## 3 Engineering Example
### Seismic Response Analysis of Shield Tunnel Based on  
#### 3.2.1 Computational ModelResponse Displacement Method  
(1) Lateral Analysis Model  
Figure 8 is selected as the typical section of shield tunnel, and a beam-spring combined model is adopted to simulate the lining segments and the surrounding soil mass of the shield tunnel. The analysis model is shown in Figure 12.  
The locations and number of monitoring sites for the shield tunnel lining segments are shown in Figure 13, in which Figure 13a is the monitoring site for the rate of diameter change in the lining segments, and Figure 13b is the joint-stretching value monitoring point for the lining segments.  
(2) Longitudinal Analysis Model  
The water-intake shield tunnel of the longitudinal analysis model is divided into 820 sections, with a total length of 984 m. There are six types of inter-ring springs which have different directions, and the stiffness coefficients are obtained from the actual bolt connection. The local model of the longitudinal response displacement method for shield tunnel is illustrated in Figure 14.  
Figure 12: Lateral Beam–spring Model of Shield Tunnel.  
Figure 13: Locations and number of the tunnel lining segment monitoring sites: (**a**) monitoring site for the rate of diameter change in lining segment; (**b**) joint-stretching value monitoring point for lining segment.  
The free-field model of the area along the shield tunnel was established using the SuperFLUSH/2D of the Japan Institute of Seismic Engineering according to the soil layer distribution and the parameters in the site's rock project reconnaissance report. Figure 15 shows the two-dimensional section model of the whole field of the shield tunnel.  
A viscous boundary is applied to the bottom and to two boundaries of the finite element model, and the seismic input takes the equivalent nodal force. In order to simulate the nonlinear characteristics of the soil mass under ground motion and achieve the equivalent linearization of nonlinear material, the shear strain of the soil mass is calculated iteratively according to the correlation between the shear modulus ratio, damping ratio, and shear strain. The calculation process is as follows: firstly, the initial shear modulus and damping ratio are used for the initial calculation; next, the maximum shear strain and equivalent shear strain of each unit are calculated from the strain time-history curve of each unit; at last, according to the equivalent shear strain of each unit, the shear modulus and damping ratio of each unit are updated and re-calculated until the errors of the two successive calculations are in the allowable range. After the nonlinear seismic response analysis of the whole field of the shield tunnel is completed, the displacement time history of the soil layer with the same buried depth as the tunnel's axis is extracted and applied to the foundation soil spring. The relative displacement of the soil layer in the whole field of the shield tunnel under an SL-2-magnitude ground motion is demonstrated in Figure 16.  
Figure 16: Relative horizontal displacement of soil layer in the whole tunnel area under action of ground motion.  
Figure 14: Local model of longitudinal response displacement method for shield tunnel.  
Figure 15: A longitudinal analysis model for the whole field of the shield tunnel.