# Effects of Corner Set--Backs on Wind Loads and Wind Induced Responses of Rectangular Tall Buildings
## 4 Correlation Factors
### Correlation Factors for Power Spectral Densities of Base Moment Coefficients  
The correlation factors are defined as follow:  
\[C_{m\;l}(f)=\frac{S_{M\;l}(f)}{S_{M\;0}(f)} \tag{14}\]  
in which, \(S_{M\;l}(f)\) and \(S_{M\;0}(f)\) are the base moment spectrum with and without corner set-backs. The subscripts \(l=D,L\) represent the along-wind and the across-wind, respectively. Figure 14 shows the correlation factors of the base moment spectra of models for a range of the reduced frequency from 0.10 to 0.25 [23]. It shows that the overall trend of correlation factor curves in corner set-backs is dispersed with reduced frequency. Compared with along-wind base moment power spectra, the correlation factors of across-wind base moment power spectra are fluctuated, especially in the range of 0.125 to 0.15. The maximum correlation factors for 10% corner set-back model and 15% corner set-back model are about 9.5 and 8.0, respectively. This phenomenon indicates that the increasing of the reduced frequency may lead to the magnification of the across-wind effects, which ought to attract the greater attention by structural designers of high-rise buildings. Table 1 lists the correlation factors in typical reduced frequencies ranged from 0.100 to 0.250, which serve as a direct reference for wind-resistant designs of similar tall buildings. In the Table 1, the along-wind correlation factors are mostly less than 1.0, and the optimal corner set-back rate is basically 20%. While the across-wind correlation factors are mostly greater than  
Figure 13: The correlation factors with different corner set-back rates. (**a**) The along-wind correlation factors for mean base moment coefficients. (**b**) The along-wind correlation factors for RMS base moment coefficients. (**c**) The across-wind correlation factors for RMS base moment coefficients.  
1.0, this is means that when calculating the across-wind equivalent static loads of similar high-rise buildings considering corner set-backs by the stipulations in the Chinese [16] and Japanese [22] design code, the expanded correlation factors amplify the resonance coefficient and thus the dynamic wind loads in across-wind are amplified. As a result, wind engineers and structural designers ought to take this into account when assessing dynamic loads in across-wind for similar recessed high-rise buildings.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Wind Load Direction**} & \multirow{2}{*}{**Corner Set-Back**} & \multicolumn{6}{c}{**Reduced Frequencies (\(\boldsymbol{fB/U_{H}}\))**} \\ \cline{3-10}  & & **Rate \(\boldsymbol{b/B}\)** & **0.100** & **0.125** & **0.150** & **0.175** & **0.200** & **0.225** & **0.250** \\ \hline \multirow{4}{*}{along-wind} & 5\% & 1.095 & 1.249 & 0.994 & 1.179 & 1.097 & 1.063 & 0.701 \\  & 10\% & 1.169 & 1.336 & 1.314 & 1.116 & 1.089 & 1.475 & 0.826 \\ \cline{1-1}  & 15\% & 0.941 & 1.005 & 0.644 & 0.955 & **0.753** & 0.953 & **0.519** \\ \cline{1-1}  & 20\% & **0.759** & **0.796** & **0.524** & **0.621** & 0.831 & **0.592** & 0.617 \\ \cline{1-1}  & 5\% & 0.340 & 2.230 & 3.345 & 2.617 & **0.827** & **1.656** & **1.049** \\ \cline{1-1}  & 10\% & 0.118 & 1.895 & 6.983 & 3.116 & 2.304 & 3.011 & 1.529 \\ \cline{1-1}  & 15\% & 0.126 & 4.105 & 3.522 & 2.728 & 1.452 & 2.508 & 1.559 \\ \cline{1-1}  & 20\% & **0.070** & **1.275** & **1.779** & **1.352** & 0.958 & 1.907 & 1.489 \\ \hline \hline \end{tabular}  
* Notes: The numbers in italics and bold are the minimum values of the correlation factors of base moment power spectra.  
\end{table}
Table 1: The correlation factors for power spectral densities of base moment coefficients in typical frequencies.  
Figure 14: The correlation factors for base moment power spectra. (**a**) Along-wind. (**b**) Across-wind.