# Machine Learning-Based Model in Predicting the Plate-End Debonding of FRP-Strengthened RC Beams in Flexure

Authors: Tianyu Hu; Guibing Li
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

Reinforced concrete (RC) beams strengthened with fiber reinforced polymers (FRPs) are structurally complex and prone to plate-end (PE) debonding. In this study, considering the extremely complicated nonlinear relationship between the PE debonding and the parameters, machine learning algorithms, namely, linear regression, ridge regression, decision tree, random forest, and neural network improved by sparrow search algorithm, are established to predict the PE debonding of RC beams strengthened with FRP. The results of reliability evaluation and parameter analysis reveal that ACI, CNR, fib-1, fib-2, and TR55-2 are a little conservative; AS and TR55-1 have the problem of overestimating the shear force; the accuracy and robustness of the SSA-BP model developed in this paper are good; the stirrup reinforcement has the greatest effect on PE debonding; and each parameter shows a complex nonlinear relationship with the shear force when PE debonding occurs.


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## Body

## 1. Introduction

The lightweight, high strength, and high corrosion resistance of fiber reinforced polymers (FRPs) make it widely used in the repair and rehabilitation of existing concrete structures [1]. Nevertheless, a large number of experimental studies have demonstrated that external FRP improves the load carrying capacity of RC beams, but due to the linear elasticity characteristics of the material itself, debonding failure often occurs after strengthening, which greatly limits its use in practical applications [2–8]. There are two main types of debonding failure in FRP-strengthened RC beams in flexure: plate-end (PE) debonding and intermediate crack (IC) debonding. For beams with small shear span, since the bending moment is minor at this time, the beam is mainly subjected to shear force and PE debonding is very likely to occur. Concrete cover separation and plate-end interfacial debonding are the two modes of PE debonding. Generally, plate-end interfacial debonding occurs merely when the width of the FRP sheet is much smaller than the width of the strengthened beam. Therefore, concrete cover separation is the more common mode of PE debonding. When the terminal of FRP is close to the support, the concrete cover separation is mainly caused by the shear crack at the end of FRP. As the load increases, the shear cracks develop to cause vertical and horizontal displacement in the concrete beam and thus generate interfacial shear stress and normal stresses. With the increase in stress, the concrete cover will separate when the crack reaches the horizontal plane of the tensile reinforcement [9, 10]. When the terminal of FRP is far away from the support, the inclined cracks are generated in the shear area, and when the inclined cracks reach the level of the tensile reinforcement, the cracks spread in the horizontal direction and lead to the splitting of the concrete cover [11, 12]. In order to solve the problem of limiting the use of FRP due to PE debonding, end anchoring is usually used to prevent it. And, in order to better design the end-anchoring system, it is necessary to first determine the shear force when PE debonding occurs.Researchers and codes have developed different computational models for PE debonding of FRP-strengthened RC beam based on the shear force of the concrete beam and the debonding strain of FRP. Most of these models are based on shear strength of the beams or based on fracture mechanics. Oehlers developed a strength model based on the shear force and bending moment acting at the plate end in 1992 [9]. Jansze proposed a plate-end debonding strength model, which was originally developed for steel-plated beams in 1997. The proposed model considered the occurrence of PE debonding failure at the onset of shear cracking in RC beams [13]. Ahmed and van Germert modified the model of Jansze considering the differences between FRP and steel properties and the effect of shear reinforcement in 1999 [14]. Smith and Teng proposed a model that is based on the concrete shear strength only in 2002 [15]. A theoretical model based on truss analogy was proposed by Colotti and others in 2004 to predict the failure mode and ultimate capacity of FRP-strengthened RC beams [16]. Yao and Teng and Teng and Yao conducted experimental and analytical investigations on FRP-strengthened beams in 2007, thus modifying the moment shear interaction expression proposed by Oehlers [17, 18]. The fib Bulletin 14 (fib, 2001) presented the model proposed by Blaschko which is based on the concrete shear strength of the beam. The Technical Report 55 (TR55) of the Concrete Society (2012), ACI 440.2 R (ACI, 2017), and Australian Standard AS 5100.8 (Standards Australia, 2017b) recommended an upper limit for the acting shear force at the plate-end region to avoid PE debonding. El-sayed and others proposed a model which is based on the concrete shear strength of the beams considering main parameters known to affect the opening of the shear cracks and consequently affect PE debonding in 2021 [19]. Nevertheless, due to the complex structure of FRP-strengthened RC beams and the nonlinear relationship between PE debonding and parameters, most of the above models suffer from low computational accuracy and poor robustness. Therefore, it is especially important to establish a more precise nonlinear mapping relationship between PE debonding and each parameter.Given the large number of parameters that affect the PE debonding, this study uses several machine learning algorithms, namely, linear regression, ridge regression, decision tree, random forest, and neural network optimized by the sparrow search algorithm to develop several intelligent prediction models for PE debonding of FRP-strengthened RC beams and then select the best prediction model from it. Based on the best model obtained, the robustness of the model and the codes are assessed and the parameters affecting PE debonding are analyzed too.


## 2. Parameter Identification and Data Collection

### 2.1. Parameter Identification

Based on relevant codes and experimental studies [13–24], concrete strength (f'c), location of FRP cut-off point (Lua/a), tensile strength of tensile reinforcement (fy), tensile strength of stirrup reinforcement (fyv), stirrup reinforcement ratio (ρsv), tensile strength of FRP (ffu), FRP stiffness (Eftf), and the ratio of FRP width to the width of the strengthened beam (bf/b) are selected as input parameters for predicting PE debonding in this study. For the convenience of establishing the model, the above parameters are denoted as X1, X2, X3 X4, X5, X6, X7, and X8, respectively, and the shear force is taken as the output of models and denoted as V.Figure 1
Range of the parameters. (a) f'c/MPa. (b) Lua/a. (c) fy/MPa. (d) fyv/MPa. (e) ρsv. (f) ffu/MPa. (g) Eftf/GPa.mm. (h) bf/b.
(a)(b)(c)(d)(e)(f)(g)(h)


### 2.2. Criteria for Collection Analysis of Parameters

(1)
PE debonding occurred in all strengthened beams [6–8, 17, 25–52].(2)
The FRP sheets are not prestressed and the end of the strengthened beams is not anchoraged.(3)
The geometric and material properties of the strengthened beams, FRP, and reinforcement are clear.(4)
The range of parameter variation is large and basically covers the case of general beams. The range of the variation of each parameter is shown in Figure1.From Figure1, we can get that the maximum interval of f’c is 30 MPa–40 MPa, accounting for 49%, and merely 6% of the f’c is more than 60 MPa, indicating that the f’c is mostly at plain level; the maximum interval of the Lua/a is 0.8–1.0, accounting for 64%, while other intervals account for less; the design value of fyv is mainly distributed from 250 MPa to 510 MPa, accounting for 69%, and only 6% of the strength is over 570 MPa, indicating that the fyv is mostly at ordinary level, and high-strength reinforcement accounts for a relatively small percentage; the maximum interval of ρsv is 0.6–0.9, accounting for 38%, and the distribution of each range is more uniform; the maximum interval of ffu is 3000 MPa–4000 MPa, accounting for 54%, and the tensile strength above 4000 MPa accounts for only 9%; the maximum interval of the fy is 350 MPa–510 MPa, accounting for 51%, but the percentage of tensile strength over 510 MPa is 49%, indicating that a larger portion of tensile reinforcement in the collected specimens is high-strength reinforcement; the maximum interval of Eftf is from 75 MPa to 125 MPa, accounting for 40%, and only 8% above 225 MPa; the maximum interval of the bf/b is > 0.9, accounting for 37%, and the distribution of each interval is relatively uniform. In addition, some of the parameters differed significantly across intervals, which would result in machine learning models having smaller prediction errors in intervals with more data than in intervals with less data.


## 2.1. Parameter Identification

Based on relevant codes and experimental studies [13–24], concrete strength (f'c), location of FRP cut-off point (Lua/a), tensile strength of tensile reinforcement (fy), tensile strength of stirrup reinforcement (fyv), stirrup reinforcement ratio (ρsv), tensile strength of FRP (ffu), FRP stiffness (Eftf), and the ratio of FRP width to the width of the strengthened beam (bf/b) are selected as input parameters for predicting PE debonding in this study. For the convenience of establishing the model, the above parameters are denoted as X1, X2, X3 X4, X5, X6, X7, and X8, respectively, and the shear force is taken as the output of models and denoted as V.Figure 1
Range of the parameters. (a) f'c/MPa. (b) Lua/a. (c) fy/MPa. (d) fyv/MPa. (e) ρsv. (f) ffu/MPa. (g) Eftf/GPa.mm. (h) bf/b.
(a)(b)(c)(d)(e)(f)(g)(h)


## 2.2. Criteria for Collection Analysis of Parameters

(1)
PE debonding occurred in all strengthened beams [6–8, 17, 25–52].(2)
The FRP sheets are not prestressed and the end of the strengthened beams is not anchoraged.(3)
The geometric and material properties of the strengthened beams, FRP, and reinforcement are clear.(4)
The range of parameter variation is large and basically covers the case of general beams. The range of the variation of each parameter is shown in Figure1.From Figure1, we can get that the maximum interval of f’c is 30 MPa–40 MPa, accounting for 49%, and merely 6% of the f’c is more than 60 MPa, indicating that the f’c is mostly at plain level; the maximum interval of the Lua/a is 0.8–1.0, accounting for 64%, while other intervals account for less; the design value of fyv is mainly distributed from 250 MPa to 510 MPa, accounting for 69%, and only 6% of the strength is over 570 MPa, indicating that the fyv is mostly at ordinary level, and high-strength reinforcement accounts for a relatively small percentage; the maximum interval of ρsv is 0.6–0.9, accounting for 38%, and the distribution of each range is more uniform; the maximum interval of ffu is 3000 MPa–4000 MPa, accounting for 54%, and the tensile strength above 4000 MPa accounts for only 9%; the maximum interval of the fy is 350 MPa–510 MPa, accounting for 51%, but the percentage of tensile strength over 510 MPa is 49%, indicating that a larger portion of tensile reinforcement in the collected specimens is high-strength reinforcement; the maximum interval of Eftf is from 75 MPa to 125 MPa, accounting for 40%, and only 8% above 225 MPa; the maximum interval of the bf/b is > 0.9, accounting for 37%, and the distribution of each interval is relatively uniform. In addition, some of the parameters differed significantly across intervals, which would result in machine learning models having smaller prediction errors in intervals with more data than in intervals with less data.


## 3. Machine Learning Models

This part uses several machine learning algorithms, namely, linear regression, ridge regression, decision tree, random forest, and neural network optimized by sparrow search algorithm to develop several intelligent prediction models for PE debonding of FRP-strengthened RC beams. The data are used from [6–8, 17, 25–52], where the percentages of training set, validation set, and testing set are 60%, 20%, and 20%, respectively.To further analyze the prediction accuracy and generalization ability of each model, the average absolute error (MAE) and goodness-of-fit (R2) of the training set, the validation set, and the testing set of the models are calculated and given in Figure2.Figure 2
Performance of the models. (a) MAE. (b) R2.
(a)(b)From Figure2, the MAE and the R2 of linear regression, ridge regression, decision tree, and random forest differ greatly in the training set, the validation set, and the testing set, indicating that the generalization ability of these models is poor, while the MAE of BP neural network is 5.42, 5.52, and 5.62 in the training set, the validation set, and the testing set, respectively, which is not only the smallest but also the most average among all machine learning algorithms. Also, the R2 of the training set, the validation set, and the testing set of the BP neural network is 0.97, 0.86, and 0.90, respectively, which is the highest among all machine learning models. In summary, the BP neural network model not only has the highest prediction accuracy but also has excellent generalization ability.


## 4. Reliability Evaluation of the Code

To further illustrate the precision and robustness of the SSA-BP, this section compares it with several international codes [13–17]. The codes and their expressions are shown in Table 1, and the comparison between the calculated and actual values of SSA-BP and the codes is shown in Figure 3.Table 1
Codes and their expressions.
CodesCalculation formulaACI440.2 RVdb,end<0.67VcAS 5100.8Vdb,end<0.67VuFib-1Vdb,end<0.15fck1/3bdFib-2εfd=α1c1kckbfct/nfrpEfrptfrplb≥lb,maxα1c1kckbfct/nfrpEfrptfrp⋅lb/lb,max2−lb/lb,maxlb<lb,maxlb,max=nfrpEfrptfrp/c2fctkb=1.062bfrp/b/1+bfrp/400≥1bfrp/b≥0.33TR55-1Vdb,end<0.67VrdTR55-2εfd=0.5kbfct/nfrpEfrptfrplb≥lb,max0.5kbfct/nfrpEfrptfrp⋅lb/lb,max2−lb/lb,maxlb<lb,maxlb,max=0.7nfrpEfrptfrp/fctCNRεfd=1/γf,d2ΓFd/nfrpEfrptfrplb≥lb,max1/γf,d2ΓFd/nfrpEfrptfrp⋅lb/lb,maxlb<lb,maxΓFd=kbkGfc′fctkb=2−bfrp/b/1+bfrp/b≥1bfrp/b≥0.251.18bfrp/b<0.25lb,max=min1/γRdfbdπ2EfrptfrpΓFd/2,200mmfbd=2ΓFd/SuFigure 3
Comparison of the calculated values and real values. (a) SSA-BP. (b) ACI. (c) AS. (d) CNR. (e) fib-1. (f) fib-2. (g) TR55-1. (h) TR55-2.
(a)(b)(c)(d)(e)(f)(g)(h)It can be visualized from Figure3 that the calculated values of SSA-BP are basically within 15% above and below the true value and are significantly superior to the calculated values given by each code. The calculated values of ACI and fib-1 are basically below 15% of the true value, indicating that these two codes are too conservative; while AS and TR55-1 have a larger portion of calculated values above 15% of the true value, which takes the risk of overestimating the shear strength in the event of PE debonding. The quantitative evaluation of the performance of BP and each code is shown in Table 2.Table 2
Performance of the models.
ModelsEvaluation indicatorsCoefficient of variation (%)Conservative (%)Nonconservative (%)SSA-BP20.35248ACI26.61000AS44.53863CNR23.47921Fib-125.11000Fib-221.28020TR55-148.33961TR55-220.5955As can be seen from Table.2, the coefficient of variation of the neural network is merely 20.3%, which is better than the calculated values of each code. The coefficients of variation of CNR and fib-1 and TR55-2 are smaller, but their conservative estimates account for 79%, 80%, and 95%, respectively, which are a little conservative compared with 50% for SSA-BP.


## 5. Parametric Study

### 5.1. Analysis of the Importance of Parameters

In Matlab software, by inputting the 'net.iw', the'net.lw', and the 'net.b', the weights and biases of the neural network can be obtained. See Table3 for details.Table 3
Weights and biases between connection layers.
Hidden layerOutputH(1 : 1)H(1 : 2)H(1 : 3)H(1 : 4)H(1 : 5)H(1 : 6)H(1 : 7)VBias.743.5292.692.061−1.476.038.776X1.0221.514.906.179.279.443−.473X2−.095−.656−.243.438.662−.188−1.296X3−.348.946−.998−.395.900−.240.362X4−.037.765−.727.350−.142−.096−1.060X5−.006.7821.731.816−.139.281−.277X6−.681−.587−.845−.428−.135−.695.584X7.2721.773−.444.053.583−.0301.619X81.225.077−.300.169.669.940.993Bias1.115H(1 : 1)−1.425H(1 : 2).381H(1 : 3)−2.224H(1 : 4).511H(1 : 5)−1.377H(1 : 6)1.147H(1 : 7).736After getting the weights and biases between layers, we can get the transfer functions through the code 'TransferFcn' in Matlab. The transfer functions are shown in equation (1) and (2).(1)yi=f⋅∑iwijxi+ϕj=21+e−2∑iwijxi+ϕj−1.(2)yi=f⋅∑iwijxi+ϕj=∑iwijxi+ϕj.The weights and biases obtained in Table1 were substituted into (1) and (2) and subsequently normalize the calculated value obtained to get the importance of each parameter on PE debonding, as shown in Figure 4.Figure 4
Importance of each parameter.As can be seen from Figure4, the importance of each parameter is X5>X7>X2>X1>X8>X4>X3>X6 in order; that is, the ρsv has the greatest effect on the PE debonding, and the ffu, fy, and fyv have less effect on it.


### 5.2. Sensitivity Analysis of Parameters

Based on the effects of each parameter on PE debonding obtained in section 5.1, the ffu, which has the least effect on PE debonding, is selected as the grouping variable for the sensitivity analysis of each parameter, and four grades of 200, 1200, 2200, and 3200 are taken based on its distribution in section 2.1. When studying the sensitivity of a single variable to PE debonding, the other variables are averaged based on the statistics in section 1.2. The sensitivity of each parameter to the shear strength is shown in Figure5.Figure 5
Sensitivity of each parameter.
(a)(b)(c)(d)(e)(f)From Figure5, we can get that (a) when the ffu is in the first two grades, the shear strength tends to increase and subsequently decrease with the increase in f’c. When it is in the last two grades, the shear strength tends to increase, subsequently decrease, and then increase as the f’c increases; (b) for the Lua/a, the shear strength increases and subsequently decreases with the increase in the Lua/a, regardless of the grade of ffu; (c) regarding the ρsv, no matter which grade the ffu is at, as the ρsv increases, the shear strength first decreases, then increases, and then decreases; (d) for the fyv, there are three patterns when the ffu is in different grades, namely, when the ffu is in two grades of 200 and 1200, the shear strength is inversely proportional to the increase in fyv; when the ffu is in grade 2200, the shear strength tends to decrease and subsequently increase with the increase in fyv; when the ffu is in grade 3200, the shear strength is proportional to the fyv; (e) for Eftf and bf/b, as they increase, the shear strength basically increases and subsequently decreases, but when the Eftf and bf/b are larger, the shear strength decreases not significantly.In summary, there is a complex nonlinear relationship between the shear force at PE debonding and each parameter. In the future, the effect of different parameters on PE debonding under different conditions needs to be considered when performing the end anchorage system.


## 5.1. Analysis of the Importance of Parameters

In Matlab software, by inputting the 'net.iw', the'net.lw', and the 'net.b', the weights and biases of the neural network can be obtained. See Table3 for details.Table 3
Weights and biases between connection layers.
Hidden layerOutputH(1 : 1)H(1 : 2)H(1 : 3)H(1 : 4)H(1 : 5)H(1 : 6)H(1 : 7)VBias.743.5292.692.061−1.476.038.776X1.0221.514.906.179.279.443−.473X2−.095−.656−.243.438.662−.188−1.296X3−.348.946−.998−.395.900−.240.362X4−.037.765−.727.350−.142−.096−1.060X5−.006.7821.731.816−.139.281−.277X6−.681−.587−.845−.428−.135−.695.584X7.2721.773−.444.053.583−.0301.619X81.225.077−.300.169.669.940.993Bias1.115H(1 : 1)−1.425H(1 : 2).381H(1 : 3)−2.224H(1 : 4).511H(1 : 5)−1.377H(1 : 6)1.147H(1 : 7).736After getting the weights and biases between layers, we can get the transfer functions through the code 'TransferFcn' in Matlab. The transfer functions are shown in equation (1) and (2).(1)yi=f⋅∑iwijxi+ϕj=21+e−2∑iwijxi+ϕj−1.(2)yi=f⋅∑iwijxi+ϕj=∑iwijxi+ϕj.The weights and biases obtained in Table1 were substituted into (1) and (2) and subsequently normalize the calculated value obtained to get the importance of each parameter on PE debonding, as shown in Figure 4.Figure 4
Importance of each parameter.As can be seen from Figure4, the importance of each parameter is X5>X7>X2>X1>X8>X4>X3>X6 in order; that is, the ρsv has the greatest effect on the PE debonding, and the ffu, fy, and fyv have less effect on it.


## 5.2. Sensitivity Analysis of Parameters

Based on the effects of each parameter on PE debonding obtained in section 5.1, the ffu, which has the least effect on PE debonding, is selected as the grouping variable for the sensitivity analysis of each parameter, and four grades of 200, 1200, 2200, and 3200 are taken based on its distribution in section 2.1. When studying the sensitivity of a single variable to PE debonding, the other variables are averaged based on the statistics in section 1.2. The sensitivity of each parameter to the shear strength is shown in Figure5.Figure 5
Sensitivity of each parameter.
(a)(b)(c)(d)(e)(f)From Figure5, we can get that (a) when the ffu is in the first two grades, the shear strength tends to increase and subsequently decrease with the increase in f’c. When it is in the last two grades, the shear strength tends to increase, subsequently decrease, and then increase as the f’c increases; (b) for the Lua/a, the shear strength increases and subsequently decreases with the increase in the Lua/a, regardless of the grade of ffu; (c) regarding the ρsv, no matter which grade the ffu is at, as the ρsv increases, the shear strength first decreases, then increases, and then decreases; (d) for the fyv, there are three patterns when the ffu is in different grades, namely, when the ffu is in two grades of 200 and 1200, the shear strength is inversely proportional to the increase in fyv; when the ffu is in grade 2200, the shear strength tends to decrease and subsequently increase with the increase in fyv; when the ffu is in grade 3200, the shear strength is proportional to the fyv; (e) for Eftf and bf/b, as they increase, the shear strength basically increases and subsequently decreases, but when the Eftf and bf/b are larger, the shear strength decreases not significantly.In summary, there is a complex nonlinear relationship between the shear force at PE debonding and each parameter. In the future, the effect of different parameters on PE debonding under different conditions needs to be considered when performing the end anchorage system.


## 6. Conclusion

In this study, a machine learning approach was used to synthesize the effects of various parameters on the PE debonding of FRP-strengthened RC beams in flexure, and the resulting models were evaluated against several codes. The following conclusions can be drawn:(1)
ACI, CNR, fib-1, fib-2, and TR55-2 are too conservative, which have 100%, 80%, 100%, 79%, and 95% values below the experimental values, respectively. AS and TR55-1 had 63% and 61% values above the experimental values, respectively, potentially overestimating the shear force in the case of PE debonding. The coefficient of variation, conservative, and nonconservative values of SSA-BP is 20.3%, 52%, and 48%, respectively, and its robustness and prediction accuracy are superior to the above codes.(2)
ρsv, Eftf, and Lua/a have more influence on the shear force at PE debonding, while fy and ffu have less influence on the shear force. Moreover, there are complex nonlinear relationships between each parameters and the shear force, and the effect of different parameters on PE debonding under different conditions needs to be considered when performing the system of end anchorage in the future.(3)
There are two problems with the model built in this paper: on the one hand, the uneven distribution of parameters in the dataset on which the model is built leads to the prediction accuracy of the model to be improved, and on the other hand, the model is complicated by considering too many parameters. In the future, more data need to be collected and the parameters in the model need to be streamlined.



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