Study on Finite Element Model Modification of Long-Span Suspension Bridge Based on BPANN-GA

Abstract

In order to improve the reliability of the finite element analysis model of long-span suspension bridges, this paper proposes a finite element model (FEM) modification method by the hybrid algorithm of backpropagation artificial neural network (BPANN) and genetic algorithm (GA) based on field measurements and vibration modal analysis. First, finite element computational data is used to train the neural network. The trained neural network is then used to predict the natural frequencies corresponding to different modal shapes under various parameters. Based on the measured values, a fitness function is constructed, and the GA is used to optimize the model parameters. These optimized parameters are subsequently applied to correct the FEM of long-span suspension bridges. Finally, the computational errors of the initial model, the BPANN corrected model, and the BPANN-GA model are compared and analyzed to verify the advantages of using BPANN-GA for bridge model modification. The results show that the average computational error of the natural frequencies for the 1st to 8th modes before modification is 7.04%. After modification using BPANN-GA, the absolute value of the computational error for the 1st to 8th modes is all below 3%, and the modal assurance criterion (MAC) values all exceed 90%. Compared to the conventional BPANN, BPANN-GA can effectively improve the modification effect, with the average computational errors of natural frequencies being 3.84% and 1.50%, respectively. For static displacement, the computational error after modification is also significantly reduced, with the average computational error decreasing from 11.4% to 5.9%. After modification using BPANN-GA, the static and dynamic responses of the structure can be better reflected by the corrected model, thus providing a more reliable computational basis for understanding the safety status of the bridge.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, bridges with high piers and large spans have become prevalent in the construction of mountainous highways in China. To study the static and dynamic characteristics of these long-span bridge structures, researchers often rely on finite element methods (FEMs). However, the FEMs based on design drawings and specifications are simplified to some extent, leading to calculation results that are not entirely satisfactory. These models can only partially reflect the theoretical mechanical characteristics of bridge structures due to various uncertainties, such as construction differences, material variations, and modeling discrepancies [1, 2]. Indeed, the accuracy of a FEM in simulating real bridge structures is highly dependent on the precision of its parameters. The structural response patterns of actual bridges under environmental factors and external loads often do not align perfectly with theoretical models. To reduce simulation errors in FEMs, it is possible to minimize these errors by performing load tests on real bridge structures to obtain measured data and then applying an inverse correction to the initial FEM established based on design data. This approach provides a more reliable model foundation for bridge calculation and evaluation [3].

Regarding the objective function used for correcting bridge models, early methods primarily relied on data obtained from static or quasi-static load tests. For example, Huang [4] introduced a method that uses strain and deflection measurements to correct FEMs. Similarly, Enevoldsen et al. [5] adjusted the stiffness of a FEM for a truss railway bridge using quasi-static deformation values measured when a train passed and then employed this refined FEM to evaluate the fatigue life of the bridge under higher axle loads. Chajes, Mertz, and Commander [6] used strain measurements under truck loads to determine the sectional characteristics and constraints of a steel girder bridge by correcting the initial model. Zhang and Fan [7] used measured static data on a cantilever beam to correct the structural stiffness and further developed a structural damage identification scheme. Deng and Rex [8] selected parameters such as rotation and curvature as static load data to be corrected, established an error objective function between the model’s calculated values and analytical values and studied the impact of different parameters on the response values at different positions of the structure. In recent years, the use of vibration modal characteristics to correct FEMs has gained notable applications. For example, Daniell and Mcdonald [9] demonstrated that by using modal characteristics as the target, manually correcting the FEM of a cable-stayed bridge can better align with the characteristic frequencies observed in ambient vibration tests, thus proving the feasibility of employing nonlinear optimization methods for FEM modification. Additionally, Zong and Xia [10] proposed an objective function for correcting the FEM of a steel frame arch bridge, incorporating both dynamic modal parameters and static displacement.

In the realm of correcting FEMs for bridges, two primary types of algorithms are employed. The first type focuses on response surface surrogate models, which are instrumental in refining these models. Among the commonly used surrogate models are support vector machines, neural networks, and Kriging models, which play a crucial role in the modification process [11–13]. In addition to the response surface surrogate models, the second type of algorithm transforms FEM modification into a mathematical optimization problem. This approach utilizes suitable intelligent algorithms to perform iterative optimization and arrive at a solution. Perera, Fang, and Rui [14] introduced a bridge model modification method that integrates particle swarm optimization (PSO) and genetic algorithms (GAs). Tang, Yue, and Hua [15] developed a method for correcting the local stiffness of a bridge FEM using an artificial bee colony (ABC) algorithm. Fu et al. [16] leveraged a bridge health monitoring system and employed a whale optimization algorithm (WOA) to correct the static and dynamic characteristics of the initial FEM of the bridge. Lv [17] utilized a backpropagation artificial neural network (BPANN) for the model modification of continuous beam bridges. Bao, Qi, and Ding [18] perturbed the parameters of the initial bridge model to create multiple computational samples, used these samples to train BPANN, and estimated the optimal modification parameters.

Currently, it is common to use ANN intelligent algorithms to update FEMs of bridges. However, ANN relies on gradient descent methods for optimization searches. Given the multiobjective and nonlinear characteristics of model modification, this approach can have a slow convergence rate, unstable system training, and in some cases, may only find a local optimum rather than the global optimum. Therefore, ANN is sometimes not ideal for modification FEMs of large and complex structures, and the computational cost of model modification can be extremely high. As a result, improving ANN has become a research focus.

The significance of this paper lies in proposing a hybrid algorithm, BPANN-GA, which leverages the global search capabilities of GAs to achieve the goal of quickly and accurately obtaining the updated model parameters. The approach involves training the neural network with existing computed data, using the trained network to predict the natural frequencies of various modes under different parameters, constructing a fitness function based on measured values, and optimizing the model parameters using a GA. The resulting model parameters are then applied to the FEM modification of a long-span suspension bridge. By comparing the calculation errors of the initial model, BPANN-corrected model, and BPANN-GA, the advantages of using BPANN-GA for bridge model modification are validated, thus providing a more reliable basis for calculations.

2. Project Overview and FEM

2.1. Project Overview

The focus of this study is a large-span suspension bridge located in Hunan Province (Figure 1). The bridge spans a total length of approximately 1073.65 m. It features a main cable span arrangement of 242 m, 1176 m, and 116 m, respectively, with a bridge deck width of 24.5 m. The entire bridge is supported by two main cables with a transverse center distance of 27 m. It incorporates a total of 71 pairs of suspenders, each spaced 14.5 m apart longitudinally. The vertical height difference between the bridge deck and the ground is nearly 330 m. The horizontal distance between the mountain slopes on both sides ranges from 900 to 1300 m, providing excellent scenic views.

[figure(s) omitted; refer to PDF]

2.2. Measured of Dynamic Characteristics

The load efficiency coefficients of the test loading vehicle and the test items meet the requirements of the Chinese “Code for Load Test of Highway Bridges” (JTG/TJ20-01-2015) [19]. The load was generated by 98 fully loaded 40-ton trucks, with a front single axle and dual wheels and double axle with four tires on each axle at the rear part. Dynamic response and static displacement were both generated by the load above.

Vibration acceleration signals were acquired using strategically placed sensors. The layout of the measurement points on the bridge is shown in Figure 2. Measurement points JSD-1 to JSD-3 measure only transverse acceleration, while measurement points JSD-4 to JSD-6 measure acceleration in the transverse, longitudinal, and vertical directions. JL1 to JL4 are static displacement measurement points situated in the center of the downstream lane, and their test data are utilized for model validation. During the vibration test, the acceleration time history curves captured by the upstream longitudinal and transverse acceleration sensors are depicted in Figures 3 and 4, respectively. However, the transverse acceleration test at JSD-4 experienced a malfunction, and therefore, Figure 5 displays the acceleration time history curve recorded by the downstream longitudinal acceleration sensor.

[figure(s) omitted; refer to PDF]

From Figures 4 to 6, it can be observed that the time history curves of longitudinal vibration on both the upstream and downstream sides are similar in shape, and the vibration times caused by vehicle load at symmetrical positions on both sides are nearly identical. The vibration amplitude of the upstream longitudinal acceleration is greater than that of the downstream longitudinal acceleration. Specifically, the range of the upstream longitudinal acceleration amplitude is approximately 0.06 m/s², while the range of the downstream longitudinal acceleration amplitude is approximately 0.008 m/s². In comparison, the range of the acceleration amplitude in the no-vehicle section is approximately 0.001 m/s². The range of the upstream transverse acceleration amplitude is approximately 0.02 m/s², while the range of the acceleration amplitude in the no-vehicle section is approximately 0.005 m/s².

[figure(s) omitted; refer to PDF]

The dynamic response of the structure is primarily influenced by its mass, stiffness, and damping characteristics. During the vibration test, acceleration changes were measured in the transverse, longitudinal, and vertical directions. These measurements were then used to perform modal parameter identification in the time domain. This analysis yielded the natural frequencies of the first 11 modes of vibration for the bridge, as detailed in Table 1.

Table 1

Measured natural vibration frequency ratio.

Order	1	2	3	4	5	6	7	8	9	10	11
Frequency(Hz)	0.0572	0.12925	0.14363	0.16505	0.23606	0.24967	0.27629	0.28181	0.2804	0.28714	0.29848

Additionally, temperature usually has some impact on the measured result [20]. However, due to the high cost of load testing, it was not feasible to conduct parallel load tests across multiple temperature ranges. The bridge is located in an area with an average annual temperature of around 16.4°C. According to statistics, the temperature variation is relatively small, with more than 240 days each year having temperatures between 10 and 20°C. The load test was conducted at temperatures close to the annual average, approximately 15–17°C, which generally reflects the bridge’s typical operating conditions.

2.3. Establishment of Initial FEM

The 3D FEM of the bridge is shown in Figures 6 and 7, and it mainly comprises four parts: the steel truss girder, the main cable, the suspenders, and the bridge tower.

[figure(s) omitted; refer to PDF]

The FEM of the main cable and the suspenders is shown in Figure 8. The FEM parameters were referenced from the bridge’s design documents. The main cable and suspenders are simulated using rod elements. The elastic modulus of the parallel steel wires of the main cable is 2.00 × 10⁵ MPa, the Poisson’s ratio is 0.3, and the density is 7850 kg/m³. The suspenders have an elastic modulus of 1.15 × 10⁵ MPa, with the same Poisson’s ratio and density as the parallel steel wires. The steel truss girder is modeled using custom sections, with more than 10 types of sections defined based on the bridge design. The elastic modulus of the steel truss girder is 2.06 × 10⁵ MPa, the Poisson’s ratio is 0.3, and the density is 7850 kg/m³. The main cable, girder, and suspenders are all connected via hinged joints. The material of the bridge tower is C55 concrete, with an elastic modulus of 3.55 × 10⁴ MPa, a Poisson’s ratio of 0.167, and a density of 2600 kg/m³.

[figure(s) omitted; refer to PDF]

2.4. Initial Analysis

Modal analysis was performed using the initial FEM. In the modal analysis, the natural frequencies corresponding to different mode shapes were extracted and arranged in ascending order, with the order representing the mode shape sequence. Figure 9 shows the typical vibration modes calculated using the FEM with the initial parameters. The natural frequencies corresponding to modes 1–8 are detailed in Table 2, which reveals a significant computational error and a low degree of model reliability.

[figure(s) omitted; refer to PDF]

Table 2

Dynamic responses of model with initial parameters.

Order	Vibration mode description	Frequency
		Measured value (Hz)	Calculated value (Hz)	Calculation error (%)
1	First-order lateral bend	0.05720	0.05520	−3.5
2	First-order vertical bend	0.12925	0.11546	−10.67
3	Second-order vertical bend	0.14363	0.13527	−5.82
4	Second-order lateral bend	0.16505	0.15711	−4.81
5	Third-order vertical bend	0.23606	0.20778	−11.98
6	Fourth-order vertical bend	0.24967	0.24193	−3.1
7	Fifth-order vertical bend	0.27629	0.25024	−9.43
8	First-order torsion	0.28181	0.26197	−4.64

3. BPANN-GA Optimization Method

BPANN is often employed to construct nonlinear mapping relationships. It has the characteristics of information forward propagation and error backward feedback, featuring excellent learning and memory capabilities. Its network structure comprises an input layer, a hidden layer, and an output layer. The BPANN structure with a single hidden layer is illustrated in Figure 10.

[figure(s) omitted; refer to PDF]

BPANN fits models based on gradient descent search. Equations (1) and (2) show the adjustment methods for the hidden layer weights w_ij and thresholds b_ij.$$\begin{array}{}\text{(1)}& w_{ij}=w_{ij}-{\eta }_1\frac{\partial Ew,b}{\partial w_{ij}}=w_{ij}-{\eta }_1{\delta }_{ij}{x}_i,\end{array}$$$$\begin{array}{}\text{(2)}& b_j=b_j-{\eta }_2\frac{\partial Ew,b}{\partial b_j}=b_j-{\eta }_2{\delta }_{ij},\end{array}$$where E is the error function; η₁, η₂ is the learning rates; and x_i is the node output value.

For complex nonlinear functions, it is difficult to solve optimization problems well through neural network training alone. Therefore, this paper attempts to combine BPANN with GA to adapt to complex nonlinear problems. The idea is to treat the trained BPANN as a prediction function, combine it with measured values to construct an objective function, i.e., a fitness function, and then use GA for global optimization search. The calculation process is as follows:

1. Train the neural network with the training set, and then use the trained neural network to generate a population containing a large number of individuals.

2. Establish a fitness function to calculate the fitness value of each individual. Through a series of genetic operations such as selection and crossover, search for the individual with the optimal fitness. For the selection operation, the selection probability p_i of individual i is as follows:

$$\begin{array}{}\text{(3)}& p_i=\frac{k/F_i}{\sum _{j=1}^{N}k/F_j},\end{array}$$

where F_i is the fitness value of individual i; k is the coefficient, typically set to 10; and N is the population size.

If two chromosomes a_k and a_l are crossed at position j, the operation method is as follows:

$$\begin{array}{}\text{(4)}& \begin{array}{c}{a}_{kj}=a_{kj}1-b+a_{lj}b\\ a_{lj}=alj1-b+a_{kj}b\end{array},\end{array}$$

where b is a random number in the range [0,1].

If the jth gene a_ij of the ith individual undergoes mutation, the operation method is as follows:

$$\begin{array}{}\text{(5)}& a_{ij}=\begin{array}{c}{a}_{\text{ij}}+a_{\text{ij}}-a_{\max }\times fg,r\ge 0.5\\ a_{ij}+a_{\min }-a_{ij}\times fg,r\le 0.5\end{array},\end{array}$$

where a_max is the upper limit of gene a_ij; a_min is the lower limit of gene a_ij; f (g) = r₂ (1−g/G_max) ²; r₂ is a random number; g is the current iteration number; G_max is the maximum number of evolutions; r is a random number between [0,1].

3. After completing the GA optimization, retrieve the corrected optimization parameters. Use these parameters to perform a new finite element simulation analysis. Finally, compare the results of this simulation with those obtained before the optimization to assess the improvements.

4. Model Modification and Verification

4.1. Determination of Modification Parameters

A single-factor sensitivity analysis method called the One-at-a-Time (OAT) method was used to identify five parameters that significantly affect the static and dynamic responses of the bridge, which were then modified. The core idea of this method is to change only one input variable at a time while keeping other input variables constant, thereby observing the changes in the output variable. The steps of the OAT method in this paper are as follows:

1. Setting the Baseline

Select the mean of the input variables as the baseline and calculate the output value at the baseline:

$$\begin{array}{}\text{(6)}& Y_0=f{X}_0,\end{array}$$

where X₀ is the baseline of the input, f is the model, and Y₀ is the output at the baseline.

2. Univariate Perturbation

Change each input variable one by one while keeping the other variables constant. The perturbation method used in this paper is to increase or decrease the variable by one standard deviation. For each variable X_i (i = 1,2,…,k), the output after perturbation is as follows:$$\begin{array}{}\text{(7)}& Y_i^{+}=f{X}_0+\Delta X_i,\end{array}$$

$$\begin{array}{}\text{(8)}& Y_i^{-}=f{X}_0+\Delta X_i,\end{array}$$

where ΔX_i is the standard deviation.

3. Calculation of Sensitivity Indices

Sensitivity is typically measured using the standard deviation or the root mean square error (RMSE) of the output changes. In this paper, the standard deviation is used. The sensitivity S_i of each input variable X_i can be calculated using the following formula:

$$\begin{array}{}\text{(9)}& S_i=\frac{Y_i^{+}-Y_0+Y_i^{-}-Y_0}{2}.\end{array}$$

4. Ranking Variables by Sensitivity

Based on the calculated sensitivities, the variables are ranked in order of their sensitivity.

More than 10 parameters were selected for the sensitivity analysis, including various parameters of bridge components, such as the elastic modulus and density. Ultimately, five critical parameters with the highest sensitivity indicators were chosen for correction, namely: the modulus of elasticity of the main beam E₁, the unit weight of the main beam γ₁, the modulus of elasticity of the bridge tower E₂, the modulus of elasticity of the parallel steel wires of the main cable E₃, and the modulus of elasticity of the suspenders E₄.

4.2. Process of Model Adjustment

Referring to the process shown in Figure 11, the program of bridge FEM modification by BPANN-GA was developed.

[figure(s) omitted; refer to PDF]

The modification process is described as follows:

1. Initially, sample input and output values, specifically the modal natural frequencies from the 1st to the 8th order, are extracted to train BPANN. Subsequently, an objective function (Equation (10)) is established and utilized as the fitness function for the GA during its iterative process

$$\begin{array}{}\text{(10)}& F=\sum _{i=1}^n{\frac{f_{ci}-f_{mi}}{f_{mi}}}^2.\end{array}$$

In the formula, f_ci represents the calculated vibration frequency, and f_mi represents the measured vibration frequency.

2. Construct the ANN structure based on the number of input and output parameters, which in turn determines the encoding length of the individuals in the GA. In this study, the ANN structure is defined as 5-10-8, indicating that the input layer consists of 5 nodes, the hidden layer comprises 10 nodes, and the output layer includes 8 nodes. This configuration results in a total of 5 × 10 + 10 × 8 = 130 weights and 10 + 8 = 18 thresholds. The training termination condition is set to a maximum of 100 training iterations.

3. Considering the uncertainty and variability in the material properties, 200 sets of inputs, including five critical parameters, are randomly generated. These random values are generated within a ±15% range of the material parameters in the initial FEM. These parameters are subsequently employed with the element model to compute the natural frequencies corresponding to the 1st–8th-order vibration modes. This yields 200 sets of input and output values, which are then utilized for training and testing the ANN.

4. The trained ANN is employed to generate 50 sets of individuals for optimization using a GA. These individuals are assessed through a fitness function, and the optimal individual is determined through selection, crossover, and mutation operations. This optimal individual signifies the corrected parameters. The crossover probability is set at 0.5, the mutation probability at 0.03, and the process concludes after a maximum of 200 iterations.

5. Validate the results of the FEM modification. Adjust the parameters of the initial FEM to obtain an optimized FEM. Conduct static and dynamic calculations on the optimized FEM and compare the results with the actual measurements from structural load tests to verify the effectiveness of the FEM modification.

4.3. Results of Model Modification

The parameter values before and after modification are presented in Table 3. Notably, the modification amplitude for the main girder’s elastic modulus is relatively large, reaching 11.7%.

Table 3

Comparison of model parameters before and after modification.

Modified parameters	Initial value	Modified values	Modified change range (%)
Elastic modulus of girder E1	2.06 × 105 MPa	2.30 × 105 MPa	11.7
Unit weight of girder γ1	7,850 kg/m3	8,117 kg/m3	3.4
Elastic modulus of bridge tower E2	3.55 × 104 MPa	3.88 × 104 MPa	9.3
Elastic Modulus of main cable E3	2.00 × 105 MPa	2.05 × 105 MPa	2.5
Elastic modulus of suspenders E4	1.15 × 105 MPa	1.23 × 105 MPa	7.0

Since the actual bridge’s main girder has a certain thickness of deck pavement above, which was not considered in the initial FEM, the modification effectively adjusts the elastic modulus of the main girder to an equivalent elastic modulus that accounts for the deck pavement. Therefore, the amplitude of the modification is relatively significant. Additionally, the modification amplitude for the tower’s elastic modulus reached 9.3%. This is attributed to the inherent variability in concrete’s elastic modulus. When the production batch quality is high, or construction requirements are strictly controlled, the actual elastic modulus value tends to be higher than the design value. Totally, the updated parameters are within a reasonable range according to previous studies [21, 22].

The corrected parameter values from Table 3 were used as material parameters in the bridge model for calculations, enabling the determination of the natural frequencies corresponding to each mode shape of the corrected model. The calculated natural frequencies for each mode shape before and after model modification using the BPANN-GA method, along with the measured natural frequencies, are presented in Table 4. As indicated in Table 4, before the modification, the average error of the calculated frequencies for modes 1–8 was 7.04%, with the highest error reaching 11.98%. After the modification, the largest error in the calculated vertical bending frequency was observed in the second-order positive symmetric vertical bending, with an error of 2.71%. The largest error in the calculated lateral bending frequency was found in the first-order antisymmetric lateral bending, with an error of 1.40%. The largest error in the calculated torsional frequency was noted in the first-order positive symmetric torsion, with an error of 1.76%. In summary, following the modification, the absolute values of the calculated errors for modes 1–8 were all below 3%. This indicates that after utilizing BPANN-GA for modification, the modal results obtained through FEM calculations closely matched the actual modal data from vibration tests.

Table 4

Comparison of dynamic responses of models before and after modification by BPANN-GA.

Order	Description of mode	Frequency					Amended MAC (%)
		Actual (Hz)	Calculation (Hz)		Calculation error (%)
			Amendment	Modified	Amendment	Modified
1	First-order symmetric transverse bending	0.05720	0.05520	0.05752	−3.5	0.56	88.9
2	First-order antisymmetric vertical bending	0.12925	0.11546	0.12625	−10.67	−2.32	92.4
3	First-order antisymmetric transverse bending	0.14363	0.13527	0.14162	−5.82	−1.40	96.4
4	First-order symmetrical vertical bending	0.16505	0.15711	0.16228	−4.81	−1.68	90.0
5	First-order symmetric torsion	0.23606	0.20778	0.23191	−11.98	−1.76	88.7
6	First-order opposition twist	0.24967	0.24193	0.25122	−3.1	0.62	97.6
7	Second-order symmetrical vertical bending	0.27629	0.25024	0.26881	−9.43	−2.71	92.7
8	Second-order antisymmetric vertical	0.28181	0.26197	0.27908	−4.64	−0.97	95.1

Abbreviations: BPANN, backpropagation artificial neural network; GA, genetic algorithm; MAC, modal assurance criterion.

Figure 12 provides a comparative analysis of the calculation outcomes from the corrected models using BPANN-GA and BPANN. It is evident that BPANN results in significantly higher calculation errors. The average calculation error for BPANN-GA is 1.50%, in contrast to 3.84% for BPANN. Notably, the calculation errors for the 5th and 7th mode shapes are particularly larger, with values of 7.5% and 5.5%, respectively.

To verify the match between the corrected FEM and the actual measured mode shapes, the modal assurance criterion (MAC) was employed. The MAC is a statistical tool used to assess the mutual independence and consistency between two modes, as defined in Equation (11):$$\begin{array}{}\text{(11)}& \text{MAC}{\phi }_a,{\phi }_e=\frac{{{\phi }_a^T{\phi }_e}^2}{{\phi }_a^T{\phi }_a{\phi }_a^T{\phi }_e},\end{array}$$where ${\phi }_a$ is the computed modal shape vector; ${\phi }_e$ is the measured modal shape vector.

The range of MAC values is [0, 1], with higher MAC values indicating more accurate calculation results. Figure 13 presents the distribution of MAC values for the first eight modes of theoretical vibration versus actual measured vibration. From Figure 13, it can be observed that the MAC values for the first eight modes are all greater than 0.9, which demonstrates that the corrected FEM can accurately reflect the actual vibration state of the bridge. This high level of agreement between the theoretical and measured mode shapes confirms the effectiveness and reliability of the corrected FEM in capturing the dynamic behavior of the bridge.

[figure(s) omitted; refer to PDF]

4.4. Model Validation Results

To enhance the validation of the model, it is essential to employ the modified model for calculating bridge characteristic parameters beyond the natural frequencies of the 1st–8th-order vibration modes. A small deviation between the calculated and measured values would confirm the model’s capability to realistically simulate the actual structure. Conversely, a significant deviation would suggest that additional factors contributing to errors have not been accounted for. The data employed for validation encompasses both the static displacement obtained from the static test, with measurement points detailed in Figure 3, and the 9th, 10th, and 11th-order natural frequencies. The inclusion of these higher-order frequencies is crucial as they provide a more robust assessment of the model’s calculation reliability. Table 5 clearly shows the impact of model modification on static displacement and the 9th–11th-order natural frequencies. Before the model was modified, there was a significant computational error in static displacement, with an average error rate of 11.4%, and the calculated values tended to overestimate the actual displacements. After the revision, the error rate was substantially reduced to an average of 5.9%, and the displacement values also saw a decrease. This improvement is due to the adjustments made to parameters such as the elastic modulus of the main beam and the elastic modulus of the parallel steel wires in the main cable, which is consistent with the patterns observed in the referenced literature [6]. Additionally, the computational error for the 9th–11th-order vibration frequencies has been decreased. Overall, after the revision, the calculated values for both static displacement and higher-order vibration frequencies have become more consistent with the actual measured data. This reflects that the BPANN-GA revision has improved the modified model’s capability to accurately represent the static and dynamic responses of the structure. It is also evident that even though the optimization was focused solely on dynamic response data, the accuracy of the modified model in calculating static parameters has been improved.

Table 5

Comparison of model validation results.

	Point number or mode	Measured values	Calculated		Calculation error (%)
			Amendment	Modified	Amendment	Modified
Static displacement	JL-1	3.5 mm	4.03 mm	3.30 mm	15	5.6
	JL-2	15.6 mm	17.2 mm	16.71 mm	10.1	7.1
	JL-3	20.1 mm	22.0 mm	21.37 mm	9.3	6.3
	JL-4	10.7 mm	11.9 mm	10.22 mm	11.2	−4.5
Natural frequency	Second-order symmetric torsion (9th order)	0.28040 Hz	0.27732 Hz	0.27816 Hz	−1.1	−0.8
	Second-order antisymmetric transverse bending (10th order)	0.28714 Hz	0.27853 Hz	0.29059 Hz	−3.0	1.2
	Second-order antisymmetric torsion (order 11)	0.29848 Hz	0.28296 Hz	0.29102 Hz	−5.2	−2.5

It is worth noting that although the modification is based on modal analysis from dynamic load tests, the modified model also demonstrates higher accuracy in simulating displacements under static load tests. This case highlights the adaptability of the BPANN-GA method to different loading conditions. However, further exploration is needed to determine the performance of BPANN-GA on more complex environments and loading conditions.

5. Discussion

Figure 14 presents a comparison of computational efficiency between the BPANN-GA method and the conventional GA method, where RMSE represents the magnitude of the error. It can be observed that the RMSE of the BPANN-GA method rapidly decreases within the 0–20 generations and stabilizes at approximately 0.013 by 40 generations, indicating that BPANN-GA can quickly obtain the correction parameters. In contrast, the GA method shows a slower reduction in RMSE, and even after 200 generations, the RMSE remains relatively high and has not stabilized, requiring further increases in the number of generations. Therefore, it is evident that the computational efficiency of the BPANN-GA method is significantly superior to that of the conventional GA method.

[figure(s) omitted; refer to PDF]

Based on the above analysis, the GA-BPANN has certain advantages in the parameter correction of FEMs for bridges. However, it also has some limitations. For example, selecting the appropriate parameters can be challenging, as GAs involve many parameters that need adjustment, such as crossover probability and mutation probability. Inappropriate parameter selection can affect the performance of the algorithm. Moreover, the GA is a general optimization method, requiring problem-specific encoding and fitness function design for different problems. Future research should focus on adaptive parameter selection methods, encoding schemes, and other related areas.

6. Conclusions

This study introduces a novel FEM modification approach, which integrates GA and BPANN, specifically tailored for long-span suspension bridge structures. The following conclusions can be drawn:

1. In the engineering case examined in this paper, the initial FEM exhibited an average calculation error of 7.04% for the natural frequencies of the 1st–8th-order vibration modes, with a maximum error of 11.98%. Following the application of the BPANN-GA revision method, the absolute values of the calculation errors for these modes were reduced to below 3%, and the MAC values for all modes exceeded 90%.

2. Compared to the commonly used BPANN model modification algorithm, BPANN-GA demonstrates significant advantages in addressing the nonlinear bridge model revision problem. Specifically, the average calculation error for BPANN-GA is 1.50%, in contrast to 3.84% for BPANN. Notably, the calculation errors for the 5th and 7th mode shapes are particularly larger, with values of 7.5% and 5.5%, respectively.

3. Regarding static displacements, the computational error following the revision process is substantially reduced, with an average of 5.9%. This reduction is a result of the enhancements in parameters like the elastic modulus of the main girder and the elastic modulus of the parallel steel wires in the main cable. Even though the optimization was focused solely on dynamic response data, the refined model now exhibits improved precision in the calculation of static parameters.

4. Following the modification using BPANN-GA, the modified model more accurately reflects both the static and dynamic responses of the structure. This provides a more dependable foundation for assessing the safety status of the bridge. It also illustrates the effectiveness of employing BPANN-GA for refining large-scale structural FEMs.

Author Contributions

Zi-Xiu Qin: conceptualization, formal analysis, and investigation. Xi-Rui Wang: formal analysis and writing–original draft. Wen-Jie Liu: conceptualization, supervision, and writing–review and editing. Zi-Jian Fan: conceptualization and supervision.

References

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