## 2 Real-Time City-Scale Nonlinear Time-History Analysis
### City-Scale Nonlinear Time-History Analysis
#### 2.4.2 Parameter Determination for Backbone Curve Based on the HAZUS Data  
Based on the HAZUS database, [PERSON] et al. [15] also proposed a modeling approach by which all parameters of the MDOF shear model can be determined from the basic building information (i.e., number of stories, height, year built, structural type, floor area, and occupancy). Specifically, the first vibration period of the building can be determined by the typical buildings presented in Tables 5.5 and 5.7 of FEMA [7]. The elastic parameters of a building, including the inter-story shear stiffness, \(k_{0}\), and the mass, \(m\), are determined using Equations (1)-(4). Subsequently, the inter-story backbone curve parameters of story \(i\) in Figure 3 are determined as follows:  
\[k_{0,i}=\lambda\frac{4\pi^{2}m}{{T_{1}}^{2}} \tag{9}\]  
\[V_{y,i}=SA_{y}\cdot\alpha_{1}\cdot m\cdot g\cdot N\cdot\Gamma_{i} \tag{10}\]  
\[\eta_{i}=\frac{SA_{\text{u}}-SA_{y}}{SD_{\text{u}}-SD_{y}}\cdot\frac{SD_{y}}{ SA_{y}} \tag{11}\]  
\[\beta_{i}=\frac{SA_{\text{u}}}{SA_{y}} \tag{12}\]  
\[\Delta_{\text{c},i}=\delta_{\text{Co}}\cdot h \tag{13}\]  
where \(g\) is the acceleration of gravity; \(\delta_{\text{Co}}\) is the inter-story drift ratio at the threshold of the complete damage state, as suggested by [PERSON] [7]; \(h\) is the story height; (\(SD_{y}\), \(SA_{y}\)) and (\(SD_{\text{u}}\), \(SA_{\text{u}}\)) are the yield capacity point and ultimate capacity point, respectively, of the capacity curve suggested by HAZUS [7], which are functions of the design intensity and year built; \(\alpha_{1}\) is the mode factor suggested by HAZUS [7]; and \(\Gamma_{i}\) is the ratio between the inter-story shear strength of the \(i\)th story, (\(V_{y,i}\)) and that of the ground story (\(V_{y,1}\)), which is calculated as follows:  
\[\Gamma_{i}=\frac{V_{y,i}}{V_{y,1}} \tag{14}\]  
A single-parameter pinching model (Figure 4) proposed by [PERSON] and [PERSON] [21] is adopted to represent the pinching behavior subjected to cyclic loads. Five damage states, ranging from none, slight, and moderate, to extensive and complete damage, are considered, which are identical to the damage states defined by HAZUS. The inter-story drift ratio is used as the threshold for each structural damage state, and the values for different structural types are based on Table 5.9 of HAZUS [7]. The classical Rayleigh damping model is adopted in the analysis. The empirical values of the damping ratio are estimated according to the structural type of the building [15]. For example, the damping ratio of steel structures equals 0.05. It should be noted that the soil-structure-interaction (SSI) effects and structural eccentricity aren't considered in this work. The reliability of the city-scale nonlinear time-history analysis method is further validated by comparing the simulation results with earthquake site investigations, experimental results, and a large number of numerical results [11]. With outstanding computational efficiency, this method can be used well for post-earthquake emergency responses. There exists an inherent uncertainty in the seismic performance of buildings, which is considered in this method by incorporating the parametric uncertainty of the building backbone curve [22]. Consequently, the proposed method can not only provide the building response using the median value of the backbone curve parameters but also provide the responses with the median value \(\pm\) one standard deviation to account for the parametric uncertainty, which is crucial for scientific decision-making. The nonlinear time-history analysis of the buildings in the target area is implemented using the ground motions obtained from the seismic network. Subsequently, the time histories of the seismic response of each story in every building can be obtained. According to the engineering demand parameters (EDPs) and the damage criteria [11], the damage state of each building in the region is determined, based on which, the destructive power of the ground motion to the target area is evaluated.