Integration of Machine Learning Models and Tiering Technique in Predicting the Compressive Strength of FRP-Strengthened Circular Concrete Columns

Abstract

This study aims to investigate the performance of the combined machine learning (ML) models and tiering technique for predicting the compressive strength of FRP-strengthened circular concrete columns. A dataset consisting of 725 experimental results has been assembled from available research studies to evaluate the prediction models. Pearson’s correlation analysis has been carried out to investigate the relationship between seven input parameters and the target parameter. The Taylor diagram has been plotted to deter-mine the best design-oriented strength model. The prediction performance of the combined ML models and tiering technique was compared with that of single ML models and ten design-oriented strength models. The research outcomes revealed that applying the tiering technique significantly improved the prediction accuracy of the ML models. It was also found that the best ML model for predicting the compressive strength of FRP-strengthened circular concrete columns was the combined random forest model and tiering technique, which outperformed single ML and design-oriented strength models.

1. Introduction

The fiber-reinforced polymers (FRPs) with the superior characteristics of light weight, high tensile strength, being corrosion-free, and ease and speed of construction, as well as their short maintenance duration, have been recognized as effective and efficient confinement materials for strengthening or retrofitting reinforced concrete columns [1,2,3,4,5]. A great amount of research attention has been devoted over the last two decades to the FRP-strengthened circular concrete columns (hereafter referred to as FRP-SCCC columns for brevity) to comprehensively understand the behavior and accurately measure the ultimate conditions (e.g., compressive strength and ultimate strain) of FRP-SCCC columns [6,7,8,9,10]. As a result of extensive research investigations, the behavior of FRP-SCC columns has been fully captured, and more than 90 stress–strain models of FRP-SCC columns have been developed [11,12,13,14,15].

The available stress–strain models have been classified into two main categories, namely analysis-oriented and design-oriented stress–strain models [16]. The first category treats the FRP-SCC column as two separate components (e.g., FRP jacket and concrete core) and the interaction between these two components is explicitly accounted for by the force equilibrium and strain compatibility. Accordingly, the stress–strain curve of FRP-SCC columns has been developed based on an incremental numerical procedure. Although the analysis-oriented stress–strain models have been recognized as versatile and accurate models in capturing the behavior of FRP-SCC columns, they may be preferred for use in numerical analysis (e.g., non-linear finite element analysis) and undesirable for direct use in design due to their complexity. In contrast to the first category, the second category uses simple closed-form expressions, directly derived from experimental results, to predict the ultimate conditions and stress–strain curves of FRP-SCC columns [17]. Due to their simplicity and convenience, the design-oriented stress–strain models are suitable for direct use in practical designs. In fact, the available design codes for concrete structures externally strengthened by FRP materials, such as ACI 440.2R-17 [18], FIB Bulletin 14 [19], and CNR-DT 200 R1/2013 [20], adopted design-oriented stress–strain models for predicting the ultimate conditions and stress–strain curves of FRP-SCC columns.

It has been found that there is a considerable discrepancy between the predicted compressive strength obtained by design-oriented strength models and experimental results [11,21]. This could be attributed to the following reasons: (1) As the design-oriented models have been directly interpolated and collaborated from experimental results, the accuracy of design-oriented models heavily depends on the database used to develop the models. In fact, the design-oriented strength models provided an accurate prediction of the compressive strength of FRP-SCC columns with the experimental database used to develop the model; however, they yielded unsatisfactory predictions of the compressive strength of FRP-SCC columns with an extensive experimental database [11]. (2) Most of the design-oriented strength models were developed by the modification of the expressions proposed by Richart et al. [22] in which the correlation between the compressive strength of FRP-SCC columns $\left(f_{cc}^{\prime }\right)$ and lateral confining pressure generated by the FRP jacket ($f_{la})$ was assumed to be linear, which was governed by the mechanical properties of FPR materials and FRP size $(f_{la}=2E_{f}n{t}_f{\epsilon }_{fe}/D)$. However, it has been found that the compressive strength of FRP-SCC columns was influenced by various parameters such as the stiffness of the FRP jacket [21,23], the ultimate strain of the control concrete [21,23], and unknown multiple parameters [24]. Thus, the design-oriented strength models could not present the influence of various parameters on the compressive strength of FRP-SCC columns. (3) The compressive strength of FRP-SCC columns heavily depends on the maximum hoop strain of the FRP jacket $\left({\epsilon }_{fe}\right)$, which was defined as a product of the reduction factor $\left(k_{\epsilon }\right)$ and FRP ultimate strain $\left({\epsilon }_{fu}\right)$ determined from the coupon test. However, the reduction factor was reported to vary in the literature [16,21,25,26,27], whereas it was taken to be a constant value in most of the design-oriented strength models. This contributes to the uncertainty of the design-oriented strength models in predicting the compressive strength of FRP-SCC columns. Because of the uncertainty of the design-oriented strength model due to the above-mentioned reasons, it is crucial to find new approaches for accurately and reliably predicting the compressive strength of FRP-SCC columns.

Recently, ML models have been widely used to solve various complex problems in civil engineering, from construction management [28,29] and geotechnics [30,31], as well as materials [32,33], to structural engineering [34,35,36,37]. This is due to ML models offering significant advantages over traditional statistical methods as they are capable of learning complex and non-linear relationships from data without requiring explicit equations [38]. The ML models have been demonstrated to be a powerful tool for predicting the compressive strength of FRP-SCC columns as they can naturally learn complex relationships between variables in high-dimensional feature space to estimate the compressive strength of FRP-SCC columns without requiring input data of the maximum hoop strain [39,40,41,42,43,44]. Cevik and Guzelbey [40], Jalal and Ramezanianpour [41], Naderpour et al. [42], and Elsanadedy et al. [43] applied an artificial neural network (ANN) in predicting the compressive strength of FRP-SCC columns. It was reported that the predicted results from the ANN model were in good agreement with test values, and the ANN model achieved higher prediction accuracy than the existing design-oriented strength models. It should be noted that, in Cevik and Guzelbey [40], the ANN model obtained satisfactory prediction accuracy with the correlation coefficient $\left(R\right)$ of 0.98 on 101 test data of the compressive strength of CFRP-strengthened circular concrete (CFRP-SCC) columns. The ANN model adopted in Cevik and Guzelbey [40] was far more accurate than the design-oriented strength models. In contrast, Jalal and Ramezanianpour [41] and Naderpour et al. [42] found that the prediction accuracy of the ANN model with an average error of 10% and 9% was slightly higher than that of the design-oriented strength model with an average error of 14% and 13% on 128 and 213 test data, respectively, of the compressive strength of CFRP-SCC columns. Elsanadedy et al. [43] obtained a similar observation, noting that the prediction accuracy of the ANN model was slightly better than that of the design-oriented strength models on a database of 272 experimental results of the compressive strength of FRP-SCC columns. Cevik et al. [45] developed empirical equations for predicting the compressive strength of CFRP-SCC columns based on the stepwise regression and genetic programming techniques using 101 test results. It has been found in Cevik et al. [45] that the proposed empirical equations provided a good prediction for the compressive strength of CFRP-CS columns, with the ratio between the predicted and actual compressive strength varying between 0.99 and 1. It was also found that the proposed empirical equations outperformed the available design-oriented strength models. Cevik [46] collected 180 test results of the compressive strength of CFRP-SCC columns and applied soft computing models consisting of genetic programming, stepwise regression, neuro-fuzzy model, and ANN in predicting the compressive strength of CFRP-SCC columns. Cevik [46] reported that the prediction performance of soft computing models was much better than that of design-oriented strength models, in which the best soft computing model was obtained by the ANN model.

The application of the GP in developing the empirical equations for the ultimate conditions (e.g., compressive strength and ultimate strain of FRP-SCC columns) was also conducted by Lim et al. [47]. In Lim et al. [47], an extensive database of 832 test results of the compressive strength of FRP-SCC columns was assembled. It was found that the proposed empirical equation obtained a slightly higher prediction accuracy than the design-oriented strength models. Mansouri et al. [48] collected an extensive database of 1079 test results of FRP-SCC columns and applied single ML models, including neuro-fuzzy, neural network, multivariable adaptive regression splines (MARS), and M5 models in predicting the ultimate conditions of FRP-strengthened concrete columns. Mansouri et al. [48] concluded that the prediction accuracy of the M5 tree and ANN model was slightly higher than the best design-oriented stress–strain models in predicting the ultimate conditions of FRP-SCC columns, while the ML models of ANFIS and M5 Tree obtained lower prediction accuracy than the best design-oriented stress–strain models. Mozumder et al. [49] assembled 238 test results of FRP-SCC columns and applied two single ML models consisting of support vector regression (SVR) and ANN in predicting the compressive strength of FRP-SCC columns. It was reported in Mozumder et al. [49] that the single ML models predicted well the compressive strength of FRP-SCC columns and outperformed the existing design-oriented strength models. It was also reported that the SVR model performed better than the ANN model. Keshtegar et al. [50] used the dynamic harmony search (DHS) algorithm and 780 test results of FRP-SCC columns in developing the five types of strength and the five types of strain models of FRP-SCC columns. The research outcomes of Keshtegar et al. [50] showed that the proposed models provided a good prediction of the compressive strength of FRP-SCC columns with root mean square error ($RMSE$) of 0.359 and average absolute error ($AAE$) of 0.24, and had higher accuracy than the existing design-oriented stress–strain models in predicting the ultimate conditions of FRP-SCC columns.

Yu and Hu [51] applied five single ML models consisting of linear regression (LR), ridge regression (RR), decision tree (DT), random forest (RF) and ANN in predicting the compressive strength of CFRP-SCC columns. Yu and Hu [51] found that, among the single ML models, the ANN model obtained the highest prediction accuracy with a goodness of fit of 0.96. Tao, et al. [44] applied extreme gradient boosting (EGBoost) and compared its performance with three ML models of multivariable adaptive regression spline (MARS), Extreme Learning Machine (EML) and Random Forest GenRator (Ranger). Tao et al. [44] concluded that EGBoost model yielded a satisfactory prediction of the compressive strength of FRP-SCC columns with $R$ of 0.90. Zeng et al. [52] used single ML models of RF, Gradient Boosting Decision Tree (GBDTR), EGBoost, and ANN in predicting the compressive strength of FRP-SCC columns. The Conditional Tabular Generative Adversarial Network (CTGAN) was adopted in Zeng et al. [52] to generate synthetic data, which were combined with the experimental data (1111 data) for training and testing the model. The single ML models selected by Zeng et al. [52] surpassed the design-oriented strength models in predicting the compressive strength of FRP-SCC columns, with the best model obtained by the GBDTR model having the $R^2$(coefficient of determination), $RMSE, \mathrm{a}\mathrm{n}\mathrm{d} MAE$ of 0.972, 3.557, and 6.303, respectively, on the synthetic and test data.

It has been revealed from the literature review that although different ML models have been adopted to predict the compressive strength of FRP-SCC columns, there are no available studies that provide a comprehensive comparison of the performance of various single ML models. A thorough comparison of the prediction accuracy of different single ML models for the compressive strength of FRP-SCC columns may help design engineers in selecting an appropriate ML model for this purpose. Also, no investigation has been conducted to investigate the prediction accuracy of the combined ML models and tiering technique in predicting the compressive strength of FRP-SCC columns. Thus, this study aims to provide a comprehensive evaluation of the performance of single ML models and the combined ML models and tiering technique in predicting the performance of FRP-SCC columns. The performance accuracy of the ML models was compared to that of design-oriented strength models available in the literature and adopted in the three design codes for concrete structures externally strengthened with FRP materials

2. Prediction Models

2.1. Machine Learning Models

2.1.1. Artificial Neural Networks

Artificial neural networks (ANNs) have been designed to mimic the way that biological neural systems in the human brain process information. The ANN models typically consist of a multiplicity of nodes considered as the neurons of the human brain, accordingly called artificial neurons or units. The nodes are interconnected to each other and divided into a number of layers in which the first layer is called the input layer, the last layer is called the output layer, and the layers between the first and the last layers (one or more layers) are called hidden layers. The number and size of hidden layers indicate the complexity of the ANN models. The strength of the connections between nodes is represented by the weights ($w_i)$.

Each node of the ANN model performs three main functions, including receiving input signals from previous layer neurons, then processing them and sending an output signal to the subsequent layer neurons. The nodes receive the input signal by computing the weighted sum of the input signals and adding a bias ($b)$, as expressed in Equation (1). The nodes process the output signals and pass them to the subsequent layer neurons using the activation function. The widely used activation function is the sigmoid function, expressed in Equation (3), which transforms a continuous value to a value between 0 and 1.

$$z={\sum }_{i=1}^n{w_{i}x}_i+b$$

(1)

$$y=f\left(z\right)$$

(2)

$$f\left(z\right)={\displaystyle \frac{1}{1+e^{-z}}}$$

(3)

where $x_i=[x_1,x_2,\dots ,x_n]$ indicates the input signals; $w_i=[w_1,w_2,\dots ,w_n]$ indicates the connection weights; and $b$ indicates the bias.

The connection weights of ANN models and biases are adjusted and optimized during the training process to improve the model performance. For the training process, the ANN models perform three main steps consisting of forward propagation, error calculation, and backpropagation. In the forward propagation step, the input data move through the input layer to the hidden layer and then to the output layer. The neuron of each layer process and generate the data by first calculating the weighted sum and bias and applying the activation function. In the error calculation step, the predicted output is compared with the actual value using a loss function. The mean squared error (MSE) is usually used as a loss function for a regression problem, as expressed in Equation (4).

$$L={\displaystyle \frac{1}{N}}\sum _{i=1}^n{(y-y^{\prime })}^2$$

(4)

In the backpropagation step, the model computes the gradients of the loss, as shown in Equation (5), and updates the weights (Equation (6)) to minimize the loss.

$${\displaystyle \frac{\partial L}{\partial w_i}}={\displaystyle \frac{\partial L}{\partial \mathrm{a}}}.{\displaystyle \frac{\partial a}{\partial z}}.{\displaystyle \frac{\partial z}{\partial w_i}}$$

(5)

$$w_i^{new}=w_i^{old}-\alpha {\displaystyle \frac{\partial L}{\partial w_i}}$$

(6)

where $\alpha$ represents the learning rate.

2.1.2. Support Vector Machine for Regression

Support vector machines, which were first proposed by Vapnik [53], have been recognized as powerful and versatile ML models for classification and regression problems. Based on the required output, the SVMs can be categorized into two main types including support vector classification (SVC) and support vector regression (SVR), in which the former is used for classification tasks and returns the output with a set of discrete values (e.g., 0 or 1), while the latter is employed for the regression tasks and returns the output with the continuous numerical values (e.g., temperature or house price). Accordingly, the aim of the SVC model is to find an optimal hyperplane to separate different classes of data points with maximum margin, while the objective of the SVR model is to find an optimal function (hyperplane in high dimensions) to fit data points within an epsilon-tube ($ϵ$-insensitive tube).

Although the SVR is considered an extension of the SVMs, this model has received significant attention in the literature for non-linear regression problems. For conducting a regression task, the SVR model first applies a fixed mapping procedure to map input data into the high-dimensional feature space then utilizes non-linear kernel functions to fit the high-dimensional feature space. The input data in the high-dimensional feature space can be separated more easily than in the original input space. In the high-dimensional feature space, a linear regression function can be found to predict a continuous target value ($y)$ using the following expression:

$$f\left(x\right)={\omega }^T\varnothing \left(x\right)+b$$

(7)

where ${\omega }^T$ represents the weight vector; $b$ represents a bias term; and $\mathsf{\varnothing }\left(x\right)$ represents a kernel function to map input data ($x)$ to a vector in the high-dimensional feature space.

The most distinctive feature of SVR is to introduce an epsilon-tube ($ϵ$-insensitive tube) of tolerance defined in the vicinity of the linear regression function, in which the data points within the $ϵ$-insensitive tube incur zero penalty, while the data points occurring outside the $ϵ$-insensitive tube, which contribute to the loss function, receive penalties. The $ϵ$-insensitive loss function is defined as follows:

$$L_{ϵ}\left(y, f\left(x\right)\right)=\mathrm{m}\mathrm{a}\mathrm{x} (0, \left|y-f\left(x\right)\right|-ϵ$$

(8)

The ${\omega }^T$ and $b$ can be found by solving the optimization problem, as expressed in Equation (9). The optimization problem has two main objectives consisting of minimizing the model complexity (or maximize the flattest) and minimizing the errors outside the $ϵ$-insensitive tube. The model complexity is represented by the norm of the weight vector ${‖\omega ‖}^2$ in which a smaller ${‖\omega ‖}^2$ implies less complexity of the model. The error outside the $ϵ$-insensitive tube is measured by the slack variables (${\zeta }_i {\mathrm{a}\mathrm{n}\mathrm{d} \zeta }_i^{\ast })$.

$${min}_{\omega ,b,\zeta ,{\zeta }^{\ast }} {\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C\sum _{i=1}^n\left({\zeta }_i+{\zeta }_i^{\ast }\right)$$

(9)

subjected to the constraints:

$$\left\{\begin{array}{c}{y}_i-\left({\omega }^T\varnothing \left(x\right)+b\right)\le ϵ+{\zeta }_i^{\ast }\\ \left({\omega }^T\varnothing \left(x\right)+b\right)-y_i\le ϵ+{\zeta }_i^{\ast }\\ {\zeta }_i,{\zeta }_i^{\ast }\ge 0\end{array}\right.$$

(10)

where C represents the regularization parameter which controls the trade-off between flatness and errors.

2.1.3. Linear Regression

The linear regression model is a supervised ML model, which models the relationship between the input features (independent variables) and the target variable (dependent variable) as a linear equation, as expressed as follows:

$$y^{\prime }={\omega }_0+{\omega }_1{x}_1+{\omega }_2{x}_2+\cdots +{\omega }_p{x}_p$$

(11)

where ${\omega }_0$ represents the bias term (intercept); and ${\omega }_i$ represents the weights (coefficients) of feature $x_i$. The optimal values of ${\omega }_0$ and ${\omega }_i$ can be found by minimizing the loss function. The typical loss function of the linear regression uses the mean square error ($MSE$), as expressed as follows:

$$MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-y_i^{\prime })}^2$$

(12)

where $y_i$ represents the actual target value; $y_i^{\prime }$ represents the predicted value; and $n$ represents the number of data points.

2.1.4. Classification and Regression Tree (CART)

Classification and Regression Trees, developed by Breiman, et al. [54], are supervised ML models, which are used for either classification or regression problems depending on the nature of the dependent variable. A CART model allows predictor fields to be used at different tree levels, handles missing data with surrogate splitting, and can incorporate misclassification costs and prior probabilities in a classification problem. By applying the logic rules, the decision tree method has more advantages compared to other modelling techniques.

Three impurity measures are available for CART models to identify optimal splits depending on the nature of the target field. The Gini index $g\left(t\right)$ is commonly used for symbolic targets, while the least-squares deviation method is automatically chosen for continuous targets without explanation. The Gini index $g\left(t\right)$ at a node ($t)$ can be found using the following expression:

$$g\left(t\right)=\sum _{j\ne i}p\left(j|t\right)p\left(i\right|t)$$

(13)

where $i$ and $j$ represent the target field categories, which are defined as follows:

$$SE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-y_i^{\prime })}^2$$

(14)

$$p\left(j,t\right)={\displaystyle \frac{\pi \left(j\right)N_j\left(t\right)}{N_j}}$$

(15)

$$p\left(t\right)=\sum _{j}p\left(j,t\right)$$

(16)

where $N_j\left(t\right)$ and $N_j$ represent, respectively, the number of records of the category $j$ of node $t$ and category $j$ of the root node.

It should be noted that when the Gini index is utilized to measure the improvement from a split in tree development, any records in the current node ($t$) or the root node that lacks valid values for the split-predictor is excluded from the computations of $N_j\left(t\right)$ and $N_j$.

2.1.5. M5P Model

The M5P model is constructed by combining a decision tree and linear regression, where the decision tree is used to partition the data into groups with similar characteristics at the tree’s nodes, while linear regression is employed to represent the characteristics of each data group at the leaves. After the tree is fully developed, the M5P model applies pruning techniques to remove branches that do not improve the model’s accuracy, thereby simplifying the model and mitigating overfitting.

The dataset is recursively partitioned by maximizing the expected reduction in the standard deviation of the target value. For a potential partition that divides a set of instances $S$ into subsets $S_i$, the standard deviation reduction $\left(SDR\right)$ is given by:

$$SDR=sd\left(S\right)-\sum _{i=1}^n{\displaystyle \frac{\left|S_i\right|}{S}}\times sd(S_i)$$

(17)

where $S$ represents the set of instances reaching the node; and $st\left(S\right)$ represents the standard deviation of the target value in the set $S$. The partition that maximizes $SDR$ is chosen.

Once the tree structure is formed, each leaf node contains a linear regression model, which is expressed as:

$$y^{\prime }={\omega }_0+{\omega }_1{x}_1+{\omega }_2{x}_2+\cdots +{\omega }_p{x}_p$$

(18)

where $y^{\prime }$ represents the predicted value; ${\omega }_o$ represents the intercept; and ${\omega }_i$ represents the weight (coefficient) of input feature $x_i$.

2.1.6. M5Rules Models

The M5Rules model is considered an improvement over the M5P model, as it combines a set of if–then rules to construct a decision tree from the root to the leaves, with linear regression models at the leaves to represent the characteristics of the data groups. The best if–then rule at the leaf node is used as the if–then rule for the M5Rules model. The general form of each rule generated by M5Rules is expressed by:

$$\mathrm{IF} \left(x_1\in R_1\right)\cap \left(x_2\in R_2\right)\cap \dots \cap (x_m\in R_m) \mathrm{T}\mathrm{H}\mathrm{E}\mathrm{N} y^{\prime }=f\left(x\right)$$

(19)

where $R_i$ represents the condition on input variable $x_1$; and $f\left(x\right)$ represents a linear regression model associated with the rule.

2.1.7. Random Forest Model

The random forest (RF) model operates by combining multiple decision trees through the averaging of their outputs. Relying on the results of multiple decision trees helps the RF model mitigate overfitting and outperform individual tree models. The RF is built based on two main principles: random sampling with replacement to construct decision trees and random feature selection. For the first principle, the RF generates multiple subsets of data from the original dataset using bootstrap sampling, with each subset used to build an independent decision tree. For the second principle, the algorithm randomly selects a subset of features rather than considering all features when constructing each node of the decision tree.

For training a tree, a bootstrap sample is drawn from the original training dataset. For a given dataset of $D={\left\{(x_i,y_i)\right\}}_{i=1}^N$, a bootstrap sample $D_b$ is determined as:

$$D_b=\left\{\left(x_{i1},y_{i1}\right),\left(x_{i2},y_{i2}\right),\dots ,(x_{iN},y_{iN})\right\}$$

(20)

where $i_k$ represents an index, which is randomly selected from $\left\{1, 2,\dots ,N\right\}$ with replacement. The bootstrap sample normally contains approximately two-thirds of the original samples, while approximately one-third of the original samples is left out.

For growing a random tree, a subset of features ${\mathrm{F}}_r\subset \mathrm{F}$ is selected at each partition node of a tree as follows:

$$\left|{\mathrm{F}}_r\right|=m, m\ll p$$

(21)

where $p$ represents the total number of input features. The best partition is selected only among these $m$ features.

The predicted values were obtained by averaging the predictions of all individual trees of the random forest as follows:

$$y^{\prime }={\displaystyle \frac{1}{T}}\sum _{t=1}^T{y}_t$$

(22)

where T represents the total number of trees; and $y_t$ represents the prediction from the $t-\mathrm{t}\mathrm{h}$ tree.

2.1.8. Random Tree Model

The random tree (RT) model is a simplified version of the RF, referring to a single tree constructed by incorporating random elements. To build a RT, the model first randomly selects a subset of the training data. At each splitting node of the RT, only a random subset of features is considered, rather than all features.

For a given training dataset of $S={\left\{({\mathit{x}}_i,y_i)\right\}}_{i=1}^N$, where ${\mathit{x}}_i=(x_{i1},x_{i2},\dots ,x_{id})\in {\mathbb{R}}^d$ is the input feature vector and $y_i$ is the corresponding target variable, a tree in the random tree is grown by recursively partitioning the original data. At each internal node, a feature $x_j$ is randomly selected from a subset of features, and a split point $s$ is randomly drawn from the range of that feature. The dataset is then divided as follows:

$$S_{left}=\left\{\left({\mathit{x}}_i,y_i\right)|x_{ij}\le s\right\}, S_{right}=\left\{\left({\mathit{x}}_i,y_i\right)|x_{ij}>s\right\}$$

(23)

This iterative process proceeds until termination conditions are fulfilled, such as attaining the maximum allowable tree depth, having an insufficient number of observations within a node, or observing no further gains in prediction accuracy.

For a regression random tree, each left node corresponds to a region $R_k$ of the feature space. A regression random tree predicts output $y^{\prime }$ for an unseen input by averaging the training responses in that region as follows:

$$y^{\prime }={\displaystyle \frac{1}{\left|R_k\right|}}{\sum }_{i:{\mathit{x}}_i\in R_k}{y}_i \mathrm{if} \mathit{x}\in R_k$$

(24)

2.2. Design-Oriented Strength Models

The performance of the ML models in predicting the compressive strength of FRP-SCC columns was compared with the performance of ten design-oriented strength models available in the literature, which have been considered the best strength models for predicting the ultimate conditions of FRP-SCC columns. It should be noted that most of the design-oriented models for the compressive strength of FRP-SCC columns was developed based on the general expressions originally proposed by Richart et al. [22] for actively confined concrete, as expressed in Equation (25). Richart et al. [55] found that this model can be directly used for steel-confined concrete, while Fardis and Khalili [56] has demonstrated that this model was also suitable for FRP-SCC columns. For the use of the Richart et al. [22] stress–strain model for FRP-SCC columns, the lateral confining pressure $\left(f_l\right)$ can be found based on the strength and amount of the FRP jacket, as illustrated in Equation (26).

$${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1+k_1{\displaystyle \frac{f_l}{f_{co}}}$$

(25)

$$f_l={\displaystyle \frac{2f_{frp}n{t}_f}{D}}$$

(26)

where $f_{cc}^{\prime }$ and $f_{co}$ are, respectively, the compressive strengths of confined and control concrete; $k_1$ is the confinement effectiveness coefficient; ${nt}_f$ is the total thickness of FRP jacket; $D$ is the diameter of FRP-strengthened concrete; and $f_{frp}$ is the tensile strength of the FRP jacket in the hoop direction.

The ten design-oriented strength models comprised the three strength models adopted in the design codes for concrete structures externally strengthened with FRP materials (e.g., ACI 440.2R-17 [18], FIB Bulletin 14 [20] and CNR-DT 200/2004 [20]) and seven strength models developed by Ozbakkaloglu and Lim [21], Teng et al. [23], Shehata et al. [57], Pham and Hadi [58], Wei and Wu [59], Youssef et al. [60], and Fallah Pour et al. [61], which have been recognized as the best strength models in predicting the ultimate conditions of FRP-SCC columns. The ten design-oriented strength models are listed in

Table 1. Available models for determining the compressive strength of FRP-SCC columns.

No.	Reference	Compressive Strength Formulation
1	ACI 440.2R-17 [18]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1+\psi 3.3{\displaystyle \frac{f_l}{f_{co}}}$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fe}}{D}}$ ${\epsilon }_{fe}=k_{\epsilon }{\epsilon }_{fu}$
2	FIB Bulletin 14 [19]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=2.254{\left(1+7.94{\displaystyle \frac{f_l}{f_{co}}}\right)}^{0.5}-2{\displaystyle \frac{f_l}{f_{co}}}-1.254$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fu}}{D}}$
3	CNR-DT 200 R1/2013 [20]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1+2.6{\left({\displaystyle \frac{f_l}{f_{co}}}\right)}^{2/3}$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fe}}{D}}$ ${\epsilon }_{fe}$$=\min \left\{ {\eta }_a{\epsilon }_{fu}/{\gamma }_f;0.004\right\}$
4	Shehata et al. [57]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1+2.0{\displaystyle \frac{f_l}{f_{co}}}$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fu}}{D}}$
5	Youssef et al. [60]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1.0+2.25{\left({\displaystyle \frac{f_{lu}^{\prime }}{f_{co}}}\right)}^{{\displaystyle \frac{5}{4}}}$ $f_{lu}^{\prime }={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fu}}{D}}$
6	Teng et al. [23]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1+3.5\left({\rho }_k-0.01\right){\rho }_{\epsilon }$$\mathrm{for} {\rho }_k\ge 0.01$ ${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1$$\mathrm{for} {\rho }_k<0.01$ ${\rho }_k={\displaystyle \frac{{2E}_f{t}_f}{D\left(f_{co}/{\epsilon }_{co}\right)}}$ ${\rho }_{\epsilon }={\displaystyle \frac{{\epsilon }_{fe}}{{\epsilon }_{co}}}$ ${\epsilon }_{fe}=k_{\epsilon }{\epsilon }_{fu}$
7	Wei and Wu [59]	${\displaystyle \frac{f_{cc}^{\prime }}{f_{co}}}=1.0+2.2{\left({\displaystyle \frac{2r}{b}}\right)}^{0.72}{\left({\displaystyle \frac{f_l}{f_{co}}}\right)}^{0.94}{\left({\displaystyle \frac{h}{b}}\right)}^{-1.9}$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fu}}{b}}$
8	Ozbakkaloglu and Lim [21]	$f_{cc}^{\prime }=c_1{f}_{co}+k_1(f_{lu,a}-f_{lo})$ $c_1=1+0.0058{\displaystyle \frac{K_l}{f_{co}}}$ $f_{lo}=K_l. {\epsilon }_{l1}$ ${\epsilon }_{l1}=(0.43+0.009{\displaystyle \frac{K_l}{f_{co}}}){\epsilon }_{co}$ $K_l={\displaystyle \frac{{2E}_f{t}_f}{D}}\ge f_{co}^{1.65}$ ${\epsilon }_{co}=\left(-0.067f_{co}^2+29.9f_{co}+1053\right)\times {10}^{-6}$ $f_{lu,a}={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fe}}{D}}$ ${\epsilon }_{fe}=k_{\epsilon }{\epsilon }_{fu}$ $k_{\epsilon }=0.9-2.3\times f_{co}\times {10}^{-3}-0.75\times E_f\times {10}^{-6}$
9	Pham and Hadi [58]	$f_{cc}^{\prime }=0.7f_{co}+1.8f_l+5.7{\displaystyle \frac{t}{D}}+13$ $f_l={\displaystyle \frac{2E_{f}n{t}_f{\epsilon }_{fu}}{D}}$
10	Fallah Pour et al. [61]	$f_{cc}^{\prime }=f_{co}+k_1{K}_l{\epsilon }_{fu}$ $K_l={\displaystyle \frac{{2E}_f{t}_f}{D}}$ $k_1=2.5-0.01f_{co}$

Note: $n$ denotes the number of FRP layers; $t_f$, ${\epsilon }_{fe},$ ${\epsilon }_{fu}$, and $E_f$, respectively, denote the thickness, effective tensile strain, ultimate tensile strain, and elastic modulus of FRP; $f_{co}$, ${\epsilon }_{co}$, and $E_c$ denote the compressive strength, axial strain at the $f_{co}$, and elastic modulus of unconfined concrete, respectively; $f_{cc}^{\prime }$ denotes the ultimate compressive strength of FRP-SCC columns; ${\rho }_j$ denotes the volumetric ratio of FRP in the concrete columns; $D$ denotes the diameter of the circular columns; ${\rho }_{\epsilon }$ and ${\rho }_k$ denote the strain and confinement stiffness ratios of FRP confined concrete.

.

3. Experimental Database

3.1. Data Collections

A dataset consisting of 725 experimental results of the compressive strength of FRP-SCC columns was used for training and testing the ML models as well as for evaluating the accuracy of design-oriented strength models. For collecting the experimental results, only the FRP-SCC columns having the fiber orientation perpendicular to the cylinder axis were considered in this study. The experimental results assembled in this study have been reported by researchers [24,26,27,60,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142], which were also summarized in Ozbakkaloglu and Lim [21]. It should be noted that the test results were obtained from the compression test of the concrete cylinders, which were treated as short columns. Different types of external confining materials, including carbon fiber-reinforced polymers (CFRPs), high-modulus CFRP (HM CFRP), ultra-HM CFRP, glass fiber-reinforced polymers (GFRPs), and aramid fiber-reinforced polymers (AFRPs), were reported in these experimental studies.

For predicting the compressive strength of FRP-SCC columns, denoted as $Y$ for the ML models and as $f_{cc}^{\prime }$ for design-oriented strength models, seven input parameters, denoted $X_1$ to $X_7$ for the ML models, were used. The input parameters consist of the column diameter $\left(D\right)$; column height $\left(H\right)$; compressive strength of concrete $\left(f_{co}^{\prime }\right)$; ultimate strain of concrete $\left({\epsilon }_{co}\right)$; elastic modulus of CFRP $\left(E_{frp}\right)$; tensile strength of CFRP $\left(f_{frp}\right)$; and the thickness of CFRP $\left(n{t}_{frp}\right)$. It should be mentioned that, in this study, the influence of FRP materials on the compressive strength of FRP-SCC columns was considered via material properties [e.g., elastic modulus $\left(E_{frp}\right)$, tensile strength $\left(f_{frp}\right)$, and total thickness of FRP $\left(n{t}_{frp}\right)]$ instead of the lateral confinement provided by the FRP jacket ($f_l)$. This approach was also adopted in Mozumder et al. [49] and Naderpour et al. [42] for predicting the $f_{cc}^{\prime }$ of FRP-SCC columns. For the specimens without reporting the ${\epsilon }_{co},$ the ${\epsilon }_{co}$ was determined using the formula proposed by Ozbakkaloglu et al. [11], as follows:

$${\epsilon }_{co}=(-0.067{f_{co}^{\prime }}^2+29.9f_{co}^{\prime }+1053)\times {10}^{-6}$$

(27)

The statistical information of the experimental data is described in

Table 2. Statistics of parameters of FRP-SCC columns.

Parameters	Symbol	Unit	Min	Max	Mean	Std.
Column diameter	$D$	mm	50.0	406.4	155.8	54.48
Column height	$H$	mm	100.0	812.8	316.8	112.66
Concrete compressive strength	$f_{co}$	MPa	6.20	55.2	35.1	10.12
Ultimate concrete strain	${\epsilon }_{co}$	(%)	0.14	0.63	0.236	0.04
FRP elastic modulus	$E_{frp}$	GPa	4.9	640.0	173.0	107.61
FRP tensile strength	$f_{frp}$	MPa	75.0	4510	2745.8	1326.80
FRP total thickness	$n{t}_{frp}$	(mm)	0.057	15.0	0.842	1.284
Compressive strength of FRP-SCC column	$f_{cc}^{\prime }$	(MPa)	17.8	275.9	74.7	30.07

. Figure 1 illustrates the distribution of the experimental data in the form of histograms. The test database used to predict the compressive strength of FPR-SCC columns was presented in Table S1 of the Supplementary Materials.

3.2. Pearson’s Correlation Analysis

Pearson’s correlation analysis has been carried out to evaluate the linear correlation between parameters, as illustrated in Figure 2. Additionally, the influence of the FRP lateral confining pressure $\left(f_l\right)$ on the compressive strength of FRP-SCC columns was evaluated. It is evident from Figure 2 that there is no strong linear correlation between the seven input parameters and the output parameter (i.e., compressive strength—$f_{cc}^{\prime }$) of FRP-SCC columns; however, a strong linear correlation between the FRP lateral confining pressure and the compressive strength of FRP-SCC columns has been obtained. Figure 2 also shows that the compressive strength of control concrete $\left(f_{co}^{\prime }\right)$, the elastic modulus $\left(E_{frp}\right)$, and tensile strength $\left(f_{frp}\right)$, as well as the total thickness of the FRP jacket $\left(n{t}_{frp}\right)$, have a positive influence on the compressive strength of FRP-SCC columns in which the compressive strength of control concrete has the highest influence, followed by the influence of FRP total thickness. In contrast, column diameter $\left(D\right)$ and height $\left(H\right)$, as well as the strain of control concrete $\left({\epsilon }_{co}\right)$, have a negative influence on the compressive strength of FRP-SCC columns in which the influence of column diameter was slightly greater than the influence of column height.

4. Results and Discussions

4.1. Statistical Metrics

The accuracy of the design-oriented strength model and single ML models in predicting the compressive strength of FRP-SCC columns was evaluated using five statistical metrics, including correlation coefficient ($R$), mean absolute percentage error ($MAPE$), root mean square error ($RMSE$), and mean absolute error ($MAE$). The statistical metrics were computed as follows:

$$R={\displaystyle \frac{n\sum y{y}^{\prime }-(\sum y\left)\right(\sum y^{\prime })}{\left(\sqrt{n(\sum y^2)-{\left(\sum y\right)}^2 }\right)\sqrt{n(\sum y^{\prime 2})-{(\sum y^{\prime })}^2}}}$$

(28)

$$MAPE={\displaystyle \frac{1}{n}}\sum _1^n\left|{\displaystyle \frac{y-y^{\prime }}{y}}\right|$$

(29)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _1^n{\left(y-y^{\prime }\right)}^2}$$

(30)

$$MAE={\displaystyle \frac{1}{n}}\sum _1^n\left|y-y^{\prime }\right|$$

(31)

where $y$ and $y^{\prime },\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$, represent the actual and predicted values; and $n$ represents the number of data samples.

The $R$ metric reflects the correlation between actual and predicted data, with values ranging from −1 to 1. As the absolute value of $R$ approaches 1, the model is considered highly accurate. Meanwhile, the $MAPE$, $RMSE$, and $MAE$ metrics measure the discrepancy between predicted and actual values; thus, the smaller these metrics, the more reliable the model’s predictive capability.

4.2. Cross-Validation

The 10-fold cross-validation was utilized in this study to evaluate the performance of the selected ML models on 725 experimental results of compressive strength of FRP-SCC columns. The 10-fold cross-validation has been proven to be an effective method for optimizing the computational time and variance [143]. For this method, the dataset of 725 experimental results was randomly divided into ten equal or nearly equal parts (referred to as folds), with each fold containing approximately 10% of the data. An iterative process of ten rounds was constructed for the application of the 10-fold cross-validation in which, for each round, nine folds were used as the training dataset and one fold as the testing dataset. For each iteration, the statistical metrics of the ML models were determined on the held-out test fold. The average values of the statistical metrics from ten iterations were used as the final statistical metrics of the ML models.

4.3. The Prediction Accuracy of the Models

4.3.1. Design-Oriented Strength Models Versus Single ML Models

The correlations between the predicted $f_{cc}^{\prime }/f_{co}$ obtained from the prediction models and the experimental $f_{cc}^{\prime }/f_{co}$ are plotted in Figure 3. The threshold lines representing 20% difference between the predicted and experimental $f_{cc}^{\prime }/f_{co}$ are also plotted in Figure 3. Figure 4, Figure 5, Figure 6 and Figure 7 and

Table 3. Statistical metrics for the strength prediction $\left(f_{cc}^{\prime }\right)$.

No.	Model	Number of Data Points	Metrics
			$\mathit{R}$	$\mathit{M}\mathit{A}\mathit{P}\mathit{E}$ (%)	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ (MPa)	$\mathit{M}\mathit{A}\mathit{E}$ (MPa)
1	ACI 440.2R-17 [18]	725	0.79	17.2	20.2	13.7
2	FIB Bulletin 14 [19]	725	0.75	31.0	25.7	21.0
3	CNR-DT 200/2004 [20]	725	0.70	24.0	28.8	20.5
4	Shehata et al. [57]	725	0.79	16.2	19.8	12.6
5	Youssef et al. [60]	725	0.72	19.2	28.0	15.0
6	Teng et al. [23]	725	0.42	40.1	52.7	31.5
7	Wei and Wu [59]	725	0.67	35.3	37.4	29.4
8	Ozbakkaloglu and Lim [21]	725	0.80	17.6	21.5	12.7
9	Pham and Hadi [58]	725	0.79	15.7	19.4	12.5
10	Fallah Pour et al. [61]	725	0.78	15.7	20.2	12.0
11	Linear regression model	725	0.44	28.5	27.0	19.6
12	ANN model	725	0.71	28.8	23.2	18.7
13	SVR model	725	0.40	25.6	28.3	18.7
14	M5Rules model	725	0.81	16.1	17.5	11.8
15	M5Rules-Tiering model	725	0.82	16.0	16.5	11.5
16	M5P model	725	0.88	14.1	14.4	10.1
17	M5P-Tiering model	725	0.84	15.9	16.2	11.4
18	Random tree model	725	0.89	11.5	13.7	8.2
19	Random Tree-Tiering model	725	0.99	1.1	1.5	0.8
20	Random forest model	725	0.94	9.4	11.0	6.6
21	Random Forest-Tiering model	725	0.99	6.0	6.4	4.3
22	RepTree model	725	0.77	16.5	19.2	12.0
23	RepTree-Tiering model	725	0.76	16.4	17.2	11.6

compare the performance of the design-oriented strength model and ML models in predicting the compressive strength of FRP-SCC columns using different statistical matrices.

As illustrated in Figure 3, the data points representing the correlation between the predicted $f_{cc}^{\prime }/f_{co}$ achieved by CNR-DT 200/2004 [20], Teng et al. [23], and Wei and Wu [59] strength models and experimental $f_{cc}^{\prime }/f_{co}$ are distributed under the diagonal and lower threshold lines while the data points of the seven remaining design-oriented strength models of ACI 440.2R-17 [18], FIB Bulletin 14 [20], Shehata et al. [57], Youssef et al. [60], Ozbakkaloglu and Lim [21], Pham and Hadi [58], and Fallah Pour et al. [61] are distributed along the diagonal lines. These distributions indicates that the CNR-DT 200/2004 [20], Teng et al. [23], and Wei and Wu [59] strength models underestimated the compressive strength of FRP-SCC columns and were less accurate than the seven remaining design-oriented strength models. Among these seven remaining design-oriented strength models, the scatter of data points obtained by the FIB Bulletin 14 [20] model exceeded the upper threshold line, indicating that this model overestimated the compressive strength of FRP-SCC columns. It is evident from Figure 3 that the distribution of the data points obtained by the strength models of Pham and Hadi [58] and Fallah Pour et al. [61] were close to the diagonal lines, indicating that these two strength models are the most accurate design-oriented strength models in predicting the compressive strength of FRP-SCC columns. Among three design codes for FRP-strengthened concrete structures, the ACI 440.2R-17 [18] model yielded the highest prediction accuracy for the compressive strength of FRP-SCC columns, while the CNR-DT 200/2004 [20] model provided the lowest prediction accuracy.

For the ML models, it can be found from Figure 3 that a large number of data points determined by the LR, ANN, and SVR models are sparsely distributed between the upper and lower threshold lines, indicating that these three models had a low accuracy in predicting the compressive strength of FRP-SCC columns. In contrast, the data points obtained by the RT and RF models were tightly clustered along the diagonal line and fell into the upper and lower threshold lines, demonstrating that the predicted compressive strength obtained by the RT and RF models was in good agreement with the test results. Compared with the distribution of data points obtained from the design-oriented strength models, the data points obtained from the ML models are closer to the diagonal line, showing that the prediction accuracy of the ML models is higher than that of the design-oriented strength models. Specifically, the number of data points located outside the upper and lower threshold lines of the two best ML models (e.g., 202 data points for the random tree model and 170 data points for the random forest model) are much lower than that of the two best design-oriented strength models (e.g., 283 data points for the Pham and Hadi [58] model and 263 data points for the Fallah Pour et al. [61] model).

As shown in Table 3 and illustrated in Figure 4, Figure 5, Figure 6 and Figure 7, the five best design-oriented strength models were Ozbakkaloglu and Lim [21], Pham and Hadi [58], Fallah Pour et al. [61], ACI 440.2R-17 [18], and Shehata et al. [57] models. However, the best design-oriented strength model depends on the statistical metric. Figure 4 reveals that, by using the R metric, the best strength model is Ozbakkaloglu and Lim [21] model ($R$ = 0.8), followed by the ACI 440.2R-17 [18], Shehata et al. [57], and Pham and Hadi [58] models, which had the same $R$ of 0.79. By using the $MAPE$ metric, the Pham and Hadi [58] and Fallah Pour et al. [61] models with the $MAPE$ of 15.7% are the best strength models, followed by Shehata et al. [57] with the $MAPE$ of 16.2%. For the use of the $RMSE$ metric, the Pham and Hadi [58] and Shehata et al. [57] models had the highest prediction accuracy. In contrast, for the use of the $MAE$ metric, the best performance models are Fallah Pour et al. [61] and Pham and Hadi [58], respectively. It is also shown in Figure 4, Figure 5, Figure 6 and Figure 7 and Table 3 that, even using different statistical metrics, the best prediction model is the RF model, followed by the RT, M5P, and M5Rules models, respectively. It should be noted that, even using different statistical metrics, the three ML models of the RF, RT, and M5P outperformed the best design-oriented strength models of Ozbakkaloglu and Lim [21], Pham and Hadi [58], Fallah Pour et al. [61], ACI 440.2R-17 [18], and Shehata et al. [57].

Due to different statistical metrics yielding the distinct best design-oriented strength models, the Taylor diagrams of these models were plotted using the testing data, as illustrated in Figure 8. The Taylor diagram combines three key statistical metrics consisting of standard deviation, correlation coefficient, and root mean square error. In the Taylor diagram, the data point, which represents the prediction model, closest to the reference point indicates the best-performing model. As can be seen in Figure 8, Shehata et al. [57] was shown to be the best strength model among the design-oriented strength models of Ozbakkaloglu and Lim [21], Pham and Hadi [58], Fallah Pour et al. [61], ACI 440.2R-17 [18], and Shehata et al. [57] in predicting the compressive strength of FRP-SCC columns.

As the best design-oriented strength model was achieved by the Shehata et al. [57] model, the prediction accuracy of the ML models was evaluated by comparison with the Shehata et al. [57] model. For the use of the R metric, the prediction accuracies of the RF, RT, and M5P models were, respectively, 19.0%, 12.7%, and 11.4% higher than that of the best design-oriented strength model of Shehata et al. [57], while, for the use of the $MAPE$ metric, this improvement was 42.0%, 29.0%, and 13.0%, respectively. It was also found that the RF, RT, and M5P models obtained 44.4%, 30.8%, and 27.3%, respectively, higher prediction accuracy than the Shehata et al. [57] model by using the $RMSE$ metric while the enhancement of these three ML models compared to the best design-oriented strength model was 47.6%, 34.9%, and 19.8%, respectively.

4.3.2. The Prediction Accuracy of the Single ML Models Incorporated the Tiering Technique

The tiering method denotes a systematic framework in which data processing and prediction are conducted across multiple hierarchical levels to enhance predictive performance, computational efficiency, and latency management. In this approach, the output of an earlier tier is used as an input for subsequent tiers. The tiering technique is particularly useful for datasets exhibiting heterogeneity, hierarchical structures, or distinct operational regimes, where the use of a single global model may be insufficient to accurately capture complex underlying relationships. The tiering technique has been applied in this study to process experimental data to improve the performance of the ML models in predicting the compressive strength of FRP-SCC columns. The tiering technique was performed by adding one input parameter, namely data classification, which was either the H letter or the L letter. For the data point with the output parameter greater than or equal to the output parameter threshold $\left(T\right)$, it was classified as $H$, whereas for the output parameter smaller than the output parameter threshold, it was classified as $L$. The output parameter threshold was defined as follows:

$$T={\displaystyle \frac{Y_{max}+Y_{min}}{2}}$$

(32)

The performance of the ML model incorporating the tiering technique has been summarized in Table 3 and Figure 9. It is evident from Figure 9 and Table 3 that applying the tiering method yielded better prediction performance for the RF, RT, M5Rules, and RepTree models, but resulted in lower prediction performance for the M5P models. It is clear that the tiering technique significantly improved the prediction performance of the RF and RT models. By incorporating the tiering technique, the prediction accuracy of the RF model was improved by 5.3%, 36.2%, 41.8%, and 34.9% corresponding to the $R$, $MAPE,$ $RMSE$, and $MAE$ metrics, respectively. Similarly, the application of the tiering technique led to an increase in the prediction accuracy of the RT model of 11.2%, 90.4%, 89.1%, and 90.2%, respectively, corresponding to $R$, $MAPE,$ $RMSE$, and $MAE$ metrics. It is also found in Table 3 that, compared to the Shehata et al. [57] model, the RF and RT models incorporating the tiering technique had 25.3% higher prediction accuracy by using the $R$ metric; 63.0% and 93.2%, respectively, higher prediction accuracy by using the $MAPE$ metric; 67.7% and 92.4%, respectively, higher prediction accuracy by using the $RMSE$ metric; and 65.9% and 93.7%, respectively, higher prediction accuracy by using the $MAE$ metric. It is interesting to know that by incorporating the tiering technique, the RT model outperformed the RF model. The improvement of the RF and RT models incorporating the tiering technique in comparison to the best design code for the FRP-strengthened concrete structure, ACI 440.2R-17 [18], was 25.3% by using the $R$ metric; 93.6% and 65.1%, respectively, by using the $MAPE$ metric; 92.6% and 68.3% by using the $RMSE$ metric; and 94.1% and 68.6% by using the $MAE$ matric.

5. Concluding Remarks

This study presents the performance of the combined ML models and the tiering technique in predicting the compressive strength of FRP-SCC columns in comparison to the single machine learning models and design-oriented strength models. Based on the research outcomes of this study, the following concluding remarks can be given:

• Among selected design-oriented strength models, the most accurate strength model was obtained by the Shehata et al. [57] model and followed by Fallah Pour et al. [61], while among three design codes for the FRP-strengthened concrete structure, the ACI 440.2R-17 [18] model yielded the highest prediction accuracy and the CNR-DT 200/2004 [20] model provided the lowest prediction accuracy in predicting the compressive strength of FRP-SCC columns. For the ML models, the best performance models for the compressive strength of FRP-SCC columns were the random forest and random tree models.
• The best single ML models were the random forest, random tree, and M5P, which outperformed the best design-oriented strength model in predicting the compressive strength of FRP-SCC columns. The prediction accuracy of the random forest model was 19.0%, 42.0%, 44.4%, and 47.6% higher than that of the best design-oriented strength models by using $R$, $MAPE,$ $RMSE$, and $MAE$ metrics, respectively. The improvement in predicting the compressive strength of FRP-SCC columns of the random forest model compared to the best design-oriented strength model was 12.7%, 29.0%, 30.8%, and 34.9% by using $R$, $MAPE,$ $RMSE$, and $MAE$ metrics, respectively. These enhancements of the M5P model compared to the Shehata et al. [57] model were 11.4%, 0.6%, 11.6%, and 6.34%, respectively, by using $R$, $MAPE,$ $RMSE$, and $MAE$ metrics.
• The incorporation of the tiering technique into the ML model significantly improved the performance of the ML models in predicting the compressive strength of FRP-SCC columns. By applying the tiering technique, the prediction accuracy of the random forest model was improved by 5.3%, 36.2%, 41.8%, and 34.9% corresponding to $R$, $MAPE,$ $RMSE$, and $MAE$ metrics, respectively. The improvement in the prediction accuracy of the random tree model due to the incorporation of the tiering technique was 11.2%, 90.4%, 89.1%, and 90.2% corresponding to $R$, $MAPE,$ $RMSE$, and $MAE$ metrics, respectively.
• Compared to the best design code for the FRP-strengthened concrete structure, ACI 440.2R-17 [18], the improvement of the RF and RT models incorporating the tiering technique was 25.3% for the $R$ metric; 93.6% and 65.1%, respectively, for the $MAPE$ metric; 92.6% and 68.3% for the $RMSE$ metric; and 94.1% and 68.6% for the $MAE$ matric.
• The effectiveness of integrating the tiering technique with ML models was limited to predicting the compressive strength of FRP-SCC columns based on the 725 test results used in this study. Therefore, further investigations into the effectiveness of combining the tiering technique with ML models for predicting the ultimate conditions of FRP-SCC columns are needed. Additionally, the effectiveness of this combination should be evaluated using a larger dataset of the compressive strength of FRP-SCC columns.