Cable Tension of Long-Span Steel Box Tied Arch Bridges Based on Radial Basis Function-Support Vector Machine Optimized by Quantum-Behaved Particle Swarm Optimization

Abstract

To investigate hanger force during the construction phase of large-span steel box tie arch bridges, the challenge of low accuracy in force identification due to multifactor coupling was addressed. An energy method was employed to derive formulas for calculating forces under different boundary conditions. Utilizing the QPSO-RBF-SVM machine learning algorithm model, predictions of bridge formation stage forces were conducted, integrating findings from actual engineering case studies. Error analysis on hanger force was performed, revealing that the quantum particle swarm optimization (QPSO) algorithm optimizes parameters in the radial basis function support vector machine (RBF-SVM). The model was trained on datasets, achieving an average relative error of 0.65% in predicted cable force values compared with measured values in the test set, with a coefficient of determination of 0.97. These results demonstrate superior accuracy compared with calculations derived from the energy method and other machine learning algorithms. This algorithmic model presents a promising approach for accurately assessing cable forces in large-span steel box tie arch bridges.

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: Zui et al. [2] proposed a practical formula for segmental fitting of the cable force solution for inclined cables; Ren 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 Liu et al. [4], and Chen 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. Yuan et al. [6] considered the influence of vibration-damping devices and applied the additional mass method and genetic algorithm to identify the hanger tension; Fu et al. [7] 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; Wang et al. [8] 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. Xiao et al. [9] 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; Ereiz et al. [10] 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; Šurdilović et al. [11] 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 [12,13,14] and has achieved good results with a high degree of accuracy and wide scope of application. Li Rui et al. [15] used a particle swarm algorithm to solve the beyond-frequency analytical equations of two-end cemented hangers to identify the cable force and bending stiffness of the hangers; Xie et al. [16] used a multilayer perceptron and long-short-term memory network to learn the nonlinear mapping between the arch rib line shape and the cable force, and utilized a DNN model to predict the desired construction cable force corresponding to the target line shape; Bai et al. [17] used a QPSO- SVM model, which accurately predicted the contact resistance increment and mass loss of SF6 circuit breakers; Ouyang et al. [18] identified the sensorless cable-stayed bridge cable-stayed force through a back-propagation neural network in combination with a finite element model; Gai et al. [19] optimized the generalized regression neural network using a sparrow search algorithm and combined it with finite element simulation to predict the cable force of the bridge, which is better compared with other algorithms. Luo et al. [20] used the BPVMM technique to amplify the small vibration of the cable, used the line tracking algorithm to measure the cable displacement of the amplified video, and then obtained the frequency of the cable to find out the cable force.

In summary, while deep learning models demonstrate strong performance in predicting bridge cable forces, their efficacy is contingent upon larger datasets and longer training times. Additionally, other algorithms face challenges related to limited model generalization ability and high computational complexity. In contrast, the QPSO-SVM model exhibits unique advantages, including stronger generalization ability, higher computational efficiency, and better robustness. As such, it holds great promise for application in the field of bridge cable force prediction. In practical engineering, the design and construction of sub-bearing cable-stayed arch bridges present numerous challenges, particularly with regard to the accuracy of cable force calculations, which directly impact bridge safety and stability. The traditional formulas for calculating cable forces often result in significant errors due to the complex interplay of factors such as materials, environment, and construction conditions on bridge structures. This study proposes a novel cable force prediction method based on QPSO-RBF-SVM, offering a fresh perspective for identifying cable forces in arch bridges and effectively reducing testing errors.

2. Derivation of Theoretical Formulae for Flexible Hanger

To illustrate the universality of the formula, take any of the inclined flexible hanger cables shown in the following Figure 1: L represents the length of the hanger, T denotes the tension applied to the hanger in the x-axis direction, θ indicates the angle of inclination of the hanger, θ is 0 when the hanger is vertical, s signifies the droop, and L₀ is defined as the vertical span of tie.

According to Rayleigh’s [21] method based on the law of conservation of energy, the following can be shown:

$$E_0+V_0=C=E_{\max }=V_{\max },$$

(1)

where E₀ and V₀ are the kinetic and potential energies at a certain time, C is a constant, and E_max and V_max are the maximum values of kinetic and potential energy. Assuming the first-order vibration function of the structure:

$$y(x,t)=\phi (x)\cos (\omega t+\theta ),$$

(2)

where φ(x) is the mode shape function, and ω is the natural frequency. Equation (2) takes the first-order derivative with respect to t, which yields the following:

$$y^{\prime }(x,t)=-\omega \phi (x)\cos (\omega t+\theta )$$

(3)

The total kinetic energy of the structure is the following:

$$E_0={\displaystyle \frac{1}{2}}{{\displaystyle {\int }_0^{1}m(y^{\prime })}}^{2}dx,$$

(4)

where m is the linear density of the hanger.

The total potential energy of the structure is the following:

$$V_0={\displaystyle \frac{1}{2}}{{\displaystyle {\int }_0^{l}EI\left[{\phi }^{″}(x)\right]}}^{2}dx+{\displaystyle \frac{1}{2}}{{\displaystyle {\int }_0^l(T+mgx\sin \theta \left[{\phi }^{\prime }(x)\right]}}^{2}dx$$

(5)

Then, the maximum value of kinetic energy is the following:

$$E_{\max }={\displaystyle \frac{1}{2}}{\omega }^2{\displaystyle {\int }_0^{l}m\left[\phi {(x)}^2\right]}dx$$

(6)

The maximum value of the potential energy of the structure is the following:

$$V_{\max }={\displaystyle \frac{1}{2}}{{\displaystyle {\int }_0^{l}EI\left[{\phi }^{″}(x)\right]}}^{2}dx+{\displaystyle \frac{1}{2}}{{\displaystyle {\int }_0^{l}T\left[{\phi }^{\prime }(x)\right]}}^{2}dx$$

(7)

The joint Equations (1), (4) and (5) are obtained as follows:

$${\omega }^2={\displaystyle \frac{{{\displaystyle {\int }_0^{l}EI\left[\phi (x)\right]}}^{2}dx+{{\displaystyle {\int }_0^l(T+mgx\sin \theta )\left[{\phi }^{\prime }(x)\right]}}^{2}dx}{{{\displaystyle {\int }_0^{l}m\left[\phi (x)\right]}}^{2}dx}}$$

(8)

When the boundary conditions at both ends are different, the vibration mode function of the structure is also different, and the formula for calculating the cable force under different boundary conditions is analyzed as follows:

(1)

Articulated at both ends

The vibration function is similar to that of a simply supported structure and can be expressed as follows:

$${\phi }_n\left(\mathrm{x}\right)=\sin {\displaystyle \frac{n\pi \mathrm{x}}{l}}$$

(9)

By substituting ${\omega }_n=2\pi f_n$ and the first and second derivatives of Equation (9) with respect to x into Equation (8) yields the following:

$$T={\displaystyle \frac{4m{f}_n^2{l}^2}{n^2}}-EI{\displaystyle \frac{n^2{\pi }^2}{l^2}}-{\displaystyle \frac{wgl\sin \theta }{2}},$$

(10)

where f_n is the nth-order fundamental frequency.

(2)

Articulated at one end and fixed at the other

Since it is difficult to find the higher-order vibration pattern of the structure, the first-order vibration pattern is set as [22]:

$${\phi }_1(x)=\{\begin{array}{l}1-\cos {\displaystyle \frac{\pi x}{l}};x\in \left[0,{\displaystyle \frac{l}{2}}\right]\\ \sin {\displaystyle \frac{\pi x}{l}};x\in \left[{\displaystyle \frac{l}{2}},l\right]\end{array}$$

(11)

Substitute Equation (11) into Equation (8) to obtain the following:

$$T={\displaystyle \frac{4m{f}_1^2{l}^2}{1.376}}-EI{\displaystyle \frac{{\pi }^2}{l^2}}-1.203g\sin \theta {\displaystyle \frac{l}{2}}$$

(12)

(3)

Fixed at both extremities

The first-order mode can be assumed as follows:

$${\phi }_1(x)=\{\begin{array}{l}1-\cos {\displaystyle \frac{\pi x}{l}};x\in \left[0,{\displaystyle \frac{l}{2}}\right]\\ 1+\cos {\displaystyle \frac{\pi x}{l}};x\in \left[{\displaystyle \frac{l}{2}},l\right]\end{array}$$

(13)

Substituting Equation (13) into Equation (8) yields the following:

$$T={\displaystyle \frac{4m{f}_1^2{l}^2}{2.205}}-EI{\displaystyle \frac{{\pi }^2}{l^2}}-g\sin \theta {\displaystyle \frac{l}{2}}$$

(14)

Equations (10), (12) and (14) are suitable for most engineering requirements. However, in practical projects, the first-order frequency of the hanger is often measured due to its complex vibration pattern and susceptibility to environmental disturbances, making it difficult to capture higher-order frequencies. As a result, the use of energy methods is significantly limited. Additionally, vibration dampers are typically installed at the fixed end of the hanger to collect vibration energy under dynamic loading and prevent fatigue damage. This installation affects frequency measurement accuracy, hanger stiffness, and calculated length, leading to errors in cable force measurement. A machine learning model can effectively capture this nonlinear relationship. The QPSO-RBF-SVM model is employed to predict hanger cable force during bridge formation stages with high accuracy while handling nonlinear relationships and avoiding errors caused by frequency order and boundary conditions.

3. QPSO-RBF-SVM Model

3.1. RBF-SVM Modeling

Support vector machine (SVM) is a statistical-based machine learning algorithm that effectively addresses the challenges of small sample sizes and nonlinearity [23]. SVM achieves linear separability in high-dimensional space by nonlinearly mapping the input samples from low-dimensional space to high-dimensional space and then constructs classification or regression functions.

The set of nonlinear samples is assumed to be $\left\{x_i,y_i\right\}(i=1,2,\dots ,m)$, where x_i represents the input vector of the i-th sample and y_i is the output; a loss function is introduced to address the equation:

$$f\left(x,\omega \right)=\left(w,\phi \left(x\right)\right)+b$$

(15)

The weight vector is denoted by $w$ in the formulations, while the bias is denoted by b.

To solve the optimization problem of finding the maximum hyperplane with respect to the loss function, the problem description formula is as follows:

$$\min \left\{{\displaystyle \frac{1}{2}}||w|{|}^2+c{\displaystyle \sum _{i=1}^m\left({\xi }_i+{\xi }_i^{\prime }\right)}\right\},$$

$$s.t.\{\begin{array}{l}{y}_i-wk\phi \left(x\right)-b\le {\xi }_i+\epsilon \\ wk\phi \left(x\right)+b-y_i={\xi }_i^{\prime }+\epsilon \\ {\xi }_i\ge 0,{\xi }_i^{\prime }\ge 0\end{array},$$

(16)

where c represents the penalty factor, ${\xi }_i,{\xi }_i^{\prime }$ is the relaxation factor, $\epsilon$ denotes the loss function, and k stands for the kernel function.

By introducing the Lagrangian factor ${\alpha }_i,{\beta }_i$ and kernel function k, we formulated the optimization problem as follows:

$$\max W\left({\alpha }_i,{\beta }_i\right)={\displaystyle \sum _{i=1}^n{y}_i\left({\alpha }_i-{\beta }_i\right)}-\epsilon {\displaystyle \sum _{i=1}^n\left({\alpha }_i+{\alpha }_i^{\prime }\right)}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1,j=1}^n\left({\alpha }_i-{\alpha }_i^{\prime }\right)}\left({\alpha }_j-{\alpha }_j^{\prime }\right)k\left(x_i-x_j^{\prime }\right)$$

$$s.t.\{\begin{array}{l}{\displaystyle \sum _{i=1}^n{\alpha }_i={\displaystyle \sum _{i=1}^n{\alpha }_i^{\prime }}}\\ {\alpha }_i\ge 0,{\alpha }_i^{\prime }\le c\end{array}$$

(17)

In this equation, W represents the dual problem corresponding to the original optimization problem.

Revised sentence: Support vector machine regression (SVR) is a crucial component of support vector machines, which utilizes kernel tricks to transform a low-dimensional linear regression problem into a high-dimensional nonlinear regression problem. It aims to minimize the loss function in order to fit the data while maximizing the margin between the predicted value and the actual value. This transformation results in a convex optimization problem that needs to be solved. The generalization ability of SVR heavily relies on the selected kernel function [24]. Common types of kernel functions include Linear Kernel Function, Polynomial Kernel Function, Multi-layer Perception Kernel Function, Radial Basis Function Kernel Function, and Convex Optimization Problem.

The RBF kernel function was utilized in this paper due to its superior capability for local approximation and better handling of nonlinearities. Its expression is given by the following:

$$K\left(x,x^{\prime }\right)=\exp \left(-\gamma {\Vert x-x^{\prime }\Vert }^2\right),$$

(18)

where x and $x^{\prime }$ are sample points, and $\gamma$ represents the width parameter of the RBF kernel.

3.2. QPSO Algorithm

The quantum particle swarm optimization (QPSO) algorithm simulates the behavior of particles in a quantum state to collaboratively find the global optimal solution, building upon the PSO algorithm, which is derived from the migration and aggregation phenomenon observed in bird foraging processes [25]. QPSO overcomes the drawbacks of local optimization and convergence speed associated with PSO, thereby enhancing parameter searching effectiveness. The particle state in QPSO is described using wave function with the expression:

$$\{\begin{array}{l}{X}_i\left(\mathrm{t}+1\right)=P_i\left(t\right)+\alpha \left|Mbes{t}_i\left(t+1\right)-X_i\left(\mathrm{t}\right)\right|\times \ln \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$u$}\right.}\right),k\ge 0.5\\ X_i\left(\mathrm{t}+1\right)=P_i\left(t\right)-\alpha \left|Mbes{t}_i\left(t+1\right)-X_i\left(\mathrm{t}\right)\right|\times \ln \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$u$}\right.}\right),k<0.5\end{array},$$

(19)

$$\alpha =w_{\max }-{\displaystyle \frac{iter}{ite{r}_{\max }}}\times w_{\min },$$

(20)

$$Mbes{t}_i\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j-1}^N{p}_j\left(t-1\right)},$$

(21)

$$P_i\left(t\right)=P_i\left(t-1\right)+\left(1-\beta \right)p_j\left(t-1\right),$$

(22)

in which $X_i\left(t\right)$ is the position of the i-th particle in generation t, $\alpha$ is the contraction coefficient, $w_{min}$ and $w_{max}$ are the minimum and maximum inertia weights, respectively, iter is the number of current iterations, $Mb\mathrm{e}s{t}_i\left(t\right)$ is the best position of all particles in generation t − 1, N is the number of particles in the population, and $P_i\left(t\right)$ is the optimal position of the i-th particle in generation t.

3.3. Model Construction Steps

The flow chart of the QPSRBF-SVM model is depicted in Figure 2.

(1) Preprocess the collected bridge sample data by removing any ambiguous, erroneous, or incomplete entries to ensure data quality and consistency. Subsequently, apply feature scaling and feature expansion to address scale differences between features and then split the data into training and testing sets.

(2) Evaluate each particle’s position by training a Support Vector Regression (SVR) model and calculating the Mean Squared Error (MSE) for all particles as their fitness values. The particle with the lowest MSE is selected as the current global best parameter configuration.

(3) Update each particle’s position by computing the update amount based on the difference between the particle’s current position and the global best position, incorporating a random factor to determine the update direction. This update amount is combined with the global best and individual best positions to adjust and determine the new particle positions. To ensure that all particle parameter values remain within the predefined range, normalization is applied.

(4) After completing all iterations of the QPSO algorithm, identify the optimal parameters by minimizing the MSE on the training data. Use these optimal parameters to retrain the model on the training set, further enhancing the model’s performance and ensuring that the obtained parameter configuration performs optimally on the training data.

(5) Utilize the optimized SVR model to make predictions on the test set. Calculate the mean relative error and the coefficient of determination (R²) of the predictions and evaluate the model’s performance on the test set. This process helps validate the effectiveness and generalization capability of the QPSO-RBF-SVM model and provides the final prediction results.

4. Modeling and Prediction Results

4.1. Project Overview

The engineering background of the study was a large-span steel-box concrete arch bridge with a span arrangement of 90 + 24.24 m and a bridge width of 43–47 m. The main bridge features a 90 m inverted-arch steel box girder with simple-span support, while the south side of the bridge utilizes 24.24 m pre-stressed concrete cast-in-place box girders. The main arch curve follows a second-order parabola, with a vertical height of 17.44 m and a vertical-horizontal ratio of 1/5. An auxiliary arch is positioned on the outside of the main arch at an angle of 11°, featuring an arch curve projection onto the vertical plane as a second-order parabola, with a vertical height of 21.8 m and a vertical-horizontal ratio of 1/4. There are also 17 horizontal link beams between the main and auxiliary arches, spaced at intervals of 3 m without any horizontal connection system between the two arch ribs. There are a total of 11 pairs of cables, with a longitudinal cable spacing of 6 m and a horizontal anchorage point spacing of 35.8 m. The cables are made of 7-mm galvanized parallel steel wire finished cables and are tensioned at one end.

The MIDAS CIVIL 2022 software was utilized for bridge modeling and calculation; the main beam was simulated by using double main beams and transverse through the diaphragm; the hanger used cable elements, which are only tensile elements, the main beam, diaphragm, main arch and auxiliary arch use beam elements, the sidewalks on both sides of the bridge use plate elements, and the general support simulation supports. The whole bridge has 835 nodes and 950 elements. The bridge model is shown in Figure 3. The side diagram of the bridge and the number of the hanger are shown in Figure 4.

4.2. Data Collection

This method utilizes the relationship between cable force and cable vibration frequency. In a known length of cable with fixed ends, mass distribution, and other parameters, a highly sensitive sensor is attached to the hanger to capture cable vibrations induced by ambient vibrations. The captured vibration signals undergo filtering, signal amplification, A/D conversion, and spectral analysis to determine the self-oscillation frequency of the hanger. By correlating this frequency with cable force, the actual cable force can be derived.

To ensure accuracy, multiple measurements were conducted on the 11 pairs of hangers on the bridge. Several datasets were collected and averaged to obtain the precise cable strength values for the bridge hangers. Figure 5 illustrates the installation of the JMM-268 dynamic measuring instrument (Kingmach Measurement & Monitoring Technology Co., Ltd., Changsha, China) by site construction personnel for cable force measurements.

4.3. Data Processing and Model Building

During the construction process of a large-span steel box concrete tie arch bridge, it is susceptible to external environmental factors, material parameters, and construction techniques, leading to discrepancies between the actual state and the design state of the bridge. Variations also exist between the measured values of bridge stress, arch rib deflection, and hanger cable strength compared with their design values. The basic parameters of the hanger before tensioning are presented in

Table 1. Bridge foundation parameters.

Spreader Bar Number	Length L (m)	Linear Density (kg/m)	Elastic Modulus E (kN/mm2)	Theoretical Cable (kN)
D1	7.149	42.20	195	1196
D2	10.100	42.20	195	1276
D3	12.385	42.20	195	1288
D4	14.014	42.20	195	1287
D5	14.990	42.20	195	1285
D6	15.312	42.20	195	1285
D7	14.981	42.20	195	1285
D8	13.994	42.20	195	1287
D9	12.348	42.20	195	1288
D10	10.046	42.20	195	1276
D11	7.102	42.20	195	1195

; as the bridge is symmetric in structure, only one side of the hanger data are provided. To predict the hanger cable force at the completion stage, this study utilized a dataset comprising 10 groups of 90 sample data measured after lifting the bridge; D1 and D11 hangers were excluded due to their short length. Input parameters for modeling include hanger length, measurement frequency, line density, and elastic modulus—all significantly impacting hanger cable force. Fuzzy data were cleaned by addressing poor frequency identification resulting from excessively short hangers.

The measured fundamental parameters of the bridge were integrated with the finite element model to obtain the sample dataset, and the cable force values obtained from the finite element simulation model were utilized as the actual measured values. The modeling environment employed was Windows 10 22H2 operating system with Intel (R) Core (TM) i5-9300 H CPU and 8 G RAM. python 3.8 is used as the environment setting.

As shown in Figure 2, the measured sample data were partitioned into training and test sets. The first nine sets of 81 sample data were used as training samples, while the remaining one set of nine sample data was utilized as test samples. Prior to model training, feature scaling and polynomial processing were conducted on the dataset. Initially, the original data underwent feature scaling using StandardScaler to normalize the features by calculating their mean and standard deviation, resulting in a mean value of 0 and a standard deviation of 1. Subsequently, the normalized features were transformed using PolynomialFeatures to create quadratic features that include squares of the original features and their products, thereby expanding the feature space and enhancing model fitting for nonlinear relationships. The model was trained by minimizing mean square error on the training set, followed by running the QPSO algorithm to determine optimal parameters (penalty factor c and kernel width γ) for the SVR model. These optimal parameters were then assigned to the RBF-SVR model for prediction.

4.4. Prediction Results of Cable Force

Following the flowchart for cable force prediction, as depicted in Figure 2, the QPSO-RBF-SVM algorithm model was trained 100 times, resulting in a stable and significantly low mean square error between the output value and measured value. The test set data were thoroughly analyzed and processed, with detailed findings presented in

Table 2. Test set data prediction results.

Data Number	Actual Measured Value (kN)	Projected Value (kN)	Relative Error
1	1261.89	1268.57	0.53%
2	1275.78	1274.01	−0.14%
3	1348.07	1341.01	−0.52%
4	1303.29	1300.34	−0.23%
5	1347.37	1339.11	−0.61%
6	1342.61	1333.67	−0.67%
7	1170.76	1169.54	−0.10%
8	1412.81	1403.64	−0.65%
9	1228.15	1257.88	2.42%

.

The results depicted in Figure 6 demonstrate the predictions generated using the QPSO-RBF-SVM model. The close match between the predicted and true values is clearly demonstrated, highlighting the high accuracy achieved by the model. The average relative error on the test set was only 0.65%, indicating that the model maintains an extremely low error on the test set.

It is also worth noting that the coefficient of determination R² value ranges from 0 to 1, quantifying the proportion of variability in the dependent variable that can be explained by the independent variables in the model, and the closer it is to 1, the better the fit is. The coefficient of determination R² value for the QPSO-RBF-SVM model was 0.97, indicating that the model fits the data very well and exhibits excellent performance. The high correlation between predicted and actual values indicates not only the accuracy of the model but also the reliability of the model in capturing potential patterns in the data set.

In order to better illustrate the predictive performance of the model, Figure 7 displays the relative error diagrams comparing the predicted and measured values of the hanger cable force on the test set. It is evident that the maximum error of the hanger cable force is 2.42%, and the overall error falls within 5%. This indicates that the model’s prediction accuracy meets the requirements for controlling hanger cable force, and its results strongly support the reliability of the practical application.

5. Comparative Analysis

5.1. Comparison with Formula

The reliability and accuracy of the QPSO-RBF-SVM model for predicting cable force were comparatively analyzed by comparing it with the formulae derived from the energy method and the magnetic flux method [26]. The basic parameters corresponding to the samples in the test set were substituted into the formula to calculate the value of cable force, and the results are presented in Figure 8.

According to Figure 8, it can be observed that the results of the different methods in the calculation of the cable force are significantly different, among which the magnetic flux method calculates the largest error in the value of the cable force, which may be due to the external interference in the magnetic field resulting in the deviation between the measured and actual values of the magnetic flux. Compared with the magnetic flux method, the accuracy of the energy method is improved, but the prediction result still has a high error with the actual value. In contrast, the QPSO-RBF-SVM model fully considers the nonlinear characteristics of the variation of the cable force, and through the combination of the optimization algorithm and the machine learning technique, it shows higher accuracy in predicting the value of the cable force and its prediction results are closer to the actual value of the cable force.

5.2. Comparison with Other Machine Learning Algorithms

Using the other two machine learning algorithm models in the same dataset for training and testing, each algorithm’s hanger cable force prediction value and the actual cable force prediction value were compared with each other. Using the other two machine learning algorithms model in the same dataset for training and testing, each algorithm had a hanger cable force prediction value and the actual cable force value to make a comparison; the results are shown in Figure 9.

As can be seen from Figure 9, both the PSO-SVM algorithm model and the SVM algorithm model fit less well than the QPSO-RBF-SVM algorithm model, in which the SVM model has the worst fit and the average absolute error is higher than that of the other two models, whereas the error of the QPSO-RBF-SVM model is kept at a very low level and has the best fit. By comparing the results of the PSO-SVM, SVM, and QPSO-RBF-SVM algorithm models for the prediction of the cable force, it can be found that the QPSO-RBF-SVM model is able to make full use of the nonlinear characteristics of the data and find the best model parameters through the optimization algorithm, which makes the QPSO-RBF-SVM model able to fit the actual data more accurately when predicting the cable force and, at the same time, maintaining a lower error level, which provides higher accuracy and stability in the prediction of the cable force.

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:

(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.