# Structural Responses of a Superall Building Subjected to a Severe Typhoon at Landfall
## 3 Methodology Statement  
### Estimation of Structural Extreme Response  
High-rise buildings subjected to strong wind excitations may vibrate excessively, which can result in comfort problems for occupants and other serviceability issues for relevant components or affiliated devices. Thus, it is conventionally required that the extreme response of building structures be restricted within an acceptable level [31; 32]. Otherwise, associated measures should be implemented for vibration reduction and control of the dynamic systems [33; 34; 35; 36]. In practice, the acceleration response is widely used as a preferred indicator for comfort/serviceability assessment of high-rise buildings, and many acceleration-response based criteria have been proposed, including those suggested in ISO 6897 [31], ISO 10137 [32], Melbourne and Palmer [37], JGJ 3-2010 [38], and AIJ-GEH-2004 [39]. Among these sources, the peak value of acceleration response is mostly utilized, although some recommend other alternatives, such as root-mean-square (RMS) value [9; 10]. Statistically, the maximum acceleration response of a system \(A_{\text{max}}\) can be estimated as:  
\[A_{\text{max}}=g\cdot\sigma_{a} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Meteorological Station** & **Geographic Coordinates** & **Anemometer Height (m)** \\ \hline Canton Tower & \(23^{\circ}6^{\prime}44^{\prime\prime}\) N,\(113^{\circ}19^{\prime}52^{\prime}\) E & 532 \\ Leatop Plaza & \(23^{\circ}7^{\prime}54^{\prime}\) N,\(113^{\circ}19^{\prime}56^{\prime}\) E & 303 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Information of meteorological stations. Figure 3: Best track of Typhoon Mangkhut and location of the study building. The sub-indices denote the time number, which is correlated with time through \(t=(k-1)/f_{s}\), where \(f_{s}\) is the sampling frequency. \(A\) is the discrete state matrix, which represents all the dynamic characteristics of the system, \(B\) is the discrete input matrix, \(C\) is the output matrix, and \(D\) is the direct transmission matrix. For simplification, the two items \(w_{k}\), \(v_{k}\) in Equation (7) can be regarded as two independently stationary zero-mean white-noise processes with the covariance matrix expressed as:  
\[E\!\left[ \begin{array}{cc}w_{i}\\ v_{i}\end{array} \right)\ \ \left( \begin{array}{cc}w_{j}^{T}&v_{j}^{T}\end{array}  \right)\ \right]=\left( \begin{array}{cc}\mathbf{Q}&\mathbf{S}\\ \mathbf{S}^{T}&\mathbf{R}\end{array} \right)\delta_{ij} \tag{8}\]  
where \(E\) is the mathematical expectation operator, \(\delta_{ij}\) is the Kronecker function whose value equals to one for \(i=j\) and to zero for \(i\
eq j\). \(w_{i}\) and \(w_{j}\) are the input noises at moment \(i\) and \(j\):\(v_{i}\) and \(v_{j}\) are the measurement noises at moment \(i\) and \(j\), respectively, and \(Q\), \(S\), _and_\(R\) belong to covariance matrices. To implement the SSI method, the so-called Hankel matrix should be constructed first based on output records. The block Toeplitz matrix can be then obtained by calculating the output covariance sequence. Finally, the state matrix \(A\) can be computed via singular value decomposition (SVD) from the block Toeplitz matrix. Conducting eigenvalue decomposition operation on \(A\):  
\[A=\psi\Lambda\psi^{-1} \tag{9}\]  
where \(\Lambda\) is the eigenvalue diagonal matrix associated with the discrete-time system, whose non-zero elements are complex values \(\lambda\), and \(\psi\) is the discrete-time eigenvector matrix. According to the relationship between the eigenvalues associated respectively with the discrete-time system \(\lambda_{c}\) and the continuous-time system \(\lambda\), one has:  
\[\lambda_{c}=\ln\lambda/\Delta t=-\xi\omega\pm j\omega\sqrt{1-\xi} \tag{10}\]  
where \(\xi,\omega\) represent damping ratio and natural frequency, respectively, and \(\Delta t\) represents sampling interval.