Abstract
The interaction between soil and tunnel lining is an important factor affecting the performance of the tunnel longitudinal structure. In this paper, a three-dimensional model (Model 1) considering the interaction between soil and lining by interface element is built to analyze the response of shield tunnel lining to local surcharge loading. In order to do a comparison, the other model (Model 2) is established without an interface element too. The parametric study of the interface element in Model 1 is carried out, in which parameters such as cohesion, friction angle, normal stiffness, and tangential stiffness are considered. The numerical models are calibrated with experimental results of the model test in the lab obtained from the literature. It is concluded that, among the parameters of the interface element, the changes of cohesion and friction angle have little influence on soil-lining interaction, and the values of normal stiffness and tangential stiffness are more important in simulating soil-lining interaction. The maximum settlement of the tunnel is generally located in the center of the load, and the settlement is symmetric about the center of the load. With the increase of stiffness of soil, the trend of the longitudinal settlement of tunnel is similar, and the maximum settlement of tunnel decreases gradually. The result of Model 1 is much closer to experiment results than Model 2. The general deformation in Model 1 is smaller than the one in Model 2, and the slippage between the soil and lining can be found clearly in Model 1. Because of the interface element in Model 1, the restrain of soil around the lining can be simulated better. Through an engineering project, it can be found that the tunnel deformation can be predicted by the three-dimensional model considering the interaction between soil and lining by interface element well.
1. Introduction
Urban rail transit has become an important component of urban transportation because of its outstanding characteristics of fast, safety and high efficiency. By the end of 2021, the total length of global urban rail transit operation has been about 36,854.20 km, and the total length of China’s urban rail transit operation has reached about 9,724.35 km, accounting for 26.4% of the global total length, ranking first in the world, and the total length of Shanghai rail transit and subway has ranked first in the global cities [1].
Usually, there are various loads around the tunnels in the service period, among which ground loading on the subway is easy to cause structural diseases in the shield tunnel lining. For example, Nanjing metro line 2 from Maqun to Jinma deformed due to excessive muck, in which the length of deformed lining was 50 m. There was a ground loading in Shanghai metro line 2 from Chuangxin Middle Road station to Huaxia East Road Station, which caused the monitored maximum horizontal convergence value to reach 19.4 cm, and the tunnel lining was seriously deformed [2]. In December 2008, there was a large mass of soil on the shield tunnel in the Shanghai subway with the maximum height was 7 meters high. It caused serious water leakage and structural damage, even causing broken concrete to fall and blots to be broken in some rings of the lining segment [3]. Obviously, ground loading is a great threat to the safety of the tunnel lining service period, so it is of great significance to analyze the influence of surface surcharge loading on tunnel lining.
At present, the main methods to analyze the influence of surface surcharge loading on the shield tunnel are field measurement method [4–6], model test method [7–9], theoretical analysis method [10, 11], and numerical analysis method [12–17]. The deformation and the actual situation of the tunnel can be obtained directly by the field measurement method, but its cost is high and the monitoring process is time-consuming. A model test considering soil-tunnel lining interaction was executed by Wang [7]. In the model test, local surcharge loading on the 30-ring stagger-jointed tunnel model was considered. The influences of soil underlayer, depth of covering soil, load position on the settlement, and bending moment of the lining were simulated and analyzed. In terms of the theoretical calculation method, a longitudinal mechanical model of shield tunnel structure under the ground loading was established by Li et al. [10] based on the elastic foundation beam theory, and the longitudinal additional settlement and internal force of the underground shield tunnel under concentrated or uniform ground load can be predicted. However, simplified conditions can only be considered in the model test method and the theoretical analysis method. While much more complicated conditions and soil layers can be easily simulated in the numerical analysis method. Therefore, most researchers use numerical analysis method to analyze the influence of the existence of ground loading on tunnel lining. Lan [13] used MADIS/GTS NX software to establish a tunnel model to analysis the influence of various factors on tunnel settlement when the soil in the longitudinal direction changed. Sun [14] and Yang et al. [16] both established the stratum-structure model to analyze the influence of ground loading on the shield tunnel lining. One of the basic assumptions in their model is that the relative displacement between tunnel lining and soil was not considered in the calculation, and the slippage between the soil and the lining is not considered.
In the research above, the simulation between the lining and soil is treated as hard contact, which fails to simulate the relative sliding between the lining and the soil. The aim of this paper is to build a three-dimensional numerical model (Model 1), in which the contact between lining and soil is simulated by the interface element in FLAC3D, to analyze the behaviors of the tunnel lining under the local surface surcharge loading. To validate the numerical modeling, a comparison between the numerical results and (lab-based) experimental results obtained from the literature [7] was conducted. And further, the other model (Model 2) with no interface element was built too. By comparison between Model 1 and Model 2, the superiority of Model 1 was shown clearly. In the analysis, the parameters of the interface element were analyzed deeply, and the deformations of the tunnel lining in the transverse and longitudinal direction were analyzed for several different soil conditions. Then, through an engineering project, the settlement of shield tunnel lining under the loading and unloading conditions is analyzed.
2. Contact Problems
At present, there are generally two methods for dealing with the contact problem between soil and structure. One is the contact boundary constraint method (also known as the contact mechanics method), in which the contact problem is transformed into the constraint problem of force and displacement boundary, such as Lagrange multiplier method, penalty function method [18, 19], and augmented Lagrange multiplier method [20]. The other is to treat the contact problem as a special element or a mechanical property problem, which is called the contact surface element method. In this method, contact surface element is set on the contact surface between soil and structure. The classic contact surface elements are the non-thickness element proposed by Goodman [21] and the thin layer element proposed by Desai C S [22].
In this paper, the Goodman interface element is adopted to simulate the interaction between soil and segment lining. In the element, the constitutive relation is described by Coulomb-shear model, which is shown in Figure 1. The mechanical behavior of the interface element is simulated by the normal spring and the tangential spring simulated of the interface element node, the contact property of the interface element is reflected on the interface element node, and the contact force is only transmitted on the node. The constitutive relationship between the normal stress σ and the shear stress τ of the interface element is as given as
In which, A is the constitutive matrix.
In which, is the normal stiffness of the interface element; is shear stiffness of the interface element.
The strain increment is obtained based on the displacement, and the stress increment is obtained based on the strain increment. The shear stress follows Mohr–Coulomb yield criterion.
In which, c is the cohesion of the interface element, and δ is the internal friction angle of the interface element.
When the tangential force on the interface element is less than the maximum tangential force , the interface element is in the elastic stage. When the tangential force on the interface element exceeds the maximum tangential force , the interface element enters the plastic stage, the sliding between the soil and the lining occurs.
3. Numerical Model
3.1. Model Test
The schematic diagram of load and soil distribution in the model test [7] is given in Figure 2. The model of tunnel containing 30 staggered rings is fixed at both ends in the box. The lining segment dimension and physical parameters are shown in Table 1. The underlying soil layer is divided into three parts containing hard soil and soft soil as shown in Figure 2(b). By changing the soft soil which are soil 1, soil 2, and soil 3, respectively, shown in Table 2, the influence of the underlying soil layer on the tunnel is studied. The load is 48.96 kPa.
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3.2. Numerical Model
To avoid excessive calculation, the uniform rigidity ring model is adopted to describe the tunnel lining. The stiffness reduction is used to deal with the effect of the transverse joint and the longitudinal joint on the rigidity of lining. In order to keep the material properties, the transverse stiffness reduction is realized by changing the lining thickness.
In which, represents the lining thickness after transverse stiffness reduction; represents the original lining thickness; is the efficiency of transvers stiffness.
The longitudinal stiffness is realized by changing the cross section moment of inertia I.
In which, represents the longitudinal stiffness of the lining segment after reduction; is the efficiency of longitudinal bending stiffness; is the longitudinal stiffness of the original lining segment.
According to references [7, 23, 24], is about 0.76 and is between 0.2 and 0.35, the stiffness reduction in transverse and longitudinal direction is given in Figure 3.
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In order to better simulate the interaction between soil and lining, the stratum-structure model is adopted in the model shown in Figure 4. The buried depth of tunnel is 12.5 m, and the thickness of the underlying soil layer is 12.5 m. To avoid the boundary effects, the width and length of the tunnel are 5 times of the diameter, which are 37.5 meters. The load is simplified as the local uniform load, which intensity is 48.96 kN/m2. The region of load is a square, which is 6.25 m × 6.25 m. Solid element is used for soil, in which coulomb model is chosen, and shell element is used for lining. The displacement of the model is fixed in the horizontal direction for the left, right, front, and back border. The bottom of the model is fixed; thus, the bottom does not undergo displacement in the horizontal or vertical directions. The top is free.
In this paper, two simulation models are built, which are called Model 1 and Model 2. In Model 1, interface element is used to simulate the contact surface between soil and tunnel lining (as shown in Figure 5). It is attached to the tunnel surface and connects the tunnel to soil. The interface element can simulate the normal and tangential deformation characteristics of the contact surface by the normal stiffness and tangential stiffness. The displacement discontinuity between the two media contact interfaces is fully considered, and the slippage and tensile crack of the contact surface can be obviously simulated. In Model 2, the interface element is not used, which is the only different aspect from Model 1.
3.3. Analysis of Interface Element Parameters
In the simulation, the two sides of the contact surface will be embedded or separated [25] if the value of parameters of interface element is not suitable. Therefore, it is necessary to analyze the suitable parameter value for the interface element in order to simulate the performance of the model better.
There are six parameters in interface element, which are normal stiffness , tangential stiffness , cohesion force (c), friction angle (δ), dilatancy angle, and tensile strength. In general, the tensile strength of soil is 0, so the tensile strength of the interface element is set to 0, without considering its influence. For the convenience of analysis, the dilatancy angle of the interface element is also set as 0 without considering the dilatancy of soil.
According to the conclusions of Potyondy [26], Acer et al. [27] et al., it is assumed that c and δ in the interface parameters are a multiple (0.5–0.8) of the friction parameters (cohesion force C and friction angle θ) of the surrounding soil. Chen and Xu [28] proposed that the normal stiffness and the tangential stiffness of pile-soil interface element should be 10 times the equivalent stiffness of adjacent soil, namely:
In which, K is the bulk modulus of soil around the interface element, G is the corresponding shear modulus, and is the minimum size of the interface element in the normal direction.
According to the above suggested value, the cohesion force c and friction angle δ in the simulation are 0.5, 0.6, 0.7, and 0.8 times of cohesion force C, friction angle θ of the soil, respectively. According to expression (6), the parameters , are multiple of () in the analysis, the multiple are 1, 10, 100, and 200, respectively. Seven cases are considered when the influence of contact surface parameters on the settlement tunnel is analyzed, as shown in Table 3.
The interface element parameters and longitudinal settlement of the lining are shown in Table 3, and the variation of tunnel settlement under different parameter combinations is shown in Figure 6.
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In Figure 6(a), it is found that the maximum settlements are 8.5 mm (for case 1), 8.49 mm (for case 2), 8.61 mm (for case 3), and 8.58 mm (for case 4) when the normal and tangential stiffness , of the interface element remain unchanged. The differences are very small, which is only 0.12 mm. It shows that cohesion force and friction angle have a little effect on tunnel settlement, which is very different from pile-soil interaction. In the pile-soil interface, the changes of c and have great influences on the force and deformation. In tunnel lining-soil case, the cohesion force and friction angle of the interface element parameters are suggested to be 0.7 times of cohesion force C and friction angle θ of the soil.
In Figure 6(b), it is found that the maximum settlements are 8.61 mm (for case 3), 8.47 mm (for case 5), 8.06 mm (for case 6), and 7.51 mm (for case 7) when the normal and tangential stiffness , of the interface element are changed, cohesion force c and friction angle remain unchanged. The difference is 1.10 mm, so it is concluded that the values of , have a great influence on the settlement of the tunnel. When the stiffness , are 10 times of (), the numerical data is closer to the experimental data, which indicates that the values of interface element , should be about 10 times of (). In Figure 6(c), it can be found that the settlement at the center of the tunnel is nonlinear with the increase of , .
Based on the above analysis, the stiffness , , in the interface element are suggested to be 10 times of (), and the cohesion force c and friction angle are recommended to be 0.7 times of corresponding parameters of soil. In the analysis of next section, the value of , , c, are 86 MPa, 86 MPa, 8.87 MPa, and 22.11°, individually. The variation of transverse deformation and longitudinal settlement of the tunnel under local static load considering the change of the underlying soil layer in the longitudinal direction will be analyzed below.
4. Analysis of the Lining under Different Working Conditions
4.1. Analysis of Tunnel Longitudinal Settlement
The variations of tunnel settlement in longitudinal direction are shown in Figure 7, and the settlement data of tunnel under different working conditions are given in Table 4.
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Since the uniform load on the top of the tunnel is local, the influence of the uniform load on the longitudinal direction of the tunnel is uneven. The additional stress in the soil near the load region is much larger than other regions, so the maximum settlement occurs in the middle portion of the tunnel where the underlying soil layer is soft.
In Figure 7(a), while the underlying soil is soft soil 1 (E = 3.77 MPa), it is found that the maximum settlement occurs on 15th ring. The trends of variation of settlement in the longitude of Model 1 and Model 2 are similar to the experiment results. And obviously the result of Model 1 is much closer to experiment results than Model 2. However, the radian of the trend in numerical simulation is smaller than experiment results. The reason of the error between Model 1 and experiment is that the both ends of the tunnel limit its vertical displacement in the experiment. For analysis in soft soil 2 (E = 3.77 MPa), soft soil 3 (E = 3.77 MPa), the conclusion is similar to above.
In Table 4, it is found that the maximum settlements are 10.01 mm (for model test), 8.61 mm (for Model 1), and 5.25 mm (for Model 2) when the underlying soil is soft soil 1. While the underlying soil is changed from soil 1 to soil 2, the maximum settlement of the tunnel is increased by 10.7% (for model test), 12.4% (for Model 1), and 34.7% (for Model 2). It can be seen that the results in Model 1 is closer to experiment results than Model 2. The reason which causes the difference between numerical simulation results and test results may be that the consolidation of soil mass is not considered completely in the experiment.
From the above analysis, the application of interface elements in the stratum-structure model will make the simulation results be closer to experiment results, and the existence of contact surface will simulate the interaction between soil and lining under different working conditions better.
4.2. Analysis of Transverse Deformation
While the underlying soil is soft soil 1 (E = 3.77 MPa), the general deformation of the 15th ring of the lining is shown in Figure 8(a), in which the dotted line is for Model 1 and the dash dot line for Model 2. The 15th ring is changed into the shape of “horizontal duck egg.” The top is sunken downward, the radius of curvature in the lower half circle at the bottom is increased, and outward expansion of the tunnel haunch is found, which means the outside surface of the haunch is in tension and the inside of the arch crown and bottom are tensioned. The difference of the displacement between the two models is small, and the main difference is shown in the haunch and bottom of the lining. In Model 1, the horizontal convergence of lining ring of Model 1 is 3.355 mm, and the vertical convergence is 3.44 mm. In Model 2, the horizontal convergence of lining ring of Model 2 is 4.325 mm, which is 0.97 mm higher than Model 1. The vertical convergence is 3.975 mm, which is 0.535 mm higher than Model 1. It is clear that the deformation in Model 1 is smaller than the one in Model 2.
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The deformation of the 15th ring lining haunch along the horizontal direction is shown in Figure 8(b), it is obvious that the deformations of the lining ring are quite different in Model 1 and Model 2. In Model 1, the deformation at the inner surface in the haunch is 2.31 mm, and it is 1.02 mm at the outer surface. The difference of the deformation is 1.29 mm in Model 1, while it is 0.35 mm in Model 2. This major difference of deformation is occurred on the outer surface, it can reflect the constraint ability of the soil around the lining. Because of the interface element in Model 1, the restraint of soil around the lining can be simulated better. For analysis in soft soil 2 (E = 3.77 MPa), soft soil 3 (E = 3.77 MPa), the conclusion is similar to above.
The local region of Model 1 and Model 2 are shown in Figure 9. In Model 1, the interface element is built between the surface of lining and the soil. When the tangential stress is greater than the maximum shearing stress of the soil, there is slippage between the soil and the lining. In Model 2, the lining element and soil element are hard connected, and there is no slippage between nodes. Therefore, the slippage can be simulated well and the interaction between soil and lining can be reflected better with the existence of the interface element.
5. Engineering Project
Because of a viaduct construction, the Xiaolaigang river channel was temporarily used as a landfill site, resulting in settlement deformation of the shield section of Zhongchun Road-Jiuting Section of Shanghai metro line 9 below the river [29]. The river channel of Xiaolaigang is 24 m wide and 3 m deep. The distance from the bottom of the river channel to the top of the subway tunnel is 5 m, and the filling height is 4.5 m. Layout of the river, filling, and tunnel is shown in Figure 10.
In this case, lining segment and physical parameters are shown in Table 5, the soil physical and mechanical parameters are shown in Table 6. A three-dimensional model considering the interaction between soil and lining by the interface element is built to analyze the settlement of shield tunnel lining (75% lateral stiffness efficiency considered only) under the loading and unloading conditions as shown in Figure 11.
The comparison of the vertical displacement between finite element results and field test values is shown in Figure 12. It is seen that the longitudinal uneven settlement of the shield tunnel beneath river channel caused by soil filling is large. The maximum settlement is 44 mm in numerical calculation and 28 mm in field test values, both of which exceed the control value of 20 mm. The range of tunnels whose settlement exceeds the control value is about 27 m according to the field test values and is about 66 m for the numerical simulation. The reason of the error between field test values and numerical simulation is that the deformation of the tunnel is the final data in numerical simulation but a certain time value in the field test.
Later, 57% of the fill is excavated, which causes local uplift of the tunnel. The comparison of the uplift data between finite element results and field test values for the unloading condition is shown in Figure 13. The maximum deformation is almost the same. After excavation, the tunnel rebounds about 22 mm which is near to the middle portion of the river. The main uplift area of the tunnel caused by excavation is about 50 meters in field test values, and which is 70 m in numerical calculation.
From the above analysis, the settlement and uplift trends of the tunnel in the numerical simulation are close to the field test, and the data are also close. Therefore, the model with interface element can simulate the interaction between soil and lining well.
6. Conclusion
The interface element in FLAC3D is used to simulate the interaction between soil and lining, and the three-dimensional numerical model (Model 1 and Model 2) are established based on an experiment test. Based on the analysis of interface element parameters, the appropriate interface element parameters between lining and soil are selected. Then, the transverse deformation and longitudinal settlement of tunnel under local static load considering the change of the underlying soil layer in the longitudinal direction are analyzed, and an engineering project is analyzed too. Based on the analysis, the following main conclusions can be drawn:(1)In the interface element parameters, cohesion and friction angle have little effect on soil-lining interaction. The cohesion and friction angle of the contact surface can be 0.7 times of the corresponding parameters of the soil, respectively. The values of the normal stiffness and the tangential stiffness of the interface element have great influence on soil-lining interaction. With the increase of the numerical value, the settlement of the tunnel center does not change linearly, and , is suitable to be 10 times of ().(2)The maximum settlement of the tunnel is generally located at the load center, and the maximum settlement is affected by the change of the underlying soil layer. The trends of variation of settlement in the longitude direction in Model 1 and Model 2 are similar to the experiment results. And the results of Model 1 are much closer to experiment results than Model 2. In the stratum-structure model, the existence of interface elements will make the simulation results be closer to real condition, and it will simulate the interaction between soil and lining under different working conditions better.(3)The deformation trend of lining ring in Model 1 and Model 2 is roughly consistent with that in common condition. The ring is changed into the shape of “horizontal duck egg.” It is clear that the deformation in Model 1 is smaller than the one in Model 2. It can be seen that the restraint of soil around the lining can be simulated better in Model 1 with the interface element.(4)In the analysis of the engineering project, the settlement and uplift trends of the tunnel in numerical simulation are close to the field test, and the data are also close. Therefore, the model with the interface element can simulate the interaction between soil and lining well.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
All the authors declare that there are no conflicts of interest regarding the publication of this paper.