Neural network‐based models versus empirical models for the prediction of axial load‐carrying capacities of FRP‐reinforced circular concrete columns

This study presents new neural‐network (NN)‐based models to predict the axial load‐carrying capacities of fiber‐reinforced polymer (FRP) bar reinforced‐concrete (RC) circular columns. A database of FRP‐reinforced concrete (RC) circular columns having outside diameter and height ranged between 160–305 and 640–2500 mm, respectively was established from the literature. The axial load‐carrying capacities of FRP‐RC columns were first predicted using the empirical models developed in the literature and then predicted using deep neural‐network (DNN) and convolutional neural‐network (CNN)‐based models. The developed DNN and CNN models were calibrated using various neurons integrated in the hidden layers for the accurate predictions. Based on the results, the proposed DNN and CNN models accurately predicted the axial load‐carrying capacities of FRP‐RC circular columns with R2 = 0.943 and R2 = 0.936, respectively. Further, a comparative analysis showed that the proposed DNN and CNN models are more accurate than the empirical models with 52% and 42% reduction in mean absolute percentage error (MAPE) and root mean square error (RMSE), respectively involved in the empirical models. Moreover, within NN‐based prediction models, the prediction accuracy of DNN model is comparatively higher than the CNN model due to the integration of neurons in each layer (9‐64‐64‐64‐64‐1) and embedded rectified linear unit (ReLu) activation function. Overall, the proposed DNN and CNN models can be utilized as paramount in the future studies.

Fiber reinforced polymers (FRPs) offer high tensile strength, which lead to the significant increase in the use of FRPs over the years in reinforced concrete (RC) structural members subjected to different loading conditions. 1,24][5][6][7][8][9] The FRPs have excellent fire resistance, which offers better structural performance compared to that of steel reinforced structural members. 3Fallah-Valukolaee et al. 4 found that using steel fiber in bilayer concrete beam reinforced with glass fiber-reinforced polymer (GFRP) rebars have superior structural response in terms of flexural resistance, flexural stiffness, toughness, fracture energy, and load-displacement as compared to beams reinforced with steel rebars. 4Also, FRP strips could be a significant paramount as stirrups reinforcements to enhance the shear performance of different types of FRP-RC beams. 5n the last few years, FRP bars have gained significant importance as an alternative longitudinal reinforcement in RC columns, replacing the steel reinforcement.Also, research studies have found that FRP bars are suitable to be used as an independent internal reinforcement as well as in combination with steel reinforcement in RC columns. 1,2,4][8][9][10][11][12][13][14][15][16] Hassan et al. 17 investigated the axial loadcarrying capacity of self-compacting RC column reinforced with GFRP bars.It was found that replacing steel rebars with GFRP rebars resulted in an increase in the axial loadcarrying capacities of columns by an average of 22%.However, a considerable difference in various approaches was observed to predict the axial load-carrying capacity of steel and GFRP-RC columns, as shown in Table 1.
Elchalakani et al. 18 investigated the axial loadbehavior of geopolymer concrete (GPC) columns reinforced longitudinally and transversely with GFRP bars.It was found that the design codes for FRP design overestimated the axial load-carrying capacities of columns when compressive strengths of the longitudinal GFRP bars were ignored.Elshamandy et al. 19 carried out a detailed experimental program to investigate the lateral deformation of FRP-RC columns.Based on the test results, GFRPs could be used in lateral resisting systems and overall GFRP-RC columns achieved good strength and load-deformation capacity.Tobbi et al. 20 examined the structural response of columns reinforced with GFRP bars in both longitudinal and transverse direction.It was reported that the average axial load-carrying capacities of columns were increased by 10% due to the contribution of GFRP bars.Further, it was recommended that GFRP bars can be effectively employed in confined concrete columns subjected to compression.
Recently, based on the experimental studies, different empirical models (Table 1) were established to predict the axial load-carrying capacities of FRP-RC columns.Consequently, the experimental and theoretical studies have achieved significant success in determining the axial load-carrying capacity of FRP-RC circular columns.However, both the experimental and theoretical approaches have some limitations.For example, the experimental investigation requires significant amount of time and resources, where the test results depend on the structural parameters and conduction of an appropriate testing methods.Similarly, the theoretical approaches are based on various fundamental assumptions to simplify the nonlinear relationships, and if the effect of individual parameter is overlooked, it leads to the low prediction accuracy and may not reflect the actual behavior of FRP-RC columns.Also, the theoretical models have limited utilization, unless a new formulation is established.As a result, a state-of the art data-driven artificial intelligence (AI)-based approach are introduced to accurately predict the axial load-carrying capacities of FRP-RC columns.
The AI-based prediction models are established using large dataset and have emerged as revolutionary in dealing with real-world problems, which involve non-linear relationships. 21,22Apart from predicting the axial load-carrying T A B L E 1 Empirical equations for GFRP-RC columns.

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Reference Empirical equations capacities of different types of FRP-RC columns, such approaches have also shown promising results in predicting other structural parameters, such as the compressive strength of concrete, 23 shear strength prediction of steel-RC beams 24 and flexural strength of concrete beams reinforced with FRP bars. 146][27][28][29][30] The NNs and other genetic computing models accurately predicted various structural parameters of FRP-RC members, such as ultimate limit state response. 31,32In such regard, Arash et al. 33 developed an artificial neural network (ANN)-based model to predict the load-carrying capacity of GFRP-RC columns.It was found that the strain in longitudinal and transverse bars, spacing between the bars, and ultimate strength were the critical parameters in the prediction model for the axial load-carrying capacity of GFRP-RC columns.Yan et al. 34 found that bar diameter, surface position, and embedded length greatly affected the prediction accuracy of GFRP-RC concrete specimens.Raza et al. 13 developed an ANN-based prediction model to predict the axial load-carrying capacity of FRP-RC columns.Further, Peng et al. 35 employed ANN and support vector regression (SVR) models to investigate the relationship between the effective parameters and predicted the mechanical performance of FRP-RC columns.
Based on the literature, the existing ML-based models developed for the prediction of the axial load-carrying capacity of FRP-RC columns cover limited range of structural parameters and type of FRPs.Further, the models developed for FRP-RC columns have limited applicability in predicting the structural response of rectangular or square cross-sections only.Also, there is no NN-based model available which can precisely predict the axial load-carrying capacities of circular concrete columns reinforced with Carbon Fiber-reinforced Polymer (CFRP) and GFRP bars.Hence, given the precision of AI-based prediction models, this research study proposes deep learning-based regression models, that is, deep neuralnetwork (DNN) and convolutional neural-network (CNN), for the prediction of axial load-carrying capacity of CFRP and GFRP-RC circular concrete columns.Following are the main contributions of this study: 1. Development of a numerical dataset of FRP-RC circular concrete columns to facilitate the implementation of NN-based models.2. Development of DNN and CNN-based models for the prediction of axial load-carrying capacity of FRP-RC circular columns and their comparison with the existing empirical models.

| NUMERICAL DATASET
The behavior of circular concrete columns reinforced longitudinally with FRP bars and transversally with FRP helices or hoops was experimentally investigated in several research studies.The dataset used in this study comprising 99 FRP-RC circular concrete columns specimens was established from the literature.The outside diameter of the columns ranged between 160 and 305 mm; and the height of the columns ranged between 640 and 2500 mm.Out of the 99 specimens, 86 specimens were reinforced with GFRP reinforcement, and 13 specimens were reinforced with CFRP reinforcement.Among the GFRP-RC specimens, 70 specimens were reinforced with longitudinal GFRP bars and GFRP helices; and 16 specimens were reinforced with longitudinal GFRP bars and GFRP hoops.Among the CFRP-RC specimens, 10 specimens were reinforced with longitudinal CFRP bars and CFRP helices, and 3 specimens were reinforced with longitudinal CFRP bars and CFRP hoops.All specimens opted for the dataset were tested experimentally under axial loading in the previous research studies.The experimental setup used in the literature varied as per the testing facilities.The detailed experimental program adopted in the individual research study can be found in the literature.  Hower, for clarification, the general details of the test setup adopted in various experimental research studies is summarized in this study.All specimens were capped at the top and bottom ends with high strength plaster to provide a uniform surface for equal distribution of applied loading.Further, the surface of the specimens was wrapped externally with CFRP, where the stress concentration was expected, to avoid premature failure during the testing.A pair of loading heads were used at both ends of the specimens to ensure that axial loading was applied at the desired loading eccentricity.The axial-deformations in specimens were monitored using different devices, such as electrical strain gauges, linear variable displacement transducers (LVDTs), and laser triangulation.All specimens were tested using displacement-controlled loading in a compression testing machine.During the test, the results were collected using a computer system attached with a data logger and internal load cell of the testing machine.
The dataset includes various parameters of the specimens, such as the cross-section details height to diameter h/d ratio of the FRP-RC circular concrete columns, compressive strength of concrete (f 0 c ), longitudinal reinforcement ratio (ρ l ), ultimate tensile strength of longitudinal FRP bars (f u ), elastic modulus of longitudinal FRP bars (E l ), diameter of the helix or hoop (d h ), spacing of the transverse reinforcement (s), tensile strength of FRP helix  or hoop (f h ), elastic modulus of FRP bar helix or hoop (E h ), and axial load-carrying capacity (P o ) of columns.Table 2 summarizes the parameters of the dataset used in this study.The developed dataset has been explored for correlation between input parameters and target variable (see Figure 1).The parameters including, diameter of the specimens, area of FRP bars and height of specimens were found most correlated to FRP_tran_pitch, indicating suitability of the dataset for the implementation of learning-based models.

| Empirical modeling
In the last few decades, the behavior of FRP-RC circular columns was extensively investigated, and various empirical models [58][59][60][61][62][63][64][65] were developed to predict the axial load-carrying capacities of the columns.Fanaradelli and Rousakis 58 proposed an analytical model to predict the peak stress and strain of rectangular FRP-RC columns tested under axial cyclic-loading.Similarly, Ibrahim et al. 59 proposed an empirical model to predict the ultimate strains under different loading conditions and extended the applications of empirical model for different types of FRP-RC columns.Cao et al. 60 investigated the experimental and theoretical study to investigate the behavior of FRP-RC columns under cyclic-compression loading.Similar findings in the light of existing research on the development of analytical models of FRP-RC columns were also reported by Mathays et al. 61 and Fanaradelli. 62In addition, several finite element models (FEM) were introduced in the literature to predict the axial loadcarrying capacity of FRP-RC columns.Fanaradelli and Rousakis 63 developed a pseudo-dynamic three-dimensional finite element (FE) model to predict the axial load-carrying capacity of different types of FRP-RC columns and explored multiple parameters, which could be effectively employed in seismic design and analysis of such members.It was found that the materials properties and loading conditions of FRP-RC columns primarily influence the accuracy of results.Fahmy et al. 64 found that using the material properties, loading ratio and stress-strain models for the development of FE model efficiently predicted the behavior of FRP-RC confined column.However, the developed FE model was not capable of predicting the local damage.The inaccuracy of FE model was attributed to the input parameters that is, materials properties and loading conditions.To cope up with the insignificant prediction of the local damage through FE models in FRP-RC confined columns, Ilki et al. 65 examined numerous types FRP-RC columns and considered additional factors, including cross-section shape, concrete strength, amount of internal transverse reinforcement, corner radius, existence of pre-damage, loading type (monotonic or cyclic), and the bonding pattern (orientation, spacing, anchorage details, additional corner supports) to investigate their effect on the ultimate strength, axial deformation, and ductility.
where f 0 cc and A g are the confined concrete strength and gross area of section, respectively.The symbols ϵ f , Ef, A f refer to the strain, modulus of elasticity and area of the FRP bars, respectively.
In Equation ( 1), f 0 cc is based on the confining pressure provided by lateral reinforcement.Several empirical models were developed to estimate the confined concrete strength in the columns reinforced with FRP helices/hoops.Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al. 45 proposed Equations ( 2)-( 5), respectively, to calculate the confining pressure provided by the lateral reinforcement in FRP-RC circular concrete columns.
where f lFRP is the confinement pressure developed by the FRP helices; E FRP and A hFRP = elastic modulus and crosssectional area of the FRP helices, respectively; f bent and d b = tensile strength of FRP helix or bent FRP bar and diameter of FRP helices; d c = diameter of the concrete core; s = pitch of FRP helices; and k e = confinement coefficient [as recommended in ACI 440.2R-08]. 70he change in pitch of FRP helices/hoops significantly affect the confinement pressure, which alters the axial loadcarrying capacities of the reinforced concrete columns.Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al. 45 developed Equations ( 6)-( 9), respectively using f lFRP to calculate the confined concrete strength (f 0 cc) of FRP-reinforced circular concrete columns.
where f co is the strength of unconfined concrete core; ψ f and k a = 0.95 and 1, respectively [as recommended in Pantelides et al. 8 ; and k c is the confinement coefficient factor (k c ¼ f co þ5f lFRP f co þ0:5f lFRP )].In the literature, the value of ϵ f was adopted based on various assumptions, for example, ε f = 0.002 corresponding to the initiation of plastic deformation, 40 Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al., 45 ϵ f = 0.003 was adopted corresponding to the strain in concrete.Hadhood et al. 45 adopted modified factors and used ϵ f = 0.0035 which resulted in reasonable predictions of the axial load-carrying capacities of FRP-RC concrete columns.The axial load-carrying capacities of the established dataset was predicted using the empirical models presented in Equations ( 2)-( 9).The empirical models proposed in Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al. 45 were used in this study to predict the axial load-carrying capacities of various specimens established in the dataset.

| Deep neural network
DNNs are learning-based AI algorithms that are formulated to function like animal brain.The structure of DNN consists of nodes, the hidden layers, and connections between nodes.As a fundamental working principle, input to the model is passed through the nodes which apply non-linear transformation and transmit it to neurons in next layer.At each hidden layer, a certain transformation is applied to input and is controlled by the weights, which update during the model training.
Neurons or nodes from one layer to other layer are connected through connections whose strength is determined by the weights.As a summary, DNN consists of an input layer, hidden layers, and an output layer. 71,72The structure of the DNN used for this research study is presented in Figure 2. It consists of four hidden layers with 64 neurons in each layer while (x 1 , x 2 , …, x 9 ) denote the input parameters and y denotes the target variable.The DNNs have the advantage of learning complex and non-linear relationships in data.However, DNNs have limitations of hyperparameters section, high computational resources requirement and lack of explanation for predictions.
F I G U R E 2 Deep neural network (DNN) structure.

| Convolutional neural network
CNNs are deep learning models specifically designed to deal with a grid type or matrix data.Since in CNN, each neuron is not connected to all the neurons in the next layer, rather inspired by the functionality of visual cortex (i.e., one neuron only responds to a stimuli within a limited receptive field), CNN models are often referred as the regularized form of DNN.A CNN model consists of convolutions layers, pooling layers, normalization layers, activation functions, and fully connected layers.Some hyperparameters for the construction of a CNN include depth of network, number of filters at each layer, stride size for filter, type of activation function, type of optimiser, type of loss function, learning rate, batch size and training epochs. 73In general, CNNs are specifically designed for the grid type data.However, their functionality of using only a limited number of neuron connections has also proven helpful in dealing with one-dimensional (1D) data.A transformed version of CNN to deal with 1D data is referred to as 1D-CNN. 74The structure of 1D-CNN used for this research study is presented in Figure 3 consisting of two convolutional layers and one fully connected layer.

| RESEARCH APPROACH
In this study, the investigation was carried out in four stages for the prediction of axial load-carrying capacity of FRP-RC circular columns.The first stage involved the development of a dataset of FRP-RC circular concrete columns, collected from the literature and experimental investigations as shown in Table 3.At the second stage, exploratory analysis was performed on the developed dataset to understand the correlations among the selected input parameters and target variable.In addition, the dataset was prepared for the learning-based models' implementation and was normalized using Standard Scalar transformation.The third stage involved the development of deep learning models (DNN, CNN) and selection of appropriate hyperparameters for training.Finally, at the fourth stage, the performance of the trained models was evaluated using standard measures [i.e., mean absolute percentage error (MAPE), root mean squared error (RMSE) score].In addition, the models were compared quantitively and graphically to assess their performances in predicting the axial load-carrying capacity of FRP-RC circular columns.

| MODELS AND EVALUATION MEASURES
Models were implemented using Python as programming language with scikit-learn, TensorFlow and Keras as supporting libraries.The NVIDIA GeForce graphical processing unit (GPU) hardware with memory of 6 GB was used for the training of models.For the reported

| Architecture of neural networks models
The computer architecture, structural configurations, number of neurons, and layers configuration significantly affect the performance of NN models.To train the available dataset and to get the suitable neural network architecture, the input parameters as shown in Table 2 were considered to establish both DNN and CNN based models.In addition, correlation map with input and target variables was established to determine the relationship of different input parameters with the output variables.Afterwards, the developed models were validated using the existing empirical models.

| RESULTS AND DISCUSSIONS
Mean absolute percentage error (MAPE), Root mean square error (RMSE) and Coefficient of determination (R 2 ) were used as evaluation measures for the assessment of implemented deep learning regression models.Brief theoretical details about each measure are summarized as: MAPE is the measure to evaluate the accuracy of regression model and is considered effective for cases involving no outliers.Mathematically, it is determined by dividing the error over the actual values as given in Equation (10).
where y a refers to the actual value, y t refers to the predicted value and n denotes the total number of data samples.RMSE is the measure to determine the spread of prediction error (residuals) around regression line and one of the common metrics for model evaluation.Mathematically, it is determined by taking squared root of mean squared error (i.e., dividing the sum of prediction error by the total number of data samples) as expressed in Equation (11).
where b Y t denotes the predicted value, Y a denotes the actual value and n denotes the total number of data samples.
The R 2 score is known as coefficient of determination and used for evaluating the capability of regression model against the input variable variations.Mathematically, it is determined by dividing the sum of squared prediction error by the sum of squared predictions (see Equation ( 12)).
Table 4 presents the comparison of empirical models and deep learning models for the prediction of axial load-carrying capacities of FRP-RC circular concrete columns.It can be observed that, among all the empirical models, the model proposed by Afifi et al. 38 predicted the load carrying capacities of all specimens in the database with the least MAPE and RMSE of 13.5 and 386.8, respectively.The model proposed in Hadhood et al. 45 predicted the axial capacities with MAPE and RMSE approximately 55% and 32% higher, respectively, than those predicted using the model proposed in Afifi et al. 38 However, the model proposed by Hadhood et al. 45 has a lowest standard deviation (SD) and coefficient of variation (COV) among all empirical models as shown in Figure 5. On the other hand, the developed deep learning models (DNN and CNN) predicted the axial load-carrying capacities of FRP-RC circular concrete columns with the least MAPE and RMSE.The developed CNN model predicted the axial capacities of the specimens with MAPE and RMSE of 9.3 and 195.9 which is approximately 36.84% and 65.52%, respectively than that of Afifi et al. 38 Similarly, the developed DNN model predicted the axial capacities of specimens with precise MAPE and RMSE and showed an improvement of 58.37% and 65.52%, respectively, compared to Afifi et al. 38 The accuracy of developed models (CNN and DNN) in comparison to the existing models to predict the axial capacities of specimens can also be observed from the R 2 values of 0.936 and 0.943.
T A B L E 4 Accuracy of the prediction models for the axial load-carrying capacities of FRP-RC circular concrete columns.

Error Empirical models
Neural network-based models Pantelides et al. 8 Afifi et al. 38 Karim et al. 40 Hadhood et al. 45 3) with a percentage difference between the experimental and predicted values of approximately 16% and 21%, respectively.This is due to the fact, the input parameters, and architecture of the NN models play a vital role to precisely predict the axial load-carrying capacities of FRP-RC concrete circular columns.Also, the established NN models can be further optimized in terms of input variables, number of neurons and hidden layers to get more closer relationship between the predicted and experimental results.The significance of features in the performance of NN-based models can be measured using multiple means.The simplest approach is to look at the correlation map (Figure 1) and identify the input features with highest correlation to the target variable.It can be observed that specimen's diameter is the top correlated feature.In addition, a more advanced approach was used to perform the SHAP (Shapley additive explanations) sensitivity analysis.It is noted that, SHAP is a powerful tool in the realm of machine learning interpretability.It provides a game-theoretic approach to explain the output of any machine learning model.With origins in cooperative game theory, SHAP assigns each feature an importance value for a particular prediction.These importance values provide a clear understanding of how the input features contribute to the model's predictions.This method not only aids in comprehending the inner workings of complex models but also enables the identification of crucial features that significantly impact the model's decisions.Figure 5 shows the summary of SHAP analysis for the CNN model.It can be observed that the specimen diameter, compressive strength of concrete (f c 0 ), and tensile strength (fu) of longitudinal FRP bars are identified as the top three influencing features and critical source of error in the accuracy of prediction models, which is aligned with the correlation map theory.Figures 6 and 7 show the comparison of the developed model with the existing models proposed by different scholars such as Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al. 45 As shown in Figure 7a, the DNN model showed a closer relationship in between ±10% with the best fitted line for results predicted using Pantelides et al. 8 of axial capacities of FRP-RC concrete circular columns.Similar observations were found in Figure 7b-d while predicting the axial capacities of specimens through DNN model to that of Afifi et al., 38 Karim et al., 40 and Hadhood et al. 45 This phenomenon can be explained that the developed DNN model has a good agreement with the experimental results and can precisely predict the axial load-carrying capacities of FRP-RC circular columns compared to the existing empirical models.Also, a satisfactory comparison was observed between DNN and CNN models to test and relate the accuracy of both models while predicting the axial capacities of specimens as showed in Figure 7e.Both DNN and CNN models satisfied the criteria for closer relationship and lie in between ±10% with the best fitted experimental line.Thus, it can be concluded, that the developed models with R 2 values closer to 1 have more significance than the existing empirical models to predict the axial load-carrying capacities of FRP-RC circular columns.

| Comparative analysis
Apart from the existing models proposed by different scholars such as Pantelides et al., 8 Afifi et al., 38 Karim et al., 40 and Hadhood et al., 45 the accuracy and precision of both developed ANN and CNN models were compared with the different ML algorithms in recently published research of Cakiroglu et al. 75 Based on R 2 values, the developed DNN and CNN models (R 2 = 0.943, and 0.936) for the prediction of axial load-carrying capacities of FRP-RC circular columns showed a good agreement with CatBoot and Lasso ML models (R 2 = 0.931 and 0.955).However, there was a slight variation between the R 2 values of the developed DNN and CNN models with the other established ML models.Further, a similar approach in terms of selection of critical parameters (see Table 2) for the prediction of axial load-carrying capacities of FRP-RC columns was adopted by Almomani et al. 76 using genetic expression programming (GEP) algorithm.The accuracy of the established DNN and CNN models (R 2 = 0.931 and 0.955) were validated and shown a reasonable agreement with the previous GEP ML models (R 2 = 0.978 and 0.992 for eccentric and concentric axialload, respectively).Based on the validation of developed ML models with the recently established ML models, these phenomena can be explained that the newly established DNN and CNN models using ML approach could be better employed to predict the axial load-carrying capacities of FRP-RC circular columns having outside diameter and height ranged between 160-305 and 640-2500 mm, respectively.

| CONCLUSION
The purpose of this research study is to propose NNbased prediction models for the prediction of the axial load-carrying capacities of FRP-RC circular columns based on an experimental dataset established from the literature.The objective of the investigation was achieved by comparing the axial load-carrying capacities of FRP-RC columns calculated using the empirical models from the previous studies with the deep learning model-based predictions.The following conclusions can be drawn from the results and analysis performed in this study: 1. Two distinct NN-based models, that is, DNN and CNN were proposed to predict the axial load-carrying capacity of FRP-RC circular columns based on a dataset of 99 FRP-RC circular columns established from the literature.2. Among the current empirical models proposed in the literature, the model proposed in Afifi et al. 38 resulted in better performance over the developed dataset for the prediction of axial load-carrying capacities of FRP-RC columns with R 2 = 0.831, MAPE = 13.5, and RMSE = 386.8. 3.In comparison with the empirical models, the proposed DNN and CNN models are more accurate with reduction in MAPE and RMSE by approximately 52% and 42%, respectively.Also, DNN and CNN models satisfied the best fitted line criteria and lie in between ±10% with the experimental results.However, CNN model slightly overpredicted the axial load capacities of few FRP-RC specimens compared to the experimental results.4. The prediction accuracy of DNN model is comparatively higher than the CNN prediction model for the axial load-carrying capacities of FRP-RC circular columns with R 2 = 0.946, MAPE = 7.4, and RMSE = 185.2.This is due to the effective integration of neurons in each layer (9-64-64-64-64-1) and embedded ReLu activation function within the structure of the model.
Both DNN and CNN models can predict the peak axial load-carrying capacities of FRP-RC columns.
However, among NN based models, DNN model is more efficient and can be employed to predict the axial loadcarrying capacities of FRP-RC concrete circular columns with high accuracy.Based on SHAP sensitivity analysis, specimen's diameter is the most influential input parameter influencing the accuracy of the neural network-based prediction models.Therefore, it is recommended to include a modified factor related to the specimen's diameter may increase the accuracy of the prediction models for FRP-RC columns in future studies.
Note: d h , diameter of the helix or hoop; E h , elastic modulus of FRP bar helix or hoop; E l , elastic modulus of longitudinal FRP bars; f 0 c , compressive strength of concrete; f h , tensile strength of FRP helix or hoop; f u , tensile strength of longitudinal FRP bars; ρ l , longitudinal reinforcement ratio; P o , axial load-carrying capacity of column; s, spacing of the transverse reinforcement.a Standard deviation.b Coefficient of variation in percentage.F I G U R E 1 Correlation mapping of input features with target variable.

6. 1 |
Validation of DNN and CNN models

Figure 4
Figure4shows the testing performance of DNN and CNN models.The developed DNN model precisely predicted the peak axial capacities of 20 specimens with respect to the experimental results.For the first 5 specimens, DNN model over predicted the axial load-carrying capacities of FRP circular columns with an average predicted axial load-carrying capacities approximately 8.7% higher compared to the experimental results.However, for the next 15 specimens, the DNN model underpredicted the peak response with an average predicted axial load-carrying capacity approximately 1.8% lower compared to the experimental results.Similarly, the developed CNN model followed almost the same pattern as DNN model with a slight overprediction of axial capacities of FRP reinforced concrete circular columns to that of experimental results.A dominant variation between the predicted and experimental results was observed for specimens C3 and C12 (as shown in Table3) with a percentage difference between the experimental and predicted values of approximately 16% and 21%, respectively.This is due to the fact, the input parameters, and architecture of the NN models play a vital role to precisely predict the axial load-carrying capacities of FRP-RC concrete circular columns.Also, the established NN models can be further optimized in terms of input variables, number of neurons and hidden layers to get more closer relationship between the predicted and experimental results.The significance of features in the performance of NN-based models can be measured using multiple means.The simplest approach is to look at the correlation map (Figure1) and identify the input features with highest correlation to the target variable.It can be observed that specimen's diameter is the top correlated feature.In addition, a more advanced approach was used to perform the SHAP (Shapley additive explanations) sensitivity analysis.It is noted that, SHAP is a powerful tool

F I G U R E 4
Testing performance of DNN and CNN.F I G U R E 5 SHAP (Shapley additive explanations) sensitivity analysis.FI G U R E 6 Performance of empirical and machine learning prediction approaches.

F I G U R E 7
Comparison of DNN with other prediction approaches.
Comparison of experimental and theoretical axial load-carrying capacities for FRP-RC circular concrete columns.
Note: SD and COV (%) refer to standard deviation and coefficient of variation in percentage, respectively.
a Designation of the specimen as referred in the original study.