## 2 The Time-Varying Performance Function of In-Service Transmission Towers
### Random Equivalent Static Wind Loads of Transmission Towers  
The dynamic reliability analysis of transmission towers under wind loads is complicated, and the corresponding theory is not yet fully developed. One of the frequently used methods is to simplify dynamic wind loads into equivalent static wind loads, and then convert dynamic reliability into static reliability. According to Ref. [9], the mean, standard deviation, and probability distribution of random equivalent static wind loads can be expressed as follows:  
\[\mu_{W_{w}}=\mu_{\beta_{z}}\mu_{W_{z}} \tag{2a}\] \[\sigma_{W_{w}}=\sqrt{\mu_{W_{z}}^{2}\sigma_{\beta_{z}}^{2}+\beta_{z}^{2}\sigma_ {W_{z}}^{2}}\] (2b) \[F_{W_{w}}(w)=\exp\biggl{\{}-\exp\biggl{[}-\frac{w-[1-0.45\delta_{W_{w}}]\mu_{W_ {w}}}{0.779\delta_{W_{w}}\cdot\mu_{W_{w}}}\biggr{]}\biggr{\}} \tag{2c}\]  
where \(\delta_{W_{w}}\) is the coefficient of variation in the random equivalent static wind loads \(W_{w}\), \(\mu_{W_{z}}\) and \(\sigma_{W_{z}}\) are mean and standard deviation of the random static wind loads \(W_{s}\), and \(\mu_{\beta_{z}}\) and \(\sigma_{\beta_{z}}\) are mean and standard deviation of the random wind vibration coefficient \(\beta_{z}\).  
Since wind loads are taken as the maximum value during the design reference period [10], the probability distribution of the maximum random equivalent static wind loads during the design reference period T can be further expressed as  
\[F_{W_{wT}}(w)=\exp\biggl{\{}-\exp\biggl{[}-\frac{w-[1-(0.45-0.779\ln T)\delta_ {W_{w}}]\mu_{W_{w}}}{0.779\delta_{W_{w}}\cdot\mu_{W_{w}}}\biggr{]}\biggr{\}} \tag{3}\]  
and the corresponding mean and standard deviation can be calculated as follows:  
\[\biggl{\{}\mu_{W_{wT}}=(1+0.779\delta_{W_{w}}\ln T)\mu_{W_{w}}\,. \tag{4}\]  
The Equivalent Description of the Time-Varying Performance Function of In-Service Transmission Towers  
The main member is the primary bearing member of the transmission tower system, and it plays a crucial role in the structural design of the tower [9]. Therefore, using the failure criterion that the stress of the main member exceeds the yield strength, the equivalent performance function of in-service transmission towers can be established based on the equivalent description method [22] and the resistance degradation model shown in Equation (1). This function can be expressed as  
\[Z(t)=\min_{i=1,2,\ldots,l}[R(t)-S_{i}(W_{setT},t)] \tag{5a}\] \[\left\{\begin{array}{l}S_{i}(W_{setT},t)=\frac{N_{c}(t)}{\varphi \cdot\varphi}+\frac{M(t)}{\omega}\text{ for compression }-\text{ bending member}\\ S_{i}(W_{setT},t)=\frac{N_{c}(t)}{A}+\frac{M(t)}{\omega}\text{ for tension }-\text{ bending member}\end{array}\right. \tag{5b}\]  
in which \(Z(t)\) denotes the time-varying equivalent extreme variable, \(R(t)\) denotes the time-varying yield strength of the main member, and \(l\) is the number of main members. \(S_{i}(W_{setT},t)\) denotes the time-varying stress of each main member under random equivalent static wind loads, \(N_{c}(t)\) is the time-varying axial compression force, \(N_{t}(t)\) is the time-varying axial tension force, \(M(t)\) is the time-varying bending moment, \(\varphi\) is the stability coefficient under axial compression, \(A\) is the area of cross-section, and \(\omega\) is the cross-section modulus [23].