## 4 Results and Discussion
### Dynamic Response Analysis  
The seismic responses of a shield tunnel can be investigated in terms of three primary aspects, namely, extension/compression, longitudinal bending and ovaling [9]. In this study, the extension/compression can be recognized through the axial forces and the bending behavior can be evaluated through the bending moments; the ovaling deformation can be evaluated by measuring the diameter variation of the tunnel. In total, 24 typical cross-sections including nine in the island part, 10 in the cross-river part and five in the land part (as shown in Figure 7) are chosen as the concerned locations. Figure 13 depicts the evolutions of dynamic internal forces and Figure 14 illustrates the evolutions of dynamic internal moments. The tunnel suffers alternately positive and negative dynamic forces and moments. Time history of internal forces of other sections show comparable results. To get a total view of internal forces, Figure 15 illustrates the variation of maximum axial forces along the tunnel and Figure 16 depicts that of the maximum bending moments. The obtained results show that both the bending moments and the axial forces change along the tunnel. Different internal forces and moments have been observed among the island part, cross-river part and land part. The differences may be attributed to the non-uniform seismic input model, the winding and different spatial positions of the tunnel, the dissimilar distributions of soil strata and the diverse buried depths of the tunnel. The comparison illustrates the distributions of both the internal moments and the internal forces of the tunnel during and after the earthquake, which can be used to aid the design of connections of different rings, connections of different segments in a ring and water-proof measures. Figure 14: Dynamic integral transverse moments at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part. Figure 13: Dynamic integral axial forces at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part. To evaluate the strength of the tunnel, the hoop stress is extracted and analyzed. As shown in Figure 17, the local coordinate system is needed in each typical cross-section to obtain the hoop stress. Figure 18 depicts the maximum compressive and tensile dynamic hoop stress distributions of cross-sections RS1 and RS5. It is shown that the distributions present similar patterns in different sections. The maximum stresses lie in the two sides of the ring, and the minimum stresses lie in the arc bottom and vault. Other sections show similar change laws. Table 3 lists the maximum hoop stress for the chosen sections. The hoop stress also changes along the tunnel, and the larger stress appears at sections IS1, IS5, IS6 and IS9 in the island part, sections RS1 and RS10 in the cross-river part, and section LS1 in the land part. The maximum compressive stress and the maximum tensile stress are 8.36 MPa and 5.32 MPa, respectively. The larger stresses may be due to the ovaling deformation of the tunnel and the sharp stiffness changes between the work shafts and the tunnel rings. Since the maximum compressive and tensile hoop stresses are smaller than the corresponding concrete strength used, the tunnel meets the strength design requirements under the specific seismic excitation. Figure 16: Maximum integral transverse moments along the tunnel. Figure 15: Maximum integral axial forces along the tunnel. Figure 17: Local coordinate system: (**a**) local cylindrical coordinate; (**b**) hoop and axial stress. Figure 18: Distributions of max dynamic hoop stresses around the tunnel lining during earthquake: (**a**) tensile; (**b**) compressive. The ovaling describes the cross-sectional distortion of the tunnel lining and can be characterized by the change of the diameter. Thus, the ovaling here is defined by both the horizontal and the vertical diameter variations.