# Effects of Corner Set--Backs on Wind Loads and Wind Induced Responses of Rectangular Tall Buildings
## 4 Correlation Factors
### Correlation Factors for Base Moment Coefficients  
The correlation factors for base moment coefficients are given by the following equations:  
\[\eta_{MD}(\gamma)=\frac{C_{MD}(\gamma)}{C_{MD}(0)},\ \eta^{\prime}_{MD}(\gamma)= \frac{C_{MD}(\gamma)}{C_{MD}(0)} \tag{9}\]  
\[\eta^{\prime}_{ML}(\gamma)=\frac{C_{ML}(\gamma)}{C_{ML}(0)} \tag{10}\]  
where \(\eta_{MD}(\gamma)\) is the correlation factor of the mean base moment coefficients in along-wind. \(C_{MD}(0)\) and \(C_{MD}(\gamma)\) are the mean along-wind base moment coefficients without and with corner set-backs, respectively, \(\eta^{\prime}_{MD}(\gamma)\) is the correlation factor for the RMS base moment coefficients in along-wind. \(C_{MD}(0)\) and \(C_{MD}(\gamma)\) are the RMS along-wind base  
Figure 12: The base moment power spectra with reduced frequency. (**a**) The along-wind base moment power spectra. (**b**) The across-wind base moment power spectra.  
moment coefficients without and with corner set-backs, respectively, and \(\eta t_{ML}(\gamma)\) is the correlation factor of the RMS across-wind base moment coefficients. \(C\prime_{ML}(0)\) and \(C\prime_{ML}(\gamma)\) are the RMS across-wind base moment coefficients without and with corner set-backs, respectively. Since the mean across-wind base moment coefficient comes to nearly zero, it is not discussed in the paper.  
Figure 13 displays the variations of correlation factors with different corner set-back rates for the mean along-wind base moment coefficients, RMS along-wind and across-wind base moment coefficients, respectively. With the increasing of corner set-back rate, the correlation factor of mean along-wind base moment coefficient decreases and then increases. The minimum correlation factor of the mean along-wind base moment coefficients is found when the corner set-back rate is 10%, which is about 0.52 times of the benchmark model. Regarding the RMS along-wind base moment coefficients, their correlation factors basically decrease by increasing the corner set-back rate. The minimum correlation factor is 0.58 times of the benchmark model. The correlation factors of the RMS base moment coefficients in across-wind also decrease by increasing the corner set-back rate. The minimum correlation factor reaches to 0.30 when the corner set-back rate is 20%.  
Figure 13: Cont.  
By fitting the correlation coefficients of the base moment coefficients to obtain an empirical formula with the corner set-back rate as the independent variable, which offers a reference for engineering applications. The fitting formulas of correlation factors for the along-wind mean base moment coefficients, along-wind and across-wind RMS base moment coefficients are shown in Equation (11) to Equation (13). The fits of the test results to the corresponding formulas are 0.922, 0.895 and 0.949, respectively. In general, the fitting formula curves match well with the test results.  
\[\eta_{{}_{MD}}(\gamma)=25.1\gamma^{2}-6.66\gamma+1\quad 0\leq\gamma\leq 0.20 \tag{11}\]  
\[\eta_{{}_{MD}}^{\prime}(\gamma)=15.97\gamma^{2}-5.47\gamma+1.04\quad 0\leq\gamma \leq 0.20 \tag{12}\]  
\[\eta_{{}_{ML}}^{\prime}(\gamma)=0.2\gamma^{2}-3.958\gamma+1.05\quad 0\leq\gamma \leq 0.20 \tag{13}\]