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Article

Prediction of Mechanical Behaviours of FRP-Confined Circular Concrete Columns Using Artificial Neural Network and Support Vector Regression: Modelling and Performance Evaluation

by
Pang Chen
1,
Hui Wang
1,
Shaojun Cao
1,* and
Xueyuan Lv
2
1
School of Civil and Transportation Engineering, Hebei University of Technology, Tianjin 300401, China
2
China Construction First Group Construction & Development Co., Ltd., Beijing 100102, China
*
Author to whom correspondence should be addressed.
Materials 2022, 15(14), 4971; https://doi.org/10.3390/ma15144971
Submission received: 7 June 2022 / Revised: 8 July 2022 / Accepted: 11 July 2022 / Published: 17 July 2022
(This article belongs to the Special Issue Modeling and Simulations of Construction Materials)
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Abstract

:
The prediction and control of the mechanical behaviours of fibre-reinforced polymer (FRP)-confined circular concrete columns subjected to axial loading are directly related to the safety of the structures. One challenge in building a mechanical model is understanding the complex relationship between the main parameters affecting the phenomenon. Artificial intelligence (AI) algorithms can overcome this challenge. In this study, 298 test data points were considered for FRP-confined circular concrete columns. Six parameters, such as the diameter-to-fibre thickness ratio (D/t) and the tensile strength of the FRP (ffrp) were set as the input sets. The existing models were compared with the test data. In addition, artificial neural networks (ANNs) and support vector regression (SVR) were used to predict the mechanical behaviour of FRP-confined circular concrete columns. The study showed that the predictive accuracy of the compressive strength in the existing models was higher than the peak compressive strain for the high dispersion of material deformation. The predictive accuracy of the ANN and SVR was higher than that of the existing models. The ANN and SVR can predict the compressive strength and peak compressive strain of FRP-confined circular concrete columns and can be used to predict the mechanical behaviour of FRP-confined circular concrete columns.

1. Introduction

Fibre-reinforced polymers (FRPs) are widely used in composite structures and for structural strengthening owing to their light weight, high strength, and good corrosion resistance [1,2,3,4,5,6,7]. As shown in Figure 1, an FRP-confined circular concrete column consists of core concrete and an external confinement of wrapped FRP. σ c is the axial compressive stress on the FRP-confined circular concrete column; σ f is the lateral constraint stress on the core concrete of the FRP-confined circular concrete column; and σ cf is the reaction force generated by the core concrete under the FRP-confined action.
In FRP-confined circular concrete columns, the core concrete is subjected to triaxial compression under axial loading, which improves the bearing capacity and ductility of the members and subsequently reduces the section size and self-weight of the members [8,9,10,11]. In addition, external confinement by the wrapping of FRP protects the core concrete from corrosion and improves the durability of the members. Relevant research has shown that the restraining action of FRP can increase the bearing capacity and ductility of core concrete by 2–46% and 14–923%, respectively [12,13,14,15]. Therefore, studying the axial compressive mechanical behaviours of FRP-confined circular concrete columns is vital in expanding their application in engineering practices.
Researchers have extensively investigated the axial compressive mechanical behaviours of FRP-confined concrete columns. They have studied the influences of the diameter-to-fibre thickness ratio (D/t), the compressive strength of the core concrete (fco), the strain corresponding to the compressive strength of core concrete (εco), the tensile strength of FRP (ffrp), the elastic modulus of FRP (Efrp), and the ultimate tensile strain of FRP (εfrp) on the axial compressive mechanical behaviours of FRP-confined circular concrete columns and proposed their axial compression constitutive models. Fardis studied the influences of the section size and compressive strength of core concrete on the axial compressive mechanical behaviours of FRP-confined circular concrete columns and proposed a corresponding axial compression constitutive model based on the test results [16]. Saadatmanesh established an axial compression constitutive model of CFRP- and GFRP-confined circular concrete columns based on the steel-confined concrete model proposed by Mander [17,18]. Nanni studied the influence of the FRP type on the axial compressive mechanical behaviours of FRP-confined circular concrete columns and proposed their hyperbolic constitutive model based on the test results [19]. Samaan studied the influence of FRP layers on the axial compressive mechanical behaviours of GFRP-confined circular concrete columns and proposed an axial compression constitutive model [20]. Lam and Teng proposed a two-stage constitutive model of FRP-confined circular concrete columns based on experimental data, which combined a parabola and a straight line [21,22,23,24]. Chastre and Silva studied the influences of the diameter-to-fibre thickness ratio, the elastic modulus of FRP, and the strength of core concrete on the axial compressive mechanical behaviours of GFRP-confined circular concrete columns and proposed an axial compression constitutive model of FRP-confined circular concrete columns based on the test results [25]. In order to reduce the error increase in the FRP constraint model due to FRP failure, Gian proposed an effective strain analysis model of FRP-confined cylinder concrete columns based on a theoretical analysis and the test data [26]. Wu and Wei studied the effects of the ultimate tensile strain of FRP and the strength of core concrete on the axial compressive mechanical behaviours of FRP-confined circular concrete columns and proposed an axial compression constitutive model of FRP-confined circular concrete columns based on the test results [27]. With the depth of research on FRP-confined concrete columns increasing, researchers have carried out a series of research works on FRP-confined non-circular concrete columns. Gian established a simplified model of FRP-confined circular concrete columns based on the iterative constraint model proposed by himself [28]. Jiang studied the effect of FRP restraint on reducing the strength degradation of square concrete columns based on a test under the eccentric loading of FRP-confined square concrete columns [29]. He reviewed the research status of FRP-confined non-circular concrete columns with modified shapes and proved the advantage of the curving section through the influence of key parameters on the FRP constraints [30]. All the existing models are semi-empirical and semi-theoretical formulations based on limited data, which do not consider the influence of all the parameters on the compressive strength and ultimate compressive strain of the FRP-confined circular concrete columns. The accuracy of the prediction needs to be evaluated; therefore, it is urgent to propose a reliable model that can accurately predict the axial compressive constitutive relationship of FRP-confined circular concrete columns.
In recent years, machine learning has shown unique advantages in solving various problems in structural engineering. A variety of machine learning methods have been widely applied in performance prediction, data classification, image recognition, structural simulation, etc. In particular, artificial neural networks (ANNs) and support vector regression have attracted the most attention [31,32,33,34,35,36,37,38].
ANNs are mathematical models that simulate the neural frame of a human to process complex information. They are widely used for data prediction [39,40]. Liu used an ANN and a swarm intelligence algorithm to predict the carbonation depth of recycled concrete through nine parameters, including temperature, recycled aggregate replacement rate, water absorption, and exposure time, and the results showed that the ANN performed better than the conventional formula [41]. Amiri used an ANN to predict the mechanical behaviours and durability of concrete containing constructional waste through the water-binder ratio, the recycled aggregate replacement rate, and other parameters. The results proved that the ANN had excellent accuracy in predicting the target value [42]. Jayasinghe established a shear test database of concrete beams without shear reinforcement and evaluated the forecast accuracy of the ANN and existing models. The results showed that the ANN was more accurate and is an effective tool for the influence analysis of a single parameter [43].
Support vector regression (SVR) is a regression model developed by Vapnik [44], who introduced insensitive loss functions into the support vector machine (SVM) based on statistical learning theory. It has better predictive ability. Ahmad used SVR, NMR (nuclear magnetic resonance), and an ANN to predict the splicing strength of reinforced concrete members based on the diameter of the rebar, the compressive strength of concrete, and the protective cover thickness. The results show that SVR has the highest forecasting accuracy [31]. Tran used SVR to predict the adhesion strength of the interface between FRP and concrete using the water–cement ratio, the recycled aggregate replacement rate, the sand rate, and other parameters. The results showed that SVR had good predictive performance [45].
ANNs and SVR have been successfully applied in the field of architecture and achieved ideal results [46,47,48,49,50]. Whether they can accurately predict the axial compressive mechanical behaviour of FRP-confined circular concrete columns needs to be evaluated. Based on this, our study collected the test data of the axial compressive mechanical behaviours of FRP-confined circular concrete columns from the relevant published literature and established a reliable database containing 298 datasets. D/t, fco, εco, ffrp, Efrp, and εfrp were set as input sets. Based on the established database, the forecast accuracy of the existing axial compression constitutive model for FRP-confined circular concrete columns was evaluated and analysed. ANN and SVR models were developed, and the forecast accuracy of the axial compressive mechanical behaviours of the FRP-confined circular concrete columns was evaluated and analysed using the ANN, SVR, and the existing axial compression constitutive models. Finally, based on the intelligent algorithm model, the affecting key factors were analysed using expanded parameters. Finally, the key factors affecting the axial compressive mechanical behaviours of the FRP-confined circular concrete columns were analysed based on an intelligent algorithm.

2. Database

The database relied on 298 specimens of tested data obtained from 13 studies [1,8,43,44,45,51,52,53,54,55,56,57,58]. The data information has been listed in Appendix A. It is crucial to collect appropriate parameters to study the axial compressive mechanical behaviour of FRP-confined circular concrete columns. Thus, such a database considers the influence of six parameters on the mechanical behaviour of FRP-confined circular concrete columns and the influence of D/t, fco, εco, ffrp, Efrp, and εfrp on the axial mechanical behaviour of FRP-confined circular concrete columns. Detailed information on the main parameters is presented in Table 1.
The range of the D/t, fco, εco, ffrp, Efrp, and εfrp are shown in Figure 2 and Figure 3. The D/t was mainly distributed in the range of 0–500. The minimum fco was 9.9 MPa, the highest was 136.3 MPa, and the data between 50 MPa and 100 MPa were the highest. The εco was concentrated in the range of 1500–3000 με. The ffrp in the range of 2000 MPa–4000 MPa was up to 65%. Most data on the Efrp were distributed in the range of 200–300 GPa. The data points of the εfrp in the range of 4.5 × 104–5 × 104 με were the least, and maximum data points were in the range of 3 × 104–4.5 × 104 με.

3. Predictive Models of the Axial Compressive Mechanical Behaviours of FRP-Confined Circular Concrete Columns

3.1. Evaluation Indices

To evaluate the predictive accuracy of the models of the axial compressive mechanical behaviours of the FRP-confined circular concrete columns, the regression coefficient (R2), the mean square error (MSE), the mean absolute percentage error (MAPE), and the integral absolute error (IAE) were used to evaluate the predictive accuracy of the models. R2 reflects the correlation between the independent and dependent variables. The closer R2 is to 1, the higher is the correlation between the predictive value and the actual value. The MSE reflects the average error. The lower the MSE, the smaller the error between the predictive value and the actual value. MAPE reflects the degree of data dispersion. The smaller the MAPE, the more the predictive value converges to the actual value. Additionally, the IAE reflects the predictive accuracy. The smaller the IAE, the higher the predictive accuracy of the data. All the indices are often used to evaluate the predictive accuracy of the neural network model [42,59,60,61,62]. The design formulae for each index are given in Equations (1)–(4).
R 2 = ( ∑ i = 1 n ( O i − O ¯ i ) ( C i − C ¯ i ) ) 2 ∑ i = 1 n O i − O ¯ i 2 ∑ i = 1 n C i − C ¯ i 2
M S E = 1 n ∑ i = 1 n C i − O i 2
M A P E = 1 n ∑ i = 1 n C i − O i C i
I A E = ∑ i = 1 n O i − C i 2 ∑ i = 1 n C i × 100 %
where O i is the actual value of the compressive strength or ultimate compressive strain (MPa/με); Ci is the predictive value of the compressive strength or ultimate compressive strain (MPa/με); O ¯ i is the average value of the actual value of the compressive strength or ultimate compressive strain (MPa/με); C ¯ i is the average value of the predictive value of the compressive strength or ultimate compressive strain; and n is the number of data points.

3.2. Evaluation of Existing Axial Compression Constitutive Models

3.2.1. Existing Axial Compression Constitutive Models

At present, there is much research on the axial compressive mechanical behaviour of FRP-confined circular concrete columns. The influence of key parameters on the axial compressive mechanical behaviours of FRP-confined circular concrete columns has been studied, and many axial compression constitutive models of FRP-confined circular concrete columns have been proposed. Among them, the Mander [16], Fardis [17], Lam [21], Bisby [63], Wu [64], and Youssef [65] models are widely used.
The Mander model was proposed as a steel-confined concrete model in 1988 [16] and was later adopted by the guidelines for the selection, design, and installation of FRP systems for externally strengthening concrete structures to calculate the axial compressive mechanical behaviours of FRP-confined circular concrete columns [66]. Fardis experimentally studied the axial compressive mechanical behaviours of 46 FRP-confined circular concrete columns and proposed axial compression constitutive models of CFRP-confined circular concrete columns [17]. Lam and Teng proposed a constitutive model for FRP-confined circular concrete columns combined with a parabola with a straight line [21]. Bisby studied the influence of the FRP type on the axial compressive mechanical behaviours of FRP-confined circular concrete columns and proposed a constitutive model applicable to medium and weak FRP-confined concrete columns [63]. Wu Gang proposed a simplified trilinear constitutive model for FRP-confined circular concrete columns [64]. Youssef proposed an axial compressive constitutive model of FRP-confined circular concrete columns considering the influence of the FRP type and the diameter-to-fibre thickness ratio [65]. Detailed information on each model is shown in Table 2.

3.2.2. Predictive Results of Existing Axial Compression Constitutive Models

Based on the established database, the predictive accuracies of the six axial compressive constitutive models of FRP-confined circular concrete columns were compared and analysed. The results of the compressive strength of each model are shown in Figure 4.
As shown in the figure, most of the data points of the Mander [16] and Fardis [17] models are above a straight line, indicating that both the models overestimate the compressive strength of FRP-confined circular concrete columns. However, most of the data points of the Bisby [63] and Youssef [65] models are below the best fit line, indicating that both these models underestimate the compressive strength of FRP-confined circular concrete columns. The predictive capability of the Lam [21] and Wu [64] models is good, and the data points are uniformly distributed on both sides of the best fit line. Among them, the Lam model [21] has a more concentrated distribution of data points, and the Lam model [21] has the highest predictive accuracy of compressive strength for FRP-confined circular concrete columns.
The predictive results of the models for strain corresponding to the compressive strength of the FRP-confined circular concrete columns are shown in Figure 5. It can be observed from the figure that most of the data points of the Mander [16], Lam [21], and Wu [64] models are above the best fit line, indicating that these models overestimate the peak compressive strain of FRP-confined circular concrete columns. Most of the data points of the Fardis [17], Bisby [63], and Youssef [65] models are below the best fit line. This indicates that these models underestimated the peak compressive strain of FRP-confined circular concrete columns. The distribution of the data points in each figure is scattered, indicating that the predictive accuracy of the peak compressive strain in the existing axial compressive constitutive models of FRP-confined circular concrete columns is poor.
The evaluation indexes (R2, MSE, MAPE, and IAE) for the predictive accuracy of the compressive strength of the above models are listed in Table 3. It can be seen from Table 3 that the predictive accuracy of all the models is good, and the R2 is above 0.5. The Lam model [21] had the highest predictive accuracy, with an R2 of up to 0.83 and the lowest average error and a small dispersion.
Table 4 summarises the evaluation indices (R2, MSE, MAPE, and IAE) of the predictive accuracy in the peak compressive strain of all the models. It can be observed from Table 4 that the predictive accuracy of the peak compressive strain is lower than that of the compressive strength for all the models. The regression coefficient of the Fardis model [17] with the highest predictive accuracy was only 0.48; it had a high dispersion and error.

3.3. Evaluation of Predictive Models Based on ANN and SVR

The predictive accuracy of the compressive strength and peak compressive strain by the existing axial compressive constitutive model of FRP-confined circular concrete columns is low and that of the peak compressive strain is especially poor. Therefore, a new predictive model is urgently required to predict the axial compressive mechanical behaviour of FRP-confined circular concrete columns.

3.3.1. Machine Learning Models

Artificial Neural Networks (ANNs)

An ANN is an information processing system that simulates the structural and functional characteristics of a biological neural system [61]. It includes input, hidden, and output layers. Each of these layers contains some nodes, which are interconnected to the elements in the subsequent layers [67]. The accuracy and the precision of the ANN is highly dependent on the structure of the developed models as well as the model parameters. These parameters contain the number of nodes in the hidden layers, the momentum rate, the learning cycle, and the learning accuracy. The basic idea is that in each hidden layer node, the weighted inputs from the previous layer are added together and the deviation is added to the system, where the weight depends on the momentum rate, and the deviation depends on the learning accuracy. This combination is then passed through a nonlinear activation function to form the output of each hidden neuron [68]. The back-propagation algorithm is often used to train ANN. Training is defined as the procedure for finding the optimal weights of the network so that the prediction error is minimized. In the learning cycle, the results are back-propagated, and the weights and the bias are adjusted in such a way that the obtained error is minimized [69].
A MATLAB-based program with a graphical user interface (GUI) was developed to train and evaluate the ANN model. The ANN model divided the established database into two parts: 80% for training (general is 70–90%) and 20% for testing (general is 10–30%). According to the predictive target, there are six and one nodes in the input and output layers, respectively; the momentum rate is 0.5; the learning cycle is 103; and the learning accuracy is 4 × 10−7.
The number of nodes in the hidden layer plays a vital role in the predictive accuracy of an ANN. To determine this, the predictive accuracies of different numbers of nodes in the hidden layers were compared and analysed. The influence of the number of nodes in the hidden layers on the predictive accuracy of the compressive strength is shown in Figure 6, and the influence of the number of nodes in the hidden layers on the predictive accuracy of peak compressive strain is shown in Figure 7. It can be seen that when the number of nodes in the hidden layers is 10, the MSE of the compressive strength and peak compressive strain is the lowest, and R2 is the highest. The predictive accuracy of the ANN model was the highest. Therefore, when predicting the compressive strength and peak compressive strain, the number of nodes in the hidden layers in the ANN model was 10.

Support Vector Regression (SVR)

The SVM is an intelligent algorithm for general classification problems, first proposed by Boser in 1992 [70]. Unlike traditional neural networks based on empirical risk minimisation, SVR is based on the principle of structural risk minimisation, which aims to minimise the upper bound of the expected risk and avoids reliance on the designer’s empirical knowledge. Vapnik introduced an insensitive loss function into the SVM to form a support vector regression (SVR) [44].
For the SVR model, the established database was divided into two parts: 80% for training and 20% for testing. The SVR model involves three parameters: ε (insensitive loss function), C (regularised constant), and g (kernel coefficient). In SVR, the main goal is to obtain a function that differs at most ε from the actual targets for all training data, while being as flat as possible [71]. The smaller the ε, the smaller the error of the regression function and the higher the degree of model fitting, where C is the regularized constant specified by the user. It is defined as the penalty factor to indicate the trade-off between the flatness of the function and the empirical error. C was mainly used to prevent overfitting. The higher the C, the more the samples with a training error greater than ε are punished, and the stronger is the predictive ability [72]. g is the kernel coefficient. The choice of kernel function is closely related to the performance of the SVR. The commonly used kernel functions in the regression include the linear kernel function, the polynomial kernel function, the radial basis function (RBF), and the sigmoid kernel function. Considering the infinite dimensional feature space corresponding to the RBF, the RBF is adopted in this study. Its expression is shown in Equation (5).
K x i , x j = e − g x j − x j 2
where x i , x j   is the input vector, and g is the key parameter of the RBF which can affect the smoothness of the function. The larger the g, the better the predictive effect of the training set.
To determine the optimal values of ε, C, and g in the SVR, each value is tested in a certain space on the premise of specifying the step. Then, based on the flow of the SVR algorithm, the parameter values of the SVR with optimal accuracy are derived: ε = 0.01, C = 20, and g = 0.3. The flow of the SVR algorithm is illustrated in Figure 8.

3.3.2. Predictive Results and Discussion of ANN and SVR

Based on the established database, the comparison between the ANN and SVR models for the predictive results of compressive strength and peak compressive strain are shown in Figure 9, Figure 10, Figure 11 and Figure 12, respectively. It can be seen from Figure 9, Figure 10, Figure 11 and Figure 12 that the ANN and SVR models can accurately predict the compressive strength and peak compressive strain of FRP-confined circular concrete columns, and the predictive accuracy is much higher than that of their existing axial compressive constitutive models.
Among the existing axial compressive constitutive models for FRP-confined circular concrete columns, the Lam [21] and Fardis [17] models have the highest predictive accuracies for the compressive strength and peak compressive strain of FRP-confined circular concrete columns, respectively. The evaluation indexes of the predictive accuracy for the ANN, SVR, Lam [21], and Fardis [17] models are listed in Table 5. It can be seen that the predictive accuracy of the SVR model for the compressive strength and peak compressive strain of FRP-confined circular concrete columns is slightly higher than that of the ANN model and far higher than that of the Lam [21] and Fardis [17] models. In the SVR model, the R2 of the compressive strength and peak compressive strain is up to 0.96 and 0.94, respectively. In conclusion, the SVR model proposed in this study can provide an approximate basis for revising and unifying the compressive strength and peak compressive strain formulas for FRP-confined circular concrete columns.

4. Parameter Analysis

An ANN was used to accurately analyse the influence of D/t, fco, ffrp, Efrp, fcc/ fco, and εcc/εco on the axial compressive mechanical behaviour of FRP-confined circular concrete columns. While analysing the influence of each parameter, the values of the other parameters were set as the average value of each parameter in the established database.
The influence of D/t on the compressive strength and peak compressive strain of the FRP-confined circular concrete columns is shown in Figure 13. It can be seen that both the compressive strength and the peak compressive strain of FRP-confined circular concrete columns decrease with an increase in the D/t of the specimen. When the D/t increased from 200 to 1000, the compressive strength decreased from 130 MPa to 60 MPa and the peak compressive strain decreased from 3 × 104 to 104. When the D/t increases and the section size of the core concrete remains unchanged, the fibre thickness decreases and the force of constraint it can provide decreases, thus reducing the bearing capacity and ductility of the specimens [73,74,75,76,77].
The influence of the Efrp on the compressive strength and peak compressive strain of the FRP-confined circular concrete columns is shown in Figure 14. It can be seen that the compressive strength of the FRP-confined circular concrete columns is positively correlated with the Efrp, whereas the peak compressive strain is negatively correlated with the Efrp. When the Efrp increases from 40 GPa to 200 GPa, the compressive strength increases from 90 MPa to 150 MPa, and the peak compressive strain decreases from 2.5 × 104 to 1.5 × 104. When the Efrp increases, the elastic deformation resistance, the brittleness of the specimens, and the bearing capacity of the specimens increase, and the ductility decreases [12,78,79].
The influence of ffrp on the compressive strength and peak compressive strain of the FRP-confined circular concrete columns is shown in Figure 15. It can be observed that the compressive strength and the peak compressive strain of the FRP-confined circular concrete columns increase with an increase in ffrp. When ffrp increases from 800 MPa to 4000 MPa, the compressive strength increases from 50 MPa to 250 MPa, and the peak compressive strain increases from 8 × 103 to 4 × 104. When ffrp increases, the force of the constraint provided by the FRP increases, and the bearing capacity and the ductility of the specimens are improved [51,80,81].
The influence of fco on the compressive strength and peak compressive strain of the FRP-confined circular concrete columns is shown in Figure 16. It can be seen that the compressive strength of the FRP-confined circular concrete columns is positively correlated with fco, while the peak compressive strain of the FRP-confined circular concrete columns is negatively correlated with fco. When fco increases from 40 MPa to 100 MPa, the compressive strength of the FRP-confined circular concrete columns increases from 90 MPa to 160 MPa, and their peak compressive strain drops from 1.6 × 104 to 4 × 103. When fco increased, the overall strength, brittleness, and the bearing capacity of the specimens increased, and the ductility decreased [82,83,84].
The influence of the mechanical behaviours of the core concrete on the FRP-confined circular concrete columns is shown in Figure 17. It can be seen that fcc/fco is negatively correlated with fco, while εcc/εco is positively correlated with εco. When fco increases from 40 MPa to 100 MPa, fcc/fco drops from 2.5 to 1.5. When εco increases from 3000 μ ε to 6000 μ ε , εcc/εco increases from 4 to 5. When fco increases, its ductility decreases, and the restraint of FRP on the core concrete weakens. Additionally, when εco increases, its ductility increases. The restraint effect of FRP on core concrete is strengthened.

5. Conclusions

This study evaluated the existing axial compressive constitutive model of FRP-confined circular concrete columns and established a predictive model for the axial compressive mechanical behaviours of FRP-confined circular concrete columns based on ANNs and SVR. The main conclusions are as follows.
(1)
A database of the axial compressive mechanical behaviours of FRP-confined circular concrete columns was established from the relevant published literature, which included 298 datasets. The effects of D/t, fco, εco, ffrp, Efrp, and εfrp on the axial mechanical behaviours of FRP-confined circular concrete columns were considered.
(2)
Comparing and analysing the existing axial compressive constitutive model of FRP-confined circular concrete columns, the Lam model [21] exhibited the highest predictive accuracy for the compressive strength of FRP-confined circular concrete columns.
(3)
Comparing and analysing the existing axial compressive constitutive models of FRP-confined circular concrete columns, the Fardis model [17] exhibited the highest predictive accuracy for the peak compressive strain of FRP-confined circular concrete columns.
(4)
The ANN and SVR models can be used to predict the axial mechanical behaviours of FRP-confined circular concrete columns. Their predictive accuracy was much higher than that of the existing axial compressive constitutive model of FRP-confined circular concrete columns, and SVR had the highest predictive accuracy.
(5)
An intelligent algorithm was used to analyse the parameters of the axial mechanical behaviour of FRP-confined circular concrete columns. The analysis results show that D/t is positively correlated with the compressive strength, whereas fco, ffrp, and Efrp are negatively correlated with the compressive strength of the FRP-confined circular concrete columns. The ffrp is positively correlated with the peak compressive strain, whereas D/t, Efrp, and fco are negatively correlated with the peak compressive strain of FRP-confined circular concrete columns.

Author Contributions

Conceptualization, S.C.; Data curation, X.L.; Formal analysis, H.W.; Project administration, P.C.; Supervision, P.C.; Validation, X.L.; Writing—original draft, H.W.; Writing—review & editing, S.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the financial support of the National Natural Science Foundation of China (Project No. 52108132), the Natural Science Foundation of Hebei Province (Project No. E2021202067), the Colleges and Universities in Hebei Province Science and Technology Research (Project No. QN2021037), and the China Construction First Bureau (Group) Co., Ltd. Technology research and development projects (Project No. KJYF-2021-26-03).

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table
Table A1. Database information.
Table A1. Database information.
ReferenceParametersTimeNumber of Data
[1]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201230
[8]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201821
[43]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP, GFRP20014
[44]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP, GFRP200116
[45]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201212
[51]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP202016
[52]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP, GFRP201680
[53]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP, GFRP201922
[54]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , GFRP20188
[55]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP200412
[56]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201317
[57]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201845
[58]D/t ,   f co ( MPa ) ,   ε co μ ε ,   f frp ( MPa ) ,   E frp ( GPa ) ,   ε frp μ ε , CFRP201015

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Materials 15 04971 g001 550
Figure 1. FRP-confined circular concrete column.
Figure 1. FRP-confined circular concrete column.
Materials 15 04971 g001
Materials 15 04971 g002a 550Materials 15 04971 g002b 550
Figure 2. The value range of each parameter.
Figure 2. The value range of each parameter.
Materials 15 04971 g002aMaterials 15 04971 g002b
Materials 15 04971 g003 550
Figure 3. The histogram of each parameter distribution.
Figure 3. The histogram of each parameter distribution.
Materials 15 04971 g003
Materials 15 04971 g004 550
Figure 4. Predictive results of compressive strength. (a) Mander Model, (b) Fardis Model, (c) Lam Model, (d) Bisby Model, (e) Wu Model, (f) Youssef Model.
Figure 4. Predictive results of compressive strength. (a) Mander Model, (b) Fardis Model, (c) Lam Model, (d) Bisby Model, (e) Wu Model, (f) Youssef Model.
Materials 15 04971 g004
Materials 15 04971 g005a 550Materials 15 04971 g005b 550
Figure 5. Predictive results of peak compressive strain. (a) Mander Model, (b) Fardis Model, (c) Lam Model, (d) Bisby Model, (e) Wu Model, (f) Youssef Model.
Figure 5. Predictive results of peak compressive strain. (a) Mander Model, (b) Fardis Model, (c) Lam Model, (d) Bisby Model, (e) Wu Model, (f) Youssef Model.
Materials 15 04971 g005aMaterials 15 04971 g005b
Materials 15 04971 g006 550
Figure 6. Influence of changes in the number of hidden layers on predictive accuracy of compressive strength: (a) the influence of changes in the number of hidden layers on MSE; (b) the influence of changes in the number of hidden layers on R2.
Figure 6. Influence of changes in the number of hidden layers on predictive accuracy of compressive strength: (a) the influence of changes in the number of hidden layers on MSE; (b) the influence of changes in the number of hidden layers on R2.
Materials 15 04971 g006
Materials 15 04971 g007 550
Figure 7. Influence of changes in the number of hidden layers on predictive accuracy of peak compressive strain: (a) The influence of changes in the number of hidden layers on MSE; (b) the influence of changes in the number of hidden layers on R2.
Figure 7. Influence of changes in the number of hidden layers on predictive accuracy of peak compressive strain: (a) The influence of changes in the number of hidden layers on MSE; (b) the influence of changes in the number of hidden layers on R2.
Materials 15 04971 g007
Materials 15 04971 g008 550
Figure 8. The SVR algorithm flow.
Figure 8. The SVR algorithm flow.
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Materials 15 04971 g009 550
Figure 9. Predictive results of compressive strength by ANN.
Figure 9. Predictive results of compressive strength by ANN.
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Materials 15 04971 g010 550
Figure 10. Predictive results of compressive strength by SVR.
Figure 10. Predictive results of compressive strength by SVR.
Materials 15 04971 g010
Materials 15 04971 g011 550
Figure 11. Predictive results of peak compressive strain by ANN.
Figure 11. Predictive results of peak compressive strain by ANN.
Materials 15 04971 g011
Materials 15 04971 g012 550
Figure 12. Predictive results of peak compressive strain by SVR.
Figure 12. Predictive results of peak compressive strain by SVR.
Materials 15 04971 g012
Materials 15 04971 g013 550
Figure 13. The influence of D/t on compressive strength and peak compressive strain.
Figure 13. The influence of D/t on compressive strength and peak compressive strain.
Materials 15 04971 g013
Materials 15 04971 g014 550
Figure 14. The influence of Efrp on compressive strength and peak compressive strain.
Figure 14. The influence of Efrp on compressive strength and peak compressive strain.
Materials 15 04971 g014
Materials 15 04971 g015 550
Figure 15. The influence of ffrp on compressive strength and peak compressive strain.
Figure 15. The influence of ffrp on compressive strength and peak compressive strain.
Materials 15 04971 g015
Materials 15 04971 g016 550
Figure 16. The influence of fco on compressive strength and peak compressive strain.
Figure 16. The influence of fco on compressive strength and peak compressive strain.
Materials 15 04971 g016
Materials 15 04971 g017 550
Figure 17. The influence of mechanical behaviours of core concrete on FRP-confined circular concrete columns.
Figure 17. The influence of mechanical behaviours of core concrete on FRP-confined circular concrete columns.
Materials 15 04971 g017
Table
Table 1. The detailed information of main parameters.
Table 1. The detailed information of main parameters.
ParameterD/tfco/MPaεco/μεffrp/MPaEfrp/GPaεfrp/με
Minimum57.39.995038321.67200
Median615.439.42600350023416,000
Maximum1500.0136.33850493324543,000
Average721.246.724403321196.616,686
Standard deviation433.125.6598106671.24967
Skewness0.31.7−0.9−1.1−1.33.9
Table
Table 2. Details of existing axial compression constitutive models.
Table 2. Details of existing axial compression constitutive models.
ModelsEquation
for Stress		Equation
for Strain		ParametersRange of
Application
Mander [16] f cc f co = 2.254 1 + 7.94 f l f co − 2 f l f co − 1.254 ε cc ε co = 5 f cc f co − 4 f co / t frp / styles of FRPCFRP/GFRP-confined concrete columns
Fardis
[17]		 f cc f co = 1 + 4.1 f frp t frp R f co ε cc = 0.002 + 0.0005 E frp t frp R f co R / styles   of   FRP CFRP-confined circular concrete columns
Lam
[21]		 f cc f co = 1 + 2 f l f co ε cc ε co = 15 f l f co + 2 f co / t frp / styles   of   FRP / f frp CFRP/GFRP-confined circular concrete columns
Bisby [63] f cc f co = 1 + 3.587 f r 0.84 ε cc ε co = 1 + 0.024 f r f co styles of FRPMedium and weak FRP-confined concrete column
Wu
[64]		 f cc f co = 1.316 + 2.098 f l f co − 0317 ( f l f co ) 2 ε cc = 3.223 ε frp ( f l f co ) 0.44 f co / t frp / styles   of   FRP / R AFRP/CFRP/GFRP-confined circular concrete columns
Youssef
[65]		 f cc f co = 1 + 2.25 f r 1.25 ε cc = 0.003368 + 0.259 f r f co f frp E frp 0.5 t frp / styles   of   FRP / R CFRP/GFRP-confined circular concrete columns
Note: f cc : compressive strength of FRP-confined circular concrete columns (MPa); f co : compressive strength of core concrete (MPa); ε cc : strain corresponding to compressive strength of FRP-confined circular concrete columns (με); ε co : strain corresponding to compressive strength of core concrete (με); f frp : tensile strength of FRP (MPa); E frp : elastic modulus of FRP (GPa); R: section radius of specimen (mm); f r : effective lateral constraint stress provided by FRP (MPa); f l : lateral constraint stress provided by FRP (MPa).
Table
Table 3. Predictive accuracy of compressive strength.
Table 3. Predictive accuracy of compressive strength.
ModelsR2MSEMAPEIAE/%
Mander [16]0.79788.50.221.64
Fardis [17]0.791078.30.211.86
Lam [21]0.83271.90.201.30
Bisby [63]0.50707.50.271.99
Wu [64]0.79393.50.191.31
Youssef [65]0.83317.80.221.45
Table
Table 4. Predictive accuracy of peak compressive strain.
Table 4. Predictive accuracy of peak compressive strain.
ModelsR2MSE/10^7MAPEIAE/%
Mander [16]0.455.450.392.36
Fardis [17]0.485.690.613.48
Lam [21]0.4663.490.594.15
Bisby [63]0.139.180.794.86
Wu [64]0.1547.100.523.97
Youssef [65]0.455.490.332.81
Table
Table 5. Predictive accuracy analysis of compressive strength and peak compressive strain.
Table 5. Predictive accuracy analysis of compressive strength and peak compressive strain.
Performance
Indices		Compressive StrengthPeak Compressive Strain
ANNSVRLam [21]ANNSVRFardis [17]
R20.920.960.830.870.940.48
MAPE0.110.090.200.180.130.61
IAE/%0.090.071.300.150.113.48
MSE84.0063.46271.90.74 × 1070.47 × 1075.69 × 107
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Chen, P.; Wang, H.; Cao, S.; Lv, X. Prediction of Mechanical Behaviours of FRP-Confined Circular Concrete Columns Using Artificial Neural Network and Support Vector Regression: Modelling and Performance Evaluation. Materials 2022, 15, 4971. https://doi.org/10.3390/ma15144971

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Chen P, Wang H, Cao S, Lv X. Prediction of Mechanical Behaviours of FRP-Confined Circular Concrete Columns Using Artificial Neural Network and Support Vector Regression: Modelling and Performance Evaluation. Materials. 2022; 15(14):4971. https://doi.org/10.3390/ma15144971

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Chen, Pang, Hui Wang, Shaojun Cao, and Xueyuan Lv. 2022. "Prediction of Mechanical Behaviours of FRP-Confined Circular Concrete Columns Using Artificial Neural Network and Support Vector Regression: Modelling and Performance Evaluation" Materials 15, no. 14: 4971. https://doi.org/10.3390/ma15144971

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