## 1 Introduction
To clarify the performance of the crossing cable, [PERSON] [15] derived the formula for the longitudinal constraint stiffness of the crossing cable to the mid tower, and proposed a simplified calculation method for the longitudinal displacement of the mid tower and the vertical displacement of the loading span deck under uniformly distributed load for three-tower cable-stayed bridges [16]. Subsequently, based on the longitudinal constraint stiffness formula of the crossing cable, the influence of crossing cable area and anchorage position on its constraint performance was analyzed [17]. Theoretical research indicates that when the height-to-span ratio of the tower to the span length ranges from 0.2 to 0.3, the maximum constraint stiffness is attained when the crossing cable anchor is located at a distance of 0.69-0.73 times the span length from the mid tower. Based on the Queensferry Bridge, [PERSON] et al. [18] compared the economic efficiency of two crossing cable schemes, namely increasing the stay cable in the mid span and changing the cable spacing, and believed that the second scheme is more economical. [PERSON] et al. [19] removed the mid tower crossing cables on the basis of setting crossing cables at the mid span, thereby increasing the range of the side tower cable supporting the deck, forming an \" unequal-size fan\" form. This cable arrangement utilizes the strong constraint of the back cables to improve the structural stiffness. [PERSON] et al. [20] optimized the steel volume of the Queensferry Bridge by setting the area and pre-tension force of each cable as variables. The optimization results revealed that increasing the length of the crossing zone could reduce the steel volume in the cables. Subsequently, [PERSON] et al. [21] optimized the crossover cable zone length by employing the non-dominated sorting genetic algorithm, with the objectives of minimizing steel consumption in crossing cables and displacement at the tower top. A solution from the Pareto front revealed that the optimal anchorage length of crossing cables was 0.28 times the main span length. All aforementioned optimizations adopted the mid-span arrangement of crossover cables. Subsequently, [PERSON] et al. [22], based on the Queensferry Bridge case study, optimized the entire cable system. Under constraints of stiffness and mechanical performance, the optimal cable arrangement with minimal steel consumption positioned crossing cables away from the mid span. As the crossing cable zone expanded, their anchorage locations progressively moved closer to the side towers. This demonstrates that anchoring crossing cables near side towers provides greater constraint stiffness. [PERSON] et al. [23] proposed an optimization strategy that combines analytical and optimization algorithms to optimize the arrangement of crossing cables with the goal of minimizing steel consumption for cables. The optimization results confirm that asymmetric arrangement has a better effect on improving structural stiffness than symmetric arrangement. The above research analyzes the influence of the crossing cables on the mechanical properties of the structure and the optimal setting of the crossing cables, with a research background focusing on cable-stayed bridges with a main span ranging from 600 to 700 m. However, in long-span cable-stayed bridges, the sag effect of cables is significant, and the influence of crossing cables on the mechanical properties of the structure is still unclear. In Section 2 of this paper, the equivalent elastic modulus is incorporated into the constraint stiffness formula of crossing cables to account for the sag effect. Based on the modified formula, the influence of crossing cable parameters on their mechanical performance is analyzed. Section 3 introduces the finite element models of two three-tower cable-stayed bridges with different spans.