Parametric modeling and interpretable machine learning prediction on load-carrying capacity of a circular hollow section X-joint

Abstract

The aim of this paper is to explore the effect of geometrical parameters on ultimate load-carrying capacity of a circular hollow section (CHS) X-joint under axial compression of the brace end. First of all, finite element (FE) model to calculate ultimate load-carrying capacity of the CHS X-joint subjected to uniaxial load of the brace is constructed, and the calculated load–displacement curves are compared to the experimental ones. After validation of the FE model, 46080 groups of FE calculation models with different geometrical parameters are generated by means of parametric modeling. Subsequently, eight variables including gusset thickness and chord thickness are set as input to predict load-carrying capacity of the CHS X-joint by four machine learning (ML) algorithms, i.e., Generalized Regression Neural Network, Support Vector Machine, random forest (RF), and Extreme Gradient Boosting (XGBoost). Finally, the constructed ML prediction models are interpreted by SHapley Additive exPlanations, to explore the impact weight of each factor on ultimate load-carrying capacity of the joint. The results show that all the four models can predict the load-carrying capacity of the subject accurately, with all the R² values greater than 0.97. In addition, RF model yields the minimum mean-square error, Root Mean Squared Error, Mean Absolute Error, and Mean Absolute Percentage Error values, and the greatest R² value, while the prediction accuracy of XGBoost is relatively worse. Among all the eight considered geometrical parameters, brace diameter has the strongest impact on load-carrying capacity of the joint, followed by chord thickness, chord ring width, chord ring thickness, brace ring width, and brace thickness, while the thicknesses of the gusset plate and brace have marginal influence on load-carrying capacity. The study of the current paper can provide guidelines for dimension design of CHS X-joints.

Graphical abstract

Keywords

machine learning / SHAP-based interpretability / load-carrying capacity / CHS X-joint / parametric modeling

1 Introduction

Steel hollow section structures first appeared in offshore platform structures [1], and with the continuous development of building industry, hollow section members have been widely applied in large building structures such as airports, exhibition halls, sports halls, and stations [2–5]. The widespread application of hollow section structures relies on their excellent torsional resistance, stability, strong bending resistance, easy construction, and beautiful appearance. With the occurrence and development of multi-dimensional digital controlled cutting technique, which solved the problems such as difficulty in cutting intersecting lines, intersecting connections have been widely used in building structures [6–8]. In practical engineering, due to the complexity of the forces, intersecting joints are not only subjected to axial forces, but also to significant shear and bending moments. While for thin-walled structures such as steel pipes, the axial stiffness is much greater than radial stiffness, and the damage always derives from the radial load transferred from the joint. Combined with the influence of connection pattern of the joint on stress distribution, damage often occurs at the panel zone. Therefore, mechanical performance of the joint is an important factor that determines overall bearing capacity of the whole structure. In practical engineering, joints are often strengthened for enhancing bearing capacity, and the strengthening approaches mainly include adding cushion plate to the chord joint, welding external stiffeners to the chord, setting ring plates to the chord, infilling concrete to the chord, thickening local wall of the chord, adhering Carbon Fiber Reinforced Polymer to external of the chord, etc. [9–18].

Choo et al. [19] introduced a novel method for defining strength in their research, which contributed significantly to the field. They delved into the impact of chord stresses on the static strength of thick-walled circular hollow section (CHS) X-joints, offering valuable insights [20]. Beyond X-joints, their work also examined how boundary conditions (BCs) and chord stresses influence the static strength of thick-walled CHS K-joints, further broadening the understanding of these structures [21]. Other notable studies in this area include Wang and Chen’s [22] investigation of the hysteretic behavior of tubular joints under cyclic loads, which is crucial for assessing their performance in dynamic or cyclically loaded scenarios. Liu et al. [23] researched the resistance and strain during tearing for tubular joints under reversed axial actions, providing important data on the durability and failure mechanisms of these joints. Chatziioannou and colleagues [24] studied the ultra-low-cycle fatigue performance of S420 and S700 steel welded tubular X-joints, shedding light on the fatigue behavior of high-strength steels in such connections. Yang and his research group explored the strength of external-ring-stiffened tubular X-joints subjected to brace axial compressive loads, revealing how factors such as the brace-to-chord diameter ratio (β), stiffening ring width factor (β_r), and stiffening ring thickness factor (τ_r) affect the ultimate strength enhancement coefficient (R_e) [25]. They found that as β increases, R_e decreases, and larger β_r and τ_r lead to a faster degradation rate of R_e with β. When β is 0.6 or greater, R_e remains constant regardless of further changes in β. Lan et al. [26,27] investigated the structural behavior and design of high-strength steel CHS X-joints and T-joints, contributing to the development of more robust and efficient designs for these types of connections. Li and colleagues [28] researched the mechanical behavior of reinforced CHS T-joints by welding collar plates, exploring methods to enhance the strength and stability of these joints. Guerra et al. [29] conducted experimental and numerical assessments on the structural behavior of thin-walled CHS-rectangular hollow section T-joints, providing a comprehensive understanding of their performance under various loading conditions. Zhao and his research group [30] studied the hysteretic behavior of CHS X-joints under in-plane bending moments, which is essential for assessing their seismic performance and resilience. Vegte and Makino [31,32] carried out ultimate strength formulations for axially loaded CHS uniplanar T-joints, as well as further research on chord length and BCs of CHS T- and X-joints, contributing to the development of more accurate and reliable design methods for these structures.

Similar to majority of other complex members, there are numerous geometric parameters that affect the load-carrying capacity of CHS X-connections, and strong nonlinearity exists between the bearing capacity and each geometrical dimension. In light of the aforementioned “multi-variable and nonlinearity” problem, it is almost impossible to propose a set of close-formed empirical formulas for design, resulting in the lack of classical empirical formula. While with the fast development of artificial intelligence, machine learning (ML) based algorithms provide an alternative solution [33–38]. They are perfectly suitable for multivariate and nonlinear problems. Commonly used ML algorithms mainly include linear regression (LR), neural network (NN), random forest (RF), Support Vector Machine (SVM), K-nearest neighbor algorithm, genetic algorithm, gray incidence analysis method, and Fuzzy clustering, etc. Each algorithm has unique characteristic and application scenarios. LR can solve simple regression problems characterized by simple calculation and fast training speed, but can’t be used to fit nonlinear data [39]. Kim et al. [40] conducted strength prediction of steel CHS X-joints via leveraging finite element (FE) method and ML solutions. Two ML models; a support vector regression and a deep neural network regression were formulated, extensively trained, tested, and validated via comparison with experimental data. Cheok et al. [41] developed a local digital twin approach for identifying, locating and sizing cracks in CHS X-joints subjected to brace axial loading. Chen et al. [42] carried out strength prediction and uncertainty quantification of welded CHS tubular joints via Gaussian process regression. Grey incidence analysis method not only needs relatively less sample data amount, but also is possible to perform correlation analysis and processing on limited and seemingly irregular data, so as to find the potential law characteristic existed in the system itself [43]. The subject to be analyzed of fuzzy clustering is unclassified group. The fuzzy similarity relationship is constructed according to the features, similarities, and close or distant relations between things, so as to classify objective things.

Before carrying out ML prediction, the most fundamental step is to prepare data set. It is evident that a larger sample size in the data set will increase the prediction accuracy, as well as boost generalization capability of the prediction model. However, it is almost impossible to obtain thousands of data sets by experiments due to expensive time and economy costs. In this case, a commonly used approach is to calibrate several FE simulation models by limited experimental data, and then the required data set can be generated by the validated FE calculation models with various parameters. Firouzi et al. [44] studied the time-dependent mechanics of membranes via the nonlinear FE method, and provided new insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers [45]. Once a benchmark FE model is created manually, other models can be generated by means of parametric modeling automatically. Another problem derives from the inherent characteristic of ML models. From the origination of ML, the denunciation of lacking physical meaning has never stopped, interpretable and universal ML algorithms or models constructed are still limited [46–48]. The aforementioned multi-variable and nonlinearity problems in load-carrying capability forecasting of CHS X-connections, together with the acquisition of data set and interpretability of ML models are the motivations of the current paper, where Section 2 first gives a brief description of the subject to be studied, and then the FE calculation model of the ultimate bearing capacity of the connection with brace under external uniaxial tension is constructed. The load–displacement curves calculated by FE is compared to the experimental ones, in which way verifies the effectiveness of the FE model. Subsequently, geometrical dimensions are changed and 46080 groups of FE calculation models are generated automatically through parametric modeling in Section 3. In Section 4, according to the combinations of geometrical variables and the FE results calculated automatically, the ultimate load-carrying capability of the connection is predicted by four ML algorithms, i.e., Generalized Regression Neural Network (GRNN), SVM, RF, and Extreme Gradient Boosting (XGBoost). Finally, the constructed models are interpreted by SHAP in Section 5, to find the influencing weight of each geometrical parameter on the output.

2 Overview of the subject

The subject of this paper is a CHS X-joint, in which the four intersection corners of the joint are stiffened with gusset plates. Simultaneously, both the chord and the ring are strengthened with external ring stiffeners. Schematic of the member is shown in Fig.1. Two nominally identical test specimens, with the labels of XP1 and XP2, with specific geometrical dimensions listed in Tab.1, were experimentally studied in Ref. [49]. In the test (Fig.2), bottom of the brace was fixed to ground by a steel bearing. An axial compression with the magnitude of 205 kN was applied at the chord end as pre-load by a hydraulic jacket. Uniaxial force was then applied on top of the brace by a hydraulic actuator. A transducer was installed at the end of the brace to gauge its distortion, while load data was read and recorded by the actuator directly. More details of the experimental setup can be found in Ref. [49].

Abaqus software is used to construct the FE model for calculating ultimate bearing capability of the CHS X-connections, as shown in Fig.3. Four reference points are created to couple the ends of the chord and the brace, to facilitate the application of force, BCs, and the output of load-displacement data. In detail, the degree of freedoms (DOFs) of both chord-ends in X direction are fixed, while those in Z direction are released to allow axial shrinkage of the chord subjected to the compressive force. As for the brace, the bottom is fixed and the DOF of the top in Y direction is released to allow axial deformation. Material properties (Tab.2) are set in accordance with those of Ref. [49]. S4R element is utilized to split up the geometrical model. According to the mesh convergence analysis in Refs. [49,50], the global mesh size is set as 3.5%D, as displayed in Fig.4. After calculation, the stress distribution and deformation mode are demonstrated in Fig.5, from which one can see that plastic damage mainly concentrates on the junction between the brace and the stiffener, and the junction between the brace and the chord. Fig.6 illustrates the comparison in load–displacement curve between experiment and FE, it can be seen that the simulated curve agrees well with tested ones, in which way the effectiveness of the FE model is verified.

3 Parametric modeling

3.1 Modeling procedure

The first step of parametric modeling is to create a benchmark FE model manually, which has been done in Section 2. During the manual modeling process, Abaqus will record each action taken by the user and save the modeling commands to an RPY file and a JNL file. The main difference between the RPY file and the JNL file is that the RPY file includes the viewpoint adjustment commands of visualization interface while the JNL file does not. To simplify the modeling commands, the JNL file is used herein. The second module of parametric modeling is to create FE models with different geometrical parameters by the aforementioned JNL file automatically. First of all, the JNL file should be modified to py file, to make it readable for python, and then the file can be edited. During the editing process, for loop is first used to define geometrical variables, and then character string combined with the variables can be utilized to define names of the models and jobs. It should be noted that the model type command is not included in the original JNL file, and it must be declared in the modified py file. After the definition of variables, the geometrical parameters in the modeling commands can be modified to variables. For example, if a circle is defined using the coordinate points of the center (0, 0) and a perimeter point (0, 201.5), it can be revised to (0, D/2) in the parametric modeling command. After modification, Abaqus can be adopted to run the modified script file, and a batch of computer aided engineering (CAE) models with different geometrical parameters can be generated. When the CAE and INP files are fully prepared, a bat file can be complied to run the INP files in sequence (Module-3). The calculated results will be recorded to post-processing files with the format of odb. For the post-processing module, the first step is to open a benchmark job of Abaqus/CAE (.odb file) and conduct result post-processing. In this step, unnecessary operations should be avoided to simplify the RPY file. Command recording mechanism of the RPY file is consistent with that of the pre-processing counterpart: Every time Abaqus/CAE is opened, a new abaqus.rpy will be generated in the working directory, and the operation actions of Abaqus/CAE will be recorded by python language. Only when Abaqus/CAE is closed, the file can be opened by modifying the suffix to .py. Step 2 is to create the required XY data, and export the data to files (excel, txt, etc.), and then in Step 3, the abaqus.rpy file should be found in the working directory. After revising the suffix as .py, one can add corresponding sentences (for loop) to related paragraphs of each operation action, followed by saving the .py script. The final step is opening Abaqus/CAE to run the .py script, and checking if the data are correct. Entire working procedures of geometrical modeling are demonstrated by Fig.7.

3.2 Input variables and correlation analysis

From Fig.1 it can be seen that a total of 11 geometrical parameters is included in the specimen, where the chord length and brace height are kept as constants to fit the overall dimension of the beam-to-column connection in the structural system, while the diameter of the chord D is fixed as 400 mm to ensure enough space for the arrangement of fluid, cables, or other necessary instruments inside the chord. The other eight geometrical variables involved in parametric modeling are summarized in Tab.3. The range of geometrical variables meets following conditions: 10 ≤ γ≤ 50, 0.3 ≤ β≤ 0.9, 8 ≤ t_r≤ 20. The range is wide enough to satisfy majority of practical design requirements. According to combinations of the parameters, a total of 3 × 5 × 4 × 4 × 3 × 4 × 4 × 4 = 46080 groups of FE simulation are carried out. It should be noted that H_g and L_g must not be less than w_r and w_br, respectively, and herein the value of H_g is set as w_r + 34, while L_g = w_br + 138, in accordance with the benchmark model.

Correlation analysis is a useful tool for detecting redundancy among input variables, and both Pearson and Spearman correlations provide different perspectives on the relationships between variables, helping to make informed decisions about variable selection and model building. Here are the steps and considerations when conducting correlation analysis (such as Pearson or Spearman correlation) to check for redundancy among input variables.

1) Pearson correlation coefficient (r)

The r between two variables X and Y is calculated using the formula

(1)r=\frac{{{\displaystyle \sum }}_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{{\displaystyle \sum }}_{i=1}^n(x_i-\overline{x}{)}^2{{\displaystyle \sum }}_{i=1}^n(y_i-\overline{y}{)}^2}},

where x_i and y_i are individual observations of variables X and Y, respectively, \overline{x} and \overline{y} are the means of X and Y, and n is the number of observations. The value of r ranges from −1 to 1. A value close to 1 indicates a strong positive linear relationship (as one variable increases, the other tends to increase linearly), a value close to −1 indicates a strong negative linear relationship (as one variable increases, the other tends to decrease linearly), and a value close to 0 indicates little to no linear relationship. If two input variables have a high absolute value of the r (e.g., |r| > 0.7 or a value depending on the context), it may suggest a significant linear redundancy between them.

2) Spearman correlation coefficient (ρ)

The ρ is based on the ranks of the data. First, the data for each variable are ranked. Then, the formula for calculating the ρ is similar to the r but applied to the ranked data. Mathematically, it can also be calculated using the formula

(2)\rho =1-\frac{6{{\displaystyle \sum }}_{i=1}^n{d}_i^2}{n(n^2-1)},

where d_i is the difference in ranks of the ith observation between the two variables and n is the number of observations. The value of ρ also ranges from −1 to 1. It measures the monotonic relationship between two variables. A high absolute value of ρ (e.g., |ρ| > 0.7) indicates a strong monotonic relationship. If two input variables have a high ρ, it implies that there is a tendency for one variable to increase (or decrease) as the other variable increases (or decreases), regardless of the linearity of the relationship.

3) Procedure for checking redundancy among input variables

Data collection: Gather the data for all the input variables of interest.

Calculation: Calculate the correlation coefficients (either Pearson or Spearman) for all pairs of input variables. This can be done using statistical software (e.g., R, Python with libraries like ‘numpy’ and ‘scipy’ for numerical calculations, and ‘pandas’ for data manipulation) or spreadsheet software (e.g., Excel with appropriate functions).

Analysis: Identify pairs of variables with high correlation coefficients. Variables with high correlation may be redundant, and in some cases (such as in regression analysis or dimensionality reduction techniques), one of the highly correlated variables may be removed to avoid issues like multicollinearity.

Based on the aforementioned theories and procedures, the correlation analysis results of input variables are illustrated in Fig.8.

3.3 Results

Fig.9 displays the CHS X-joint models with typical geometrical dimensions obtained by parametric modeling approach. Fig.10 presents the deformation modes of CHS X-joints with typical geometrical parameters. The images show the joints under axial compression, and the deformation is scaled 10 times for better visibility. Each sub-figure corresponds to a different set of parameter values, including gusset plate thickness, chord thickness, brace thickness, and more. A clear trend emerges where the main deformation occurs as a depression at the connection between the chord and the brace. This indicates that this region is highly stressed under axial loads. For example, in all the shown cases, the area where the chord and brace meet experiences significant distortion. This observation is crucial as it helps in understanding the structural behavior of the joint and identifying the areas that are most likely to fail under load. It also provides a visual basis for further analysis of how different geometrical parameters affect the overall stability and load-carrying capacity of the CHS X-joint.

Fig.11 displays the load–displacement curves of the CHS X-joint with typical geometrical parameters. By comparing these curves, important conclusions can be drawn about the influence of various parameters on the joint’s load-bearing capacity. It is evident that among the considered parameters, gusset plate thickness and brace thickness have a relatively weaker impact compared to the other six parameters. For instance, changes in chord thickness, chord ring width, and others result in more significant shifts in the load–displacement curves.

Fig.12 illustrates load–displacement curves of the joint with typical geometrical parameters and the variation of load–displacement curves with w_br. It can be primarily concluded from the figures that when the brace diameter is greater than 240 mm (0.6D, Fig.12(a) and 12(b)), brace ring width w_br has slight impact on the ultimate bearing capacity. While when d < 0.6D (Fig.12(c)–12(h)), the load-carrying capacity increases with the increase of w_br. But if w_br is increased to 60 mm (0.3d–0.5d), the increase in load-carrying capacity will stop, i.e., it will be useless to further increase w_br for the purpose of improving load-carrying capability of the joint. This reminds us that the upper boundary of w_br should be set to 0.3d–0.5d when conducting engineering design. The impact of each parameter on bearing capability of the CHS X-connection will be further discussed in following sections by interpretable ML.

4 Machine learning prediction

One of the significant purposes of conducting FE calculation is to provide guidelines for engineering design according to the simulated results. As mentioned in Section 1, however, in light of the multi-variable and nonlinearity problems in the prediction on ultimate load-carrying capacity of CHS X-joints, it’s almost impossible to propose a set of close-formed empirical design formulas. Therefore, ML based algorithms must be introduced. Herein, the eight geometrical parameters, i.e., t_g, T, t, d, w_br, t_r, w_r, and t_br are set as input, and ultimate load-bearing capability of the connection is set as output. Notably, during the entire loading process, the member in question does not exhibit a significant peak load, indicating a relatively uniform stress distribution and stable performance under increasing loads. This observation is particularly important when considering the member’s integration with other kinds of hollow structural sections (HSS) connections, which are generally recognized for their exceptional flexibility and adaptability in various structural applications. Given the inherent flexibility of these HSS connections, researchers have proposed ultimate deformation limits to define when a connection is deemed to have “failed.” Notably, Yura et al. [51], Korol and Mirza [52], and Lu et al. [53] have all contributed to this field by suggesting specific deformation thresholds. In the context of the current discussion, the axial load corresponding to an ultimate deformation of 3% of the chord diameter (0.03D) for the chord face is defined as the ultimate load-carrying capacity of the joint. This definition aligns with established guidelines and standards outlined in Ref. [54], providing a clear and quantifiable metric for assessing the structural integrity and performance of the connection under load. It should be noted that it’s not easy to get the local depression displacement of the intersecting joint precisely, but according to Fig.10, the local depression displacement approximately equals to that of the loading point. So as a simplification, each calculation is stopped when the loading displacement reaches 3%D. In this way, it is easy to select the peak force on the load–displacement curve as the ultimate load-carrying capability. In the predictions, 20% of the data set (9216 groups) are set as testing set, and 80% (36864 groups) are defined as training set.

4.1 Generalized regression neural network

GRNN algorithm is a prediction method that developed on the basis of conventional NN. GRNN algorithm mainly consists of four layers of network structures [55–64], and the fundamental architecture of GRNN algorithm is illustrated in Fig.13.

GRNN algorithm is proposed on the basis of mathematical nonlinear regression theory. In the process of learning and predicting data, the first step is to assume load-bearing capability as variable y, and governing factors as variables x₁,x₂,…,x_n. Multiple influencing factors and variables form X together, and the probability density function between variables y and X is defined as f (X, y). The nonlinear regression result of variables y and X can be calculated by Eq. (3).

(3)\hat{Y}=E(y/\mathit{X})=\frac{{\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}yf(\mathit{X},y)\mathrm{d}y}}{{\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(\mathit{X},y)\mathrm{d}y}}.

\hat{Y} in Eq. (3) can be used to represent the prediction results of GRNN. In the above equation, however, the precise value of probability density function f (X, y) between variables y and X is difficult to be solved, and only can be obtained by means of estimation. The estimation equation is shown in Eq. (4).

(4)\begin{array}{rl}\hat{f}(\mathit{X},y)=& \frac{1}{n_{\mathrm{s}}{(2\pi )}^{\frac{p+1}{2}}{\sigma }_{\mathrm{g}}{}^{p+1}}\sum _{i=1}^{n_{\mathrm{s}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{\left(\mathit{X}-{\mathit{X}}_i\right)}^{\mathrm{T}}\left(\mathit{X}-{\mathit{X}}_i\right)}{2{\sigma }_{\mathrm{g}}{}^2}\right]\\ & \cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{\left(\mathit{X}-Y_i\right)}^2}{2{\sigma }_{\mathrm{g}}{}^2}\right],\end{array}

where X_i is the row matrix generated by the data of the ith row in variable X, i.e., the various influencing factor data of the ith group sample. Y_i is the load-carrying capacity data corresponds to the ith group. n_s is the amount of sample. p is the dimension of sample (the amount of influencing factor). σ_g is smooth factor, when the value of σ_g is larger, the predicted load-carrying capacity will approach the average of all input load-carrying capacity data. While if the value of σ_g is smaller, the prediction error will be relatively larger. Therefore, the selection of smoothing factor is the key to the use of GRNN algorithm. To ensure the accuracy of prediction result, cross validation is used to optimize the smoothing factor in the current investigation.

4.2 Support vector machine

SVMs are a set of supervised learning methods used for classification, regression, and outlier detection [65]. The standard formulation of a linear SVM for a linearly separable data set with N training examples (x_i, y_i), where x_i∈\mathbb{R}^d and y_i∈ {− 1, 1} is as follows [66].

The goal of SVM is to minimize the objective function {\displaystyle \frac{1}{2}}\Vert \mathit{w}{\Vert }^2+C{{\displaystyle \sum }}_{i=1}^N{\xi }_i subject to the constraints y_i(w^Tx_i + b) ≥ 1 – ξ_i, ξ_i≥ 0, i = 1,2,…,N, here, w is the weight vector, b is the bias term, ξ_i are the slack variables which allow some data points to be on the wrong side of the margin or even the separating hyperplane (Fig.14) in the case of a nonlinearly separable data set, and C > 0 is a regularization parameter that controls the trade-off between maximizing the margin (minimizing {\displaystyle \frac{1}{2}}\Vert \mathit{w}{\Vert }^2) and minimizing the classification error (minimizing the sum of ξ_i). If the data set is linearly separable, the slack variables ξ_i = 0 for all i, and the goal is to minimize {\displaystyle \frac{1}{2}}\Vert \mathit{w}{\Vert }^2 subject to the constraints y_i(w^Tx_i + b) ≥ 1 for i = 1,2,…,N. The geometric interpretation of w and b is that the hyperplane w^Tx + b = 0 separates the two classes of data points, and the margin of the SVM is given by {\displaystyle \frac{2}{\Vert \mathit{w}\Vert }}. Minimizing {\displaystyle \frac{1}{2}}\Vert \mathit{w}{\Vert }^2 is equivalent to maximizing the margin of the separating hyperplane.

As for nonlinear case, the kernel trick is used. A kernel function K(x_i, x_j) = ϕ(x_i)^Tϕ(x_j) is defined, where ϕ is a nonlinear mapping from the original feature space \mathbb{R}^d to a higher-dimensional feature space. In the dual form of the SVM, the optimization problem is formulated in terms of the kernel function instead of the original features. Radial Basis Function kernel function is used herein

(5)K(x,y)=\exp (-\gamma \Vert x-y{\Vert }^2),

where x, y are character vectors of inputs, representing the coordinates of samples in the character space. As the inputs of the kernel function, they calculate the similarity between samples, and hence construct the decision boundary in the high-dimensional space. γ governs the width of the kernel function and determines the influence scope of individual sample on the model. A greater γ leads to a narrower kernel function, a more complex model, the possibility of overfitting, and vice versa. Herein, grid search technique is leveraged for tuning the hyperparameters [67–72], and the best parameters are found as C = 100, γ = 0.01.

4.3 Random forest

RF is an ensemble learning algorithm proposed in 2001 for supervised classification (and regression). Repeated sampling approach is used to select the sample with the same dimension of the training set from the original sample, and the decision tree is constructed. During the testing process, each decision tree as a weak learner is able to generate a response (vote) when presenting a new set of features, and the response result is treated as the output result [41]. The selection fashion of decision point is similar to random selection of data. Each splitting process in RF is selected from the entire decision points, which means all the decision points can be selected. This makes the decision tree in RF the optimal for the current data set, thereby improving classification performance. The framework of RF model is illustrated by Fig.15, in which D represents the data set, Bagging means random sampling, and voting denotes weight assignment on the result.

4.4 Extreme gradient boosting

XGBoost algorithm is a kind of integrated algorithm that combines basis function and weight, to form good fitting effect of data [73]. Unlike traditional gradient boosting decision tree, XGBoost added normalization term to the loss function. XGBoost is more efficient when processing large scale data sets and intricate models, and also performs well in preventing overfitting and improving generalization ability. The design concept of XGBoost algorithm is demonstrated in Fig.16.

4.5 Predicted results

Data normalization or transformation techniques should be used before training ML models, to Eliminate dimensional differences between features, as well as enhance convergence speed and prediction accuracy of the models. Herein, maximum minimum normalization method is leveraged, with the main function of transferring the original data linearly to the scope of [0,1]. The normalization formula is as follow

(6)x^{\ast }=\frac{x-x_{min}}{x_{max}-x_{min}},

where x_max and x_min are the maximum and minimum values of the sample data, respectively.

The values of fitting coefficient R² and mean-square error (MSE) are utilized to evaluate the fitting accuracy

(7)R^2=1-\frac{{{\displaystyle \sum }}_{k=1}^{nt}({\hat{y}}_k-y_k{)}^2}{{{\displaystyle \sum }}_{k=1}^{nt}({\hat{y}}_k-\overline{y_k}{)}^2},

(8)MSE=\frac{1}{nt}\sum _{k=1}^{nt}{\left(y_k-{\hat{y}}_k\right)}^2,

where {\hat{y}}_k represents the fitting value, y_k denotes the tested value, {\overline{y}}_k is the mean value of y_k, and nt stands for the amount of data.

Calculation results for ultimate bearing capacity of the CHS X-joint under the aforementioned four ML algorithms are shown in Fig.17, and multiple error metrics of the four prediction models are further summarized in Tab.4. It can be seen that all the four models can predict the load-carrying capacity of the subject accurately, with all the R² values greater than 0.97. In addition, RF model yields the minimum MSE, Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) values, and the greatest R² value, while the prediction accuracy of XGBoost is relatively worse.

5 Interpretability of the models

5.1 Interpretability analysis method

To deeply excavate the decision mechanism of the prediction models and enhance interpretability, SHapley Additive exPlanations (SHAP) method is adopted to study the analysis framework of the interpretability. Contribution degrees of single prediction factor and multi-factor combination on prediction results can be assessed by calculating SHAP values. Thus, attribution analysis of prediction results can be conducted. For a certain ML prediction model which has been well trained, the SHAP value of each prediction factor can be calculated by the following equation

(9){\beta }_i=\sum _{S\subseteq M\mathrm{\setminus }\{i\}}\frac{|S|!(|M|-|S|-1)!}{M}\left[f_{S\cup \{i\}}(x_{S\cup \{i\}})-f_S(x_S)\right],

where β_i is the SHAP value of prediction factor i, M is set of the entire prediction factors, |M| is amount of the entire prediction results, S is any subset of forecast factors without factor i, |S| is the amount of prediction factors in the subset, S∪{i} is the subset of any prediction factors that include factor i, f_S∪{i}(x_S∪{i}) is the prediction result obtained by subset S∪{i} of the prediction factors, f_S(x_S) is the prediction result obtained by prediction factor subset S.

Besides, SHAP ML interpretability analysis framework also can decompose the SHAP main effect value of individual prediction factor and the interactive effect SHAP value of two prediction factors.

(10)\begin{array}{rl}{\beta }_{i,j}=& \sum _{T\subseteq M\mathrm{\setminus }\{i,j\}}\frac{|T|!(|M|-|T|-2)!}{2(M-1)!}\\ & \cdot \left[f_{T\cup \{i,j\}}(x_{T\cup \{i,j\}})-f_{T\cup \{i\}}(x_{T\cup \{i\}})-f_{T\cup \{j\}}(x_{T\cup \{j\}})+f_T(x_T)\right],\end{array}

(11){\beta }_{i,i}={\beta }_i-\sum _{j\ne i}{\beta }_{i,j},

where β_i,j is the SHAP interaction values of forecast factors i and j, T is the subset of any prediction factors excluding factors i and j, T∪{i,j} is the subset of any prediction factors including factors i and j, f_T∪{i,j}·(x_T∪{i,j}) is prediction result of model obtained by prediction factor subset T∪{i, j}, f_T(x_T) is prediction result of the model obtained by prediction factor subset T, β_i,i is the SHAP main effect value of forecast factor i.

According to the values, main effect values, and interactive effect values of SHAP, and from the aspects including contribution degree of each forecasting factor on prediction results, dependent relationship of prediction result on forecasting factors, and the interaction between each forecasting factors, interpretability study of the constructed load-carrying capacity prediction models is conducted. Taking RF model as an example, calculation results of SHAP value for each geometrical factor are displayed in Fig.18 and Fig.19, from which one can see that among all the eight considered geometrical parameters, brace diameter has the strongest impact on load-carrying capacity of the joint, followed by chord thickness, chord ring width, chord ring thickness, brace ring width, and brace thickness, while the thicknesses of the gusset and brace have marginal influence on load-carrying capacity. In addition, the relationships between all of the geometrical parameters and ultimate load-bearing capability are positive. Calculation results of SHAP values remind us that increasing diameter of the brace is the most effective way to enhance load-carrying capacity of CHS X-joints, while there is no need to increase t_g and t_br. Tab.5 shows the calculation of feature importance rankings from RF, it is evident that the rankings calculated by feature importance and SHAP are in good agreement with each other, and thus the parameter influence is validated.

Fig.20 demonstrates the correlations between two input variables. Taking Fig.20(a) for example, no matter how chord thickness changes, the load-bearing capability always increases with the increase of brace thickness, and vice versa, indicating that the chord thickness and brace thickness are two independent variables that are not correlated with each other. While for the width of the brace ring and the brace diameter (Fig.20(b)), when d < 150 mm (0.375D), load-carrying capacity of the joint increases with the increase of the brace ring width, but if d > 0.375D, increasing the width of the brace ring will make no contribution for increasing ultimate load-carrying capacity of the joint. Analogous to the interaction between the width of the brace ring and the diameter of the brace, interaction also exists between brace thickness and the diameter of the brace. When d > 0.375D, brace thickness will loss the impact on ultimate load-carrying capacity. For engineering design, it is recommended that w_br and t should not be greater than 60 and 5.5 mm, respectively, for the purpose of saving material.

5.2 Comparison between SHapley Additive exPlanations-based parameter rankings and existing design guidelines

The SHAP-based parameter rankings show both consistencies and some potential areas of divergence with existing design guidelines for CHS X-joints. Overall, they offer valuable insights that can complement and enhance current design practices.

1) Consistent emphasis on key parameters

The SHAP analysis in the study found that the brace diameter has the strongest impact on the load-carrying capacity of CHS X-joints, followed by chord thickness. Existing design guidelines also recognize the significance of these parameters. For example, in practical engineering, increasing the brace diameter can effectively enhance the load-bearing capacity as it provides more material to resist the applied load. A larger-diameter brace can better transmit the axial load from the brace to the chord, reducing stress concentration at the joint. Chord thickness is also crucial as a thicker chord has a higher moment of inertia and can better withstand the bending moments and shear forces transferred from the brace. This aligns with the importance placed on these parameters in traditional design approaches.

2) Agreement on secondary parameter influence

The SHAP results indicate that chord ring width, chord ring thickness, and brace ring width also have a relatively strong influence on the load-carrying capacity. In design guidelines, these parameters are also considered in the design process. A wider chord ring can increase the effective area of the chord near the joint, improving load distribution and enhancing the stability of the chord at the joint region. Similarly, the width of the brace ring can affect the load-carrying capacity, especially when the brace diameter is relatively small. This is in line with the SHAP-based rankings, suggesting that the SHAP analysis accurately reflects the influence of these parameters as recognized in existing design guidelines.

3) Differences in minor parameter consideration

The SHAP analysis shows that the thicknesses of the gusset plate and brace have marginal influence on the load-carrying capacity. However, existing design guidelines may still require a certain minimum thickness for these components for other reasons, such as ensuring constructability or preventing local buckling. While the SHAP-based rankings focus solely on the impact on load-carrying capacity, design guidelines take a more comprehensive approach, considering factors like manufacturing feasibility and long-term durability. This difference highlights the need to balance the insights from SHAP analysis with other design requirements.

4) New insights for design optimization

The SHAP-based rankings provide new insights into the interaction between parameters. For example, it was found that when the brace diameter is greater than 150 mm (0.375D), the brace ring width has a slight impact on the ultimate load-carrying capacity. And when d < 0.375D, the load-carrying capacity increases with the increase of the brace ring width until it reaches 60 mm (0.3d–0.5d), after which further increase is ineffective. This kind of detailed parameter interaction information is not always explicitly stated in traditional design guidelines. Designers can use these insights to optimize the design, for example, by carefully choosing the dimensions of the brace ring based on the brace diameter to achieve the best performance-cost ratio.

As a summary, the current work provides a comprehensive study for the influence of geometrical parameters on ultimate load-carrying capability of brace axial loaded CHS X-connections strengthened with external stiffeners and gusset plates. ML prediction models for bearing capacity of the CHS X-connections are constructed, which can be used in parametric design directly. In addition, the constructed ML prediction models are interpreted by SHAP. However, due to the requirement of FE simulations with large number, the current study is still insufficient. For example, the influence of chord axial compression ratio is not considered, which can be a further research interest in the further. Alongside quantitative analysis methods, introducing sensitivity analysis in future work could offer a clearer understanding of the importance of various input parameters [74,75]. Unfortunately, the trained models are only responsible for the geometric configurations inside the given data set, i.e., it is unclear whether the models perform well for new geometric configurations outside the given data set. So further research should be focused on the external validation on an unseen data set. Once more data are added in future studies, the models can be updated efficiently by re-training. For future work, a promising approach in ML involves integrating physical constitutive equations with artificial neural networks to develop physics-informed neural networks [76].

6 Conclusions

In this paper, FE model to calculate ultimate load-bearing capability of a CHS X-joint under uniaxial load of the brace is constructed, and the simulated load–displacement curve is compared to the experimental ones. After validation of the FE model, 46080 groups of FE calculation models with different geometrical parameters are generated by means of parametric modeling. Subsequently, eight variables including gusset thickness and chord thickness are set as input to predict load-carrying capacity of the CHS X-joint by four ML algorithms, i.e., GRNN, SVM, RF, and XGBoost. Finally, the constructed ML forecasting models are interpreted by SHAP, to explore the impact weight of each geometrical variable on ultimate load-carrying capacity of the joint. According to analysis results, following conclusions can be drawn.

1) All the four models can predict the load-carrying capacity of the subject accurately, with all the R² values greater than 0.97. In addition, RF model yields the minimum MSE, RMSE, MAE, and MAPE values, and the greatest R² value, while the prediction accuracy of XGBoost is relatively worse.

2) Among all the eight geometrical parameters of interest, brace diameter has the strongest impact on load-carrying capacity of the joint, followed by chord thickness, chord ring width, chord ring thickness, brace ring width, and brace thickness, while the thicknesses of the gusset and the brace have marginal influence on load-carrying capacity.

3) Interactions exist among the width of the brace ring, the diameter of the brace, and brace thickness: When the brace diameter is greater than 150 mm (0.375D), brace ring width has slight impact on the ultimate load-carrying capacity. While when d < 0.375D, bearing capacity increases with the increase of w_br, but if w_br is increased to 60 mm (0.3d–0.5d), load-carrying capacity stops increasing, i.e., it will be useless to further increase w_br for the purpose of enhancing load-carrying capability of the connection. This reminds us that the upper boundary of w_br should be set to 0.3d–0.5d when conducting engineering design. If d > 0.375D, brace thickness will loss the effect on ultimate bearing capacity.

4) Due to the requirement of larger number FE simulations, the current study is still insufficient. For example, the influence of chord axial compression ratio is not considered, which can be regarded as a further research interest in the further.

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