Development of deep neural network model to predict the compressive strength of FRCM confined columns

Abstract

The present study describes a reliability analysis of the strength model for predicting concrete columns confinement influence with Fabric-Reinforced Cementitious Matrix (FRCM). through both physical models and Deep Neural Network model (artificial neural network (ANN) with double and triple hidden layers). The database of 330 samples collected for the training model contains many important parameters, i.e., section type (circle or square), corner radius r_c, unconfined concrete strength f_co, thickness n_t, the elastic modulus of fiber E_f , the elastic modulus of mortar E_m. The results revealed that the proposed ANN models well predicted the compressive strength of FRCM with high prediction accuracy. The ANN model with double hidden layers (APDL-1) was shown to be the best to predict the compressive strength of FRCM confined columns compared with the ACI design code and five physical models. Furthermore, the results also reveal that the unconfined compressive strength of concrete, type of fiber mesh for FRCM, type of section, and the corner radius ratio, are the most significant input variables in the efficiency of FRCM confinement prediction. The performance of the proposed ANN models (including double and triple hidden layers) had high precision with R higher than 0.93 and RMSE smaller than 0.13, as compared with other models from the literature available.

Graphical abstract

Keywords

FRCM / deep neural networks / confinement effect / strength model / confined concrete

1 Introduction

In recent decades, many studies have focused on upgrading existing structures due to aging and introducing more stringent design requirements [1,2]. In the area of seismic design, for example, the performance of structures designed based on old seismic standards requires an upgrade to satisfy standards required by current regulations (i.e., Eurocodes). Therefore, the engineer has gradually shifted to use advanced structural materials to achieve the sustainability and cost-effectiveness of strengthening methods. Over the last decade, fabric-reinforced cementitious matrix (FRCM) materials, also recognized as textile-reinforced mortars (TRMs) or textile-reinforced concrete (TRC), have become an attractive retrofitting solution. They enhance the deformation capacity and the strength of critical plastic regions of reinforced concrete (RC) members designed with a small volume content of transverse reinforcement. FRCM is a cement-based composite material that is comprised of high-strength fibers (i.e., carbon, glass, or basalt) in the form of textiles, in combination with inorganic binders such as cement or hydraulic lime-based mortars. The use of the FRCM system as an external means to strengthen existing RC columns to enhance their axial load capacity as well as ductility, has emerged in recent years with good potential results [3–6]. As demonstrated in these previous studies, the mechanical behavior of FRCM-strengthened concrete relates to many phenomena dependent on the properties of the FRCM system, the geometrical shape of the specimens, and the installation procedure. In addition, the random nature of the reinforcing system and its strong dependence on those phenomena lead to a challenge to determine an analytical model, which can forecast the mechanical behavior for all types of FRCM-confined concrete columns. Most of the existing empirical models for estimating the confined compressive strength of the FRCM-wrapped columns were developed by employing a few experimental databases and curve-fitting techniques with limited curve fitting functions [7,8]. The proposed empirical models using regression analysis do not have a good prediction of the behavior of FRCM confined concrete columns. This is because the behavior of FRCM-confined concrete is dependent on many variables, which have a complex relationship with each other resulting in noisy experimental data. Thus, further studies using a wider range of experimental data adopting a more precise approach is needed to achieve a more general empirical model.

Thanks to the development of science and technology, machine learning (ML) and artificial intelligence have been developed and applied in many fields. For example, it has become popular to apply deep learning to many engineering problems [9–14]. It was reported that deep learning could be used to predict the compressive strength of recycled concrete and rubber concrete, with 74 sets of concrete blocks and 223 experimental results, respectively [15,16]. The results in these previous studies indicated that deep learning achieved higher prediction accuracy in comparison with the classical supervised models.

Previous studies have employed artificial neural networks (ANN) to estimate the confined compressive strength of fiber reinforced polymer (FRP) concrete [17–21]. They indicated that ANN is a promising technique to evaluate the confined compressive strength of FRP concrete using material and structural properties. For example, Naderpour et al. [17] employed ANN to estimate the confined compressive strength of FRP concrete using six input variables (related to the size of the specimen, the thickness of FRP, tensile strength and elasticity of FRP, and unconfined compressive strength of concrete). They found that ANN models performed better than other existing linear, nonlinear, and second-order models. Although ANN has been considered as a good technique in predicting the compressive strength of concrete (including FRP concrete), up to now, there is no study using ANN to estimate the compressive strength of the FRCM-confined concrete. The ANN is one of the ML techniques which can deal with a large number of datasets, and this approach has proven itself. Thus, it is believed that ANN can tackle the limitations of traditional physical models as well as empirical approaches.

As a result, this is the first study that employed the ANN approach to examine the strength prediction of the FRCM-confined concrete using a dataset of 330 concrete columns, externally confined with FRCM, from different literature. This novel research contributes to the literature and practice, in several important directions. First, the ANN models in this study included a large dataset of 330 samples with 15 input variables related to geometric properties, mechanical properties of concrete, and mechanical properties of the fiber reinforcement. Second, the proposed ANN models were compared with various design codes and empirical models. Third, this study surveyed different sets of input variables to examine the relative importance of input variables in estimating the compressive strength of FRCM-confined concrete. Finally, based on the results of the proposed ANN model, this study can provide a graphical user interface for preliminary prediction of the strength of confined concrete columns FRCM system, which can provide a convenient platform for the practical design of the FRCM reinforcement.

2 Physical models/theories

At present, there are few models, which have been proposed to predict the behavior of FRCM confined concrete elements. Triantafillou et al. [22] proposed a simple confinement model with the hypothesis that the confined compressive strength and ultimate strain depend on the confining stress at failure. The investigation considered eight cylindrical samples with 150 mm in diameter and 300 mm in height and six short columns with a square cross-section of 250 mm × 250 mm, a corner radius of 15 mm, and a height of 700 mm. Colajanni et al. [23] studied 30 specimens with cross-sections of circular and square columns subjected to monotonic axial compression to assess the efficiency of a p-Phenylene Benzobis Oxazole (PBO) fiber. The authors of this work also proposed a theoretical model for PBO-FRCM confined concrete elements on the basis of the iterative formula suggested Spoelstra and Monti [24] for FRP confined concrete elements. This model took into account the interaction of the cementitious matrix of PBO-FRCM by a simple stress sum and was verified by the laboratory with regard to strength and ductility. Ombres [25] evaluated the effects of the number and orientation of fiber layers of the PBO-FRCM system compression on the strength and ductility of the confined specimens from twenty cylindrical concrete columns. The author also developed a semi-empirical model to estimate axial peak strength associated with the strain of PBO FRCM confined concrete. A design-oriented estimation model concerning both peak strength and the axial strain was proposed and validated using a large experimental database including 152 results of compression tests performed on FRCM-confined, plain concrete cylinders by Ombres and Mazzuca [7]. Fossetti et al. [26] proposed the simplified analytical models acquired from data collected from different experimental works for concrete columns confined by the FRCM system. The main advantages were that the models do not require the definition of the lateral confinement pressure to estimate the strength, ductility, and dissipated energy enhancement.

Recently, Gonzalez-Libreros et al. [27] tested 60 concrete columns to give experimental stress−strain curves under monotonic axial loading. The types of fibers, i.e., carbon and glass fibers, section geometries with two corner radius values, and two different pre-existing damage levels were employed as the input variables. Faleschini et al. [28] conducted twelve slender RC columns under uniaxial compressive loading by considering the different configurations of section geometry, amount of transverse steel reinforcement, and the number of fiber layers. Kadhim et al. [29] compiled an experimental database containing 137 specimens of FRCM confined concrete columns and investigated a review of four existing stress−strain models. Toska and Faleschini [30] conducted tests on 37 columns, in which 26 of them were confined to employing FRCM, and 11 were used as reference. The fiber type, number of layers, cross-section geometry, and loading protocol were used as the input variables.

This section will summarize some existing confinement models of concrete elements strengthened by FRCM to predict the increase of axial compressive strength through f_cc/f_co. Several parameters related to geometry, mechanical material properties of the specimens, and the FRCM system were considered. The existing empirical strength models consist of some conventional parameters such as the efficiency coefficient of the geometry (i.e., k_e used in Refs. [1–4] and κ_a, κ_b as used in ACI Committee 549 [31]) and the effective confining pressure. These coefficients depend on section shape (circle, rectangular), the corner radius r_c, the side dimensions of gross cross-section b, d, and the section areas A_e, A_c, A_g of the effective confined concrete, cross-section area the core concrete, respectively (Fig.1).

On the other hand, the parameters related to the specimens’ mechanical properties are the axial compressive strength (f_co), strain at the peak stress of unconfined specimens (ε_co), the elastic modulus of the fiber E_f, the fiber’s strength f_fu, the fiber’s ultimate strain ε_fu, the quantity of fiber layers n_f, and the thickness of the FRCM system t_f, and the fiber volumetric ratio {\rho }_{\text{f}}={\displaystyle \frac{4n_{\text{f}}{t}_{\text{f}}}{L}}. Besides that, Ombres and Mazzuca [7] considered the fiber’s orientation coefficient k_{\text{θ}}={\displaystyle \frac{1}{1+3\tan \theta }}; where \theta is the angle between the longitudinal fibers direction and the axis of the specimen. The ACI 549 [31] and Ortlepp’s research [32] considered the influence of the fiber’s cross-sectional area/textile layer a_f of the FRCM. The formulations for the effective confining pressure f_lu are presented in Tab.1.

It’s important to note that these models were built with limited points databases, and the mortar properties were not included in the presented model. In this study, additional parameters such as elastic modulus of mortar E_m and the compressive strength of mortar f_cm were used to propose a better prediction model.

3 Artificial neural networks

Nowadays, ANNs have attracted much research for the modeling of various problems relating to the structural engineering field [17,18,33–37]. Neural networks imitate the nervous systems of both humans and animals, which interprets the vital information in the brain [1–3]. ANN is a developed algorithm inspired by the human brain to deal with complex and time-consuming problems. The computer algorithm can carry out duties similar to the ones of the human brain, such as learning, making decisions, recalling, and concluding. ANN can learn from input and output and can model the relationships between them and based on results it can predict outputs on unseen input data. Although ANN shows analogies to only some of the human brain’s operations, they were originally inspired and established by modeling the structure of the brain. These networks are implemented to evaluate the goals according to the extensive delivered information. In general, ANNs can learn, classify, summarize, and predict the appointed task. ANN models comprise a number of connected layers, each of which includes the interconnected neuron system. As shown in Fig.2, ANNs have three layers, including input, hidden, and output layers. The detailed description of ANNs process presented in Fig.2 can be found in Refs. [38,39]. The cross-validation technique is used to reduce the error by dividing the input data into three sub-sets of data, as discussed in the previous works [38,39]. Equation (24) expresses the mathematical algorithm of the artificial neuron.

(24)O=f\left(\sum x_{\text{z}}{w}_{\text{z}}+b\right),

where O is the estimated value from ANN, w_z is the weight link values, x_z is the given assigned input, b is the value of bias.

The outcomes of the activation are then assigned to the next layer, as exhibited in Fig.2. The final selected weights are calculated from the available information. Hyperbolic and tanh activation functions are used in the present study between the input layer and intermediate layer (processing layer). These hyperbolic functions are used in the processing layer, and the output (target) layer (Eq. (25)) can tabulate the inaccuracy that occurs during this process. After finishing the training process, the result is then compared to the given target values using Eq. (25).

(25)E\left(w\right)=\frac{1}{2}{\sum }_i{\left(T-O\right)}^2,

where T is the target and O is predicted.

4 Database description and analysis

4.1 Database of FRCM for modelling

In ML models, the accuracy and reliability prediction of the output variables depend significantly on the input dataset. This study considered all parameters related to geometrical properties (section types, corner radius), mechanical properties (compressive strength, modulus of elasticity, ultimate tensile strength), and mechanical properties of fiber reinforcement. The target output variable is the confined compressive strength. A total of 330 experimental datasets were collected from previous studies [23,25,27,33,40–58] and summarized in Tab.2.

Fig.3 shows the frequency distribution of each input variable used for modeling. It can be seen that a high frequency of area was found in a range from 0 to 5 ×10⁴ mm² and mostly located around 2 ×10⁴ (Fig.3(a)), the l/h ratio mainly concentrates at the value of 0.5 (Fig.3(b)). The value of θ (° ) is almost at 90° with some values smaller than 50° (Fig.3(c)). For the properties of core concrete, the f_co and f_uo are observed in a range from 10 to 50 MPa and from 0 to 35 MPa, respectively (Fig.3(d) and Fig.3(f)). There are only a few f_co values bigger than 40 MPa, a few f_uo values smaller than 5 MPa, or larger than 30 MPa. Besides that, the ε_co is distributed from 0 to 0.75%, corresponding to the ε_uo is distributed from 0 to 1%. While the ε_co mainly concentrates at the value of 0.2 then the ε_uo concentrates at the value of 0.25 (Fig.3(e) and Fig.3(g)). For matrix, the f_cm is distributed over a fairly wide region from 2 to 80 MPa with very few values less than 10 MPa or greater than 70 MPa (Fig.3(h)). The f_tm ranges from 0 to 13 MPa and is mostly less than 10 MPa (Fig.3(i)); the E_m varies from 6 to 35 GPa (Fig.3(j)). For fibers, the f_fu was found in a range from 600 to 5800 MPa (Fig.3(k)) and the E_f ranging 50 to 300 GPa (Fig.3(l)). The a_f varies from 1 to 600 mm²/m (Fig.3(o)), however, there are some values greater than 500 mm²/m. These fibers are made from many different materials such as carbon, aramid, steel, glass, etc. The ρ ratio mainly concentrates at the value of 1 (Fig.3(m)); ρ_f ratio ranges from 0 to 0.02 but is mostly less than 0.01 (Fig.3(n)).

4.2 Detail of the hyper-parameter of LM

The following hyper-parameters are used in this work for the Levenberg–Marquardt training of the ANN models, as described in Matlab. These parameters will be used for remodeling the ANN models for predicting the strength of FRCM and shown in Tab.3.

4.3 Pre-processing phase

The standard data point’s process significantly impacts the ANN training process because all the different inputs are common between the same values regardless of the various units [59–61]. Thus, in this study, all the critical parameters are then normalized between the new limits of 0.1 to 0.9 using Eq. (26).

(26)X=\frac{\mathrm{\Delta }X}{\mathrm{\Delta }x}x+\left(X_{max}-\frac{\mathrm{\Delta }X}{\mathrm{\Delta }x}{x}_{max}\right),

where x is the present value, X denotes the normalized value and Δ indicates the difference of limits. In the current case, the value of X_max = 0.9 and ΔX = 0.8 are adopted against the new limits of 0.1−0.9.

4.4 Cross-validation for over-fitting and local minima

The problem of local minima (as illustrated in Fig.4(a)) is associated with the convergence of the iterative process to the solution which is not representative of the whole database. As a result, the training/calibration process stops early, without processing the whole database, and caused the problem of over-fitting or over-prediction (as illustrated in Fig.4(b)) of the ANN model compared to their counterpart target values. To avoid problems associated with local minima and over-fitting, the database is divided into three subsets used for training, validation, and testing purposes instead of just two (used for training and testing purposes only) [38,39,62].

As the authors used these cross-validation methods for predicting the strength of CFRP cylinders and GFRP RC members [38,39,62]. Through this method, during the process of the ANN model, the over-fitting problem can be avoided, and also the error will not be struck in the local minima, as illustrated in Fig.4(a).

To enhance the performance and accurate predictions of the ANN, the above-mentioned normalized data is initially divided into three sub-sets: (training, validation, and testing purposes). Matlab software is used to develop the ANN models and to randomly divide the database into three sub-sets: 80% of each database is used for training, 10% for validation, and another 10% for testing purposes [63,64].

Cross-validation (Fig.5) is a resampling procedure used to estimate the skill of a ML model on unseen data, widely used to evaluate ML models on a limited data sample [65]. A single parameter called k refers to the number of groups that a given data sample is to be split into. The procedure starts by shuffling the dataset randomly before slipping into k groups. For each unique group, a group will be considered a test data set to evaluate the model and the remaining groups as a training data set to train the model. The procedure allows each sample to be used as the hold-out set 1 time and train the model k – 1 times.

4.5 Criteria for the optimized models

The values of Pearson’s correlation coefficient (R) are used and determined using Eq. (27) [66–68] to make the different combinations of the parameters. The input variable has greater values of |R| of the output, indicating its importance for the output parameter. The choice of the optimized ANN model is evaluated by: 1) The error is calculated by the correlation coefficient (R); 2) mean absolute error (MAE); 3) mean squared error (MSE); and 4) root mean squared error (RMSE) [69,70], which are systematically expressed by Eqs. (27)–(30), respectively.

(27)R=\frac{{\displaystyle \sum _{i=1}^n\left[\left(T_i-\overline{T}\right)\left(O_i-\overline{O}\right)\right]}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(T_i-\overline{T}\right)}^2\cdot {\displaystyle \sum _{i=1}^n{\left(O_i-\overline{O}\right)}^2}}}},

(28)MAE=\frac{{\displaystyle \sum _{i=1}^n\left|T_i-O_i\right|}}{n},

(29)MSE=\frac{{\displaystyle \sum _{i=1}^n{\left(T_i-O_i\right)}^2}}{n},

(30)RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(T_i-O_i\right)}^2}}{n}},

where O_i is the target results and T_i denotes estimated results from ANN; the ratio of test values (T_i) and predicted values (O_i) and n the number of all samples in the databank. The ANN model has the highest R value and the lowest values of RMSE, MSE, and MAE is known as the optimized model.

5 Results of proposed ANN models

5.1 ANN modeling using all available 15 inputs parameters

In the first phase, the ANN models were trained by using the all available 15 input parameters i.e., A, l/h, θ, f_co, e_co, f_uo, e_uo, f_cm, f_tm, E_m, f_fu, E_f, ρ, ρ_f, a_f for predicting the compressive strength of the FRCM specimens. The terminology of the models is selected as all parameter double layer (APLD) and all parameter triple layer (APTL). For the Architecture of ANN (i.e., number of the hidden layers, different numbers of hidden neurons, type of activation function between layers, selection of input parameters, and training of ANN models), the author used the same procedures and guidelines as suggested in previous works [38,39,62]. As ANN is the global problem solver having no guideline for the training. Therefore, against each model, the number of hidden neurons is increasing from the equal number of input parameters, (as in APDL-1) to the double number of input parameters (as in APDL-6), indicated in the previous works [38,39,62]. Tab.4 describes the architecture details of the double layer ANN models with the available 15 input parameters, and Tab.5 describes the architecture details of the triple-layer ANN models.

The main features of the learning process are implemented using a multi-layer ANN model coded in the MATLAB environment using the Levenberg-Marquardt with MLFNN method [71]. The results of prediction using ANN models for all datasets for double layers and triple layers are shown in Fig.6. A higher value of R indicates a better prediction ability. In general, both ANN models for both double layers and triple layers performed well. The result of the compressive strength prediction of FRCM confined columns is comparable with the result of compressive strength prediction of FRP concrete in Refs. [17–19]. This indicates that the ANN model can predict well the compressive strength of FRCM confined columns.

The errors of ANN models in predicting FRCM for double layers and triple layers are shown in Fig.7. The results of MSE, MAE, and R of the ANN model with double layers are shown in Fig.7(a). All series exhibit a high performance of prediction ability with the value of R larger than 0.80. Among six series, the APDL-6 (series 6) achieved the highest prediction ability with the highest value of R (R = 0.93) and smallest values of MSE and MAE (MSE = 0.111 and MAE = 1.923), while the smallest prediction accuracy was found for the APDL-2 (series 2) with the lowest value of R (R = 0.84) and the highest values of MSE and MAE (MSE = 0.239 and MAE = 3.310). Fig.7(b) presents the results of MSE, MAE, and R of the ANN model with triple layers. Similar to the case of double layers, all cases of triple layers also possess a high prediction accuracy with a value of R greater than 0.76. Among six series, the APDL-2 (serie 2) obtained the highest prediction accuracy with the highest value of R (R = 0.90) and smallest values of MSE and MAE (MSE = 0.16 and MAE = 2.52). Whereas the lowest prediction accuracy was found for the serie 6 with the lowest value of R (R = 0.76) and the highest values of MSE and MAE (MSE = 0.51 and MAE = 5.24). Overall, the results indicate that the model for double layers has a higher prediction ability than that for triple layers. In summary, among the 12 trained models, the APDL-6 ANN model exhibited the best results with the highest value of R and lowest values of MAE and MSE, as a result, this model is the best optimized ANN model.

5.2 ANN modelling using different input parameters

As the above predictions of ANN models exhibited reasonable results, however, they used all available 15 input parameters, i.e., A, l/h, θ, f_co, e_co, f_uo, e_uo, f_cm, f_tm, E_m, f_fu, E_f, ρ, ρ_f, a_f for predicting the compressive strength of the FRCM specimens. Therefore, more investigation was carried out to optimize the accuracy of results while using the reduced number of input parameters. For this purpose, only some of the parameters are selected from the available parameters, based on the underlying theories for predicting the response of the FRCM cylinders.

The multi-correlation of the input parameters and output using Pearson’s Correlation is shown in Fig.8. Pearson’s correlation determines the strength and direction of the monotonic relationship between two variables rather than the strength and direction of the linear relationship between two variables. This is what Pearson’s correlation determines. The different color indicates the different values of correlation. It can be seen that among the input variables, there exists some strong correlation between them. For example, a strong correlation is observed for the case of f_uo and f_co. In addition, there is also a strong correlation between ε_uo and ε_co. A high correlation is also found for between f_cm and f_tm, E_m. The strong correlations are also observed between E_f and f_fu and between a_f and ρ_f. The strong correlations between inputs and output were revealed between a_f, ρ_f and f_cc/f_co (output).

This section examines the effect of the input variables on the confined maximum compressive stress prediction by extracting the relative importance of each input variable from the XGBoost model. Like ensembles of decision tree methods, the XGBoost can automatically provide estimates of the feature importance from a trained predictive model. In other words, the model explicitly calculates each attribute in the dataset and provides a score indicating how valuable each feature was in constructing the boosted decision tree. The higher the score of an input variable indicates a more significant impact on the prediction model. With the XGBoost algorithm, we can evaluate the importance of each input from three primary factors: gain, frequency, and cover [72], and the gain factor was used in this study. Further details related to this algorithm and the definition of the three primary factors can be found in Zheng et al. [73] and Breiman et al. [74]. Fig.9 shows the scores of the fifteen input variables. This result reveals that the fiber volumetric ratio is the critical factor governing confined compressive stress. Moreover, the angle between the longitudinal fibers direction and the axis of the specimen, the slender ratio l/h, and the tensile modulus of elasticity of fiber plays an essential role in predicting the confined compressive stress.

Based on the results of multi-correlation, it can be observed that some input variables have a low correlation with output. Thus, to reduce the number of input variables for a practical application, we have tried to use different numbers of input variables for modeling to obtain optimized models. The detail of different cases using different input variables for double layers and triple layers are shown in Tab.6 and Tab.7, respectively.

The results of ANN prediction for double and triple layers are shown in Fig.10. Regarding the case of double layers, all models obtained high prediction accuracy with R > 0.75 except for the case of SPDL-5 (R = 0.54) (Fig.10(a)). Regarding the case of triple layers, all cases also achieved a good prediction accuracy with R > 0.60. Overall, from Fig.10, it can be observed that the models for double layers have a higher prediction accuracy than those for triple layers.

The ANN model errors of the FRCM compressive prediction for double layers are shown in Fig.11(a). In general, the higher value of R and the lower values of MSE and MAE indicate better prediction accuracy of the model. The model SPDL-5 has the highest values of MSE and MAE (MSE = 0.65 and MAE = 5.52) and the lowest value of R (R = 0.54). This implies that the SPDL-5 model has the lowest prediction ability. In opposite, the SPDL-1 model has the lowest values of MSE and MAE (MSE = 0.19 and MAE = 2.42) and the greatest value of R (R = 0.89). This means that the SPDL-1 model attains the highest prediction accuracy. Besides, it can be observed that the SPDL-2 and SPDL-6 models have a comparable prediction accuracy in comparison with the SPDL-1 model. This also indicates that the SPDL-6 model with 5 input variables can be alternatively used to estimate properly f_cc/f_co.

Fig.11(b) shows the ANN model errors of the FRCM compressive prediction for triple layers. It can be observed that the SPTL-2 model has the highest prediction accuracy with R = 0.77, MSE = 0.37, and MAE = 4.15. The SPTL-3 model also achieved a comparable accuracy of prediction to that of the SPTL-2. The lowest prediction accuracy was found for the SPTL-1 model with MSE = 0.61, MAE = 5.22, and R = 0.60. In comparison with the result of the model using 15 input variables (APDL-6 model in Fig.7), the result of the SPTL-1 model, in this case is comparable. In summary, from the results of prediction accuracy for double layers and triple layers, the SPDL-1 model (with 13 input variables) has the highest prediction accuracy. In parallel, the SPDL-6 model (with 5 input variables) also achieved a high prediction accuracy in comparison with SPDL-1. Thus, it is suggested that the SPDL-6 (double layer with 5 input variables) can be used for estimating f_cc/f_co.

6 Comparison of ANN models with other models

In this section, the authors compared the two best ANN models (D-ANN and T-ANN) with other 4 classical models including Decision Tree, SVM, Gaussian Process, and XGBoost, and also with the other empirical physical models.

6.1 Comparison of ANN models with other classical models

For validation purposes, the two best ANN models (D-ANN and T-ANN) with 4 classical models including Decision Tree, SVM, Gaussian Process, and XGBoost, as described in Tab.8. In this study, we don’t compare the performance of the ANN models with the state-of-the-art ensemble learning models such as LightGBM and CatBoost. Their limitation is that it does perform well on the small dataset, and it mostly overfits the small datasets (rows less than 10000).

Fig.12 shows the results of four classical models including decision tree, SVM, Gaussian process, and XGBoost. In general, it can be observed that four models performed well with high accuracy of prediction. Among these models, the XGBoost model exhibited the highest value of R² (R² = 0.88).

The comparison between D-ANN and T-ANN with four classical models is presented in Tab.9. All models achieved a high accuracy of prediction, in which, the D-ANN and T-ANN performed better than other models for both training, testing, and all dataset. The decision tree model has the lowest prediction accuracy, while the D-ANN model obtained the highest prediction accuracy.

6.2 Comparison between ANN models and physical models

The strength models proposed by Colajanni’s model [23], Ortlepp’s model [32], Triantafillou’s model [22], ACI 549.4-13 [32], Ombres’s model [7], de Caso’s model [33], Fossetti’s model [26], were evaluated using the database to compare with the strength model using the ANN approach. These models were chosen because of their simple implementation and are extensively used in the literature as summarized in Tab.1. Because D-ANN and T-ANN models with 15 input variables achieved high prediction accuracy, thus these models with 15 input variables were employed for comparison. The preliminary evaluations were done using two indices; one is R and the second is RMSE given by Eqs. (27) and (29), respectively. The comparison of predictions between the ANN model and physical models for the compressive strength of FRCM is presented in Fig.13. It can be observed that both D-ANN and T-ANN models performed very well with a high value of R (R ≥ 0.90). Among ANN models and physical models, the D-ANN model achieved the lowest value of RMSE (RMSE = 0.11) and the highest value of R (R = 0.93), and the T-ANN model has ranked the second with a low value of RMSE (RMSE = 0.13) and high value of R (R = 0.90).

In contrast, other physical models have a low prediction accuracy with a high value of RMSE and a low R-value. All physical models obtained a value of R less than 0.73. The Ortlepp’s Model attained the lowest value of R (R = 0.51), while the highest value of R (R = 0.71) was found for the Ombres’s Model. Furthermore, it can be seen that all data points of D-ANN and T-ANN models locate inside ±20% error line, while all existing physical models contain many data points located out of ±30%. These results indicate that D-ANN and T-ANN performed well and achieved higher prediction accuracy in comparison with existing physical models. In summary, the D-ANN model achieved the highest prediction accuracy among ANN and physical models.

Fig.14 illustrates the normal distribution (ND) of the V_EXP/V_PRED ratios, as for ANN (calculated as V_EXP/V_ANN) generated the least standard deviation of 1.05, better than the other empirical NDs. The ND curves for the Current Design Codes (CDCs) (i.e., EC2 and ACI) have a wide range from 0 to 2.0 on the x-axis with a peak value smaller than 1.5. From Fig.14, it can be seen that underestimated and overestimated values were found for the physical models. Two models namely De Caso and Colajanni show overestimated values (i.e., most values of V_EXP/V_PRED smaller than 1.0), while other physical models give underestimated values (i.e., values of V_EXP/V_PRED larger than 1.0). The predicted values of CDCs (i.e., ACI and EC2) are mainly smaller than experimental results (V_EXP/V_CDCs) greater than 1.0. Furthermore, Fig.14 shows that 90% of ANN models locates in a range of 0.755 to 1.255, in which the D-ANN model has a higher peak. In summary, the ANN models give a better result of the normal distribution curve.

The range limits of f_cc/f_co ratio for the FRCM are presented in Fig.15. From the figure, it can be observed that f_cc/f_co ratio is mostly located in two ranges of 1.0–1.25 and 1.25–1.50. For all ranges of f_cc/f_co, the number of samples generated from two ANN models fits well with the experimental results, particularly the D-ANN model. For the range 1.0–1.25, Ortlepp, ACI 594.4, Ombres, and Triantafillou models overestimate the number of samples; while de Caso and Colajanni’s models underestimate the number of samples. Regarding the range of 1.25–1.50, four physical models (Ortlepp, ACI 594.4, Ombres, and Triantafillou) underestimate the number of samples, however, two models de Caso and Colajanni’s models overestimate. In summary, the results confirm that two ANN models performed better than physical models, and the D-ANN model (double layers model) has the highest prediction ability. As a result, it can be concluded that these ANN models can be applied for practical applications.

7 Conclusions

This study evaluated the compressive strength of FRCM confined concrete using ANN models in comparison with the existing physical models. Some main conclusions drawn from this study are given as follows.

1) The performance of the proposed ANN models (including double and triple hidden layers) was in high accuracy with R greater than 0.93 and RMSE less than 0.13. Compared with the ACI design code and five empirical models, the ANN model with double hidden layers was shown to be the best to predict the compressive strength of FRCM confined columns.

2) ANN models were compared with other classical models (decision tree, SVM, Gaussian process, and XGBoost). The results show that all models performed well in predicting the compressive strength of FRCM. The D-ANN model has the highest prediction accuracy among six models.

3) The proposed ANN models can estimate the compressive strength of FRCM-confined concrete with high precision, which can be useful for engineers in the design of the compression members of confined concrete.

4) The proposed ANN models for double hidden layers with 15 input variables (APDL-1) achieved the highest prediction accuracy. Besides, the results also indicate that the ANN model for double hidden layers with 5 input variables (SPDL-6) also obtained a comparable prediction accuracy. This indicates that f_co, f_fu, E_f, ρ, ρ_f (i.e., the compressive strength of unconfined concrete, type of fiber mesh for FRCM, type of section, and the corner radius ratio) are the most important factors for modeling.

5) A graphical user interface for predicting the compressive strength of FRCM confined columns was established using the ANN model to give a convenient platform for designing FRCM reinforcement.

The finding of this study can provide an easy method to estimate the compressive strength of FRCM confined columns. The result of this study can contribute to literature as well as help an engineer in calculating and designing. However, further efforts using different machine algorithms should be made to compare as well as enhance the performance of the design models and provide an accurate theoretical confinement model for FRCM-wrapped columns for design and evaluation purposes. Finally, to measure the relationship between variables, some other metrics, such as Distance Correlation, Mutual information, and the Maximal information coefficient should be conducted in further studies.

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