# Seismic Vulnerability Assessment of a Shallow Two-Story Underground RC Box Structure
## 2 Analytical Procedure for Constructing Fragility Curves
### Definition of Damage State of Underground Subway Station  
In the pushover analyses, a lateral force is applied incrementally to produce displacements from zero to the ultimate value in order to observe the nonlinear behavior of the structure. The type of lateral force applied on the structure should be selected based on the characteristics of interaction between the soil and structure. For deep tunnels, shear force developed at the exterior surface of \begin{table}
\begin{tabular}{c c c c c} \hline
**Soil Layer** & **Unit** & **Sandy** & **Weathered Soil** & **Rock** \\ \hline Mass density & kg/m\({}^{3}\) & 1800 & 2000 & 2300 \\ Shear wave velocity & m/s & 275 & 500 & 1500 \\ Poisson’s ratio & - & 0.35 & 0.35 & 0.25 \\ Thickness & m & 6.0 & 16.6 & - \\ \hline \end{tabular}
\end{table} 
Table 1: Physical properties of soil. Figure 4: Artificial ground motion time histories with peak ground acceleration (PGA) of (**a**) 0.02 g and (**b**) 1.3 g.  
Figure 3: Typical spectral response acceleration. the roof is the primary cause of the structural racking deformation. Thus, the racking effects of a deep rectangular tunnel can be simulated by applying a point/concentrated loading at the top of the wall. On the contrary, for a shallow tunnel with a thin overburden soil cover, the racking deformation may be developed along the side walls of the tunnel owing to the normal earth pressures. Therefore, the racking deformation should be imposed by applying an inverse triangular displacement loading along the wall depth [20]. When the structure is buried relatively deep in the soil, both two types of pseudo-static lateral force models (concentrated and inverse triangular force) are expected to be used to produce a unit displacement of the tunnel station, as depicted in Figure 5. The tunnel station is considered as a frame structure, and the plastic hinges are assumed to be developed at the end of the frames. The plastic hinge properties used are those suggested by FEMA 356 [21]. The evaluation of damage is normally based on the deflection response of the structure. The damage states can be divided into three performance levels based on ATC 40 and FEMA 356 [21; 22; 23], i.e., (a) immediate occupancy (IO), (b) life safety (LS), and (c) collapse prevention (CP), which indicate the correlation between the material nonlinearity and deterministic projections for the structural damage sustained. They can be classified as minor, moderate, and extensive damage states, respectively. To demonstrate the behavior of a structure, a two-story reinforced concrete tunnel station is modeled by using SAP2000 [24] with the dimensions and details of reinforcement bars depicted in Figure 6. In this program, a post-yield behavior is assigned by concentrated plastic hinges to frames. The inelastic behavior is obtained through integration of the plastic strain and plastic curvature which suggested in FEMA 356. One of the main factors that control the uncertainty of the response of a reinforced concrete structure is the variability of the material strength. The mean and coefficient of variation are used to describe the statistical variation of the material properties. Thus, uncertainties in the design parameters are considered as random variables, and their statistical properties are given in Table 2[25; 26]. In this study, the Latin hypercube approach is applied to simulate the uncertainty and randomness of the random variables. The reason of using this method is that Latin hypercube sampling stratifies the input probability distributions, in which it divides the cumulative curve into equal intervals on the cumulative probability scale, then takes a random value from each interval of the input distribution [27]. Therefore, with a small amount of sample, it shows a good representative of the real variability comparing to the traditional technique such like Monte Carlo Sampling, without sacrificing much accuracy.