caused by soil unloading is studied.

To verify the calculation results, the contrast between the measured datum and the simulation result is shown in the following section. In the in situ deformation measure, the absolute deformation on the tunnel crown and bottom was conducted. The deformation of the whole tunnel in the soil unloading range was measured during the construction, and the simulation section was in the measuring range. The interval length between the measuring section was about 5.5 m.

1) Heaving deformation on the tunnel CRown

Figures 14(1)a,(2)a show the contrast of the heaving deformation on the tunnel crown between the calculation result and the in site measured datum.

From the figures, it can be seen that after the soil excavation construction, deformation occurs on the tunnel crown, which is in agreement with the in situ measured data. The resulting agreement can be reflected in three aspects. First, on one hand, given the heaving deformation of the tunnel crown, the calculated maximum deformation is 2.1 mm and the measured maximum deformation is 2.2 mm, which indicates that they are consistent. Secondly, on the other hand, from the perspective of longitudinal tunnel deformation, due to the different thickness of the unloading soil, the heaving deformation values of the tunnel crown at different positions should be different, and the numerical simulation results accord with the distribution trend of the measured deformation of the tunnel. Finally, because the right line of the subway tunnel is closer to the maximum thickness of the unloading soil, the value of the heaving deformation of the right line tunnel should be larger than that of the left line tunnel, and the numerical simulation results are consistent with the measured data.

2) Heaving deformation on the tunnel bottom

Figures 14(1)b,(2)b show the contrast of the heaving deformation on the tunnel bottom between the calculation result and the in site measured datum.

From the Figures, it can be seen that:

1) Due to the soil unloading construction on the ground, the rebound deformation of the foundation soil caused the heaving deformation of the subway tunnel. Due to the relatively deep depth at the bottom of the tunnel and the large rigidity of the tunnel lining structure, the heaving deformation at the bottom of the tunnel was significantly smaller than that at the top of the tunnel.

Figure 15 shows the vertical clearance convergence deformation of the subway tunnel in the longitudinal direction. The positive and negative values of vertical clearance convergence deformation of the tunnel are defined as: the positive value of deformation represents the stretching of the tunnel section, and the negative value represents the contraction of the tunnel section.

These figures indicate that:

1) Due to the soil unloading construction on the ground, the heaving deformation occurs on the soil and the tunnel. Although there is heaving deformation at the tip and bottom of the tunnel, the heaving deformation on the crown of the tunnel is larger than that the bottom. Therefore, the vertical section of the tunnel is stretched and its shape tends to become elliptical.

According to the ‘Technical code for safety protection of urban rail transit structures (CJJ/t202-2013’ of China, the metro tunnel deformation must be limited to 5 mm and the ellipticity ratio (calculated by the formula (2)) of the tunnel must be limited within 5‰. E l l i p t i c i t y R a t i o = ( D max − D min ) / D d e s i g n (2) Where, D _max and D _min are respectively the maximum and minimum diameters of the deformed tunnel and D _design is the designed diameter of the tunnel.

Figure 16 shows the ellipticity ratio of the two metro tunnels of the soil unloading construction. It can be seen that the ellipticity ratio of the two metro lines is all smaller than the mandatory standard. The maximum ellipticity ratios of the two tunnels are respectively 1.6‰ and 1.42‰. From the deformation aspect, the values of the horizontal and vertical deformation of the two tunnels are all within the demand of the code. Therefore, although the soil unloading construction would have a certain influence on the metro tunnels, the tunnels are still in accordance with the relevant regulations, meaning the tunnels can be deemed safe.