## 4 Structural Energy Response Spectrum of the Tunnel Blasting Vibration
### Influence of blasting Vibration Characteristics on the Hysteretic Energy Consumption Spectrum  
The biggest defect of the energy response spectrum of the elastic SDOF system structure is that it cannot reflect the cumulative effect of blasting vibration. In tunnel blasting, the cumulative effect is primarily manifested in multiple repeated blasting and blasting with a long duration. Because the tunnel face must be continuously drilled, the same tunnel structure must be affected by multiple blasting vibrations. Multiple detonators are usually used in tunnel blasting, and the duration will be longer. Because the damping ratio typically appears in the form of a fixed value in the calculation process, the energy consumed by plastic deformation is not involved in the conversion of energy and is finally dissipated completely. Therefore, when evaluating the cumulative damage of engineering structures [30; 31], it is of practical importance to measure the cumulative damage of structures with hysteretic energy consumption. Excluding the influence of the dominant frequency, duration, structural parameters, and restoring force model parameters, the influence of the PPV on the hysteretic energy spectrum is analysed. Using the three acceleration time history curves in Figure 5, the hysteretic energy spectrum is shown in Figure 17 with a damping ratio of 0.05, a yield strength coefficient of the bilinear restoring force model of 0.3, and a stiffness reduction coefficient after yielding of 0.02. Figure 17 shows that the increase in the peak acceleration has no effect on the shape of the input energy spectrum. The square of the increase of acceleration is the increase of hysteretic energy consumption of the system, which indicates that the increase of acceleration will lead to the rapid increase of the input energy of the system. The same method is used to investigate the influence of the dominant frequency on the hysteretic energy spectrum. Figure 18 shows that the natural period of the structure corresponding to the peak of hysteretic energy dissipation decreases with the increasing dominant frequency of the blasting seismic wave. There are multiple peaks on the hysteresis energy spectrum of the blasting seismic wave with a dominant frequency of 50 Hz, which is the same as the influence of the dominant frequency on the input energy spectrum. There is a close relationship between the dominant frequency of the tunnel-blasting seismic wave and the shape of the hysteretic energy dissipation spectrum. Figure 19 shows the influence curve of duration on the hysteretic energy spectrum. The peak hysteretic energy increases as the duration increases, which shows that the hysteretic energy consumption is a process of time accumulation. The longer the duration of the vibration is, the greater the value is. Therefore, hysteretic energy consumption can accurately reflect the cumulative effect of duration on structural plastic failure. The vibration factor of duration is included in the scope of plastic c  
Figure 17: Influence of the acceleration peak on the hysteretic energy consumption. structure; thus, the hysteretic energy dissipation as the criterion of cumulative damage can fully describe the vibration's characteristics. Influence of Structural Parameters and Restoring Force Model Parameters on the Hysteretic Energy Consumption Spectrum  
To analyse the influence of the damping ratio on the hysteretic energy spectrum, the influence of the blasting vibration characteristics and restoring force model parameters should be excluded. The acceleration signal in Figure 5a is selected, and the peak value is normalised. The yield strength coefficient of the bilinear restoring force model is 0.3. The stiffness reduction coefficient after yielding is 0.02. The structural damping ratio is set as 0.02, 0.05, and 0.08. Figure 20 shows that the shape of the hysteretic energy spectrum curve does not change with increasing structural damping ratio. The peak energy of the hysteretic energy spectrum increases when the damping ratio decreases.