# Effects of Corner Set--Backs on Wind Loads and Wind Induced Responses of Rectangular Tall Buildings
## 5 Case Study
In this study, the equivalent static wind loads \(\hat{F}(z)\), the mean wind loads \(\overline{F}(z)\) and the inertial wind loads \(\widetilde{F}(z)\) are calculated using the below formulas:  
\[\hat{F}_{l}(z)=\overline{F}_{l}(z)+\widetilde{F}_{l}(z) \tag{17}\]  
\[\overline{F}_{l}(z)=w_{r}\sum_{i=1}^{N}{\overline{C_{p}}(i)A(i)} \tag{18}\]  
\[\widetilde{F}_{l}(z)=gm(z_{i})w_{1}^{2}\phi(z_{i})(\int_{0}^{\infty}{|H(n)|}^{ 2}{S_{Fl}(n)dn})^{1/2} \tag{19}\]where \(w_{r}\) is the pressure at the height of \(z\) (182.88 m), \(m(z_{i})\) is the mass of the floor at height of \(z_{i}\), \(w_{1}\) is the first modal frequency of the building, \(\phi(z_{i})\) is the coordinate of the mode shape, and \(H(n)\) is the frequency response function. Figure 16 illustrates the mean along-wind wind loads. The mean along-wind wind load of high-rise buildings decreases first and then increases with increasing of the corner set-back rates. The maximum reduction of mean along-wind wind loads happens at a corner set-back rate of 10%, and the overall along-wind mean wind loads is decreased by about 50%. This phenomenon is consistent with the variation trend of the mean drag force coefficients and mean along-wind base moment coefficients. As the mean across-wind wind loads are almost zero, and the across-wind mean wind loads is not discussed in this study. Figure 15: The peak acceleration at the top floor of the buildings. (**a**) Along-wind direction. (**b**) Across-wind direction. Figure 17 represents the results of the inertial wind loads in along-wind and across-wind of high-rise buildings. In Figure 17a, the along-wind inertial wind load decreases by increasing the rate of corner set-back. The variation trend is consistent with the RMS drag force coefficients and RMS base moment coefficients in along-wind. When the rate of corner set-back exceeds 5%, the along-wind inertial wind load is basically constant with the increase of the corner set-back rate, and the along-wind inertial wind load on the top of high-rise buildings is reduced by approximately 40%. As seen in Figure 17b, inertial wind load in across-wind decreases with increasing of the corner set-back rate. The across-wind inertial wind load on the top of the building is decreased by about 67% for a corner set-back rate of 20%. In addition, the across-wind inertial wind load is greater than the along-wind inertial wind load when the corner set-back rate is less than 15%, while the along-wind inertial wind load is greater than the across-wind inertial wind load when the corner set-back rates are 15% and 20%. Figure 16: Along-wind mean wind loads. Figure 17: _Cont._  
As the across-wind mean wind loads are close to zero at the unfavorable wind direction of 0\({}^{\circ}\), the equivalent static wind loads in across-wind are equal to the inertial wind loads. Therefore, only the equivalent static winds at along-wind direction are presented and discussed in this study. Figure 18 displays the along-wind equivalent static wind load of high-rise buildings. The equivalent static wind load first decreases and then increases with the corner set-back rate increases. Once the rate of corner set-back exceeds 5%, the reductions caused by corner set-backs are almost the same for different rate of corner set-back including 10%, 15%, and 20%. The maximum reduction occurs at the corner set-back rate of 10%, and the overall loads is decreased by about 45%. In total, the corner set-back modification effectively reduces the equivalent static wind load in along-wind.