1. Introduction
Since the 1990s, with the sustained and rapid development of China’s economy, bridge construction has entered a 30 year period of rapid development. As of the end of 2021, China has built 961,100 highway bridges and 73.8021 million meters, of which over 80% are small and medium-sized bridges. With the rapid increase in the number of bridges and the emergence of innovative and breakthrough bridges, concerns about their safe operation are increasing. In the United States, approximately 40% of bridges require repair or reconstruction, and according to estimates from the Federal Highway Administration (FHWA),
$90 billion is needed to address these issues. In China, with the increase in traffic flow, design and construction defects, overloading, insufficient operation and maintenance, and natural aging of bridges in recent years, the number of old and dangerous bridges remains high, but they operate with “diseases” all year round, posing huge safety hazards [
1,
2].
In response to the huge safety hazards existing in existing bridges, many scholars have attempted to detect and eliminate safety hazards based on structural health monitoring systems and have achieved certain results [
3,
4]. One of the main purposes of structural health monitoring is to obtain accurate finite element models, which can enable effective structural safety assessments [
5].
Traditional finite element model updates have two shortcomings: (1) Sensitivity-analysis-based finite element model updates require iterative calculations, and each calculation requires calling a finite element model, which results in a large computational load and is not conducive to its application in engineering; (2) In the updating of finite element models of complex structures, there are many parameters that need to be modified, so a large number of finite element calculations are essential and not easy to implement on computers [
6]. Finite element model updating based on the response surface method is a new method that differs from traditional models. Its basic concept [
7,
8] is that in the design space of variables, regression analysis methods will be used to fit the response values or test values of sample points to obtain surface responses that simulate real limit states. This will replace the finite element model or make the design or calculation of other complex models more effective. Ren et al. [
9,
10] introduced the updating of finite element models based on dynamic and static response surfaces. Fang et al. [
11] used D-optimal design and first-order response surface models to predict the dynamic response and damage identification of intact and damaged systems. The effectiveness of this method was verified by the results of reinforced concrete frame model tests and I-40 bridge tests. Chen and Zhang et al. [
12,
13] introduced the uncertainties and correlation between reinforcement corrosion and concrete cracking, and a case study was employed to discuss the life-cycle modeling of concrete cracking and reinforcement corrosion. Zong et al. [
14] used the central composite experimental design method (CCD) and second-order response surface model to update the finite element model based on the health monitoring of a large-span continuous rigid frame bridge, proving that finite element model updating based on a second-order response surface has a high accuracy [
15,
16,
17]. In recent years, finite element modeling and correction technology based on artificial intelligence has attracted more and more attention; for instance, wavelet convolutional neural networks and deep-learning neural networks have been used for wind-induced vibration modeling and stress distribution prediction [
18,
19,
20].
In this paper, a continuous beam bridge is employed as the engineering background, the FE model updating of the bridge was conducted based on the visual inspection and ambient vibration testing and third-order response surface model, after which the FE model can be further applied in bridge health monitoring and a safety evaluation.
2. Basic Methodology
The selection of the response surface function form, i.e., the response surface model, is an important part of the application of the response surface method. It should meet two requirements: (1) the expression of the response surface function should be as simple as possible while basically describing the relationship between the system input parameters and output response; (2) The number of undetermined coefficients in the response surface function expression should be minimized to reduce the number of system experiments or calculations [
21].
Response surface models include complete and incomplete polynomial models, Kriging models, BP neural network models, radial basis functions (RBFs), and multivariate adaptive regression spline functions (MARSFs). The response surface model in this article adopts a polynomial response surface model [
22].
Assuming that the system response for the dependent variable, , is the design parameters, which is selected through the analysis of the variance method, the polynomial response surface model form is as follows:
In Equation (1), , , are the upper boundary and lower boundary of the design parameter values, and are undetermined coefficients.
The fitting of the response surface function is the process of solving the undetermined coefficients in the response surface function. The least squares method is the basic method for solving undetermined coefficients, and its steps are as follows:
- (1)
Determine parameters and their range of values, and determine sample points (calculation points) through the experimental design;
- (2)
Calculate the response values of sample points through finite element analysis to obtain sample data;
- (3)
Substitute the sample data into Equation (1), and then use regression analysis to calculate the undetermined coefficients ;
- (4)
Perform response surface model validation. If the accuracy of the response surface model meets the requirements, this response surface model can be used for model correction. If the accuracy of the response surface model does not meet the requirements, go back to step (1) and redo the experimental design until the accuracy meets the requirements.
After calculating the unknown parameters of each response surface function, it is necessary to verify the accuracy of the unknown parameters. The initial definition of model validation provided by the American Computer Simulation Association was that model validation is the process of validating conceptual models expressed by computational models within a given accuracy range. Among them, the conceptual model refers to the finite element analysis model, while the computational model is the response surface model of regression.
Based on the calculation of the finite element model and the response surface model, the standards for testing the accuracy of the response surface model include a normal distribution test of residuals, mean of residuals, relative root mean square error (RMSE) and
test. For more complex models and response surface models with multiple responses, the latter two standards are usually used, and their expressions are shown in Equations (2) and (3), respectively [
23,
24,
25].
where
represents the calculated value of the response surface model,
represents the corresponding finite element analysis calculation results,
represents the average value of the finite element analysis calculation results, and
represents the number of inspection points in the design space.
The values of and represent the difference between the response surface and the finite element analysis calculation, both taking values between 0 and 1. The closer the value of is to 1, the more accurately the response surface model of the regression describes the relationship between the system input and output in the experimental design space. On the other hand, the value of is the opposite, and the closer the value of is to 0, the more accurate the model is.