## 2 Analysis of Sag Effect Influence  
The structural form of setting crossing cables in the mid span of the three-tower cable-stayed bridge is shown in Figure 1. [PERSON] et al. [15; 16] analyzed the mechanical mechanism by which the crossing cables provide restraint to the bridge towers. When an unbalanced horizontal force is applied to the top of the mid tower, the change in the cable forces of a pair of crossing cables is shown in Figure 2. In the figure, the crossing cables are anchored at a distance \(a\) from the mid tower, with the main span length denoted as \(L\) and the tower height above the deck as \(h\). When the mid tower deflects, the crossing cables of the mid tower are tensioned, resulting in increased cable forces that restrain the tower's deflection. Meanwhile, the cable forces in the crossing cables of the side towers decrease. A portion of the dead load on the main girder originally borne by the side towers' crossing cables is transferred to the mid tower's crossing cables. This load redistribution maintains the force equilibrium of the main girder and enhances the structural stiffness. Figure 1: Crossing cables are set in the mid span of three-tower cable-stayed bridge. [PERSON] et al. [15; 16; 17] derived the analytical formula of the longitudinal constraint stiffness of the crossing cable to the mid tower based on the action mechanism of the crossing cable:  
\[k=\frac{a^{2}}{\frac{1}{E_{s}^{3}A_{s}}+\frac{1}{E_{m}^{2}A_{m}}} \tag{1}\]  
where \(a\) represents the distance from the anchorage position of the crossing cable to the mid tower and \(l\) and \(A\) are the length and cross-sectional area of the cable, respectively. \(E^{\prime}\) is the elastic modulus of the cable. The subscripts \(s\) and \(m\), respectively, represent the side tower and the mid tower. When the sag effect is not considered, the anchorage position with the maximum constraint stiffness of the crossing cable is located near 0.7 times the span length from the mid tower [17]. Under actual loading conditions, stay cables develop sag along the span due to their self-weight [24]. The presence of sag causes the stay cables to deviate from a straight-line configuration, leading to a reduction in their axial stiffness. As the cable length increases, the sag effect exerts a more pronounced influence on the axial stiffness of stay cables [25; 26]. The Ernst formula [27] accounts for the sag effect by modifying the elastic modulus of the cable. The equivalent elastic modulus is calculated as shown in Equation (2), and this formula has been widely adopted [28; 29; 30]. \[E^{\prime}=\frac{E}{1+\frac{\gamma^{2}d^{2}E}{12\sigma^{3}}} \tag{2}\]  
where \(\gamma\) represents the weight of the cable, \(d\) denotes the horizontal projected length of the cable, \(E\) is the elastic modulus of the cable, and \(\sigma\) is the initial stress of the cable. Under dead load, the stress level of the crossing cable is \(\sigma\). Substituting Equation (2) into Equation (1), the longitudinal constraint stiffness \(k\) of the crossing cable considering the sag effect can be obtained, which is expressed as follows:  
\[k=\frac{12 Ea^{2}\sigma^{3}}{\frac{\left(h^{2}+\left(L-a\right)^{2}\right)^{ \gamma_{1}}\left(12\sigma^{3}+\gamma^{2}E(L-a)^{2}\right)}{A_{s}}+\frac{\left( h^{2}+a^{2}\right)^{\gamma_{1}}\left(12\sigma^{3}+\gamma^{2}Ea^{2}\right)}{A_{m}}} \tag{3}\]  
Let the ratio of the cross-sectional areas of the mid-tower crossing cable and the side tower crossing cable be \(A_{m}/A_{s}=t\), where \(A_{s}=A\). Equation (3) can be expressed as follows:  
\[k=\frac{12 EAa^{2}\sigma^{3}}{\left(h^{2}+\left(L-a\right)^{2}\right)^{\gamma _{2}}\left(12\sigma^{3}+E\gamma^{2}(L-a)^{2}\right)+\frac{\left(h^{2}+a^{2} \right)^{\gamma_{2}}\left(12\sigma^{3}+E\gamma^{2}a^{2}\right)}{t}} \tag{4}\]  
The ratio of tower height to main span length, denoted as \(h/L=\alpha\), is referred to as the height-to-span ratio. The ratio of the length \(a\) from the crossing cables to the mid tower  
Figure 2: Diagram of crossing cable force change.