# Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses
## 2 Refinement of Structural Wind-Induced Response Assessment Method
### The Full Three-Component Expressions in the Frequency Domain  
The average response of a high-rise building under wind loads is  
\[\overline{r}(z)=\int_{0}^{H}\overline{P}(z^{\prime})i(z,z^{\prime})dz^{\prime} \tag{5}\]  
where \(\overline{P}(z^{\prime})\) is the average wind load at height \(z^{\prime}\), and \(i(z,z^{\prime})\) is the influence function (including the influence of displacement, bending moment and shear force), which represents the response generated at position \(z\) when a unit load is applied at position \(z^{\prime}\)[23]. The frequency domain method is mostly used for the analysis of the structural wind vibration response, given its advantages of clear physical concepts and speed. When the wind speed at one spatial point reaches a maximum, it usually does not reach a maximum at another point in a certain range at the same time. Therefore there is a certain spatial correlation between the wind speeds at two points in the space [35]. From Equation (4), the structure's generalized wind load mutual spectrum \(S_{FiFj}(f)\) can be expressed as:  
\[S_{FiFj}(f)=\sum_{k=1}^{N}\sum_{l=1}^{N}\phi_{ik}\phi_{jl}\rho^{2}C_{D}{}^{2 }A_{k}\overline{U}_{k}A_{l}\overline{U}_{l}R_{X}R_{Z}\sqrt{S_{ik}(f)S_{ul}(f)} \tag{6}\]  
where \(R_{X},R_{Z}\) are the horizontal and vertical coherence function, \(\overline{U}_{i}\) is the average wind speed at the \(i\)th node, and \(S_{ui}(f)\) is the pulsating wind speed spectrum. According to Equation (2), if the generalized wind load mutual spectrum \(S_{FiFj}(f)\) is known, then the total pulsation response spectrum \(S_{qi}(f)\) of the modal generalized coordinates \(q_{j}\) can be expressed as Equation (7), and then using the modal superposition principle (Equation (3)), the response spectrum \(S_{qi}(f)\) at node \(i\) of the structure can be obtained, as in Equation (8). \[S_{qi}(f)=\sum_{i=1}^{N}S_{FiFj}(f)H_{i}^{*}(f)H_{j}(f) \tag{7}\]\[S_{yi}(f)=\sum_{j=1}^{N}\sum_{k=1}^{N}\phi_{ji}\phi_{ki}H_{j}^{*}(f)H_{k}(f)S_{FjFk }(f) \tag{8}\]  
\[H_{k}(f)=\frac{1}{K_{k}^{*}}\cdot\frac{1}{1-\left(f/f_{k}\right)^{2}+\mathrm{i}2 \zeta_{k}(f/f_{k})} \tag{9}\]  
where \(H_{k}\)(\(f\)) is the kth modal complex frequency response function, and \(H_{j}^{*}(f)\) is the conjugate of \(H_{j}(f)\). The response spectra of the bending moment and shear force can be obtained by replacing the vibration displacement \(\phi\) in Equation (8) with bending moment or shear force. \[\sigma_{Ti}= \ (\int_{0}^{\infty}\big{|}S_{yi}(f)\big{|}df)^{1/2} \tag{10}\]  
The total pulsation response \(\sigma_{Ti}\) can be calculated according to Equation (10), and it contains three components: the background component \(\sigma_{B}\), the resonance component \(\sigma_{R}\), and the background-resonance coupling component \(\sigma_{BR}\). Because of the complicated nature of the calculation for \(\sigma_{T}\), researchers often choose to determine each component, and then calculate \(\sigma_{T}\) according to Equation (11). The expressions for each component are as follows.