A Study on Seismic Isolation of Shield Tunnel Using Quasi-Static Finite Element Method

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, Hubei 430071, China University of Chinese Academy of Sciences, Beijing 100049, China Graduate School of Engineering, Civil and Environment Engineering Course, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi 400-8511, Japan Graduate School of Interdisciplinary Research, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi 400-8511, Japan Disaster and Sustainable Administration Research Center, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi 400-8511, Japan


Introduction
Shield tunnels are widely used for water supply, public transport, communication, sewerage, and other infrastructures.Tunnels do not cause self-excited vibration under earthquakes but rather are controlled by the surrounding ground deformation.erefore, it is generally recognized that tunnels are not easily affected by earthquakes.In recent years, with the increase in the number of underground structures and the frequent occurrence of seismic damage to underground structures, the antiseismic issue of underground structures is increasingly attracting high attention from seismologists around the world.
As shown in Figure 1, the seismic isolation layer applied to a tunnel is a technology to reduce the seismic response of the tunnel by isolating the tunnel body from deformation of the ground in the outer periphery of the structure.e effectiveness of the seismic isolation layer has been verified by a series of numerical simulations and experiments [1][2][3][4].Seismic response analysis of tunnels is divided into crosssectional seismic response analysis of tunnels and seismic response analysis of tunnels as a whole.At present, there are many methods for seismic response analysis of tunnels; typical ones include the Bessel function approximate solution, the response displacement method, and the finite element dynamic analysis method.
e Bessel function approximate solution is mainly suitable for cross-sectional seismic response analysis of tunnels, without dynamic analysis of the surrounding ground, and establishment of a new structural model of the tunnel.e approximate solution of the tunnel lining internal force of a circular cross section can be obtained using a theoretical formula.e analytic target and external force are based on ideal assumption.In the case of a sudden change in ground conditions, the approximate analytical Bessel function solution for general ground cannot give the internal force of the tunnel lining correctly.Under this condition, it is necessary to use other analytic methods [5,6].
e response displacement method is suitable for cross-sectional seismic response analysis of tunnels and seismic responses of tunnels as a whole at the same time.e tunnel lining is simulated as a ring beam for the cross-sectional seismic analysis or the straight beam for the whole longitudinal seismic analysis.
e ground spring, showing the interaction between the tunnel lining and surrounding ground, is placed around the tunnel lining, and the surrounding ground response displacement and shear stress are input into one end of the ground spring.is method is practical.However, due to the complexity of ground conditions, it is difficult to set up the ground spring, and the beam-spring model used for seismic design of underground pipelines does not necessarily give a good approximation for tunnels with large diameters, so it has limitations [7,8].In the finite element dynamic nonlinear analytic method, it is needed to consider various nonlinear questions, and the modeling process is complicated.It requires very large calculation resources and time, especially in three-dimensional analysis.Meanwhile, it is difficult to provide strict dynamic boundary conditions.Hence, it is not suitable for practical seismic isolation design of tunnels [9,10].e behavior of a tunnel subjects to deformations imposed by the surrounding ground.At present, the research on seismic isolation of tunnels mainly focuses on tunnel cross sections.However, as a linear underground structure, the key point is to study the seismic isolation of tunnel as a whole.e whole longitudinal behavior of a tunnel subjected to deformations imposed by the surrounding ground can be divided into two types [11]: (1) compressive and tensile deformations along the longitudinal direction (Figure 2(a)) and (2) bending deformation along the transverse direction (Figure 2(b)).In this paper, a quasi-static method based on an axisymmetric finite element model for seismic response analysis of seismically isolated tunnels is used to research the seismic isolation effect and mechanism of seismic isolation layer of a shield tunnel along both the longitudinal and transverse directions.

Quasi-Static Method for Seismic Response
Analysis of Seismically Isolated Tunnels

Outline of the Quasi-Static Method.
e quasi-static approach adopting an axisymmetric FEM and the response acceleration method is used for practical design of seismically isolated tunnels; this approach eliminates the troublesome processing of boundary conditions related to reflected waves and modeling complex ground and structural conditions.
It is well known that the response acceleration method provides slightly better evaluations than the finite element dynamic analysis approach; that is, the method presents more reliable and safer evaluation results.
is method has been adapted to the seismic design of underground ducts, shafts, and rock caverns.In this method, the 2 Shock and Vibration surrounding ground is modeled by means of finite elements, and accelerations are calculated in a free field.e calculated accelerations are then applied to the finite element model, including an underground structure [12][13][14].
Figure 3 illustrates a schematic representation of the axisymmetric modeling.In the upper part of the figure, a tunnel is constructed through a soil deposit, while the lower part represents the axisymmetric modeling of the ground and tunnel conditions shown in the upper part; here, the centerline of the tunnel is set as an axis of symmetry.It is essential that special considerations should be taken when modeling the effects of the ground surface, boundary conditions, and seismic load.Accordingly, if a method can be defined to convert the seismic load in the axisymmetric model, thereby equalizing the ground displacements around the tunnel shown in the upper and lower parts of Figure 3, a simplified procedure can be developed to evaluate the 3-D interaction effects around a tunnel.

Method to Convert the Seismic Load.
e fundamental theory regarding the method employed to determine the seismic load in the axisymmetric model will be described in this section.e corresponding underground and loading conditions are shown in Figure 4, where H denotes the thickness of the soil deposits, z denotes the coordinate originating from the bedrock, and h c denotes the height from the bedrock to the tunnel center under actual tunnel and ground conditions.In the axisymmetric model, an axis of symmetry is located at the height of the upper outer surface of the actual tunnel, and the outer radius R denotes the distance from the axis to the bedrock.e outer surface of the axisymmetric model corresponding to the outer surface of the tunnel lining is located at the radius r � r 0 , where r 0 denotes the outer radius of the tunnel lining.
en, h c and R can be related by h c � R − r 0 .e seismic load considered here is an inertial force originating from the ground acceleration due to seismic ground motions.
e static loading method, in which the inertial force due to an earthquake is loaded statically while ignoring the damping term in the equations of motion, is employed herein.
Figure 5 illustrates a schematic diagram to describe the method of applying an inertial seismic load on the ground.
In the proposed model, the seismic load is divided into two components: an inertial force denoted P(z), which is given in equation (1), acting on the ground beneath the tunnel and a concentrated force S, which is given in equation ( 2), acting on the outer surface of the tunnel body: Shock and Vibration 3 e conversion of the seismic load in the axisymmetric model is conducted as shown in Figure 5.

Seismic Load in the Longitudinal Direction.
e conversion of the seismic load in the longitudinal direction is conducted in the following manner [15,16].e displacement of the ground at z with the unit width is defined as u when the horizontal force P is acting at a height z. e displacement at r (�R − z) of a hollow cylindrical disk with a unit thickness, the outer surface of which is constrained, is defined as u * when the horizontal force P * is acting at r. Placing the displacements u * and u equal to each other, P * can be expressed using P, R, and r as follows: e concentrated force S in equation ( 2) can also be replaced by S * as shown below: Figures 6(a) and 6(b) represent the relation between the concentrated force and ground shear deformation in the actual ground and in the axisymmetric model, respectively; the case of a soil column is shown in Figure 6(a), and the case of a hollow cylindrical disk is shown in Figure 6(b).
e shear displacement values at the location r � r 0 in Figure 6(a) and the shear displacement values at the location z � h c in Figure 6(b) are coincident with each other.However, the shape of the vertical shear displacement distribution in the hollow cylindrical disk in Figure 6(b) is not straight and is widely different from that in the soil column displayed in Figure 6(a).To make the vertical distribution of the shear displacement in the proposed model equal to that in the actual ground, the following method is adopted.
When the concentrated load S * acts on r � r 0 , the shear displacement u c is However, as shown in Figure 7, assuming that the acceleration is uniformly distributed in the vertical direction, the inertial force When the inertial force s * i acts on a hollow cylindrical disk with a unit thickness ranging from r i−1 to r i , the shear displacement u c,i produced by the inertial force is en, the shear displacement produced at r � r 0 is e uniformly distributed acceleration, which can be derived from equation (8), is given by equation the following equation: In numerical analysis, earthquake loads acting at r are calculated by multiplying the mass matrix by the ground acceleration at r. us, the acceleration at r used in such calculations can be given by equation (10), which is derived from equations ( 3) and ( 9); this acceleration denoted a(r) used herein is defined as a modified acceleration coefficient: Finally, by applying the inertial force, which is the product of the mass m(r) and acceleration a(r) of each node as the seismic load, the displacement generated in the actual ground can be reproduced.In the analysis of the longitudinal direction, the load acts in the longitudinal direction on each hollow cylindrical disk as an axisymmetric load.

Seismic Load in the Transverse Direction.
e conversion of the seismic load in the transverse direction is conducted in the following manner.e displacement of the ground at z with the unit width is defined as u when the horizontal force P is acting at a height z. e displacement at r (�R − z) of a hollow cylindrical disk with a unit thickness, the outer surface of which is constrained, is defined as u * when the horizontal force P * is acting at r. Placing the displacements u * and u equal to each other, P * can be expressed using P, R, and r as follows: e concentrated force S in equation ( 2) can also be replaced by S * as follows: Figures 8(a) and 8(b) represent the relation between the concentrated force and ground shear deformation in the actual ground and the axisymmetric model, respectively; the case of a soil column is shown in Figure 8(a), and the case of a hollow cylindrical disk is shown in Figure 8(b).e shear displacement values at the location r � r 0 in Figure 8(b) and the shear displacement values at the location z � h c in Figure 8(a) are coincident with each other.However, the vertical shear displacement distribution in the hollow cylindrical disk displayed in Figure 8(b) is not straight and is widely different from that in the soil column illustrated in Figure 8(a).To make the vertical distribution of the shear displacement in the proposed model equal to that in the actual subsurface, the following method is adopted.
When the concentrated load S * acts on r � r 0 , the shear displacement u c is However, as shown in Figure 9, assuming that the acceleration is uniformly distributed in the vertical direction, the inertial force s * i is When the inertial force s * i acts on a hollow cylindrical disk with a unit thickness ranging from r i−1 to r i , the shear displacement u c,i produced by the inertial force is en, the shear displacement produced at r � r 0 is e uniformly distributed acceleration a, which can be derived from equation (16), is given by the following equation: In numerical analysis, an earthquake load acting at r is calculated by multiplying the mass matrix by the ground acceleration at r. us, the acceleration at r used in such calculations can be given by equation (18), which is derived from equations ( 11) and (17); the acceleration a(r) used herein is defined as a modified acceleration coefficient: Finally, by applying the inertial force, which is the product of the mass m(r) and acceleration a(r) at each node as the seismic load, the displacement generated in the actual ground can be reproduced in the axisymmetric model.In the analysis of the transverse direction, the load acts in the transverse direction on each hollow cylindrical disk as an asymmetrical load.

Modeling of Shield Tunnels and Method for Determining Calculation Parameters.
A shield tunnel is a structure formed by segments, which are fastened by joints.For reasonably showing the shield tunnel characteristics, joint effects should be included when modeling a shield tunnel.
An equivalent stiffness beam model of the shield tunnel is often used in the actual shield tunnel design.As shown in Figure 10, based on the equivalent stiffness beam model, the calculation parameters of the shield tunnel finite element in the axisymmetric finite element model are determined.e equivalent stiffness of the shield tunnel can be calculated using the following equation: where (EA) C eq is the equivalent compression stiffness, (EA) T eq is the equivalent tension stiffness, (EI) eq is the equivalent bending stiffness, l s is the segment width, A s is the crosssectional area of the segment, E s is the elastic modulus of the segment, K j is the sum of the tension stiffness of the spring of ring joints, I s is the moment of inertia of the segment  Shock and Vibration section, and φ is the angle corresponding to where the neutral axis of segment section exists.erefore, the elastic modulus of the finite elements of shield tunnel lining in the axisymmetric finite element model is as follows: where E C is the compression elastic modulus, E T is the tension elastic modulus, and E I is the bending elastic modulus.

Research on the Seismic Isolation Effect of Isolation Layers.
As shown in Figure 11, to study the seismic isolation effect of the isolation layer, seismic response analyses are conducted on cases in which an isolation layer is applied to a shield tunnel buried in irregularly bounded surface soil deposits, and the results of these analyses are compared in this paper.
A shield tunnel with an outer diameter of 5.1 m made of reinforced concrete segments with a thickness of 25 cm constructed in the place where ground conditions change sharply is considered.Because the outer diameter of the shield machine was 5.2 m, the thickness of the isolation layer was set to 10 cm, which equals the thickness of the tail void.Table 1 reports the equivalent stiffness of the shield tunnel.
Figure 12 shows the discretization and boundary conditions of the axisymmetric finite element mesh.In the axisymmetric model, the mesh is discretized into 9 layers in the circumferential direction.e lining and isolation layer are each modeled by a layer.As shown in Figure 13, the input earthquake motion for Level 2 earthquake motion was chosen from among standard waves used for seismic design of road bridges [17].Waves obtained by performing amplitude of the EW component at JR Takatori Station were used as Type II earthquake motion for category II ground, which is a motion originated from a near-fault earthquake.
Seismic isolation materials suitable for underground structures should have certain physicochemical properties, and the most important thing to consider is whether the existence of an isolation layer will affect the static stability of underground structures.A softer isolation layer has a more evident seismic isolation effect; however, considering the uneven settlement of ground, Poisson's ratio of the isolation material should be close to 0.5.When Poisson's ratio of the isolation material is close to 0.5, uneven settlement of the ground can be avoided effectively, even if the isolation material is very soft [18].In this paper, silicone material (SISMO), recommended for seismic isolation design of shield tunnels, is used as the isolation material for numerical simulations.Its shear elastic modulus is relatively small, and its Poisson's ratio is close to 0.5.It meets requirements for grouting material.After being hardened, there is no harmful ingredient, nor does it pollute underground water.Table 2 reports the parameters for the SISMO silicone isolation material.
One-dimensional site response analysis based on the multireflection theory for layered soil (termed as the multireflection analysis) is conducted using the equivalent linear technique in this paper.e site responses as a free field are calculated, and the equivalent seismic rigidity and damping of the layered soil are obtained.e equivalent seismic rigidity of the soil deposits in Table 3 is used in every seismic analysis hereafter.
e calculated accelerations are then applied to the axisymmetric finite element model.14(a) and 14(b) show a comparison of the analysis results for the tunnel lining axial strains and stresses.Evidently, the axial peak strain and stress decrease after implementing the isolation layer.Table 4 indicates a decreased rate of axial strain and stress corresponding to different shear modulus.

Longitudinal Direction. Figures
e lower the shear modulus of the isolation material, the better the isolation effect of the isolation layer.

Transverse Direction.
Figure 15(a) shows a comparison of the analysis results for the tunnel lining axial strains.After the isolation layer is added, the axial peak tension strain decreases slightly, and the axial peak compression strain increases slightly.
Figure 15(b) shows a comparison of the analysis results for the tunnel lining axial stresses.After the isolation layer is added, the axial peak tension stress decreases slightly, and the axial peak compression stress almost remains the same.Shock and Vibration 7 erefore, when Poisson's ratio of isolation material is close to 0.5, the seismic isolation effect of isolation layer is not evident in the transverse direction.

Research on Seismic Isolation Mechanism of Isolation
Layers.A spring is used to simulate the interaction between the seismically isolated tunnel and ground.Figures 16(a e calculation formula is as follows: wherein K x is the spring stiffness coefficient of the isolation layer along the longitudinal direction of a tunnel, K y is the spring stiffness coefficient of the isolation layer along the transverse direction of a tunnel, R is the isolation layer outer diameter, r is the isolation layer inner diameter, G is the shear modulus of the isolation layer, v is Poisson's ratio of the isolation layer, and t is the thickness of the isolation layer.
Figure 17 shows how the shear modulus of isolation material affects the spring stiffness when Poisson's ratio is 0.48.As is known, the spring stiffness increase for both transverse and longitudinal tunnels with increased shear modulus.Figure 18 shows how Poisson's ratio affects the spring stiffness when the shear modulus is 0.3 MPa.When the shear modulus is constant, Poisson's ratio does not affect the spring stiffness of the isolation layer in the longitudinal direction, but the spring stiffness of the isolation layer in the transverse direction increases with increased Poisson's ratio.When Poisson's ratio is greater than 0.4, the increasing amplitude of spring stiffness increases sharply with increased Poisson's ratio.
For isolation material, the shear modulus should be relatively small, and Poisson's ratio is close to 0.5, avoiding ground settlement caused by the isolation layer.When the shear modulus is constant, Poisson's ratio does not affect the spring stiffness of the isolation layer along the   longitudinal direction of a tunnel.When an isolation material with a small shear modulus is used, the tunnel is isolated from the deformation of its peripheral ground through the shear deformation of the isolation layer.A good isolation effect along the longitudinal direction of a tunnel is achieved.When Poisson's ratio is close to 0.5, the spring stiffness is still great when an isolation material with small shear modulus is used, verifying that ground settlement can be avoided effectively when Poisson's ratio is close to 0.5.However, the tunnel cannot be isolated from deformation of its peripheral ground through the compression deformation of the isolation layer.erefore, a good isolation effect along the transverse direction of a tunnel cannot be achieved.

Research on Seismic Isolation System of Shield Tunnel
Based on the above concept, the material used for the isolation layer in shield tunnels should have relatively low shear modulus, and its Poisson's ratio should be approximately 0.5. the isolation layer cannot provide good isolation effect along the transverse direction of a tunnel.
Considering that an earthquake would lead to concentration of stress at shaft joints and the positions at which ground conditions sharply change, flexible segments are correspondingly employed to increase the flexibility of the shield tunnel and improve its bending deformation capacity.However, there exist some difficulties in the layout of flexible   Shock and Vibration segments owing to the uncertainty regarding the strain distribution in a shield tunnel when an earthquake occurs.us, a shield tunnel isolation system combining an isolation layer and flexible segments is proposed in this paper.As is shown in Figure 19, flexible segments are set at both ends of an isolation layer.e seismic responses of the shield tunnel along the longitudinal direction of the tunnel are shown in Figure 20(a).It is observed that when the seismic isolation system is built, the peak tension stress decreases to 47.9% of its original value and the stress distribution becomes smoother along the axial direction of the tunnel.On the contrary, the seismic responses of the shield tunnel along the transverse direction of the tunnel are represented in Figure 20(b).It is observed that when the axial peak tension stress decreases to 16.7% of its original value, the peak compression stress drops to 47.5% of its original value, and the stress distribution becomes smoother along the axial direction of the tunnel.erefore, the proposed isolation system could have obvious effects on both longitudinal and transverse directions of a shield tunnel.

Conclusions
In this paper, a quasi-static method based on an axisymmetric finite element model for seismic response analysis of seismically isolated tunnels is used to research the seismic isolation effect and mechanism of seismic isolation layer of a shield tunnel along both longitudinal and transverse directions.
e conclusions derived in this paper can be summarized as follows:    Shock and Vibration (1) Along the longitudinal direction of a tunnel, the seismic isolation effect of the elastic isolation layer is mainly affected by the shear modulus.e smaller the shear modulus of the isolation material, the better the isolation effect.is result occurs because the tunnel can be isolated from deformation of its peripheral ground through shear deformation of the isolation layer.(2) Along the transverse direction of a tunnel, the seismic isolation effect of the elastic isolation layer is mainly affected by the shear modulus and Poisson's ratio.When Poisson's ratio is approximately 0.5, the seismic isolation effect is not obvious.is result occurs because the tunnel cannot be isolated from the deformation of its peripheral ground through the compression deformation of the isolation layer.(3) A seismic isolation system for a shield tunnel in which flexible segments are arranged at both ends of an isolation layer is proposed.Based on the seismic response analysis, it is concluded that the system could have obvious effects on both the longitudinal and transverse directions of a tunnel.

Figure 1 :
Figure 1: Concepts of a seismic isolation layer for shield tunnels.(a) Traversing the boundary between a soft soil deposit and a stiff soil deposit; (b) traversing a fault fracture zone; (c) underlying boundary between a soft soil deposit and a stiff soil deposit; (d) junction with a vertical shaft.

Figure 5 :
Figure 5: Method employed to apply a seismic load.

Figure 6 :Figure 7 :
Figure 6: Relationship between the concentrated force and ground shear deformation along the longitudinal direction of a tunnel in (a) the actual ground and (b) the axisymmetric model.

Figure 8 :Figure 9 :Figure 10 :
Figure 8: Relationship between the concentrated force and ground shear deformation along the transverse direction of a tunnel in (a) the actual ground and (b) the axisymmetric model.

Figure 11 :
Figure 11: Ground and structure conditions in numerical simulations.
) and16(b)  show calculation models of the isolation layer spring stiffness along the longitudinal and transverse directions of the tunnel, respectively.

Figure 12 :Figure 13 :
Figure 12: Discretization and boundary conditions of axisymmetric finite element mesh.(a) Longitudinal direction and (b) transverse direction.

Figure 14 :
Figure 14: Analysis results along the longitudinal direction.(a) Axial strain and (b) axial stress.

Figure 15 :
Figure 15: Analysis results along the transverse direction.(a) Axial strain and (b) axial stress.

Figure 16 :Figure 17 :Figure 18 :
Figure 16: e model for calculating the isolation layer spring stiffness in (a) the longitudinal direction and (b) the transverse direction.

Figure 19 :
Figure 19: Shield isolation system for the shield tunnel.

Table 1 :
Equivalent stiffness of the shield tunnel.

Table 2 :
Mechanical parameters of SISMO material.

Table 3 :
e equivalent seismic rigidity of soil deposits.

Table 4 :
Decrease rate of axial peak strain and stress corresponding to different shear moduli.