## 1 Introduction  
The cable-stayed bridge is one of the commonly used bridge types among long-span bridges. Since the birth of the first modern cable-stayed bridge, the Stromsend Bridge, it has undergone a transformation from a sparse cable system to a dense cable system [1]. With the application of high-strength materials, the spanning capacity of cable-stayed bridges has been continuously improved, evolving from an initial main span of 182.6 m to the current 1208 m. Some representative long-span cable-stayed bridges include the following; The Sutong Yangtze River Highway Bridge (2008) [2] is the world's first cable-stayed bridge with a span exceeding 1000 m, featuring a main span of 1088 m and a cable strength of 1770 MPa. The Russky Island Bridge (2012) [3] holds the record for the world's largest span of a highway cable-stayed bridge, with a main span of 1104 m. The Changtai Yangtze River Bridge (expected to be completed by 2025) [4] boasts the world's largest span for a dual-purpose highway and railway cable-stayed bridge, spanning 1208 m and featuring a strengthened cable of 2100 MPa. Another notable railway cable-stayed bridge is the Ma'anshan Yangtze River Bridge (expected to be completed by 2026) [5], a three-towerstructure with a main span of 2 \(\times\) 1120 m. Compared to double-tower cable-stayed bridges, multi-tower cable-stayed bridges have greater spanning capacity, but they suffer from issues such as low structural rigidity [6]. The development trend of cable-stayed bridges is towards larger spans, and taking effective measures to enhance structural stiffness is a key technical issue in designing large-span multi-tower cable-stayed bridges. The mid tower of multi-tower cable-stayed bridges lacks the longitudinal constraint of backstays, resulting in insufficient overall structural resistance to deformation. Under live load, the deformations of the deck and towers are relatively large [7]. Measures to enhance the structural rigidity of multi-tower cable-stayed bridges primarily fall into two categories [8; 9]: one is to increase the bending stiffness of the bridge towers; the other is to add stiffening cables to the main span. The Rion-Antirion Bridge [10] (with a main span of 3 \(\times\) 560 m), the Millau Bridge [11] (with a main span of 6 \(\times\) 342 m), and the Ma'anshan Yangtze River Bridge have adopted the first measure, which involves increasing the bending stiffness of the bridge towers to reduce the displacement of the towers and deck. The second measure mainly involves two schemes: setting stabilizing cables at the mid tower and setting crossing cables at the main span. The Ting Kau Bridge [12] (with two main spans of 448 + 475 m) has stabilizing cables set up along the bridge direction on both sides of the mid tower to increase the rigidity of the mid tower. The stabilizing cables at the mid tower need to span the entire main span. When the main span is long, its sag effect becomes more pronounced, greatly reducing its restraining effect. The Queensferry Bridge [13] (with two main spans of 2 \(\times\) 650 m) and the Huangmaohai Bridge [14] (with two main spans of 2 \(\times\) 720 m) have crossing cables installed at main spans to enhance the overall rigidity of the structure, which proves that this method can effectively improve the structural stiffness. However, as the length of the main span increases, the sag effect of the cables will reduce the axial stiffness of the crossing cables. For three-tower cable-stayed bridges with main spans up to the kilometer scale, there is still a lack of relevant research on the influence of the crossing cable on the structural performance and its optimal setting method. [PERSON] [9] proposed a scheme to enhance the structural rigidity of multi-tower cable-stayed bridges by installing crossing cables on the main span. This scheme redistributed the dead load of the deck among the crossing cables, thereby providing effective longitudinal restraint to the mid tower.