## 2 Seismic Resilience Evaluation Framework
### Modelling Soil-Tunnel Configurations  
Steps (a), (b), and (c) in Figure 1 are introduced in detail in this section on Modelling Soil-Tunnel Configurations. A typical shield tunnel, located in Shanghai city of China, is used to perform a seismic resilience assessment, as shown in Figure 2. The circular tunnel is built in the soft soil profile, at a depth of 100 m, and lies on the elastic bedrock. This type of tunnel has a radius of 3.10 m, and a lining thickness of 0.35 m. Moreover, its elastic modulus (\(E_{c}\)) is 3.55 GPa, and its Poisson's ratio (\(v_{c}\)) is 0.2. As indicated in Figure 2, this tunnel has a buried depth (_h_) of 9.0 m; therefore, the tunnel overburden depth ratio (_h_/_d_) is equal to 1.45. This is a typical case of a shallow tunnel.  
Three soil profiles are chosen and classified as soil type D according to EC8 [24]; they are further referred to as soil deposits D1, D2, and D3 in this work. For all three selected soil profiles, the detailed developments of shear wave velocities (\(V_{s}\)), density (\(\rho\)), cohesion (\(c\)), and friction angle (\(\varphi\)) along the soil height are presented in Figure 3, respectively. Additionally, Figure 4a,b show the typical \(G\)-\(\gamma\)-\(D\) curves for claycy and sandy layers, respectively. The author's previous work could be used to check more details about the selected soil profiles [23].  
Figure 1: Proposed framework to evaluate the seismic resilience of tunnels.  
Figure 2: Diagrammatic sketch for the examined circular tunnel (not to scale).  
Figure 5 shows a detailed two-dimensional (2D) finite element model of the studied soil-tunnel configuration generated by the platform ABAQUS. This model has a height of 100 m and a width of 400 m. As for the base boundary of the model, it is simulated as an 'elastic bedrock' using the method of [PERSON] and Kuhlemeyer [25] by setting appropriate dashpots, where the selected ground motions are inputted. For the lateral boundaries of the model, a kinematic tie constraint is set to let the opposite vertical sides move at the same time. The tunnel-soil interface is appropriately simulated by a finite sliding hard contact algorithm provided by ABAQUS. The tangential behavior is reflected based on the penalty friction formulation by introducing a commonly used Coulomb frictional model with a friction parameter (\(\mu\)) equal to 0.6. The response of tunnel lining is modeled by a simple linear elastic model. A uniform tunnel lining is adopted in the numerical model, focusing the computational efficiency in the analyzed cases. A Mohr-Coulomb model is utilized to capture the non-linear behavior of soil in the dynamic analysis. Each analysis is performed in two consequent steps, i.e., a geostatic step followed by a dynamic analysis step. The first step is conducted to build the geostatic stress equilibrium within the numerical model. In contrast, the acceleration time history is introduced via dashpots at the model's base in the second step. More details about the numerical model development could be checked in the author's previous work [23].  
Figure 4: \(G\)-\(\gamma\)-\(D\) curves for (**a**) clayey layers and (**b**) sandy layers [23].  
Figure 3: Geotechnical parameters of the examined soil profiles.  
It is very important to choose a set of ground motions that cover a high level of uncertainty within the seismic fragility analysis of any element at risk so as to perform an accurate seismic risk assessment. Table 1 shows the general properties of the selected 12 seismic ground motions in this work, which are provided by the Pacific Earthquake Engineering Research Strong Motion Database ([[https://ngawest2.berkeley.edu](https://ngawest2.berkeley.edu)]([https://ngawest2.berkeley.edu](https://ngawest2.berkeley.edu)) accessed on 10 April 2020). These ground motions are scaled from 0.1 g to 1.0 g, with a gradient of 0.1 g, and then applied in the numerical modeling. More information about the selection of the adopted seismic records can be found in the author's work for the PBD-IV [23].