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

Authors: Juan Liu; Xuewei Bai
Journal: Computational Intelligence and Neuroscience
 (2023)

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## 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.


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## Body

## 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 areN variables:(1)X=X1,X2,…,XN.With one out putX1 and N−1 inputs Xii=2,3,…,N. These two kinds of sequences have strong correlations with each other. For each variable Xii=1,2,…,N, we assume that the sequence length is n, that is,(2)Xi=xi1,xi2,…xin,i=1,2,⋯,N.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:(3)x¯i=1n∑k=1nxik,σi=1n−1∑k=1nxik−x¯i,i=2,3,⋯,N,where x¯i is the mean of each inputs sequence, and σi is the variance of each inputs sequence.Then, we obtain(4)yik=xik−x¯iσi,i=2,3,⋯,N,where yik is the i-th input sequence after normalization.The difference sequences of the relevant factors are found as follows:(5)∆yik=y1k−yik,i=2,3,⋯,N.The correlation coefficient is calculated between the sequences as follows:(6)λx1k,xik=τmaximaxkx1k−xik+miniminkx1k−xikτmaximaxkx1k−xik+Δyik,0≺τ≺1,where i=2,3,⋯,N and k=1,2,⋯,n and τ is the resolution coefficient.The gray relational degree was calculated as follows:(7)λX1,Xi=1n∑k=1nλx1k,xik.The gray relational degrees of all the influencing factors were calculated and ranked theλi (i=2,3,⋯,N) from high to low, as λi2>λi3>⋯>λiN, and the gray relational coefficient is between 0 and 1. The greater the relational degree, the stronger the relation, the correlation order is λi2,λi3,⋯,λiN. 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 Figure1 [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.Figure 1
Structure of a BPNN with a single hidden layer [18].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 thatx1,x2,⋯,xn represents the input of the neurons 1,2,⋯,n, respectively; wj1,wj2,⋯,wjn represent the connection weights of 1,2,⋯,n and the j-th neuron, respectively; and bj is the threshold.Suppose the net input valueSj of the jth neuron is as follows:(8)Sj=∑i=1nwjixi+bj=WjX+bj,where X=x1,x2,⋯,xnT,Wj=wj1,wj2,⋯,wjnIfx0=1 and wj0=bj we obtain X=x0,x1,x2,⋯,xnT,Wj=wj0,wj1,wj2,⋯,wjnThen,(9)Sj=∑i=0nwjixi=WjX.After the net inputSj passes through the transfer function f• (this function is a monotonic rising function; there must be a maximum value), the output value yj of the j-th neuron is obtained:(10)yj=fsj=f∑i=0nwji⋅xi=FWjX.


#### 2.2.2. BP Network

Assuming that the input layer, hidden layer, and output layer of the BP network haven, 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 vki and wjk, respectively. The transfer function of the hidden layer is f1•, then,(11)zk=f1∑i=0nvkixi,k=1,2,⋯,q.The transfer function of the output layer isf2•, and an output value is obtained in accordance with the group of weights and thresholds.(12)yj=f2∑k=0qwjkzk,j=1,2,⋯,m.


#### 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:(13)E=12∑j=1n∑i=1mdi−yi2,where di is the expected output value, and E 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 areN variables:(14)X0=X10,X20,…,XN0.With one out putX10 and N−1 inputs Xi0i=2,3,…,N. These two kinds of sequences have strong correlations with each other. For each variable Xi0i=1,2,…,N,we assume that the sequence length is n, that is,(15)Xi0=xi01,xi02,…xi0n,i=1,2,⋯,N.The 1-AGO sequences ofXi0 are defined as follows:(16)Xi1=xi11,xi12,…,xi1n,i=1,2,⋯,N.The mean sequences generated by consecutive neighbors ofXi1 are defined as follows:(17)Zi1k=0.5xi1k−1+xi1k,k=2,3,⋯,n,i=1,2,⋯,N.The expression of the GM (1, N) model is as follows [21]:(18)dx11tdt+ax11t=∑i=2Nbixi1t,where a is the development coefficient of the sequence, ∑i=2Nbixi1t is the deriving term, and bi is the driving coefficient.The equation (18) can be regarded as a system of linear equations with respect to the parameters p=a,b2,...,bmT, that is,(19)Bp=Y,where(20)B=x212⋯xm12−z112x213⋯xm13−z113⋯x21n⋯xm1n−z11n,Y=x102x103⋮x10m.Using the ordinary least-squares estimate (OLSE) method, the parameters P can be obtained as follows:(21)p=a,b2,⋯,bmT=BTB−1BTY.Thus, the time response function of the GM(1,N) model can be derived as:(22)x^11t+1=x101−1a∑i=2mbixi1t+1e−at+1a∑i=2Nbixi1t+1.It can get the predicted value as follows:(23)x^10t+1=x^11t+1−x^11t.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 ∑i=1Nbixi1t 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 sequencesX10 and Xi0i=2,3,…,N be defined as in equations (15) and (16). Whereas the 1-AGO sequences be defined as:(24)Zi1k=γ−1xi1k−1−γxi1k,k=2,3,⋯,n,i=1,2,⋯,N,where, the parameter γ can be adjusted according to the simulation accuracy.Assume that theX11 sequence approximates the exponential change law, and its influence factor sequence is X21,X31,⋯,Xm1; then, the Xi1 sequence should satisfy the following first-order linear differential equation:(25)dX11dt=ax11+ρ2x21+ρ3x31+⋯+ρmxm1.The above formula is discretized, and a linear correction termck−1 is added; then, the relationship between data points changes in the dependent variable sequence ϕi. Subsequently, the differential equation expression of OGM1,N is obtained as follows:(26)x10k+az11k=∑i=2nρixi1k+ck−1+ϕ.Upon discretizing it, we obtain(27)x10k+γax11k−1+x11k=ρ2x21k+ρ3x31k+⋯+ρmxm1k+ck−1+ϕ,where k=1,2,⋯,n, and ck−1 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 termck−1 is introduced in (27) to improve the structure of the OGM (1, N) model.In the new OGM (1, N) model (27), N+2 parameters, i.e., P=a,ρ2,⋯,ρn,c,ϕT need to be estimated. These parameters can be estimated by solving the system of linear equations:(28)Bp=Y,where:(29)B=x212⋯xm12−z11221x213⋯xm13−z11331⋯x21n⋯xm1n−z11nn1,Y=x102x103⋮x10m,(30)P=aρ2⋯ρncϕT=BTB−1BTY.Substituting (30) into (27), the time response function of the GM (1, N) model can be derived as follows:(31)x^11k=∑t=1k−1η1∑i=2nη2t−1ρixi1k−t+1+η2k−1x^111+∑j=0k−2η2jk−jη3+η4,k=1,2,⋯,n,where(32)η1=11+γa,η2=1−a1+γa,η3=c1+γa,η4=ϕ1+γa,x^111=x101.Which is referred to as theOGM1,N model.It can get the predicted value as follows:(33)x^10k=x^11k−x^11k−1,k=1,2,⋯,n.


## 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 areN variables:(1)X=X1,X2,…,XN.With one out putX1 and N−1 inputs Xii=2,3,…,N. These two kinds of sequences have strong correlations with each other. For each variable Xii=1,2,…,N, we assume that the sequence length is n, that is,(2)Xi=xi1,xi2,…xin,i=1,2,⋯,N.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:(3)x¯i=1n∑k=1nxik,σi=1n−1∑k=1nxik−x¯i,i=2,3,⋯,N,where x¯i is the mean of each inputs sequence, and σi is the variance of each inputs sequence.Then, we obtain(4)yik=xik−x¯iσi,i=2,3,⋯,N,where yik is the i-th input sequence after normalization.The difference sequences of the relevant factors are found as follows:(5)∆yik=y1k−yik,i=2,3,⋯,N.The correlation coefficient is calculated between the sequences as follows:(6)λx1k,xik=τmaximaxkx1k−xik+miniminkx1k−xikτmaximaxkx1k−xik+Δyik,0≺τ≺1,where i=2,3,⋯,N and k=1,2,⋯,n and τ is the resolution coefficient.The gray relational degree was calculated as follows:(7)λX1,Xi=1n∑k=1nλx1k,xik.The gray relational degrees of all the influencing factors were calculated and ranked theλi (i=2,3,⋯,N) from high to low, as λi2>λi3>⋯>λiN, and the gray relational coefficient is between 0 and 1. The greater the relational degree, the stronger the relation, the correlation order is λi2,λi3,⋯,λiN. 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 Figure1 [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.Figure 1
Structure of a BPNN with a single hidden layer [18].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 thatx1,x2,⋯,xn represents the input of the neurons 1,2,⋯,n, respectively; wj1,wj2,⋯,wjn represent the connection weights of 1,2,⋯,n and the j-th neuron, respectively; and bj is the threshold.Suppose the net input valueSj of the jth neuron is as follows:(8)Sj=∑i=1nwjixi+bj=WjX+bj,where X=x1,x2,⋯,xnT,Wj=wj1,wj2,⋯,wjnIfx0=1 and wj0=bj we obtain X=x0,x1,x2,⋯,xnT,Wj=wj0,wj1,wj2,⋯,wjnThen,(9)Sj=∑i=0nwjixi=WjX.After the net inputSj passes through the transfer function f• (this function is a monotonic rising function; there must be a maximum value), the output value yj of the j-th neuron is obtained:(10)yj=fsj=f∑i=0nwji⋅xi=FWjX.


### 2.2.2. BP Network

Assuming that the input layer, hidden layer, and output layer of the BP network haven, 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 vki and wjk, respectively. The transfer function of the hidden layer is f1•, then,(11)zk=f1∑i=0nvkixi,k=1,2,⋯,q.The transfer function of the output layer isf2•, and an output value is obtained in accordance with the group of weights and thresholds.(12)yj=f2∑k=0qwjkzk,j=1,2,⋯,m.


### 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:(13)E=12∑j=1n∑i=1mdi−yi2,where di is the expected output value, and E is the error objective function.


## 2.2.1. BP Neuron

Assuming thatx1,x2,⋯,xn represents the input of the neurons 1,2,⋯,n, respectively; wj1,wj2,⋯,wjn represent the connection weights of 1,2,⋯,n and the j-th neuron, respectively; and bj is the threshold.Suppose the net input valueSj of the jth neuron is as follows:(8)Sj=∑i=1nwjixi+bj=WjX+bj,where X=x1,x2,⋯,xnT,Wj=wj1,wj2,⋯,wjnIfx0=1 and wj0=bj we obtain X=x0,x1,x2,⋯,xnT,Wj=wj0,wj1,wj2,⋯,wjnThen,(9)Sj=∑i=0nwjixi=WjX.After the net inputSj passes through the transfer function f• (this function is a monotonic rising function; there must be a maximum value), the output value yj of the j-th neuron is obtained:(10)yj=fsj=f∑i=0nwji⋅xi=FWjX.


## 2.2.2. BP Network

Assuming that the input layer, hidden layer, and output layer of the BP network haven, 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 vki and wjk, respectively. The transfer function of the hidden layer is f1•, then,(11)zk=f1∑i=0nvkixi,k=1,2,⋯,q.The transfer function of the output layer isf2•, and an output value is obtained in accordance with the group of weights and thresholds.(12)yj=f2∑k=0qwjkzk,j=1,2,⋯,m.


## 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:(13)E=12∑j=1n∑i=1mdi−yi2,where di is the expected output value, and E 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 areN variables:(14)X0=X10,X20,…,XN0.With one out putX10 and N−1 inputs Xi0i=2,3,…,N. These two kinds of sequences have strong correlations with each other. For each variable Xi0i=1,2,…,N,we assume that the sequence length is n, that is,(15)Xi0=xi01,xi02,…xi0n,i=1,2,⋯,N.The 1-AGO sequences ofXi0 are defined as follows:(16)Xi1=xi11,xi12,…,xi1n,i=1,2,⋯,N.The mean sequences generated by consecutive neighbors ofXi1 are defined as follows:(17)Zi1k=0.5xi1k−1+xi1k,k=2,3,⋯,n,i=1,2,⋯,N.The expression of the GM (1, N) model is as follows [21]:(18)dx11tdt+ax11t=∑i=2Nbixi1t,where a is the development coefficient of the sequence, ∑i=2Nbixi1t is the deriving term, and bi is the driving coefficient.The equation (18) can be regarded as a system of linear equations with respect to the parameters p=a,b2,...,bmT, that is,(19)Bp=Y,where(20)B=x212⋯xm12−z112x213⋯xm13−z113⋯x21n⋯xm1n−z11n,Y=x102x103⋮x10m.Using the ordinary least-squares estimate (OLSE) method, the parameters P can be obtained as follows:(21)p=a,b2,⋯,bmT=BTB−1BTY.Thus, the time response function of the GM(1,N) model can be derived as:(22)x^11t+1=x101−1a∑i=2mbixi1t+1e−at+1a∑i=2Nbixi1t+1.It can get the predicted value as follows:(23)x^10t+1=x^11t+1−x^11t.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 ∑i=1Nbixi1t 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 sequencesX10 and Xi0i=2,3,…,N be defined as in equations (15) and (16). Whereas the 1-AGO sequences be defined as:(24)Zi1k=γ−1xi1k−1−γxi1k,k=2,3,⋯,n,i=1,2,⋯,N,where, the parameter γ can be adjusted according to the simulation accuracy.Assume that theX11 sequence approximates the exponential change law, and its influence factor sequence is X21,X31,⋯,Xm1; then, the Xi1 sequence should satisfy the following first-order linear differential equation:(25)dX11dt=ax11+ρ2x21+ρ3x31+⋯+ρmxm1.The above formula is discretized, and a linear correction termck−1 is added; then, the relationship between data points changes in the dependent variable sequence ϕi. Subsequently, the differential equation expression of OGM1,N is obtained as follows:(26)x10k+az11k=∑i=2nρixi1k+ck−1+ϕ.Upon discretizing it, we obtain(27)x10k+γax11k−1+x11k=ρ2x21k+ρ3x31k+⋯+ρmxm1k+ck−1+ϕ,where k=1,2,⋯,n, and ck−1 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 termck−1 is introduced in (27) to improve the structure of the OGM (1, N) model.In the new OGM (1, N) model (27), N+2 parameters, i.e., P=a,ρ2,⋯,ρn,c,ϕT need to be estimated. These parameters can be estimated by solving the system of linear equations:(28)Bp=Y,where:(29)B=x212⋯xm12−z11221x213⋯xm13−z11331⋯x21n⋯xm1n−z11nn1,Y=x102x103⋮x10m,(30)P=aρ2⋯ρncϕT=BTB−1BTY.Substituting (30) into (27), the time response function of the GM (1, N) model can be derived as follows:(31)x^11k=∑t=1k−1η1∑i=2nη2t−1ρixi1k−t+1+η2k−1x^111+∑j=0k−2η2jk−jη3+η4,k=1,2,⋯,n,where(32)η1=11+γa,η2=1−a1+γa,η3=c1+γa,η4=ϕ1+γa,x^111=x101.Which is referred to as theOGM1,N model.It can get the predicted value as follows:(33)x^10k=x^11k−x^11k−1,k=1,2,⋯,n.


## 2.3.1. Traditional Gray Model GM (1, N)

For a given system, assume that there areN variables:(14)X0=X10,X20,…,XN0.With one out putX10 and N−1 inputs Xi0i=2,3,…,N. These two kinds of sequences have strong correlations with each other. For each variable Xi0i=1,2,…,N,we assume that the sequence length is n, that is,(15)Xi0=xi01,xi02,…xi0n,i=1,2,⋯,N.The 1-AGO sequences ofXi0 are defined as follows:(16)Xi1=xi11,xi12,…,xi1n,i=1,2,⋯,N.The mean sequences generated by consecutive neighbors ofXi1 are defined as follows:(17)Zi1k=0.5xi1k−1+xi1k,k=2,3,⋯,n,i=1,2,⋯,N.The expression of the GM (1, N) model is as follows [21]:(18)dx11tdt+ax11t=∑i=2Nbixi1t,where a is the development coefficient of the sequence, ∑i=2Nbixi1t is the deriving term, and bi is the driving coefficient.The equation (18) can be regarded as a system of linear equations with respect to the parameters p=a,b2,...,bmT, that is,(19)Bp=Y,where(20)B=x212⋯xm12−z112x213⋯xm13−z113⋯x21n⋯xm1n−z11n,Y=x102x103⋮x10m.Using the ordinary least-squares estimate (OLSE) method, the parameters P can be obtained as follows:(21)p=a,b2,⋯,bmT=BTB−1BTY.Thus, the time response function of the GM(1,N) model can be derived as:(22)x^11t+1=x101−1a∑i=2mbixi1t+1e−at+1a∑i=2Nbixi1t+1.It can get the predicted value as follows:(23)x^10t+1=x^11t+1−x^11t.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 ∑i=1Nbixi1t 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 sequencesX10 and Xi0i=2,3,…,N be defined as in equations (15) and (16). Whereas the 1-AGO sequences be defined as:(24)Zi1k=γ−1xi1k−1−γxi1k,k=2,3,⋯,n,i=1,2,⋯,N,where, the parameter γ can be adjusted according to the simulation accuracy.Assume that theX11 sequence approximates the exponential change law, and its influence factor sequence is X21,X31,⋯,Xm1; then, the Xi1 sequence should satisfy the following first-order linear differential equation:(25)dX11dt=ax11+ρ2x21+ρ3x31+⋯+ρmxm1.The above formula is discretized, and a linear correction termck−1 is added; then, the relationship between data points changes in the dependent variable sequence ϕi. Subsequently, the differential equation expression of OGM1,N is obtained as follows:(26)x10k+az11k=∑i=2nρixi1k+ck−1+ϕ.Upon discretizing it, we obtain(27)x10k+γax11k−1+x11k=ρ2x21k+ρ3x31k+⋯+ρmxm1k+ck−1+ϕ,where k=1,2,⋯,n, and ck−1 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 termck−1 is introduced in (27) to improve the structure of the OGM (1, N) model.In the new OGM (1, N) model (27), N+2 parameters, i.e., P=a,ρ2,⋯,ρn,c,ϕT need to be estimated. These parameters can be estimated by solving the system of linear equations:(28)Bp=Y,where:(29)B=x212⋯xm12−z11221x213⋯xm13−z11331⋯x21n⋯xm1n−z11nn1,Y=x102x103⋮x10m,(30)P=aρ2⋯ρncϕT=BTB−1BTY.Substituting (30) into (27), the time response function of the GM (1, N) model can be derived as follows:(31)x^11k=∑t=1k−1η1∑i=2nη2t−1ρixi1k−t+1+η2k−1x^111+∑j=0k−2η2jk−jη3+η4,k=1,2,⋯,n,where(32)η1=11+γa,η2=1−a1+γa,η3=c1+γa,η4=ϕ1+γa,x^111=x101.Which is referred to as theOGM1,N model.It can get the predicted value as follows:(33)x^10k=x^11k−x^11k−1,k=1,2,⋯,n.


## 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 Tables1 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].Table 1
Results of indoor accelerated corrosion test of reinforced concrete.
Data groupξ (mm/a)t (min)D (mm)c (mm)ω (mm)i (mm)d (mm)10.012218590.424.90.080.07812.0620.014298590.424.90.130.07812.1230.027418590.424.90.150.07812.3140.028498590.424.90.210.07812.3350.032588590.424.90.450.07812.3660.041665590.424.90.620.07812.3970.011665588440.060.01812.180.042781590.424.90.710.07812.490.015781588440.070.01812.15100.026913488440.080.01812.3110.052918590.424.90.830.07812.43120.0411125488440.140.01812.39130.1011129590.424.90.930.07812.5140.1182173590.424.90.980.07812.63150.132173588441.420.01812.76160.1333230088441.610.01812.87170.1223292590.424.91.010.07812.65180.1413813188441.620.01813.01190.1283813590.424.91.130.07812.68200.1426138590.424.91.620.07813.08210.1486138588442.910.01813.25220.1638764590.424.93.690.07813.65230.1738764588443.240.01813.71Table 2
The practical engineering data (Liaoning Benxi and Shanxi steel plant).
Data groupη (mm)ω (mm)fcu (MPa)c (mm)d (mm)16.840.1821.211522214.510.2521.631720324.530.33326.583018435.640.66729.863022515.70.522.345530640.7531.761430735.30.66721.213025814.81.125.681520918.60.66720.3112251021.10.66723.881625117.040.851817.312129.621815.412134.21.81812.1121410.920.51814.812159.70.5202081614.450.22520208177.330.3204012188.520.35102012196.370.25201012206.721.151815.712214.880.32020122221.20.728.2130252313.10.522.3133.530247.440.221.2120202520.6121.6430282614.050.630.541812277.680.91813.112288.321.51811.712295.850.3202016In Table1, the steel bar corrosion rate ξ and the steel bar corrosion extent η are included in the main sequence. The crack width ω, corrosion time t, corrosion current i, protective layer thickness c, steel bar diameter d, and steel bar spacing D 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:(34)λi=λω,λt,λi,λc,λd,λD=0.7704,0.7516,0.6268,0.6224,0.6021,0.5928.Table2 shows the crack width ω, concrete strength grade fcu, and steel bar diameter d. The thickness of the protective layer c 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:(35)λi=λd,λfcu,λc,λω=0.7943,0.7648,0.7604,0.6783.


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

Using the accelerated corrosion data in Table1, 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 beG=a,ρ2,⋯,ρnT=−0.1001,1.2176e−6,0.0272,0.0298T. After repeated trials, when γ=0.05, the fitting and prediction results of the OGM (1, 3) model were ideal, and the calculated prediction parameters are as follows:(36)G=a,ρ2,⋯,ρn,c,ϕT=0.0513,0,0.0332,0.0103,0.001,0.0037T,η1=1.0026,η2=1.0515,η3=0.001,η4=0.0037.According to practical engineering data presented in Table2, 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 G=aρ2⋯ρnT=0.0643,−8.3266,0.5578,−0.2714,0.0427T. After repeated trials, when γ=0.75, the simulation and prediction results of the OGM (1, 4) model were satisfactory. The calculated prediction parameters are as follows:(37)G=a,ρ2,⋯,ρn,c,ϕT=−4.0305,0.1797,−0.0306,−0.0849,−0.0112,0.6459,15.7473Tη1=1.0085,η2=1.0113,η3=0.6514,η4=15.8810.


### 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 Figures2–5. And the prediction results and errors of the different models are listed in Tables 3 and 4.Figure 2
Results of the different models for rapid corrosion test.Figure 3
Results of the BPNN and OGM (1, N) models for rapid corrosion test.Figure 4
Results of the different models for project data.Figure 5
Results of the BPNN and OGM (1, N) models for project data.Table 3
Comparison of the average fitting errors between models.
BPNN (%)GM (1, N) (%)OGM (1, N) (%)Accelerated corrosion test data9.3787.315.48Project data10.3094.2149.98Table 4
Comparison of prediction values and errors between models.
Actual valuesBPNNGM (1, N)OGM (1, N)Accelerated corrosion test data0.1480.151−1.590.1430.1630.1470.3860.1620.1730.15560.1570.17The average prediction error7.29%>100%1.9%Project data7.688.381−32.3955.9988.327.943103.2062.3145.858.780.9965.21The average prediction error8.2%>100%48%


## 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 Tables1 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].Table 1
Results of indoor accelerated corrosion test of reinforced concrete.
Data groupξ (mm/a)t (min)D (mm)c (mm)ω (mm)i (mm)d (mm)10.012218590.424.90.080.07812.0620.014298590.424.90.130.07812.1230.027418590.424.90.150.07812.3140.028498590.424.90.210.07812.3350.032588590.424.90.450.07812.3660.041665590.424.90.620.07812.3970.011665588440.060.01812.180.042781590.424.90.710.07812.490.015781588440.070.01812.15100.026913488440.080.01812.3110.052918590.424.90.830.07812.43120.0411125488440.140.01812.39130.1011129590.424.90.930.07812.5140.1182173590.424.90.980.07812.63150.132173588441.420.01812.76160.1333230088441.610.01812.87170.1223292590.424.91.010.07812.65180.1413813188441.620.01813.01190.1283813590.424.91.130.07812.68200.1426138590.424.91.620.07813.08210.1486138588442.910.01813.25220.1638764590.424.93.690.07813.65230.1738764588443.240.01813.71Table 2
The practical engineering data (Liaoning Benxi and Shanxi steel plant).
Data groupη (mm)ω (mm)fcu (MPa)c (mm)d (mm)16.840.1821.211522214.510.2521.631720324.530.33326.583018435.640.66729.863022515.70.522.345530640.7531.761430735.30.66721.213025814.81.125.681520918.60.66720.3112251021.10.66723.881625117.040.851817.312129.621815.412134.21.81812.1121410.920.51814.812159.70.5202081614.450.22520208177.330.3204012188.520.35102012196.370.25201012206.721.151815.712214.880.32020122221.20.728.2130252313.10.522.3133.530247.440.221.2120202520.6121.6430282614.050.630.541812277.680.91813.112288.321.51811.712295.850.3202016In Table1, the steel bar corrosion rate ξ and the steel bar corrosion extent η are included in the main sequence. The crack width ω, corrosion time t, corrosion current i, protective layer thickness c, steel bar diameter d, and steel bar spacing D 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:(34)λi=λω,λt,λi,λc,λd,λD=0.7704,0.7516,0.6268,0.6224,0.6021,0.5928.Table2 shows the crack width ω, concrete strength grade fcu, and steel bar diameter d. The thickness of the protective layer c 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:(35)λi=λd,λfcu,λc,λω=0.7943,0.7648,0.7604,0.6783.


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

Using the accelerated corrosion data in Table1, 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 beG=a,ρ2,⋯,ρnT=−0.1001,1.2176e−6,0.0272,0.0298T. After repeated trials, when γ=0.05, the fitting and prediction results of the OGM (1, 3) model were ideal, and the calculated prediction parameters are as follows:(36)G=a,ρ2,⋯,ρn,c,ϕT=0.0513,0,0.0332,0.0103,0.001,0.0037T,η1=1.0026,η2=1.0515,η3=0.001,η4=0.0037.According to practical engineering data presented in Table2, 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 G=aρ2⋯ρnT=0.0643,−8.3266,0.5578,−0.2714,0.0427T. After repeated trials, when γ=0.75, the simulation and prediction results of the OGM (1, 4) model were satisfactory. The calculated prediction parameters are as follows:(37)G=a,ρ2,⋯,ρn,c,ϕT=−4.0305,0.1797,−0.0306,−0.0849,−0.0112,0.6459,15.7473Tη1=1.0085,η2=1.0113,η3=0.6514,η4=15.8810.


## 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 Figures2–5. And the prediction results and errors of the different models are listed in Tables 3 and 4.Figure 2
Results of the different models for rapid corrosion test.Figure 3
Results of the BPNN and OGM (1, N) models for rapid corrosion test.Figure 4
Results of the different models for project data.Figure 5
Results of the BPNN and OGM (1, N) models for project data.Table 3
Comparison of the average fitting errors between models.
BPNN (%)GM (1, N) (%)OGM (1, N) (%)Accelerated corrosion test data9.3787.315.48Project data10.3094.2149.98Table 4
Comparison of prediction values and errors between models.
Actual valuesBPNNGM (1, N)OGM (1, N)Accelerated corrosion test data0.1480.151−1.590.1430.1630.1470.3860.1620.1730.15560.1570.17The average prediction error7.29%>100%1.9%Project data7.688.381−32.3955.9988.327.943103.2062.3145.858.780.9965.21The average prediction error8.2%>100%48%


## 4. Discussion

It can be observed from Figure2, 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 Figure3 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 Figures4 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 Tables3 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].



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*Source: 2695142-2023-01-05.xml*