## 3 Uncertain FEMs of Tower-Line Systems  
In this paper, a 220 Kv transmission line with one tower and two span lines is employed as the research object. The total height of the transmission tower is 50.043 m, with a nominal height of 33.043 m, and the base size is 10 \(\times\) 10 m. The main members of the tower are composed of Q345-type equilateral angle steel, and the diagonal and secondary members are composed of Q345-type and Q235-type equilateral angle steel. The employed tower is shown in Figure 2. The conductor and ground wire are 2-bundled JL/LB20A-630/45 and OPGW-105 optical cables, respectively, and the design parameters are shown in Table 1.  
Figure 2: The size of the tower. During the design state of the structure, all model parameters are considered to be constant. However, in practical engineering cases, deviations from these designed values can occur during production, transportation, and installation processes, affecting material strength, geometric dimensions, and other structural parameters. Thus, the geometric size, material strength, boundary conditions, and other parameters of the whole tower-line system model are uncertain. Since the soil-structure interaction is not considered in this paper, the foot of the transmission tower is fixed. In addition, compared with the size of the tower, the span of the transmission line is larger, and the section size is smaller [40]. Therefore, the uncertainty of material parameters (yield strength, elastic modulus, and Poisson's ratio) and geometric dimensions (limb width and limb thickness of the angle steel) of the members in the tower are considered. According to the Chinese code [41], for structural material strengths that conform to a lognormal distribution, the standard strength value is equivalent to the 0.05 quantile of the probability distribution, which is calculated using Equation (8). The standard values for the elastic modulus and Poisson's ratio of the material correspond to the 0.5 quantile of the probability distribution, which is the mean value. Table 2 presents the probability distribution of member material parameters. Regarding the uncertainty of geometric dimensions, existing research has demonstrated that the geometric dimensions of angle steel conform to a normal distribution, with the specific probability distribution as shown in Table 3 [40]. \[f_{k}=\mu_{f}\mathrm{exp}\left(-1.645\delta_{f}\right) \tag{8}\]  
where \(f_{k}\) is the material strength standard value; \(\mu_{f}\) is material strength standard value; \(\delta_{f}\) is the variable coefficient of material strength. After establishing the probability distribution of uncertainty parameters, the LHS method is employed to extract 30 groups of member parameters. Then, 30 uncertain FEMs (U1-U30) and 1 certain FEM (C1) of the transmission tower-line system are established by ANSYS (the version is ANSYS V19). The BEAM188 element is used to simulate the transmission tower, and the LINK10 element is used to simulate the transmission line [42]. The Bilinear isotropic hardening plasticity model is used to describe the constitutive model of the steel member [34]. The foot of the transmission tower and the two ends of the transmission lines are fixed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Diameter (mm)** & **Mass (kg/m)** & **Area (mm2)** & **Elastic Modulus (MPa)** \\ \hline JL/LB20A-630/45 & 33.60 & 2.0072 & 666.55 & 65,000 \\ OPGW-105 & 11.40 & 0.4170 & 68 & 132,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of conductors and ground wires.