## 6 Conclusions  
To enhance the accuracy of predicting hanger cable-stay forces in large-span steel box concrete tie arch bridges during practical engineering and to address issues arising from low measurement accuracy due to construction interference and external factors at the bridge completion stage, this paper proposes a prediction method based on the QPSO-optimized RBF-SVM model. The developed machine learning model significantly improves hanger cable force prediction accuracy, meeting practical engineering requirements. The following conclusions were drawn:  
Figure 9: Comparison of the predicted and measured values of the cable tension for each algorithm in the test set.  
(1) Utilizing the principle of energy conservation, the Rayleigh method was employed to derive the calculation formula for hanger cable force under diverse boundary conditions. Essential parameters of hanger cable-stay forces were measured during the completion stage of large-span steel box tie arch bridges in actual projects. A prediction method was then established using an optimized radial basis function kernel of the quantum particle swarm optimization algorithm combined with a support vector machine regression model.  
(2) The actual measurement of the collected hanger length, line density, frequency, and bending stiffness as the feature input, combined with the finite element model to derive the measured value after the QPSO-RBF-SVM algorithm model training in the test set of the predicted value of the cable force and the measured value of the average relative deviation of 0.65%, the coefficient of determination was 0.97, the maximum error was only 2.42%, the overall error did not exceed 5%.  
(3) Compared with cable force calculation formulas derived from the energy method, machine learning demonstrates superior prediction accuracy across all scenarios. Within the same dataset, the QPSO-RBF-SVM algorithm model outperforms the PSO-SVM algorithm model on all evaluation indices. Specifically, the PSO-SVM model shows a coefficient of determination of 0.58 and an average relative error of 2.47%, indicating lower prediction accuracy and goodness of fit compared with the QPSO-RBF-SVM algorithm model.  
(4) In the actual project, in order to solve the problem of low accuracy of the hanger cable-stay force measurement of the large-span steel box tie arch bridge due to various factors during the bridge completion stage, a cable-stay force prediction method based on the QPSO-RBF-SVM algorithm model is proposed, which effectively reduces the cable-stay force error caused by the external environment and provides a new way of thinking for the measurement of the hanger cable-stay force of the arch bridge to safeguard the structure of the bridge and the safety of the construction. Looking forward, in consideration of the intricate stress conditions experienced by the short hanger, which concentrates the deformation resulting from both the temperature and force of the arch ribs and further bears a greater degree of shear deformation compared with the long hanger, rendering it significantly more susceptible to the impact of dynamic loads, it becomes imperative to conduct further research into the problem of predicting cable forces for the short hanger.  
Writing--original draft preparation, software, H.S.; writing--review and editing, Conceptualization, methodology, M.S.; validation; data curation, software, W.X. All authors have read and agreed to the published version of the manuscript.  
This study is supported by the National Key Scientific Project of China (No. 51978249).  
Not applicable.  
The data used in this study can be obtained from the corresponding author.  
The authors want to thank the editor and anonymous reviewers for their valuable suggestions for improving this paper.  
The authors declare no conflict of interest.