The selection of updating parameters constitutes a crucial step in the successful updating of FEM. The suitability of the selected update parameters significantly impacts both the correction accuracy and the correction efficiency of the model update. This step mainly depends on understanding structural principles, sound engineering judgment, and test objectives. Therefore, the selection of updating parameters is mainly grounded in prior experience and sensitivity analysis of parameters. We usually choose the modulus of elasticity, density, Poisson’s ratio, damping ratio, friction coefficient, and bearing stiffness of each part of the bridge materials as the updating parameters.

Currently, the commonly used experimental design methods in civil engineering include full-factorial design, uniform design, central composite design, and Latin hypercube design. The full-factorial design method is a basic experimental method with relatively intuitive logic. As the number of parameters to be corrected increases, the number of experiments increases exponentially (if there are n parameters to be corrected and the number of levels for each component is L, the number of experiments is m = Lⁿ), so the full-factorial design method does not apply to complex FEM. The uniform design method focuses on the uniform distribution of experimental sample points within the research interval, and other factors may be ignored. Regarding the Latin hypercube design method, the number of sample points must exceed that of parameters, which is likely to lead to a relatively large workload for generating the sample set.

The central composite design method is one of the most commonly used design methods in RSM, and it combines the partial factorial experimental design method with the interpolated node method to increase the number of sample points within a distance of ±δ on the principle that they are over the center point and parallel to the coordinate axis. The center composite test consists of center, cubic, and axial points with k parameters to be corrected. The number of cubic points is 2k; the cubic points are used to estimate the linear and interaction terms but not the bending. The axial points are on the axis of the test space with a total number of 2k, and the (0, 0) coordinates are the center points, which are used to check the bending, and the two-factor center composite test is shown in Fig. 2.

The central composite design method features a uniform distribution of test points, which overcomes the deficiency of only outputting linear relationships between variables and factors. It can effectively expand the design space to obtain higher-order information, with the advantages of simple design and fewer trials. The central composite method design test has a very high advantage over other test design methods, making the test results more accurate. Hence, the test design method used in this paper is the central composite method.

The choice of the mathematical functions for the RSM is crucial to the overall response surface analysis method, and the type of function chosen determines the accuracy of the subsequent corrections [22]. For large-scale structures, finite element calculations’ structural inputs and outputs are often complex and exhibit a certain degree of nonlinearity. Thus, the RSM is generally required to reflect the structure of the input-output relationship and simplify the calculation process. The RSM method smartly substitutes the traditional FEM for repeated iterative calculations by constructing a response surface model, and it is especially vital to express the relationship between design parameters and the response using a reasonable function mapping within an acceptable accuracy range. The RSM demands the selection of an appropriate RSF, which is directly related to the success of the problem solution.

Polynomial functions are more widely used in response surfaces owing to their strong nonlinear ability and simple forms. Two aspects need to be considered when choosing a suitable response surface function. Firstly, the function expression should comprehensively reflect the correlation between the input parameters and the output values. Secondly, a relatively straightforward function expression should be chosen as it can reduce the workload associated with the parameters to be corrected and regression analysis. Generally, second-order and third-order polynomials meet the accuracy requirements for representing the relationship between the parameters to be corrected and the response values. In the finite element model updating of bridges, the most prevalently utilized response surface function models are first-order polynomials and second-order polynomials.

First-order polynomials are mostly used for fitting simple linear relationships. Higher-order polynomials are mostly used for fitting complex nonlinear relationships between structural responses and parameters. Among them, second-order polynomials are the most widely adopted because they can achieve a good balance between fitting accuracy and computational efficiency. Li et al. [23] fitted the response surface equations of the parameters to be corrected and the objective function, with and without considering the parameter cross terms, respectively, and obtained the explicit equations between them and the optimal values of the parameters to be corrected. Their results indicated that the updating effect was better when the parameter cross terms were considered during the updating process. Therefore, this paper chose second-order complete polynomials as the RSF, as shown in Eq. (1). (1)y^=β0+∑i=1kβixi+∑i=1kβiixi2+∑i∑jβijxixj+ε,where y^ is the response surface function; β0 is the constant term to be determined; βi is the primary term to be coefficient; βii and βij are the quadratic terms to be determined; xi and xj denote the correction parameters of the structure; k denotes the number of coefficients to be determined; ε is the higher order error and ε∼N(0,σ2).

In this paper, according to the bridge design drawings, the cable-stayed bridge FEM was established and analyzed with the finite element software ANSYS based on the actual situation. The entire bridge deck was modeled in the form of a fishbone girder, as shown in Fig. 7, where the longitudinal girder and the transverse stiffeners are crossed in the shape of a “fishbone.” The transverse rigid arm connects the centroid of the beam’s cross-section and cable end to transfer the load. The advantage of the “fishbone” mode is that the mass and the stiffness of the bridge deck system can be concentrated in the central node. Connecting the center of gravity of the beam section and the cable node through the stiffening arm makes the stiffness modeling of the main girder more accurate.

The main material parameters and their corresponding components of the initial FEM are shown in Table 1.

The main girder, cable tower, and pier of the bridge are simulated by the BEAM4 element, which is a uniaxial force unit that can withstand tension, compression, bending, and torsion, and this unit has six degrees of freedom at the nodes, which are the linear and angular displacements in the X, Y, and Z directions. The cable is simulated by the LINK8 element, a three-dimensional rod unit that can only withstand tension and compression, and the unit has three degrees of freedom at each node: displacement in the X, Y, and Z directions. The initial FEM of the cable-stayed bridge is shown in Fig. 8.

This paper calculated the first ten modes of the cable-stayed bridge. The first five vibration modes are shown in Fig. 9, and the first ten vibration frequencies are listed in Table 2.

The monitoring contents of the bridge monitoring system used in this paper include vibration monitoring, cable force monitoring, and deflection monitoring. There are nine unidirectional acceleration sensors, and the layout diagram of sensor measuring points is shown in Fig. 10.

The traditional Bayesian Fast Fourier Transform (FFT) method is susceptible to adverse conditions and may fail to converge under such circumstances. Li et al. [24] devised an Expectation-Maximization (EM) algorithm for calculating the second-order Taylor approximation of the logarithmic posterior probability density function (PDF) at the local maximum. When the traditional algorithm fails to converge in modal analysis, this method can substitute for the conventional one and enhance computational efficiency. Compared with the traditional algorithm, the improved Bayesian Fast Fourier Transform (FFT) algorithm can identify structural responses with greater accuracy and higher efficiency.

In light of the theory underlying the improved Bayesian FFT algorithm, the acceleration monitoring data of the cable-stayed bridge from the first quarter of 2021, which is presently accessible and closest to the bridge’s initial state, was chosen to perform modal analysis on the measured vibrations of the cable-stayed bridge.

The modal analysis was conducted using ANSYS to obtain the first 10 orders of modes of the cable-stayed bridge. The measured and calculated frequencies of the first 10 orders of modes of cable-stayed bridges and their comparison are shown in Table 2, except for the third-order modes with the largest relative error of 13.211%, and the fourth-order modes with a relative error of 10.382%. The relative error of the rest of the modes is less than 10%.

In this paper, we initially select certain parameters based on experience. Subsequently, through parameter sensitivity analysis, we select the parameters with higher sensitivity to update.

When multiple correction parameters are within the structure, their units are not identical. Eq. (5) can be employed to calculate the dimensionless sensitivity values to avoid this kind of interference. This facilitates an intuitive evaluation of the degree of influence the updated parameters exert on the state variables. (5)∂F∂p=(F″−F′)Δp/(p0+Δp),where p represents the parameter to be corrected, p0 is the initial value of the parameter to be corrected, Δp is the slight change amount of the parameter to be corrected, and F is the state parameter of the structure. When p=p0, the state variable F′ in this case can be calculated through ANSYS. Keeping other parameters unchanged and making a small change to the parameter to be corrected p, that is, p=p0+Δp, the state variable F′, in this case, can also be calculated through ANSYS.

Initially, the elastic modulus and density of main girders, bridge towers, abutments, and diagonal cables were chosen as the parameters to be updated. Specifically, the elastic modulus and density of the main girder are E1 and D1, and the elastic modulus and density of the bridge tower are E2 and D2. The elastic modulus and density of the bridge abutment are E3 and D3. The elastic modulus and density of the diagonal cable are E4 and D4. By employing the parameter perturbation method, each of the eight preliminarily screened parameters to be corrected was increased by 10% to analyze their sensitivities to the first 10 orders of frequencies. The original values and perturbed values of the parameters to be updated are shown in Table 3. The frequency sensitivity analyses of each parameter are shown in Fig. 11. When the parameter analysis is based on sensitivity, the parameter with high sensitivity is usually selected as the parameter to be updated, because the high sensitivity parameter controlled the objective function. In contrast, the low sensitivity parameter tends to increase the model error. Consequently, the elastic modulus and density of the main girder, bridge tower, and abutment are selected as the parameters to be updated.

By comparison, the frequency of cable-stayed bridges exhibits a higher sensitivity to the parameters of the main girders. Therefore, the variation rates of the density and elastic modulus of the main girders are 1±0.2, while those of the remaining parameters are 1±0.3, and parameters to be updated and their respective variation rates are shown in Table 4.

To establish the response surface equations, the central composite design method is used to devise the experimental conditions for the response surface. Given that there are six parameters to be corrected, the number of factors is thus six. There are 64 cubic points and 12 axial points, with the value of α being 2.828. The finite element analysis results for each representative test case are shown in Table 5, where cases 1 to 64 are cubic points, cases 65 to 76 are axial points, and cases 77 to 82 are center points. The calculated results in Table 5 are used to construct the response surface equations.

According to the calculation and analysis results of the FEM under each test case, the least squares method is adopted to regress the undetermined coefficients βi in the response surface equations, and the βl˙ of the RSFs for each order of the frequencies are shown in Table 6.

This paper utilized the measured first 10 order frequencies of the bridge structure and the corresponding first 10 order frequencies obtained from the response surface calculation. It calculated the sum of the squares of their relative errors as the objective function. The relative error of the frequency was adopted as the objective function rather than the absolute error. This was mainly due to the consideration that the magnitudes of frequencies of each order vary, aiming to avoid the error resulting from inconsistent weighting magnitudes. The constructed objective function was given as in Eq. (6). (6)y=min∑i=18(fAi−fEifEi)2.

Here, fEi is the measured value of the frequency, and fAi is the calculated value of the response surface of the frequency, and i=1,2,…,8. Once the objective function has been constructed, iterative optimization can be carried out to seek the objective function’s minimum value under the parameters’ constraints. This paper conducted optimization iteration on the MATLAB platform using the sequential quadratic programming method. The percentage error of the objective function values between two adjacent model updates during the optimization iteration process was used to judge whether convergence had been achieved. If the percentage error of the objective function value was less than 1% for two consecutive times and this situation occurs twice in a row, it was deemed that convergence has been reached. Then, the optimal values of the design parameters can be obtained. The comparison of the model updating parameters before and after correction is shown in Table 7.

The comparison between the pre-updating frequency and the post-updating frequency is shown in Table 8. It could be seen that after the model update, the vibration frequency values calculated by the finite element method (FEM) were closer to those monitored from the actual bridge structure. Through the model updating, the absolute error values between the vibration frequency values calculated by the FEM and the monitored ones changed from ranging between 1.426% and 15.222% to a range of 1.162% and 5.216%. The most significant error occurred in the third-order vibration frequency, which decreased from 15.222% to 2.643%, thus showing better agreement with the actual vibration frequency value.