Numerical and machine learning modeling of GFRP confined concrete-steel hollow elliptical columns | Scientific Reports

Scientific Reports volume 14, Article number: 18647 (2024) Cite this article

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This article investigates the behavior of hybrid FRP Concrete-Steel columns with an elliptical cross section. The investigation was carried out by gathering information through literature and conducting a parametric study, which resulted in 116 data points. Moreover, multiple machine learning predictive models were developed to accurately estimate the confined ultimate strain and the ultimate load of confined concrete at the rupture of FRP tube. Decision Tree (DT), Random Forest (RF), Adaptive Boosting (ADAB), Categorical Boosting (CATB), and eXtreme Gradient Boosting (XGB) machine learning techniques were utilized for the proposed models. Finally, these models were visually and quantitatively verified and evaluated. It was concluded that the CATB and XGB are standout models, offering high accuracy and strong generalization capabilities. The CATB model is slightly superior due to its consistently lower error rates during testing, indicating it is the best model for this dataset when considering both accuracy and robustness against overfitting.

Fiber reinforced polymer (FRP) members have been extensively implemented in the construction industry attribute to their durability and extraordinary strength to weight ratio. Recently, they have been widely combined with traditional construction materials to form hybrid members. Hybrid or double-skin tubular members (DSTM) were initially proposed by Teng et al. in 20041,2. The authors introduced a new composite column that has an inner steel tube and an outer FRP tube confinement with concrete placed in between. The cross section of the external and internal tubes can be altered to meet any requirements; therefore, they are considered design-friendly members. These members can serve as columns or even beams only by adjusting their design and mechanical properties3. The authors also assure that these members are construction-friendly, as they can be precast or cast in situ to satisfy different construction site circumstances.

Hybrid double-skin tubular members (DSTM) possess many desired properties, as the exterior FRP layer provides the utmost corrosion resistance, higher ductility due to the internal and external confinement of concrete, and a void, resulting in a light-weight section4. They also have high mechanical and shear resistance, and both tubes are perfectly utilized as stay-in-place formwork, providing higher ductility, impact and fatigue resistance5,6. Moreover, the internal steel tube serves as a substitute for the primary longitudinal reinforcement. These sections require less periodic maintenance, and they have low thermal conductivity because of the heat-sink effect of concrete7. All of these previously mentioned advantages will definitely result in cost-cutting8.

Many articles have studied the mechanical performance of DSTMs, investigating their flexural behavior if they are serving as beams or their compression or seismic behavior if they are serving as columns. Also, many papers were investigating the behavior of hybrid double-skin tubular columns (DSTCs), considering specific outer cross sections. Circular, rectangular, square, and elliptical cross sections were all investigated previously; the circular section was investigated the most.

Ozbakkaloglu and Idris9 investigated the behavior of FRP high-strength concrete-steel columns under both axial compression and cyclic-lateral loading. The investigated parameters in their study were type and amount of FRP, cross section’s shape, steel tube’s thickness, axial load level, and the characteristic strength of the concrete. Fanggi and Ozbakkaloglu tested 16 hollow DSTCs to study the impact of the shape of the internal steel tube on the compression performance of square DSTCs. The authors assured that the circular steel tube endure higher axial strains and stresses than square-shaped tubes, and the presence of concrete filling developed a great enhancement to the square-shaped performance10.

Some authors investigated the pros developed by utilizing a specific type of FRP in the hybrid sections. Yu and Teng11 discussed the usage of CFRP and confirmed that CFRP increased the columns ductility and carrying load capacity and delayed the steel tube local buckling. Another study12 experimentally investigated the behavior of hybrid columns under uniaxial compression loading, considering the diameter of steel tubes and layers of glass fiber-reinforced polymers. They reported that the investigated columns had improved deformation performance and load capacity due to the presence and combination of steel and GFRP. Furthermore, a study reported a 47% improvement in the concrete confined axial stress attributable to using stiffened steel tubes in hybrid DSTCs13.

Considering different aspects, the authors of this article experimentally discussed the behavior of large-scale DSTCs subjected to compression loading. They also investigated the fabrication techniques of FRP, including both wet lay-up and filament wound techniques14. Few studies have tried using recycled aggregates in long DSTCs creating FRP-recycled aggregate concrete-steel long columns FRSLCs, as Zeng et al.15, the authors assured that recycled aggregate concrete is very applicable to be used in structural members and proposes an environmentally friendly alternative to the construction industry as the FRSLCs had a comparable axial performance to DSTLCs. They added that the FRSLCs with proper dimensions provided lesser cost, higher ductility, reduced weight, and better capacity. Increasing the FRP thickness increased the ultimate strength, as increasing the thickness of the steel tube, which resulted in a 15% ultimate strength improvement, as reported by Youssf et al.7. Also, the authors assured that a higher diameter void ratio can significantly provide several advantages, such reducing the weight of the member, covering and holding the utilities and services of the building, and definitely improving the strength of the column.

Many studies in the past twenty years have examined the behavior of hybrid members with respect to several factors. Yu et al.16 considered both the thickness of the steel tube, and the external FRP confinement, and the inner void ratio as well. Then, they introduced a stress strain model for circular DSTCs. Another study also focused on circular DSTCs2, altering the GFRP thickness and discussing their behavior under flexural and axial loading. The majority agreed on having axile deformations accompanying the internal steel tube buckling, which affects the structural integrity of the DSTCs, especially at utilizing slender tubes, in addition to possessing slightly low shear enduring17. Consequently, adding stiffeners will definitely enhance the previously mentioned drawbacks. Zakir et al. also proved that stiffened DSTCs had improved ductility and higher axial load capacity by 32%18.

Another research by Zakir et al.19 experimentally studied the compression behavior of DSTCs with different characteristics of the stiffeners using polyethylene terephthalate (PET) FRP and GFRP, stating that the PET FRP had triple the rupture strain of GFRP. Prior to rupture, hybrid DSTCs have superior energy dissipation capabilities, as also mentioned by Han et al., Xue, and Gong.20,21. But after that, a dramatic decrease will happen to all DSTCs’ properties. As a result, the authors confirm the necessity of further investigation to propose a numerical model and develop a parametric study to carefully consider each constituent of the DSTCs.

Recent studies as Chen et al.22 discussed using different types of concrete. The authors stated that hybrid DSTCs provide superior seismic resistance and ductility, but their durability and service life are major drawbacks when using normal-strength concrete. The authors investigated the behavior of DSTCs utilizing normal and ultra-high-strength concrete samples in order to compare the results.

As previously discussed, most studies focused on using different FRP with varying thickness and number of layers, using diverse concrete types such as recycled aggregate concrete, normal-concrete, or high- or ultra-high-strength concrete. Also, some studies considered different internal steel sections. Most of the studies concentrated on circular DSTCs and then square and rectangular cross sections. DSTCs with elliptical cross sections were hardly discussed23,24. Even though having a curved and smooth figure similar to a circular cross section and short and long characteristics of axes compared to rectangular sections, that makes them very appealing to designers as they can maintain different moment and stiffness capacities around their axes25. Due to the curvature of sections, sharp edges can easily be avoided by utilizing elliptical sections24. There is no stresses concentration at the corners as well, which results in enhanced hoop strain-carrying capacity of the FRP tube. It was also proved that even inner steel tubes with elliptic sections provided enhanced strength, ductility, and better mechanical performance compared to rectangular steel tubes25,26,27. The authors also assure that no clear relation was investigated considering elliptical steel tube, concrete, and elliptical FRP tube previously23. Consequently, numeric modeling and parametric studies must be developed to investigate and introduce further insights regarding the compressive behavior of elliptical DSTCs.

The incorporation of Artificial Intelligence (AI) in civil engineering has garnered considerable interest from researchers28. AI algorithms, known for their capability to address complex problems, offer a viable alternative to conventional modeling approaches29. Within structural engineering, Machine Learning (ML), a branch of AI, has been extensively utilized to forecast the behavior of structural components30,31,32,33, identify damage34, and determine buckling loads35,36. Numerous studies have applied ML techniques to predict the axial load capacity of columns30,37,38,39,40,41,42,43,44.

Highlighting the research gaps by studying more than 40 references, as shown in Table 1, a score was calculated to represent the frequency of studying each parameter. It was concluded that elliptical DSTCs need further investigation, considering different parameters. From this analysis, it is evident that elliptical cross-sections are significantly underrepresented in the current research. The elliptical cross-section in hybrid columns offers structural and performance advantages over traditional circular and rectangular sections. It mitigates fiber stress concentrations at corners, ensuring uniform stress distribution and enhancing load-bearing capacity23,45. This shape optimizes structural performance, material efficiency, and ease of construction, resulting in higher ultimate loads and strains46. Thus, the elliptical design is superior for managing fiber stresses and improving overall column performance23,47. Despite their potential advantages they have not been thoroughly explored. The table shows that only 3 studies have investigated elliptical cross-sections, compared to 32 studies on circular sections, 8 on square sections, and 5 on rectangular sections. Moreover, none of these studies utilized ML models. There is a significant gap in the literature regarding the application of ML in this area. Incorporating ML could provide enhanced predictive capabilities and optimize the design and analysis processes for these complex structures. This study aims to address these gaps by integrating ML models into the investigation of GFRP confined concrete-steel hollow elliptical columns. ML models can significantly enhance predictive capabilities by learning complex patterns from the data, which may not be easily captured by FE models alone. They provide rapid predictions once trained, which can be beneficial in scenarios requiring real-time or multiple quick evaluations. Additionally, ML models can complement FE analysis by identifying trends and interactions in the data that might not be apparent from FE results alone, thus providing a more comprehensive understanding of the structural behavior. While FE analysis provides accurate results based on physical and material properties, ML models can quickly learn from existing data, offering rapid evaluations and identifying complex patterns that FE models might miss. This combination of methodologies offers a more robust and comprehensive approach to understanding and predicting the behavior of hybrid DSTCs. The ML models developed in this study use six input features: the area of the steel tube, the yield strength of the steel tube, the area of the concrete core, the aspect ratio of the column cross-section, the total thickness of FRP multiplied by the elastic modulus of FRP, and the unconfined concrete strength. The outputs of these models are the confined ultimate strain and the ultimate load of confined concrete at the rupture of FRP tube. This approach aims to improve the predictive accuracy and optimize the structural analysis of hybrid DSTCs.

The methodology of this research was carried out in four stages. After studying the research gaps, the authors gathered relevant experimental results and extended them to obtain the parametric study dataset. Then, numerical modeling was conducted using ABAQUS software99, followed by the machine learning modeling stage, which considered various machine learning techniques (Decision Tree DT, Random Forest RF, Adaptive Boosting ADAB, Categorical Boosting CATB, and eXtreme Gradient Boosting XGB). Finally, the considered ML models were visually and quantitatively evaluated at the final stage of this study. A flowchart of the research methodology is introduced in Fig. 1.

Research methodology flowchart.

In order to deeply investigate the behavior of elliptical hybrid DSTCs, the data experimentally developed by Chen et al.23 was adopted for this article. These specimens are elliptical DSTC stub columns with a height of 500 mm, tested under axial loading conditions. The grouping of specimens and the considered parameters are illustrated in Table 2.

Initially, 8 specimens from Chen et al.23, labeled as S1 to S8, were utilized and referred to as Group 1 (G1). These specimens served as the basis for the parametric numerical modeling, where the yield strength of the steel tube was varied. This extended set was still designated as G1 with an additional note "PARAMETRIC" for clarity.

To further investigate the effects of varying steel tube thickness, Group 2 (G2) was created by duplicating G1, resulting in 16 specimens, and modifying the steel tube thickness. For a comprehensive analysis, Groups 3 (G3), 4 (G4), 5 (G5), and 6 (G6) were formed, where the strength of concrete and the number of FRP layers were adjusted to maintain a similar lateral confinement pressure ratio \(({f}_{lf}/{f}_{c}{\prime})\). Additionally, Groups 7 (G7), 8 (G8), and 9 (G9) were created by removing the steel tube from the corresponding groups to study the effects of GFRP confinement alone. This extensive parametric study allowed for a systematic analysis of the impact of various parameters on the axial load-bearing capacities of hybrid DSTCs.

In total, nine groups comprising 116 specimens were created to conduct a comprehensive parametric study. Table 3 shows the specifications of all specimens, including both original and parametric.

Hybrid DSTCs consist of three different materials, as previously illustrated. Each material was identified with its own specified properties and characteristics. Then, the three materials were assembled and meshed, as shown in Figs. 2 and 3. This study involved a detailed parametric analysis using ABAQUS software99, focusing on these key components: the concrete core, steel tubes, and FRP composites. Each material was assigned specific element types to accurately reflect their unique physical and mechanical characteristics.

FEM of elliptical DSTC components using ABAQUS software99.

Details of FEM.

The concrete core utilized C3D6 elements, which are 6-node linear triangular prisms. These elements were selected for their ability to effectively capture the nonlinear and inelastic behavior of concrete under various loading conditions and to handle complex boundary constraints proficiently. Steel tubes were modeled using C3D8R elements, 8-node linear brick elements with reduced integration and hourglass control. These elements are optimal for depicting the behavior of steel, particularly under linear elastic and plastic conditions, which are crucial for structural analysis and design. FRP composites were represented by S4R elements, 4-node quadrilateral stress/displacement shell elements, ideal for thin shell structures. The reduced integration and large-strain formulation of these elements ensure an accurate simulation of the mechanical responses of FRP composites, especially under high strain conditions.

To simulate the interactions between the different components accurately, several constraints were applied at the interface regions. Surface-to-surface tie constraints were used to model the interfaces between the concrete and FRP, as well as between the concrete and steel tubes. These constraints captured both tangential and normal interactions, providing a thorough simulation of interface dynamics. The normal interaction was characterized as "hard," indicating a perfectly bonded state, while the tangential interaction employed the "penalty" method with a friction coefficient of 0.2 to realistically model the frictional forces. Additionally, a coupling constraint was implemented at the extremities of the specimen, particularly at the start and end faces, to restrict all degrees of freedom. This approach replicates the realistic boundary conditions that these faces would encounter in practical applications.

The methodologies utilized in this analysis align with recent advancements in the field23,39,46,70,71,72, as documented in contemporary literature. These references validate the modeling techniques and element choices, providing a robust framework for assessing the predictive accuracy and reliability of the simulation results.

Hybrid DSTCs consist of three different materials, as previously illustrated. The stresses and strain values for each material were calculated by the FEMs, as shown in Figs. 4 and 5. The confined ultimate strain and the ultimate load of confined concrete at the rupture of FRP tube for each specimen were then calculated to form the dataset for the machine learning models. Figures 6 and 7 illustrate the comparison between the FEMs and experimental tests on hybrid DSTCs. Figure 6 shows the correlation between the loads applied to the FEMs and those recorded in experimental tests, with a high coefficient of determination (\(R^{2}\)\(= 0.9812\)) demonstrating that the FEM accurately predicts the experimental load capacities of the hybrid DSTCs. This strong agreement between the modeled and experimental results highlights the precision of the FEM approach. Figure 7(a) and (b) present the axial load versus axial strain curves for the FEMs compared to the experimental tests under axial compression. The close agreement between the experimental data and the FEM predictions further validates the efficacy of the finite element approach. These results underscore the reliability of finite element modeling in accurately simulating the experimental behavior of hybrid DSTCs.

Stresses and stain values calculated by ABAQUS software99 illustrated in horizontal sections.

Stresses and strain values calculated by ABAQUS software99 illustrated in vertical direction.

Tested load versus FEM load.

Comparison of Axial Load vs. Axial Strain between Experimental Tests and FEM for Hybrid DSTC Models.

The current study developed multiple ML models to estimate the confined ultimate strain and the ultimate load of confined concrete at the rupture of FRP tubes using a final dataset of 116 data points with six input features. The area of the steel tube (\({A}_{s}\), mm2), the yield strength of the steel tube (\({f}_{y}\), MPa), the area of the concrete core (\({A}_{g}\), mm2), the aspect ratio of column cross-section (\(2a/2b\)), the total thickness of FRP multiplied by the Elastic Modulus of FRP (\({t}_{f}{E}_{f}\), mm MPa), and unconfined concrete strength (\({f}_{c}{\prime}\), MPa).

These variables were designated as Input 1 (X1), Input 2 (X2), Input 3 (X3), Input 4 (X4), Input 5 (X5), and Input 6 (X6), respectively. The confined ultimate strain (\({\varepsilon }_{cc}\), mm/mm), and the ultimate load of confined concrete at the rupture of FRP tube (\({P}_{cc}\), kN) were considered as output variables to develop the models and were respectively denoted as Output 1 (Y1), and Output 2 (Y2). Input and output parameters are displayed with descriptive statistics in Table 4, which indicates the diverse range of the dataset.

Following the descriptive study, a statistical analysis was also performed using a heat-map to establish the degree of correlation between the variables using the Pearson correlation. It is clear from Fig. 8 that Y1 and Y2 are highly correlated with X4 and X6, respectively. For output 1, (Y1), X1, X2, and X3 show a negative correlation, whereas X4, X5, and X6 depict a positive correlation. For output 2 (Y2), all six inputs demonstrate significant correlation, with X4 showing a negative correlation, whereas X1, X2, X3, X5, and X6 depict a positive correlation.

Pearson correlation between features with heat-map.

To enhance predictive accuracy, six distinct input combinations were developed based on the correlation heatmap (Fig. 8), aiming to identify the combination that most effectively predicts the outputs. It is posited that parameters of higher importance exert a more significant influence on the prediction of the outputs. The proposed input combinations for predicting the two outputs are illustrated in Table 5. Additionally, the efficacy of these combinations was evaluated through the R2 and RMSE metrics.

This research considered five ML models as introduced in Fig. 9: Decision Tree (DT), Random Forest (RF), Adaptive Boosting (ADAB), Categorical Boosting (CATB), and eXtreme Gradient Boosting (XGB). The progression of DT models to XGB is illustrated in Fig. 10.

Applied ML models in this study.

Progression of DT to XGB.

DT is a simple way to decide how to classify things, and it works well for making predictions73. Each DT branch is a possible answer to that question, and each endpoint of the tree gives us a final category. Decision Trees are great because they're easy to understand and use, and they don't need the data to be in any special form. They're also flexible enough to work with different types of data. However, one issue with Decision Trees is that they can sometimes get too complicated during training, which means they might not work well with new, unseen data. RF model, which was initially introduced by74 and further developed by75, is a versatile technique applicable to both categorizing data and predicting outcomes. The model operates by creating multiple variations of the original data through a process called bootstrap resampling76. In essence, it creates several subsets of the original dataset to form individual decision trees, with each tree's predictions eventually being averaged to produce a final result.

ADAB was first introduced by Freund77. It is an ensemble ML model, where the base learner trains with the training set to update the weights based on the performance of the prior iterations. The output of the ADAB model is the weighted combination. However, the final prediction model can only be robust by this procedure when the performance of each weak learner is significantly better than random guessing. CATB is a powerful machine learning algorithm designed for handling categorical features and producing accurate predictions. It is a variant of gradient boosting that can handle both categorical and numerical features without requiring preprocessing like one-hot encoding or label encoding. CATB employs its built-in encoding system called "ordered boosting" to process categorical data directly, resulting in faster training and better model performance78. It is particularly useful for regression tasks where the goal is to predict a continuous target variable. CATB is known for its speed, accuracy, and ease of use, especially in situations involving structured data with many categorical features79. It also offers feature relevance rankings that help with feature selection and understanding model choices. The XGB is a prominent ensemble algorithm that integrates a vast array of weak learners to generate a strong learner80. The XGB model represents a version of gradient-boosted decision trees optimized for fast execution and high efficiency. It is a highly flexible and versatile algorithm known for its efficiency in handling sparse data and its ability to perform well on a wide range of regression and classification problems. The XGB model has been used successfully in numerous machine learning competitions due to its scalability and ability to produce highly competitive predictive models.

Generally, an optimal combination of hyperparameters can improve the performance of the ML model81,82,83. Grid Search (GS), Random Search (RS), and Bayesian Optimization (BO) algorithms are commonly used for fine-tuning the hyperparameters in the ML models81. However, GS and RS are not guaranteed to find the optimal solution due to the inherent variability in their search processes. Moreover, both the GS and RS methods require many different trials and can be time-consuming. Meanwhile, the BO algorithm searches for optimal solutions by considering past evaluations more efficiently84,85,86. Besides, the BO algorithm identifies the best parameters in significantly less time than the other methods. Therefore, the BO algorithm is used to find the optimal parameters of the predictive models in this study.

In addition, the cross-validation (CV) method is used to ensure that the ML model can perform well on unseen data (the test set) and avoid overfitting. Accordingly, the BO algorithm integrated with a fivefold CV is used to find the optimal hyperparameters of the predictive models. Additionally, the average negative mean squared error (neg-MSE) from the 5 folds is considered the fitness value.

Assessing the efficacy of each predictive model is essential, as the verification of a model’s predictive accuracy is fundamental to ensuring its practicality and scientific credibility76,87. The training datasets used during the construction phase of a model merely indicate how well the model conforms to the particular dataset in question. To validate the predictive models, testing datasets were therefore utilized. Model evaluations and their subsequent comparisons predominantly engage two approaches: visual and quantitative methods.

Methods based on visualization include the use of scatter plots, violin boxplots, and Taylor diagrams. These visual tools are beneficial for providing immediate, compelling comparative analyses, offering a swift assessment of the model’s predictive accuracy regarding statistical values such as the maximum, minimum, median, and quartiles. They surpass quantitative metrics, which might overlook these elements, but visual tools also fall short in interactive comprehensive model performance data88,89. To address this, three quantitative metrics were adopted: the coefficient of determination (\({R}^{2}\)), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Mean Bias Error (MBE), with their performance equations and optimal values presented in Table 6.

These metrics were selected to ensure a comprehensive evaluation of machine learning models, as each provides unique insights into different aspects of model performance90. \({R}^{2}\) is used to assess the goodness of fit of the model, indicating how well the model's predictions match the actual data. An \({R}^{2}\) value closer to 1 signifies that the model explains a large portion of the variance in the outcome variable. RMSE measures the average magnitude of the errors between predicted and actual values, giving higher weight to larger errors. RMSE highlights the presence of significant prediction errors, thus assessing the model’s prediction accuracy.

MAE provides a measure of the average magnitude of errors in a set of predictions, offering a clear indication of the average prediction error. It assesses the average absolute error between predicted and actual values, making it easy to interpret. MAPE expresses prediction accuracy as a percentage, facilitating easier understanding and comparison across different datasets and contexts. MAPE provides insight into the average percentage error of predictions, which is particularly useful when comparing accuracy across different scales. MBE measures the average bias in the predictions, indicating whether the model tends to overestimate or underestimate the actual values. MBE assesses the directional bias of the model’s predictions, with a positive MBE indicating a tendency to overestimate and a negative MBE indicating a tendency to underestimate.

In summary, the combination of \({R}^{2}\), RMSE, MAE, MAPE, and MBE ensures a comprehensive evaluation of machine learning models. These metrics capture different dimensions of model performance, including goodness of fit, prediction accuracy, error magnitude, relative error, and bias. This multifaceted approach provides a robust framework for understanding and improving model performance.

To analyze the sensitivity and interpret ML models on both a wide-scale and a more detailed level76,91, researchers use the SHapley Additive exPlanations (SHAP) approach92, which draws on principles from cooperative game theory93. The SHAP method was employed to gauge the comparative impact of input variables on the prediction process. As an advanced method within the realm of explainable artificial intelligence, SHAP helps to clarify the complex interactions between the input variables and the predictions of the model. It offers critical insights by identifying which features are most influential on predictions and how they modify the predicted results94,95.

In Fig. 11 illustrates the axial load-bearing capacities of hybrid GFRP-confined concrete-steel hollow elliptical columns for both the original test specimens (S1-S8) and those with increased yield strength (S9-S16).

Effect of steel tubes’ yield strength.

The results show a noticeable increase in the axial load capacity for specimens with increased yield strength of the steel tube. Specifically, for specimen S1/S9, the axial load capacity increased from 3797.41 to 4163.50 kN, demonstrating a significant enhancement of approximately 9.65%. Similarly, specimen S2/S10 exhibited an increase from 3151.38 to 3484.91 kN, a 10.6% improvement. For specimens S3/S11 and S4/S12, the axial load increased from 2708.71 to 3129.54 kN and from 2433.98 to 2816.58 kN, respectively. This indicates increases of 15.5% and 15.7%.

This trend continues across all specimens, indicating that increasing the yield strength of the steel tube from an original range of 337.6–364.5 MPa to a new yield strength of 500 MPa consistently enhances the axial load-bearing capacity of the hybrid columns. This enhancement can be attributed to the improved confinement effect provided by the steel tube, which delays the onset of buckling and allows the concrete core to sustain higher loads before failure. These findings suggest that optimizing the yield strength of steel tubes is a viable strategy for enhancing the mechanical performance and durability of such composite columns in practical applications.

Figure 12(a) and (b) illustrate the axial load-bearing capacities of hybrid DSTCs with varying steel tube thicknesses.

Effect of steel tubes’ thickness.

Figure 12(a) shows the axial load capacities for the original test specimens (G1) and those with increased steel tube thickness (\({t}_{s}= 8 mm\)). The results indicate a significant improvement in load-bearing capacity with increased steel tube thickness. For instance, the axial load capacity for specimen S1 increased from 3797.41 to 4349.77 kN, representing a 14.55% increase. Similarly, specimen S2 showed an increase from 3151.38 to 3603.02 kN, approximately 14.33%. This trend is consistent across all specimens, with increases ranging from 8.87 to 24.32%.

The enhanced performance can be attributed to the greater confinement effect provided by the thicker steel tube, which enhances the structural integrity and delays the buckling of the column. Consequently, the concrete core can sustain higher loads before failure, making this an effective strategy for improving column strength.

Conversely, Fig. 12(b) presents the axial load capacities for the original test specimens (G1) and those with decreased steel tube thickness (\({t}_{s} = 3.5 mm\)). The results indicate a reduction in load-bearing capacity with decreased steel tube thickness. For instance, the axial load capacity for specimen S9 decreased from 4163.50 to 3760.49 kN, representing a 9.68% decrease. Specimen S10 showed a reduction from 3484.91 to 3123.09 kN, approximately 10.39%. This pattern is observed across all specimens, with reductions ranging from 7.35 to 20.12%.

The decreased performance can be attributed to the reduced confinement effect provided by the thinner steel tube, which is less effective in enhancing the structural integrity and delaying buckling. As a result, the concrete core reaches its failure load more quickly, highlighting the importance of optimizing steel tube thickness for achieving desired structural performance.

The figures provided (Figs. 12 and 13) illustrate the ultimate axial load and ultimate strain for hybrid DSTCs with different concrete strengths and FRP layer configurations. The comparison is made between columns with high concrete strength (\({f}_{c}{\prime} = 72.40 \text{MPa}\)) and those with reduced strengths (43.45 MPa and 28.95 MPa), while maintaining a fixed lateral confinement pressure ratio (\({f}_{lf}/{f}_{c}{\prime}\)).

Effect of steel tubes’ concrete strength on the axial load capacity at the same confinement ratio.

Figure 13(a) to (d) show the ultimate axial load capacities for the specimens. Figure 13(a) compares G1 (\({f}_{c}{\prime} = 72.40 \text{MPa}\)) and G3 (\({f}_{c}{\prime} = 43.45 \text{MPa}\)). The results demonstrate a consistent reduction in axial load capacity when the concrete strength is decreased. For instance, specimen S1’s axial load capacity decreased from 3797.41 to 2613.91 kN, and specimen S9’s capacity decreased from 4163.50 to 2980.83 kN. Similarly, Fig. 13(d) compares G2 (\({f}_{c}{\prime} = 72.40 \text{MPa}\)) and G6 (\({f}_{c}{\prime} = 28.95 \text{MPa}\)). The trend continues with significant reductions in axial load capacities, such as S1’s load decreasing from 4349.77 to 2589.27 kN, and S9’s from 3760.49 to 1998.30 kN. These results indicate that higher concrete strength significantly enhances the axial load-bearing capacity of the columns. Decreasing the concrete strength, even while maintaining a fixed lateral confinement pressure ratio, results in lower load capacities, highlighting the crucial role of concrete strength in the overall performance of these hybrid columns.

Figure 14(a) to (d) depict the ultimate strain for the different specimen groups. Figure 14(b) compares G2 (\({f}_{c}{\prime} = 72.40 \text{MPa}\)) and G4 (\({f}_{c}{\prime} = 43.45 \text{MPa}\)). Similar observations are made where ultimate strain values are marginally reduced for lower concrete strengths, such as S1’s ultimate strain dropping from 0.0170 to 0.0164 mm/mm, and S9’s from 0.0171 to 0.0164 mm/mm. Similar trends are observed in Fig. 14(d) that compares G2 (\({f}_{c}{\prime} = 72.40 \text{MPa}\)) and G6 (\({f}_{c}{\prime} = 28.95 \text{MPa}\)), with ultimate strain values decreasing for specimens with lower concrete strength, such as S1’s ultimate strain dropping from 0.0170 to 0.0160 mm/mm, and S9’s from 0.0171 to 0.0162 mm/mm. These findings demonstrate that both the ultimate axial load and ultimate strain are significantly influenced by the concrete strength. The decrease in both parameters with reduced concrete strength, despite maintaining a fixed lateral confinement pressure ratio, highlights the critical importance of concrete strength in determining the mechanical performance and durability of hybrid GFRP-confined concrete-steel hollow elliptical columns. Adjusting both concrete strength and the number of FRP layers, while keeping the lateral confinement pressure ratio constant, allows for a comprehensive understanding of their combined effects on column performance.

Effect of steel tubes’ concrete strength on the ultimate strain at the same confinement ratio.

Figure 15(a) to (c) present the ultimate axial load capacities of hybrid DSTCs with and without an inner steel tube. These comparisons highlight the significant impact of the presence of an inner steel tube on the load-bearing capacity of the columns.

Effect of inner steel tube.

Figure 15(a) compares G1 (with steel tube) and G7 (no steel tube). The results show a considerable reduction in the axial load capacity when the steel tube is removed. For example, specimen S1’s axial load capacity decreased from 3797.41 to 2781.23 kN, and specimen S2’s from 3151.38 to 2211.69 kN. Similar reductions are observed for all specimens, indicating the crucial role of the steel tube in enhancing the structural capacity. Figure 15(b) compares G3 (with steel tube) and G8 (no steel tube). The trend of reduced axial load capacity without the steel tube continues. Specimen S1’s load dropped from 2613.91 to 1618.10 kN, and specimen S2’s from 2206.07 to 1288.64 kN. These reductions are consistent across all specimens, underscoring the significant contribution of the steel tube to the overall load-bearing performance. For Fig. 15(c), the results again indicate a substantial decrease in axial load capacities when the steel tube is absent. For instance, S1’s capacity fell from 2027.92 to 1032.15 kN, and S2’s from 1726.41 to 824.35 kN. All specimens showed similar trends, reinforcing the importance of the steel tube.

The data clearly indicate that the presence of an inner steel tube significantly enhances the axial load capacity of the hybrid columns. The steel tube provides additional confinement to the concrete core, preventing premature buckling and enhancing the column’s load-bearing capacity. When the steel tube is removed, the columns exhibit significantly lower axial load capacities, demonstrating its critical role in the structural performance.

In this study, five ML models: DT, RF, ADAB, CATB, and XGB, were used to predict the confined ultimate strain and the ultimate load of confined concrete. The optimal hyperparameters of the adopted ML models based on the BO method are summarized in Table 7. The evaluation of the ML models visually and quantitatively, input combination analysis, BO k-fold CV analysis, and SHAP feature importance analysis are explained in the following sections.

Figures 16 and 17 show the performance of the adopted ML models during the TS stage at the different input combinations based on \({R}^{2}\) and RMSE metrics for Y1 and Y2, respectively. The figure consists of two bar charts illustrating the performance of different ML models during the testing stage, evaluated across various input combinations based on the R2 metric. For Y1, the \({R}^{2}\) values range from 0.292 to 0.920, showing varying degrees of model performance across different input combinations. Input 1 (X4) exhibits relatively low \({R}^{2}\) values, while Input 5 (X4, X3, X5, X6, X1) and Input 6 (X4, X3, X5, X6, X1, X2) demonstrate significantly higher \({R}^{2}\) values, indicating better model performance. The CatBoost and XGBoost models consistently perform well, particularly with more complex input combinations. The RMSE values for Y1 follow a similar trend, with lower values observed for Input 5 and Input 6, suggesting improved prediction accuracy.

Performance of the adopted ML models during the testing stage at the different input combinations based on R2 metric for (a) Output 1 and (b) Output 2.

Performance of the adopted ML models during the testing stage at the different input combinations based on RMSE metric for (a) Output 1 and (b) Output 2.

For Y2, the \({R}^{2}\) values also exhibit significant variation across input combinations, ranging from 0.003 to 0.970. The simpler input combinations (e.g., Input 1: X6) show very low \({R}^{2}\) values, indicating poor model performance. In contrast, Input 5 (X6, X5, X1, X2, X3) and Input 6 (X6, X5, X1, X2, X3, X4) yield much higher \({R}^{2}\) values, with models like CatBoost and XGBoost achieving superior results. The RMSE values for Y2 reflect this trend, with significantly lower errors observed for Input 5 and Input 6, further highlighting the effectiveness of these input combinations in improving model predictions.

In summary, the analysis indicates that more complex input combinations, particularly Input 6 for both outputs Y1 and Y2, result in higher \({R}^{2}\) values and lower RMSE values, signifying better model performance. This suggests that incorporating a larger number of relevant features can significantly enhance the predictive accuracy of ML models.

Figure 18 presents a comparison of the performance of the models in training and fivefold CV. Performance is evaluated based on the RMSE metric. Lower values for both metrics indicate better predictive accuracy. Figure 18(a) compares the RMSE values of regression models including DT, RF, ADAB, CATB, and XGB for Output 1. The RMSE for training is depicted in blue, and the fivefold CV is in red. CATB and XGB demonstrate the lowest RMSE, indicating superior performance on this output, with good generalization evidenced by the similarity in training and CV values. Figure 18(b) presents RMSE comparisons for Output 2 on a larger scale (kN). CATB shows the lowest RMSE during CV, closely followed by XGB, suggesting high predictive accuracy for Output 2. The discrepancy between training and CV RMSE for certain models, notably DT, suggests possible overfitting.

Comparison of the performance of the ML models in the training and tenfold CV stages based on RMSE values for (a) Y1 and (b) Y2.

For Output 1, CATB and XGB emerge as the best-performing models, while for Output 2, CATB stands out as the most accurate. Overall, CATB exhibits exceptional performance across both outputs, maintaining low RMSE values in the training and CV phases. This consistency across different scales and outputs underlines CATB's robustness, making it the preferred model for general application.

Figure 19 presents the predictive performance of various ML models for estimating the confined ultimate strain, showcasing both training (TR) and testing (TS) stages. Figure 19(a) shows that in the DT model, both TR and TS data points are plotted against actual values, with TR performing notably better than TS, as seen by the proximity to the equality line and tighter error bounds. Figure 19(b) shows that the RF model indicates improved TS performance compared to the DT model, with higher R2 and lower errors, suggesting better generalization. Figure 19(c) shows that the ADAB model reveals a performance drop in TS, with a lower R2 than RF, showing susceptibility to overfitting.

Scatter plots for Output 1 and deviation error lines of ± 10% during the TR and TS stages based on (a) DT, (b) RF, (c) ADAB, (d) CATB, and (e) XGB models.

Figure 19(d) shows that the CATB model represents an impressive match between predicted and actual values, with the highest R2 and lowest errors among the models for TS. Lastly, Fig. 19(e) shows that the XGB model demonstrates excellent TR results but a slight performance reduction in TS when compared to the CATB model. Considering TS as the true performance indicator, the CATB model emerged as the best model with superior predictive accuracy and the tightest error margins. The XGB model, while strong, slightly lags behind the CATB model in TS. Overall, the CATB model demonstrated the best balance of high accuracy and low error rates in the testing stage, confirming its robustness as the best model among the alternatives.

Figure 20 presents the predictive performance of various ML models for estimating the ultimate load of confined concrete at the rupture of FRP tube, showcasing both training (TR) and testing (TS) stages. Figure 20(a) shows that the DT model had a noticeable deviation from the equality line in the testing stage, indicating reduced prediction accuracy compared to training. Despite the RF model showing a tighter grouping around the equality line, particularly during the training stage (see Fig. 20b). This model has high \({R}^{2}\) values for both training and testing, with relatively small increases in RMSE and MAE from training to testing, suggesting good generalization without severe overfitting. Figure 20(c) shows that the ADAB model exhibited a pattern similar to the RF model, with a dense cluster near the equality line during training. However, during testing, ADAB's predictions show a bit more spread, with a noticeable increase in RMSE and MAE, indicating that while the model generalizes well, it is slightly less accurate than RF.

Scatter plots for Output 2 and deviation error lines of ± 10% during the TR and TS stages based on (a) DT, (b) RF, (c) ADAB, (d) CATB, and (e) XGB models.

Figure 20(d) shows that the CATB model presents a strong alignment with the equality line during both the training and testing stages, signifying high predictive accuracy. With the smallest increases in RMSE and MAE from training to testing among the models shown, the CATB model exhibits excellent generalization and is arguably the best-performing model for this dataset. Figure 20(e) shows that the XGB model also demonstrates high accuracy, with a dense concentration of points near the equality line and impressive \({R}^{2}\) values. However, the increase in RMSE from training to testing, although small, is larger compared to the CATB model, suggesting that while the XGB model is highly accurate, the CATB model has a slight edge in performance. Overall, the CATB and XGB are standout models, offering high accuracy and strong generalization capabilities. The CATB model is slightly superior due to its consistently lower error rates during testing, indicating it is the best model for this dataset when considering both accuracy and robustness against overfitting.

Violin boxplots combine elements of boxplots and kernel density plots to provide a more detailed representation of the data distribution76. When comparing actual and predicted values, they can show the spread and density of errors, giving insights into the variance and bias of the model's predictions. Figure 21(a) shows violin plots of Output 1. The CATB and XGB models show a narrower distribution, which suggests more precise predictions with less variability. Figure 21(b) shows violin plots of Output 2. For Output 2, the ADAB and CATB plots are narrower, especially in the interquartile range, suggesting these models have more consistent predictions for this output. Considering both outputs together, the CATB model appears to perform consistently well, with relatively narrow interquartile ranges and less extreme data points outside the 1.5 IQR range, indicating it might be the overall best model in terms of consistent performance and reliability.

Violin boxplots during the testing phase for (a) Y1 and (b) Y2.

Taylor diagrams are a specialized graphical representation used to quantify the similarity between actual and predicted values96,97. These diagrams plot the correlation, the standard deviation, and the root mean square error of predictions on a single chart. This provides a comprehensive view of a model's accuracy, variability, and overall performance compared to the actual observations. Figure 22 shows Taylor diagrams for two different outputs: (a) Output 1 and (b) Output 2, during the testing phase of various models or stages, which are represented by different colored dots on the diagrams. Figure 22(a) shows the Taylor diagram of Output 1.

Taylor diagrams during the testing phase for (a) Output 1 and (b) Output 2.

Each dot represents a different model's performance with respect to the reference point, which is the centered dot labeled TS-Output 1. The radial distance from this point indicates the standard deviation of the model outputs relative to the standard deviation of the observed data. The closer a dot is to the reference point, the better the model matches the observed data's variability. The concentric arcs represent correlation coefficients between the model outputs and the observed data, with a correlation of 1 being perfect and represented by the outer arc. For Output 1, the CATB model has the highest correlation, followed by the XGB and RF models. The ADAB and DT models show less correlation and a higher standard deviation compared to the observed data. For Output 2, the same interpretation applies. Figure 22(b) shows the Taylor diagram of Output 2. The centered dot represents TS-Output 2. It appears that the XGB and CATB models have the best correlation with the observed data, followed by the RF model. The ADAB and DT models have lower correlations and higher standard deviations.

Table 8 compares ML models' performance for Output 1 across training and testing stages, focusing on \({R}^{2}\), RMSE, MAE, MAPE, and MBE metrics. During training, the DT model showed excellent fit with the highest R2 but revealed overfitting with a significant drop in testing \({R}^{2}\). The CATB model showed high \({R}^{2}\) with minimal performance loss from training to testing, also maintaining the lowest RMSE, MAE, MAPE, and MBE in testing, indicating it as the top model for Output 1. Table 9 shows the models’ performance for Output 2, contrasting training and testing stages with the same metrics. Although the RF and ADAB models presented high training \({R}^{2}\), they exhibited a performance decline in testing. The CATB model maintained an exceptional testing R2 and lower error metrics, closely followed by XGB, which offers a slightly higher \({R}^{2}\) but with marginally higher error values.

Overall, considering both outputs and focusing on testing stage metrics, the CATB model consistently exhibits strong performance, with the best \({R}^{2}\) for Output 1 and competitive for Output 2, alongside low error metrics, making it the best model. It successfully generalizes with lower error rates, which is key for robust predictive models. The XGB model closely competes, with a slight advantage in \({R}^{2}\) for Output 2, but the CATB model's balanced performance across all metrics secures its position as the preferable model for these outputs.

Figure 23 shows the SHAP summary plots for the two outputs. These plots were created based on the best predictive model for Y1 and Y2: the CATB model. In the SHAP summary plot for Output 1 (Fig. 23a), it is evident that the features X3, X5, X4, X2, X1, and X6 have varying degrees of impact on the model output. The feature X3 exhibits SHAP values mostly concentrated around zero, indicating a minor impact, with higher values of X3 slightly contributing positively. X5 displays a distribution of SHAP values on both sides of zero, showing both positive and negative impacts on the output. Higher values of X5 tend to contribute positively, while lower values have a negative effect. Similarly, X4 shows SHAP values near zero, with higher values contributing marginally positively. X2's impact is mixed, with SHAP values spread across both sides of zero; higher values of X2 positively affect the output, while lower values contribute negatively. X1 has a very minimal effect, as indicated by SHAP values clustered around zero. Lastly, X6 also exhibits SHAP values near zero, indicating an insignificant effect on the output. Overall, features X2 and X5 are the most influential for Output 1, with varying impacts based on their values.

SHAP summary plots based on CATB model for (a) Output 1 and (b) Output 2.

Figure 23(b) shows the SHAP summary plot for Output 2. The figure reveals that the features X1, X5, X6, X2, X3, and X4 have a more pronounced impact compared to Output 1. X1 significantly affects Output 2, with SHAP values widely spread across a large range. Higher values of X1 are associated with a strong positive impact, while lower values tend to have a negative effect. Similarly, X5 shows a substantial impact, with higher values leading to positive SHAP values and lower values contributing negatively. X6 also has a noticeable impact, with a wide distribution of SHAP values; higher values result in positive contributions, while lower values have negative impacts. X2 displays a mix of positive and negative contributions, with higher values positively affecting the output and lower values contributing negatively. X3 exhibits considerable impact, with higher values generally showing a positive effect and lower values a negative one. Lastly, X4 shows significant impact, with SHAP values widely spread; higher values of X4 are associated with positive contributions, while lower values negatively impact Output 2. In summary, for Output 2, all features have a pronounced influence, with X1, X5, X6, and X4 showing particularly strong contributions.

Finally, to address the practical needs of engineers and designers in efficiently applying ML models to their needs, this section introduces a significant advancement. Despite the complex requirements of database assembly, model training, and validation hindering the seamless adoption of ML in everyday design tasks, a novel solution has been crafted. A Python web application has been developed, integrating a model equipped with optimized hyperparameters via an intuitive graphical user interface (GUI). This GUI is specifically designed for predicting the outputs, as illustrated in Fig. 24. Both outputs are directly depicted by clicking the Calculate button. The GUI was built based on the CATB model (the best predictive model for both outputs). It was designed with the Tkinkter package98. The GUI can be freely accessed at https://github.com/mkamel24/FRP-tubes_strain_load.

GUI for predicting Output 1 and Output 2.

This research aimed to investigate the load-carrying capacities of hybrid FRP-Concrete-Steel columns with an elliptical cross-section, using both finite element modeling and machine learning techniques. The study concluded with several key findings, summarized as follows:

The FEM demonstrated a robust \(R^{2}\) value of 0.9812, indicating that approximately 98.12% of the variance in load capacities could be accurately predicted, highlighting its precision and reliability in modeling hybrid DSTCs.

Increased yield strength and thickness of the steel tube significantly enhance the axial load capacity of hybrid columns. Stronger steel provides better confinement, delaying buckling and allowing the concrete core to sustain higher loads.

Higher concrete strength also leads to a substantial increase in both ultimate axial load and strain. Conversely, lower concrete strength results in decreased load capacities and strain values, underscoring the importance of concrete strength for performance and durability.

The presence of an inner steel tube significantly boosts the axial load capacity by providing additional confinement to the concrete core and preventing premature buckling. Removing the inner steel tube results in notably lower load capacities, highlighting its essential role in structural performance.

Considering the visual evaluation of the ML models, it can be concluded that:

Utilizing the Scatter plots, the RF model indicates improved performance compared to the DT model, with higher \(R^{2}\) and lower errors, suggesting better generalization. The XGB model demonstrates excellent training results but a slight performance reduction in test results when compared to the CATB model. The CATB model represents an impressive match between predicted and actual values, with the highest \(R^{2}\) and lowest errors among the models.

Utilizing Violin boxplots and considering both outputs together, the CATB model appears to perform consistently well, with relatively narrow interquartile ranges and less extreme data points outside the 1.5 IQR range, indicating it might be the overall best model in terms of consistent performance and reliability.

Utilizing Taylor diagrams, it appears that the XGB and CATB models have the best correlation with the observed data, followed by the RF model. The ADAB and DT models have lower correlations and higher standard deviations.

Considering the quantitative evaluation of the ML models, the CATB model successfully generalizes with lower error rates, which is key aspect for robust predictive models. The XGB model closely competes, with a slight advantage in \(R^{2}\), but the CATB model's balanced performance across all metrics secures its position as the preferable model for these outputs.

The yield strength of the steel tube indicates a strong positive influence on the prediction models. In contrast, the area of the steel tube appears to have a neutral effect, and unconfined concrete strength has the least impact, with SHAP values clustered close to zero.

Applying optimization techniques to determine the optimal configuration of hybrid hollow columns in terms of the dimensions and materials used by conducting numerical simulations or using genetic algorithms to find the most efficient and effective design parameters.

Studying the environmental conditions, such as temperature, moisture, and chemical exposure, on the mechanical properties and performance of the hybrid members to investigate the durability of hybrid FRP-Concrete-Steel columns.

Implementing advanced machine learning techniques or alternative algorithms to improve the accuracy and efficiency of the predictive models developed in this study.

Investigating the environmental sustainability aspects of using hybrid hollow members compared to conventional construction materials and methods.

Data associated with the present study will be available upon request from the corresponding authors.

Fiber reinforced polymer

Decision tree

Random forest

Adaptive boosting

Categorical boosting

EXtreme gradient boosting

Double-skin tubular members

Glass fiber-reinforced polymers

Carbon fiber-reinforced polymers

FRP-recycled aggregate concrete-steel long columns

Polyethylene terephthalate

Lateral confinement pressure ratio

Coefficient of determination

The area of the steel tube, mm2

Yield strength of the steel tube, MPa

The area of the concrete core, mm2

The aspect ratio of column cross-section

The total thickness of FRP multiplied by the Elastic Modulus of FRP, mm MPa

Unconfined concrete strength, MPa

The confined ultimate strain, mm/mm

Ultimate load of confined concrete at the rupture of FRP tube, kN

Root mean square error

Grid search

Random search

Bayesian optimization algorithms

Negative mean squared error

Mean absolute error

Mean absolute percentage error

Mean bias error

SHapley additive exPlanations

Steel tube thickness, mm

Training stage

Testing stage

Interquartile range

Graphical user interface

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School of Applied Technologies, Qujing Normal University, Qujing, 655011, Yunnan, China

Haytham F. Isleem & Tang Qiong

Structural Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt

Mostafa M. Alsaadawi

Civil Engineering Department, Faculty of Engineering, Horus University-Egypt, New Damietta, 34517, Egypt

Mostafa M. Alsaadawi & Mohamed Kamel Elshaarawy

Structural Engineering and Construction Management Department, Faculty of Engineering and Technology, Future University in Egypt (FUE), Cairo, Egypt

Dina M. Mansour

Building Engineering & Construction Management, Rajshahi University of Engineering & Technology, Rajshahi, Bangladesh

Faruque Abdullah

Department of Civil and Water Engineering, Laval University, Quebec City, Quebec G1V 0A6, Canada

Ahmed Mandor

Department of Civil Engineering, University of Garmian, Kalar, Kurdistan Region, 46021, Iraq

Nadhim Hamah Sor

Faculty of Engineering, University of Balamand, P.O. Box 100, Tripoli, Lebanon

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Project Administration, Supervision, Funding Acquisition: H.F.I., T.Q., & M.M.A.; Conceptualization, Formal Analysis, Methodology, Investigation, Visualization, Software, Resources, Validation, Data Curation, Writing – original draft, Writing – review & editing: H.F.I., T.Q., M.M.A., M.K.E., D.M.M., F.A., A.M., N.H.S., & A.J. All authors have read and agreed to the published version of the manuscript.

Correspondence to Haytham F. Isleem, Tang Qiong or Mostafa M. Alsaadawi.

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Isleem, H.F., Qiong, T., Alsaadawi, M.M. et al. Numerical and machine learning modeling of GFRP confined concrete-steel hollow elliptical columns. Sci Rep 14, 18647 (2024). https://doi.org/10.1038/s41598-024-68360-4

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Received: 12 June 2024

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DOI: https://doi.org/10.1038/s41598-024-68360-4

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