# Wind-Induced Dynamic Performance Evaluation of Tall Buildings Considering Future Wind Climate
## 2 Basic Settings for the Simulations
### Performance Evaluation of the Building Under Wind Loads
#### 2.2.3 Wind-Induced Dynamic Response Analysis of Tall Building  
Dynamic response analysis for tall buildings involves studying structural behavior and performance under wind loads or other dynamic forces. This analysis helps assess the building's stability, evaluate its response to wind-induced vibrations, and ensure its safety and functionality. The wind-induced response analysis of the tall building can be computed in the time domain. Many past studies have developed various methods and models. One approach involves a time-domain method for predicting wind-induced responses of tall buildings by analyzing the dynamic equations of motion numerically, taking into account the mass, stiffness, and damping properties of the building to determine the building response over time. Generally, the floor diaphragm of the tall building structure is considered rigid. These multi-story structures, featuring rigid floor diaphragms, are represented by a concentrated mass at each floor level. Each floor's movement is confined to three degrees of freedom, encompassing two translational movements and one rotational movement around the vertical axis. For a building with \(n\) stories, considered as a lumped mass system with \(3n\)-degree of freedom, the equation of motion can be expressed in matrix form as in Equation (19):  
\[[M]\{\ddot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=[F] \tag{19}\]  
where \([M]\), \([C]\), \([K]\), and \([F]\) are the mass matrix, damping coefficient matrix, stiffness matrix, and external forces of time. \([x]\)_is_ displacements, which is used generally to represent \(x,y\) displacements, and \(z\) rotation. Equation (20) represents the matrix form of Equation (20) as follows [43]:  
\[ \begin{pmatrix}M&0&0\\ 0&M&0\\ 0&0&I\end{pmatrix}\begin{pmatrix}\ddot{X}\\ \ddot{Y}\\ \ddot{\Theta}\end{pmatrix} + \begin{pmatrix}C_{XX}&0&C_{XB}\\ 0&C_{YY}&C_{YB}\\ C_{XB}^{T}&C_{YB}^{T}&C_{\Theta\Theta}\end{pmatrix}\begin{pmatrix}\dot{X}\\ \dot{Y}\\ \dot{\Theta}\end{pmatrix} + \begin{pmatrix}K_{XX}&0&K_{XB}\\ 0&K_{YY}&K_{YB}\\ K_{XB}^{T}&K_{YB}^{T}&K_{\Theta\Theta}\end{pmatrix}\begin{pmatrix}X\\ Y\\ Z\end{pmatrix} = \begin{pmatrix}F_{X}\\ F_{Y}\\ F_{\Theta}\end{pmatrix}  \tag{20}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Location** & **Floor** & **Height (m)** & **Drag Coefficients, \(C_{d}\)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 5: Aerodynamic coefficients of the representative floors of the building. where M = diag [\(m_{i}\)] is the mass submatrix, where \(m_{i}\) is the lumped mass at the \(i\)th floor. I = diag [\(I_{i}\)] is the mass moment of the inertia matrix of each floor's diaphragm, where \(I_{i}=m_{i}\tau_{i}^{2}\) and \(r_{i}\) is the radius of the gyration of each floor. X = (\(x_{1}\), \(x_{2}\),...., \(x_{n}\))T, Y = (\(y_{1}\), \(y_{2}\),...., \(y_{n}\))T, \(\Theta\) = (\(\theta_{1}\),\(\theta_{2}\),....,\(\theta_{n}\))T represent the displacement response sub-vectors. The damping submatrices C\({}_{XX}\), C\({}_{YY}\), C\({}_{\Theta\Theta}\), C\({}_{X\Theta}\), and C\({}_{Y\Theta}\) and stiffness submatrices K\({}_{XX}\), K\({}_{YY}\), K\({}_{\Theta\Theta}\), K\({}_{X\Theta}\), and K\({}_{Y\Theta}\) define the dynamic properties of the building. The wind load forces in the X and Y directions and the torsional wind moment about the vertical axis are represented by the sub-vectors F\({}_{X}\) = (\(F_{x1}\), \(F_{x2}\),...., \(F_{xn}\))T, F\({}_{Y}\) = (\(F_{y1}\), \(F_{y2}\),...., \(F_{yn}\))T, and F\({}_{Q}\) = (\(T_{01}\), \(T_{02}\),...., \(T_{0n}\))T respectively. The system of coupled differential equations can be transformed into a set of uncoupled equations using normal modes. By applying the orthogonality conditions of the mode shapes and assuming classical damping, the governing dynamic equations of motion can be transformed into a system of \(j\) = 1, 2,..., \(n\) uncoupled equations as shown in Equation (21):  
\[ \begin{pmatrix}m_{1}&0&0\\ 0&\ddots&0\\ 0&0&m_{n}\end{pmatrix}\begin{pmatrix}\tilde{q}_{1}(t)\\ \vdots\\ \tilde{q}_{n}(t)\end{pmatrix} +