# Study on the Seismic Response of Shield Tunnel Structures with the Preload Loss of Bolts
## 4 Result Analysis
### Acceleration Response Analysis  
Five monitoring points, named the arch vault, spandrel, foot and bottom, are selected at 45-degree intervals along the circumferential centre line. Acceleration time-history curves are extracted for each operating condition. Figure 10 shows the time-history curves of acceleration at various points under a seismic wave with a PGA of 0.5 g when the bolt preload force is 280 kN. In terms of time, the peak acceleration at each monitoring point appears sequentially from the arch bottom to the arch foot, waist, spandrel and vault, following the direction of seismic wave propagation. Based on the observed patterns of peak acceleration under different levels of preload loss, it becomes apparent that peak acceleration cannot provide an intuitive reflection of the impact of varying preload loss levels on the structure. According to existing research, what controls earthquake-induced damage is not the peak values recorded in seismic records but the combined effects of the spectral components, duration, and amplitudes of seismic motion. To further evaluate the extent of structural damage during seismic loading, the Arias intensity \(I_{\text{a}}\) was introduced as a measure in this research [39; 40; 41; 42; 43]. The Arias intensity, introduced by Arias in 1970, represents the energy input to a single-degree-of-freedom system per unit mass. It encapsulates both the amplitude and duration information of seismic motion and has a strong correlation with the destructive effects of seismic events and finds widespread application in seismic hazard analysis. It can be calculated by Equation (6). \[I_{a}=\frac{arccos\lambda}{g\sqrt{1-\lambda^{2}}}\int_{0}^{T}a_{x}^{2}(t)dt \approx\frac{\pi}{2g}\int_{0}^{T}a_{x}^{2}(t)dt \tag{6}\]  
Figure 10: The acceleration of the monitoring points. where \(a_{x}(t)\) is the acceleration time-history in the x-direction; \(T\) is the record time; \(g\) is gravity; \(\lambda\) is the damp ratio of structure, 0.05. According to the compiled results of the Arias intensity at various monitoring points, it can be observed that the maximum values occur at the arch spandrel for all conditions. The Arias intensity at the arch spandrel is shown in Figure 11a-c. As the level of preload loss increases, the values under the influence of seismic waves with various PGA all exhibit an increase trend. When the PGA is 0.1 g, at lower levels of preload loss, the increase is gradual, but it gradually becomes more significant as the level of loss increases. The early-stage impact of preload loss is smaller compared to the later stages. At a PGA of 0.3 g, there is a relatively uniform increase in the trend with the loss of preload. When the PGA is 0.5 g, with increasing level of preload loss, the growth rate gradually decreases, indicating that the early-stage impact of preload loss is more pronounced. Additionally, based on the results extracted from monitoring points at different positions in the structure, it can be observed that some points may exhibit a decreasing trend with the increase in preload loss level. Therefore, it is considered to take the average of the Arias intensity at the five reference points, as shown in Figure 11d-f. When the PGA is 0.1 g, the average Arias intensity at each monitoring point gradually decreases with the increasing level of preload loss. This is likely due to the fact that as preload is lost to a certain extent, the overall stiffness of the structure is reduced, which hinders the transfer of seismic energy. With an increase in the level of preload loss, the Arias intensity at the arch spandrel observation point exhibits a gradual decrease in trend. Numerically, this difference is 2.055 \(\times\) 10\({}^{-5}\). A similar trend is also observed when the PGA is 0.3 g and 0.5 g, where the Arias intensity gradually increases, with corresponding differences of 1.030 \(\times\) 10\({}^{-3}\) and 1.565 \(\times\) 10\({}^{-3}\), respectively. This leads to an increase in the Arias intensity of 0.088% and 0.048%, respectively. Figure 11: Cont.