## 2 Modeling of Stay Cable  
The model (Figure 4) represents a stay cable comprised of\"n\"equal parallel strands, isolated from the bridge stayed structure, where the system length is denoted as \(\mathrm{l}_{k}\), the cross-sectional area of each strand as \(\mathrm{A}_{u}\), and the modulus of elasticity as \(\mathrm{E}_{u}\). Due to the tensioning of the cable, the anchor nodes A and B are displaced to positions A' and B', where \(\delta_{\mathrm{A}}\) and \(\delta_{\mathrm{B}}\) represent the displacement vectors. Figure 3: Isotension method [27]: (**a**) principle diagram; (**b**) scheme of installation of the parallel strand cables with the isotension method. Within the framework of the observed bridge's structural system, the stay cable is modeled as a linear system (single member) with appropriate boundary conditions. The following assumptions are made: the change in the distance between anchor nodes A and B (Figure 4a), or the shortening (\(\xi_{\text{k}}\)) along the system axis, is a linear function of the cable tension force \(Z_{\text{k}}\), the difference between the length of the straight cable axis (Figure 4) and the real, curved elastic axis due to cable sag, is considered negligible. Additionally, it is assumed that the temperature remains constant during cable tensioning. After tensioning the cable, considering the influence of its self-weight (\(\text{g}_{\text{k}}\)) and the force \(Z_{\text{k}}\) applied through the active anchor A (Figure 4), the support tension forces along the cable axis will be:  
\[\text{R}_{\text{A}^{\prime}}=Z_{\text{k}} \tag{1}\]  
\[\text{R}_{\text{B}^{\prime}}=Z_{\text{k}}+\text{g}_{\text{k}^{\prime}}(\text{ l}_{\text{k}}-\text{g}_{\text{k}})\sin(\alpha+\psi) \tag{2}\]  
where \(\text{R}_{\text{A}^{\prime}}\) and \(\text{R}_{\text{B}^{\prime}}\) denote the support tension forces at anchor nodes A\({}^{\prime}\) and B\({}^{\prime}\), respectively, \(Z_{\text{k}}\) denotes the tension force applied to the cable, \(\text{g}_{\text{k}}\) denotes the cable self-weight, \((\text{l}_{\text{k}}-\text{g}_{\text{k}})\) denotes the cable length after deformation, and \(\alpha\) and \(\psi\) denote the angles of cable inclination. For a long cable length and small cable force, the cable sag should be considered in the estimation of the cable force [30]. In that calculation, as a rule, the parallel modulus of elasticity of the cable should be introduced, for example, according to [PERSON]'s Formula (3). Subjected to its own dead load and axial tensile force, the cable, supported at its ends, naturally forms a catenary curve due to sag. Therefore, when utilizing a straight cable element to analyze the entire stay cable, it becomes essential to consider the sag effect, as the cable's axial stiffness varies with changing sag. Taking into account the nonlinear sag phenomenon in inclined cable stays, employing an equivalent straight cable element with an equivalent modulus of elasticity proves beneficial for accurately representing the catenary behavior of the cable. This approach is based on the equilibrium equation of the deformed state of an infinitely flexible cable under the influence of its own weight or a uniformly distributed load along a specified direction, assuming a parabolic deformed configuration, and was developed by [PERSON] in 1965 [31]. The equivalent elastic modulus of a stay cable can be described by [PERSON] formula:  
\[\text{E}_{\text{eq}}=\frac{\text{E}_{0}}{1+\frac{(\gamma\text{U})^{2}}{12 \sigma^{3}}\text{E}_{0}} \tag{3}\]  
where \(\text{E}_{\text{eq}}\) is the equivalent modulus of elasticity, \(\text{E}_{0}\) is the elastic modulus of the cable, \(\sigma\) is the tensile stress, \(\text{I}\) is the length of the stay cable, and \(\gamma\) is the weight per unit length of the stay cable [32]. According to Equation (3), it can be demonstrated that the difference between the modulus of elasticity and the effective modulus of elasticity is negligible for cables of short length and significant tensile stress [5].