# Seismic Resilience Evolution of Shield Tunnel with Structure Degradation
## 2 Resilience Assessment Model
### Introduction of Resilience Assessment Model  
A typical resilience assessment frame is shown in Figure 1. During the long-term operational phase of shield tunnels, the resilience evolution of the shield tunnel can be divided into three states: the normal state, the affected state, and the recovered state. The normal state refers to the gradual evolution of the durability of the shield tunnel over time under the influence of various factors, as indicated by the curve Q(t) in the figure. The affected stage signifies a sudden external force acting on the shield tunnel at a specific moment, leading to a rapid decline in structural performance, as indicated by the red arrow in the figure. In this study, the affected stage specifically refers to the impact of seismic loads. Specifically, the affected state can be subdivided into (a) disruption stage (t\({}_{i}\) is the start time of seismic loads, and F(t) represents the curve of tunnel performance reduction caused by seismic loads during t\({}_{i}-\)t\({}_{d}\)); (b) evolution stage: (t\({}_{d}-\)t\({}_{c}\), where S(t) represents the curve of tunnel performance evolution after seismic loads occurred but before any measures were taken); (c) recovery stage (t\({}_{c}-\)t\({}_{r}\), where R(t) represents the recovery curve of tunnel performance after implementing restoration measures). The downward value d\({}_{1}\) during the disturbance stage can be referred to as the vulnerability of the structure, while the residual value d\({}_{2}\) reflects the robustness of the tunnel structure.  
Regarding the resilience indicator Re of tunnel structures, both references [14; 24] employ the area enclosed by the disturbance-induced performance curve and the normal decay curve divided by the coordinate axes. However, this method does not account for the degradation in structural performance caused by environmental erosion before the disturbance, leading to an overestimation of the resilience indicator. [PERSON] et al. [21] solved this problem by considering the area of the dark-shaded region between the normal performance decay curve and the bounding straight line Q\({}_{0}\), as shown in Figure 1. On the other hand, [PERSON] et al. [22] use a resilience indicator for tunnel structures defined by weighted coefficients to reduce the calculation error. In this study, a resilience assessment model for tunnel structures under seismic loads was established based on extensive studies [20; 21; 22; 24]. The seismic resilience is defined as follows:  
\[\text{Re}=\frac{F+S+R}{Q_{0}\Delta T} \tag{1}\]  
where Q\({}_{0}\) is the initial performance of shield tunnel; \(\Delta T\) is the duration of the affected stage; F, S, and R, respectively, represent the areas enclosed by the tunnel structure performance decay curves, and the coordinate axes during the disruption stage, evolution stage, and recovery stage. They can be calculated by Equations (2)-(4).  
\[F=\int_{t_{i}}^{t_{d}}f_{n}(t)dt \tag{2}\]  
Figure 1: Framework of shield tunnel resilience evaluation.  
\[S =\int_{t_{d}}^{t_{e}}s_{n}(t)dt \tag{3}\] \[R =\int_{t_{e}}^{t_{r}}r_{n}(t)dt \tag{4}\]