# Seismic Resilience Evolution of Shield Tunnel with Structure Degradation
## 2 Resilience Assessment Model
### Resilience Assessment Indicators  
The selection of appropriate structural performance indicators for shield tunnels is a prerequisite for conducting resilience assessments. The Technical Code for the Protection Structures of Urban Rail Transit [25] recommends the use of a series of safety control indicators for shield tunnels, including the displacement of tunnel structures, radial convergence deformation, track height difference, tunnel deformation curvature radius, segment joint opening, segment crack width, and additional loads. However, it is hard to consider all the above parameters comprehensively in practical engineering. It is common to select certain parameters as important measurement indicators to ensure the safety of tunnel structures. Because under seismic loads, they have significant correlation with the loss of shield tunnel lining performance. In this study, the ellipticity and the joint opening of the shield tunnel are chosen as the resilience assessment indicators separately. Additionally, the damage to segments is an irreversible process, and resilience assessment indicators should reflect the tunnel's most adverse conditions throughout its entire operational period. Therefore, when the ellipticity is used as the resilience assessment indicator, the resilience can be calculated by Equation (5). \[Q(t)=\left\{ \begin{array}{cc}\frac{1}{1+\frac{1}{36}\times\left(\frac{T(t) +T_{\text{max,bis}}}{2}\right)^{2}},&T(t)\leq T_{\text{max,bis}}\\ \frac{1}{1+\frac{1}{36}\times T(t)^{2}},&T(t)\leq T_{\text{max,bis}}\end{array} \right. \tag{5}\]  
where \(Q(t)\) is the resilience value; \(T(t)\) is the ellipticity of the shield tunnel at time t; \(\Delta T_{\text{max,bis}}\) is the historical maximum horizontal ellipticity of the shield tunnel. The variation of structural performance with ellipticity is shown in Figure 2. According to the Tunnel Construction and Acceptance Code for Shield Tunnelling [26], the specified limit for ellipticity is 6%, which is considered conservative. In this study, it is assumed that the lining performance is divided into four levels: high, medium, low, and extreme. The boundary between high-medium performance is 3%, and the boundary between medium-low performance is 6%. Additionally, it has been observed based on existing full-scale segment experiments and numerical simulation results that tunnel enters a stage of rapid deformation development when the ellipticity of a shield tunnel exceeds 12%, indicating an extreme performance. The lining performance undergoes a recovery stage after seismic loads, but it cannot fully return to the initial state. As evident from the lining performance curves under different historical maximum convergence deformations in Figure 2, it can be seen that deformation callback measures can restore the performance to the previous level at most when the lining performance declines to a certain level. Resilience is \(Q(t)=1\) when the shield tunneling ellipticity \(T(t)=0\), indicating good structural performance. Resilience \(Q(t)\) decreases with \(T(t)\) increases, implying a deterioration in structural performance. Resilience \(Q(t)=0\) when \(T(t)=T_{\text{max,his}}\), signifying the complete loss of structural performance. This method overcomes the issue of varying structural resilience under the same convergence deformation [27], and it possesses a certain degree of universality. When the segment joint opening is used as the resilience assessment indicator, the resilience can be calculated by Equation (6). \[Q(t)=\left\{ \begin{array}{cc}\frac{1}{1+0.25\times\left(\frac{P(t)+P_{\text {max,bis}}}{2}\right)^{2}},&P(t)\leq P_{\text{max,his}}\\ \frac{1}{1+0.25\times P(t)^{2}},&P(t)\leq P_{\text{max,his}}\end{array} \right. \tag{6}\]where \(\Delta P_{\text{max,his}}\) is the historical maximum ellipticity of the shield tunnel. The Technical Specifications for Structural Safety Protection of Urban Rail Transit [28] has specified warning values and control values.