# Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses
## 3 The Assessment Method for the Equivalent Static Wind Load of Arbitrary Structural Response  
### Theoretical Analysis  
As in previous studies, for high-rise buildings and towering structures, this paper still assumes that the equivalent static wind load on the structure is equal to the product of the mean wind load and the gust load factor, which can be expressed as follows:  
\[\hat{P}(z)=GLF(z)\cdot\overline{P}(z) \tag{23}\]  
\[GLF(z)=1+\frac{\sigma_{T}(z)}{\overline{r}(z)}=1+\sqrt{\frac{\sigma_{B}(z)^{2} }{\overline{r}(z)^{2}}+\sum\limits_{i}^{n}\frac{\sigma_{Ri}(z)^{2}}{\overline {r}(z)^{2}}}=1+\sqrt{G_{B}(z)^{2}+\sum\limits_{i}^{n}G_{Ri}(z)^{2}} \tag{24}\]  
\[\sigma_{B}(z)=g_{u}\bigg{(}\int_{0}^{\infty}\int_{0}^{H}\int_{0}^{H}\int_{0}^{ W}\int_{0}^{W}\frac{p(x_{1},z_{1},f)p(x_{2},z_{2},f)i(z,z_{1})i(z,z_{2})dx_{1}dx_{2} dz_{1}dz_{2}df\bigg{)}^{1/2} \tag{25}\]  
\[\sigma_{Rj}(z)=\int_{0}^{H}\hat{P}_{Rj}(z^{\prime})i(z,z^{\prime})dz^{\prime} \tag{26}\]  
\[\hat{P}_{Rj}(z)=m(z)(2\pi f_{j})^{2}\sigma_{YRj}\varphi_{j}(z) \tag{27}\] \[=g_{R}\bigg{(}S_{j_{f}}(f_{j})\frac{\pi f_{j}}{\overline{\chi}_{ j}^{2}}\bigg{)}^{1/2}\varphi_{j}(z)\] \[=g_{R}\bigg{(}\sum\limits_{k=1}^{N}\sum\limits_{l=1}^{N}\hat{ \varphi}_{lk}\hat{\varphi}_{jl}p^{2}C_{D}{}^{2}A_{k}\overline{l}_{k}A_{l} \overline{l}_{l}R_{X}R_{Z}\sqrt{S_{u}(f_{j})S_{u}(f_{j})}\frac{\pi f_{j}}{ \overline{\chi}_{j_{i}}^{2}}\bigg{)}^{1/2}\varphi_{j}(z)\]  
where \(\hat{P}_{Rj}(z)\) is the \(j\)th order modal resonant equivalent wind load at height \(z\) of the structure, and its value is equal to the inertia force caused by the \(j\)th order modal resonant displacement [38]. The background response \(\sigma_{B}(z)\) and resonant response \(\sigma_{R}(z)\) can be obtained according to Equations (25) and (26), respectively, and then bringing them into the \(GLF\) expression (Equation (24)) can give the expressions of the \(G_{B}(z)\) and \(G_{R}(z)\) at any position on the structure. The background gust load factor \(G_{B}(z)\) of the structure's response when under wind load at any location is  
\[G_{B}(z)= \frac{\sigma_{B}(z)}{\overline{r}(z)} \tag{28}\] \[=\frac{g_{B}\sqrt{\int_{0}^{\infty}\int_{0}^{H}\int_{0}^{H}\int_ {0}^{W}\int_{0}^{W}\overline{p(x_{1},z_{1},f)p(x_{2},z_{2},f)i(z,z_{1})i(z,z_{2 })dx_{1}dx_{2}dz_{1}dz_{2}df}}}{\int_{0}^{H}\overline{P}(z^{\prime})i(z,z^{ \prime})dz^{\prime}}\] \[=\frac{2g_{B}I_{H}H^{\alpha}}{\int_{0}^{h}z^{\prime 2\alpha i}i(z,z^{ \prime})dz^{\prime}+\int_{h}^{H}z^{\prime 2\alpha i}i(z,z^{\prime})dz^{\prime}} \sqrt{\int_{0}^{\infty}S_{u}^{*}(f)|K_{Z}(\alpha,z,f)|^{2}|I_{X}(f)|^{2}df}\]  
\[GLF(z)=1+\frac{\sigma_{T}(z)}{\overline{r}(z)}=1+\sqrt{\frac{\sigma_{B}(z)^{2} }{\overline{r}(z)^{2}}+\sum\limits_{i}^{n}\frac{\sigma_{Ri}(z)^{2}}{\overline {r}(z)^{2}}}=1+\sqrt{G_{B}(z)^{2}+\sum\limits_{i}^{n}G_{Ri}(z)^{2}} \tag{24}\]0 \[GLF(z)=1+\frac{\sigma_{T}(z)}{\overline{r}(z)}=1+\sqrt{\frac{\sigma_{B}(z)^{2} }{\overline{r}(z)^{2}}+\sum\limits_{i}^{n}\frac{\sigma_{Ri}(z)^{2}}{\overline {r}(z)^{2}}}=1+\sqrt{G_{B}(z)^{2}+\sum\limits_{i}^{n}G_{Ri}(z)^{2}} \tag{24}\]1 \[GLF(z)=1+\frac{\sigma_{T}(z)}{\overline{r}(z)}=1+\sqrt{\frac{\sigma_{B}(z)^{2} }{\overline{r}(z)^{2}}+\sum\limits_{i}^{n}\frac{\sigma_{Ri}(z)^{2}}{\overline {r}(z)^{2}}}=1+\sqrt{G_{B}(z)^{2}+\sum\limits_{i}^{n}G_{Ri}(z)^{2}} \tag{24}\]2 \[GLF(z)=1+\frac{\sigma_{T}(z)}{\overline{r}(z)}=1+\sqrt{\frac{\sigma_{B}(z)^{2} }{\overline{r}(z)^{2}}+\sum\limits_{i}^{n}\frac{\sigma_{Ri}(z)^{2}}{\overline {r}(z)^{2}}}=1+\sqrt{G_{B}(z)^{2}+\sum\limits_{i}^{n}G_{Ri}(z)^{2}} \tag{24}\]3  
Following the method of \(GLF\) based on displacement (DGLF), [PERSON] [15] defined the integral term in the structural force spectrum with respect to height z as a vertical joint receiver function. To make the above expression formally identical to the DGLF method,this paper also draws on this idea, and defines the integral term with respect to height z as a new function (Equation (29)), but the difference is that we use four integral terms.