# Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses
## 4 Example and Analysis  
[PERSON] [20] used a high-rise building as an example to illustrate the correctness of their proposed method of basal bending moment gust wind load factor (MGLF). In the following, the \(GLF\), \(G_{\mathrm{B}}\) and \(G_{Rj}\) of the same structure are calculated according to the method given in this paper. The physical parameters of the structure are: dimensions \(H\times W\times D=200\times 50\times 40\) m, natural frequency \(f_{1}=0.22\) Hz; damping ratio \(\zeta=0.01\); first order vibration mode \(\varphi_{1}(z)=(z/H)^{\beta}\); mass distribution \(m(z)=m_{0}(1-\lambda(z/H))\); \(m_{0}=5.5\times 10^{5}\) kg/m; section drag coefficient \(C_{\mathrm{D}}=1.3\). The wind environment parameters are: fundamental wind speed \(\overline{U}_{10}=30\) m/s; wind profile power index \(\alpha=0.15\); and turbulence \(I_{10}=0.2\), while the wind spectrum type is Davenport wind spectrum, and the coherence function parameter \(C_{X}=C_{Z}=11.5\). The following four operating conditions are obtained by adjusting the values of the vibration index \(\beta\) and the mass discount factor \(\lambda\), respectively (Table 1). [PERSON] (2001) calculated the values of \(GLF\), \(G_{\mathrm{B}}\) and \(G_{\mathrm{R1}}\) via the DGLF method proposed by [PERSON], as well as the QGLF method, and the MGLF method, which was proposed by him. The DGLF method directly takes the first-order vibration mode as linear, i.e., the vibration index \(\beta=1\) (of course, the DGLF method can also obtain results when \(\beta\
eq 1\)), but the DGLF method itself cannot consider the mass discount of the structure along with the height. Therefore, [PERSON] could only derive the result of one of the above four conditions using the DGLF method. The MGLF method also divides the structure \(\sigma_{T}\) into two parts: the \(\sigma_{B}\) is calculated directly based on the wind load and influence function, and the \(\sigma_{R}\) is calculated by the indirect method, which distinguishes the influence of \(\beta\) and \(\lambda\), so the results of the four conditions are different. The results are presented in reference [20]. According to Equations (24), (28) and (29), the \(GLF\), \(G_{\mathrm{B}}\) and \(G_{Rj}\) values at arbitrary positions are calculated, and when the height is taken as 0 or \(H\), the \(GLF\), \(G_{\mathrm{B}}\) and \(G_{\mathrm{R1}}\) values at these positions can also be obtained--the results are shown in Table 2. The values in parentheses are the results relative to the results of [PERSON] [20]. It can be seen that the \(GLF\) values of the base bending moments obtained by the method proposed in this paper are identical to those obtained by [PERSON] [20]. The values of base shear force \(GLF\) and top displacement \(GLF\) deviate slightly, but the maximum deviation is only 0.6%, meaning they are consistent with each other. The expressions of the basal bending \begin{table}
\begin{tabular}{c c c} \hline
**Condition** & **Vibration Index \(\beta\)** & **Mass Discount Factor \(\lambda\)** \\ \hline
1 & 1.0 & 0.0 \\
2 & 1.6 & 0.0 \\
3 & 1.0 & 0.2 \\
4 & 1.6 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: The values of the parameters for the four working conditions [20]. moment \(GLF\) and top displacement \(GLF\) obtained according to the method of this paper are identical to those of [PERSON] [20]. The deviations in the result may be caused by numerical integration. Due to the different values derived for the respective integration step and the upper integration limit when numerical integration is performed, the integration results will be somewhat deviant. The specific values of these parameters are not given by [PERSON] [20]. The first-order frequency of this high-rise building is \(f_{1}\) = 0.22 Hz; the trapezoidal integration formula is used in this paper for the numerical integration calculation, and the frequency integration step is \(df\) = 1/1000 Hz, while the upper integration limit is \(f_{MAX}\) = 6 Hz.