## 5 Conclusions  
This study verifies the application effect of crossing cables in super-kilometer multi-tower cable-stayed bridges, and provides a reasonable structural scheme for long-span cable-stayed bridges. The main conclusions are as follows:  
1. The analytical formula of the longitudinal constraint stiffness of crossing cables based on the equivalent elastic modulus has high accuracy and can be used to determine the anchorage position with the maximum constraint stiffness of the crossing cables in long-span three-tower cable-stayed bridges. 2. As the main span length increases, the sag effect becomes significant, causing the anchorage position with the maximum constraint stiffness of the crossing cables to shift toward the mid span. Taking a three-tower cable-stayed bridge with a height-to-span ratio of 0.25 as an example, when the cable stress level reaches 600 MPa, expanding the span from 700 m to 1500 m reduces the anchorage span ratio of the crossing cables from 0.69 to 0.63. Increasing the initial stress of the crossing cables helps improve their constraint stiffness, while the anchorage position with maximum constraint stiffness moves closer to the side towers. 3. When the steel volume of the crossing cable is constant, increasing the ratio of the crossing cable area between the mid tower and the side tower can make full use of the restraining effect of the ordinary cables near the crossing cable of the side tower on the deck, so as to improve the restraining effect of the crossing cable on the mid tower. 4. For multi-tower cable-stayed bridges with main spans exceeding 1000 m, the installation of crossing cables can still effectively enhance structural stiffness and reduce bending moments of the tower bottom. In the case study of this paper, after adding six pairs of crossing cables to three-tower cable-stayed bridges with main spans of 1000 m and 1500 m, the vertical displacement of the deck decreased by 55% and 40%, respectively, while the bending moment at the base of the mid tower was reduced by 44% and 46%. 5. When the tower height below the deck is increased, the crossing cables can still effectively improve the deformation of the deck and the bending moment of the mid tower. Therefore, for a three-tower cable-stayed bridge built in the deep-water area, the setting of crossing cables in the main span is still an effective measure to improve the structural stiffness. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Main Span**} & \begin{tabular}{c} **Tower Height** \\ **Below Deck** \\ \end{tabular} & \begin{tabular}{c} **0 Pairs of** \\ **Crossing Cables** \\ \end{tabular} & \
\begin{tabular}{c} **6 Pairs of** \\ **Crossing Cables** \\ \end{tabular} &
\begin{tabular}{c} **Bending Ree** \\ \end{tabular} \\ \hline \multirow{2}{*}{\(L=1000\) m} & 70 & 629.46 & 351.49 & 44.16\% \\  & 120 & 436.87 & 275.83 & 36.86\% \\  & 170 & 304.71 & 220.75 & 27.55\% \\ \hline \multirow{2}{*}{\(L=1500\) m} & 70 & 5190.73 & 2778.71 & 46.47\% \\  & 120 & 4443.35 & 2347.47 & 47.17\% \\ \cline{1-1}  & 170 & 3816.81 & 1992.38 & 47.80\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Bending moment of tower bottom under different tower heights (MN.m). Future work will explore the local stress situation of crossing cable three-tower cable-stayed bridges, with a focus on key positions such as crossing cable anchorage zones and bridge decks, and optimize local structural design by establishing refined solid models. Conceptualization, S.C.; Methodology, S.C. and K.H. ; Software, K.H. ; Validation, S.C.; Formal analysis, K.H. ; Investigation, X.W. ; Writing--original draft, K.H. ; Writing--review & editing, S.C. and X.W. ; Supervision, X.W. ; Funding acquisition, S.C. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (Grant No. 52178166). The authors express their gratitude for the financial assistance. Not applicable. The original contributions presented in this study are included in the article.