Proposed numerical and machine learning models for fiber-reinforced polymer concrete-steel hollow and solid elliptical columns

Abstract

This study employs a hybrid approach, integrating finite element method (FEM) simulations with machine learning (ML) techniques to investigate the structural performance of double-skin tubular columns (DSTCs) reinforced with glass fiber-reinforced polymer (GFRP). The investigation involves a comprehensive examination of critical parameters, including aspect ratio, concrete strength, number of GFRP confinement layers, and dimensions of steel tubes used in DSTCs, through comparative analyses and parametric studies. To ensure the credibility of the findings, the results are rigorously validated against experimental data, establishing the precision and trustworthiness of the analysis. The present research work examines the use of the columns with elliptical cross-sections and contributes valuable insights into the application of FEM and ML in the design and evaluation of structural systems within the field of structural engineering.

Graphical abstract

Keywords

elliptical column / fiber-reinforced polymer / machine learning / finite element method / ABAQUS

1 Introduction

In the realm of structural engineering, innovation is constantly sought to improve the performance and efficiency of building elements. One such advancement is the utilization of fiber-reinforced polymer (FRP) composites; these have emerged as a promising alternative to conventional construction materials. FRP composites offer benefits such as a high strength-to-weight ratio and excellent corrosion resistance [1–3]. They have found applications in retrofitting of existing reinforced concrete structures as well as in constructing new structures, often hybridized with traditional materials [4–9].

External confinement using FRP jackets has become widely accepted as a cost-effective solution for enhancing the strength and ductility of reinforced concrete columns [1,8,10]. This technique significantly improves the compressive strength and ductility of circular concrete columns [11–13]. However, the effectiveness of FRP confinement varies depending on the shape of the column section. Circular columns exhibit notable improvements in the strength and ductility with FRP confinement, while rectangular columns experience limited benefits due to the non-uniform distribution of confining pressure [14–19]. To address this issue, corner rounding is often recommended [15]. Another area of interest is the development of hybrid FRP-concrete-steel double-skin tubular columns (DSTCs). These columns combine an inner steel tube, an annular layer of concrete, and an outer FRP tube to optimize their performance [20,21]. Hybrid DSTCs offer advantages such as cost reduction; the outer FRP tube acts as an in-situ mold for concrete casting and provides excellent corrosion resistance [20,21]. They are particularly efficient for use in seismic regions due to their superior energy dissipation capabilities during earthquakes. The choice of concrete material in the design of FRP-reinforced concrete structures plays a critical role in their performance. High-strength concrete (HSC) has gained attention due to its enhanced mechanical properties, but its behavior when confined by FRP composites requires further investigations [22,23]. Challenges associated with HSC in FRP-confined columns include non-uniform confinement stress distribution and reduced FRP strain capacity near corners [24–26]. To address these challenges and improve the performance of FRP-confined HSC columns, modification of rectangular sections into elliptical columns with continuously varying curvature has been proposed as a potential solution [27]. However, there is a lack of research on elliptical DSTCs, even though they offer design flexibility and structural advantages [27–30].

Application of finite element method (FEM) and machine learning (ML) techniques in the field of structural engineering have gained appreciable attention and exhibit great potential in accurate prediction of structural forces and reactions [31–40]. FEM simulations enable accurate analysis and prediction of complex behaviors in various structural engineering applications, such as FRP-confined structures [41–43], composite material [44], seismic-resistant structures [45], pre-stressed structures [46,47], etc. ML techniques involve development of algorithms and models that allow computers to learn from data and make predictions or decisions without being explicitly programmed. They offer a novel approach to understanding and predicting the behavior of complex structures [48,49], complementing traditional analysis methods like FEM [50–53]. ML is especially advantageous when it comes to handling large amounts of structural data and identifying complex patterns within those data. By training ML models on historical data obtained from physical experiments, numerical simulations, or sensor measurements, it is possible to create data-driven models that capture the underlying relationships between input variables and output structural responses and thereby to predict the behavior of structures under different conditions. Deep collocation method is one such method for solving diverse problems, offering simplicity, accuracy, and good computational efficiency [54–57].

This study presents a comprehensive investigation on the behavior of elliptical DSTCs and double-skin filled tubular columns (DSFTCs) confined using glass fiber-reinforced polymer (GFRP). The research approach combines FEM simulations using the ABAQUS software with ML techniques to gain insights into the structural performances of these columns. The FEM simulations conducted in ABAQUS provide a detailed understanding of how the columns behave under parameter variations, while the ML models enhance the prediction of the structural responses of DSTCs and DSFTCs. The research methodology involves a diligent comparative analysis between DSTCs and DSFTCs using 54 specimens modeled employing ABAQUS. Furthermore, five different ML algorithms are evaluated to identify the most suitable algorithm for predicting the structural response of these columns. To ensure the accuracy and reliability of the analysis, the results obtained from the FEM simulations are validated by comparing them to experimental findings [58]. An in-depth parametric study is conducted to investigate the effects of key parameters, including aspect ratio (AR), concrete strength (C), number of confinement layers of FRP (G), and variations in the dimensions of steel tubes along the major and minor axes. Through a systematic examination of these parameters, the present research work provides a comprehensive understanding of the structural behavior and performance of the elliptical DSTCs and DSFTCs. The application of ML techniques in predicting the responses of these columns is also explored, thereby contributing to the advancement of knowledge in this nascent field. The findings contribute to the overall understanding of the behavior of FRP-confined elliptical columns and can inform future design and analysis approaches in structural engineering. A flowchart of the integrated process is demonstrated in Fig.1.

2 Methodology

The methodology employed in this investigation has two primary components. The initial segment entails FEM modeling of DSTCs and DSFTCs utilizing the ABAQUS software. The subsequent segment centers on the utilization of ML algorithms to predict axial loads and axial strains in DSTCs and DSFTCs. These two approaches are elaborated upon extensively in the subsequent sections of this article.

2.1 Finite element modeling

2.1.1 Finite element model development

The study employs the ABAQUS software to create FEM models of DSTCs and DSFTCs. These columns have a fixed height (H) of 500 mm and feature four different AR values as 1.0, 1.3, 1.7, and 2.0, corresponding to major axis × minor axis dimensions of 250 mm × 250 mm, 250 mm × 192 mm, 250 mm × 147 mm, and 250 mm × 125 mm, respectively. Both DSTCs and DSFTCs consist of an outer GFRP layer, followed by concrete and an inner steel tube. In DSTCs, the inner section remains hollow, while in DSFTCs, it is filled with concrete, as displayed in Fig.2. To model the interaction between the GFRP layers and concrete, tie constraints are applied, with the GFRP layer serving as the master surface. The steel tube inside concrete is constrained using surface-to-surface contact with a penalty contact method and a friction coefficient of 0.6, where the steel tube acts as the master surface.

Steel is represented by solid elements, and it presents a bilinear stress–strain curve. In the first region, starting from the origin and extending to the yield point, it exhibits the elastic behavior. Within this region, it possesses a modulus of elasticity (E) of 200 × 10³ N/mm² and a Poisson’s ratio of 0.3. In the plastic region, the modulus of elasticity of steel is assumed to be 1% of its value in the elastic region. The GFRP layers are modeled, and their stiffness is adjusted using a reduction factor (R) based on the confinement stiffness ratio and AR of the columns. This relationships between R and confinement stiffness ratio and between R and AR are inverse, and they take into account the behavior of the GFRP layers. For concrete material properties, the models incorporate two types: normal strength concrete (NSC) and HSC with compressive strengths of 50 and 72.4 MPa, respectively. Elastic modulus values of 35.35 and 42.54 GPa are chosen to represent the stiffness of NSC and HSC, respectively. The peak strain ({\epsilon }_{\mathrm{c}1}) and failure compressive strain ({\epsilon }_{\mathrm{c}\mathrm{u}}) are defined utilizing Eq. (1).

(1)\begin{array}{l}{\epsilon }_{\mathrm{c}1}=0.0014\left[2-e^{-0.024f_{\mathrm{c}\mathrm{m}}}-e^{-0.140f_{\mathrm{c}\mathrm{m}}}\right]\\ {\epsilon }_{\mathrm{c}\mathrm{u}}=0.004-0.0011\left[1-e^{-0.0215f_{\mathrm{c}\mathrm{m}}}\right]\end{array}.

The thickness of the GFRP layers is chosen as 0.354 mm per layer. The interactions between GFRP, concrete, and steel layers are given using appropriate modeling approaches and mathematical equations, considering the specific material properties and failure criteria. The column specimens and their associated parameters are listed in Tab.1.

2.1.2 Loading and boundary conditions

The loading and boundary conditions applied to the column models are carefully considered to accurately simulate the axial behavior of the specimens. The column specimens are subjected to the axial loading, and the boundary conditions are designed to maintain the homogeneity of the composite specimens during loading. To ensure a consistent loading scenario, kinematic coupling is used to connect the steel tube, GFRP, and concrete layers at the top and bottom of the columns. Kinematic coupling facilitates the transfer of displacements and rotations between the layers, maintaining their integrity during the loading process. Additionally, reference points are created to establish the coupling connections between the different material layers. At these reference points, axial displacements and axial loads are specified to accurately simulate the loading conditions. Fig.3 illustrates the constraints, loading, and boundary conditions applied to the column models. The axial displacements assigned to the composite specimens are set at 0.05H to ensure a realistic loading scenario and to accurately capture the structural response under axial compression. This modeling approach ensures that the composite specimens remain stable during the loading process and prevents any unwanted lateral or rotational movements.

2.1.3 Meshing, element types, and analysis

The column models are meshed to ensure an accurate representation of the geometry and to capture the complex behavior of the composite materials under the axial loading. The analysis is performed using the ABAQUS/Explicit solver, which is known for its robust capabilities in dynamic simulations. To achieve efficient computation and maintain numerical stability, the central difference integration scheme in ABAQUS/Explicit is employed. This scheme utilizes small time increments to approximate the structural response. It also ensures that the analysis output considers stable known results or parameters from the previous time step, effectively capturing the time-dependent behavior of the column models [59]. To optimize computational efficiency, a mass scaling technique is applied at the beginning of the analysis step, using a scaling factor of 10. This technique effectively balances the kinetic energy and internal energy components of the model, reducing computational demands without compromising accuracy. Additionally, step smoothing is implemented to minimize the velocity in the model, resulting in lower kinetic energy levels. By controlling the kinetic energy, the analysis focuses on capturing the static response of the column models under the axial loading. The kinetic energy of the entire model (ALLKE) is constrained to a maximum of approximately 5% of the total internal energy (ALLIE), as recommended for the static analysis [59].

For the discretization of the column models in ABAQUS, appropriate element types are selected based on the material properties and structural characteristics of the components. The steel tubes and GFRP layers are represented utilizing four-node shell elements (S4R), which are well-suited for modeling thin-walled structures. These shell elements accurately capture the bending and membrane behaviors of the composite layers, considering the anisotropic properties of GFRP. Not only do shell elements provide an accurate representation, but they also contribute to computational efficiency by reducing the computational cost compared to solid elements. The concrete layers are modeled using solid, homogeneous elements known as hexahedrons (C3D8R). These elements provide an effective representation of the concrete’s response to compression and confinement effects. The C3D8R elements account for the material’s nonlinear behavior, accurately capturing the stress distribution and deformation characteristics. To ensure computational efficiency and consistency, a uniform mesh size of 25 mm is adopted for all elements in the column models. This mesh size selection is based on the understanding that smaller mesh sizes control the time increment and yield accurate results in ABAQUS simulations. By avoiding large variations in the mesh size, a consistent and reliable analysis is achieved, providing confidence in the obtained results. Fig.4 depicts the mesh configuration of the column models, showcasing the discretization of concrete, steel tube, and GFRP layers. The DSTC and DSFTC specimens modeled in the present study are detailed in Tab.2.

2.2 Machine learning methodology

This study involves the integration of FEM simulations and ML techniques to analyze and predict the behavior of DSTCs and DSFTCs. The following sections describe the data pre-processing, exploratory data analysis, and the application of seven different ML algorithms.

2.2.1 Data pre-processing and normalization

The data set used in this study consists of 70 entries, with input parameters including AR of the column cross-section (mm/mm), total thickness of FRP (mm) multiplied by the elastic modulus (MPa), concrete strength (MPa), cross-sectional area of the inner steel tube (mm²) multiplied by the yield strength in GPa, and area of confined concrete (mm²). These parameters are denoted as Input 1, Input 2, Input 3, Input 4, and Input 5, respectively. The output parameters are the load-carrying capacity (kN) and confined ultimate strain (mm/mm), designated as Output 1 and Output 2, respectively. To ensure the effectiveness of the proposed model and minimize errors, the input and output parameters undergo a normalization process using a min-max technique. This technique transforms the actual parameter values (X) to normalized values (X_N) based on the minimum (X_min) and maximum (X_max) values observed in the input and output data sets. The normalization equation is given by Eq. (2):

(2)X_{\mathrm{N}}=\frac{X-X_{min}}{X_{max}-X_{min}}.

Descriptive statistics of the data set are analyzed to gain insights into data set characteristics. Tab.3 presents the descriptive statistics, indicating the diverse range of the experimental data. It is evident that the columns in the table follow different distributions. The primary data set was partitioned into three subsets: training (70%), validation (15%), and testing (15%). To visualize the relationships between the variables, a pair-plot as shown in Fig.5 is generated. Histogram plots for the output parameters are created as displayed in Fig.6, revealing positive skewness and slight deviation from the normal distribution. Skewness and kurtosis values are calculated for Output 1 and Output 2, providing further information on the data distribution. Skewness and kurtosis for Output 1 are 1.734662 and 2.405, respectively. Output 2 has skewness and kurtosis of 3.479036 and 13.219, respectively. Output 1 ranges from 2314.25 to 9034.53 kN, while Output 2 ranges from 0.0173 to 0.0378. To improve the performance of the ML models, feature engineering, which involves the development of new variables or transformations, is a crucial part of data analysis. This study evaluates the use of several methods to quantify the connections between attributes, concentrating on both linear and nonlinear correlations. Pearson correlation coefficient is employed to establish the degree of correlation between the Input and Output parameters. Fig.7 illustrates the correlation analysis results, depicting that Input 4 and Input 5 reveal a higher positive correlation with Output 1, while Input 1 exhibits a negative correlation. For Output 2, Input 1, Input 3, and Input 4 demonstrate a negative correlation, whereas Input 2 and Input 5 present a positive correlation.

Pearson correlation measures the strength of linear relationships between two variables, while distance correlation quantifies both linear and nonlinear dependencies, offering scale insensitivity and flexibility in capturing complex relationships. As evident from the distance correlation matrix (Fig.8), Input 4, Input 5, and Input 1 indicate strong correlations with Output 1, while Input 5 holds the dominant correlation with Output 2. In Fig.8, the values of the coefficient are represented by the bright blue-green lines. The dashed lines represented the zero-point reference values. Mutual information (MI), a technique for evaluating the mutual dependence between variables, has been utilized in this study to establish information-theoretic feature selection criteria. The MI matrix is shown in Fig.9. Although Input 2 and Input 3 substantiates low independent MI values for Output 1, their intricate interactions with other variables may enhance the model’s overall predictive power. Additionally, certain parameters may hold minimal influence but could gain significance over time as new information or events emerge. Therefore, all the five Input variables are considered for predicting Output 1 and Output 2.

Having analyzed the correlation between the input and output parameters, seven ML algorithms have been exercised enabling us to make precise predictions and conduct thorough analyses of DSTCs and DSFTCs.

1) Adaptive Boosting (AdaBoost)

Introduced by Schapire [60], the AdaBoost algorithm is a ML technique based on boosting. It creates a strong learner by combining multiple weak learners selected randomly from the data set. Weak learners are trained using various ML algorithms. The algorithm assigns weights to each sample observation and adjusts these weights in each training iteration based on the accuracy of the predictions. Incorrect predictions receive higher weights, allowing subsequent weak learners to focus on challenging instances. In regression tasks, the algorithm considers the absolute value error of an instance. The ensemble prediction of individual weak learners is obtained through either the median or a weighted average approach [61]. AdaBoost offers the advantage of adjusting sample weights to improve the prediction performance by linearly combining the weak learners into a strong predictor [62].

2) Light Gradient Boosting Machine (LightGBM)

LightGBM is a framework that implements the gradient boosting decision tree (GBDT) algorithm [63]. It offers several advantages, including efficient parallel training, faster training speed, lower memory consumption, improved accuracy, and distributed support for large-scale data sets. LightGBM employs a leaf-wise algorithm with depth restrictions, different from the traditional level-wise approach in most GBDT tools. This algorithm selects and splits the leaf with the highest split gain among all current leaves, resulting in improved accuracy with an equivalent number of splits. Nevertheless, one drawback of LightGBM is that it may lead to the growth of deeper decision trees, increasing the risk of overfitting, as displayed in Fig.10. To mitigate overfitting, LightGBM incorporates a histogram-based algorithm and a leaf-wise growth strategy with a maximum depth constraint. Fine-tuning hyperparameters such as “num_leaves”, “max_depth”, and “learning_rate” further optimize its performance [64,65]. Finding an appropriate range for these hyperparameters is crucial to achieving improved optimization results using the LightGBM algorithm.

3) Categorical Boosting Regressor (CatBoost)

CatBoost, introduced by Dorogush et al. [66], is an enhanced GBDT toolkit similar to extreme gradient boosting (XGB). It resolves issues related to gradient bias and prediction shift [67] by employing Bayesian estimators and ranking features based on prediction value change (PVC), which also calculates feature value change, or loss function change. CatBoost handles model overfitting by training each base estimator utilizing a random subset of features. This approach improves the performance and prevents overfitting of the data set. A set of input features, denoted as F, is used in the ML model along with a numeric factor {\beta }_i and a prediction step P. The representation is illustrated by Eq. (3) [68]. Equation (4) represents the prediction value P_i, which is obtained by substituting the numeric factor {\beta }_i. Here, F_j designates a specific feature from the given feature set.

(3)F=\left\{f_1,f_2,f_3,\dots ,f_n\right\},

(4)P_i={\beta }_i{F}_j.

Equation (5) exhibits P_{i+1}, which demonstrates the prediction value obtained by modifying the numeric factor {\beta }_{i+1}for the specific feature F_j. The modified numeric factor is denoted as {\beta }_{i+1}. This feature becomes relevant when PVCs due to a modification in the numeric factor, as indicated by Eq. (6).

(5)P_{i+1}={\beta }_{i+1}{F}_j,

(6)P_{i=0}\ne P_i\ne P_{i+1}.

4) Random Forest (RF)

RF is a bagging-based algorithm that combines multiple randomized decision trees to make predictions. It has achieved success as a general-purpose classification and regression tool. The algorithm trains several base learners on the data set and combines their outputs by summing their individual predictions. RF is effective in situations where the number of variables exceeds the number of observations. By averaging the predictions of the individual trees, it addresses complex issues and handles a range of learning tasks. The number of trees and number of features in each split need to be determined to apply RF [69,70]. While default values often yield satisfactory results [71], additional trees can provide a stable outcome of varying significance [72]. Further, Breiman [69] has argued that using many trees may not be essential.

5) Extra Tree Regressor (ETR)

The ETR algorithm, an extension of RF, adopts a different approach to selecting cut points and splits. It randomly selects cut points and splits based on individual attributes, rather than choosing the most discriminative split. Similarly to RF, it trains each base estimator utilizing a random subset of features. Prediction is determined by the decision made by the nodes above the leaf node. ETR has shown to obtain higher accuracy compared to support vector machines and artificial neural networks methods [73]. It trains each regression tree using the entire training data set and randomly selects significant features and stable values for node splitting [74–76]. This approach enhances the performance and mitigates risk of overfitting.

6) XGB

In recent years, XGB has emerged as the predominant and extensively adopted ML framework for scaling tree boosting algorithms. XGB employs gradient descent to optimize the loss function, resulting in a predictive model in the form of a boosting ensemble composed of weak regression trees [77,78]. This efficiency enhancement benefits both regression and classification tasks. Chen and Guestrin [79] proposed a classification of XGB algorithm settings into three categories: general settings, task-specific settings, and booster settings. To mitigate overfitting, the XGB loss function incorporates a regularization term within the objective function, which has the effect of smoothing the final learning weights. It optimizes the loss function by utilizing gradient statistics of both first and second orders. Beyond the inclusion of regularization terms, XGB also leverages row and column sampling to address the issue of overfitting. The parallel and distributed computing capabilities enable faster learning, facilitating quicker exploration of models.

7) Deep Neural Network (DNN)

DNN is a multilayer feed-forward neural network, as defined by Han et al. [80]. DNN is composed of an input layer, several hidden layers, and an output layer. In this network, information progresses solely in a forward direction, passing through the hidden nodes from the input nodes to output nodes. There are no loops or cycles within the network. Each neuron in a lower layer can directly communicate with every neuron in the layer above it. The back propagation learning algorithm is used to adjust the network’s weights [81]. DNN, through a sequence of layered transformations, maps the input to the desired or expected output. These transformations are acquired by exposing the network to training examples. DNN adapts its weights based on user-defined criteria, ensuring that, after training, it can produce the desired output for a given input. The adjustments made by each layer to its input are determined by its weights, which essentially represent a set of numerical values. The objective function quantifies the disparity between the network’s predicted output and actual experimental value for a given case.

In the present study, these seven ML algorithms are applied, and their performance is compared, to identify the approach that provides a comprehensive framework for analyzing and predicting the behavior and performance of DSTCs and DSFTCs.

3 Finite element model validation study

To assess the accuracy and reliability of the FEM models developed using the ABAQUS software, a comprehensive validation study is conducted by comparing the simulation results to the experimental findings of Chen et al. [58]. In their study, they performed a comparative analysis of 32 elliptical stub columns under axial compression. For the validation of the present study, 12 Hybrid FRP-concrete-steel DSTC specimens are selected, aligning their height with that in the experimental setup of Chen et al. [58] at 500 mm. HSC is utilized as the primary material for these columns to ensure consistency between the experimental and numerical investigations. The experimental setup employed by Chen et al. [58] is depicted in Fig.11, illustrating the test configuration used to subject the elliptical stub columns to axial compression.

The ultimate axial load and ultimate axial strain of DSTCs are considered as the primary performance indicators for the validation process. The validation results outlined in Tab.4 present a high level of agreement between the FEM simulations and experimental findings. The ultimate load predictions obtained from the FEM models closely match the experimental values, with a maximum deviation of 11.71% and a minimum deviation of 1.07%. Similarly, the ultimate strain predictions exhibit satisfactory agreement, with a maximum deviation of 20.69% and a minimum deviation of 0.55%. These deviations demonstrate that the FEM models tend to be slightly conservative, providing values that are slightly higher than the experimental results. A detailed comparative analysis is documented in Tab.4, displaying the calculated deviations for each DSTC between the FEM models and experimental data of Chen et al. [58]. Fig.12 indicates the stress and strain data of 5G-1.0-C70H, 5G-1.3-C70H, 5G-1.7-C70H, 5G-2.0#-C70H columns obtained from ABAQUS modeling, while Fig.13 shows the load-strain curves of the same columns, comparing the FEM modeling to experimental results. The close agreement between the simulation results and experimental findings validates the accuracy and reliability of the developed FEM models in capturing the behavior of DSTCs and DSFTCs.

The successful validation of the FEM models against the experimental data reinforces their suitability for subsequent parametric study and for exploring the effects of various parameters on DSTCs and DSFTCs.

4 Finite element method numerical illustrations and discussion

After confirming the accuracy of the FEM model, an extensive parametric investigation on elliptical DSTCs and DSFTCs is performed. The objective of this study is to compare the ultimate axial load capacity and ultimate axial strain of DSTCs and DSFTCs while examining the influence of various factors, such as AR, C, G, and variations in the dimensions of steel tubes along the major and minor axes on their performance. Tab.2 lists a comprehensive set of specimens that present a wide range of parameter variations. These variations include different AR values ranging from 1.0 to 2.0, G ranging from 5 to 7, and two concrete types, namely, NSC and HSC. Furthermore, the investigation considers alterations in the dimensions of steel tubes along the major and minor axes. This comparison involves DSTCs and DSFTCs with identical AR, G, and C. The percentage increase in the ultimate axial load capacity in relation to the percentage increase in the quantity of concrete due to the filling process is evaluated. Tab.5 illustrates the specimen-wise comparison for changes in the ultimate axial strain, ultimate axial load, and the percentage increase in the load and quantity of concrete. Fig.14 depicts the load–strain curve of the compared specimens.

The first notable finding is that DSFTCs reveal a higher ultimate axial strain compared to the corresponding DSTCs. The increase in axial strain in DSFTCs is approximately 0.0020, demonstrating that filling concrete inside the steel tube has a positive effect on the overall strain capacity of the columns. Further, it is observed that the percentage increase in the ultimate axial load of DSFTCs decreases as AR increases. For an AR of 1.0, the percentage increase is approximately 100%, while for AR values of 1.3, 1.7, and 2.0, the percentage increases are roughly 70%, 45%, and 25%, respectively. This diminishing trend suggests that the impact of increasing AR on the load-carrying capacity diminishes as AR increases. It is also seen that the percentage increase in the quantity of concrete used in DSFTCs decreases from 55.73% to 40.35% as AR increases from 1.0 to 1.7. However, there is a remarkable increase to 49.64% for an AR of 2.0. This finding suggests that for an AR of 1.7, the additional concrete required owing to the filling process has a minimal impact, while for an AR of 2.0, a significant increase is observed. Furthermore, it is deduced that both DSTCs and DSFTCs with HSC signifies a greater percentage increase in the ultimate load capacity compared to those with NSC. This observation highlights the significance of the concrete strength in determining the overall performance of the columns and suggests that NSC models may not be suitable for HSC models.

4.1 Effect of aspect ratio

An analysis of the load-strain curves displayed in Fig.15 reveals that as AR varies from 1.0 to 2.0, a substantial decrease in the ultimate axial load at failure can be seen. However, on the contrary, the difference in the axial load in the elastic region is much less pronounced. For instance, considering the columns labeled as 5G-AR-C70F, the axial load at the elastic limit for AR of 1.0 is 4093.74 kN, whereas for AR values of 1.3, 1.7, and 2.0, the respective values are 3311.66, 2540.53, and 2016.12 kN. This demonstrates differences of 782.08, 1553.21, and 2077.62 kN between AR of 1.0 and 1.3, 1.7, and 2.0, respectively. Similarly, the ultimate axial load at failure for AR of 1.0 is 8192.75 kN, for 1.3 is 5752.35 kN, for 1.7 is 4081.71 kN, and for 2.0 is 3253.39 kN, exhibiting differences of 2440.4, 4111.04, and 4939.36 kN between AR of 1.0 and 1.3, 1.7, and 2.0, respectively. However, when comparing DSFTCs to DSTCs, the difference in the axial load at both the elastic limit and ultimate failure is less pronounced for DSTCs than for DSFTCs. For example, considering the columns labeled as 5G-AR-C70-H, the axial load at the elastic limit for AR of 1.0 is 2428.03 kN, while for AR of 1.3, 1.7, and 2.0, the respective values are 2081.36, 1825.23, and 1553.87 kN. This represents differences of 346.67, 602.8, and 874.16 kN between AR of 1.0 and 1.3, 1.7, and 2.0, respectively. Similarly, the ultimate axial load at failure for AR of 1.0 is 3980.33 kN, for 1.3 is 3323.22 kN, for 1.7 is 2804.86 kN, and for 2.0 is 2529.97 kN, showing differences of 657.11, 1175.47, and 1450.36 kN between AR of 1.0 and 1.3, 1.7, and 2.0, respectively. Similar trends can be observed for other DSFTCs and DSTCs, as indicated in Fig.15. Further, it is worth noting that the slope of the load-strain curve increases as AR transitions from 2.0 (representing an elliptical shape) to 1.0 (representing a circular shape) in the post-elastic zone.

4.2 Effect of concrete strength

The impact of the concrete strength (NSC and HSC) on the behavior of the columns is examined through the load–strain curves presented in Fig.16. The data in Fig.16 specifically consider DSTCs and DSFTCs with a configuration of 5 GFRP confinement layers and an AR of 1.7. From the analysis of the graphs, it is evident that the columns made with HSC reveal considerably higher ultimate axial loads compared to those made with NSC. In DSFTCs, the ultimate axial load for the specimen (labeled as 5G-1.7-C70-F) is 4081.7 kN, whereas for the specimen (labeled as 5G-1.7-C50-F), it is 3622.37 kN. Similarly, in DSTCs, the ultimate axial load for the specimen (labeled as 5G-1.7-C70-H) is 2804.86 kN, whereas for the specimen labeled as (5G-1.7-C70-H), it is 2500.59 kN. Consistent results are obtained for all other specimens. These findings demonstrate the superior load-carrying capacity of HSC in comparison to NSC. Thus, replacing NSC with HSC in columns with a similar cross-section can lead to a significant increase in the overall load-carrying capacity. It should be noted that Fig.16 only displays results for 5G and 6G GFRP confinement layers, but similar trends are also observed for the columns with 7G GFRP confinement layers.

4.3 Effect of confinement layers

The influence of the number of layers of GFRP confinement on the behavior of the columns is illustrated in Fig.17. It is observed that increasing the number of confinement layers enhances the ultimate axial load of the columns at failure, while the axial load at the elastic limit remains relatively unchanged. For instance, in Fig.17, which assesses columns with an AR of 1.7 and a compressive strength of 50 N/mm² (NSC), the ultimate axial load at failure is 5987.29 kN for 7G confinement, 5463.44 kN for 6G confinement, and 5072.3 kN for 5G confinement. However, at the elastic limit, the axial loads for 5G, 6G, and 7G confinement remain nearly constant at 2693.59 kN. Similar trends are witnessed for other specimens. From this point, it can be inferred that increasing the number of confinement layers leads to an improvement in the ultimate load-carrying capacity of the columns, while having negligible effects on the load capacity up to the elastic limit.

4.4 Effect of variations in dimensions of steel tube

The investigation focused on the effect of variations in the dimensions of the steel tubes along the major and minor axes in elliptical DSTCs. Designations of the DSTC specimens which include ‘#’ symbol have different AR values for the inner and outer steel tubes. These ‘#’ specimens are designed with a different AR for the inner steel tube (a/b = 3.0) while maintaining the outer AR of the columns (a/b = 2.0). The analysis specifically examines DSTCs to evaluate the impact of changing AR of the inner steel tube while keeping the outer AR constant. Fig.18 depicts the results for DSTCs with 5G, 6G, and 7G confinement, all having an outer AR of 2.0, and two specific cases: one with an inner AR of 2.0 and another with an inner AR of 3.0 (hereinafter known as 2.0#). It can be observed that, for a given concrete type and confinement, there is minimal difference between columns with an AR of 2.0 and those with an AR of 2.0#. Both the elastic limit axial load and ultimate axial load demonstrate similar trends. For example, considering the specimen 7G-2.0-C50H, the axial load at the elastic limit is 1483.51 kN, and the ultimate axial load is 2620.13 kN. In comparison, the similar specimen with an AR of 2.0# (7G-2.0#-C50H) has an elastic limit axial load of 1438.68 kN and an ultimate axial load of 2571.07 kN. Similar results are obtained for HSC as well. From these observations, it can be deduced that the outer AR has a dominant influence on the behavior of DSTCs, while variations in the inner AR within the examined range do not considerably affect the load-carrying capacity. This suggests that optimizing the outer dimensions and reinforcing elements of the column is critical for enhancing its strength.

5 Machine learning results and discussion

The FEM study provides insights into the behavior of the columns, including the influence of factors such as the concrete strength, confinement layers, and dimensions of steel tubes. These findings serve as a foundation for designing and training the ML models. By incorporating the knowledge gained from the FEM study, the ML models are able to learn and capture the relationships between input parameters and output predictions. By comparing the ML predictions to the FEM results, it is possible to determine the effectiveness of the ML models in approximating the load-carrying capacity and ultimate strain of the columns. This allows for a comprehensive evaluation of the ML model’s performance and their ability to replicate the behavior seen in the FEM analysis.

5.1 Details of performance indices and their relation to finite element method study

To evaluate the performance of the ML models, six distinct performance metrics are calculated [82–85]. These metrics aims to assess the accuracy and predictive capabilities of the models. The ideal values and parametric equations for these metrics are delineated in Tab.6. It is expected that a perfect prediction model exhibits values that closely match their ideal counterparts. By comparing the ML predictions to the FEM results, it is possible to determine the effectiveness of the ML models in approximating the load-carrying capacity and confined ultimate strain of the columns. This allows for a comprehensive evaluation of the ML model’s performance.

5.2 Hyper-parametric configurations

70% of the primary data set are randomly selected as the training subset, with 15% of the data reserved for the validation subset. The validation subset is used for hyperparameter tuning through 5-fold cross-validation, aimed at achieving an optimized predictive model. To determine the appropriate hyperparameters, network architectures, and functions for the models, a trial-and-error tuning approach is employed during the training and validation phases. Subsequently, the model is trained utilizing the hyperparameters that produce the highest average prediction accuracy across the entire validation set, as determined through this trial-and-error process. Tab.7 presents the optimal hyperparameter values for the six individual regression models (AdaBoost, LightGBM, CatBoost, RF, ETR, and XGB). For DNN, the optimized hyperparameters are as follows: hidden layer size = 4, batch size = 1, epochs = 350, number of neurons in each layer = 64, activation function = ReLU, and optimizer = Adam. The hyperparameters optimized via trial and error are then applied to predict a new set of data, constituting 15% of the main data set, which serve as the testing data set. The performance of 5-fold cross-validation, as measured by the R² value, is shown in Fig.19 and Fig.20 for Output 1 and Output 2, respectively, where F1 to F5 denote fold 1 to fold 5. Tab.8 outlines the average R² and average RMSE values achieved on the validation data set using the best-optimized hyperparameters for each respective model.

5.3 Statistical details of results

This study involves the development of predictive models designed to estimate load-carrying capacity (kN) and confined ultimate strain (mm/mm) as Output 1 and Output 2, respectively. Tab.9 and Tab.10 illustrate assessments of the models’ performance in predicting Output 1 and Output 2, respectively. It is worth mentioning that the goodness of fit values for the developed models are evaluated based on the models’ performance with the validation subset. The results of the experiments reveal that the ETR model demonstrates the highest R² = 0.9956 and the lowest RMSE = 0.033 in the testing phase for Output 1, as indicated in Tab.9. RF emerges as the second-best model for Output 1. CatBoost and XGB advocates comparable performance, while LightGBM displays the weakest predictive performance for Output 1, with an R² of 0.8804 and RMSE of 0.1192. To select the best model from the seven models, a comprehensive ranking analysis is conducted based on six statistical indices, as described in Tab.6. This ranking approach assigns higher scores to indicators associated with better model performance [86]. Subsequently, the models are ranked utilizing this method, and the results for Output 1 during the testing stage are presented in Tab.9. The ETR model provides superior accuracy among the developed ML models for Output 2, as depicted in Tab.10. The RMSE and R² values for the ETR model are 0.0221 and 0.9967, respectively. For Output 2, XGB is identified as the second-best model with an RMSE of 0.0459, MAE of 0.0182, PI of 1.86, and VAF of 94.67. Although CatBoost exhibits higher R² and Adj.R² values compared to XGB, XGB outperforms CatBoost in terms of error and trend parameter values. Tab.9 and Tab.10 represent a comprehensive overview of other performance indicators, allowing for a thorough comparison of the models for Output 1 and Output 2. The best-performing model is expected to have data points closely aligned with the y = x line on the corresponding plots. Examination of Fig.21(a) reveals that observed and predicted values for Output 1 in the testing phase are closer to the y = x line for the ETR, RF, XGB, and CatBoost models, whereas LightGBM demonstrates deviated data points. Similar trends are observed in the regression plots for Output 2 (Fig.22(a)). Error plots for Output 1 and Output 2 during testing are displayed in Fig.21(b) and Fig.22(b), highlighting the noticeable errors associated with the LightGBM model.

The final selection of the best model among the seven models considered is determined through a ranking analysis, employing various statistical indices. Models are evaluated and ranked based on their performance in testing stages [86]. Tab.9 and Tab.10 indicate the ranking analysis results for Output 1 and Output 2, respectively. For Output 1, the ETR model achieves the highest rank of 42, followed by RF (35), CatBoost (27), XGB (24), AdaBoost (18), DNN (16), and LightGBM (6). Likewise, for Output 2, the ETR model secures the top rank of 42, with XGB (34), DNN (27), AdaBoost (23), CatBoost (19), RF (16), and LightGBM (7) following in the ranking. Notably, LightGBM obtains the lowest rank for both Output 1 and Output 2. Thus, ETR, the top-ranking model in terms of the overall evaluation for both Output 1 and Output 2, is considered the best model in this investigation. The relative importance of input parameters (Input 1, Input 2, Input 3, Input 4, and Input 5) in predicting Output 1 is assessed utilizing the RF method and is visualized in Fig.23(a) through a doughnut diagram. The diagram clearly indicates that Input 5 has the most substantial influence on predictions, followed by Input 4 and Input 1. In contrast, Input 2 and Input 3 illustrate minimal contributions to Output 1 prediction, aligning with the MI values of these inputs for Output 1, as characterized in Fig.9. For the prediction of ultimate strain (Output 2), Input 2, Input 4, and Input 5 are identified as the major contributing factors, as vouched by their relative importance values in Fig.23(b).

5.4 External validation

The estimation of the load-carrying capacity and ultimate strain by an algorithm requires external validation to ensure accuracy for new and different cases. When conducting an external validation, several methods need to be considered. Following the guidance of Golbraikh and Tropsha [87], the reliability of models for estimating the load-carrying capacity and ultimate strain is assessed by examining the gradients of regression lines (k or k') at the origin, which should be close to 1. Also, the performance indices ‘m’ and ‘n’ should be lower than the threshold value of 0.1, and the confirmed indicator (R_m) should exceed 0.5, according to the suggestions of Roy and Roy [88]. The results of the external validation, along with the mathematical formulations, and ideal conditions, are given in Tab.11 and Tab.12.

These results show that the proposed models are suitable for determining the load-carrying capacity and ultimate strain of DSTCs and DSFTCs. Based on the results provided in Tab.11 (Output 1) and Tab.12 (Output 2), it can be concluded that the AdaBoost, LightGBM, CatBoost, RF, ETR, XGB, and DNN models are highly effective in estimating the load-carrying capacity and ultimate strain. The LightGBM model, although ranking lower, still fulfils all requirements and is considered acceptable.

5.5 Random forest model-based web application for ultimate load prediction

The practical application of ML techniques in routine design work has been hindered by complex database development, model training, validation, and other tasks. To address these challenges, a Python web application incorporating the RF model and its optimized hyperparameters is developed. The graphical user interface (GUI) displayed in Fig.24 facilitates the prediction of confined ultimate load (kN). Input parameters include the depth and width of the elliptical cross-section (mm), total area of the confined concrete core (mm²), unconfined concrete strength (MPa), area of the steel tube (mm²) multiplied by its yield strength (MPa), and total thickness of the FRP wraps (mm) multiplied by its elastic modulus (MPa). For online access to GUI for predicting the confined ultimate load (kN), Github_Rakesh can be followed.

6 Conclusions

This study presents an investigation of GFRP reinforced DSTCs and DSFTCs using a combination of FEM simulations and ML techniques. 54 column specimens with varying parameters, including AR, concrete strength, number of the GFRP confinement layers, and variations in the dimensions of the steel tubes along the major and minor axes, modeled in the ABAQUS software, are analyzed and compared. Several key findings are obtained. Seven different ML algorithms are evaluated, and it has been found that all seven algorithms achieve good reliability in predicting the axial loads and axial strains of the columns. The key findings of the study are as follows.

1) DSTCs vs. DSFTCs. DSFTCs exhibit higher ultimate axial strain than DSTCs, indicating the positive effect of concrete filling. The increase in the ultimate axial load percentage diminishes with higher AR values. HSC enhances the percentage increase in the ultimate load capacity for both DSTCs and DSFTCs. The NSC models may not be suitable for the HSC models.

2) AR. As AR is varied from 1.0 to 2.0, a significant decrease in the ultimate axial load at failure can be observed, but the difference between the axial load in the elastic region is much less pronounced.

3) Concrete strength. HSC demonstrates higher load-carrying capacity than NSC, suggesting the significance of using strong concrete in the column design for substantial increases in the overall load-carrying capacity with a similar cross-section.

4) Confinement layers. Increasing the number of the confinement layers enhances the ultimate load-carrying capacity of the columns while having negligible effects on the load capacity up to the elastic limit. The presence of the confinement layers improves the structural performance and provides higher resistance to the deformation.

5) Steel tube dimensions. The dimensions of the steel tubes used for the confinement play a significant role in the behavior of the confined columns. Optimal dimensions need to be considered to achieve the desired load capacity and strain characteristics.

6) ML models. The developed ML models, including AdaBoost, LightGBM, CatBoost, RF, ETR, XGB, and DNN, all exemplify the ability to accurately predict the load-carrying capacity and confined ultimate strain of columns.

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