# 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
#### 2.2.1 Generation of Wind Speed Time History  
The time-varying wind speed is calculated by summing the mean and time-varying component from the mean using Equation (1):  
\[U_{z}(t)=\overline{U}_{z}+u_{z}(t) \tag{1}\]  
where \(U_{z}(t)\) is time-varying wind speed, \(\overline{U}_{z}\) is the mean wind speed, and \(u_{z}(t)\) is the fluctuating wind speed component due to turbulence at z height of the building. The wind speed changes from zero to a maximum at a certain height from the ground surface. The maximum wind speed height can be different depending on the roughness of the ground surface. This process generally treats wind speeds in the atmospheric boundary layer as stationary random processes [34]. The case study building was divided into 20 representative floors for wind load simulation. Mean wind speed at the representative floors of the building, \(\overline{U}_{z}\), was computed from the power law using Equation (2):  
\[\overline{U}_{z}=\overline{U}_{10}(\frac{z}{10})^{\alpha} \tag{2}\]  
where \(\overline{U}_{10}\) is reference speed, which is generally measured at the height of 10 m from the ground surface. \(\alpha\) is a dimensionless power-law exponent that depends on the terrain roughness. In accordance with the provisions of the Code for Loading of Building Structures (GB50009-2012) [39], the case study building falls under the Class B landform, with a value of approximately 0.15. Figure 2: Framework of dynamic performance evaluation and assessment of failure probability. Turbulence intensity (\(I\)) is calculated by using Equation (3):  
\[I_{z}=\frac{\sigma_{z}}{\overline{U_{z}}} \tag{3}\]  
where \(\sigma_{z}\) is the standard deviation, given by Equation (4):  
\[\sigma_{z}=\sqrt{6u_{*}} \tag{4}\]  
where \(u_{*}\) represents the frictional velocity of the wind flow. Mean wind speed and turbulence intensity profiles are shown in Figures 3 and 4, respectively. Figure 3a-c represent the comparison of mean wind speed profiles for historic, mid-future, and late-future wind speeds of return periods of 10 years, 50 years, and 100 years, respectively. Figure 4: Turbulence intensity profile. Figure 3: Mean wind speed profile comparison of historic, mid-future, and late-future for return period: (**a**) 10-year; (**b**) 50-year; (**c**) 100-year [15]. [PERSON] and [PERSON] [40] and [PERSON] [41] suggested a method for the generation of time history using the power spectral density function. A power spectral density matrix \(\mathbf{S}(\omega)\) for an \(n\)-story building with natural frequency \(\omega\) is given by Equation (5):  
\[\mathbf{S}(\omega)= \begin{pmatrix}S_{11}(\omega)&\dots&S_{1n}(\omega)\\ \vdots&\ddots&\vdots\\ S_{n1}(\omega)&\dots&S_{nn}(\omega)\end{pmatrix}  \tag{5}\]  
The elements of the \(\mathbf{S}(\omega)\) matrix are calculated by Equation (6):  
\[s_{ji}= \begin{cases}S_{j}(\omega),&j=i\\ \sqrt{S_{j}(\omega)S_{i}(\omega)}\gamma_{ji}(\omega),&j\
eq i\end{cases}  \tag{6}\]  
where \(S_{j}(\omega)\) is the Kaimal power spectral density function for the \(j\)th level, given by Equation (7):  
\[S_{j}(\omega)=\frac{200}{2\pi}u_{*}^{2}\frac{z}{\overline{U}_{j}}\Bigg{(}1+50 \frac{\omega z}{2\pi\overline{U}_{j}}\Bigg{)}^{\frac{-5}{3}} \tag{7}\]  
and \(u_{*}\) can be computed by using Equation (8):  
\[u_{*}=\frac{0.4\overline{U}_{j}}{ln(z/z_{0})} \tag{8}\]  
where \(z\) represents the height of the respective floors and \(z_{0}\) is the terrain roughness height. The value of \(z_{0}\) is 0.05 m for terrain roughness category B as per China wind loading code GB50009-2012 [39]. \(\gamma_{ji}(\omega)\) is the coherence function between the wind speed turbulence at two different floors, \(j\)th and \(i\)th, at heights \(z_{j}\) and \(z_{i}\), given by Equation (9):  
\[\gamma_{ji}(\omega)=exp\Bigg{[}-\frac{\omega}{2\pi}\frac{C_{z}|z_{j}-z_{j}|}{ \frac{1}{2}(\overline{U}_{j}+\overline{U}_{i})}\Bigg{]} \tag{9}\]  
where \(C_{z}\) is a constant called a correlation coefficient that can be taken as 10 for structural design purposes.