## 2 Real-Time City-Scale Nonlinear Time-History Analysis
### City-Scale Nonlinear Time-History Analysis
#### 2.4.1 Parameter Determination Method for Buildings in China  
The parameter determination method for buildings in China is based on the design codes and statistics of extensive experimental and analytical results [12; 13]. Because the buildings usually have a regular layout along with the height, the stiffness and masses of different stories are assumed to be uniform [14]. The elastic parameters of a building can be represented by the inter-story shear stiffness, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Number of stories & 1 & 2–3 & 4–6 & 7–9 & \(>\)9 \\ Proportions & 20.6\% & 10.6\% & 36.3\% & 4.9\% & 27.6\% \\ \hline Year built & \(<\)1990 & & 1990–1999 & & \(>\)1999 \\ Proportions & 30.3\% & & 29.8\% & & 39.9\% \\ \hline Structural type & Steel and reinforced concrete & Masonry & Wood & Other structures \\ Proportions & 47.9\% & 40.3\% & 11.6\% & 0.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical data of the buildings for Haidian District, Beijing. Figure 2: (**a**) Multiple-degree-of-freedom (MDOF) shear model; (**b**) MDOF flexural-shear model; and (**c**) trilinear backbone curve adopted in the MDOF model. \(k_{0}\), and the mass, \(m\), of each story. Equations (1) and (2) show the global stiffness [\(K\)] and mass matrices [\(M\)] of a structure with a uniform stiffness and mass along the height [15]. \[[K]=k_{0}\left[ \begin{array}{ccccc}2&-1&&&\\ -1&2&-1&&\\ &-1&\ddots&\ddots&\\ &&\ddots&2&-1\\ &&&-1&1\end{array} \right]=k_{0}[A] \tag{1}\]  
\[[M]=m\left[ \begin{array}{ccccc}1&&&\\ &1&&\\ &&1&&\\ &&&\ddots&\\ &&&1\end{array} \right]=m[I] \tag{2}\]  
The mass of each story, \(m\), in Equation (2) can be determined based on the area of each story, \(A_{1}\), and the mass per unit area, \(m_{1}\) (Equation (3)) [16]; \(m_{1}\) can be estimated according to the occupancy of each story. \[m=m_{1}A_{1} \tag{3}\]  
The relationship among the stiffness, mass, and first vibration period, \(T_{1}\), can be expressed using Equation (4). \[k_{0}=m\omega_{1}2^{\left(\frac{[\Phi_{1}]^{T}[I][\Phi_{1}]}{[\Phi_{1}]^{T}[A][ \Phi_{1}]}\right)}=\frac{4\pi^{2}m}{{T_{1}}^{2}}\frac{[\Phi_{1}]^{T}[I][\Phi_{ 1}]}{[\Phi_{1}]^{T}[A][\Phi_{1}]}\right) \tag{4}\]  
where [\(\Phi_{1}\)] is the first mode vector. Given the stiffness matrix [\(K\)] and mass matrix [\(M\)], [\(\Phi_{1}\)] can be computed using a generalized eigenvalue analysis. As shown in Equation (4), \(m\) and \(T_{1}\) are required to obtain \(k_{0}\). The vibration periods of different types of structures can be estimated using empirical equations. For example, the fundamental period of a reinforced concrete (RC) frame can be calculated using the empirical equation (Equation (5)) specified in the Chinese Code [17]. \[T_{1}=0.25+0.00053H^{2}/\sqrt[3]{B} \tag{5}\]  
where \(H\) and \(B\) are the height and width of an RC frame, respectively. For other types of structures (e.g., RC tall buildings, masonry structures, steel frames), the corresponding empirical vibration period can also be adopted to determine the inter-story stiffness. For engineering-designed structures (e.g., RC frames and reinforced masonry structures), the design strength can be estimated according to the seismic design code. Therefore, an equivalent lateral force analysis can be used to calculate the design shear force, \(V_{\text{design, }i}\), of each story, where \(i\) is the story number [18; 19]. Subsequently, according to related statistics of extensive experimental and analytical results, the yield point, peak point, and softening point on the backbone curve can be further obtained. For example, the yield point, peak point, and softening point of an RC frame can be determined by Equations (6)-(8) as follows:  
\[V_{\text{yield, }i}=\Omega_{1}V_{\text{design, }i} \tag{6}\]  
\[V_{\text{peak, }i}=\Omega_{2}V_{\text{design, }i} \tag{7}\]  
\[V_{\text{ultimate, }i}=V_{\text{peak, }i} \tag{8}\]  
where \(V_{\text{yield, }i}\), \(V_{\text{peak, }i}\), and \(V_{\text{ultimate }i}\) are the yield strength, peak strength, and ultimate strength, respectively.