Rectangular Concrete-Filled Steel Tube Rational Dimensions under Uniaxial Eccentric Compression

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1. Introduction The paper investigates rectangular concrete filled steel tube (CFST) columns under uniaxial eccentric compression. The widespread use of such structures in modern construction is explained by a number of key advantages: the synergistic effect of the combined resistance of the steel tube and concrete filling [1], increased fire resistance [2], high energy capacity under dynamic loads and economic efficiency at all stages of construction and operation of the structure. Due to their high spatial rigidity and ability to resist combined loads, these structures are widely used in construction as load-bearing elements both in Russia [3] and abroad [4]. However, their widespread implementation faces serious difficulties due to the lack of reliable calculation methods that adequately take into account the complex nature of the interaction between the steel shell and concrete filling under various types of loading. The foundation of modern research in the field of CFST column analysis is experimental data obtained during field and laboratory tests [5; 6]. This data serves as the basis for the development of more accurate analytical methods that take into account the nonlinear behavior of materials [7]. For a detailed stress analysis, researchers widely use numerical modeling, such as the finite element method. To work effectively with growing amounts of information, specialized databases are created to systematize experimental results [8-10]. Since the start of the 21st century, the use of artificial intelligence and machine learning has been actively developing. These technologies are applied to predict the load-bearing capacity of structures and optimize their parameters. Taken together, all these areas allow to comprehensively study the behavior of composite structures and develop improved calculation methods. The development of machine learning methods for designing CFST columns is largely driven by fundamental limitations of both normative and numerical calculation methods. On one hand, traditional normative approaches (Eurocode 4[2], AISC 360-16[3], SP 266.1325800.2016[4]), despite their widespread use in design practice, have significant drawbacks: they are only applicable to a narrow range of material characteristics, do not allow for reverse design tasks, are limited to certain types of cross-sections, and do not take into account complex loading cases. On the other hand, numerical methods implemented in existing finite element software (ABAQUS, ANSYS, SCAD, LIRA, etc.), although they provide more accurate modeling of nonlinear behavior of structures [11], require significant computing resources, labor-intensive model calibration, and are insufficiently effective for practical use in design work [12; 13]. These limitations of the traditional approaches have created the conditions for the introduction of innovative analysis technologies, in particular artificial neural networks and other artificial intelligence methods capable of adequately accounting for complex nonlinear interactions in structures. An artificial neural network for calculating the load-bearing capacity of square CFST columns under axial compression was first used in a study by H. Gao [14], where a three-layer feedforward network trained on experimental data showed high prediction accuracy when validated on an independent sample. This study confirmed the fundamental possibility of using neural network models as an effective auxiliary tool for engineering calculations. Subsequent studies [15-19] significantly expanded the scope of application of artificial neural networks for CFST column analysis. These papers present improved neural network architectures trained on both experimental data and numerical simulation results. Alongside the development of neural network approaches, significant progress has been made in the field of gradient boosting algorithms, which demonstrate comparable accuracy with greater computational efficiency and interpretability of results. In [20], a comparative study was conducted on the accuracy of predicting the load-bearing capacity of reinforced concrete columns using five machine learning algorithms: AdaBoost, GBR, XGBoost, LightGBM, and CatBoost. The analysis used a database of experimental data, including, in particular, 401 tests of rectangular eccentrically compressed columns, containing the following parameters: geometric characteristics (cross-sectional dimensions, wall thickness, element length), physical and mechanical properties of materials, eccentricity values, and ultimate load-bearing capacity values. Statistical analysis confirmed the representativeness of the data for a wide range of parameters. This database was previously validated by Thai et al. [21] for the evaluation of normative methods (AISC 360, Eurocode 4, AS/NZS 2327[5]), which attests to its reliability. A comparative analysis showed that CatBoost provides the highest prediction accuracy for eccentrically loaded rectangular columns, while LightGBM demonstrates slightly lower accuracy, but outperforms it in training speed by 1.5-2 times. Although the LightGBM algorithm is characterized by high training speed when working with large amounts of data due to the optimization of the tree construction process and supports distributed computing, CatBoost was preferred in this study. The key factor in the choice was the need to simultaneously predict two interrelated geometric parameters: the height and width of the crosssection of CFST columns. Unlike LightGBM, CatBoost has built-in mechanisms for multidimensional regression, which eliminates the need to create complex user-defined loss functions. An important advantage of CatBoost is its resistance to overfitting, provided by a combination of ordered boosting and automatic L2 regularization, which is especially important when working with synthetic data. Thus, despite the absence of categorical features in the analyzed data and the potential advantages of LightGBM in processing speed, the choice of CatBoost proved to be methodologically reasonable, as it best suits the specifics of the problem at hand, given that for the dataset under consideration, the gain in processing speed is not a determining factor, while the advantages of CatBoost in terms of accuracy and prediction stability become of primary importance. It is also worth noting that despite the significant number of studies devoted to CFST columns, most of them focus on verification calculations in determining the load-bearing capacity [22; 23], while issues related to determining the required cross-sectional dimensions remain insufficiently studied. Moreover, while a few studies do address design tasks, they are mainly limited to cases of central compression of round columns, while the issues of optimal design of rectangular CFST columns under eccentric loading are practically not considered, despite the importance of such calculations. In this regard, it seems particularly relevant to develop new approaches to solving these problems for rectangular cross-sections under eccentric compression. This article is a continuation of the research [24-28] aimed at obtaining the most convenient and reliable tools for performing verification and design calculations for CFST columns. The model proposed in this article can be used for automated design of CFST columns, reducing calculation time and increasing the economic efficiency of structural solutions. The results of the study are of interest to structural designers involved in the development and implementation of CFST columns, as well as for further scientific research in the field of optimization design of these structures. 2. Methods Short columns are considered, for which deformations are small and do not lead to a significant change in the eccentricity of the longitudinal force. The following values were taken as input parameters for the machine learning model: 1) Concrete compressive strength class B, MPa according to the Interstate Standard in force in Russia GOST 18105-2018[6]; 2) Compressive force magnitude F, kN; 3) Tube wall thickness t, mm; 4) Compressive force eccentricity e, mm; The output parameters of the model are width b, mm and height h, mm of the cross-section. Synthetic data was generated to train the model. The concrete class ranged from B15 to B80. The compressive load varied from 500 to 10000 kN in increments of 100 kN, the wall thickness varied from 3 mm to 22 mm in increments of 1 mm, and the eccentricity of the longitudinal force varied from 10 to 250 mm in increments of 10 mm. The selection of these concrete class ranges was determined by the possibility of purchasing concrete mixes with the declared strength from mass producers (ultra-highstrength concrete and low-grade concrete were not considered). The selected wall thickness range is determined by the rolled steel sections available in the Russian Federation. The minimum eccentricity value of 10 mm in the training set corresponds to the minimum value of random eccentricity according to SP 63.13330.2018[7]. The upper eccentricity limit of 250 mm is justified by the fact that at large eccentricities of the longitudinal force, when the structure resists mainly bending rather than compression, the use of CFST structures becomes irrational. For each set of values , , , the problem of determining the optimal values of dimensions and ℎ was solved under minimum cost of the structure taking into account the requirement to satisfy the strength condition. Since most rectangular tube sections manufactured for the Russian market are made from grade 09G2S steel, yield strength Ry of the tube material is assumed to be constant. For this grade of steel, it averages 345 MPa. The cost of 1 linear meter of the structure is determined by the formula: S = S bhb + Ssρs 2(b+ h t) → min , (1) where Sb is the cost of 1 m3 of concrete; Ss is the cost of 1 tonne of steel; ρs = 7.85t/m³ is the density of steel. The value of Ss was assumed to be 86800 rubles per tonne as an average value for tube sections of grade 09G2S steel according to the data of Metallotorg JSC[8] as on 27.03.2025 for the city of Rostov-onDon, Russian Federation. The value of Sb depends on the concrete class. Table 1 contains the values of Sb for the considered concrete classes in this study according to the data of Astragal company[9]. This table also shows the values of the design strength of concrete depending on its class based on the Russian code SP 63.13330.2018[10]. Table 1. The cost of 1 m3 of concrete depending on its compressive strength class, as well as the design compressive strength Concrete class B The cost of 1 m3 of concrete, Sb, rub Design compressive strength Rb , MPa B15 5300 8.5 B20 5700 11.5 B25 6250 14.5 B30 6650 17 B35 6900 19.5 B40 7700 22 B45 8700 25 B50 9200 27.5 B55 10200 30 B60 10800 33 B70 12700 37 B80 13400 41 S o u r c e: made by S.H. Al-Zgul. For designing the cross-section, it is necessary to write the strength condition of the CFST element under eccentric compression. This article considers the case when bending moment acts only in one plane. The effect of strength increase due to the resistance of concrete in confined conditions is not taken into account. The wall thickness of the tube is assumed to be small compared to the cross-sectional dimensions b and h. The external dimensions of the column are considered approximately equal to the dimensions of the concrete core. The limit equilibrium method is used to determine the ultimate load under eccentric compression. Previously, this method for CFST columns was validated using experimental data in [29]. The stress in the compressed zone of the concrete core is assumed to be equal to Rb . The stress in the tension zone of the concrete core is not taken into account. The stresses in the compression and tension zones of the steel tube are assumed to be equal to Ry with the corresponding signs (Figure 1). Figure 1. Diagram for determining the maximum load S o u r c e: made by S.H. Al-Zgul. The equation of equilibrium between internal and external forces projected onto the longitudinal axis of the column is as follows: F = R bb y0 + 2R ty y0 + R tby - R tby -2R t hy ( - y0). (2) From (2), parameter is expressed as: F +2R thy R bb +4R ty y0 = . (3) The equation of equilibrium between internal and external moments with respect to the x-axis in the limit state is written as: Fe = R byb 0 y0 - y0 - h + 2R tby h + 2R tyy 0 y0 - y0 - h + 2 2 2 2 2 + 2R t hy ( - y0) y0 - +h h- y0 . 2 2 Or, after simplification: (4) Fe =1 R byb 0 (h- y0)+ R t bhy ( +2y0 (h- y0)). (5) 2 Substituting (3) into (5) yields a quadratic equation with respect to , from which ultimate load Fult is determined as the minimum positive root of this equation satisfying the condition: Fult +2R thy R bb +4R ty y0 = ≤ h. (6) If none of the positive roots of the quadratic equation satisfy condition (6), this indicates that the compressed zone covers the entire concrete core and the neutral line runs either at the boundary between the concrete core and the lower side of the tube (Figure 2) or directly at the lower side of the tube. These cases are possible when the longitudinal force has a small eccentricity. Figure 2. The case when the compressed zone covers the entire concrete core S o u r c e: made by S.H. Al-Zgul. For the case shown in Figure 2, the ultimate load is determined by the formula: Fult2 = R bhb +2R hty . (7) If the neutral line is inside the lower side of the pipe, then Fult will take an intermediate value between Fult2 and Fult0, corresponding to the case of central compression: Fult0 = R bhb + 2Ry (h+b t) . (8) If when calculating it was obtained that y0 >h,the ultimate load was determined using formula (7), which was included in the factor of safety. The problem of finding the values of b and h from the minimum condition of cost function S limited by F≤Fult was solved as a nonlinear optimization problem using the interior point method [30] in the MATLAB R2024b environment (Optimization Toolbox package, fmincon function). Besides the limitation of F≤Fult , the minimum and maximum values of dimensions b and h were also constrained. These constraints are dictated by the possibility of laying concrete inside the tube (limit on minimum dimensions) and the current range of rectangular tube sections (limit on maximum dimensions): 100 mm ≤b ≤500 mm; (9) 100 mm ≤ h ≤500 mm. Only the data for which a solution to the optimization problem could be found under the specified constraints were included in the training dataset. The total volume of the training data set ultimately amounted to 504841 rows. A fragment of the training dataset is shown in Table 2. Table 2. A fragment of the training dataset No. B, MPa F, kN t, mm e, mm b, mm h, mm 1 15 500 3 10 100 171 2 15 500 3 20 100 171 3 15 500 3 30 100 171 4 15 500 3 40 100 178 5 15 500 3 50 100 191 6 15 500 3 60 100 204 7 15 500 3 70 100 217 8 15 500 3 80 100 229 9 15 500 3 90 100 240 10 15 500 3 100 100 252 … … … … … … … 504832 80 10000 22 160 259 500 504833 80 10000 22 170 272 500 504834 80 10000 22 180 285 500 504835 80 10000 22 190 299 500 504836 80 10000 22 200 312 500 504837 80 10000 22 210 326 500 504838 80 10000 22 220 341 500 504839 80 10000 22 230 355 500 504840 80 10000 22 240 369 500 504841 80 10000 22 250 384 500 S o u r c e: made by S.H. Al-Zgul. In this study, a program was developed in Python language with support of version 3.11+, implementing the gradient boosting algorithm (Catboost) for predicting two physically interrelated geometric characteristics (width and height) of CFST column cross-sections based on four initial characteristics (design compressive strength of concrete, magnitude of compressive force, tube wall thickness, eccentricity of compressive force). Since the presented model has a small number of parameters, it can be trained and used on most modern (as of 2025) consumer GPUs or even in cloud environments with free quotas (e.g., Google Colab, Kaggle). When setting the regression problem, the MultiRMSE (Composite RMSE) metric was used - a generalized case of the standard RMSE for simultaneous optimization of the loss function for two target variables - the width (b) and height (h) of the cross-section (10): MultiRMSE = 1 iN=1 (b bi -µi )2 + -(h hi µi )2 , (10) 2N where N is the number of records (columns) in the sample; 2 is the number of target variables; bi is the true width of the i-th column section, mm; hi is the true height of the i-th column section, mm; bµi is the predicted width of the i-th column section, mm; hµi is the predicted height of the i-th column section, mm. As part of the study, the initial dataset was divided into three independent subsamples. The initial split was carried out in a 80% to 20% ratio, with 20% of the data being set aside as an isolated test sample for the final evaluation of the model quality. The remaining 80% of the data, constituting the training pool, was further divided: 25% was allocated for the validation sample (corresponding to 20% of the total data volume), and the remaining 75% (60% of the total volume) formed the final training sample. The following control metrics were selected: the composite MultiRMSE metric on the validation dataset to evaluate the overall predictive power of the model, as well as individual metrics for each variable: MAE to evaluate the average accuracy of the model (in understandable, interpretable units), RMSE - allows detecting the influence of outliers when selecting the final model, and the coefficient of determination - R2 demonstrates the advantage of the current model compared to the average prediction. The split was performed by fixing the random seed parameter using the train_test_split function from the scikit-learn library. This makes the procedure completely deterministic and reproducible. To verify the quality of the split, a statistical test for the homogeneity of distributions (Kruskal - Wallis for categorical/continuous features) between samples was performed based on key features. The test results (p_value > 0.05) did not reveal any statistically significant differences, which confirms the correctness of the random split. 3. Results and Discussion The model training graph (Figure 3) demonstrates virtually identical behavior of the training and validation curves. At the initial stage (before ~500 iterations), there is a sharp decrease in RMSE from ~1.2 to ~0.1, after which the process enters a phase of smooth optimization, reaching a stable plateau after ~2000 iterations. Figure 3. Graph of the learning curve for training and validation samples S o u r c e: made by S.H. Al-Zgul. The analysis of the model accuracy (Figures 4, 5) showed that the coefficient of determination reaches R² = 0.999033 for the cross-section width (b) and R² = 0.999211 the cross-section height (h), which indicates that the predicted values correspond almost perfectly to the target values. It is worth to note the uniform distribution of errors across the entire range of the studied dimensions (100-500 mm). This fact, along with the small dispersion, confirms the high reliability of the algorithm. It should be emphasized that the difference in prediction accuracy between the parameters does not exceed 2%, which indicates the balance of the predictive power of the model for the two target variables. Since the application of the Catboost algorithm is essentially a solution to the problem of multidimensional nonlinear interpolation, the limits of the model application are determined by the range of input parameters in the training dataset. Figure 4. Accuracy of prediction of section width (b) S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. Figure 5. Accuracy of prediction of section height (h) S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. To evaluate the contribution of individual features to the final result, it was decided to use SHAP analysis. This method, based on the principles of game theory, allows to accurately determine the independent contribution of each feature, unlike the standard feature importances, which can give distorted results due to the possible uneven distribution of Gain when splitting initially balanced subsamples. SHAP-value calculation requires a trained model and a test dataset. SHAP values were calculated for the entire test sample. TreeExplainer was used, which employs the TreeSHAP algorithm for efficient and accurate calculation. As the background distribution, which is required to determine the basic mathematical expected value E[f(x)] of the model, 100 randomly selected objects from the training sample were used (100 objects is the optimal compromise between accuracy and computational efficiency). Feature importance was estimated as the mean absolute SHAP-value (mean |SHAP|) across all objects in the test sample. The results of the feature importance analysis using SHAP-values demonstrate a clear, physically justified hierarchy of the influence of input parameters (or, more accurately, features) on the geometric characteristics of the cross-section (Figures 6, 7). At the same time, the SHAP dependence graphs demonstrate the nature of the influence of individual variables on the target value (Figures 8, 9). Figure 6. Analysis of the importance of features for predicting the section width (b) S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. Figure 7. Analysis of the importance of features for predicting the section height (h) S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. Figure 8. SHAP-analysis of the influence of factors on the width of the section S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. Figure 9. SHAP-analysis of the influence of factors on the height of the section S o u r c e: made by A.S. Chepurnenko, S.H. Al-Zgul. Data analysis revealed the dominant influence of the longitudinal force (F) on the geometric parameters of the cross-section, which is confirmed by the maximum values of SHAP indicators (0.79 for the width and 0.75 for the height). This pattern is fully consistent with the fundamental principles of calculating the load-bearing capacity of compressed elements. It is worth noting that the character of the influence of the load differs for the parameters under consideration: the section width demonstrates a more pronounced dependence with the SHAP-value in the range ≈ ±3 compared to the height in the range ≈ ±2, which is explained by the aspects of stress distribution in a rectangular section under eccentric compression. The results show a positive dependence between the increase in load and the increase in the geometric parameters of the cross-section (width b and height h), which is confirmed by the data in Figures 8 and 9. The analysis reveals a significant difference in the degree of influence of the eccentricity (e) on the section parameters. Although eccentricity is significant for both dimensions, its effect on height is more pronounced (2nd in importance) than on width (3rd in importance). This is explained by the quadratic dependence of the moment resistance on the height of the cross-section, which makes an increase in h more effective in counteracting eccentric loads (Figures 6, 7). Graphical visualization confirms the need for |a proportional increase of both dimensions as eccentricity increases, which corresponds to the mechanics of the structure resistance to the combined action of bending moments and longitudinal forces. The study using SHAP-value revealed a greater influence of wall thickness (t) on the cross-sectional width (SHAP~0.51) compared to the height (SHAP~0.25) (Figures 6, 7). According to the results, an increase in wall thickness leads to a decrease in both geometric parameters (b and h), which is expected, since an increase in the wall thickness increases the proportion of forces absorbed by the steel shell. Concrete class (B) demonstrates a statistically significant but relatively weak influence on the geometric parameters of the cross-section, which is particularly pronounced for the height (Figures 6, 7). This behavior is fully consistent with the mechanics of composite CFST structures, where the concrete core is mainly involved in the resistance at the ultimate stages of loading. The analysis revealed a clear inverse relationship between the concrete class and the cross-sectional dimensions: as the strength characteristics of the concrete increased, there was a regular decrease in both the width and height of the cross-section (Figures 8, 9). The results of the analysis reveal significant dependencies between the structural parameters (wall thickness), mechanical properties of materials (concrete class, steel strength), and loading conditions (longitudinal force, eccentricity) on one hand, and the geometric characteristics of CFST columns on the other. 4. Conclusion The following scientific results were obtained based on the findings of the study: 1. A regression model based on the CatBoost algorithm has been developed and validated for predicting the optimal geometric parameters (width b and height h) of rectangular concrete filled steel tube columns under eccentric compression. The model demonstrates high prediction accuracy with a total composite metric MultiRMSE ≈ 3.6 mm (on the validation dataset), as well as MAE = 2.503 mm for width b, MAE = 2.467 mm for height h (on the test dataset) with a coefficient of determination R² > 0.999 for both target parameters. The design values of the height and width of the cross-section obtained using the algorithm are for reference only. The final selection of the geometric parameters must comply with the requirements of GOST, SP, or other regulatory documents and is agreed upon by the structural designer based on detailed calculations and a technical and economic justification. 2. A multi-criteria optimization method has been developed that takes into account strength conditions, design constraints, and economic efficiency by minimizing the cost function S. 3. The interior point method has proven its effectiveness in solving this nonlinear problem, providing an optimal balance between load-bearing capacity, overall dimensions, and economic parameters of the structure, as well as allowing design requirements and technological constraints to be taken into account at the stage of forming the training sample. 4. SHAP-analysis allowed to establish a clear hierarchy of importance of the input parameters, where the longitudinal force demonstrates dominant influence (SHAP-values in the range of 0.75-0.79), which quantitatively confirms its key role in determining the dimensions of the structure. At the same time, a differentiated influence of eccentricity on the geometric parameters was revealed - its influence on section height h was more pronounced compared to width b. The obtained distribution of feature importance fully corresponds to the fundamental theoretical principles of structural mechanics, which confirms the physical validity of the developed model. The developed approach ensures optimal design of rectangular concrete filled steel tube structures under eccentric compression while meeting reliability and cost-effectiveness requirements within the range of input parameters on which the model was trained. The obtained results form the scientific and methodological basis for the automation of concrete filled steel tube structure design processes, opening up promising areas for further research, including: expanding the model to account for long-term loads and complex loading conditions, integration with BIM technologies, and the development of regulatory recommendations for the implementation of machine learning methods in design practice. These areas will significantly improve the efficiency of structural design while ensuring the required reliability and cost-effectiveness.

About the authors

Anton S. Chepurnenko

Don State Technical University

Author for correspondence.
Email: chepurnenk@mail.ru
ORCID iD: 0000-0002-9133-8546
SPIN-code: 7149-7981

Doctor of Technical Sciences, Professor of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

Samir H. Al-Zgul

Don State Technical University

Email: samiralzgulfx@gmail.com
ORCID iD: 0000-0001-6182-786X
SPIN-code: 4483-8340

Postgraduate student of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

Batyr M. Yazyev

Don State Technical University

Email: ps62@yandex.ru
ORCID iD: 0000-0002-5205-1446
SPIN-code: 5970-5350

Doctor of Technical Sciences, Professor of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

References

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