# Data-Driven Modeling Algorithms for Cable-Stayed Bridges Considering Mechanical Behavior
## 4 Definition of a Mechanical Behavior of a Cable  
In the design stage of a cable-stayed bridge, only the deformed shape of a cable structure under dead loads is predetermined. The deformed shape of a bridge under dead loads is referred to as the target configuration under dead loads (TCUD) or the initial equilibrium configuration [23; 24; 25; 26]. In this paper, a practical determination method, which was suggested by [PERSON] and [PERSON] [24], is adopted for the calculation of cable forces. The initial cable forces were used as reference values for the convergence criterion of the Newton-Raphsons in the TCUD analysis and derived unstrained lengths of the cables. The model generation algorithm added the calculation of TCUD analysis to derive cable forces from the deformation of the cables. In the TCUD analysis, temporary supports at the anchorages in the stiffening girder are assumed, as described in Figure 12. Vertical reactions at each support are obtained by the three moment theorem. Considering cable geometry and the obtained reaction forces, cable forces and horizontal reactions are calculated. The assumptions cause large deformation at the location of the pylon because the ratio of span length between a main span and side spans and loading conditions are different. In order to solve this issue, temporary support conditions are additionally assumed to minimize lateral deformation of the pylon. An elastic catenary cable element by [PERSON] [25] was used for calculation of cable forces and unstrained lengths considering the dead load in TCUD analysis. Equation (1) presents the compatibility equations of the catenary cable element. \[\mathrm{x(L_{0})}=-\frac{\mathrm{F_{1}L_{0}}}{\mathrm{EA}}-\frac{\mathrm{F_{1 }}}{\mathrm{w}}\bigg{\{}\sinh^{-1}\bigg{(}\frac{\mathrm{F_{2}+wL_{0}}}{ \mathrm{F_{1}}}\bigg{)}-\sinh^{-1}\bigg{(}\frac{\mathrm{F_{2}}}{\mathrm{F_{1 }}}\bigg{)}\bigg{\}} \tag{1}\]  
\[\mathrm{z(L_{0})}=-\frac{\mathrm{F_{2}L_{0}}}{\mathrm{EA}}-\frac{\mathrm{wL_{ 0}^{2}}}{2\mathrm{EA}}-\frac{1}{\mathrm{w}}\bigg{\{}\sqrt{\mathrm{F_{1}^{2}+ \left(F_{2}+wL_{0}\right)^{2}}}-\sqrt{\mathrm{F_{1}^{2}+F_{2}^{2}}}\bigg{\}} \tag{2}\]  
where F\({}_{1}\) and F\({}_{2}\) denote x- and y-component of nodal forces, L\({}_{0}\) denotes the unstrained length, w is the weight of the cable per unit length, E is the elastic modulus and A is the area of the cable. In the TCUD analysis, the unstrained lengths and cable forces are unknown. Figure 13 presents a flow chart of the analysis under the initial equilibrium state. After assuming the initial values, an iterative analysis using the cable compatibility conditions, flexibility  
Figure 12: Temporary support method. matrix, and stiffness matrix, is performed to derive the incremental cable forces. The assumed variables are adjusted by the incremental process [15]. In this analysis, the convergence criteria for the cable force and the unstrained length were \(10^{-5}\) and \(10^{-7}\), respectively. The unstrained length is used for the initial equilibrium state. A bridge example is shown in Figure 14[27]. This was selected for this study to apply the proposed process for the interoperability between the digital models and the TCUD analysis. The bridge has a total length of 540 m and main span length is 300 m. There are 28 cables, including two back-stay cables. Table 1 summarizes the material properties, section properties, and loading conditions. The digital models for this bridge were generated based on the proposed modeling algorithm. A 2D frame model was also generated using the information of the digital models. Figure 13: Flow chart for initial equilibrium state of cable-supported bridges. As explained in this study, the TCUD analysis was conducted by the iterative process of the Newton-Raphson's algorithm.