# Investigation on the Displacement Ductility Coefficient of Reinforced Concrete Columns Strengthened with Textile-Reinforced Concrete

Authors: Boxue Wang; Shiping Yin; Ming Liu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

To evaluate the seismic performance of reinforced concrete (RC) columns strengthened with textile-reinforced concrete (TRC), based on the ABAQUS numerical analysis results of 15 TRC-strengthened RC columns, the grey correlation theory was used to determine the input variables of the model, and the accuracy of the numerical simulation results is verified by some experiments. Then, according to FEM data, a neural network prediction model was established for the displacement ductility coefficients of TRC-strengthened columns, and a formula was proposed for calculating the displacement ductility coefficient. The results showed that the BP (backpropagation) neural network model had good rationality and accuracy and that the ductility coefficients of the strengthened columns calculated by the model agreed well with the experimental values. Therefore, the model can be applied for predicting the displacement ductility coefficients of TRC-strengthened columns and can be used as a reference for engineering design.


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## Body

## 1. Introduction

Ductility is an important index for evaluating the seismic performance of concrete structures, and good ductility could avoid brittle failure of components and provide a certain safety reserve for components under accidental overload [1]. Reinforced concrete (RC) has been widely used in structural engineering applications all over the world [2], and as an important load-bearing component of structures, RC columns should provide sufficient strength, stiffness, and ductility so that the building can withstand earthquakes [3]. Therefore, studying the ductility of RC columns and evaluating their seismic performance are very important topics.There are still a large number of buildings around the world that are not considered for seismic design and are likely to be affected by earthquakes [4–6]. To reduce the adverse effects of earthquakes on existing structures, several strengthening methods have been proposed and widely applied in practical projects over the past few decades. Because these traditional strengthening methods adopt concrete or steel [6, 7], these methods have shortcomings, such as increasing the structural weight, easily developing rust, and changing the structural appearance. In recent years, with the development of composite materials, fibre-reinforced polymers (FRPs), a new type of reinforcement material, have been used to strengthen and repair existing structures. To evaluate the seismic performance of strengthened structures, many scholars have conducted a series of studies [8–12]. The results in the literature show that FRPs can enhance the ductility and energy dissipation capacity of strengthened columns. Although FRPs have been widely used for effective reinforcement, FRPs are not suitable for use in humid environments because of their poor compatibility with concrete substrates and their lack of vapour permeability due to the use of epoxy resin as an adhesive [13]. More importantly, it is difficult to examine and evaluate the status of the repaired structures, as the structures are obscured by the FRP reinforcement. To this end, Triantafillou et al. [13] introduced the concept of combining textile fibres with inorganic matrix (such as cement mortar). Due to the different matrix strengths and fibre types used by different scholars, this new type of cement-based material also has different names, such as textile-reinforced mortar (TRM) [13, 14], textile-reinforced concrete (TRC) [15], basalt-reinforced mortar (BRM) [16], and fabric-reinforced cementitious matrix (FRCM) [17]. The significant difference between FRP and cement-based matrix composites is the free edge stress concentration of FRP confinement, which will cause premature failure of the reinforcement system [18].Bournas et al. [19, 20] studied the seismic performance of RC columns strengthened by TRM under earthquake action. Their results showed that TRM can improve the seismic performance and deformation capacity of strengthened columns. Yin et al. [21] explored the influence of different factors on the seismic performance of TRC-strengthened columns, and their results indicated that the ductility coefficient of strengthened columns increased with increasing reinforcement layers and stirrup ratio or decreasing axial compression ratio. Al-Salloum et al. [22] studied the effect of TRM on the shear strength and ductility of beam-column joints with deficient seismic performance. Their results showed that TRM can effectively improve the shear strength and ductility of beam-column joints with defects. In addition, the results in [23, 24] indicate that although the ductility coefficient of TRC-strengthened columns decreases to some extent under environmental erosion, the overall seismic performance of the strengthened columns is good.Although TRC has received much attention in the field of seismic reinforcement, relatively few studies have focused on the ductility and deformation capacity of specimens after strengthening, and the relationships between various factors and the ductility coefficient have not been quantified. In addition, many problems in strengthening concrete column structures are nonlinear. The factors affecting the ductility of concrete columns are very complex, and there are certain coupling effects between the factors and the ductility of components. However, artificial neural networks have the ability to perform associative reasoning, simulation thinking, and self-adaptive recognition, which can be found in input-output mapping through learning [25, 26]. Therefore, artificial neural networks (ANN) are suitable for nonlinear reasoning and prediction in structural engineering. Backpropagation (BP) neural networks are the most widely used artificial neural networks, as BP neural networks have low requirements and limitations on known data and can make full use of existing data to analyze and study the evolution, development, and movement rules of objects [27, 28]. It has been successfully applied in civil engineering, such as design [29], prediction [30–33], optimization [34], and damage detection and identification [35, 36].Therefore, based on the team’s previous research results [37, 38], a BP neural network is applied to obtain the mapping relationship between ductility and various influencing factors of TRC-strengthened RC columns. Based on the mapping relationship, a ductility coefficient formula for reinforced columns is deduced, which enables accurate and efficient ductility calculations and provides a scientific basis for the evaluation of seismic performance of components and guidance for engineering design.


## 2. Grey Correlation Analysis

The grey system theory is based on the sequence operator to realize the law of things in the process of transformation. The original irregular data is generated into regular sequence, and then the differential equation method is used to predict and analyze the development trend of things objectively and scientifically. As the core content of grey system theory, grey correlation analysis is a method to predict the unknown information by using the known information. The grey correlation analysis is used to calculate the grey correlation degree of several factors. The larger the value of the correlation degree is, the greater the influence of the factor on the evaluation index is.


### 2.1. Significance of Grey Correlation Analysis

In this paper, grey system theory is applied to analyze the factors affecting the displacement ductility coefficient of TRC-strengthened columns, including the strength grade of existing concrete, axial compression ratio, shear span ratio, stirrup space, and number of textile layers. Finite element simulation results of TRC-strengthened columns were adopted as the sample data, and the following grey correlation analysis was carried out to obtain the ranking results of the correlation degree of various factors. The dominant factors affecting the displacement ductility coefficient of TRC-strengthened columns reflect the deformation capacity and seismic performance of TRC-strengthened columns to some extent. Therefore, the results of this study can also provide important references for future theoretical research and design applications of the displacement ductility coefficient of TRC-strengthened columns.


### 2.2. Correlation Analysis

In this paper, all the relevant correlation coefficients are gathered into one value, and the calculated average value is used to process the information. The correlation coefficient of each factor is calculated (i.e., the average value of all corresponding correlation coefficients), which is a quantitative analysis of the correlation degree between various factors and TRC-strengthened columns. The specific calculations and analyses are described hereafter.


#### 2.2.1. Determination of Correlation Factors

According to the results of existing research, the main factors affecting the displacement ductility coefficient of strengthened RC columns are the strength grade of existing concrete, axial compression ratio, shear span ratio, stirrup space, and strengthening layers. The main content of this paper is to determine the influence of strengthening layers, axial compression ratio, shear span ratio, and other parameters on the ductility coefficient. Therefore, the following correlation factors are selected as the comparison parameter sequence of the system: (1) reinforcement layersx1 (0, 1, 2, 3, 4), (2) axial compression ratio x2 (0.2, 0.3, 0.4, 0.5), (3) shear span ratio x3 (2, 3.8, 5.6), (4) stirrup space x4 (50 mm, 100 mm, 150 mm), and (5) concrete strength grade x5 (C30, C40, C50, C60), namely, {xi(k) | i = 1, 2, 3, 4, 5; k = 1, 2, 3, 4, 5}. The displacement ductility coefficient is obtained from 15 TRC-strengthened columns based on ABAQUS earthquake simulations in [37, 38] used as the generating sequence of the system. The specific geometric dimensions and reinforcing bars of the specimens are shown in Figure 1. And according to [37, 38], the numerical and experimental results match well (shown in Figure 2 [38]), which shows that the established model and the finite element analysis results are rational and accurate. The correlation factors and displacement ductility coefficient involved in the numerical analysis are shown in Table 1.Figure 1
Specimen size and reinforcement details.Figure 2
Hysteresis curves with different axial compression ratios of TRC-strengthened RC columns. (a) Axial compression ratios with 0.25. (b) Axial compression ratios with 0.15.
(a)(b)Table 1
Original data of the specimens.
Serial numberDuctility coefficientx0kReinforcement layerx1kAxial compression ratiox2kShear span ratiox3kStirrup reinforcement ratiox4k (%)Concrete strength gradex5k16.7610.33.80.344024.3600.33.80.344036.9120.33.80.344046.5030.33.80.344056.6040.33.80.3440610.5820.23.80.344075.3020.43.80.344083.7320.53.80.344093.3820.32.00.3440106.6920.35.60.3440117.5920.33.80.6740125.3220.33.80.2240133.7920.33.80.3430145.0320.33.80.3450155.0020.33.80.3460


#### 2.2.2. Dimensionless Data

The original data are initialized according to equation (1) [39], and the initialized data are shown in Table 2.(1)xi′k=xikxi1,i=0,1,2,3,4,5;k=1,2,…,15.Table 2
Initialized data of the specimens.
Serial numberDuctility coefficientx0′kReinforcement layerx1′kAxial compression ratiox2′kShear span ratiox3′kStirrup reinforcement ratiox4′kConcrete strength gradex5′k11.001.001.001.001.001.0020.640.001.001.001.001.0031.022.001.001.001.001.0040.963.001.001.001.001.0050.984.001.001.001.001.0061.572.000.671.001.001.0070.782.001.331.001.001.0080.552.001.671.001.001.0090.502.001.000.531.001.00100.992.001.001.471.001.00111.122.001.001.001.971.00120.792.001.001.000.651.00130.562.001.001.001.000.75140.742.001.001.001.001.25150.742.001.001.001.001.50


#### 2.2.3. Calculation of Absolute Difference

According to equation (2), the absolute difference between the parent sequence and subsequence in Table 2 (i.e., the absolute difference value results for Δi(k)) is shown in Table 3.(2)Δik=|x0′k−xi′k|,i′=1,2,3,4,5;k=1,2,…,15,where x0′k is for the parent sequence (dimensionless) and xi′k is for the subsequence (dimensionless).Table 3
Absolute difference of the specimens.
Serial numberDuctility coefficientΔ0kReinforcement layerΔ1kAxial compression ratioΔ2kShear span ratioΔ3kStirrup reinforcement ratioΔ4kConcrete strength gradeΔ5k110000020.640.640.360.360.360.3631.020.980.020.020.020.0240.962.040.040.040.040.0450.983.020.020.020.020.0261.570.430.90.570.570.5770.781.220.550.220.220.2280.551.451.120.450.450.4590.51.50.50.030.50.5100.991.010.010.480.010.01111.120.880.120.120.850.12120.791.210.210.210.140.21130.561.440.440.440.440.19140.741.260.260.260.260.51150.741.260.260.260.260.76


#### 2.2.4. Calculation of the Correlation Coefficient

Equation (3) is used to calculate the correlation coefficient γ0ik, and the results are shown in Table 4.(3)γ0ik=m+ZMΔik+ZM,Z∈0,1,i=1,2,3,4,5;k=1,2,3,…,15,where γ0ik is the correlation coefficient; Z is the resolution coefficient, which has a value that is generally between 0 and 1 and is typically taken as 0.5; m is the minimum second-order difference, which is 0 according to m=miniminkΔik; and M is the maximum second-order difference, which is 3.02 according to m=maximaxkΔik.Table 4
Correlation coefficient data of the specimens.
Serial numberγ01kγ02kγ03kγ04kγ05k11.001.001.001.001.0020.700.810.810.810.8130.610.990.990.990.9940.430.970.970.970.9750.330.990.990.990.9960.780.630.730.730.7370.550.730.870.870.8780.510.570.770.770.7790.500.750.980.750.75100.600.990.760.990.99110.630.930.930.640.93120.560.880.880.920.88130.510.770.770.770.89140.550.850.850.850.75150.550.850.850.850.67


#### 2.2.5. Calculation of the Correlation Degree

The correlation degree between the subsequencexi and the parent sequence x0 is calculated according to equation (4), which is expressed as follows:(4)γi=1n∑k=1nγ0ik,n=15,i=1,2,3,4,5;k=1,2,3,…,15.The calculated correlation degree isγi=0.59,0.85,0.88,0.86,0.86|i=1,2,3,4,5.


#### 2.2.6. Ranking of Correlation Degree

By sorting the correlation degree obtained from the calculation, the results show that the influence degree of each factor on the displacement ductility coefficient of TRC-strengthened columns is shear span ratio > (axial compression ratio, stirrup ratio, concrete strength) > TRC layers. In addition, combined with the value of the correlation degree, these five factors have a great impact on the ductility performance of TRC-strengthened columns. Therefore, based on these five factors, a BP neural network model was established by MATLAB to predict the ductility change trend of the TRC-strengthened columns.


## 2.1. Significance of Grey Correlation Analysis

In this paper, grey system theory is applied to analyze the factors affecting the displacement ductility coefficient of TRC-strengthened columns, including the strength grade of existing concrete, axial compression ratio, shear span ratio, stirrup space, and number of textile layers. Finite element simulation results of TRC-strengthened columns were adopted as the sample data, and the following grey correlation analysis was carried out to obtain the ranking results of the correlation degree of various factors. The dominant factors affecting the displacement ductility coefficient of TRC-strengthened columns reflect the deformation capacity and seismic performance of TRC-strengthened columns to some extent. Therefore, the results of this study can also provide important references for future theoretical research and design applications of the displacement ductility coefficient of TRC-strengthened columns.


## 2.2. Correlation Analysis

In this paper, all the relevant correlation coefficients are gathered into one value, and the calculated average value is used to process the information. The correlation coefficient of each factor is calculated (i.e., the average value of all corresponding correlation coefficients), which is a quantitative analysis of the correlation degree between various factors and TRC-strengthened columns. The specific calculations and analyses are described hereafter.


### 2.2.1. Determination of Correlation Factors

According to the results of existing research, the main factors affecting the displacement ductility coefficient of strengthened RC columns are the strength grade of existing concrete, axial compression ratio, shear span ratio, stirrup space, and strengthening layers. The main content of this paper is to determine the influence of strengthening layers, axial compression ratio, shear span ratio, and other parameters on the ductility coefficient. Therefore, the following correlation factors are selected as the comparison parameter sequence of the system: (1) reinforcement layersx1 (0, 1, 2, 3, 4), (2) axial compression ratio x2 (0.2, 0.3, 0.4, 0.5), (3) shear span ratio x3 (2, 3.8, 5.6), (4) stirrup space x4 (50 mm, 100 mm, 150 mm), and (5) concrete strength grade x5 (C30, C40, C50, C60), namely, {xi(k) | i = 1, 2, 3, 4, 5; k = 1, 2, 3, 4, 5}. The displacement ductility coefficient is obtained from 15 TRC-strengthened columns based on ABAQUS earthquake simulations in [37, 38] used as the generating sequence of the system. The specific geometric dimensions and reinforcing bars of the specimens are shown in Figure 1. And according to [37, 38], the numerical and experimental results match well (shown in Figure 2 [38]), which shows that the established model and the finite element analysis results are rational and accurate. The correlation factors and displacement ductility coefficient involved in the numerical analysis are shown in Table 1.Figure 1
Specimen size and reinforcement details.Figure 2
Hysteresis curves with different axial compression ratios of TRC-strengthened RC columns. (a) Axial compression ratios with 0.25. (b) Axial compression ratios with 0.15.
(a)(b)Table 1
Original data of the specimens.
Serial numberDuctility coefficientx0kReinforcement layerx1kAxial compression ratiox2kShear span ratiox3kStirrup reinforcement ratiox4k (%)Concrete strength gradex5k16.7610.33.80.344024.3600.33.80.344036.9120.33.80.344046.5030.33.80.344056.6040.33.80.3440610.5820.23.80.344075.3020.43.80.344083.7320.53.80.344093.3820.32.00.3440106.6920.35.60.3440117.5920.33.80.6740125.3220.33.80.2240133.7920.33.80.3430145.0320.33.80.3450155.0020.33.80.3460


### 2.2.2. Dimensionless Data

The original data are initialized according to equation (1) [39], and the initialized data are shown in Table 2.(1)xi′k=xikxi1,i=0,1,2,3,4,5;k=1,2,…,15.Table 2
Initialized data of the specimens.
Serial numberDuctility coefficientx0′kReinforcement layerx1′kAxial compression ratiox2′kShear span ratiox3′kStirrup reinforcement ratiox4′kConcrete strength gradex5′k11.001.001.001.001.001.0020.640.001.001.001.001.0031.022.001.001.001.001.0040.963.001.001.001.001.0050.984.001.001.001.001.0061.572.000.671.001.001.0070.782.001.331.001.001.0080.552.001.671.001.001.0090.502.001.000.531.001.00100.992.001.001.471.001.00111.122.001.001.001.971.00120.792.001.001.000.651.00130.562.001.001.001.000.75140.742.001.001.001.001.25150.742.001.001.001.001.50


### 2.2.3. Calculation of Absolute Difference

According to equation (2), the absolute difference between the parent sequence and subsequence in Table 2 (i.e., the absolute difference value results for Δi(k)) is shown in Table 3.(2)Δik=|x0′k−xi′k|,i′=1,2,3,4,5;k=1,2,…,15,where x0′k is for the parent sequence (dimensionless) and xi′k is for the subsequence (dimensionless).Table 3
Absolute difference of the specimens.
Serial numberDuctility coefficientΔ0kReinforcement layerΔ1kAxial compression ratioΔ2kShear span ratioΔ3kStirrup reinforcement ratioΔ4kConcrete strength gradeΔ5k110000020.640.640.360.360.360.3631.020.980.020.020.020.0240.962.040.040.040.040.0450.983.020.020.020.020.0261.570.430.90.570.570.5770.781.220.550.220.220.2280.551.451.120.450.450.4590.51.50.50.030.50.5100.991.010.010.480.010.01111.120.880.120.120.850.12120.791.210.210.210.140.21130.561.440.440.440.440.19140.741.260.260.260.260.51150.741.260.260.260.260.76


### 2.2.4. Calculation of the Correlation Coefficient

Equation (3) is used to calculate the correlation coefficient γ0ik, and the results are shown in Table 4.(3)γ0ik=m+ZMΔik+ZM,Z∈0,1,i=1,2,3,4,5;k=1,2,3,…,15,where γ0ik is the correlation coefficient; Z is the resolution coefficient, which has a value that is generally between 0 and 1 and is typically taken as 0.5; m is the minimum second-order difference, which is 0 according to m=miniminkΔik; and M is the maximum second-order difference, which is 3.02 according to m=maximaxkΔik.Table 4
Correlation coefficient data of the specimens.
Serial numberγ01kγ02kγ03kγ04kγ05k11.001.001.001.001.0020.700.810.810.810.8130.610.990.990.990.9940.430.970.970.970.9750.330.990.990.990.9960.780.630.730.730.7370.550.730.870.870.8780.510.570.770.770.7790.500.750.980.750.75100.600.990.760.990.99110.630.930.930.640.93120.560.880.880.920.88130.510.770.770.770.89140.550.850.850.850.75150.550.850.850.850.67


### 2.2.5. Calculation of the Correlation Degree

The correlation degree between the subsequencexi and the parent sequence x0 is calculated according to equation (4), which is expressed as follows:(4)γi=1n∑k=1nγ0ik,n=15,i=1,2,3,4,5;k=1,2,3,…,15.The calculated correlation degree isγi=0.59,0.85,0.88,0.86,0.86|i=1,2,3,4,5.


### 2.2.6. Ranking of Correlation Degree

By sorting the correlation degree obtained from the calculation, the results show that the influence degree of each factor on the displacement ductility coefficient of TRC-strengthened columns is shear span ratio > (axial compression ratio, stirrup ratio, concrete strength) > TRC layers. In addition, combined with the value of the correlation degree, these five factors have a great impact on the ductility performance of TRC-strengthened columns. Therefore, based on these five factors, a BP neural network model was established by MATLAB to predict the ductility change trend of the TRC-strengthened columns.


## 2.2.1. Determination of Correlation Factors

According to the results of existing research, the main factors affecting the displacement ductility coefficient of strengthened RC columns are the strength grade of existing concrete, axial compression ratio, shear span ratio, stirrup space, and strengthening layers. The main content of this paper is to determine the influence of strengthening layers, axial compression ratio, shear span ratio, and other parameters on the ductility coefficient. Therefore, the following correlation factors are selected as the comparison parameter sequence of the system: (1) reinforcement layersx1 (0, 1, 2, 3, 4), (2) axial compression ratio x2 (0.2, 0.3, 0.4, 0.5), (3) shear span ratio x3 (2, 3.8, 5.6), (4) stirrup space x4 (50 mm, 100 mm, 150 mm), and (5) concrete strength grade x5 (C30, C40, C50, C60), namely, {xi(k) | i = 1, 2, 3, 4, 5; k = 1, 2, 3, 4, 5}. The displacement ductility coefficient is obtained from 15 TRC-strengthened columns based on ABAQUS earthquake simulations in [37, 38] used as the generating sequence of the system. The specific geometric dimensions and reinforcing bars of the specimens are shown in Figure 1. And according to [37, 38], the numerical and experimental results match well (shown in Figure 2 [38]), which shows that the established model and the finite element analysis results are rational and accurate. The correlation factors and displacement ductility coefficient involved in the numerical analysis are shown in Table 1.Figure 1
Specimen size and reinforcement details.Figure 2
Hysteresis curves with different axial compression ratios of TRC-strengthened RC columns. (a) Axial compression ratios with 0.25. (b) Axial compression ratios with 0.15.
(a)(b)Table 1
Original data of the specimens.
Serial numberDuctility coefficientx0kReinforcement layerx1kAxial compression ratiox2kShear span ratiox3kStirrup reinforcement ratiox4k (%)Concrete strength gradex5k16.7610.33.80.344024.3600.33.80.344036.9120.33.80.344046.5030.33.80.344056.6040.33.80.3440610.5820.23.80.344075.3020.43.80.344083.7320.53.80.344093.3820.32.00.3440106.6920.35.60.3440117.5920.33.80.6740125.3220.33.80.2240133.7920.33.80.3430145.0320.33.80.3450155.0020.33.80.3460


## 2.2.2. Dimensionless Data

The original data are initialized according to equation (1) [39], and the initialized data are shown in Table 2.(1)xi′k=xikxi1,i=0,1,2,3,4,5;k=1,2,…,15.Table 2
Initialized data of the specimens.
Serial numberDuctility coefficientx0′kReinforcement layerx1′kAxial compression ratiox2′kShear span ratiox3′kStirrup reinforcement ratiox4′kConcrete strength gradex5′k11.001.001.001.001.001.0020.640.001.001.001.001.0031.022.001.001.001.001.0040.963.001.001.001.001.0050.984.001.001.001.001.0061.572.000.671.001.001.0070.782.001.331.001.001.0080.552.001.671.001.001.0090.502.001.000.531.001.00100.992.001.001.471.001.00111.122.001.001.001.971.00120.792.001.001.000.651.00130.562.001.001.001.000.75140.742.001.001.001.001.25150.742.001.001.001.001.50


## 2.2.3. Calculation of Absolute Difference

According to equation (2), the absolute difference between the parent sequence and subsequence in Table 2 (i.e., the absolute difference value results for Δi(k)) is shown in Table 3.(2)Δik=|x0′k−xi′k|,i′=1,2,3,4,5;k=1,2,…,15,where x0′k is for the parent sequence (dimensionless) and xi′k is for the subsequence (dimensionless).Table 3
Absolute difference of the specimens.
Serial numberDuctility coefficientΔ0kReinforcement layerΔ1kAxial compression ratioΔ2kShear span ratioΔ3kStirrup reinforcement ratioΔ4kConcrete strength gradeΔ5k110000020.640.640.360.360.360.3631.020.980.020.020.020.0240.962.040.040.040.040.0450.983.020.020.020.020.0261.570.430.90.570.570.5770.781.220.550.220.220.2280.551.451.120.450.450.4590.51.50.50.030.50.5100.991.010.010.480.010.01111.120.880.120.120.850.12120.791.210.210.210.140.21130.561.440.440.440.440.19140.741.260.260.260.260.51150.741.260.260.260.260.76


## 2.2.4. Calculation of the Correlation Coefficient

Equation (3) is used to calculate the correlation coefficient γ0ik, and the results are shown in Table 4.(3)γ0ik=m+ZMΔik+ZM,Z∈0,1,i=1,2,3,4,5;k=1,2,3,…,15,where γ0ik is the correlation coefficient; Z is the resolution coefficient, which has a value that is generally between 0 and 1 and is typically taken as 0.5; m is the minimum second-order difference, which is 0 according to m=miniminkΔik; and M is the maximum second-order difference, which is 3.02 according to m=maximaxkΔik.Table 4
Correlation coefficient data of the specimens.
Serial numberγ01kγ02kγ03kγ04kγ05k11.001.001.001.001.0020.700.810.810.810.8130.610.990.990.990.9940.430.970.970.970.9750.330.990.990.990.9960.780.630.730.730.7370.550.730.870.870.8780.510.570.770.770.7790.500.750.980.750.75100.600.990.760.990.99110.630.930.930.640.93120.560.880.880.920.88130.510.770.770.770.89140.550.850.850.850.75150.550.850.850.850.67


## 2.2.5. Calculation of the Correlation Degree

The correlation degree between the subsequencexi and the parent sequence x0 is calculated according to equation (4), which is expressed as follows:(4)γi=1n∑k=1nγ0ik,n=15,i=1,2,3,4,5;k=1,2,3,…,15.The calculated correlation degree isγi=0.59,0.85,0.88,0.86,0.86|i=1,2,3,4,5.


## 2.2.6. Ranking of Correlation Degree

By sorting the correlation degree obtained from the calculation, the results show that the influence degree of each factor on the displacement ductility coefficient of TRC-strengthened columns is shear span ratio > (axial compression ratio, stirrup ratio, concrete strength) > TRC layers. In addition, combined with the value of the correlation degree, these five factors have a great impact on the ductility performance of TRC-strengthened columns. Therefore, based on these five factors, a BP neural network model was established by MATLAB to predict the ductility change trend of the TRC-strengthened columns.


## 3. Establishment of Neural Network Prediction Model

Based on MATLAB, the BP neural network model is established to predict the displacement ductility coefficient of TRC-strengthened columns. The specific operation steps are as follows.


### 3.1. BP Neural Network Construction

The structure of BP neural network mainly includes the number of neural network layers and neurons in each layer. Because the three-layer BP neural network has strong nonlinear mapping ability and can approach any nonlinear function, the three-layer BP neural network was used to model in this paper. The numbers of input layers, hidden layers, and output layers are all one. Based on the analysis mentioned above, BP neural network includes five input parameters, i.e., concrete strength grade, axial compression ratio, shear span ratio, stirrup space, and strengthening layers, and the output parameter is displacement ductility coefficient. The number of input layer and output layer neurons in the corresponding model is 5 and 1, respectively. The number of hidden layer neurons is determined by formula (5) [40] and multiple training calculations. The structure of BP neural network model is shown in Figure 3.(5)D=Di+D0+a,where D is the number of nodes in the hidden layer, Di is the number of input nodes, D0 is the number of output nodes, and a is between 1 and 10.Figure 3
Architecture of three-layer backpropagation (BP) neural network model.In addition, the essence of BP learning algorithm is to get the minimum value of error function. Because it uses the steepest descent method in nonlinear programming to modify the weights according to the negative gradient direction of error function, it usually has shortcomings such as low learning efficiency, slow convergence speed, and easy fall into local minimum state. Several optimization techniques are proposed to speed up the convergence of the BP learning algorithm [41, 42]. The results indicate that the Levenberg-Marquardt (L-M) algorithm was very efficient when training networks. The flowchart of L-M algorithm is shown in Figure 4.Figure 4
A flowchart of Levenberg-Marquardt algorithm.


### 3.2. Determine Sample Data

A small number of experiments have been designed to prove the accuracy of the finite element results. After verification, the finite element simulation results are in good agreement with the test results, which proves that the finite element simulation results are accurate. In order to better perform parametric analysis, in this paper, the training sample data (10 groups) and test sample data (5 groups) for the model are determined according to the data obtained from simulation calculations. Moreover, considering the calculation efficiency and error of the model, all the sample data are normalized according to equation (6) before the sample is input into the model. The relevant linear transformation algorithm is expressed as follows:(6)y=x−minmax−min,where x is the input vector, min is the minimum value of x, max is the maximum value of x, and y is the output vector.At the same time, the parameterεi (weight coefficient) is introduced to obtain the true correlation between the influencing factors and TRC-strengthened columns, and the specific formula is shown in equation (7). The weight coefficient is combined with the data in Table 5 to obtain the input data for the final model. The specific results are shown in Table 6.(7)εi=γi∑k=15γ0i,i=1,2,3,4,5,where γi is the correlation value between each factor and displacement ductility coefficient calculated in Section 1 and Section 2 of this paper.Table 5
Normalized data of the specimens.
Serial numberReinforcement layerAxial compression ratioShear span ratioStirrup reinforcement ratioConcrete strength grade10.000.330.500.270.3320.250.330.500.270.3330.500.330.500.270.3340.750.330.500.270.3351.000.330.500.270.3360.500.000.500.270.3370.500.670.500.270.3380.501.000.500.270.3390.500.330.000.270.33100.500.331.000.270.33110.500.330.501.000.33120.500.330.500.000.33130.500.330.500.270.00140.500.330.500.270.67150.500.330.500.271.00Table 6
Sample input sample data.
Serial numberReinforcement layerAxial compression ratioShear span ratioStirrup reinforcement ratioConcrete strength grade10.000.070.110.060.0720.040.070.110.060.0730.070.070.110.060.0740.110.070.110.060.0750.150.070.110.060.0760.070.000.110.060.0770.070.140.110.060.0780.070.210.110.060.0790.070.070.000.060.07100.070.070.220.060.07110.070.070.110.210.07120.070.070.110.000.07130.070.070.110.060.00140.070.070.110.060.14150.070.070.110.060.21In this paper, the five associated valuesγi=0.59,0.85,0.88,0.86,0.86|k=1,2,3,4,5 obtained in the previous section are substituted into (7), and the weight coefficients of each factor are obtained as follows: 0.146, 0.210, 0.218, 0.213, and 0.213. Then, the weight coefficients are multiplied by the normalized data to obtain the input data. The detailed results are shown in Table 6.


### 3.3. Simulation of the Neural Network

The sample data obtained above (shown in Table6) were input into the established and trained neural network model, and the accuracy of the model was verified by comparing and analyzing the results of the neural network model and the simulated calculation.


### 3.4. Neural Network Predicted Value

Table7 and Figure 5 show a comparison between the predicted value of the displacement ductility coefficient calculated by the BP neural network model and the displacement ductility coefficient of the TRC reinforced column obtained by finite element analysis in previous studies.Table 7
Comparison of measured and predicted values.
Serial numberDisplacement ductility coefficientMeasured valuePredicted valueMeasured/predicted value14.364.520.96426.765.911.14436.916.421.07646.56.401.01656.66.491.016610.5810.620.99675.35.231.01483.735.690.65693.383.390.998106.696.770.989117.597.561.004125.325.271.009133.795.740.661145.035.120.9821554.991.002Figure 5
Ratio of measured value and predicted value of displacement ductility coefficient (Table7).In addition, data analysis software, Statistical Product and Service Solutions (SPSS) [43], was used to compare the predicted value of displacement ductility coefficient calculated in the above BP neural network model with the experimental value obtained by numerical analysis, and the correlation between the value predicted by the BP neural network and the experimental value was obtained. The specific parameters are shown in Table 8.Table 8
BP neural network predicted value fit.
ModelRR2AdjustedR2Standard deviation error10.9110.8300.8170.696Table8 shows that the R2 of the fitting degree between the predicted value of displacement ductility coefficient calculated based on the BP neural network model and the measured value calculated by the above simulation is 0.83. Therefore, the results show that the BP neural network model based on the grey correlation analysis method has a good prediction effect on the displacement ductility coefficient of TRC-strengthened RC columns.


## 3.1. BP Neural Network Construction

The structure of BP neural network mainly includes the number of neural network layers and neurons in each layer. Because the three-layer BP neural network has strong nonlinear mapping ability and can approach any nonlinear function, the three-layer BP neural network was used to model in this paper. The numbers of input layers, hidden layers, and output layers are all one. Based on the analysis mentioned above, BP neural network includes five input parameters, i.e., concrete strength grade, axial compression ratio, shear span ratio, stirrup space, and strengthening layers, and the output parameter is displacement ductility coefficient. The number of input layer and output layer neurons in the corresponding model is 5 and 1, respectively. The number of hidden layer neurons is determined by formula (5) [40] and multiple training calculations. The structure of BP neural network model is shown in Figure 3.(5)D=Di+D0+a,where D is the number of nodes in the hidden layer, Di is the number of input nodes, D0 is the number of output nodes, and a is between 1 and 10.Figure 3
Architecture of three-layer backpropagation (BP) neural network model.In addition, the essence of BP learning algorithm is to get the minimum value of error function. Because it uses the steepest descent method in nonlinear programming to modify the weights according to the negative gradient direction of error function, it usually has shortcomings such as low learning efficiency, slow convergence speed, and easy fall into local minimum state. Several optimization techniques are proposed to speed up the convergence of the BP learning algorithm [41, 42]. The results indicate that the Levenberg-Marquardt (L-M) algorithm was very efficient when training networks. The flowchart of L-M algorithm is shown in Figure 4.Figure 4
A flowchart of Levenberg-Marquardt algorithm.


## 3.2. Determine Sample Data

A small number of experiments have been designed to prove the accuracy of the finite element results. After verification, the finite element simulation results are in good agreement with the test results, which proves that the finite element simulation results are accurate. In order to better perform parametric analysis, in this paper, the training sample data (10 groups) and test sample data (5 groups) for the model are determined according to the data obtained from simulation calculations. Moreover, considering the calculation efficiency and error of the model, all the sample data are normalized according to equation (6) before the sample is input into the model. The relevant linear transformation algorithm is expressed as follows:(6)y=x−minmax−min,where x is the input vector, min is the minimum value of x, max is the maximum value of x, and y is the output vector.At the same time, the parameterεi (weight coefficient) is introduced to obtain the true correlation between the influencing factors and TRC-strengthened columns, and the specific formula is shown in equation (7). The weight coefficient is combined with the data in Table 5 to obtain the input data for the final model. The specific results are shown in Table 6.(7)εi=γi∑k=15γ0i,i=1,2,3,4,5,where γi is the correlation value between each factor and displacement ductility coefficient calculated in Section 1 and Section 2 of this paper.Table 5
Normalized data of the specimens.
Serial numberReinforcement layerAxial compression ratioShear span ratioStirrup reinforcement ratioConcrete strength grade10.000.330.500.270.3320.250.330.500.270.3330.500.330.500.270.3340.750.330.500.270.3351.000.330.500.270.3360.500.000.500.270.3370.500.670.500.270.3380.501.000.500.270.3390.500.330.000.270.33100.500.331.000.270.33110.500.330.501.000.33120.500.330.500.000.33130.500.330.500.270.00140.500.330.500.270.67150.500.330.500.271.00Table 6
Sample input sample data.
Serial numberReinforcement layerAxial compression ratioShear span ratioStirrup reinforcement ratioConcrete strength grade10.000.070.110.060.0720.040.070.110.060.0730.070.070.110.060.0740.110.070.110.060.0750.150.070.110.060.0760.070.000.110.060.0770.070.140.110.060.0780.070.210.110.060.0790.070.070.000.060.07100.070.070.220.060.07110.070.070.110.210.07120.070.070.110.000.07130.070.070.110.060.00140.070.070.110.060.14150.070.070.110.060.21In this paper, the five associated valuesγi=0.59,0.85,0.88,0.86,0.86|k=1,2,3,4,5 obtained in the previous section are substituted into (7), and the weight coefficients of each factor are obtained as follows: 0.146, 0.210, 0.218, 0.213, and 0.213. Then, the weight coefficients are multiplied by the normalized data to obtain the input data. The detailed results are shown in Table 6.


## 3.3. Simulation of the Neural Network

The sample data obtained above (shown in Table6) were input into the established and trained neural network model, and the accuracy of the model was verified by comparing and analyzing the results of the neural network model and the simulated calculation.


## 3.4. Neural Network Predicted Value

Table7 and Figure 5 show a comparison between the predicted value of the displacement ductility coefficient calculated by the BP neural network model and the displacement ductility coefficient of the TRC reinforced column obtained by finite element analysis in previous studies.Table 7
Comparison of measured and predicted values.
Serial numberDisplacement ductility coefficientMeasured valuePredicted valueMeasured/predicted value14.364.520.96426.765.911.14436.916.421.07646.56.401.01656.66.491.016610.5810.620.99675.35.231.01483.735.690.65693.383.390.998106.696.770.989117.597.561.004125.325.271.009133.795.740.661145.035.120.9821554.991.002Figure 5
Ratio of measured value and predicted value of displacement ductility coefficient (Table7).In addition, data analysis software, Statistical Product and Service Solutions (SPSS) [43], was used to compare the predicted value of displacement ductility coefficient calculated in the above BP neural network model with the experimental value obtained by numerical analysis, and the correlation between the value predicted by the BP neural network and the experimental value was obtained. The specific parameters are shown in Table 8.Table 8
BP neural network predicted value fit.
ModelRR2AdjustedR2Standard deviation error10.9110.8300.8170.696Table8 shows that the R2 of the fitting degree between the predicted value of displacement ductility coefficient calculated based on the BP neural network model and the measured value calculated by the above simulation is 0.83. Therefore, the results show that the BP neural network model based on the grey correlation analysis method has a good prediction effect on the displacement ductility coefficient of TRC-strengthened RC columns.


## 4. Ductility Formula

The displacement ductility coefficient reflects the plastic deformation capacity of the structural member. It is the ratio of the failure displacement of the specimen to the yield displacement, namely,(8)μ=ΔuΔy,where Δu is the ultimate displacement of the specimen, which is the horizontal displacement when the horizontal load drops to 85% of the peak load; Δy is the yield displacement of the specimen, which is the corresponding horizontal displacement when the specimen reaches the initial yield point.According to the correlation analysis and grey correlation analysis of the factors influencing the seismic performance of TRC-strengthened RC column mentioned in Section1 and Section 2, the displacement ductility coefficient of the TRC-strengthened RC column changes with changes in the shear span ratio, axial compression ratio, stirrup space, and concrete strength grade and TRC layers. Therefore, the influences of the shear span ratio, axial compression ratio, stirrup space, and concrete strength grade are considered when deducing the displacement ductility coefficient of the reinforced column.In this section, the influence of the shear span ratio, axial compression ratio, stirrup ratio, fibre braided mesh layout layers, concrete strength grade, and TRC layers on the displacement ductility coefficient of TRC-strengthened RC columns is considered to establish a function according to [44]. The function form is expressed in the following equation: (9)μ=a+cλ−dλv+eλs/g+λin+j,where μ is the displacement ductility coefficient, λs is the characteristic value of the fibre volume, and λ is the shear span ratio. The a, c, d, e, g, i, j are regression coefficients to be determined.The characteristic values of stirrup volume and fibre volume can be calculated according to equations (10) and (11), respectively:(10)λv=ρvfyvfc=nAslAcorsfyvfc=Asvlgl1l2sg×fyvfc,(11)λs=ρsfsfc=τslsbh×fsufc,where λv is the characteristic value of the stirrup volume; fc is the compressive strength of the concrete; ρv is the volume matching rate; n is the axial compression ratio; s the sectional area of tensile steel bars; l is the length of the stirrup; Acor is the concrete core area; As is the stirrup space; fyv is the stirrup design strength; Asv is the cross-sectional area of stirrup; lg is the perimeter of the stirrups; l1 and l2 are the lengths of two sides of the rectangular stirrups; sg is the stirrup spacing; λs is the characteristic value of the fibre volume; τs and ls are the section thickness and total perimeter of the carbon fibres, respectively; b and h are the cross-sectional width and height of the column; and fsu is the ultimate tensile strength of the carbon fibres.Relevant parameters of TRC-strengthened columns that consider the shear span ratio, axial compression ratio, stirrup ratio, TRC layers, and concrete strength grade in Table9 and the displacement ductility coefficient obtained by numerical calculation are used as the input sample (ten sets of sample data and five sets of test data). Then, the data are substituted into MATLAB software for nonlinear regression to obtain the value of the regression coefficient. Then the regression coefficient is substituted into equation (9) to obtain equation (12). The results of the calculation and analysis are compared as shown in Table 10.(12)μ=0.66+1.24λ−1.49λv+3.16λs/0.65+λn−0.07.Table 9
Raw sample data.
Serial numberDuctility coefficientμReinforcement layerλsAxial compression rationShear span ratioλStirrup reinforcement ratioλvConcrete strength grade14.360.000.33.80.0374026.760.080.33.80.0374036.910.150.33.80.0374046.500.230.33.80.0374056.600.300.33.80.03740610.580.150.23.80.0374075.300.150.43.80.0374083.730.150.53.80.0374093.380.150.32.00.03740106.690.150.35.60.03740117.590.150.33.80.07440125.320.150.33.80.02440133.790.200.33.80.05030145.030.120.33.80.03150155.000.100.33.80.02660Table 10
Comparison of the measured and predicted value.
Serial numberDisplacement ductility coefficientMeasured valuePredicted valueMeasured/predicted value14.365.270.8326.765.511.2336.915.731.2146.55.981.0956.66.191.07610.5810.131.0475.33.991.3383.733.061.2293.384.180.81106.697.460.90117.596.721.13125.325.261.01135.646.270.90145.035.430.931555.190.96A further comparison between the calculated and experimental displacement ductility coefficient was performed with the data analysis software SPSS, and the predictive value fit of the proposed formula was assessed, as shown in Table10 and Figure 6. Table 11 shows that the R2 is 0.805, the adjusted R2 value after considering the number of independent variables is 0.790, and most of the errors between the predicted and simulated values calculated by the formula are within 15%. Therefore, the calculated and experimental results are in good agreement, indicating that the proposed equation can provide a reference for engineering design and seismic performance evaluation.Figure 6
Ratio of measured value and predicted value of displacement ductility coefficient (Table10).Table 11
Predicted value fit of the proposed formula.
ModelRR2AdjustedR2Standard deviation error10.8970.8050.7900.810


## 5. Conclusions

Based on the numerical analysis results, the correlation degree between the TRC layers, axial compression ratio, shear span ratio, stirrup space, concrete strength grade, and ductility of TRC-strengthened columns was analyzed. Furthermore, a BP neural network prediction model based on grey system theory was established in MATLAB, and then a formula was proposed for the displacement ductility coefficient of TRC-strengthened columns. The specific conclusions of this study are as follows:(1)
The influence of various factors on the ductility coefficient of reinforced columns is ranked as follows: shear span ratio > (axial compression ratio, stirrup space, and concrete strength grade) > TRC layers. This ranking was obtained by calculating the correlation value between the displacement ductility coefficient and various factors of TRC-strengthened columns. These results indicate that the factors selected in this paper have a substantial impact on the ductility of reinforced columns, and the correlations of these factors are strong.(2)
A BP neural network model with three layers was established to predict the displacement ductility coefficient of TRC-strengthened columns. The results show that the model can be used to predict the ductility coefficient of TRC-strengthened columns.(3)
Based on the developed BP neural network prediction model, which is based on grey system theory, a formula for calculating the displacement ductility coefficients of TRC-strengthened columns is established by considering the influences of the shear span ratio, axial compression ratio, stirrup space, textile layers, and concrete strength grade on the displacement ductility coefficient. A verification process with relevant data shows that the calculated values are in good agreement with the experimental values, which indicates that the proposed ductility formula has a certain accuracy and rationality and that the formula can be used for predicting displacement ductility coefficients of TRC-strengthened columns and can provide a certain reference for engineering design.



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*Source: 3152619-2021-12-07.xml*