Machine Learning-Based Strength Prediction of Round-Ended Concrete-Filled Steel Tube

Abstract

Round-ended concrete-filled steel tubes (RECFSTs) present very different performances between the primary and secondary axes, which renders them particularly suitable for use as bridge piers and arches. In recent years, research into RECFST heavily relies on experimental procedures restricting the parameter range under consideration, which narrows the far-reaching applicability of RECFST. This study employs advanced machine learning methods to predict the axial load-bearing capacity of RECFST with a wide parameter range. Firstly, a machine learning database comprising 2400 RECFSTs is established, which covers a wider range of commonly used material strengths and cross-sectional dimensions. Three machine learning prediction models of this database are then developed, respectively, using different algorithms. The robustness of the machine learning models is evaluated by predicting the axial load-bearing capacity of 60 RECFST specimens from existing references. The results demonstrated that the machine learning models provided superior predictive accuracy compared to theoretical or code-based formulas. A graphical user interface (GUI) is ultimately developed based on the machine learning prediction models to predict the axial load-bearing capacity of RECFST. This tool facilitates rapid and accurate RECFST design.

1. Introduction

A round-ended concrete-filled steel tube (RECFST) not only retains the excellent load-bearing capacity, ductility, and ease of construction associated with traditional circular concrete-filled steel tubes, but it is also suitable for applications requiring enhanced resistance along a single axis of the section. Compared to rectangular concrete-filled steel tubes [1,2], RECFSTs offer lower fluid resistance, making them advantageous in structures such as bridges exposed to fluid loads. Additionally, the combination of “arc” and “straight” edges enhances both versatility and esthetics in structural design, which has led to its increasing use in practical engineering. Notable examples in China include the Houhu Bridge in Wuhan, the Shennong Bridge (formerly Weihe Bridge) in Baoji, and the Qixia Bridge in Nanjing, as illustrated in Figure 1.

Figure 1. Application of RECFST: (a) Houhu Bridge; (b) Weihe Bridge; (c) Qixia Bridge.

Lu et al. [3,4,5,6] were among the pioneers in studying the performance of RECFST. Their research primarily focused on the axial performance, eccentricity performance, design methodology, and construction techniques of RECFST columns, which were implemented in the Houhu Bridge.

Subsequently, numerous studies have been conducted on this topic. Wang et al. [7,8] investigated the axial and bending performance of RECFST. Ding et al. [9,10,11] extended this work by performing a reliability analysis of the bearing capacity in addition to their research on axial and bending performance. Wang et al. [12,13,14,15,16] conducted a comprehensive series of studies on the axial, eccentric, hysteretic, and node connection performance. Zhao et al. [17,18,19,20] contributed with studies on axial and bending performance under impact, slender columns under compression, and ongoing research on the axial performance of RECFST components utilizing recycled aggregate concrete. Ren et al. [21] focused on the axial and eccentric performance of short- and medium-length columns, examining the influence of cross-sectional parameters on component performance. Hassanein and Patel [22] conducted a finite element study on axial performance, while Piquer et al. [23] performed experimental research on eccentric performance, incorporating an analysis of high-strength concrete. To improve the efficiency of finite element calculations, researchers such as Patel [24], Zhang [25], and Ahmed [26,27] developed a finite element model for RECFST based on the fiber beam element.

Axial performance, as a fundamental aspect of RECFST, has received significant attention in the aforementioned research. Wang et al. [7], Ding et al. [10], Ren et al. [21], and Hassanein et al. [22] proposed calculation formulas for the axial compression-bearing capacity of RECFST. However, these formulas are limited by specific parameter ranges, and their prediction accuracy outside these ranges requires further investigation. Consequently, there is a need for a new calculation method for RECFST bearing capacity that can accommodate a wider range of parameters.

With the advent of the big data era, machine learning, a key branch of artificial intelligence, is widely used to solve various prediction, classification, and clustering problems [28]. In recent years, machine learning has increasingly been applied in structural engineering [29,30], with numerous researchers utilizing these techniques to predict the bearing capacity of concrete-filled steel tubes (CFSTs). While most machine learning studies focus on circular CFST [31,32,33,34,35,36], other cross-sectional shapes, such as square, rectangular, and elliptical, have also been investigated [37,38,39,40,41,42,43,44]. These studies primarily address axial performance, though some also explore other loading conditions [45,46,47,48,49,50], including eccentric compression, bending, and torsion. The research encompasses various CFST types, such as concrete-filled double-skin tubes [51,52], FRP-CFST [53,54], CFST with recycled aggregate concrete [55], and concrete-encased CFST [56]. Most machine learning studies [31,32,33,34,35,36,37,38,39,40,41,42,44,46,47,49,53,55] use experimental data for training, while some incorporate both experimental data and finite element models [43,48,50,51,52,54], or rely solely on finite element models [45]. The integration of machine learning with the mechanical performance of CFST has opened new avenues for the field, offering fresh insights into the study of RECFST performance.

This study employs three advanced machine learning techniques to predict the axial load-bearing capacity of RECFST. Using a validated finite element modeling (FEM) method and within the commonly used parameter range for RECFST structures, 2400 FEM models were developed to determine their axial load-bearing capacities. These models formed the basis of a machine learning database, with 80% of the data allocated for training and 20% for testing. Hyperparameter optimization for the three machine learning techniques was performed using grid search methods combined with fivefold cross-validation. The optimized prediction models were subsequently applied to predict the axial load-bearing capacity of 60 collected RECFST specimens. The results were compared with those derived from the existing standards and theoretical formulas, demonstrating superior accuracy. Finally, a graphical user interface (GUI) was created, incorporating the optimal prediction models to facilitate the rapid and accurate calculations of the axial load-bearing capacity of RECFST structures. The research flow of this study is illustrated in Figure 2.

Figure 2. Research flow.

2. Database Establishment

As is known, the performance of the predictive model in machine learning is better with a larger sample size. The number of experimental samples of RECFSTs currently available for retrieval is relatively limited because research on RECFSTs has only begun to gain traction in recent years. By selecting an appropriate parameter range, a database of 2400 RECFST finite element (FE) models was established, and the axial compression-bearing capacity data were obtained to facilitate machine learning for predicting the load-bearing capacity.

2.1. Finite Element Model

This study selects specimens from different references as reference samples, namely WST1-A from Ref. [10], RR-1-180-6 from Ref. [21], and RCFST-1 from Ref. [7]. The cross-sectional dimensions and material strength information of the above three specimens are shown in Figure 3 and Table 1. The finite element models of these three specimens were established, and the validity of the finite element modeling method was verified by comparing them to the experimental results.

Figure 3. Cross-section of RECFST.

Table 1. Cross-sectional information of selected specimens.

Specimens	fcu	fy	B	D	t	Nu_test	Nu_FE	Nu_FE/Nu_test
WST1-A	40.4	327.7	47	252	3.75	3429	3395.56	0.990
RR-1-180-6	31	290	150	160	6	2319	2388.66	1.030
RCFST-1	38.06	324.6	51.5	117	2.86	925	988.265	1.068

2.1.1. Selection of Elements and Mesh Division

RECFST can be divided into exterior steel tubes and core concrete. In this paper, the shell element S4R is used to simulate the exterior steel tubes, and the solid element C3D8R is used to simulate the core concrete. Both the concrete and the steel tube have the same mesh size settings. To quickly establish a parametric finite element database, the mesh size (l_mesh) is set to 1/10 of the short side of the section, i.e., l_mesh = 0.1 × min (H, D), indicating that at least 10 grids have been divided in one direction of the specimen section. The mesh division of the specimen is shown in Figure 4. To ensure mesh size convergence, the mesh number was doubled, and the results were compared. The small difference between the results (within 1%) suggests that the mesh size (l_mesh) satisfies the accuracy requirements.

Figure 4. Mesh division of the specimen.

2.1.2. Material and Constitutive Properties

(1)

Concrete

In this study, the Concrete Damaged Plasticity (CDP) model in the ABAQUS software (Version 6.14) is used for simulating the concrete. The specific parameters in the CDP model include dilation angle, eccentricity, f_b0/f_c0, K, and viscosity parameter, which are calibrated and set as 23, 0.1, 1.16, 0.667, and 0.0001, respectively.

The confinement of the enclosing steel tube greatly enhances the compression behavior and strength of the core concrete in CFST. This enhancement necessitates a reliance on confined concrete constitutive models for accurate representation. The strength of confined concrete depends on the level of confinement, which in CFST primarily comes from the shape of the outer steel tube. Circular CFSTs provide substantial confinement through the interaction between the steel tube and concrete, significantly enhancing the strength of the core concrete. In contrast, square or rectangular CFSTs offer less confinement, resulting in a more modest increase in core concrete strength. The cross-sectional shape of RECFST is intermediate between circular and rectangular configurations. According to Wang’s study [14], when the aspect ratio (H/D) exceeds 1.44, the strength evolution of the core concrete in RECFST resembles that of rectangular CFST. In contrast, when the aspect ratio is 1.44 or lower, the strength progression of the core concrete is similar to that of circular CFST.

The concrete constitutive model in this study is classified into two scenarios: when H/D > 1.44, the confined concrete constitutive for rectangular CFST is adopted; when H/D ≤ 1.44, the circular CFST model is applied. The constitutive relationships for confined concrete, referenced as [57], are presented in Equations (1)–(7).

$$y=\left\{\begin{array}{ll}2x-x^2\hfill & (x\le 1)\hfill \\ {\displaystyle \frac{x}{{\beta }_0{(x-1)}^{\eta }+x}}\hfill & (x>1)\hfill \end{array}\right.$$

(1)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_0}}\hspace{1em}y={\displaystyle \frac{\sigma }{{\sigma }_0}}$$

(2)

$${\sigma }_0=f_{\mathrm{c}}^{\prime }$$

(3)

$${\epsilon }_0={\epsilon }_{\mathrm{c}}+800{\xi }^{0.2}\times {10}^{-6}$$

(4)

$${\epsilon }_{\mathrm{c}}=\left(1300+12.5f_{\mathrm{c}}\right)\times {10}^{-6}$$

(5)

$$\eta =\left\{\begin{array}{ll}2\hfill & (H/D<1.44)\hfill \\ 1.6+1.5/x\hfill & (H/D\ge 1.44)\hfill \end{array}\right.$$

(6)

$${\beta }_0=\left\{\begin{array}{ll}{\left(2.36\times {10}^{-5}\right)}^{\left[0.25+{(\xi -0.5)}^7\right]}\times {\left(f_c^{\prime }\right)}^{0.5}\times 0.5\ge 0.12\hfill & (H/D<1.44)\hfill \\ {\displaystyle \frac{{\left(f_c^{\prime }\right)}^{0.1}}{1.2\sqrt{1+\xi }}}\hfill & (H/D\ge 1.44)\hfill \end{array}\right.$$

(7)

For the tensile constitutive model of concrete, the concrete fracture energy (G_f)–cracking displacement (u_t) relationship model provided in the CDP model has better computational convergence [58]. Therefore, the fracture energy cracking criterion (GFI) is used in this paper to describe the tensile constitutive relationship of concrete.

The fracture energy of concrete refers to the energy required to extend the crack per unit area. The concrete fracture energy G_f is calculated according to the following equation [59]:

$$G_{\mathrm{f}}=a\cdot ({\displaystyle \frac{f_{\mathrm{c}}^{\prime }}{10}}{)}^{0.7}\times {10}^{-3}$$

(8)

$$a=1.25d_{\max }+10$$

(9)

where d_max is the particle size of the coarse aggregate in the concrete; f_c′ is the compressive strength of the concrete cylinder.

The peak tensile stress σ_p of the concrete is calculated according to the following formula [60]:

$${\sigma }_{\mathrm{p}}=0.26(1.5f_{\mathrm{ck}}{)}^{2/3}$$

(10)

In which, f_ck is the standard value of the axial compressive strength of concrete (MPa); the relationship between stress and displacement after the concrete cracks is shown in Figure 5.

Figure 5. Relationship between G_f and u_t.

(2)

Steel

Assuming that steel is an isotropic material, for its elastic stage performance, it is necessary to define Young’s model and Poisson’s ratio. In this paper, Young’s modulus E_s and Poisson’s ratio v are taken to be 206 GPa and 0.3, respectively.

In the event of steel transitioning into the plastic working state, researchers currently adopt various stress–strain models, encompassing the ideal elastic–plastic model [61] and the elastic–plastic model featuring a strain hardening section [62,63]. For this study, Han’s five-stage stress–strain relationship [57] was implemented as the constitutive model for steel, as shown in Equations (11)–(13). This model encompasses elasticity, elasto-plasticity, plasticity, hardening, and fracture stages, as graphically represented in Figure 6.

$${\sigma }_{\mathrm{s}}=\left\{\begin{array}{ll}{E}_{\mathrm{s}}{\epsilon }_{\mathrm{s}}& {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{e}}\\ -A{{\epsilon }_{\mathrm{s}}}^2+B{\epsilon }_{\mathrm{s}}+C& {\epsilon }_{\mathrm{e}}<{\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{e}1}\\ f_{\mathrm{y}}& {\epsilon }_{\mathrm{e}1}<{\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{e}2}\\ f_{\mathrm{y}}[1+0.6{\displaystyle \frac{{\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{e}2}}{{\epsilon }_{\mathrm{e}3}-{\epsilon }_{\mathrm{e}2}}}]& {\epsilon }_{\mathrm{e}1}<{\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{e}2}\\ 1.6f_{\mathrm{y}}\hfill & {\epsilon }_{\mathrm{s}}>{\epsilon }_{\mathrm{e}3}& \end{array}\right.$$

(11)

$${\epsilon }_{\mathrm{e}}=0.8f_{\mathrm{y}}/E_{\mathrm{s}},\hspace{1em}{\epsilon }_{\mathrm{e}1}=1.5{\epsilon }_{\mathrm{e}},\hspace{1em}{\epsilon }_{\mathrm{e}2}=10{\epsilon }_{\mathrm{e}1},\hspace{1em}{\epsilon }_{\mathrm{e}3}=100{\epsilon }_{\mathrm{e}1}$$

(12)

$$A=0.2f_{\mathrm{y}}/{\left({\epsilon }_{\mathrm{e}1}-{\epsilon }_{\mathrm{e}}\right)}^2,\hspace{1em}B=2A{\epsilon }_{\mathrm{e}1}\hspace{1em},\hspace{1em}C=0.8f_{\mathrm{y}}+A{\epsilon }_{\mathrm{e}}^2-B{\epsilon }_{\mathrm{e}}$$

(13)

Figure 6. Constitutive relation model of steel.

2.1.3. Contact and Boundary Conditions

In the finite element simulations of steel structures or concrete-filled steel tube structures, it is sometimes necessary to simulate welds [64] in order to accurately represent the structural form. However, when the focus of the study is not on the welds, for the sake of convenience, it is often assumed that no failure occurs at the welds. As a result, welds are not considered in the model, simplifying the finite element model. This paper follows the latter approach. For the contact between the steel tube and the concrete, normal “hard” contact and tangential penalty functions can usually be set for modeling. Additionally, there is also a method to directly “tie” the inside surface of the steel tube to the outside surface of the concrete. In the axial compression of CFST stub columns, the interaction between the steel tube and concrete primarily involves mutual squeezing, with tangential relative slippage occurring rarely. According to Yu’s research [65], little difference exists in the axial load-bearing capacity of CFST using the above two methods for simulating contact between steel and concrete. Consequently, in this paper, the “tie” contact is used.

Reference points, termed as RP-Top and RP-Bottom, are established at the top and bottom of the component, respectively. The RP-Top is rigidly attached to the top steel tube and concrete node, and similarly, the RP-Bottom is rigidly fastened to the bottom steel tube and concrete node. This configuration establishes a relationship between the degrees of freedom at the component’s top and bottom and their respective reference points, RP-Top and RP-Bottom. Through the allocation of the degrees of freedom for RP-Top and RP-Bottom, the boundary conditions for the component are effectively established. The finite element models of the three specimens are shown in Figure 7. In experiments, the base of the specimen is typically fixed, while the top is either hinged or free. According to Refs. [66,67], these boundary conditions have minimal impact on the axial bearing capacity of the stub column. Therefore, in this study, the finite element model is uniformly defined with a fixed bottom and a free top. In this paper, a general static analysis type is used, applying axial displacement loads to the reference point at the top of the column (RP-Top) to simulate the axial compression process of the RECFST.

Figure 7. Finite element models of the three selected specimens.

2.2. Verification of FE Model

The load–deformation curves of the three specimens obtained from the finite element model are compared with the measured curves in the Refs. [7,10,21], as shown in Figure 8. As can be seen from the figure, apart from some differences in the descending segment, the finite element curve fits well with the measured curve, including its initial stiffness and peak load, which are quite close.

Figure 8. Comparison between FE model and tests.

Table 1 shows a comparison of the measured peak loads of the three specimens and the results calculated from the finite element model. The measured peak loads of the specimens WST1-A, RR-1-180-6, and RCFST-1 are, respectively, 3429 kN, 2319 kN, and 925 kN, while the finite element calculation results are 3395.56 kN, 2388.66 kN, and 988.27 kN, with errors of −0.96%, 3.00%, and 6.84%, respectively. Judging from the load–deformation curves and the comparison results of the peak load, the established finite element model can well simulate the axial compression process of RECFST, especially in calculating its axial bearing capacity with better accuracy.

2.3. Establishment of Machine Learning Database

This paper employs a validated finite element modeling method to create numerous finite element models of RECFST under axial compression for machine learning purposes. According to Chinese code GB50017-2017 [68], commonly used steel grades in mainland China include Q235, Q355, Q390, Q420, and Q460, corresponding to the yield strengths of f_y = 235, 355, 390, 420, and 460 MPa, respectively. Likewise, commonly used concrete strengths [69] are f_cu = 30, 40, 50, 60, 70, and 80 MPa. Thus, the established finite element database in this study should include the aforementioned material strengths.

The selected cross-sectional dimensions for RECFST columns are D = 100, 300, 500, and 700 mm; B = 100, 300, 500, and 700 mm; and t = 2, 4, 6, 8, and 10 mm. These dimensions are widely used in engineering applications and encompass most current engineering data. Ref. [70] notes that as the slenderness ratio of a component increases, its failure mode shifts from sectional strength failure to overall instability under axial load, significantly reducing its axial bearing capacity. This scenario differs from the strength concerns primarily addressed in this paper. Thus, to avoid overall instability when establishing the finite element database, the component length is uniformly set to L = 3 × min (D, H).

In summary, the parameters changing in the finite element model database established in this paper for the RECFST stub column are f_cu = 30, 40, 50, 60, 70, 80 MPa; f_y = 235, 355, 390, 420, 460 MPa; D = 100, 300, 500, 700 mm; B = 100, 300, 500, 700 mm; t = 2, 4, 6, 8, 10 mm. A grid combination, demonstrated in Figure 9, is utilized, and all the parameters have been extensively examined. This has resulted in the establishment of 6 × 5 × 4 × 4 × 5 = 2400 finite element models, thereby forming the database for machine learning around the axial compression-bearing capacity of RECFST. By using the core language Python 2.7 in ABAQUS 6.14, parameterized modeling code was developed to automate the modeling of the 2400 finite element models, significantly reducing the time required for manual operations.

Figure 9. Statistical distributions of the parameters for the RECFSTs: (a) f_cu; (b) f_y; (c) B; (d) D; (e) t.

3. Machine Learning

3.1. Sensitivity Analysis

The key parameters of the RECFST stub columns are f_cu, f_y, D, B, and t. These parameters collectively define all the properties of a RECFST, encompassing the cross-sectional shape and material strength, thereby determining the axial compression-bearing capacity of the RECFST. Hence, these variables are known as the primary parameters. In addition, there are some parameters that also have a significant impact on the axial compression bearing capacity of RECFST. These parameters are formed by the combination of the primary parameters and are referred to as the secondary parameters, such as A_s, A_c, A_sf_y, A_cf_cu, A_sf_y/(A_cf_cu), and D/B. A sensitivity analysis of both the primary and secondary variables regarding the axial compression-bearing load is conducted to improve the prediction accuracy of the machine learning model, aiding in the selection of parameters for the final training model.

Correlation analysis is a statistical method that can quantify the correlation between two variables. The Pearson correlation coefficient [71] is defined as follows:

$$r={\displaystyle \frac{n{\displaystyle \sum _{i=1}^n{x}_{\mathrm{i}}}{y}_{\mathrm{i}}-{\displaystyle \sum _{i=1}^n{x}_{\mathrm{i}}}{\displaystyle \sum _{i=1}^n{y}_{\mathrm{i}}}}{\sqrt{n{\displaystyle \sum _{i=1}^n{x}_{\mathrm{i}}^2}-{\left({\displaystyle \sum _{i=1}^n{x}_{\mathrm{i}}}\right)}^2}\sqrt{n{\displaystyle \sum _{i=1}^n{y}_{\mathrm{i}}^2}-{\left({\displaystyle \sum _{i=1}^n{y}_{\mathrm{i}}}\right)}^2}}}$$

(14)

where x_i and y_i represent the two calculation parameters, and n is the total number of samples. The closer r is to 1, the stronger the positive correlation; the closer r is to 0, the weaker the correlation; and the closer r is to −1, the stronger the negative correlation.

Figure 10 shows the Pearson correlation coefficient between each parameter, in the figure, a deeper shade of green indicates a stronger correlation. In addition to the primary parameter that determines the properties of RECFST, the second parameters such as A_cf_cu, A_c + A_s, A_c, A_s, A_sf_y, etc., all show a large correlation with the axial compressive capacity N_u, with their Pearson coefficients being 0.968, 0.909, 0.904, 0.694, and 0.684, respectively, as shown in Figure 11. This is relatively in line with the calculation formulas of the standards of various countries [72,73,74] and the empirical formulas [10,21,22]. Most of these calculation formulas contain the abovementioned secondary parameters, as listed in Table 2.

Figure 10. Correlation analysis between different parameters.

Figure 11. Correlation coefficient between ultimate capacity and different parameters.

Table 2. Comparison of input parameters between the empirical formulas and existing standards.

Calculation Model	As	Ac	Acfcu or Acfc or Acfc’	Asfy	A = As + Ac	Asfy/Acfcu
Ding [10]	×	√	√	×	×	√
Ren [21]	×	√	√	×	×	√
M.F. Hassanein [22]	×	×	×	×	×	×
ACI318-11 [72]	×	×	√	√	×	×
GB 50936-2014 [73]	×	×	×	×	√	√
AISC 360-16 [74]	×	×	√	√	×	×

Based on the outcomes of the sensitivity analysis and insights from the existing calculation methods, this study employs the following parameters as input for machine learning training: f_cu, f_y, D, B, t, A_cf_cu, and A_sf_y. The primary parameters, which define the cross-sectional geometry and material strength of RECFST, are f_cu, f_y, D, B, and t. A_cf_cu and A_sf_y serve as the secondary parameters, representing the independent compression capacity of concrete and steel tube, respectively.

3.2. Selected Algorithm

3.2.1. Gradient Boosting Regression (GBR)

Gradient Boosting Regression (GBR) is an ensemble learning method which builds a powerful predictive model by combining multiple weak learners. In its simplest form, these weak learners can be decision trees, but other learning algorithms can also be used. The core idea of this method is that each new model strives to correct the prediction errors of all the previous models. The main advantage of GBR is that it can handle various types of prediction problems (including classification, regression, etc.), and can naturally handle mixed types of features (i.e., different types of features such as categories and numbers in the same data set). In addition, GBR has strong robustness to missing values and outliers.

3.2.2. Random Forest Regression (RFR)

The random forest regression (RFR) algorithm is an important application branch of random forest. The random forest regression model obtains prediction results in parallel by randomly sampling and extracting features to build many unrelated decision trees. Each decision tree can obtain a prediction result through the extracted samples and features, and the regression prediction result of the entire forest can be obtained by averaging the results of all the trees. The main advantage of RFR is that it hardly requires input preprocessing, can handle high-dimensional (i.e., a lot of features) data, and can handle missing values. Moreover, because of the use of ensemble learning, RFR is generally not prone to overfitting, and the model has strong robustness.

3.2.3. Extreme Gradient Boosting (XGB)

Extreme Gradient Boosting (XGB) is one of the most widely used machine learning algorithms. After Friedman [75] proposed this tree-based gradient boosting algorithm, it was improved and optimized by Chen and Guestrin [76], making XGB more efficient to operate and apply to a variety of problems.

The main architecture of XGB can be summarized into three parts: feature selection, tree building, and model optimization. The workflow of the entire architecture mainly consists of the following two stages: one is setting goals according to the loss function and making predictive output using tree-based models; the other is providing feedback based on the gap between the actual output and the expected output, and model optimization is carried out using gradient boosting algorithms.

One of the main features of XGB is its high degree of customizability and flexibility. In addition to the built-in objective functions, users can also set custom objective functions for optimization. At the same time, XGB incorporates regularization parameters to help prevent model overfitting and uses pruning technology. The impacts caused by model overfitting can be directly ignored, reducing the difficulty of model tuning.

3.3. Evaluation Metrics and Hyperparameter Optimization

3.3.1. Evaluation Metrics

The evaluation of the model’s predictive performance generally involves the following parameters. In these equations [77], $N_{\mathrm{u}}^{\mathrm{FE}}$ and $N_{\mathrm{u}}^{\mathrm{ML}}$ are the axial compression bearing capacity RECFST obtained by the FE model and machine learning model, respectively.

(1)

Coefficient of determination (R²)

$$R^2={\displaystyle \frac{{\left[{\displaystyle \sum _{i=1}^n\left(N_{\mathrm{ui}}^{\mathrm{FE}}-\overline{N_{\mathrm{u}}^{\mathrm{FE}}}\right)}\left(N_{\mathrm{ui}}^{\mathrm{ML}}-\overline{N_{\mathrm{u}}^{\mathrm{ML}}}\right)\right]}^2}{\left[{\displaystyle \sum _{i=1}^n{\left(N_{\mathrm{ui}}^{\mathrm{FE}}-\overline{N_{\mathrm{u}}^{\mathrm{FE}}}\right)}^2}\right]\left[{\displaystyle \sum _{i=1}^n{\left(N_{\mathrm{ui}}^{\mathrm{ML}}-\overline{N_{\mathrm{u}}^{\mathrm{ML}}}\right)}^2}\right]}}$$

(15)

R² is between 0 and 1. The closer R² is to 1, the greater the correlation between the predicted value and the FE value, reflecting that the performance of the prediction model is better.

(2)

Mean absolute error (MAE)

$$MAE={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left|N_{\mathrm{ui}}^{\mathrm{ML}}-N_{\mathrm{ui}}^{\mathrm{FE}}\right|}}{n}}$$

(16)

MAE shows the error between the ML predicted value and the FE value and avoids the mutually offsetting problem between over-estimating errors and under-estimating ones. The smaller the MAE, the better the performance of the prediction model.

(3)

Root mean square error (RMSE)

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(N_{\mathrm{ui}}^{\mathrm{ML}}-N_{\mathrm{ui}}^{\mathrm{FE}}\right)}^2}}{n}}}$$

(17)

RMSE is somewhat similar to MAE, both reflecting the error between the ML predicted value and the FE value. Because squaring amplifies the weights of larger errors, it is more sensitive to outliers. The smaller the RMSE, the better the predictive performance of the model.

(4)

Mean (MEAN)

$$MEAN={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(N_{\mathrm{ui}}^{\mathrm{ML}}/N_{\mathrm{ui}}^{\mathrm{FE}}\right)}}{n}}$$

(18)

MEAN is the average of the ratio of the ML predicted value to the FE value. The closer it is to 1, the better the prediction performance of the model.

(5)

Coefficient of variation (COV)

$$COV=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left[N_{\mathrm{ui}}^{\mathrm{ML}}/N_{\mathrm{ui}}^{\mathrm{FE}}-MEAN\right]}^2}}{n}}}/MEAN$$

(19)

COV reflects the degree of dispersion of the ratio of the predicted value to the FE value. The smaller the COV, the lower the degree of data dispersion. The closer the MEAN is to 1 and the smaller the COV, the better the predictive performance of the model.

3.3.2. Hyperparameter Optimization

To avoid overfitting, underfitting, or achieving local optimum problems for the above models, the grid search method combined with fivefold cross-validation is adopted to optimize the hyperparameters of the above models. The objective of optimization is with the smallest MAE. Taking the GBR model as an example, the key hyperparameter learning_rate is set as 0.01, 0.10, 0.5, and 1; n_estimators is set as 100, 200, 300, and 10,000; and max_depth is set as 2,3, 5, 10, and 20. The training set is randomly divided into 5 folds, of which 4 folds are used as the training set, and the remaining 1 fold is used as the validation set. All the combinations of the above key hyperparameters are traversed to obtain the best hyperparameters. The selection of key hyperparameter ranges and the optimal hyperparameter values for the three models are shown in Table 3.

Table 3. Optimization of key hyperparameters.

ML Model	Key Hyperparameters Ranges	Best Key Hyperparameters
GBR	learning_rate: 0.01, 0.10, 0.5, 1	learning_rate: 0.1
	n_estimators: 100, 200, 300, 10,000	n_estimators: 300
	max_depth: 2,3, 5, 10, 20	max_depth: 5
RFR	n_estimators:10~1000	n_estimators:43
	max_depth:1~20	max_depth: 20
	min_samples_split: 2~11	min_samples_split: 2
XGB	learning_rate: 0.01, 0.10, 0.5, 1	learning_rate: 0.1
	n_estimators: 100, 200, 300, 10,000	n_estimators: 10,000
	max_depth: 2, 3, 5, 10, 20	max_depth: 5

3.4. Comparison of Results

The machine learning process implemented in this paper is all based on the Python platform, and all three models randomly select 80% of the sample size in the database as the training set, while the remaining 20% of the samples serve as the test set.

The calculated evaluation indexes of each model are shown in Table 4. The comparison of the coefficient of determination of each model is shown in Figure 12. For the training set, the R² of the three models all exceed 0.999. As for the test set, the XGB model achieves the best result, followed by GBR, and the R² of the RFR model is the smallest, at 0.9985. Overall, whether it is the test set or the training set, the R² of the three models is quite close to one.

Table 4. Evaluation index of the machine learning models.

ML Model	Evaluation Index
	Coefficient of Determination (R2)	Mean Absolute Error (MAE)	Root Mean Square Error (RMSE)	Mean (MEAN)	Coefficient of Variation (COV)
	Training Set /Testing Set	Training Set /Testing Set	Training Set /Testing Set	Training Set /Testing Set	Training Set /Testing Set
XGB	0.9998/0.9991	129/230	200/411	1.0011/1.0013	0.0091/0.0180
GBR	0.9995/0.9989	208/269	304/429	1.0005/1.0004	0.0188/0.0224
RFR	0.9997/0.9985	114/292	195/506	1.0002/0.9999	0.0091/0.0226

Figure 12. Coefficient of determination of different models.

Figure 13 shows the comparison of the MAE and RMSE of each model’s training set and test set. It can be seen that the XGB model achieves better results in both the training set and test set, with the smallest MAE and RMSE in the test set among the three models. The RFR model performs best in the training set but performs worst in the test set (its MAE and RMSE in the training set are the smallest among the three models, but the largest in the test set). The GBR model performs worst in the training set but performs relatively well in the test set.

Figure 13. Mean absolute error and root mean square error of different models: (a) MAE; (b) RMSE.

Figure 14 shows the comparison of the MEAN and COV of each model. The MEAN of the training and test sets of XGB and GBR are both greater than one. RFR has the smallest COV in the training set, but the largest COV in the test set, indicating that its test set has the largest dispersion of prediction results, and the difference between the results of the RFR training set and test set is the largest, which is caused by its algorithm characteristics.

Figure 14. Mean and coefficient of variation of different models.

The errors between the prediction results and the true values of the training set and test set of the three models are counted, where error = ($N_{\mathrm{u}}^{\mathrm{ML}}$−$N_{\mathrm{u}}^{\mathrm{FE}}$)/$N_{\mathrm{u}}^{\mathrm{FE}}$. The statistical results are shown in Table 5. It can be seen that overall, the prediction results of the three models are satisfactory. The error of the GBR training set, RFR training set, XGB training set, GBR testing set, RFR testing set, and XGB testing set are 2.4%, 0.1%, 0%, 2.7%, 3.5%, and 1.9%, respectively, and the proportion of errors exceeding 5% is less than 5%. In addition, it is worth noting that the results of the training set and test set of the XGB model show a significant difference: all the errors of the predicted values in the training set are within 0.5%, while about 65.8% of the predicted values in the test set have errors exceeding 0.5%.

Table 5. Predicted errors of different models.

ML Model		Percentage of Error (%)
		≤0.5%	0.5%~1%	1%~5%	≥5%
GBR	Training set	28.2	21.9	47.6	2.4
	Testing set	20.8	18.3	58.1	2.7
RFR	Training set	55.3	24.7	19.9	0.1
	Testing set	24.0	18.5	54.0	3.5
XGB	Training set	100	0	0	0
	Testing set	34.2	21.7	42.3	1.9

4. Comparison with Existing Calculation Methods

The three trained predictive models featured in this study can be employed to swiftly predict the axial load-bearing capacity of new RECFST members. These models can quickly generate results for RECFST members in seconds, offering a significant advantage over finite element models that may require dozens of minutes or more for each load-bearing capacity calculation.

In this study, 60 RECFST specimens from existing references [7,10,21,78,79,80,81,82,83] have been collected along with their axial load test results. The axial load-bearing capacity for these specimens is determined through three machine learning-based predictive models. Predictions from these models were then compared with actual experimental outcomes, as collated in Table 6. In the table, N_u is the test result, and $N_{\mathrm{u}}^{\mathrm{RFR}}$, $N_{\mathrm{u}}^{\mathrm{XGB}}$, and $N_{\mathrm{u}}^{\mathrm{GBR}}$ are the results predicted, respectively, by the RFR model, XGB model, and GBR model.

Table 6. Comparison between test results and results of prediction models.

Ref.	Specimens	${\mathit{N}}_{\mathbf{u}}$ (kN)	${\mathit{N}}_{\mathbf{u}}^{\mathbf{RFR}}$ (kN)	$\frac{{\mathit{N}}_{\mathbf{u}}^{\mathbf{RFR}}}{{\mathit{N}}_{\mathbf{u}}}$	${\mathit{N}}_{\mathbf{u}}^{\mathbf{XGB}}$ (kN)	$\frac{{\mathit{N}}_{\mathbf{u}}^{\mathbf{XGB}}}{{\mathit{N}}_{\mathbf{u}}}$	${\mathit{N}}_{\mathbf{u}}^{\mathbf{GBR}}$ (kN)	$\frac{{\mathit{N}}_{\mathbf{u}}^{\mathbf{GBR}}}{{\mathit{N}}_{\mathbf{u}}}$
[10]	WST1-A	3429	3326	0.97	3182	0.93	3327	0.97
	WST1-B	3338	3326	1.00	3182	0.95	3327	1.00
	WST2-A	4162	3790	0.91	3549	0.85	3570	0.86
	WST2-B	4168	3760	0.90	3537	0.85	3570	0.86
	WST3-A	3929	3712	0.94	3586	0.91	3701	0.94
	WST3-B	4158	3686	0.89	3729	0.90	3960	0.95
	WST4-A	4492	4381	0.98	4361	0.97	4405	0.98
	WST4-B	5530	5013	0.91	5031	0.91	5048	0.91
	WST5-A	5620	5431	0.97	5158	0.92	5506	0.98
	WST5-B	5500	5484	1.00	5158	0.94	5629	1.02
	WST6-A	3240	3655	1.13	3341	1.03	3618	1.12
	WST6-B	2993	3655	1.22	3332	1.11	3681	1.23
	WST7-A	4826	5437	1.13	5457	1.13	5305	1.10
	WST7-B	4944	5437	1.10	5457	1.10	5305	1.07
	WST8-A	6521	6873	1.05	6345	0.97	6478	0.99
	WST8-B	6493	6838	1.05	6229	0.96	6478	1.00
	WST9-A	4203	4709	1.12	4832	1.15	4829	1.15
	WST9-B	4180	4705	1.13	4846	1.16	4892	1.17
	WST10-A	7201	6847	0.95	6484	0.90	7165	1.00
	WST10-B	6905	6465	0.94	6483	0.94	6120	0.89
	WST11-A	9065	8730	0.96	8160	0.90	9338	1.03
	WST11-B	8799	8851	1.01	9219	1.05	9338	1.06
[78]	CFRT1-A	3429	3326	0.97	3182	0.93	3327	0.97
	CFRT1-B	3338	3326	1.00	3182	0.95	3327	1.00
	CFRT2-A	4162	3790	0.91	3549	0.85	3570	0.86
	CFRT2-B	4168	3760	0.90	3549	0.85	3570	0.86
	CFRT3-A	3929	3712	0.94	3556	0.91	3701	0.94
	CFRT3-B	4158	3724	0.90	3556	0.86	3901	0.94
	CFRT4-A	4492	4434	0.99	4361	0.97	4345	0.97
	CFRT4-B	5530	5013	0.91	5031	0.91	5048	0.91
	CFRT5-A	5620	5431	0.97	5158	0.92	5629	1.00
	CFRT5-B	5500	5484	1.00	5158	0.94	5629	1.02
[79]	C1	1339	1256	0.94	1251	0.93	1158	0.86
	C2	1444	1465	1.01	1493	1.03	1372	0.95
	C3	1755	1814	1.03	1787	1.02	1756	1.00
	C4	1825	1822	1.00	1748	0.96	1840	1.01
	C5	2125	2187	1.03	2190	1.03	2285	1.08
	C6	2319	2904	1.25	2696	1.16	2976	1.28
	C7	1623	1655	1.02	1618	1.00	1610	0.99
	C8	1954	2145	1.10	2158	1.10	2043	1.05
[80]	RCFST-A-0	2094	2118	1.01	2028	0.97	1948	0.93
[7]	RCFST-1	925	1047	1.13	947	1.02	1026	1.11
	RCFST-2	1215	1153	0.95	1357	1.12	1127	0.93
	RCFST-3	1635	1902	1.16	1917	1.17	2047	1.25
	RCFST-4	1658	1696	1.02	1566	0.94	1778	1.07
	RCFST-5	2091	2342	1.12	2009	0.96	2231	1.07
[21]	RR-1-180-4	1755	1826	1.04	1885	1.07	1756	1.00
	RR-1-180-6	2319	2630	1.13	2608	1.12	2618	1.13
	RR-0.5-180-6	1954	1997	1.02	2017	1.03	2038	1.04
	RR-0.5-180-4	1623	1465	0.90	1493	0.92	1372	0.85
[81]	RRCFST-A-180-1	2026	2214	1.09	2338	1.15	2250	1.11
	RRCFST-A-180-2	1915	2214	1.16	2334	1.22	2316	1.21
	RRCFST-C-180-1	1574	1655	1.05	1761	1.12	1704	1.08
[82]	PY1-180-e0-fy235	1780	1720	0.97	1735	0.97	1736	0.98
	PY1-180-fy345	2100	1928	0.92	2076	0.99	2036	0.97
	PYRE1-180-fy345	2060	1928	0.94	2076	1.01	2036	0.99
[83]	RCC1-4-180	1755	1814	1.03	1729	0.99	1756	1.00
	RCC1-6-180	2319	2523	1.09	2529	1.09	2503	1.08
	RCC0.5-4-180	1623	1459	0.90	1251	0.77	1351	0.83
	RCC0.5-6-180	1954	1904	0.97	1823	0.93	1865	0.95
	Mean			1.01		0.99		1.01
	Standard deviation			0.09		0.10		0.10

In this study, six theoretical methods are employed to calculate the axial compression-bearing capacity of RECFST, including formulas proposed by Ding et al. [10], Ren et al. [21], and Hassanein et al. [22], as well as formulas specified in different national standards (ACI 318-11 [72], GB 50936-2014 [73], and AISC 360-16 [74]), totaling from Equations (20)–(28). These methods are utilized to predict the axial load-bearing capacity of the aforementioned 60 RECSFT specimens, and their predictive performance is compared with that of three machine learning models used in this study. The predicted results from theoretical formulas and standard formulas are, respectively, presented in Table A1 and Table A2.

(1)

Ding’s formula

Ding et al. [10] conducted a series of tests and FE studies on the RECFST under compression, and proposed a formula to calculate the compression-bearing capacity considering the difference in the strong and weak confinement regions of the core concrete, as shown in Equation (20).

$$N_{\mathrm{u}}^{\mathrm{Ding}}=A_{\mathrm{c}}{f}_{\mathrm{c}}[1+(0.8+0.9K)\Phi ]$$

(20)

where K is the sectional factor and Φ is the confinement factor, with details reported in Ref. [10].

(2)

Ren’s formula

$$N_{\mathrm{u}}^{\mathrm{Ren}}=f_c^{\prime }{A}_c[1+(1.75-0.691\kappa +0.795\theta -0.263{\theta }^2)\Phi ]$$

(21)

where Φ is the confinement factor, κ is the aspect ratio of the middle rectangular part, and θ is central angle of the round ends, with details reported in Ref. [21].

(3)

M.F. Hassanein’s formula

$$N_{\mathrm{u}}^{\mathrm{Hassanein}}={\gamma }_s{f}_y{A}_{s,RE}+\left({\gamma }_c{f}_c^{\prime }+4.1f_{rp}^{\prime }\right)A_{c,RE}+f_y{A}_{s,RP,eff}+{\gamma }_c{f}_c^{\prime }{A}_{c,RP}$$

(22)

where A_s,RE and A_c,RE are, respectively, the cross-sectional areas of the steel tube and the concrete in the round ends of the columns, while A_s,RP,eff is the effective cross-sectional area of the steel in the rectangular part of the column, and A_c,RP is the cross-sectional area of the concrete in the middle rectangular parts. γ_s is used to account for the effects of strain hardening and D/t ratio on the yield stress of the steels of circular tubes. The detailed information can be found in Ref. [22].

(4)

Formula of code ACI 318-11

$$N_{\mathrm{u}}^{\mathrm{ACI}}=A_{\mathrm{s}}{f}_{\mathrm{y}}+0.85A_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime }$$

(23)

(5)

Formula of code AISC 360-16

$$N_{\mathrm{u}}^{\mathrm{AISC}}=A_{\mathrm{s}}{f}_{\mathrm{y}}+C_2{A}_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime }$$

(24)

where $C_2=\left\{\begin{array}{c}0.95\mathrm{for}\mathrm{circular}\mathrm{cross}\text{-}\sec \mathrm{tion}\\ 0.85\mathrm{for}\mathrm{rectangular}\mathrm{cross}\text{-}\sec \mathrm{tion}\end{array}\right.$

(6)

Formula of code GB 50936-2014

$$N_{\mathrm{u}}^{\mathrm{GB}}=A_{\mathrm{sc}}{f}_{\mathrm{sc}}$$

(25)

$$f_{\mathrm{sc}}=\left(1.212+B\Phi +C{\Phi }^2\right)f_{\mathrm{c}}$$

(26)

$$\Phi ={\alpha }_{\mathrm{sc}}{\displaystyle \frac{f_{\mathrm{y}}}{f_{\mathrm{c}}}}$$

(27)

$${\alpha }_{\mathrm{sc}}={\displaystyle \frac{A_{\mathrm{s}}}{A_{\mathrm{c}}}}$$

(28)

where B and C are the influence coefficient of cross-section shape on confinement effect; the specific values are referenced in Ref. [73].

Figure 15 and Figure 16 display the ratio of the predicted values to the tested values for each calculation model. The mean ratios of the predicted to tested values for the RFR, XGB, and GBR machine learning models are 1.01, 0.99, and 1.01, respectively. The corresponding standard deviations are 0.09, 0.10, and 0.10. Meanwhile, the mean ratios of the bearing capacity values calculated by Ding’s, Ren’s, and Hassanein’s empirical formulas and the ACI, GB, and AISC standard formulas to the tested values are 1.07, 0.94, 1.02, 0.88, 1.09, and 0.90, respectively. Their corresponding standard deviations are 0.10, 0.13, 0.12, 0.08, 0.08, and 0.07, respectively.

Figure 15. Ratios of the values predicted by the machine learning models to the tested values.

Figure 16. Ratios of the values predicted by the existing methods to the tested values.

The errors between the tested values and the calculation results of various calculation methods are counted, where error = $(N_{\mathrm{u}}^{\mathrm{Pre}}-N_{\mathrm{u}})/N_{\mathrm{u}}$, as shown in Figure 17. It can be seen that the calculation results of the ACI and AISC codes are conservative, and the percentage of specimens where the predicted values are lower than the tested values reached 93.55% and 91.94%, respectively. The observed conservatism in the calculation results of the ACI and AISC standards could be attributed to the fact that these two standards have uniformly reduced the bearing capacity of the concrete. Contrarily, the GB standard tends to overestimate the load-bearing capacity, with the rate of specimens having predicted values exceeding the measured values reaching 88.71%.

Figure 17. Proportion of Error < 0 and Error > 0 of different methods.

Among the above calculation methods, the GBR model, RFR model, and XGB model in machine learning achieved better prediction results, as shown in Figure 18. The proportion of predictions with an error less than 5% (0 < Error < 5% or 0 > Error > −5%) are 43.55%, 40.32%, and 33.87%, respectively, which are significantly higher than the rest of the theoretical models or standard formulas. The proportions of predictions where the error exceeds 15% (Error < −15% or Error > 15%) are only 14.52%, 9.68%, and 14.52%, respectively. These results show that the predictions made using the above three calculation methods for the axial compression-bearing capacity of RECFST are satisfactory.

Figure 18. Proportion of errors of different methods.

Based on the Python platform, these three pre-trained prediction models are integrated into a graphics user-friendly interactive interface (GUI), as illustrated in Figure 19. Within this GUI, the key parameters of the round-ended steel tube concrete, namely B, D, t, f_cu, and f_y, can be directly inputted. By simply clicking the calculate button, the axial load-bearing capacities as forecasted by the three models can be instantly displayed. This method proves to be highly accurate compared to the existing theoretical formulas and standards, and, in comparison to finite element models, offers a significant advantage in computational efficiency. Thus, it is highly meaningful for the rapid design of RECFST components.

Figure 19. GUI for calculation of compression capacity of RECFST by advanced ML methods.

5. Conclusions

The present study employs advanced machine learning methods, namely GBR (Gradient Boosting Regressor), RFR (Random Forest Regressor), and XGB (Extreme Gradient Boosting), to predict the axial load-bearing capacity of round-ended concrete-filled steel tube (RECFST). These methods are characterized by high computational efficiency, a wide range of parameters, and high accuracy, playing an important role in the design of RECFST. The following conclusions are drawn:

(1)

Using the finite element method, a machine learning database comprising 2400 RECFSTs is established. This database encompasses commonly used material strengths and cross-sectional dimensions of RECFST, effectively addressing the issue of insufficient experimental sample size in the current machine learning practices.

(2)

Through sensitive analysis, in addition to the primary parameter, the second parameters such as A_cf_cu, A_c + A_s, A_c, A_s, and A_sf_y also show a large correlation with the axial load-bearing capacity N_u, with their Pearson coefficients being 0.968, 0.909, 0.904, 0.694, and 0.684, respectively. Identifying these fundamental parameters closely related to axial load-bearing capacity is an essential foundation for applying machine learning to predict load-bearing capacity.

(3)

The three predictive models for the axial load-bearing capacity of RECFST, utilizing advanced machine learning methods, demonstrate higher accuracy compared to the existing theoretical or code-based calculation formulas. Furthermore, they exhibit higher efficiency compared to finite element methods. The development of a graphical user interface (GUI) based on these machine learning prediction models enables the rapid and accurate prediction of the axial load-bearing capacity of RECFST. This development holds significant importance for the engineering applications of RECFST.

In this paper, only the axial compression performance of RECFST stub columns was considered. In future studies, more attention should be paid to the influence of important parameters such as slenderness ratio on the axial compression performance of RECFST members. In addition, machine learning methods also have great potential in predicting the bending, shear, or torsion resistance of RECFST members. In the future, the author plans to pursue related research.