## 2 Modeling of Stay Cable
Figure 4: Stay cable: (**a**) disposition prior to and after tensioning, (**b**) mathematical model of discretized cable. Over the years, a significant number of researchers [33; 34; 35] have been focused on improving [PERSON]'s formula [36] to better understand the influence of cable deformation on the external load configuration. Assuming that the influence of the cable's self-weight can superimpose with the effects of tensioning, considering the latter dominant influence, the cable can be represented by the model described in Figure 4. The stiffness coefficients of the elastic springs \(\mathrm{k_{A}}\) and \(\mathrm{k_{B}}\) (which simulate the shortening of the cable axis) are determined based on the actual displacements of the anchor points (nodes A and B of the model in Figure 4) in the considered bridge structural system. Furthermore, this model can be simplified while maintaining identical analytical outcomes. Such a model is illustrated in Figure 5, wherein elastic springs are substituted with a fictitious rod of length \(\mathrm{l_{f}}\), a modulus of elasticity \(\mathrm{E_{f}}\), and a cross-sectional area \(\mathrm{A_{f}}\). The cable model (Figure 5) comprises \"n\" rods interconnected at anchor points (1 and 2), exhibiting identical local axes (x, y, z), and equivalent boundary conditions. At the passive anchor (node 1), all strands are anchored, and the anchor is displaceable. Upon tensioning the i-th strand (i = 1, 2,..., n), the support (anchor of the i-th strand) at the active anchor (node 2) is removed in the x-axis direction. The cable (Figure 5), composed of \"n\" identical strands, possesses a modulus of elasticity \(\mathrm{E_{u}}\) and a cross-sectional area of each strand \(\mathrm{A_{u}}\). The cross-sectional area of the fictitious rod, for the adopted length \(\mathrm{l_{f}}\), and a modulus of elasticity \(\mathrm{E_{f}}=\mathrm{E_{u}}\), is determined based on the deformation compatibility conditions (shortening) of the cable axis within the bridge system, and can be described as:  
\[\mathrm{A_{f}}=\frac{\mathrm{Z_{k}}\mathrm{l_{f}}}{\mathrm{E_{u}}\mathrm{\xi_ {k}}} \tag{4}\]  
where \(\mathrm{Z_{k}}\) denotes cable tension force, and \(\mathrm{\xi_{k}}\) denotes the shortening of the cable axis due to cable tensioning. The intensities of forces \(\mathrm{Z_{i}}\) in the strands where tensioning occurs depend on the applied procedure: tensioning with the same force in the strands over several cycles and the procedure of successive equalization of force in the strands (isotension procedure), which is performed in only one cycle. Analytical solutions for forces and deformations for individual tensioning of strands in one or more cycles, according to the model depicted in Figure 4, are presented in the papers [26; 29]. The tension force applied to an undeformed system of a stay cable (as an external force) remains within the deformed system as a system (internal) force. In order to ensure the existence of the applied force as a system force in the undeformed system (Figure 5a), in addition to the tension force, the corresponding deformation of the strands or cable must be added. In other words, displacements that change the position of one end of the cable relative to the other must be considered. Calculations for both options of individual tensioning of the strands in a stay cable are depicted through algorithms (Figures 6-8). Figure 5: Mathematical model of stay cable for computer analysis: (**a**) prior to tensioning, (**b**) after tensioning.