Compressive behavior of elliptical concrete-filled steel tubular short columns using numerical investigation and machine learning techniques | Scientific Reports

Scientific Reports volume 14, Article number: 27007 (2024) Cite this article

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This paper presents a non-linear finite element model (FEM) to predict the load-carrying capacity of three different configurations of elliptical concrete-filled steel tubular (CFST) short columns: double steel tubes with sandwich concrete (CFDST), double steel tubes with sandwich concrete and concrete inside the inner steel tube, and a single outer steel tube with sandwich concrete. Then, a parametric and analytical study was performed to explore the influence of geometric and material parameters on the load-carrying capacity of elliptical CFST short columns. Furthermore, the current study investigates the effectiveness of machine learning (ML) techniques in predicting the load-carrying capacity of elliptical CFST short columns. These techniques include Support Vector Regressor (SVR), Random Forest Regressor (RFR), Gradient Boosting Regressor (GBR), XGBoost Regressor (XGBR), MLP Regressor (MLPR), K-nearest Neighbours Regressor (KNNR), and Naive Bayes Regressor (NBR). ML models accuracy is assessed by comparing their predictions with FE results. Among the models, GBR and XGBR exhibited outstanding results with high test R2 scores of 0.9888 and 0.9885, respectively. The study provided insights into the contributions of individual features to predictions using the SHapley Additive exPlanations (SHAP) approach. The results from SHAP indicate that the eccentric loading ratio (e/2a) has the most significant effect on the load-carrying capacity of elliptical CFST short columns, followed by the yield strength of the outer steel tube (\(\:{f}_{yo}\)) and the inner width of the inner steel tube (\(\:2{a}_{ii}\)). Additionally, a user interface platform has been developed to streamline the practical application of the proposed ML.

Concrete-filled steel tubular (CFST) members are widely used in bridge engineering, large-span structures, and high-rise buildings due to their high compressive strength, good ductility, excellent seismic performance, and ease of construction1,2,3,4,5. Over the past several decades, numerous studies have examined the performance of CFST members with square, rectangular, and circular Sects6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25. Thus, the development of mechanical characteristics and design theories for standard CFST columns has significantly spurred their application in engineering practice26. Nevertheless, the rapid evolution of composite structures and the growing emphasis on architectural aesthetics have led scholars and engineers to increasingly explore special-shaped CFST members. Circular CFSTs are known for their uniform distribution of stresses around the circumference, leading to efficient load-bearing and resistance to bending and buckling. The circular shape allows for a more evenly distributed axial load, making these columns effective in withstanding compressive forces, and minimizing local buckling. Additionally, circular CFSTs often provide better seismic performance due to their symmetry and consistent behavior under lateral loads. Square CFSTs, on the other hand, experience different stress distributions due to their corners. These corners can be points of high stress concentration, which may lead to increased risk of local buckling and reduced load-bearing capacity compared to circular CFSTs. However, square CFSTs are easier to connect with other structural elements and can offer advantages in architectural design where square or rectangular shapes are preferred. The drawbacks of both circular and square CFSTs have led to the exploration and application of elliptical columns. Their geometry allows for reducing stress concentrations found in square columns while offering improved load distribution compared to circular columns. Concrete-filled elliptical steel tubular (CFEST) members, featuring a blend of circular and rectangular sections, appeal to engineers and architects for their architectural aesthetics, flexible axis distribution, and low flow resistance coefficient27. Consequently, CFEST members are widely utilized in significant engineering projects. For example, the elliptical composite columns were designed in Terminal 5 of London Heathrow Airport28, as seen in Fig. 1.

Given the promising future of CFEST members in engineering, there is a growing need to explore their mechanical properties and operational mechanisms. Numerous studies by scholars and researchers have conducted experiments on elliptical-section CFST short columns27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46. Zhao and Packer30 studied short elliptical CFST columns loaded axially to failure. They found that these columns failed due to elephant foot buckling or shear. Increasing the tube thickness enhanced the capacity of elliptical columns. Yang et al.29 conducted experiments on short CFST elliptical columns, varying tube thickness (3 mm to 6.3 mm) and concrete strength (30.5 MPa to 102.2 MPa), and considering concrete shrinkage. Results showed significant improvements in ductility and strength compared to hollow sections. Surprisingly, concrete shrinkage had minimal impact on the behavior of these columns. Jamaluddin et al.33 examined the effects of steel ratio, member slenderness, and concrete grades on CFST elliptical columns under axial loads. They found that short columns failed due to concrete crushing and tube buckling, while slender columns failed due to overall buckling. Yang et al.43 studied the structural performance of cold-formed elliptical CFST short columns with a/b ratios between 1.5 and 2.5. They discovered that concrete confinement in elliptical CFST columns is intermediate between that of circular and rectangular CFST columns. Lam et al.47 reported the only experimental results on concentrically loaded elliptical CFST short columns. They studied the effects of normal and high-strength concrete under axial compression. The common failure mode was outward buckling combined with concrete crushing in the buckling regions. Their findings revealed that columns made with normal-strength concrete exhibited better ductility compared to those with high-strength concrete.

The 5th terminal in London Heathrow28.

Furthermore, to reduce the weight and enhance the performance of CFST structures, the concrete-filled double-skin steel tube (CFDST) was developed. This composite member consists of an inner and outer steel hollow section filled with concrete. Compared to traditional CFST members, CFDST members are lighter, stronger, and possess greater bending stiffness48,49,50,51,52. Reviewing the history of studies on elliptical hollow steel tubes, both with and without in-filled concrete, reveals various mechanical investigations of CFEST and elliptical hollow steel tubes (EHS) with diameter-to-thickness ratios ranging from 28 to 40. First, Yang et al.29, Zhao and Packer30 as well as Chan and Gardner53 conducted tests on CFEST and EHS stub columns under centric compressive loading and discussed their design equations. Secondly, Chan and Gardner54,55 and Theofanous et al.56 conducted bending-shear tests on elliptical steel and stainless steel beams using symmetric three- or four-point loading configurations. Thirdly, Ruiz and Gardner57 as well as Gardner and Chan58 studied the buckling behavior of long columns made from elliptical hollow steel (EHS). Additionally, Chan et al.59 provide a comprehensive review of recent research on the behavior of concrete-filled elliptical hollow sections. Han et al.60 conducted experiments to examine the performance of stub columns made from elliptical stainless steel, concrete, and carbon steel double-skin tubular structures. Recently, Ipek et al.61 investigated the axial compressive performance of elliptical concrete-filled double-skin steel tubular columns.

While experimental tests provide invaluable data and are crucial for validating finite element (FE) analyses, constraints related to time and cost can sometimes limit their feasibility. Hence, nonlinear simulations are frequently used to study the structural behavior of axially compressed circular and rectangular CFST short columns62,63,64. However, there has been relatively limited numerical study on eccentrically loaded elliptical CFST beam-columns. Dai and Lam31 first conducted a finite element analysis to evaluate the performance of elliptical short CFST columns. They proposed a confining pressure model for elliptical CFST columns constructed from hot-rolled steels, validated with experimental results from Yang et al.29. Patel et al.65 additionally performed numerical simulations using both finite element (FE) and fiber element techniques to analyze the performance of elliptical CFST columns. Furthermore, Liu et al.40 employed finite element (FE) simulations to study the inelastic behavior of elliptical short CFST columns with a/b ratios ranging from 1.5 to 2.5. Their study did not incorporate nonlinear strain hardening of hot-rolled steel in the FE model. Sheehan et al.32 conducted the first FE analysis of elliptical short CFST members under eccentric loading, presenting a numerical method for computing axial load-moment interaction diagrams. Hassanein et al.66 conducted a study on elliptical CFST columns filled with high-strength concrete (HSC). Using Abaqus, numerical simulations explored their behavior under axial compression and eccentric loading. Parametric studies varied EHS slenderness, steel yield strength, and concrete compressive strength. A new design formula was proposed, accounting for effective confined concrete strength to optimize structural design.

To design elliptical CFST columns, current composite construction codes like AS5100-201767, AISC 360 − 1068, ACI 318 − 1169 and Eurocode 470 do not detail specific design methods for elliptical CFST columns. They suggest applying circular or rectangular CFST rules. Conversely, GB50936-2014 71 offers guidelines for estimating the ultimate capacity of elliptical CFST columns with varying a/b ratios. Furthermore, Cai et al.44 discovered that the predictions for the compressive resistance of elliptical CFT stub columns based on current international design specifications were generally conservative. Additionally, a few scholars have proposed methods for predicting the load-carrying capacity of elliptical CFST columns under pure axial compression and eccentric loading66. Therefore, there is an urgent need for a reliable model to accurately predict the load-carrying capacity of elliptical CFST columns under different eccentric loading conditions. Nowadays, machine learning (ML) algorithms provide effective solutions to complex problems, handling multiple factors without assumptions, and making them superior in addressing these concerns. ML has proven effective in solving various structural engineering problems72,73,74,75,76, including structural health monitoring77,78, damage detection79,80,81,82,83, performance evaluation5,84,85,86,87, and structural parameter identification88,89. For this reason, many research studies have been carried out to estimate the structural capacity of CFST columns using the ML approaches90,91,92,93. Ahmadi et al.94,95 used an artificial neural network (ANN) to predict the axial capacity of short CFST columns. Moon et al.96 have successfully developed a fuzzy logic model for predicting the strength of circular CFST short columns. Ipek and Güneyisi97 derived a gene expression programming model to predict the load-bearing capacity of circular CFST columns. The results showed that the ANFIS model could predict the grain yield with good accuracy. Ren et al.98 employed support vector machine and particle swarm optimization to determine the axial capacity of square CFST columns. Tran et al.99 compared three advanced data-driven models for predicting the axial compression capacity of CFDST columns. The comparative results showed that three data-driven models achieved more accuracy than existing equations. Chen et al.100 investigated the prediction of mechanical behaviors of FRP-confined circular concrete columns under axial loading using ANN and Support Vector Regression (SVR). Analyzing 298 test data points, they found that the predictive accuracy of ANN and SVR surpassed existing models. Le101 introduced a hybrid ML approach, RCGA-ANFIS, to predict load carrying capacity of elliptical CFST short columns under axial load. Combining Adaptive Neuro Fuzzy Inference System (ANFIS) with Real Coded Genetic Algorithm (RCGA), the model achieved an \(\:{\text{R}}^{2}\) of 0.974, surpassing conventional gradient descent (GD) techniques (\(\:{\text{R}}^{2}\) = 0.952). Compared to literature benchmarks (\(\:{\text{R}}^{2}\) = 0.776 or 0.768), RCGA-ANFIS demonstrated superior accuracy. ML-based models have shown promise over traditional formulas. though research has primarily focused on circular and rectangular columns. Finally, the majority of ML-based studies have focused on columns with circular and rectangular cross-sections. Therefore, additional research is needed to explore the potential applications of ML-based models in analyzing the axial behavior of elliptical CFST short columns the load-carrying capacity of elliptical CFST short columns under different eccentric loading conditions.

In the design of elliptical CFST short columns, the ultimate axial load is a critical parameter. However, challenges remain in developing a generalized analytical formula to accurately predict their ultimate load-carrying capacity under various eccentric loading conditions. Design codes for composite construction, such as AS5100-2017 67, AISC 360 − 10 68, ACI 318 − 11 69 and Eurocode 4 70, lack specific methods for designing elliptical CFST short columns. Moreover, although machine learning (ML) approaches have shown promise in various fields, no studies have yet applied them to estimate the load-carrying capacity of elliptical CFST short columns under different eccentric loading conditions. Therefore, this study uses ABAQUS software to perform numerical simulations of three different configurations (see Fig. 2) of elliptical concrete-filled steel tubular (CFST) short columns under pure axial compression and eccentric loading. The validity of the Finite Element (FE) model was confirmed by comparing it with literature test results. Subsequently, 180 numerical FE datasets of elliptical CFST short columns were generated. Based on these datasets, two design formulas were proposed to estimate the load-carrying capacity of elliptical CFST short columns under both concentric and eccentric loading conditions. Furthermore, seven ML algorithms—SVR, RFR, GBR, XGBR, MLPR, KNNR, and NBR—are employed to predict the load-carrying capacity of elliptical CFST short columns under different eccentric loading conditions. The study provided insights into the contributions of individual features to predictions using the SHapley Additive exPlanations (SHAP) approach. Using the best-trained ML model, a graphical user interface programmed in Python for practical use in designing CFST columns under different eccentric loading conditions.

Configurations of elliptical columns: (a) double steel tubes with sandwich concrete (CFDST), (b) double steel tubes with sandwich concrete and concrete inside the inner steel tube, and (c) single outer steel tube with sandwich concrete.

In this study, the finite element (FE) software ABAQUS102 was utilized to develop a model for determining the ultimate axial capacity of CFST columns. The developed FE model was then employed to generate a total of 180 FE datasets, as detailed in Table A1. Table A1 presents the cross-sectional dimensions, material properties, eccentric loading values of the models, and the ultimate load capacity of the columns (\(\:{P}_{u,FE}\)). The FE model considered three different configurations of elliptical columns, as shown in Fig. 2. The details of the sectional dimension of the elliptical CFDST column is shown in Fig. 3, where\(\:{2a}_{o}\) and \(\:{2b}_{o}\) stand for the long and short axes of the outer diameters of the outside elliptical steel tube (EST), and \(\:{t}_{o}\) represents the thickness of the outer EST. Similarly,\(\:\:{2a}_{i}\) and \(\:{2b}_{i}\) stand for the long and short axes of the outer diameters of the inner EST, and \(\:{t}_{i}\) represents the thickness of the inner EST. The following sections provide a detailed explanation of the material definition, especially the nonlinear behavior, surface identification and interaction, element type and mesh selection, as well as the loading and boundary conditions.

The sectional dimension of the elliptical CFDST column.

The CFST columns modeled in this study consist of concrete and steel tubes. Therefore, it is essential to first define the material characteristics of both steel and concrete. The steel properties identified in this study were applied to both the tubes and the endplates used for loading and boundary conditions, while the concrete properties were used exclusively for the concrete infill.

In this study, the steel material was characterized by bilinear elastic-plastic behavior with isotropic hardening. Following Han and Huo103, a two-region stress-strain relationship was employed to model steel, as depicted in Fig. 4. The elastic behavior of the steel material was specified in the first region, starting with a modulus of elasticity of 200 × 10³ N/mm² and a Poisson’s ratio of 0.3. It was assumed that 1% of the elastic modulus of steel represented the modulus in the plastic region. The following expressions were employed to define the elastic and plastic regions63:

where \(\:{\sigma\:}_{i}\), \(\:{E}_{s}\), \(\:{\epsilon\:}_{s}\), \(\:{f}_{sy}\), and \(\:{\epsilon\:}_{sy}\) are the desirable strength, elastic modulus, strain, yield strength, and the yield strain of steel, respectively. Therefore, in ABAQUS, the behavior of steel was defined as elastic (from the origin to yield strain) and plastic (from yield to ultimate strains). A true stress-strain curve needed to be defined in ABAQUS. Hence, the engineering stress-strain curve was converted to the true stress-strain curve using the following Eq. 104:

where\(\:{f}_{tr}\) and \(\:{f}_{norm}\) are the true and nominal steel stresses, respectively, \(\:{\epsilon\:}_{tr}^{pl}\) is the true plastic strain and \(\:{\epsilon\:}_{norm}\) is the nominal strain.

Stress-strain curve for steel end plates and tubes103.

Stress-strain curves for the confined and unconfined concrete105,106.

Concrete, naturally exhibiting brittle behavior, undergoes different failure mechanisms such as cracking under tensile loading and crushing under compressive loading, respectively106. However, using concrete as infill material in the steel tube alters its behavior, meaning that the steel surrounding the concrete provides extra strength to the concrete and reduces its brittleness, thereby improving its ductility107. The level of confinement provided by the steel tube is assigned regarding the diameter-to-thickness ratio of the steel tube, yet it should be known that when the diameter-to-thickness ratio of the steel tube is higher than 150, the confinement effect ensured by the steel tube can be discounted61. In this context, two different stress-strain relationships for concrete, such as confined and unconfined, can be defined to model the concrete in ABAQUS. In the current model, the stress-strain relationship for confined concrete proposed by Hu et al.105 was adopted, with a modification in the softening region as shown in Fig. 5. Hu et al.105 describe three regions: elastic, strain hardening, and linear softening. However, this study introduces a strain-softening region, differing from Hu et al.105, as the steel tube confinement of the concrete increases the peak strain. Therefore, the function suggested by Binici108 was utilized to define the descending branch of confined concrete. Consequently, as shown in Fig. 5, the proposed model in this study comprises three distinct regions: an elastic region from the origin (Point O) to the proportional limit stress (Point A), a strain-hardening region from the proportional limit (Point A) to the confined concrete stress (Point B), and a strain-softening region from the confined concrete stress (Point B) to the plateau (Point C).

In Fig. 5, \(\:{f}_{c\:}\), \(\:{f}_{cc}\), and \(\:{f}_{r\:}\) represent the unconfined compressive strength of cylinder concrete (80% of cube concrete compressive strength), confined compressive strength, and residual stress, respectively. The strains corresponding to unconfined and confined concretes are denoted by \(\:{\epsilon\:}_{c\:}\)and \(\:{\epsilon\:}_{cc}\), respectively. The compressive strength of cylinder concrete can be determined experimentally, with the unconfined concrete strain set at 0.003 as recommended by ACI69. Additionally, the strength and strain values of the confined concrete can be calculated using the formulas suggested by Mander et al.109, as detailed below:

where \(\:{k}_{1\:}\) and \(\:{k}_{2\:}\)are constant parameters taken as 4.1 and 20.4, respectively, based on the recommendation by Richart et al.110. Based on the results of numerical simulations, Hu et al.105 proposed two empirical equations for f l / f y as follows:

where \(\:De\), to, and \(\:{f}_{y\:}\) represent the equivalent diameter, thickness, and yield strength of the outer elliptical steel tube, respectively. Due to limited test data, Eqs. 7 and 8 were not obtained using the least square method. Instead, Hu et al.105 formed these Equations by connecting the data points at D/t = 22, 47 and D/t = 47, 150. The equivalent diameter of an elliptical steel tube can be determined using the given expression

where \(\:a\) and \(\:b\) denote half of the outer dimensions of elliptical sections in the major and minor axes, respectively. In this study, following Hu et al.105, half of the confined concrete compressive strength, \(\:{0.5\:f}_{cc}\), was used as the proportional limit stress. This stress was then divided by the modulus of elasticity of the confined concrete, \(\:{E}_{cc}\), to determine the corresponding strain. The formula from ACI69 (Eq. (10)) was used to determine the acceptable modulus of elasticity for the confined concrete. Moreover, the Poisson’s ratio \(\:{v}_{c}\) equal to 0.2 for the confined concrete.

The strain hardening region of the stress-strain curve for confined concrete was determined using the formula proposed by Sanez111, as follow:

where \(\:{f}_{l}^{{\prime\:}}\) and \(\:{\epsilon\:}_{l}^{{\prime\:}}\) represent the stress and strain, respectively, in the strain hardening region of the confined concrete. \(\:R\) and \(\:{R}_{E}\) are coefficients determined by the expressions provided below:

where \(\:{R}_{\sigma\:}\) and \(\:{R}_{\epsilon\:}\) are set to be equivalent to 4 pursuant to the recommendation by Hu and Schnobrich112.

To determine the stress using the formula recommended by Sanez111, the strain ε is the only unknown variable. To determine stress values in the strain hardening region (between points A and B), presumed strain values are used between proportional strain (strain at point A) and confined strain (strain at point B). Binici’s function108 is applied to derive stress and strain values for the strain-softening region in confined concrete

The stress \(\:{f}_{2}^{{\prime\:}}\) and strain \(\:{\epsilon\:}_{2}^{{\prime\:}}\) in the strain-softening region of confined concrete are defined by parameters α and β, shaping the descending branch. Equations (14) and (15), dependent on \(\:{\zeta\:}_{c}\), are used to determine residual stress (\(\:{f}_{r}\)) and α, respectively. Parameter β is typically set to 1.2 through iterative fitting processes, as \(\:{f}_{r}\)​, α, and β cannot be directly measured62:

where \(\:{\zeta\:}_{c}\) is the confinement factor and to be computed by the expression given below:

where \(\:{f}_{ck}\) and \(\:{\alpha\:}_{n}\) are, respectively, the characteristic compressive strength of the concrete and the nominal steel ratio of CFDST short column with the EHS that can be calculated by the following equations:

where \(\:{f}_{cu}\) is the characteristic compressive strength of cube concrete

where \(\:{A}_{so}\) and \(\:{A}_{c,nominal}\) are, respectively, the cross-sectional area of the outer elliptical steel tube and the nominal cross-sectional area of the concrete and to be determined by the following equations:

where \(\:{p}_{m}\) is the mean perimeter of the elliptical section and to be determined by the approximate formula proposed by Almkvist et al.113 as follows:

in which

ABAQUS assigned a material with elastic and plastic regions to describe confined concrete, using the concrete damage plasticity (CDP) model for defining its plastic properties. The CDP model parameters for concrete include dilation angle ψ, tensile/compressive meridian ratio \(\:{K}_{c}\)​, flow potential eccentricity \(\:e\), biaxial/uniaxial compressive strength ratio \(\:{f}_{bo}/{f}_{c}^{{\prime\:}}\), and viscosity parameter \(\:\nu\:\). In this study, values were 30°, 2/3, 0.1, 1.16, and 0.0001, respectively114,115. The fracture energy-based method was used to simulate the tensile behavior of concrete. Up to 10% of \(\:{f}_{c}^{{\prime\:}}\), the tensile response was assumed linear64. Beyond this point, the tensile response was modeled as follows:

where \(\:{G}_{f}\) is the fracture energy of concrete and \(\:{d}_{max}\) is the maximum coarse aggregate size (in mm).

Concrete sandwiched between steel tubes was simulated using eight-node solid elements (C3D8R) with reduced integration, while the steel tubes utilized four-node shell elements (S4R). Solid elements are chosen for their ability to model complex three-dimensional structures and capture detailed deformations and stress distributions in concrete. Sensitivity analysis revealed Do/15 as the optimal mesh size for CFST columns in both directions. Figure 6 depicts the typical meshing, boundary conditions, and loading of the concentrically-loaded elliptical CFST column.

Typical meshing, boundary conditions, and loading of the concentrically-loaded elliptical CFST column.

Rigid body constraints were used to tie the top and bottom surfaces to their respective reference points (RPs) at the lower and upper surfaces. Boundary conditions were applied to RP1 and RP2 (see Fig. 6). At RP2, all displacements and rotations were constrained (i.e., \(\:{d}_{x}\)=\(\:{d}_{y}\)=\(\:{d}_{z}\)= 0.0 and \(\:{\theta\:}_{x}\), \(\:{\theta\:}_{y}\), \(\:{\theta\:}_{z}\)=0.00). At RP1, all were constrained except vertical displacement (\(\:{d}_{z}\)), which was free. The load was simulated by applying a downward displacement to RP1 equivalent to 1/6th of the column’s length63.

Interactions between sandwiched concrete and steel tubes were simulated using surface-to-surface contact. The concrete surfaces were slave, while the steel tube surfaces were master63. Hard contact behavior was applied in the normal direction, and the Coulomb friction model with a friction coefficient of 0.6 was used in the tangential direction for the interaction model116. In elliptical CFST columns, hard contact in the normal direction prevents surface penetration or overlap between concrete and steel. The Coulomb friction model, widely used in finite element analysis, accurately simulates frictional interactions between these materials.

The most crucial aspect of modeling such columns using the finite element technique is the validation and calibration of the proposed model. Generally, a wide range of datasets from experimentally investigated columns is used to verify the developed model. However, for the elliptical CFDST short column section chosen in this study, it was not possible to compile an extensive dataset. This section compares the structural behavior of elliptical CFDST short columns, specifically their ultimate axial capacities and load-deflection responses predicted by FE models, with selected experimental results reported by Li et al.117 and Han et al.60.

Li et al.117 conduct a laboratory test on six elliptical CFDST short column. The sectional dimensions of the outside elliptical steel tube (EST) are 280 mm * 140 mm * 6 mm (\(\:{2a}_{o}*{2b}_{o}*{t}_{o}\)). Similarly, the sectional dimensions of the inside EST are 95 mm * 53 mm * 4 mm (\(\:{2a}_{i}*{2b}_{i}*{t}_{i}\)). The average cubic compressive stresses of the infilled concrete were 36.8 MPa and 58.3 MPa with average elastic rigidities of 24,062.5 MPa and 32,060.4 MPa, respectively. The column height (L) is 500 mm, making L/\(\:{2b}_{o}\) approximately 3.5. The yield strengths of the outer and inner steel tubes are presented in Table 1.

In the study by Han et al.60, two types of CFDST short columns with elliptical hollow sections (EHS) were manufactured and tested, with two specimens from each type. Table 1 details their properties. The outer and inner steel tubes had yield strengths of 319.6 MPa and 380.6 MPa, respectively, and the concrete’s compressive strength was 65.6 MPa. The columns were 720 mm tall, with outer steel tube dimensions of 240 mm (major diameter) and 160 mm (minor diameter), and a thickness of 3.62 mm. Both column types used the same outer tube but differed in the inner tube dimensions. Column E1 had inner tube diameters of 186 mm and 106 mm, while E2 had 142 mm and 62 mm, both with a thickness of 3.72 mm. This study validated the proposed finite element (FE) model using these columns.

Table 1 presents the ultimate axial strength values from both experimental testing and finite element (FE) analysis, where \(\:{P}_{u,FE}\) represents the predicted ultimate axial capacity from the FE models and \(\:{P}_{u,exp}\) represents the ultimate axial capacity from experimental testing. the mean value (µ) of\(\:{P}_{u,exp}/{P}_{u,FE}\:\) ratio is 1.067, with a coefficient of variation (CoV) of 0.046. This indicates that the FE model based on Hu’s σ-ε model105 accurately predicts the ultimate axial capacity of elliptical CFDST columns. The superior performance of this FE model is likely due to its consideration of the confinement effect induced by the inner steel tube. Therefore, Hu ‘s σ-ε model105 was used in this study to model 180 elliptical CFST columns.

Figure 7 illustrates a comparison of the axial load versus axial displacement curves derived from FE analyses based on Hu’s σ-ε model105 and those obtained from experimental measurements. The parameters of the specimens depicted in the figure are provided in Table 1. In general, both FE models effectively predict the initial stiffness of the specimens. Moreover, the FE model incorporating Hu’s σ-ε model shows superior performance in predicting the peak load of the specimens. The discrepancy between FE model predictions and experimental results can be attributed to several factors. Firstly, the boundary conditions in FE models may not accurately replicate those in experiments, potentially influencing the behavior observed. Additionally, variations in material properties among experimental specimens, such as concrete compressive strength and steel yield strength, are not fully accounted for by FE models which typically use average material properties. Moreover, experimental variability due to factors like fabrication tolerances, measurement errors, and environmental conditions further contributes to discrepancies118,119.

Validation of FEM models with experimental models of Han et al.60 and Li et al.117.

Given the reliable prediction of the FE model, the proposed modeling strategy provided an opportunity to expand the available data on the behavior of elliptical CFST columns. The parametric study conducted in this research aimed to visualize the influences of concrete compressive strength, inner tube thickness and yield strength, outer tube yield strength, sectional hollow ratio, and eccentric loading ratio (e/2a) on ultimate axial strength of elliptical CFST short columns.

In this study, three concrete compressive strength values—28.4 MPa, 45.9 MPa, and 52.7 MPa, classified as normal and high-strength concrete—were used in the FE modeling of elliptical CFST columns. The FE analysis results for configuration (a) indicate that increasing the concrete compressive strength from 28.4 MPa to 45.9 MPa results in a 10–12% increase in the peak strength of the elliptical CFST columns, as detailed in Table A1. However, it is observed that as the thickness of the inner tube increases, the rate of increment due to changes in concrete strength also rises. In summary, using concrete with higher compressive strength enhances the load-carrying capacity of elliptical columns. Figure 8 compares axial load versus axial displacement curves for elliptical CFST columns with varying inner steel tube thicknesses and concrete compressive strengths. Columns with lower concrete strength show a gradual descent in the strain-softening region, while higher strength results in a steeper decline. This highlights that higher concrete strength improves ultimate load capacity but reduces ductility. Wang et al.120 found similar results for circular hollow section columns, noting that increased concrete strength generally decreases ductility. Thus, concrete strength is crucial for the performance of elliptical CFST columns, enhancing load capacity but negatively affecting ductility.

Axial load versus axial displacement of elliptical CFST columns with various concrete compressive strengths.

To demonstrate the influence of inner tube thickness on the behavior and performance of elliptical CFST columns, three thickness values (4, 8, and 12 mm) were used in this study. The ultimate axial strength results from the FE analysis are provided in Table A1. The results for configuration (b) indicate that the inner tube thickness significantly impacts the ultimate axial strength of the elliptical CFST columns. Increasing the inner tube thickness from 4 to 12 mm enhances the peak strength by approximately 13–18%. Figure 6 shows the variation in axial load versus axial displacement of columns with various inner tube thicknesses and yield strengths. The figure indicates that increasing inner tube thickness reduces the diameter-to-thickness (D/t) ratio, which enhances the confinement effect provided by the inner steel tube. This improved confinement increases the compressive strength and load-bearing capacity of the column.

Furthermore, to illustrate the influence of inner tube yield strength, three steel yield strength values—276.2 MPa, 380.6 MPa, and 433.7 MPa, classified as normal and high yield strength—were used in the FE modeling of elliptical CFST columns, as shown in Table A1. The ultimate axial strength results for configuration (b) from the FE analysis indicate that increasing the yield strength from 276.2 MPa to 433.7 MPa enhances the peak strength by approximately 4.6–10%. Figure 9 shows that higher yield strength of the inner tube leads to greater confinement of the concrete core, further enhancing the load-bearing capacity of the elliptical CFST columns.

Axial load versus axial displacement of elliptical CFST columns with various inner tube thicknesses and yield strengths.

Similarity, to explore the impact of outer tube yield strength on elliptical CFST columns, three steel yield strength values—270.3 MPa, 319.6 MPa, and 421.5 MPa, categorized as normal and high strength—were employed in FE simulations (see Table A1). Analysis of configuration (a) revealed that increasing the yield strength from 270.3 MPa to 421.5 MPa enhances peak strength by approximately 21.7–25%. Figure 10 illustrates that higher outer tube yield strength results in increased confinement of the sandwich concrete, thereby augmenting the load-bearing capacity of the columns. Hence, outer tube yield strength significantly influences the overall axial load resistance of elliptical CFST columns.

Axial load versus axial displacement of elliptical CFST columns with various outer tube yield strengths.

In configuration (c), which features a single outer steel tube with sandwich concrete, this study explored the influence of the sectional hollow ratio (\(\:X={a}_{i}{b}_{i}/{a}_{o}{b}_{o}\)) on the behavior and performance of elliptical CFST columns. Three ratios (13%, 23%, and 50%) were investigated, as detailed in Table A1. The findings for configuration (c) highlight that the sectional hollow ratio significantly affects the ultimate axial strength of the elliptical CFST columns. Increasing the sectional hollow ratio from 23 to 50% results in a decrease of approximately 27% in peak strength. Figure 11 displays the variation in axial load versus axial displacement of elliptical CFST columns with different sectional hollow ratios, illustrating that a higher sectional hollow ratio decrease the ductility index and initial rigidity of the elliptical CFST columns.

Axial load versus axial displacement of elliptical CFST columns with various sectional hollow ratios.

In this study, four eccentric loading ratio (e/2a) values along the longer axis—0.0%, 36%, 42%, and 50%—were used in the FE modeling of elliptical CFST columns. The FE analysis results indicate that increasing the eccentric loading ratio from 0.0 to 50% results in an 83% decrease in the peak strength of the elliptical CFST columns, as detailed in Table A1. Figure 12 compares axial load versus axial displacement curves for elliptical CFST columns with varying eccentric loading ratio (e/2a) values, illustrating that a higher eccentric loading ratio decreases the ductility index and initial rigidity of the elliptical CFST columns.

Axial load versus axial displacement of elliptical CFST columns with various eccentric loading ratio (e/2a).

To date, international design specifications do not address the design strengths of elliptical CFST short columns. For the three current configurations, new design formulas for both concentric and eccentric loading conditions are proposed and compared with the FE results, as shown in Table 2.

To validate the accuracy of the proposed formulas, Fig. 13a compares the FE results of 60 elliptical CFST columns under concentric axial compression with predictions from Eq. (18). The mean \(\:{\text{P}}_{\text{u},\text{F}\text{E}}/{\text{P}}_{\text{u},\text{c}\text{o}\text{n}}\)​​ ratio is 0.998, with a CoV of 0.025, and \(\:{\text{R}}^{2}=98.66\text{\%}\), indicating the formula’s accuracy for the three configurations. Similarly, Fig. 13b compares FE results of 120 elliptical CFST columns under eccentric axial compression with predictions from Eq. (19). This formula significantly improves load-carrying capacity predictions, with a mean \(\:{\text{P}}_{\text{u},\text{F}\text{E}}/{\text{P}}_{\text{u},\text{e}\text{c}\text{c}}\) ratio of 1.018 and a CoV of 0.115, and \(\:{\text{R}}^{2}=98.98\text{\%}\), demonstrating its effectiveness for the same configurations.

Comparisons between the ultimate load-carrying capacity of elliptical CFST columns using the proposed formulas and FE results for: (a) concentric loading condition and (b) eccentric loading condition.

The dataset, consisting of 180 samples generated through finite element modeling as detailed in Table A1, meticulously curates a comprehensive set of 14 variables, including 13 distinctive features and one output parameter. These variables serve as crucial metrics in the assessment of load-carrying capacity column. Each variable holds a unique significance in the context of concrete and steel tube configurations, contributing to a holistic understanding of the structural performance. The features includes the total concrete area (Ac), concrete strength of standard cylinders (fc’), outer depth of internal steel tube (2aoi), outer width of internal steel tube (2boi), inner depth of internal steel tube (2aii), Inner width of internal steel tube (2bii), yield strength of internal steel tube (fyi), outer depth of external steel tube (2aoo), outer width of external steel tube (2boo), inner depth of external steel tube (2aio), inner width of external steel tube (2bio), yield strength of external steel tube (fyo), eccentric loading ratio (e/2a) and load carrying capacity (Pu), encapsulate diverse aspects of the concrete and steel tube geometry, dimensions, and material properties. The statistical descriptive analysis of the dataset is presented in Table 3.

To further enhance our understanding, visual representations have been crafted. Figures 14 and 15 showcase the frequency distribution histogram plot, providing insights into the distribution of data points across the variables. Moving to Figs. 16 and 17, scatter plots are presented, illustrating the relationship between each feature and the output variable. These visualizations offer a quick and intuitive understanding of how changes in concrete area, strength, steel tube dimensions, and eccentric loading may impact structural performance. The plots serve as efficient tools for identifying potential correlations and guiding further analysis, enabling a swift assessment of key factors influencing load-carrying capacity of column.

Frequency distribution histogram plot of Ac, fc’, 2aoi, 2boi, 2aii, 2bii, fyi, and 2aoo.

Frequency distribution histogram plot of 2boo, 2aio, 2bio, fyo, e/2a, and Pu.

For a deeper exploration of correlations, Figs. 18 and 19 present Pearson121 and Spearman correlation122 plots, respectively. Pearson correlation measures the strength and direction of a linear relationship between two continuous variables, whereas Spearman correlation assesses the strength and direction of a monotonic relationship and is more flexible. Pearson uses the actual data values to compute correlation, while Spearman relies on the ranks of the data, making it suitable for non-linear relationships. These figures offer a visual depiction of the relationships between variables, aiding in the identification of potential dependencies. In the Pearson correlation results, it has observed that relatively low correlation coefficients, ranging from − 0.186 to 0.207. Notably, the eccentric loading ratio (e/2a) exhibits a strong negative correlation (-0.906), suggesting a substantial impact on load capacity. On the other hand, features like concrete area (Ac) and yield strength of internal steel tube (fyi) show positive correlations, albeit of moderate strength. This indicates a nuanced interplay between these variables and the structural outcome.

Turning to the Spearman correlation results, we find slightly higher coefficients, ranging from − 0.238 to 0.245. Like Pearson, the eccentric loading ratio maintains a strong negative correlation (-0.929), underscoring its significance in influencing load capacity. Other features, such as concrete area (Ac) and outer width of internal steel tube (2boi), exhibit notable correlations, reinforcing the importance of these factors in the structural context.

Univariate analysis of Ac, fc’, 2aoi, 2boi, 2aii, 2bii, fyi, 2aoo with Pu.

Univariate analysis of 2boo, 2aio, 2bio, fyo, e/2a with Pu.

Pearson correlation matrix of features and output.

Spearman rank correlation coefficient of features and output.

Moreover, Fig. 20 includes a box plot for outlier detection, providing a valuable tool for identifying anomalous data points that may significantly impact structural analysis.

Box plot with outliers, quartile and median of features in the dataset.

In the quest of constructing an efficient and robust predictive ML model for load-carrying capacity (Pu), the methodology comprises several key steps to ensure robustness and accuracy as shown in flowchart of the study in Fig. 21.

The initial phase involves meticulous dataset preprocessing, where outliers are handled by replacing with median of feature and normalization of features within a standardized range of 0 to 1, calculated using Eq. (20). This ensures that the dataset is well-conditioned and suitable for training diverse predictive models.

Here, \(\:{X}_{i,\:unscaled}\) is the \(\:{i}^{th}\) original feature, \(\:{X}_{i,\:min}\) and \(\:{X}_{i,\:max}\) are the minimum and maximum of the original \(\:{i}^{th}\) feature.

Moving forward, the model selection phase is crucial, and the study draw inspiration from the work of previous researchers who have demonstrated the efficacy of certain models in similar contexts. The selected models include Support Vector Regressor (SVR)123, Random Forest Regressor (RFR)124, Gradient Boosting Regressor (GBR)125, XGBoost Regressor (XGBR)126, MLP Regressor (MLPR)127, K-nearest Neighbours Regressor (KNNR)128, and Naive Bayes Regressor (NBR)129. These models collectively form a diverse ensemble, providing a comprehensive exploration of the dataset’s predictive capabilities. Once the models are identified, the pre-processed dataset undergoes 5-fold cross-validation with each model130. This step ensures that the models are rigorously tested and evaluated, preventing overfitting, and enhancing generalization to unseen data. Cross-validation is a widely adopted technique that aids in assessing a model’s performance by partitioning the dataset into multiple subsets or folds. In the study, the dataset is divided into five equal parts, with each part serving as a validation set while the model is trained on the remaining four folds as shown in Fig. 22.

Flowchart of the machine learning approach to predict Pu.

K-fold cross validation dataset splitting architecture.

This iterative process is crucial for mitigating the risk of overfitting, where a model may perform exceptionally well on the training data but struggle with new, unseen data. Cross-validation ensures a more robust evaluation by exposing the model to different subsets of the dataset in multiple rounds, effectively simulating its performance on various data configurations. By averaging the performance metrics across these folds, study obtain a more reliable estimation of the model’s generalization capabilities. This approach provides a more realistic indication of how well the model is likely to perform on unseen data, enhancing its practical utility. The use of 5-fold cross-validation in the methodology adds a layer of rigor to the model selection process, contributing to the overall reliability and credibility of our predictive model for load-carrying capacity.

The performance of each model is then assessed using multiple metrics131,132,133,134 including \(\:{R}^{2}\) (Co-efficient of determination) calculated using Eq. (21), \(\:MSE\) (Mean Squared Error) calculated using Eq. (22), \(\:RMSE\) (Root Mean Squared Error) calculated using Eq. (23), \(\:MAE\) (Mean Absolute Error) calculated using Eq. (24), and MAPE (Mean Absolute Percentage Error) calculated using Eq. (25). These metrics offer a nuanced understanding of the models’ predictive accuracy and their ability to capture the intricacies of load-carrying capacity.

Here, \(\:{y}_{i}\) is the actual \(\:{i}^{th}\) output from dataset, \(\:{\widehat{y}}_{i}\) is the model predicted output against \(\:{y}_{i}\), and \(\:N\) is the total number of instances in the dataset.

The performance measuring metrices, used in the study, are described as below:

R2 measures the proportion of the variance in the dependent variable (load-carrying capacity, Pu) that is predictable from the independent variables (features). A higher R2 value, closer to 1, indicates a better fit of the model to the data. It serves as a valuable indicator of the model’s explanatory power.

MSE calculates the average squared difference between predicted and actual values. Lower MSE values signify better predictive accuracy, as the squared differences between predictions and actual outcomes are minimized. It penalizes larger errors more significantly than smaller ones.

RMSE is derived from MSE but provides an interpretable scale as it represents the square root of the average squared differences. Similar to MSE, lower RMSE values indicate more accurate predictions, offering a measure of the average magnitude of errors in the model.

MAE computes the average absolute differences between predicted and actual values. It provides a straightforward measure of the average error magnitude, irrespective of direction. Lower MAE values indicate better predictive accuracy.

MAPE expresses the average percentage difference between predicted and actual values. It provides a relative measure of accuracy and is particularly useful for understanding the magnitude of errors in the context of the actual values. Lower MAPE values indicate a smaller percentage difference and better predictive performance.

Following the thorough evaluation, the model exhibiting superior performance is crowned as the winner and selected as the final predictive model for Pu.

To delve deeper into the impact of individual features on predictions, SHAP (SHapley Additive exPlanations) analysis135, is conducted. SHAP analysis is a powerful technique employed to interpret the output of machine learning models, shedding light on the contribution of individual features to each prediction. Derived from cooperative game theory, SHAP values allocate a value to each feature, indicating its impact on the difference between the model’s prediction and the average prediction. The SHAP value for a particular feature quantifies its marginal contribution to the prediction. Positive SHAP values suggest an increase in the prediction, while negative values indicate a decrease. The sum of SHAP values for all features, along with the average prediction, equals the model’s prediction for a specific instance. In principle, SHAP analysis allows us to understand the relative importance and influence of each feature in the model’s decision-making process. This interpretability is crucial for gaining insights into the factors driving the predictions, especially in complex machine learning models.

The SHAP value (\(\:\varphi\:\)), calculated using Eq. (26), for a feature in a specific prediction is calculated using the Shapley value, which represents the average marginal contribution of a feature across all possible feature orders.

Here, \(\:N\) is the set of all features, \(\:S\) is a subset of \(\:N\), \(\:i\) is the feature for which the SHAP value is being calculated, and \(\:f\left(S\right)\) represents the model’s prediction given the set of features \(\:S\).

Finally, to make the predictive model accessible and user-friendly, a Graphical User Interface (GUI) has been developed for easy utilization of the predictive model, enabling users to predict load-carrying capacity (Pu) based on the 13 provided inputs. The GUI interface is accessible on a GitHub repository, ensuring convenience and transparency for users.

The performance of various predictive models in estimating the load-carrying capacity (Pu) on both training and test set are presented in Table 4. The evaluation metrics, including R2 Score, MSE, RMSE, MAE, and MAPE, offer valuable insights into the effectiveness of each model. Among the models, GBR and XGBR exhibited outstanding results with high test R2 scores of 0.9888 and 0.9885, respectively. Random Forest Regressor closely followed with a test R2 score of 0.9801. These models demonstrated low test MSE and RMSE values, indicating accurate predictions. Notably, Naive Bayes Regressor (NBR) displayed a challenging performance with negative R2 and considerably higher test MSE and RMSE values. Figure 23 provides a visual representation of the actual versus predicted Pu values from the machine learning models on the test set. The plots showcase the model predictions aligning closely with the actual load-carrying capacities, particularly evident in the models with higher R2 scores. Furthermore, Fig. 24 presents a visual insight into the models’ error in predicting Pu on the test set. The plots illustrate the distribution of errors, helping identify patterns and potential areas of improvement in the models.

Actual versus predicted Pu from ML models on test set.

Actual versus models’ error in prediction of Pu on test set.

Radar plots, depicted in Figs. 25 and 26, offer a comprehensive overview of each model’s performance on the training and testing sets, respectively. These plots visualize multiple performance metrics simultaneously, providing a holistic understanding of the strengths and weaknesses of each model.

In summary, the evaluation results highlight the effectiveness of Gradient Boosting Regressor and XGBoost Regressor in predicting load-carrying capacity. The visualizations complement the quantitative metrics, offering valuable insights into the models’ predictive capabilities and areas for refinement. These findings contribute to informed decision-making in structural analysis and design, emphasizing the importance of model selection for accurate load predictions.

Radar plot of model’s performance on training set.

Radar plot of model’s performance on testing set.

The SHAP analysis conducted on the predictive model elucidates the relative importance of each input feature in influencing the model’s output for Pu. The mean SHAP values offer quantitative insights into the magnitude and direction of the impact each feature has on the predictions. The results demonstrate that the eccentric loading ratio \(\:(e/2a)\) emerges as the most influential feature, with a mean SHAP value of 900.2548. This emphasizes the significant role that eccentric loading plays in determining load-carrying capacity. Following closely, yield strength of outer steel tube (\(\:{f}_{yo}\)) and inner width of internal steel tube (\(\:2{a}_{ii}\)) exhibit notable mean SHAP values of 184.6745 and 127.8848, respectively. Concrete strength (\(\:{f}_{c}{\prime\:}\)) and concrete area (\(\:{A}_{c}\)) also contribute significantly with mean SHAP values of 101.9914 and 73.6558, respectively.

Figure 27 depicts a summarized visualization of the mean SHAP values, providing a clear overview of each feature’s importance in influencing the model output. The plot reinforces the dominance of the eccentric loading ratio in shaping predictions. For a more detailed exploration, Fig. 28 presents a SHAP beeswarm plot, allowing a nuanced understanding of the distribution and spread of individual SHAP values for each feature. This visual representation aids in identifying specific instances where features exert a pronounced impact on predictions. The dominance of the eccentric loading ratio aligns with engineering intuition, underlining its pivotal role in structural behaviour. The substantial contributions of yield strength, inner dimensions, and concrete properties underscore the holistic nature of the predictive model, encapsulating various facets of structural configuration.

Mean SHAP summary plot of feature importance on model output.

SHAP beeswarm plot of feature importance on model output.

To facilitate the seamless utilization of the developed predictive model for load-carrying capacity (Pu), a user-friendly GUI has been meticulously crafted as shown in Fig. 29. This GUI allows users to effortlessly input values for all the 13 crucial features, including concrete area (\(\:{A}_{c}\)), concrete strength (\(\:{f}_{c}\)’), dimensions of internal and external steel tubes, yield strengths, and the eccentric loading ratio (\(\:e/2a\)). The GUI has been designed using the Tkinter library136, in Python 3 137 ensuring a responsive and intuitive interface for users. Tkinter provides a versatile set of tools for creating graphical interfaces, making it an ideal choice for building an interactive and user-friendly platform. For the convenience of users, the GUI tool is freely accessible on GitHub. Users can download, explore, and utilize the GUI to predict load-carrying capacity based on their specific structural configurations. The GUI streamlines the process of inputting parameters, making the predictive model accessible to users irrespective of their technical expertise. This user-friendly interaction enhances the practicality and applicability of the predictive model in real-world scenarios. The GUI tool for predicting load-carrying capacity (Pu) is available for free use on GitHub at the following link: https://github.com/tipu0003/Load-Carrying-Capacity-GUI.git. Users are encouraged to visit the repository to download the GUI, explore its features, and leverage the predictive model for structural analysis and design decisions.

GUI for the prediction of Pu.

This paper introduces a non-linear finite element model (FEM) for predicting the load-carrying capacity of three configurations of elliptical concrete-filled steel tubular (CFST) short columns. The configurations include: double steel tubes with sandwich concrete (CFDST), double steel tubes with sandwich concrete and concrete inside the inner steel tube, and a single outer steel tube with sandwich concrete. The model considers both pure axial compression and eccentric loading conditions. Using the validated FE model, 180 FE datasets were generated. a parametric study was performed to explore the influence of geometric and material parameters on the load-carrying capacity of elliptical CFST short column. Subsequently, two design formulas were developed to estimate the load-carrying capacity of elliptical CFST short columns under both concentric and eccentric loading conditions. Furthermore, seven ML algorithms—SVR, RFR, GBR, XGBR, MLPR, KNNR, and NBR—are employed to predict the load-carrying capacity of elliptical CFST short columns under different eccentric loading conditions. The SHapley Additive exPlanations (SHAP) approach is employed to pinpoint the influential input features that affect the predicted load-carrying capacity of elliptical CFST short columns. a user-friendly GUI tool has been developed that can be used by practicing engineers to calculate the load-carrying capacity of elliptical CFST short columns. The following outcomes are obtained from this study.

Based on a parametric study, higher concrete strength improves the ultimate load capacity but reduces ductility. Increasing the thickness and yield strength of the inner steel tube enhances the confinement effect, thereby increasing the compressive strength and load-bearing capacity of the column.

Increasing outer tube yield strength enhances sandwich concrete confinement, boosting column load capacity. However, increasing the hollow ratio from 23 to 50% reduces peak strength by 27%. Similarly, increasing the eccentric loading ratio from 0.0 to 50% decreases peak strength by 83% in elliptical CFST columns.

The proposed formulas significantly enhance load-carrying capacity predictions for three elliptical CFST column configurations compared to FE results. Mean \(\:{P}_{u,FE}/{P}_{u,pred}\) ratios are 0.998 and 1.018, with coefficients of variation of 0.025 and 0.115 for concentric and eccentric loading, respectively.

Gradient Boosting Regressor and XGBoost Regressor demonstrated outstanding predictive performance with high R2 scores. However, Random Forest Regressor closely followed with a commendable R2 score of 0.9801.

The results from SHAP indicate that the eccentric loading ratio (e/2a) has the most significant effect on the load-carrying capacity of elliptical CFST short columns, followed by the yield strength of the outer steel tube (\(\:{f}_{yo}\)) and the inner width of the inner steel tube (\(\:2{a}_{ii}\)).

A user-friendly GUI was developed using Tkinter, providing a convenient platform for users to input parameters and predict Pu.

All data, models, and code generated or used during the study appear in the submitted article.

Outer diameters of the outside elliptical steel tube in the long axe

Outer diameters of the inner elliptical steel tube in the long axe

Outer diameters of the outside elliptical steel tube in the short axe

Outer diameters of the inner elliptical steel tube in the short axe

Half of the outer dimensions of elliptical sections in the major axe

Total concrete area (mm2)

Nominal cross-sectional area of the concrete

Inner steel tube area (mm2)

Cross-sectional area of the outer elliptical steel tube

Half of the outer dimensions of elliptical sections in the minor axe

Equivalent diameter

Maximum coarse aggregate size

Flow potential eccentricity

Modulus of elasticity of the confined concrete

Elastic modulus

Stress in the strain hardening region of the confined concrete

Stress in the strain-softening region of confined concrete

Biaxial compressive strength

Unconfined compressive strength of cylinder concrete

Confined compressive strength

Characteristic compressive strength of the concrete

Characteristic compressive

Lateral confining pressure provided by the steel tube

Nominal steel stresses

Residual stress

Yield strength

Yield strength of the steel tube

True and nominal steel stresses

Yield strength of the inner steel tube

Yield strength of the outer steel tube

Fracture energy of concrete

Constant parameters taken as 4.1

Constant parameters taken as 20.4

Tensile/compressive meridian ratio

Mean Absolute Error

Mean Absolute Percentage Error

Mean Squared Error

Total number of instances in the dataset

Mean perimeter of the elliptical section

Ultimate load capacity of the columns from FE model

Ultimate load-carrying capacity of elliptical CFST columns under

Ultimate load-carrying capacity of elliptical CFST columns under eccentric loading conditions

Co-efficient of determination

Root Mean Squared Error

Thickness of the inner EST

Thickness of the outer EST

Actual \(\:{i}^{th}\) output from dataset

Predicted output against \(\:{y}_{i}\)

Nominal steel ratio

Strain in the strain hardening region of the confined concrete

Strain in the strain-softening region of confined concrete

Strains corresponding to unconfined concrete

Strains corresponding to confined concrete

Nominal strain

Strain

Yield strain of steel

True plastic strain

Confinement factor

Viscosity parameter

Desirable strength

Dilation angle

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College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, 350002, Fujian, China

Hazem Samih Mohamed

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

Tang Qiong

Jadara Research Center, Jadara University, Irbid, Jordan

Haytham F. Isleem

Department of Computer Science, University of York, York, YO10 5DD, UK

Haytham F. Isleem

Department of Civil Engineering, School of Engineering & Technology, K. R. Mangalam University, Gurugram, 122103, Haryana, India

Rupesh Kumar Tipu & Ramy I. Shahin

Department of Civil Engineering, Higher Institute of Engineering and Technology, Kafrelsheikh, Egypt

Saad A. Yehia

University Centre for Research and Development, Chandigarh University, Gharuan, 140413, Mohali, India

Pradeep Jangir

Department of CSE, Graphic Era Hill University, Graphic Era Deemed To Be University, Dehradun, 248002, Uttarakhand, India

Pradeep Jangir

Applied Science Research Center, Applied Science Private University, Amman, 11931, Jordan

Pradeep Jangir

Department of Biosciences, Saveetha School of Engineering. Saveetha Institute of Medical and Technical Sciences, Chennai, 602 105, India

Arpita

Department of Electrical Engineering, Imam Khomeini Naval Science University of Nowshahr, Nowshahr, Iran

Mohammad Khishe

Innovation Center for Artificial Intelligence Applications, Yuan Ze University, Taoyuan, Taiwan

Mohammad Khishe

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Mohamed, H.S., Qiong, T., Isleem, H.F. et al. Compressive behavior of elliptical concrete-filled steel tubular short columns using numerical investigation and machine learning techniques. Sci Rep 14, 27007 (2024). https://doi.org/10.1038/s41598-024-77396-5

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Received: 19 July 2024

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DOI: https://doi.org/10.1038/s41598-024-77396-5

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