# Seismic Damage Analysis of Box Metro Tunnels Accounting for Aspect Ratio and Shear Failure
## 4 Results and Discussion
### Updated Damage States and DIs of Box Tunnels  
In this section, the effects of shear failure of the interior column and aspect ratio B/H on the damage index (DI) and free-field shear strain are presented. We used the definitions of [PERSON] et al. (2018) for both the damage states and DIs but additionally account for the effect of B/H and shear failure. It is noted that DI is expressed as the ratio of the bending moment demand to the yield moment at the critical sections of tunnel frames (i.e., M/M\({}_{\text{y}}\)). Damage states are defined based on the number of plastic hinges occurred in the tunnel frame and the corresponding DI. Figure 5: Development of plastic hinges for double box with the width to height ratio (B/H) = 1.0  
Figure 6: Development of plastic hinges for triple box with B/H = 1.0. Figure 7: Capacity curves of single box tunnels with different B/H ratios and soil types. Figure 8 shows the comparison of DIs for various aspect ratios of three box tunnels. All makers in the lines denote the sequence of plastic hinges occurred in the tunnel frames. It is again to remember that flexural plastic hinges occurred in single box tunnels only. While, after four flexural plastic hinges developed in the outer frame of multi-box tunnels, a shear plastic hinge occurred at the interior column. DI at which the shear failure occurs in the internal column for double and triple box tunnels was shown to be negatively correlated to B/H. A tunnel with a higher B/H was more susceptible to a structural collapse because the shear demand was lower for the tunnel with lower B/H while the shear capacity of the structural members was invariable regardless of B/H. For double and triple box tunnels, this state at which the inner column undergoes a shear failure is defined as the collapse state. At this stage, there is a high possibility of a brittle structural failure. Moreover, thresholds of DI are specified in Figure 8 to define damage states of tunnels. For single box, three levels, which were 1.0, 1.4, and 2.3 were thresholds of minor, moderate, and extensive damage states, respectively. Meanwhile, four damage states including minor, moderate, extensive, and collapse were proposed for double and triple box tunnels associated with 1.0, 1.2, 2.0, and 3.5, respectively. Figures 9 and 10 show the increment ratios of DI and free-field shear strain with the number of plastic hinges (PHs), respectively. In both Figures 9 and 10, only the increment ratios up to fourth plastic hinges are shown. For all tunnels, DI is shown to increase with B/H up to the fourth plastic hinge. It is demonstrated that a tunnel with a higher B/H has a higher resistance to structural damage. However, the amount of increase in B/H was not significant, resulting in up to 6% increment for B/H = 2.0. Therefore, for a conservative estimate, the effect of B/H on DI may be ignored in a seismic design. B/H had relatively a higher influence on free-field shear strain at which plastic hinges were  
Figure 8: Comparison of damage indices (Dis) for various B/H for single, double and triple box tunnels. formed, as shown in Figure 10. The increment was up to 16% for single tunnels. However, it ranged from 23% to 27% for double and triple tunnels, respectively. This illustrates that a higher level of free field ground motion is needed to damage tunnels with a high B/H. Figure 9: Increment ratio of DI with B/H vs. number of plastic hinges (PHs) of box tunnels for various shear wave velocities (\(V_{s}\)) values. Table 1 shows the updated damage states, number of plastic hinges (NPH), and DIs corresponding to the respective damage states. For single box tunnels, the upper limits of DIs for minor and moderate damage states increased from 1.2 and 2.0 to 1.4 and 2.3, respectively. The cause for this increment was not because of the shear failure mode, which did not occur in single box tunnels, or the influence of B/H. It is because from [PERSON] et al. [21], conservative DIs were assigned such that they are identical for single, double, and triple boxes.