The Prediction of Steel Bar Corrosion Based on BP Neural Networks or Multivariable Gray Models

Liu, Juan; Bai, Xuewei

doi:https://doi.org/10.1155/2023/2695142

Computational Intelligence and Neuroscience

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Abstract Introduction Results Discussion Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 2695142 | https://doi.org/10.1155/2023/2695142

The Prediction of Steel Bar Corrosion Based on BP Neural Networks or Multivariable Gray Models

Juan Liu¹and Xuewei Bai²

Academic Editor: Sheng Du

Received19 Nov 2022

Revised27 Dec 2022

Accepted30 Dec 2022

Published05 Jan 2023

Abstract

The corrosion of steel bars in concrete has a significant impact on the durability of constructed structures. Based on the gray relational analysis (GRA) of the accelerated corrosion data and practical engineering data using MATLAB, a back propagation neural network (BPNN) model, a multivariable gray prediction model (GM (1, N)), and an optimization multivariable gray prediction model (OGM (1, N)) of steel corrosion were established by using a sequence of the key affecting factors. By comparing the prediction results of the three models, it is found that the GM (1, N) model has larger fitting and prediction errors for steel corrosion, while the OGM (1, N) model has smaller prediction errors in the accelerated corrosion data; the BPNN model offers more accurate predictions of the practical engineering data. The results show that the BPNN and OGM (1, N) models are all suitable for the prediction of steel bar corrosion in concrete structures.

1. Introduction

The corrosion of steel bars induces corrosion cracks in concrete structures. The appearance of cracks makes it easier for the corrosive media (H₂O and O₂) to reach the surface of the steel bars, which accelerates the corrosion rate of the steel bars. Corrosion reduces the cross section of steel bars and severely affects the bond strength between the steel bars and the concrete, resulting in structural failure. The reduced load capacity has a great impact on the durability and reliability of the structure [1]. It is difficult to measure the corrosion of steel bars in practice, especially those that are in service. In recent years, theoretical and empirical models have been proposed to estimate the extent of corrosion of steel bars after rust swelling and cracking of concrete structures. Bazant [2] and Zhang and Cheung [3] proposed a prediction model for the extent of steel corrosion according to its physical and chemical processes. Isgor and Razaqpur [4], Zhang et al. [5], Zheng et al. [6], and Xu et al. [7] conducted simulation tests in the laboratory to establish an empirical model of the corrosion rate of steel bars changing with environmental temperature, humidity, and other parameters. Based on a long-term exposure test and actual engineering durability test data, Guo et al. [8] proposed a formula for predicting the loss rate of steel bars. The theoretical model can reflect the physical and chemical processes of steel corrosion in concrete structures, and the influencing factors are comprehensive. However, many parameters in the model are difficult to determine. The empirical model can be closely linked with reality, but there are many complex factors in the model, which cannot be fully considered, resulting in certain inconsistencies with reality. Therefore, other methods are needed to predict the amount of steel corrosion.

As a method to determine whether or not variables are correlated and to determine the degree of their correlation, GRA provides a comprehensive assessment model. It was also applied to analyze the effects of the influencing factors on the steel corrosion which involves multiple variables with comprehensive correlations. Artificial neural network has been applied in the research of reinforcement corrosion [9, 10]. An et al. [11] combined the GRA and BPNN methods to predict the corrosion of steel bars; the results proved that this method can predict well. Luo et al. [12, 13] developed a hybrid enhanced Monte Carlo simulation and a dynamical adaptive enhanced simulation method coupled with support vector regression, which showed strong capability for application in the fatigue assessment of turbine bladed disks and structural reliability. Muiga et al. [14] adopted a gray prediction model (GM (1, 1)) to evaluate the carbonization of long-span reinforced concrete bridges. The accuracy of the prediction model was within a reasonable range and met the requirements of mathematical modeling. However, because of the complex corrosion mechanism of steel bars in concrete structures and the coupling relationship with crack width, protective layer thickness, and steel bar diameter, the GM (1, 1) model has a poor prediction effect sometimes, and the accuracy of its prediction is debatable [15]. In this study, a BPNN model, a multivariable gray model (GM (1, N)), and an optimized multivariable gray prediction model (OGM (1, N)) of steel corrosion are established by using the sequence of the key affecting factors after the gray relational analysis of the accelerated corrosion data. By comparing the calculation results of the three models with the practical engineering data, the applicability of the models is verified. The calculation results show that the BPNN and OGM (1, N) models perform well in terms of the prediction of the corrosion of steel bars in concrete structures and can provide some reference values for the evaluation of structural durability.

2. Prediction Methods

2.1. Gray Relational Analysis

Gray relational analysis refers to the degree of similarity between the curve geometry formed by the studied sequence and the change analysis of the influence factor sequence in the development process of the system. It helps to determine whether the connection is close by indicating the degree of connection between the curves. If changes in the trend of the two factors are consistent, the correlation between them will be greater; if the change in trend is inconsistent, the correlation will be lower [16, 17]. The gray correlation between sequences is reflected by the gray relational degree, which refers to the measurement of the correlation between the dependent variables over time or different objects, considering the relevancy between the factors so as to distinguish each factor. The greater the correlation between each influencing factor for the system, the closer relationship between them.

For a given system, assume that there are variables:

With one out put and inputs . These two kinds of sequences have strong correlations with each other. For each variable , we assume that the sequence length is n, that is,

Data standardization is used to deal with the problems of inconsistent units among various sequence factors and inconsistent physical meanings. The mean and variance of each impact factor sequence are calculated as follows:where is the mean of each inputs sequence, and is the variance of each inputs sequence.

Then, we obtainwhere is the i-th input sequence after normalization.

The difference sequences of the relevant factors are found as follows:

The correlation coefficient is calculated between the sequences as follows:where and and is the resolution coefficient.

The gray relational degree was calculated as follows:

The gray relational degrees of all the influencing factors were calculated and ranked the () from high to low, as , and the gray relational coefficient is between 0 and 1. The greater the relational degree, the stronger the relation, the correlation order is . According to the correlation order, the standby schemes can be sorted and scientific foundations for decision-making are offered.

2.2. BP Neural Network

The back propagation neural network (BPNN) can learn and store vast amounts of data because it is inspired by the structure of neurons, as illustrated in Figure 1 [18]. The BPNN is a feedforward multilevel neural network, which uses the network’s adaptive mapping ability to carry out back propagation and is able to realize any non-linear operation from input to output [19, 20]. From the analysis of the network structure, the BPNN includes an input layer, an output layer, and a hidden layer (which can also be multiple layers). Variables are read from the input layer through the network’s adaptive learning ability, and weights are calculated to determine the network output. The output result is compared with the target value, and the error is calculated. Through feedback and calculation for many iterations, the result can be output until the overall error of the network meets the requirements of the project.

The key to using the BPNN algorithm for fitting the amount of steel corrosion is to select the corrosion data set as the training set and to construct the mapping relationship between each corrosion value in the corrosion data set and the data of influential factors to allow effective training. Due to approximating ability to arbitrary nonlinear mapping, the BPNN has broad application in fitting steel corrosion.

2.2.1. BP Neuron

Assuming that represents the input of the neurons , respectively; represent the connection weights of and the j-th neuron, respectively; and is the threshold.

Suppose the net input value S_j of the jth neuron is as follows:where

If and we obtain

Then,

After the net input passes through the transfer function (this function is a monotonic rising function; there must be a maximum value), the output value of the j-th neuron is obtained:

2.2.2. BP Network

Assuming that the input layer, hidden layer, and output layer of the BP network have n, q, and m nodes, respectively, the weight values between the input layer and hidden layer and that between the hidden layer and output layer are and , respectively. The transfer function of the hidden layer is , then,

The transfer function of the output layer is , and an output value is obtained in accordance with the group of weights and thresholds.

2.2.3. Error Back Propagation

In error back propagation, the output error of each layer of neurons is calculated through the output layer, the weight and bias value of the hidden layer of the grid are adjusted according to the error gradient descent method, and the parameters are continuously modified to reduce the error during the training process. The final error objective function is as follows:where is the expected output value, and is the error objective function.

2.3. Gray Model GM (1, N)

2.3.1. Traditional Gray Model GM (1, N)

For a given system, assume that there are variables:

With one out put and inputs . These two kinds of sequences have strong correlations with each other. For each variable ,we assume that the sequence length is n, that is,

The 1-AGO sequences of are defined as follows:

The mean sequences generated by consecutive neighbors of are defined as follows:

The expression of the GM (1, N) model is as follows [21]:where is the development coefficient of the sequence, is the deriving term, and is the driving coefficient.

The equation (18) can be regarded as a system of linear equations with respect to the parameters , that is,where

Using the ordinary least-squares estimate (OLSE) method, the parameters P can be obtained as follows:

Thus, the time response function of the GM(1,N) model can be derived as:

It can get the predicted value as follows:

From the above discussion, we can see that the GM (1, N) model has some obvious defects, such as the mean coefficient of the sequence in equation (18) and the driving term in equation (18) are all constants, which may lead to poor predictive precision.

2.3.2. OGM (1, N) Model

Zhai et al. [21] and Kaki et al. [22] proposed an optimized multivariable gray prediction model OGM (1, N), which is different from the traditional gray model GM (1, N). The calculation steps are as follows:

Let the two variable sequences and be defined as in equations (15) and (16). Whereas the 1-AGO sequences be defined as:where, the parameter can be adjusted according to the simulation accuracy.

Assume that the sequence approximates the exponential change law, and its influence factor sequence is ; then, the sequence should satisfy the following first-order linear differential equation:

The above formula is discretized, and a linear correction term is added; then, the relationship between data points changes in the dependent variable sequence . Subsequently, the differential equation expression of is obtained as follows:

Upon discretizing it, we obtainwhere , and reflects the linear relationship between the dependent variable and the independent variable.

Compared with the traditional gray model GM (1, N), an additional linear correction term is introduced in (27) to improve the structure of the OGM (1, N) model.

In the new OGM (1, N) model (27), parameters, i.e., need to be estimated. These parameters can be estimated by solving the system of linear equations:where:

Substituting (30) into (27), the time response function of the GM (1, N) model can be derived as follows:where

Which is referred to as the model.

It can get the predicted value as follows:

3. Results

3.1. Calculation of Gray Relational Degree

The calculation of the gray relational degrees is based on the corrosion test data of steel bars presented in Tables 1 and 2. The data in Table 1 is the accelerated corrosion data of reinforced concrete indoors collected by Liu and Wan [23]; and the data in Table 2 is the practical engineering data collected by Chen et al. [9].

In Table 1, the steel bar corrosion rate and the steel bar corrosion extent are included in the main sequence. The crack width , corrosion time , corrosion current , protective layer thickness , steel bar diameter , and steel bar spacing are included in the reference sequence for calculating steel bar corrosion rate. The gray relational analysis of all influencing factors is carried out, and the gray relational degrees is obtained as follows:

Table 2 shows the crack width , concrete strength grade , and steel bar diameter . The thickness of the protective layer serves as the reference sequence for the amount of steel corrosion. The gray correlation analysis of all influencing factors is carried out, and the gray relational degrees is obtained as follows:

3.2. Establishment of Prediction Models Based on BPNN, GM (1, N), and OGM (1, N) Models

Using the accelerated corrosion data in Table 1, according to the results of the correlation analysis, the BPNN model selects the crack width, corrosion time, corrosion current, and protective layer thickness to form the input vector; the steel corrosion rate forms the output vector. A BPNN with four nodes in the input layer and one node in the output layer is established. Based on multiple fitting trials, the hidden layer is set to eight layers, and the learning rate is 0.035. In order to prevent the network from over-fitting, the noise intensity is set as 0.01. The target error value is specified as 0.65 × 10⁻³.

In GM (1, N) and OGM (1, N) modeling, crack width, corrosion time, and corrosion current with a greater correlation were included in the correlation sequences, and GM (1, 3) and OGM (1,3) models were established. The GM (1, 3) model predicted the parameters to be . After repeated trials, when , the fitting and prediction results of the OGM (1, 3) model were ideal, and the calculated prediction parameters are as follows:

According to practical engineering data presented in Table 2, gray relational degrees is relatively high; therefore, both the BPNN and gray multivariable model choose all influencing factors for modeling. The BPNN modeling process is the same as that using Table 1. The predicted parameters obtained by calculation using the GM (1, 4) model are . After repeated trials, when , the simulation and prediction results of the OGM (1, 4) model were satisfactory. The calculated prediction parameters are as follows:

3.3. Comparison of Calculation Results

The above three models are all modeled and fitted with N-3 groups of data, and the remaining 3 groups of data are predicted. The fitting results are shown in Figures 2–5. And the prediction results and errors of the different models are listed in Tables 3 and 4.

4. Discussion

It can be observed from Figure 2, for the corrosion data of the accelerated corrosion data, among those of the three calculation methods, the fitting error of the GM (1, N) model is relatively large, while the other two models have relatively small errors.

It can be seen from Figure 3 that the fitting error of the BPNN is much smaller than that of OGM (1, N). Because steel corrosion is affected by many factors, there is a highly nonlinear relationship between steel corrosion and each influencing factor, and the available data is limited. It is especially suitable for BPNN fitting; however, the BPNN may also exhibit over-fitting, resulting in regularity distortion and the predicted value error being relatively large.

It can be seen from Figures 4 and 5, for the measured corrosion data of practical engineering buildings, due to the large discretization of the measured values, the fitting and prediction errors obtained using the GM (1, N) and OGM (1, N) models are both large, while the predicted results obtained using the BPNN model are close to the real values.

It can be seen from the comparative analysis of the prediction results in Tables 3 and 4, the average fitting errors of the OGM (1, N) model for both the fast test and the engineering scatter data are larger than those of the BPNN model, whereas the predicted value of the OGM (1, N) for the accelerated corrosion data is closer to the real value with minimal error. However, the BPNN model has good prediction precision for practical engineering data.

5. Conclusions

The corrosion of steel bars has a great influence on the safety and durability of reinforced concrete structures. There are many factors that influence the corrosion of steel bars in practical engineering, such as the crack width of the concrete structure, strength grade of concrete, diameter of steel bar, thickness of the concrete protective layer, and environmental factors of the project. In this paper, the BPNN, GM (1, N), and OGM (1, N) are used to fit and predict the accelerated corrosion data and practical engineering data of the concrete structure, and the following conclusions are drawn:(1)Compared with the traditional gray model GM (1, N), the OGM (1, N) model exhibits a higher fitting and predicting accuracy.(2)Compared with the GM (1, N) and OGM (1, N) models, the BPNN model exhibits a higher fitting accuracy for the two kinds of data (the accelerated corrosion data and practical engineering data(3)The BPNN has a higher prediction accuracy for the practical engineering data, while the OGM (1, N) model has a higher prediction accuracy than the BPNN for the accelerated corrosion data.(4)In the practical engineering data, there are several influencing factors for the corrosion of steel bars. In this study, we only analyze a few influencing factors that cause greater correlation and do not consider the atmospheric parameters of the service environment of the actual engineering structure. More relevant data can be collected for model forecasting in actual engineering applications [23].

Data Availability

The indoor accelerated corrosion test of reinforced concrete data and actual engineering test data used to support the findings of this study are included within the articles “Predict Corrosion Degree of Steel Bars in Reinforcing Concrete Based on ABC-BP Neural Network” and “Assessment on corrosive degree of reinforcement in concrete by artificial neural networks,” respectively.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

Bai, X.W conducted Methodology and Software. Liu, J was responsible for writing, reviewing, and editing.

Acknowledgments

The authors would like to thank Editage (https://www.editage.cn) for English language editing. This research was funded by the Liaoning Provincial Education Department (Grant nos. LSNQN201904 and LJKZ0425), China and by Liaoning Petrochemical University (Grant no. 1100130241).

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Copyright © 2023 Juan Liu and Xuewei Bai. 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.

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