# Effects of Corner Set--Backs on Wind Loads and Wind Induced Responses of Rectangular Tall Buildings
## 5 Case Study  
Prototype tall buildings with the same dimension of CAARC standard tall building (with a length of 45.72 m, a width of 30.48 m, and a height of 182.88 m) are selected to study the effects of corner set-backs on wind loads and wind-induced responses. The buildings are concrete structures, for which the densities are about 300 kg/m\({}^{3}\). The first two natural frequencies of the five models are assumed to be 2.743 s and 2.743 s, respectively [24]. When calculating wind loads, the design wind pressure is assumed to be 0.50 \(\text{KN}/\text{m}^{2}\) with a damping ratio of 5% [16], while when calculating acceleration, the design wind pressure is assumed to be 0.35 \(\text{KN}/\text{m}^{2}\) with a damping ratio of 2% [25; 26; 27; 28]. Based on the linear mode [29], the first-order vibration mode function is \(\phi(z_{i})=(z_{i}/H)^{1.4}\), where \(z_{i}\) is the height of the \(i\) th floor from the ground, and \(H\) is the height of the tall building. The tall buildings are evenly divided into 60 floors along the height and the height of each floor is H/60. Due to the greater impact (uncomfortable vibration) caused by wind-induced acceleration rather than wind-induced displacement for high-rise buildings, this study takes wind-induced acceleration of the top floor as the typical parameter to analyze the influences of corner set-backs on wind-induced response reduction of rectangular high-rise buildings. Due to the low efficiency and long calculation time of the time-domain method to calculate the structural wind load and response, and the results obtained by the time-domain method are not very convenient for engineering designers. Therefore, the frequency domain method is used to analyze the wind load and response of the structure in this paper. In accordance with the theory of random vibration and structural mechanics and wind tunnel test results, the peak acceleration and RMS acceleration of the top floor can be determined by Equations (15) and (16). \[\widetilde{a}_{l}(z)=g\hat{a}_{l}(z) \tag{15}\]  
\[\hat{a}_{l}(z)=(\int_{0}^{\infty}{(2\pi n)}^{4}{S_{Fl}(n)dn})^{\frac{1}{2}} \tag{16}\]  
in which, the subscript \(l\) can be taken as \(D\) and \(L\), representing the along-wind direction and the across-wind direction, respectively, \(\widetilde{a}(z)\) and \(\hat{a}(z)\) are the peak acceleration and RMS acceleration at the height of \(z\), respectively, \(g\) is peak factor, this paper take 3.5, \(S_{FD}(n)\) and \(S_{FL}(n)\) is the along-wind and across-wind generalized force spectrum, respectively, and \(n\) represents the natural frequency. Figure 15 shows the values of the along-wind and across-wind peak acceleration at the top floor of the five different corner set-back high-rise buildings. Figure 15a shows that the peak acceleration in along-wind direction gradually decreases by increasing the corner set-back rate. The minimum peak acceleration 0.0477 m/s\({}^{2}\) is observed when the rate of corner set-back is 20%. In Figure 15b, the peak acceleration in across-wind direction decreases rapidly by increasing the corner set-back rate. The across-wind peak acceleration reaches the minimum value 0.0369 m/s\({}^{2}\) at the corner set-back rate of 20%. The Chinese design load assumed that the RMS wind loads can be taken regard as the inertial wind loads and combine them with the mean wind loads to be the equivalent static wind loads for structural design of high-rise buildings.