## 1 Introduction
According to the fixed point of the saddle, the position equation of the saddle is obtained through the geometry and force relationship between the saddle and the main cable in the state of the empty cable. By using the Newton Raphson method, the Jacobian matrix was calculated and the initial value was modified to meet the requirements of the saddle position [10]. Although the influence of friction on the saddle position was considered, the definition and derivation of the nominal friction coefficient were lacking. Moreover, the Jacobian matrix was of order 11, which is costly to calculate and requires more accurate initial values and constraints to ensure convergence. [PERSON] et al. studied the requirements of anti-sliding coefficient and bending span ratio in a multi-tower suspension bridge and reached the conclusion that the longitudinal stiffness in the tower must be controlled within a certain range [11]. The paper focuses on the stiffness of the tower but does not make a detailed study on the friction coefficient between the main cable and the saddle. [PERSON] et al. proposed two mechanical models to analyze the frictional resistance between the cable saddles. On this basis, an analytical formula of the friction resistance of the saddle with a vertical or horizontal friction plate was derived [12]. The friction analysis model of the main cable saddle based on slip compatibility condition was established to accurately evaluate the friction between the main cable and the saddle [13]. The simulation of friction between the cable and the auxiliary cable pipe in the saddle anchorage of a low tower cable-stayed bridge was analyzed [14]. The contact and sliding behaviors between saddles were analyzed, and the mechanical and friction properties of the saddles under different friction coefficients and unbalanced cable forces were studied [15]. The transverse force between the main cable and saddle was studied [16]. The concept of anti-sliding of horizontal friction plate was proposed and an analysis model considering interlayer sliding was established [17]. [PERSON] et al. presented an analytical model for calculating the frictional resistance between cable saddles. A new saddle was proposed to accommodate two main cables, and the friction resistance was analyzed by mechanical analysis verified by finite element analysis. A simple and effective model is proposed as a tool to analyze the friction resistance between single strand and saddle. They also gave a numerical method for calculating the frictional resistance between the main cable and saddle. The effectiveness of this method was verified by a large-scale cable saddle model test applied to the friction resistance analysis of the multi span suspension bridge [18; 19; 20; 21]. However, there is no discussion about the influence of the change in friction coefficient on the structure. After considering the change in saddle pre-deflection caused by friction, there is less discussion on the influence of the datum strand shape [22]. The control of the datum strand shape of a suspension bridge is the key link of construction control. At present, the commonly used method for calculating the datum strand shape is to use analytical software to calculate the strand shape under the condition of empty cable. Neither the parabola formula nor the catenary formula can consider the change in tangent points of cable saddle. In fact, due to the existence of friction, the position of the tangent points of the cable strand in the saddle groove will also change. Therefore, it is very important to establish the calculation formula of the cable strand based on catenary theory and consider the change in tangent points of the saddle. [PERSON] et al. proposed a nonlinear analysis method of suspension bridge structure considering the influence of the pylon saddle. By determining the expression of the cable attachment point on the saddle, the saddle can be automatically considered in the analysis process [23]. [PERSON] et al.