# Effects of Corner Modification on the Wind-Induced Responses of High-Rise Buildings
## 4 Wind-Induced Response of the Building  
### Acceleration and Displacement  
The equation of motion in the modal coordinate is expressed as:  
\[q^{\prime\prime}{}_{j}(t)+2\xi_{j}\omega_{j}q^{\prime}{}_{j}(t)+\omega_{j}^{2}q _{j}(t)=\frac{P_{j}(t)}{M_{j}} \tag{1}\]  
where \(q^{\prime\prime}{}_{j}\), \(q^{\prime}{}_{j}\) and \(q_{j}\) are the generalized acceleration, velocity and displacement of \(j\)-th mode, respectively; \(\xi_{j}\) is the structural damping ratio of \(j\)-th mode and set as 0.04 [26]; \(\omega_{j}\) is the circular natural frequency of \(j\)-th mode; and \(P_{j}\) is the generalized aerodynamic forces of \(j\)-th mode.  
\(M_{j}\) is the generalized mass of \(j\)-th mode, which can be calculated as:  
\[M_{j}=\int_{0}^{H}m(z)\Phi_{j}^{2}(z)dz \tag{2}\]  
where \(m\) is the structural mass along the building height; \(\Phi_{j}\) is the mode shape of \(j\)-th mode.  
The variance of displacement and acceleration under wind excitation can then be obtained as:  
\[\sigma_{q_{j}}^{2}=\int\limits_{0}^{\infty}S_{q_{j}}(f)\mathrm{d}f =\frac{1}{\left(2\pi f_{j}\right)^{4}M_{j}{}^{2}}\int_{0}^{\infty} \left|H_{j}(f)\right|^{2}S_{P_{j}}(f)\mathrm{d}f \tag{3a}\] \[\sigma_{q_{j}}^{2}=\int\limits_{0}^{\infty}(2\pi f)^{4}S_{q_{j}}(f)\mathrm{d}f =\frac{1}{\left(2\pi f_{j}\right)^{4}M_{j}{}^{2}}\int_{0}^{\infty} \left(2\pi f\right)^{4}\left|H_{j}(f)\right|^{2}S_{P_{j}}(f)\mathrm{d}f \tag{3b}\]  
where \(S_{p_{j}}\) is the spectrum of the \(j\)-th modal force spectrum and \(f_{j}\) is the natural frequency of the \(j\)-th mode.  
The wind-induced acceleration \(a\) and displacement \(d\) can then be determined on the basis of the dynamic structural properties of the building. According to the structural design, the first-order frequency \(f_{1}\) equals 0.140 Hz in the \(x\)- and \(y\)- direction. The mass properties along the height are obtained from the design procedure.  
Figure 6 shows the standard deviation of the acceleration \(a\) atop the building with various corner modifications. For the acceleration in the \(x\)-direction, azimuth \(0^{\circ}\) is the most unfavorable angle, corresponding to the cross-wind direction. At azimuth \(0^{\circ}\), the standard deviation of acceleration decreases with the sequence of Ref, RO5%, RE5%, RE 10%, and RO10%. Obviously, the RO10% model shows the smallest values among the five models. For the acceleration in the \(y\)-direction, azimuth 270\({}^{\circ}\) is the most unfavorable angle, which corresponds to the cross-wind direction. The standard deviation of acceleration at azimuth 270\({}^{\circ}\) decreases with the sequence of Ref, ROS%, RE 10%, RE5%, and RO10%. As expected, the RO10% model shows the smallest values among the five models. Figure 7 illustrates the standard deviations of the displacement \(d\) atop the building. It can be found that the distribution of the displacement is quite similar to that of the acceleration with respect to the wind azimuth and models. It can also be seen that the RO10% model is the smallest among the five models for the displacement of the building.  
Figure 6: Standard deviations of accelerations in different azimuths: (**a**) \(x\)-direction. (**b**) \(y\)-direction.