## 1 Introduction  
The suspension rod is a critical component of medium and under-bearing arch bridges, and accurately measuring the cable strength of these rods is crucial during the bridge construction process, directly impacting structural safety. In the testing of cable forces for large-span steel box-stayed arch bridges, common methods include the hydraulic pressure gauge method, pressure sensor method, magnetic flux method, and frequency method [1]. Among these, the frequency method is widely preferred in engineering due to its economic feasibility, simplicity, and practical benefits. In practical applications, several factors influence the accuracy of hanger cable force measurements, such as the bending stiffness of the hanger, boundary conditions, linear density, and vibration dampers. At present, many scholars have studied and analyzed the hanger cable force: [PERSON] et al. [2] proposed a practical formula for segmental fitting of the cable force solution for inclined cables; [PERSON] et al. [3] took into account the effect of cable drape and bending stiffness on the frequency and established a practical formula for calculating the cable force using the fundamental frequency by adopting the energy method and the curve-fitting method; and [PERSON] et al. [4], and [PERSON] et al. [5] took into account the boundary conditions to derive the theoretical and practical formulas for sling tension. However, the application scope of these early traditional calculation formulas is limited, and their accuracy is inadequate. Scholars have enhanced the precision of cable force calculations by integrating computer numerical models, thereby further augmenting the accuracy of such calculations. [PERSON] et al. [6] considered the influence of vibration-damping devices and applied the additional mass method and genetic algorithm to identify the hanger tension; [PERSON] et al. (2017) used the principle of minimizing the bending strain energy of the structure to accurately calculate the reasonable rope force of the cable-stayed bridge at the bridge completion stage based on the elasticly supported continuous girder model; [PERSON] et al. (2018) introduced the influence matrix method into the tie arch bridge to establish the construction and bridge completion stage. The optimization model reduces the number of hanger tensioning times and obtains the hanger tension force. [PERSON] et al. (2019) obtained the wave component coefficients based on the Timoshenko beam model by fluctuation analysis and least squares fitting d and identified the cable force and flexural stiffness with the objective of minimizing the fitting residuals; [PERSON] et al. (2019) introduced the boundary condition coefficients by combining the experimental determination of the dynamic properties and the correction of the numerical model and determined the cable force of the hanger based on the modal vibration pattern of the hanger; [PERSON] et al. (2019) developed a computer-aided calculation algorithm by means of a computational model of the sling to determine the required tension and its corresponding deformation during tensioning of the wire, discussing two tensioning methods and demonstrating the effectiveness of the proposed procedure by means of a numerical example. Despite the integration of numerical simulations and computational models, which has resulted in some enhancement in the accuracy of cable force calculations, there are still certain limitations and scopes of application that necessitate consideration. In recent years, the rapid development of artificial intelligence has had a significant impact on all aspects of human daily life. As a result, machine learning is gradually being applied to the field of bridge engineering ([PERSON] et al., 2018; [PERSON] et al., 2019; [PERSON] et al., 2020) and has achieved good results with a high degree of accuracy and wide scope of application. [PERSON] et al.