Prediction of the axial capacity of various polygon-type section concrete-filled steel (CFST) columns utilising deep neural network and machine learning regressor

Abstract

The advantages of concrete-filled steel tubular (CFST) have made it a popular choice for structural applications. For structural design and safety, the precise estimation of the axial capacity of the CFST is a crucial element. However, design codes currently cover only conventional CFST parts, such as square, circular, and rectangular sections, and aren’t applicable to polygonal sections, such as hexagonal, octagonal, etc. In this study, four data-driven models were developed to predict the axial capacity of polygonal CFST columns, including an optimized multilayer perceptron (MLP) neural network, Support Vector Regressor (SVR), Random Forest Regressor (RFR), and Gradient Boosting Regressor (GBR). This study uses a dataset containing 835 experimental and numerical test samples from the literature. All samples will be analysed using machine learning models, and the best predictors will be determined. For further evaluation, ML model predictions will be compared to existing design codes (such as Eurocode 4, ANSI/AISC 360, Chinese Code GB50936, etc.) and proposed equations from the literature. The results showed that DNN was the best predictor and outstanding (which can be applied to all polygonal CFST) when compared to design codes and proposed equations from the literature. Finally, a GUI-based polygonal CFST application predictor is offered to integrate engineering applications.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-49985-z.

Introduction

Concrete-filled steel tubular (CFST) is a composite structural system consisting of an outer steel tube filled with concrete. It is currently receiving a lot of attention due to its advantages, including high ductility and stiffness, high strength, high seismic absorbance, and high fire resistance^1–3. CFST can also serve as permanent formwork and primary reinforcement, which reduces construction time and labour⁴. The use of CFST is no longer restricted to square, rectangular, or circular shape sections; it may now be applied to a wide variety of shape sections, including triangular, pentagonal, hexagonal, heptagonal, and octagonal CFST, or polygonal CFST. Since current design codes, such as Eurocode 4 (EC4)⁵, and ANSI/AISC 360⁶ only offering simplified square and circular sections, other shapes of CFST lack a trustworthy and precise method for determining their axial capacity⁷.

Over recent years, there have been several experimental and numerical studies on the behaviour of CFST columns with a variety of shapes and geometrical aspects, among them Chen and Chan⁸ studied octagonal CFST utilising the circumcircle approach of EC4 and ANSI/AISC 360, as well as the Chinese code GB 50,936 − 2014, and obtained excellent predictions. Ayough et al.⁹ also explored hexagonal and octagonal CFST utilising EC4 (using the square and circle CFST equations), and the ANSI/AISC 360. As a result, ANSI/AISC 360 forecasts with less accuracy than EC4 (with the mean values of N_{u, exp}/N_u, and pre of 1.09 using the square equation, and 0.96 and 0.99 using the circular equation for hexagonal and octagonal, respectively). However, research has yet to establish the use of machine learning to predict the axial capacity of polygonal CFST, in contrast to design codes and proposed formulas in the literature.

The limitations of the design regulations for calculating axial capacity of CFST are outlined in Table 1. AISC 360 − 16 and Eurocode 4 limit the concrete compressive strength, denoted by (f’_c) the steel yield strength, denoted by (f_y), whereas ACI 318 limits only the concrete compressive strength and does not impose a direct limit on steel strength. In comparison, GB50936 established restrictions for both f’_c and f_y. AISC 360 established limits for f’_c between 21 and 69 MPa, f_y of steel composites below 525 MPa, and f_y of reinforcing bars below 550 MPa. ACI318 established restrictions for f’_c above 17.2 MPa. Eurocode 4 restricts f’_c at 20–50 MPa and f_y to 230–460 MPa. Meanwhile, GB50936 established restrictions for f’_c below 60 MPa and f_y below 500 MPa.

Table 1.

Overview of dataset variables.

Variable name	Unit	Description
	MPa	Compressive strength of standard concrete cylinders
	mm	Height of the column
	mm	Perimeter of the column cross-section
	Degrees	First angle between two sides of the column cross-section
	Degrees	Second angle between the following two sides of the column cross-section
	mm	Base of the column cross-section (Outer-to-Outer)
	mm	Depth of the column cross-section (Outer-to-Outer)
	mm	Thickness of outer steel tube
	MPa	Yield strength of the outer steel tube
	mm	Thickness of inner steel tube
	MPa	Yield strength of the inner steel tube
	mm	Depth of inner steel tube (Outer-to-Outer)
	mm²	Area of inner hollow section (inner void)
	mm/mm	Eccentric loading ratio
	kN	Load carrying capacity

Compared with traditional approaches, machine learning (ML) methods are more efficient, as conventional methods are time-consuming and expensive¹⁰. Researchers have undertaken extensive studies on machine learning to predict axial CFST. Zhang et al.¹¹ studied predictors of circular CFST reinforced with basalt fiber-reinforced polymer (BFRP) axial capacity using Linear Regression (LR), Support Vector Regression (SVR), Bayesian Ridge Regression (BRR), Random Forest (RF), XGBoost, and CatBoost algorithms. It was discovered that LR, SVR, and BRR had accuracy R²s of approximately 0.81, while CatBoost had the highest at 0.98, and RF and XGBoost had R²s of 0.97. Lai et al.¹² studied circular CFST using ML models, and the best predictors were Natural Gradient Boosting (NGBoost), Artificial Neural Network (ANN), and Extreme Gradient Boosting (XGBoost). Wang et al.¹³ explored rectangular CFST using ANN and obtained an R² accuracy of 0.97. However, the research covers only circular and rectangular CFST, with no studies on polygonal CFST. Very recent studies have shown that advanced machine-learning and deep-learning models can also predict the capacity of other non-conventional CFST and closely related composite columns, including round-ended CFST columns and elliptical concrete-filled steel tubular short columns^1,4.

Recent studies have further confirmed the effectiveness of data-driven methods for predicting the axial capacity of steel–concrete composite coluns. Bayesian optimization has been used to improve machine-learning hyperparameter selection for CFST axial-capacity prediction¹⁴. ANN, GEP, and MLR have been applied to recycled-aggregate concrete-filled circular steel tube columns¹⁵, while ANN, random forest, and extreme gradient boosting approaches have been used for high-strength CFST columns¹⁶. Metaheuristic-assisted neural-network models have also been reported for square concrete-filled steel tube columns¹⁷ and for concrete–steel columns more broadly¹⁸, showing that optimization-based learning frameworks can improve predictive accuracy. Recent hybrid studies have also combined machine learning with finite-element modelling for concrete-filled PVC tube columns under uniaxial eccentric compression and for elliptical FRP-confined concrete columns under axial compression, further supporting the use of data-driven frameworks for composite-column capacity assessment^2,3. Similar soft-computing strategies have also been extended to GFRP-reinforced concrete columns¹⁹. Machine learning has also been extended to elliptical FRP-concrete-steel double-skin columns for axial load-capacity prediction, indicating that accurate data-driven strength estimation is feasible even for more complex composite-column configurations²⁰. Despite these advances, studies focused specifically on polygon-type CFST columns remain very limited^{8,9,19,22,42,55}.

Aside from that, there is a neural network method known as a Deep Neural Network (DNN) that has more hidden layers between the input and output, which improves neural network learning and mapping skills and outperforms ANNs. DNN is a viable option for fault tolerance and robustness^21,22. Xu et al.²³ conducted a recent study utilising a DNN to forecast stress-strain curves for ultra-high-performance concrete (UHPC). They found that the results were exceptionally accurate in complex studies. As an outcome, the authors will propose using DNN for machine learning analysis.

Thus, this study uses machine learning models, including the Deep Neural Network (DNN), Support Vector Regressor (SVR), Random Forest Regressor (RFR), and Gradient Boosting Regressor (GBR), to examine the axial capacity of polygonal CFST. All ML models will be assessed to determine the best predictor of polygonal CFST. ML models will be compared with design codes and proposed formulas from the current literature to make a comprehensive decision. Lastly, an application-based GUI predictor for polygonal CFST is developed, integrating engineering applications to simplify prediction.

Machine learning approach

Data collection and preprocessing

Dataset collection

The dataset utilized in this study comprises 835 samples from 41 studies^8,9,24–62, meticulously compiled from existing literature and finite element method (FEM) analyses in the current study. This comprehensive dataset encompasses a range of input variables (X1 to X14) and a single output variable (Y), each playing a critical role in predicting the load-carrying capacity of composite columns. Table 1 provides a detailed overview of these variables, including their names, units, and descriptions. All sample details were illustrated in Fig. 1.

Fig. 1.

Sample details illustration of this study.

Because the database combines published experimental records and finite element (FE) results generated in the current study, leakage control was treated as a key requirement. Each sample was tagged byits source study, and FE samples from the current investigation were additionally grouped by parametric family so that closely related cases derived from the same base configuration remained in a single group. All train-validation splits were then performed at the group level rather than at the individual-row level. This grouping strategy prevented similar specimen families or FE variants from appearing simultaneously in training and validation subsets.

The input variables were selected to represent the main geometric, material, and loading parameters governing the axial behavior of polygonal concrete-filled steel tubular columns. X1 denotes the compressive strength of standard concrete cylinders. X2–X7 describe the specimen geometry, including column height, cross-sectional perimeter, polygon angles, base width, and section depth. X8–X13 characterize the steel components and inner hollow section through the thickness and yield strength of the outer and inner steel tubes, the depth of the inner steel tube, and the area of the inner void. X14 represents the eccentric loading ratio and captures non-uniform loading conditions that commonly arise in practice.

The output variable, Y, is the load-carrying capacity in kN. Together, these variables provide a physically meaningful description of each specimen and include the main factors affecting axial strength. This variable set supports the development of a robust prediction model for polygonal CFST columns with different shapes, material strengths, and loading conditions. This careful selection and detailed characterization of variables facilitates the creation of an accurate and reliable dataset, which is essential for the successful application of finite element methods and machine learning approaches to predict structural performance.

Dataset analysis

A comprehensive analysis of the dataset presented in Supplementary Table S1 was conducted to understand the underlying characteristics and relationships among the variables. Table 2 presents a detailed statistical description of each variable, including measures such as mean, standard deviation, minimum and maximum values, quartiles, skewness, and kurtosis. The statistical analysis reveals significant variability across the variables.

Table 2.

Statistical description analysis of the dataset.

	Mean	Std. Dev.	Min	Q1	IQR	Q3	Max	Skewness	Kurtosis
	50	26	13	30	40	65	112	0.70	2.43
	813	492	249	595	650	1200	3810	2.27	11.73
	960	504	327	532	680	1327	2651	0.72	2.38
	122	17	60	120	120	135	135	– 1.34	4.01
	121	18	60	120	120	135	135	– 1.26	3.72
	309	172	100	169	222	434	866	0.69	2.29
	290	159	50	150	213	433	866	0.84	2.85
	5.6	3.4	1.7	3.1	4.7	6.6	16	1.33	4.42
	385	128	219	311	350	435	771	1.59	5.38
	1.01	1.1	0	0	0	0	7.5	2.24	7.84
	75	143	0	0	0	0	711	1.60	4.31
	18	37	0	0	0	0	203	2.13	7.46
	1192	3336	0	0	0	0	29,877	4.95	33.62
	0	0	0	0	0	0	1.1	5.35	32.98
	7201	7747	281	1625	3058	11,567	36,722	1.46	4.53

For instance, X1 has a mean of 50.4 MPa and a standard deviation of 26.6 MPa, indicating a broad range of concrete strengths in the dataset. The X2 shows considerable variation, with values ranging from 248.8 mm to 3810 mm and a high standard deviation of 491.99 mm, suggesting diverse column sizes are represented. The X3 also displays substantial variability, reflecting the inclusion of various column shapes and sizes. Angles X4 and X5 have skewness of -1.34 and − 1.26, respectively, indicating left-skewed distributions, with most data points clustering toward higher angle values. This is consistent with typical column designs, in which angles are often standardized within a specific range. Geometric dimensions such as X6 and X7 exhibit positive skewness (0.69 and 0.84, respectively), suggesting that while most columns have smaller base and depth dimensions, there are instances of significantly larger sizes. Material properties, including X8 to X13, show varying degrees of skewness and kurtosis, highlighting the diversity in steel tube thicknesses, yield strengths, and hollow section areas. Notably, X13 and X14 exhibit extreme skewness and kurtosis, indicating the presence of outliers and a heavy-tailed distribution.

Figure 2 displays histogram distributions for various variables in the dataset, providing a visual representation of each variable’s density over specified ranges. This visualization is essential for understanding the distribution characteristics, such as central tendencies, spread, and the presence of outliers or skewness. Variables X1, X2, X3, X6, and X7 demonstrate a range of distribution shapes, with some showing significant skewness. For example, X2 and X12 exhibit pronounced right skewness, which is consistent with their high skewness values of 2.2687 and 2.1253, respectively, as noted in Table 2. This characteristic suggests heavy tails and peakedness, distinct from a normal distribution. The histograms corroborate this by showing a long tail extending toward higher values.

Fig. 2.

Statistical descriptive analysis of various variables, captured across four distinct charts of mean, standard deviation, skewness, kurtosis, and variance.

Variables X4 and X5 exhibit left-skewed distributions, as evidenced by their negative skewness values of -1.3403 and − 1.2625. The histograms for these variables indicate that most data points are concentrated at higher values, with fewer occurrences of lower values, aligning with the quartile measurements listed in Table 2. Variables X8 and X9 also exhibit right-skewed distributions, with histograms showing longer right tails. This aligns with their skewness and kurtosis values, indicating a concentration of data points near a mode with fewer outliers extending toward larger values.

Variables X10 and X11, notable for their extreme positive skewness and high kurtosis, suggest the presence of significant outliers or a non-uniform distribution in which most data points cluster near zero, while some extend to much larger values. Their histograms underscore this distribution pattern, with the majority of data concentrated near zero and sporadic larger values appearing infrequently. Lastly, the output variable Y shows considerable variability and skewness of 1.4619, with its histogram illustrating a broad spread of values, peaking at lower ranges and then tapering off. This pattern matches its high standard deviation and range as reported in Table 2.

The mean plot indicates that most variables have relatively low mean values, except for variable Y, which exhibits a substantially higher mean. This suggests that variable Y operates on a different scale or exhibits distinct behavior compared to the other variables in the dataset. The standard deviation plot shows that variables Y and X13 have significantly higher standard deviations than the other variables. This increased variability indicates a broader range of data points, suggesting more significant fluctuations or greater dispersion around the mean for these variables.

In the skewness and kurtosis plot (Fig. 3), both metrics are displayed for each variable, where skewness (green) measures the symmetry, or lack thereof, of the data distribution and kurtosis (blue) indicates the tendency of the distribution to show heavier tails and more extreme values. Notably, variables X10, X11, X13, X14, and especially Y demonstrate considerable skewness and kurtosis. These values suggest non-symmetric, heavy-tailed distributions for these variables, which are essential considerations for the selection of statistical methods and models, as such distributions may influence the validity of standard statistical tests. The variance plot aligns with observations from the standard deviation data, with variables Y and X13 showing exceptionally high variances. This indicates that data points for these variables span a wider range of values, which could be crucial for applications in predictive modeling and risk assessment.

Fig. 3.

Histogram distribution plot of variables in the dataset.

Figures 4 and 5 present box plots that illustrate the distribution, quartiles, and outlier handling of the input variables used in the dataset. These figures provide insights into the preprocessing steps necessary for statistical analysis and model building.

Fig. 4.

Box plot of the input variables with outliers and quartiles.

Fig. 5.

Box plot of the input variables after handling outliers and normalizing.

Figure 4 shows the original box plot of the input variables with outliers and quartiles. This plot is crucial for understanding the initial state of each variable, revealing the median (central line in the box), interquartile range (IQR, the box itself), and the presence of outliers (indicated by dots outside the whiskers). Several variables, such as X11, X12, and X13, display a significant number of outliers, suggesting that these variables contain extreme values that could potentially skew the analysis. The wide spread in variables like X13 also indicates high variability within the dataset.

Figure 5, on the other hand, presents the box plot of the input variables after handling outliers and normalizing the data. This plot is essential for demonstrating the effects of the preprocessing techniques used to mitigate issues observed in Fig. 4. Normalization adjusts the scales of the variables, bringing them into a comparable range, typically between 0 and 1, which is evident as all the variables in Fig. 5 are confined within these bounds. The normalization process is particularly important when variables are measured in different scales and units, as it prevents any single variable from dominating the analysis due to its scale. Additionally, the reduction in the number of outliers and the more compact interquartile ranges indicate that the data has been standardized, reducing the influence of extreme values.

Figures 6 and 7 present Pearson⁶³ and Spearman correlation⁶⁴ heatmaps, respectively, which are instrumental in identifying the relationships between variables in the dataset. These figures allow for a detailed examination of how variables correlate, revealing potential dependencies and interactions that are vital for further analysis and model building.

Fig. 6.

Pearson correlation heatmap.

Fig. 7.

Spearman rank heatmap.

Figure 6 shows the Pearson correlation heatmap, which measures the linear relationship between variables. In this heatmap, several high correlations are apparent, such as between X3 and X6, and between X7 and X3. Notably, a strong positive correlation exists between the output variable Y and several inputs, such as X3, X6, X7, and X8. This suggests that these input variables increase linearly with Y. On the other hand, features including X10, X11, and X12 are found to negatively impact the output Y linearly. Lastly, feature X9 was found to have a negligible impact on the output Y; however, it was positively correlated with feature X14, so it was retained in the dataset for model training.

Figure 7 presents the Spearman rank correlation heatmap, which measures the monotonic relationship between variables. This method is more robust to non-normal distributions and can capture non-linear relationships that the Pearson method might miss. Similar to the Pearson heatmap, the Spearman heatmap shows significant correlations, such as between X11 and X12 and between X10 and X13. The Spearman correlations are generally more substantial and more widespread across the variables, indicating that many variables exhibit strong monotonic relationships, even when the relationship is not linear. For instance, Y shows a high monotonic correlation with X3 and X8, suggesting that as X3 and X8 increase or decrease, Y tends to move in a predictable direction, regardless of the rate of change.

Overall, the dataset statistical analysis underscores the complexity and multidimensionality of factors affecting the load carrying capacity of composite columns. The identified statistical properties, distribution patterns, and inter-variable correlations provide valuable insights for developing effective machine learning models. By understanding these underlying characteristics, the study ensures that the predictive models are built on a solid foundation, capable of capturing the intricate relationships and variability inherent in composite column designs.

Feature selection

The selection of relevant features is pivotal for the development of effective machine learning models. In this research, an Analysis of Variance (ANOVA) F-test⁶⁵ was employed to identify the most influential features impacting the response variable, thus enhancing model training and performance. The ANOVA F-test discerns features that significantly explain variance within the dataset, which are crucial for accurate predictions. Table 3 details the selected features based on the ANOVA F-test results. This table highlights features alongside their respective F-scores and P-values, providing a quantitative measure of their impact and statistical significance. To avoid information leakage, ANOVA-based feature scoring was computed only on the training partition of each fold and was then applied to the corresponding validation partition.

Table 3.

Selected features based on ANOVA F-test.

Feature	F-score	P-value	Feature	F-score	P-value
X3	3927.2	1.5832 × 10 − 317	X10	84.398	3.147 × 10 − 19
X6	3802.8	9.8025 × 10 − 313	X2	43.38	8.0449 × 10 − 11
X7	3551.1	1.2394 × 10 − 302	X13	34.767	5.3982 × 10 − 9
X8	712.03	7.3333 × 10 − 114	X5	34.406	6.4501 × 10 − 9
X1	156.46	5.0597 × 10 − 33	X4	27.622	1.8754 × 10 − 7
X11	105.52	2.1796 × 10 − 23	X14	19.213	1.3187 × 10 − 5
X12	85.231	2.1468 × 10 − 19	X9	0.92151	0.33736

Figure 8 illustrates these results, elucidating the statistical impact and significance of each feature. This barplot (on left Y-axis) illustrates the F-scores of the features derived from the ANOVA analysis, highlighting the statistical effect of each feature on the response variable. In addition, the barplot (on the right Y-axis) displays the P-values associated with the F-scores from the ANOVA test, demonstrating the statistical significance of the selected features. From the figure, it is clear that variables X3, X6, and X7 possess the highest F-scores, suggesting they are the most significant in explaining the variance in the target variable. This is corroborated by their extremely low P-values, as shown by the ANOVA P-value, which provides strong evidence against the null hypothesis of no effect, thereby confirming their importance in the model. The plots confirm the relevance of features such as X3, X6, and X7, which exhibit the highest F-scores and the lowest P-values, indicating their substantial roles in variance explanation within the model.

Fig. 8.

ANOVA F-Scores and P-value Barplot.

Further insights are gleaned from the Fig. 9 which visualizes the relationships between these significant features and the target variable. This scatter plot aids in identifying the nature of the relationships, whether linear or non-linear, and in spotting potential outliers that could affect the model’s predictive accuracy. Each plot illustrates how changes in feature values correspond to variations in the target variable, thereby aiding understanding of feature relevance and interaction dynamics within the dataset.

Fig. 9.

Scatter plot of top features with Target (Y).

X3 and X6 both demonstrate a strong positive linear relationship with the target variable, as evidenced by the upward trend in the scatter plots. The data points are closely clustered around the trend line, indicating a robust correlation and suggesting that these features are highly predictive of the target variable. The linear nature of these relationships implies that simple linear models could effectively capture and utilize these dependencies for prediction. X7 and X8 show a slightly more scattered distribution, but still maintain a generally positive trend with the target. This indicates some degree of linear correlation, although the presence of outliers or wider spreads suggests that additional data preprocessing or the use of more complex models might be necessary to fully leverage these variables.

X1 and X11 display a less pronounced linear trend with the target, indicating a weaker relationship. In the case of X1, the scatter plot shows a horizontal trend for a wide range of values, implying that X1 might have a limited predictive power on its own. Similarly, X11 presents a very flat slope, suggesting only a marginal change in the target variable with changes in X11, which might limit its utility in predictive models unless combined with other features. X12 and X10 scatter plots exhibit very low variability in target values for a broad range of feature values, and a significant number of outliers, especially for X12. This pattern may complicate the use of these features in straightforward linear predictive models without transformations or robust outlier handling strategies.

X2, X5, and X13 are interesting as they show distinct clusters or groupings of data points. For X2 and X13, there are distinct vertical bands where the target variable changes sharply at specific intervals of the feature values, indicating possible categorical or threshold effects within these features. This could suggest that non-linear models or classification approaches might be more suitable for capturing the predictive dynamics of these features. X14 demonstrates a relatively flat relationship with the target across its range, interspersed with a few outliers. This pattern suggests that X14 may not be very influential in predicting the target variable, as changes in X14 do not correspond to clear or consistent changes in the target. X9 presents a particularly scattered plot with a minimal trend, indicating an almost negligible linear relationship with the target. This implies that X9 has limited predictive value, at least in a linear sense, which is corroborated by its high P-value in the ANOVA test, suggesting that it does not significantly affect the target variable.

Deep learning model

In this study, a Deep Neural Network (DNN)⁶⁶ model is employed as the primary deep learning approach to predict the load carrying capacity of composite columns. The selection of a DNN is motivated by its ability to capture complex, nonlinear relationships within the dataset, which are often inherent in structural engineering applications. The architecture of the proposed DNN comprises multiple hidden layers, each containing a substantial number of neurons to ensure sufficient capacity for learning intricate patterns from the input variables.

The DNN architecture begins with an input layer that receives the fourteen input features (X1 to X14), each representing critical geometric, material, and loading parameters of the composite columns. Following the input layer, the network includes several fully connected hidden layers. Each hidden layer applies a linear transformation to its inputs, followed by a nonlinear activation function, the Rectified Linear Unit (ReLU), as shown in Fig. 10. The activation function introduces nonlinearity into the model, enabling it to learn and represent complex relationships between the input variables and the output variable, Y (load carrying capacity).

Fig. 10.

Architecture of the DNN model employed in the study to predict Y.

Mathematically, the transformation performed by each neuron in a hidden layer can be described by the Eq. (1).

1

where represents the activations of layer , and are the weight matrix and bias vector for layer , respectively, and ReLU is the activation function defined as .

The depth of the network, determined by the number of hidden layers, and the width, determined by the number of neurons per layer, are critical hyperparameters that are optimized to balance model complexity and computational efficiency. To prevent overfitting, regularization techniques such as dropout and L2 regularization are integrated into the model. Dropout randomly deactivates a fraction of neurons during training, which helps in reducing interdependencies among neurons and promotes the learning of robust features. L2 regularization adds a penalty term to the loss function based on the squared magnitude of the weights, thereby discouraging overly complex models (given by Eq. (2)).

2

Where, is the regularization parameter that controls the amount of regularization applied, and is the summation of the squares of the weights, ​, across all weights in the model.

The output layer of the DNN consists of a single neuron that produces the predicted load-carrying capacity, Y. Since the task is a regression problem, the output layer uses a linear activation function to provide continuous output values. The model is trained by minimizing the Mean Squared Error (MSE) loss function, which quantifies the difference between the predicted and actual load-carrying capacities, given by Eq. (3).

3

where is the actual load capacity and ​ is the predicted value for the sample.

To optimize the network parameters, the Adam optimization algorithm is utilized due to its efficiency and ability to handle sparse gradients. Adam combines the advantages of both Adaptive Gradient Algorithm (AdaGrad) and Root Mean Square Propagation (RMSProp), adapting the learning rate for each parameter and accelerating convergence. The training process involves iterating over the dataset for a predefined number of epochs, using mini-batch gradient descent to update the weights incrementally based on subsets of the data.

The methodology also incorporates techniques including early stopping and learning rate scheduling to enhance training performance. Early stopping monitors the validation loss and halts training when no significant improvement is observed, thereby preventing overfitting. Learning rate scheduling dynamically adjusts the learning rate during training, allowing the model to make large updates initially and fine-tune the weights as it approaches convergence.

Once trained, the DNN model’s performance is evaluated using metrics namely the coefficient of determination (R²), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) on a separate validation set. These metrics provide insights into the model’s accuracy and its ability to generalize to unseen data. Additionally, feature importance analysis is conducted to interpret the influence of each input variable on the predicted load carrying capacity, leveraging SHAP (SHapley Additive exPlanations) values.

For comparative analysis, the DNN model’s performance is benchmarked against other machine learning models, including Random Forests, Support Vector Machines, and Gradient Boosting Machines. This comparison highlights the strengths and potential limitations of the DNN approach in the context of predicting structural performance metrics. The overall research methodology flowchart is depicted in Fig. 11.

Fig. 11.

Overall flowchart of the deep learning-based methodology for predicting Y.

Hyperparameter optimization

Hyperparameter optimization was applied to all predictive models considered in this study. In this study, we employ Bayesian Optimization⁶⁷ optimization was used to identify the best-performing hyperparameter settings within the training data of each outer fold, thereby avoiding optimistic bias during model selection.

Bayesian Optimization is a sequential model-based optimization method that builds a probabilistic surrogate model of the objective function and uses it to select the most promising hyperparameters to evaluate in the true objective function. Unlike grid search, which exhaustively explores the hyperparameter space, or random search, which samples hyperparameters randomly, Bayesian Optimization strategically balances exploration and exploitation to identify optimal hyperparameters with fewer evaluations. This efficiency is particularly advantageous given the computationally intensive nature of training DNNs.

The Bayesian Optimization process begins with the selection of a prior distribution over the objective function, typically modeled using a Gaussian Process (GP). The GP serves as the surrogate model, capturing our beliefs about the objective function based on the observed data. Mathematically, the GP is defined by its mean function and covariance function , where and represent different hyperparameter configurations as given by Eq. (4).

4

As hyperparameter evaluations are performed, the GP is updated with the observed performance metrics, refining the surrogate model’s predictions. The acquisition function, such as Expected Improvement (EI) or Upper Confidence Bound (UCB), is then utilized to determine the next set of hyperparameters to evaluate. The acquisition function quantifies the utility of evaluating a new point in the hyperparameter space by balancing the potential for improvement (exploitation) against the uncertainty in the surrogate model’s predictions (exploration), calculated using Eq. (5).

5

where is the best observed value so far. The hyperparameter configuration that maximizes the acquisition function is selected for the subsequent evaluation, and the process iterates until a predefined stopping criterion is met, such as a maximum number of iterations or convergence of the objective function.

In the context of this study, the hyperparameters optimized using Bayesian optimization include number of hidden layers (determines the depth of the DNN, influencing its capacity to capture complex patterns), number of neurons per layer (affects the model’s ability to learn representations of varying complexity), learning rate (controls the step size during the optimization process, impacting convergence speed and stability), dropout rate (mitigates overfitting by randomly deactivating neurons during training), batch size (influences the gradient estimation and training dynamics).

The optimization workflow commenced with an initial exploration phase, where a diverse set of hyperparameter configurations was evaluated to establish a foundational understanding of the objective function. Subsequently, Bayesian Optimization iteratively refined the search based on the surrogate model’s predictions and the acquisition function’s guidance. This method allowed for efficient identification of hyperparameter settings that yielded optimal performance metrics, specifically minimizing the MSE and maximizing the R² on the validation dataset.

To implement Bayesian Optimization, we utilized the scikit-optimize library, which provides robust tools for sequential model-based optimization. The optimization was constrained within reasonable bounds for each hyperparameter to ensure computational feasibility and practical relevance. For instance, the number of hidden layers was restricted between 2 and 5, neurons per layer between 10 and 100, and the learning rate between 1 × 10^− 5 and 1 × 10^− 1.

Model training and validation

The training, tuning, and validation of all predictive models were conducted using a nested grouped cross-validation framework. To achieve a robust evaluation, the study employs a 5-fold cross-validation strategy coupled with comprehensive validation metrics, namely the R², RMSE, MAE, and MAPE.

Model training with 5-fold cross-validation

Model development followed a nested grouped cross-validation workflow to obtain an unbiased estimate of predictive performance. First, the full dataset was divided using an outer 5-fold GroupKFold procedure, where grouping was defined by source study and FE parametric family. In each outer iteration, one grouped fold was held out for testing and the remaining grouped folds were used for model development.

Within each outer-training partition, hyperparameter tuning was carried out using an inner grouped cross-validation loop. The same workflow was applied to all four models considered in this study, namely the MLP neural network, Support Vector Regressor, Random Forest Regressor, and Gradient Boosting Regressor, thereby ensuring a fair comparison. After selecting the best hyperparameter configuration from the inner loop, the corresponding model was retrained on the full outer-training data and evaluated on the untouched outer-test fold.

Final model performance was reported by aggregating the outer-fold results using R2, RMSE, MAE, and MAPE. The actual-versus-predicted plots were generated from out-of-fold predictions collected across the outer test folds. Complete step wise steps is depicted in Fig. 12.

Fig. 12.

Architecture of 5-fold cross-validation.

Mathematically, if we denote the dataset as ​, the 5-fold cross-validation procedure can be described as follows:

1. Partitioning: Divide into five subsets , , , , of approximately equal size.

2. Iteration: For each :

• Use ​ as the validation set.

• Combine the remaining subsets ​ to form the training set.

• Train the DNN model on the training set.

• Evaluate the model on the validation set using the selected metrics.

• 3.Aggregation: Compute the average performance across all five iterations to obtain the final evaluation metrics.

This method ensures that the model is exposed to various subsets of data, thereby mitigating the risk of overfitting and providing a more reliable estimate of its predictive capabilities.

Validation metrics

To comprehensively evaluate the performance of the trained DNN model, four key metrics are employed: R², RMSE, MAE, and MAPE. Each metric offers unique insights into different aspects of the model’s predictive accuracy and reliability.

1. Coefficient of Determination (R²): The R² metric quantifies the proportion of the variance in the dependent variable (load carrying capacity, ) that is predictable from the independent variables (features X1 to X14​). It is given by Eq. (6).

   6

   where ​ is the actual load capacity, ​ is the predicted value, and is the mean of the actual values. An R² value closer to 1 indicates a strong predictive power of the model.

2. Root Mean Squared Error (RMSE): RMSE measures the square root of the average of the squared differences between predicted and actual values. It is given by Eq. (7):

   7

   ​RMSE provides a clear indication of the model’s prediction accuracy, with lower values signifying better performance.

3. Mean Absolute Error (MAE): MAE calculates the average of the absolute differences between predicted and actual values, as given by Eq. (8):

   8

   This metric offers a straightforward interpretation of the average prediction error, making it easy to understand the model’s typical deviation from actual values.

4. Mean Absolute Percentage Error (MAPE): MAPE expresses the prediction error as a percentage, facilitating the comparison of model performance across different datasets or contexts. It is calculated using Eq. (9).

   9

   ​​​MAPE is particularly useful for assessing the model’s performance in terms of relative error, although it can be sensitive to small actual values of .

Implementation and optimization

During the training phase, the DNN model undergoes multiple epochs of learning, adjusting its weights through backpropagation to minimize the chosen loss function i.e., MSE. The hyperparameters optimized using Bayesian Optimization, as previously described, play a crucial role in enhancing the model’s performance by fine-tuning aspects such as learning rate, number of neurons, and dropout rates. All preprocessing operations, including scaling and feature selection, were fit exclusively on the training data within each fold.

To ensure that the model generalizes well to unseen data, early stopping mechanisms and regularization techniques are employed. Early stopping monitors the validation loss and halts training when no significant improvement is observed, thereby preventing overfitting. Regularization methods, such as dropout and L2 regularization, further enhance the model’s ability to generalize by reducing the complexity of the network.

Feature importance

Understanding the relative importance of input features is essential for interpreting the DNN model’s predictions and for gaining insights into the underlying factors that influence the load carrying capacity of composite columns. In this study, we employ SHAP analysis^68,69 to quantify and visualize the contribution of each feature to the model’s output. SHAP is a game-theoretic approach that assigns each feature an importance value for a particular prediction, providing both global and local interpretability of the model.

SHAP values are grounded in cooperative game theory, where each feature is considered a “player” contributing to the “game” of making a prediction. The SHAP value for a feature represents the average marginal contribution of that feature across all possible feature coalitions. Mathematically, the SHAP value for a feature iii is given by Eq. (10).

10

where is the predictive model, is the set of all features, and is a subset of features excluding iii. This equation ensures that the SHAP values satisfy desirable properties such as local accuracy, missingness, and consistency, making them a robust choice for feature importance analysis.

The methodology for applying SHAP analysis in this study involves several key steps. First, the trained DNN model is integrated with the SHAP library, which leverages the model’s prediction capabilities to compute SHAP values for each feature across the dataset. Given the complexity of DNNs, which consist of multiple hidden layers and nonlinear activations, SHAP effectively decomposes the model’s predictions into contributions from each input feature, despite the model’s inherent opacity.

Once the SHAP values are computed, they are aggregated to derive global feature importance metrics. This aggregation involves averaging the absolute SHAP values across all samples, providing a ranking of features based on their overall impact on the model’s predictions. Features with higher average SHAP values are deemed more influential in determining the load carrying capacity. Additionally, SHAP summary plots and dependence plots are generated to visualize the distribution and interaction of SHAP values, offering deeper insights into how specific features affect the predictions across different data points.

Results and discussion

DNN model performance & comparative analysis

The performance of the DNN model, alongside traditional machine learning models, Support Vector Regressor (SVR), Random Forest Regressor (RFR), and Gradient Boosting Regressor (GBR), was meticulously evaluated to predict of composite columns. The comparative analysis of these models is summarized in Table 4, which presents performance metrics employed in the study, namely the R², RMSE, MAE, and MAPE.

Table 4.

Outer-fold predictive performance of different models in predicting Y under nested grouped cross-validation.

Model	R²	RMSE (kN)	MAE (kN)	MAPE (%)
Support Vector Regressor	0.737	4102.71	2111.74	30.51
Random Forest Regressor	0.984	1015.16	538.15	8.95
Gradient Boosting Regressor	0.962	1560.01	899.61	24.03
DNN	0.998	362.83	253.07	6.61

As illustrated in Table 4 and in Fig. 13, the DNN model outperformed the other models across all evaluated metrics. The DNN achieved an impressive R² value of 0.9979, indicating that approximately 99.79% of the variance in the load carrying capacity () is explained by the model. This is substantially higher compared to SVR (R² =0.7367), RFR (R² =0.9839), and GBR (R² =0.9619), underscoring the superior predictive capability of the DNN in capturing complex nonlinear relationships inherent in the data.

Fig. 13.

Model’s evaluation metrics to predict .

In terms of RMSE, the DNN recorded the lowest value of 362.83 kN, signifying minimal average deviation between the predicted and actual values. This contrasts sharply with SVR’s RMSE of 4102.71 kN, RFR’s 1015.16 kN, and GBR’s 1560.01 kN, highlighting the DNN’s enhanced accuracy. Similarly, the MAE for the DNN was 253.07 kN, which is considerably lower than that of SVR (2111.74 kN), RFR (538.15 kN), and GBR (899.61 kN). The DNN also demonstrated the best MAPE of 6.61%, indicating that the predictions are, on average, within 6.61% of the actual values. This metric further confirms the DNN’s reliability and precision in forecasting the load-carrying capacity. The robustness of these findings is visually corroborated by the Actual vs. Predicted plots for each model, depicted in Figs. 14, 15, 16, and 17. Figure 14 showcases the DNN model’s predictions, where the points closely align with the reference diagonal, reflecting the high R² and low error metrics. In contrast, Fig. 15 for SVR reveals a more scattered distribution of points, indicating less accurate predictions and higher error rates. Figures 16 and 17 illustrate the performance of RFR and GBR, respectively, where RFR shows a tighter clustering around the diagonal compared to GBR, yet still not as precise as the DNN.

Fig. 14.

Actual versus DNN model predicted .

Fig. 15.

Actual versus SVR model predicted .

Fig. 16.

Actual versus RFR model predicted .

Fig. 17.

Actual versus GBR model predicted .

Furthermore, a Radar plot of Fig. 18 provides a holistic comparison of the models’ performances across all metrics. The DNN’s profile dominates the radar, extending significantly in all directions, which visually reinforces its superior performance. The radar plot effectively highlights the DNN’s balanced excellence across R², RMSE, MAE, and MAPE, whereas SVR’s limited extension underscores its comparatively weaker performance.

Fig. 18.

Radar plot of all the models’ performance across metrics.

Additionally, Fig. 19, the Taylor plot, offers another dimension of comparative analysis by simultaneously displaying the correlation coefficient, centered RMS error, and the standard deviation of each model’s predictions against the observations. The DNN’s plot point is positioned closest to the ideal point, indicating high correlation and minimal error, further validating its exceptional performance relative to SVR, RFR, and GBR.

Fig. 19.

Taylor plot of model’s performance comparison.

In summary, the DNN model not only surpasses traditional machine learning models in predictive accuracy but also demonstrates superior generalization capabilities, as evidenced by the comprehensive set of evaluation metrics and visualizations. The integration of advanced neural architectures enables the DNN to effectively model the intricate dynamics of composite column load capacities, making it a highly valuable tool for structural engineering applications.

Hyperparameter optimization

Optimizing hyperparameters is a crucial step in enhancing the performance of machine learning models, as it fine-tunes the algorithms to better capture the underlying patterns in the data. In this study, Bayesian Optimization was employed to systematically identify the most effective hyperparameter configurations for each of the selected models: Support Vector Regressor (SVR), Random Forest Regressor (RFR), Gradient Boosting Regressor (GBR), and the Deep Neural Network (DNN) represented by the Multilayer Perceptron (MLP) Regressor. The optimized hyperparameters for each model are consolidated in Table 5.

Table 5.

Optimized hyperparameters for various models.

Model	Parameter	Best value
Support Vector Regression	C	100
	epsilon	1
	gamma	scale
	kernel	linear
Random Forest Regressor	n_estimators	500
	max_depth	15
	min_samples_split	2
	min_samples_leaf	1
	max_features	sqrt
Gradient Boosting Regressor	n_estimators	377
	learning_rate	0.105651
	max_depth	4
	min_samples_split	9
	min_samples_leaf	3
	max_features	log2
MLP Regressor (DNN)	hidden_layer_sizes	100
	activation	relu
	solver	adam
	alpha	0.01
	learning_rate	constant
	learning_rate_init	0.01
	max_iter	1500

As depicted in Table 5, each model’s hyperparameters were meticulously tuned to achieve optimal performance.

For the SVR, the hyperparameter C was set to 100, indicating a higher penalty for misclassified samples, thereby promoting a model that fits the training data more closely. The epsilon parameter was optimized to 1, which defines the margin of tolerance where no penalty is given to errors. A gamma value of scale allows the algorithm to automatically adjust the influence of each feature based on the data, while the choice of a linear kernel suggests that the relationship between the input features and the target variable is predominantly linear.

The RFR benefited from a substantial number of trees, with n_estimators set to 500, enhancing the ensemble’s ability to generalize by reducing variance. A max_depth of 15 strikes a balance between underfitting and overfitting, allowing trees to capture intricate patterns without becoming overly complex. The minimal values for min_samples_split and min_samples_leaf (2 and 1, respectively) ensure that the trees are flexible enough to model the data accurately. Additionally, setting max_features to sqrt limits the number of features considered for splitting, promoting diversity among the trees and improving model robustness.

In the case of the GBR, the n_estimators was optimized to 377, indicating a sufficient number of boosting stages to minimize residual errors effectively. The learning_rate of 0.105650842 moderates the contribution of each tree, preventing overfitting while ensuring steady convergence. A shallow max_depth of 4 restricts the complexity of individual trees, fostering a model that captures essential patterns without becoming excessively intricate. The parameters min_samples_split and min_samples_leaf were tuned to 9 and 3, respectively, providing a balance between model flexibility and generalization. Setting max_features to log2 further enhances the model’s ability to generalize by considering a logarithmic number of features at each split.

The DNN showcased a robust architecture with a single hidden layer comprising 100 neurons. The ReLU activation function (activation) introduces non-linearity, enabling the network to model complex relationships effectively. Utilizing the Adam optimizer (solver) facilitates efficient training by adapting the learning rate based on first and second moments of the gradients. The regularization parameter alpha was set to 0.01 to prevent overfitting by penalizing large weights. A constant learning rate (learning_rate) with an initial value of 0.01 ensures a steady and controlled update of weights during training. Finally, a generous max_iter of 1500 iterations guarantees sufficient training epochs for the network to converge to an optimal solution.

The optimization process, leveraging Bayesian Optimization, systematically explored the hyperparameter spaces to identify configurations that minimize the MSE and enhance the model’s predictive accuracy. This method proved to be highly effective, as evidenced by the superior performance metrics attained by each model post-optimization, particularly the DNN model. The chosen hyperparameters not only improved individual model performances but also contributed to the overall robustness and generalizability of the predictions.

Feature importance analysis

Understanding the influence of various features on the predictive models is crucial for interpreting model behavior and enhancing decision-making processes. In this study, SHAP values were employed to determine the impact of each feature on the output of the GBR and RFR. The SHAP summary plots for both models are illustrated in Figs. 20 and 21, providing a visual interpretation of feature contributions across individual predictions.

Fig. 20.

SHAP summary plot of GBR model.

Fig. 21.

SHAP summary plot of the RFR model.

Gradient boosting regressor feature importance

Figure 20 presents the SHAP summary plot for the Gradient Boosting Regressor. This visualization clearly depicts the distribution of the impact each feature has on the model output, with the color scale indicating feature value (red high, blue low). It is evident from the plot that X3 (Perimeter of the column cross-section) and X8 (Thickness of the outer steel tube) are the most influential features, showing substantial positive and negative impacts on the model output, respectively. High values of X3 tend to increase the predicted load carrying capacity significantly, which is intuitive as larger perimeters typically suggest greater structural strength.

Conversely, increased values of X8 generally lead to a decrease in the load carrying capacity. This might indicate that thicker outer tubes could be introducing stiffness that does not proportionally contribute to bearing more load under the conditions simulated. Other significant features such as X1 (Compressive strength of standard concrete cylinders) and X6 (Base of the column cross-section) also show notable variability in their effects, suggesting complex interactions between the material properties and geometric dimensions of the columns.

Random forest regressor feature importance

Figure 21 displays the SHAP summary plot for the Random Forest Regressor. Similar to the GBR, X3 appears as a predominant feature, reinforcing the importance of the perimeter in determining the structural capacity of the columns. The plot also highlights the significant role of X7 (Depth of the column cross-section) and X6, with high values generally correlating with increased predictive outputs. This aligns with structural engineering principles where deeper and broader columns can support greater loads.

In contrast to the GBR, the RFR demonstrates a more distributed influence across features, indicating a model that relies on a broader array of inputs to make predictions. This behavior is typical of Random Forest models, which integrate diverse decision paths across many decision trees, thereby utilizing a broader range of feature information.

The comparison between Figs. 20 and 21 reveals distinct patterns of feature importance between the two models, underscoring the different mechanisms by which these algorithms process the same input data. While GBR shows concentrated effects from fewer features, RFR distributes importance more evenly among a larger set of features. This distinction can guide practical applications: for scenarios where specific features are known to be more reliable or easier to measure, GBR might be preferable due to its focused sensitivity. Conversely, RFR might be more suitable for applications requiring robustness to variations across a broader set of features.

In summary, the SHAP analysis provides a compelling narrative about how different features drive complex models’ predictions in the context of structural engineering. By understanding these dynamics, engineers and researchers can better design experiments, interpret model predictions, and make informed decisions about structural design and assessment. The insights drawn from these analyses not only enhance trust in the machine learning models but also enrich our understanding of the fundamental factors influencing composite column behavior.

Graphical user interface

The Graphical User Interface (GUI) of the application, as depicted in Fig. 22, serves as a user-friendly platform enabling users to easily input parameters, predict outcomes, and visualize the structure of the data inputs and outputs. This section delves into the design and functionality of the GUI, emphasizing its integral role in facilitating the practical application of the machine learning model developed to predict the load carrying capacity of composite columns.

Fig. 22.

Screenshot of the graphical user interface.

The GUI layout is divided into two main sections: the input panel on the left and the information panel on the right. The input panel consists of a series of text fields where users can enter values for each of the 14 features (X1 to X14) that influence the prediction model. These features include compressive strength of standard concrete cylinders, height of the column, perimeter of the column cross-section, and other geometric and material properties that are critical in calculating the column’s load carrying capacity.

Each input field is clearly labeled with the corresponding variable name and unit (e.g., X1 (MPa), X2 (mm)), which not only aids in data entry but also reduces potential errors due to unit mismatches or misinterpretations. After inputting the data, the user can initiate the prediction by clicking the ‘Predict’ button. This triggers the backend predictive workflow to process the inputs using the same preprocessing pipeline adopted during model development and then pass them through the trained optimized MLP neural-network model selected as the best performer in the comparative analysis to generate the predicted load-carrying capacity (), which is displayed in a popup message box.

For transparency, the comparative benchmark models remain available in the archived code package, but the default GUI prediction corresponds to the final selected MLP model reported in Tables 4 and 5.

The information panel, shown on the right side of Fig. 22, provides a detailed overview of each variable used in the model. It lists the variable name, unit, and a brief description, thereby educating the user about the nature of each input and its relevance to the prediction task. This section is particularly useful for new users or stakeholders who may not be familiar with the specific terminologies or the structural engineering domain, as it enhances transparency and understanding of how the model makes its predictions.

Upon submission of the input values, the prediction result is immediately displayed in a dialog box, as seen in the screenshot where the predicted load carrying capacity is shown. The prediction, expressed in kilonewtons (kN), is based on the user-provided inputs, processed through the predictive model. This instant feedback loop makes it practical for engineers or decision-makers to use the application in real-time scenarios, such as during planning stages of construction projects or in educational settings for instructional purposes.

The GUI is intended as a practical implementation tool and does not replace the methodological details presented in Sect.  2.1–2.5. One can reproduce the prediction workflow directly from the manuscript through the reported dataset variables, preprocessing procedure, DNN architecture, hyperparameter optimization settings, and validation protocol. To further support transparency and reuse, the developed GUI, trained model files, and source code are provided through the Code and GUI Availability statement.

Comparison with design codes and literature

Table 6 lists the equations and proposed formulas from the literature, and Table 7 compares the present ML study with design codes. The comparison revealed that ML models can predict all shapes of polygon CFST, implying that the best method for designing CFST considering all polygon shapes is with DNN, which is the best predictor among other ML models. Finite Element (FE) analysis is the second-best predictor available for all shapes of polygon CFST, with an R² range of 0.9112 to 1 (it returned 1 due to the short sample quantity). All of the suggested equations predict polygonal shapes with high accuracy for each type, with R² greater than 0.8, except for EC4, for which Hilo et al.⁵⁵ reported R² values between 0.7699 and 0.7868 for hexagonal, heptagonal, and octagonal CFST, and the equation proposed by Yu et al.⁵², which yielded R² values of 0.5362 and 0.7174 for square CFST.

Table 6.

Equations of design codes and literatures.

Codes	Shape	Equation
EC45	General CFST	(11)
EC45	Octagonal CFST short column	(12) (13) (14) where, the relative slenderness ratio is
GB5093670	General CFST	(15) (16) (17) (18)
ACI318-99 and AISC 360 − 165,71	General CFST	(19)
AISC 360 − 166	Octagonal CFST short column	(20) , where (21) (22) (23) (24)
DBJ72	General CFST	(25) (26) (27)
AIJ-200873	General CFST	(28)
Equation 10 of Ding et al.26	Octagonal CFST	(29)
Equation 14 of Ding et al.29	Hexagonal CFST	(30)
Equation of Yu et al.59	Solid, Holow, Circle, and Polygonal CFST	(31) where, (32)
Equation of Patel et al.25,74	Octagonal CFST	(33) where, (34)
Equation Li et al.30	Rectangular CFST	(35) (36) (37) (38)
Equation of Ahmed and Liang40	Octagonal CFST short column	(39)
Equation Nic J. Chen and T. M. Chan8	Octagonal CFST	(40)
Equation J. Chen and T. M. Chan8	Octagonal CFST	(41)
Equation Zhu and Chan60	Octagonal CFST	(42) where, is insribed diameter of octagonal section; and are the modification factors based on the inscribed circularsection of an octagonal section.
Equation Liu et al.60,75	Octagonal CFST	(43) (44) where, is confining pressure and is hoop stress in steel tube.
Equation Ramadan et al.28	Hexagonal CFST	(45)
Equation Nu with fscy modified MOHURD28,76	Hexagonal CFST	(46) (47) (48) (49) (50)
Equation Nu with fscy modified Han et al.28,77	Hexagonal CFST	(51) (52) (53) (54) (55) (56) (57)
Equation Nu with fscy modified Shen et al.28,31	Hexagonal CFST	(58) (59) (60) (61) (62)

Table 7.

Comparison ML models with design codes and literatures.

No.	Authors	Shape of CFST	Type of comparison	FEM description	R 2	RMSE (kN)	MAE (kN)	MAPE (%)
1	Present study	All shapes of polygon CFST	SVR	–	0.7367	4102.71	2111.74	30.51
			RF		0.9839	1015.16	538.15	8.95
			GB		0.9619	1560.01	899.61	24.03
			DNN		0.9979	362.83	253.07	6.61
2	Ren et al.34	Triangular CFST	DBJ13-51-2010 – EXP	–	0.8498	98.40	76.50	5.83
			Eurocode 4 – EXP		0.7543	79.08	71.17	5.21
3	Zhu and Chan49	Triangular CFST	FE – EXP	Observed by themselves or the authors of the work.	1	60.12	43.50	3.60
4	Liu and Williamns36	Square CFST	FE – EXP	Observed by Ayough9	0.9999	155.49	144.60	6.69
			AISC 360 – EXP	-	0.9314	1270.23	863.33	29.26
			Eurocode 4 – EXP		1	119.39	119.33	6.33
			ACI – EXP		1	278.88	262	12.25746
5	Huang et al.37	Square CFST	FE – EXP	Observed by Ayough9	The sample is only one available and not possible to count R2.
6	Schneider35	Square CFST	FE – EXP	Observed by Ayough9	0.9880	109.04	83.76	5.99
7	Zhu and Chan39	Square CFST	Eurocode 4 – EXP	–	0.9963	308.74	260.14	4.73
			ACI – EXP		0.9964	701.97	601.14	10.72
8	Zhong55	Square CFST	Eq. M. Yu et al.59 – EXP	–	0.5362	313.37	245.89	10.83
9	Liu57	Square CFST	Eq. M. Yu et al.59 – EXP	–	0.8787	194.24	164.50	7.42
10	Liu et al.56	Square CFST	Eq. M. Yu et al.59 – EXP	–	0.9949	98.04	81.13	3.87
11	Zhang et al.56	Square CFST	Eq. M. Yu et al.59 – EXP	–	0.9436	248.62	218.59	11.79
12	Yu et al.59	Square CFST	Eq. M. Yu et al.59 – EXP	–	0.7174	90.82	76.33	4.84
13	Zhu et al.60	Square CFST	AISC 360 – EXP	–	0.9974	1225.85	1201.75	38.60
			Eurocode 4 – EXP		0.9974	1581.80	1529.25	48.38
14	Hilo et al.62	Square CFST	Eurocode 4 – FE	Parametric study of FEM	0.8206	134.02	105.03	7.80
15	Ayough et al.9	Square, Hexagon, & Octagon CFST	AISC 360 – EXP	–	Comparison with literatures is only show the mean of Nu formula/Nu EXP.
			Eurocode 4 – EXP
16	Li et al.30	Rectangular CFST	FE – EXP	Observed by Ami and Patel31	0.9735	78.43	64.75	3.96
			AISC 360 – EXP	–	0.9615	571.62	568.50	34.17
			GB50936 – EXP		0.9598	240.69	234.05	13.94
			CECS 28:90 - EXP		0.9682	205.93	200.13	12.01
			Eq. Li et al.30 – EXP		0.9506	61.44	55.83	3.48
17	Hilo et al.62	Rectangular CFST	Eurocode 4 – FE	Parametric study of FEM	0.8080	247.48	232.44	17.40
18	Hilo et al.62	Pentagon CFST	Eurocode 4 – FE	Parametric study of FEM	0.8047	205.81	186.08	13
19	Ding et al.29	Hexagon CFST	FE – EXP	Observed by themselves or the authors of the work.	0.9320	289.22	278.13	4.54
			Eq. M. Yu et al.59 – EXP	–	0.9261	491.50	419.88	7.08
			Eq. Ding et al.29 – EXP		0.9263	513.81	445.75	7.49
20	Ding et al.29	Hexagon CFST	FE – EXP	Observed by Ami and Patel31	0.9805	195.20	155.75	2.57
21	Ding et al.29	Hexagon CFST	FE – EXP	Observed by Ayough9	0.9112	444.75	382.16	6.07
22	Xu et al.47	Hexagon CFST	FE – EXP	Observed by themselves or the authors of the work.	0.9122	99.71	73.50	4.25
			DBJ13-51-2010 – EXP	–	0.9008	108.58	83.83	4.54
			Eurocode 4 - EXP		0.8957	86.25	74.83	4.33
			AIJ 2008 – EXP		0.8969	162.58	141.83	7.51
23	Liu et al.48	Hexagon CFST	FE – EXP	Observed by Ahmed and Liang50	Not possible to count R2.
24	Zhu and Chan49	Hexagon CFST	FE – EXP	Observed by themselves or the authors of the work	1	26	26	1.38
			FE – EXP	Observed by Ahmed and Liang50	1	31.02	29	1.54
			AISC 360 – EXP	–	Not possible to count R2.
			GB50936 – EXP		Not possible to count R2.
			Eurocode 4 - EXP		Not possible to count R2.
25	Shen et al.52	Hexagon CFST with/without inner steel tube	FE – EXP	Observed by themselves or the authors of the work.	0.9851	582.02	393.46	4.83
26	Ramadan et al.28	Hexagon CFST with/without inner steel tube	FE – EXP	Observed by themselves or the authors of the work.	0.9887	271.75	235.20	5.15
27	Ramadan et al.28	Hexagon CFST	Nu with fscy modified MOHURD76 – FE	Parametric study of FEM	0.9659	244.53	170.71	8.03
			Nu with fscy modified Han et al.77 – FE		0.9919	268.35	212.58	9.90
			Nu with fscy modified Shen et al.33 – FE		0.9961	61.76	48.54	2.63
			Ramadan et al.28 – FE		0.9946	172.00	149.25	7.84
28	Hilo et al.62	Hexagon CFST	Eurocode 4 – FE	Parametric study of FEM	0.7868	240.55	222.28	15.20
29	Hilo et al.62	Heptagon CFST	Eurocode 4 – FE	Parametric study of FEM	0.7832	256.58	239.40	16.20
30	Ding et al.26	Octagon CFST	FE – EXP	Observed by themselves or the authors of the work.	0.9868	501.08	469.25	4.22
			FE – EXP	Observed by Hassanein et al. (2018)	0.9764	490.71	427.88	3.87
			FE – EXP	Observed by Ayough9	1	691.77	591.20	5.68
			Eq. Ding et al.26 – EXP	–	0.9881	1234.93	1197.25	10.46
			Eq. M. Yu et al.59 – EXP		0.9883	1238.86	1225.75	11.06
31	Tomii et al.24	Octagon CFST	FE – EXP	Observed by Ding et al.26	0.9231	33.36	33.33	3.00
			FE – EXP	Observed by Ayough9	0.9440	48.34	47.48	4.59
			FE – EXP	Observed by Hassanein et al.25	0.9714	43.10	39.75	3.92
			Eq. Ding et al.26 – EXP	–	0.9478	214.52	212.00	18.88
			Eq. M. Yu et al.59 - EXP		0.9490	58.62	52.33	4.73
32	Fang et al.38	Octagon CFST	FE - EXP	Observed by themselves or the authors of the work.	0.9978	51.34	43.45	2.25
33	Zhu and Chan39	Octagon CFST	Eurocode 4 - EXP	–	0.9984	605.10	604.57	17.41
			ACI - EXP		0.9984	868.31	862.71	24.37
34	Ahmed and Liang40	Octagon CFST	Ahmed and Liang50 - FE	Parametric study of FEM	0.9908	1497.27	1161.33	4.77
35	Chen and Chan8	Octagon CFST	Eurocode 4 - EXP	–	0.9948	243.10	215.60	6.14
			Nu ic - EXP		0.9937	110.60	83.20	2.27
			Nu cc - EXP		0.9928	111.93	87.45	2.38
36	Hassanein et al.25	Octagon CFST	Eq. Ding et al.26 - FE	Parametric study of FEM	0.9910	720.02	559.60	3.44
			Eq. M. Yu et al.59 - FE		0.9754	3453.61	2986.69	16.97
			Eq. Patel et al., 2016 - FE		0.9905	2344.07	2093.38	12.31
37	Fang et al.42	Octagon CFST	FE - EXP	Observed by themselves or the authors of the work.	0.9937	133.69	118.50	3.24
38	Manigandan et al.43	Octagon CFST with/without inner steel tube	FE - EXP	Observed by themselves or the authors of the work.	0.9938	37.97	36.75	4.51
39	Alqawzai et al.41	Octagon CFST with/without inner steel tube	Parametric study of FEM and not compared to design codes/proposed equations of literatures
40	Ami and Zahrai51	Octagon CFST with/without inner steel tube	Parametric study of FEM and not compared to design codes/proposed equations of literatures
41	Zha53	Octagon CFST	Eq. M. Yu et al.59 - EXP		Not possible to count R2.
42	Zhang et al.54	Octagon CFST	Eq. M. Yu et al.59 - EXP	–	0.9958	376.72	346.82	13.63
43	Zhu et al.60	Octagon CFST	AISC 360 - EXP		0.9830	888.67	865.57	22.96
			Zhu and Chan39 - EXP		0.9832	381.03	341.57	8.65
			Liu et al.75 – EXP		0.9829	255.21	222.86	6.76
44	Hilo et al.62	Octagon CFST	Eurocode 4 - FE	Parametric study of FEM	0.7699	327.96	313.27	20.40
45	Almamoori et al.27	Triangular, Square, Rectangular, Rhombic, Pentagon, Hexagon, and Octagon CFST	Not compared to design codes/proposed equations of literatures
46	Bai et al.32	Square CFST	Not compared to design codes/proposed equations of literatures
47	Samir et al.61	Hexagon, Square, & Triangle CFST	Not compared to design codes/proposed equations of literatures
48	Hassanein et al.45	Hexagon	Not compared to design codes/proposed equations of literatures
49	Evirgen et al.46	Hexagon CFST	Not compared to design codes/proposed equations of literatures

Conclusions

This study developed and compared four data-driven models for predicting the axial capacity of polygon-type concrete-filled steel tubular columns using a dataset of 835 samples collected from 41 studies. Among the evaluated models, the DNN delivered the best overall performance under the adopted validation framework, with , RMSE = 362.83 kN, MAE = 253.07 kN, and MAPE = 6.61%. These results were substantially better than those of SVR, RFR, and GBR, which confirms the stronger ability of the DNN to capture the nonlinear interaction among geometric, material, and loading variables.

The interpretation analysis further showed that the perimeter of the column cross-section and the thickness of the outer steel tube were among the most influential predictors, while section depth and base dimensions also made strong contributions to capacity estimation. The comparison with existing design-code equations and published formulas showed that the proposed framework provides broader applicability across polygonal CFST shapes, where conventional provisions remain limited or shape-specific. The developed GUI translates the final optimized model into a practical prediction tool for engineering use. Overall, the study demonstrates that the proposed DNN-based framework provides an accurate and practical approach for axial-capacity prediction of polygon-type CFST columns.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Conceptualization, Supervision, Software, Resources, Data Curation, Writing – review & editing: H.F.I., M.K.; Conceptualization, Project Administration, Formal Analysis, Validation, Methodology, Investigation, Visualization, Writing – original draft: H.F.I., Y.C., B.R.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data availability

All data generated or analysed during this study are included in this published article and its supplementary information files. The source code, trained DNN model, preprocessing files, and graphical user interface (GUI) developed in this study are openly available in the Zenodo repository at https://doi.org/10.5281/zenodo.19312445.

Code availability

The source code, trained DNN model, preprocessing files, and graphical user interface (GUI) developed in this study are available at 10.5281/zenodo.19312445. The version associated with this manuscript can be accessed by all readers.

Declarations

Competing interests

The authors declare no competing interests.