Using a quasistatic method based on an axisymmetric finite element model for seismic response analysis of seismically isolated tunnels, the seismic isolation effect of the isolation layer is studied, and the seismic isolation mechanism of the isolation layer is clarified. The results show that, along the longitudinal direction of the tunnel, the seismic isolation effect is mainly affected by the shear modulus of the isolation material. The smaller the shear modulus is, the more evident the seismic isolation effect is. This is due to the tunnel being isolated from deformation of its peripheral ground through shear deformation of the isolation layer. However, along the transverse direction of the tunnel, the seismic isolation effect is mainly affected by the shear modulus and Poisson’s ratio of the isolation material. When Poisson’s ratio is close to 0.5, a seismic isolation effect is not evident because the tunnel cannot be isolated from deformation of its peripheral ground through compression deformation of the isolation layer. Finally, a seismic isolation system comprising a shield tunnel in which flexible segments are arranged at both ends of an isolation layer is proposed, and it is proved that the seismic isolation system has significant seismic isolation effects both on the longitudinal direction and on the transverse direction.
Shield tunnels are widely used for water supply, public transport, communication, sewerage, and other infrastructures. Tunnels do not cause selfexcited vibration under earthquakes but rather are controlled by the surrounding ground deformation. Therefore, 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
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.
The 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. The whole longitudinal behavior of a tunnel subjected to deformations imposed by the surrounding ground can be divided into two types [
Deformation modes of tunnels due to seismic waves. (a) Longitudinal direction and (b) transverse direction.
The quasistatic 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. This method has been adapted to the seismic design of underground ducts, shafts, and rock caverns. In this method, the surrounding ground is modeled by means of finite elements, and accelerations are calculated in a free field. The calculated accelerations are then applied to the finite element model, including an underground structure [
Figure
Schematic illustration of axisymmetric modeling.
The fundamental theory regarding the method employed to determine the seismic load in the axisymmetric model will be described in this section. The corresponding underground and loading conditions are shown in Figure
Coordinates of the axisymmetric model in comparison with the coordinates under actual conditions.
Figure
Method employed to apply a seismic load.
The conversion of the seismic load in the axisymmetric model is conducted as shown in Figure
The conversion of the seismic load in the longitudinal direction is conducted in the following manner [
The concentrated force
Figures
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.
When the concentrated load
However, as shown in Figure
Conversion of the concentrated load
When the inertial force
Then, the shear displacement produced at
The uniformly distributed acceleration, which can be derived from equation (
In numerical analysis, earthquake loads acting at
Finally, by applying the inertial force, which is the product of the mass
The conversion of the seismic load in the transverse direction is conducted in the following manner. The displacement of the ground at
The concentrated force
Figures
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.
When the concentrated load
However, as shown in Figure
Conversion of the concentrated load
When the inertial force
Then, the shear displacement produced at
The uniformly distributed acceleration
In numerical analysis, an earthquake load acting at
Finally, by applying the inertial force, which is the product of the mass
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
Schematic diagram of the equivalent stiffness beam model for a shield tunnel.
The equivalent stiffness of the shield tunnel can be calculated using the following equation:
Therefore, the elastic modulus of the finite elements of shield tunnel lining in the axisymmetric finite element model is as follows:
As shown in Figure
Ground and structure conditions in numerical simulations.
Equivalent stiffness of the shield tunnel.
Segment ring width 

m  1.0 
Segment sectional area 

m^{2}  3.81 
Segment sectional inertia moment 

m^{4}  11.23 
Segment elastic modulus 

kPa  3.75 × 10^{7} 
Sum of tension stiffness of ring joints 

kN/m  4.83 × 10^{6} 
Equivalent compression stiffness 

kN  1.43 × 10^{8} 
Equivalent tension stiffness 

kN  4.67 × 10^{6} 
Equivalent bending stiffness 

kN·m^{2}  3.59 × 10^{15} 
Discretization and boundary conditions of axisymmetric finite element mesh. (a) Longitudinal direction and (b) transverse direction.
As shown in Figure
Timeacceleration curve.
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 [
Mechanical parameters of SISMO material.
Material  Density (kg/m^{3})  Elastic modulus (MPa)  Poisson’s ratio 

SISMO1  1200  0.1  0.48 
SISMO2  1200  0.3  0.48 
SISMO3  1200  0.5  0.48 
Onedimensional 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. The site responses as a free field are calculated, and the equivalent seismic rigidity and damping of the layered soil are obtained. The equivalent seismic rigidity of the soil deposits in Table
The equivalent seismic rigidity of soil deposits.
Material  Density (kg/m^{3})  Elastic modulus (MPa)  Poisson’s ratio 

Soft soil  1800  52.2  0.45 
Stiff soil  2000  896  0.4 
Figures
Analysis results along the longitudinal direction. (a) Axial strain and (b) axial stress.
Decrease rate of axial peak strain and stress corresponding to different shear moduli.
Elastic modulus (MPa)  Decrease rate of peak strain (%)  Decrease rate of peak stress (%) 

0.1  50.5  49.4 
0.3  43.4  40.5 
0.5  39.4  35.7 
Figure
Analysis results along the transverse direction. (a) Axial strain and (b) axial stress.
Figure
A spring is used to simulate the interaction between the seismically isolated tunnel and ground. Figures
The model for calculating the isolation layer spring stiffness in (a) the longitudinal direction and (b) the transverse direction.
Figure
Effect of the shear modulus of the isolation material on the isolation layer spring stiffness.
Effect of Poisson’s ratio of the isolation material on the isolation layer spring stiffness.
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. Therefore, a good isolation effect along the transverse direction of a tunnel cannot be achieved.
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. However, 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 segments owing to the uncertainty regarding the strain distribution in a shield tunnel when an earthquake occurs. Thus, a shield tunnel isolation system combining an isolation layer and flexible segments is proposed in this paper. As is shown in Figure
Shield isolation system for the shield tunnel.
The seismic responses of the shield tunnel along the longitudinal direction of the tunnel are shown in Figure
Seismic response of shield tunnel. (a) Longitudinal direction and (b) transverse direction.
In this paper, a quasistatic 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. The conclusions derived in this paper can be summarized as follows:
Along the longitudinal direction of a tunnel, the seismic isolation effect of the elastic isolation layer is mainly affected by the shear modulus. The smaller the shear modulus of the isolation material, the better the isolation effect. This result occurs because the tunnel can be isolated from deformation of its peripheral ground through shear deformation of the isolation layer.
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. This result occurs because the tunnel cannot be isolated from the deformation of its peripheral ground through the compression deformation of the isolation layer.
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.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors acknowledge the financial support received from the National Program on Key Basic Research Project of China (973 Program) (Grant no. 2015CB057906) and the National Key Research and Development Program of China (Grant no. 2018YFF01014204).