Interpretable machine learning models for predicting probabilistic axial buckling strength of steel circular hollow section members considering discreteness of geometries and material

Abstract

Steel circular hollow section (CHS) members are widely utilized as axial force-resisting structural members in civil engineering structures. The buckling strength under axial loads is one of the critical parameters to determine the performance of the steel CHS members, which is significantly affected by the discreteness introduced by geometries, material, and initial imperfections. However, the reduction factor employed in the modern design codes (i.e. Chinese codes and EC3) only accounts for the reduction caused by all kinds of discreteness and does not reflect the impacts of every single discreteness and imperfection. To fill the gap, this paper proposed an interpretable machine-learning method to provide the probabilistic axial buckling strength of steel CHS members prediction result in a distribution form with the consideration of detailed discreteness. The model to predict the nominal axial buckling strength of steel CHS members was first developed utilizing ten machine learning algorithms after sufficient numerical simulations, where the numerical model was verified using test results. The artificial neural network (ANN) was selected for developing the prediction model due to its highly reliable performance in testing. The developed ANN models were further interpreted utilizing Shapley Additive exPlanations (SHAP) to determine the interrelationship of different parameters. Then, the probabilistic axial bucking strength prediction model was established based on the developed ANN models, where the Latin hypercube sampling method was applied to address the discreteness of geometries, material, and initial imperfections. The generated probabilistic axial bucking strength prediction model’s effectiveness was verified by the evidence that the machine learning prediction results can highly match the numerical results' probability density function and the result from codes while significantly reducing the computation time. Finally, the design parameters’ impact on the axial buckling strength’s discreteness was evaluated using the global sensitivity analysis (GSA) method. The result shows that the discreteness of design parameters substantially influences the distribution of the axial buckling strength of the steel CHS members and the proposed prediction model can provide an accurate probabilistic distribution prediction.

Keywords

artificial neural network discreteness probabilistic axial buckling strength machine learning steel circular hollow section members

Introduction

Steel circular hollow section (CHS) members, a prevalent form of steel component, exhibit notable torsional and axial compressive stiffness, along with favorable stress distribution characteristics. The advantages of steel CHS members include ease of manufacturing, cost-effectiveness, and straightforward installation (Guo et al., 2016; Cai et al., 2016). Consequently, CHS members find extensive utility in various applications, including supporting columns, steel frame supports, grid structures, power transmission, offshore platform construction, and grid shells. Many of the structures employing steel CHS members, such as grid structures (Tian et al., 2018), grid shells and steel frame supports (Stamatopoulos, 2015), are inherently statically indeterminate, featuring intricate stress patterns. Despite rigorous design calculations for CHS members to ensure strength requirements within the operational lifespan, engineering practice reveals vulnerability to buckling instability due to the introduction of initial imperfection during production and installation (Šmak and Straka, 2012). For steel CHS members, bending (out of plane) and uneven wall thickness are the most common forms of imperfection, the presence of such initial imperfection can change the steel CHS member’s buckling behavior, increasing the likelihood of the transition from strength-based failure to buckling instability-based failure when subjected to external loads or deformations (Mirzaie et al., 2018). In addition to the imperfection of steel CHS members, it is noteworthy that the material properties and geometric dimensions of steel CHS members exhibit a degree of discreteness due to factors such as material composition, manufacturing processes, loading conditions, environmental influences, and internal damage. The discreteness primarily manifests itself in the presence of a discrepancy between the nominal values and the actual values of material and dimensional parameters, this discrepancy contributes to the difference between the nominal and actual axial buckling strength (Szalai and Papp, 2009).

Many scholars have conducted extensive research on the calculation of buckling strength and steel CHS’s axial buckling strength. From the 1960s, a large number of research works about behavior, analysis, and design (including buckling) were carried out. There are three main research methods: theoretical analysis, experiments, and finite element models. A review of the related theories can be found in (Atsuta and Chen, 2007) and the main shortage is that the simplified assumptions proposed for theoretical derivation will have an inevitable impact on the accuracy of axial buckling strength calculation. Lots of experiments have also been made to observe and analyze the failure mode of the steel CHS members (Janns et al., 1992; Galambos, 1998; Fukumoto and Itoh, 1983; Bjoerhovde, 1972; Meng and Gardner, 2021; Wang et al., 2024). Comprehensive overviews of different experimental data about steel tubular CHS members are available in the research (Kulak, 1996; Dorey et al., 2000; Linzell et al., 2003; Hayeck et al., 2018; Pournara et al., 2017; Meng and Gardner, 2022; Meng and Gardner, 2020; Ma et al., 2019; O'Shea and Bridge, 1997; Prion and Birkemoe, 1992). Due to the limitations of time and cost, it’s impossible to measure the axial buckling strength of all the specimens. An alternative approach involves utilizing the finite element method, a cost-effective numerical experimental method, to estimate the steel CHS member’s axial buckling strength (Pournara et al., 2017; Meng & Gardner, 2020, 2022; Prion and Birkemoe, 1992).

However, previous research has not comprehensively considered the interrelationship between axial buckling strength and the discreteness of material, geometries, and initial imperfections. Investigating this interrelationship via physical experiments and finite element methods typically needs a large number of samples and cases for parameter analysis. Consequently, related experiments entail significant time and cost expenditures. In recent years, machine learning (ML) has attracted lots of researcher’s interest and widely applied in civil engineering (Hu et al., 2022, 2024; Hu and Zhu, 2023). The research and practical applications of ML in civil engineering can be found in the reviews (Salehi and Burgueño, 2018; Sun et al., 2021; Ben Chaabene et al., 2020; Mirrashid and Naderpour, 2021). In the field of component performance prediction and performance assessment, Jeon et al. (Jeon et al., 2014) introduced ML algorithms to forecast the reinforced concrete’s connection strength precisely. Le et al. (Le and Le, 2021) ventured into developing ML models for strength prediction of structural members composed of CFST. Wang et al. (Wang and Chan, 2023) employed three machine learning methods (SVR, RFR, and NN) to develop predictive models for the ultimate strength of CFSTs under eccentric loading. Also, Faridmehr et al. (Faridmehr and Nehdi, 2022) focused on developing a surrogate ML model to estimate the axial capacity of both circular and rectangular CFST columns under centric or eccentric loading conditions. Zhou et al. (Zhou et al., 2023) introduced the ML method to assess the circular CFST’s axial strength with localized corrosion. Fu et al. (Fu et al., 2023) developed a dual generative adversarial network to automatically design steel-braced structure members. Mangalathu et al. (2021) embarked on an endeavor about various machine learning models, employing diverse algorithms to forecast reinforced concrete flat slabs’ strength. Viet-Hung Truong et al. (Truong et al., 2022) developed twelve ML models to assess the load-bearing capacity of semi-rigid steel structures. Minh-Tu Cao et al. (Cao et al., 2022) introduced a novel artificial intelligence model named IMNNIM aiming at the quick and accurate estimation of axial pile bearing capacity, taking into account nonlinear geotechnical factors and pile dimensions. Yang et al. (Yu et al., 2022) proposed a 2D convolutional neural network optimized by an improved bird swarm algorithm, in evaluating the torsional capacity of reinforced concrete (RC) beams. Huang et al. (Huang and Burton, 2019) proposed ML models to classify the failure mode of reinforced concrete frames. The above research results prove that with sufficient samples, ML models can be trained to predict the required output rapidly and accurately. However, the effects of the initial imperfections on predicting the components’ performance have not been fully considered in previous studies. To address this problem, Hu et al. (Hu et al., 2022) proposed an ML framework that considers the effects of the uncertainties of material and geometric through two-stage analysis, and the framework was validated by RHS-CHS T-joints and extended for steel shear dampers (Hu et al., 2023).

ML models for probabilistic axial buckling strength of steel CHS members with the consideration of the detailed discreteness of geometries, material, and initial imperfections are in lack of research. To fill the research blank in this field, an interpretable machine-learning method is proposed to provide the probabilistic axial buckling strength of steel CHS members prediction result in a distribution form with the consideration of detailed discreteness. The major contributions of this paper include:

(a) The machine learning model for predicting the probabilistic axial buckling strength of steel CHS members is developed and the prediction can be provided in a distribution form.

(b) Comparison among the calculation from the FE method, codes (i.e. Chinese code and EC3), and the proposed machine learning prediction model is made to verify the effectiveness of the proposed method.

(c) Investigate the influence of discreteness of geometries and material on the performance of steel CHS members through the proposed model.

The workflow of this paper mainly includes the following steps: In Section 2, a predictive model for the nominal axial buckling strength of steel CHS members has been developed by combining common machine learning algorithms with results from finite element analysis. The machine learning model’s efficacy is assessed by the test data, and its performance is benchmarked against that of the finite element model. In Section 3, the parameter importance analysis and Shapley Additive exPlanations (SHAP) method are used to explain the developed prediction model. According to previous research, the discreteness of geometries, material, and initial imperfections are sensitive to extreme values when integrated into parameter combinations. Therefore, the Latin hypercube sampling method (LHS) is employed in Section 4 to enhance the probabilistic axial buckling strength model, which can bolster the model’s generalization performance and stability. Finally, the design parameter’s effects on the output results are investigated through the developed machine learning model in Section 5 and the global sensitivity analysis (GSA) method is used to enable a comprehensive quantitative analysis of the design parameter’s impact.

Nominal axial buckling strength of steel CHS members

Finite element models of steel CHS member

The calculation results of the finite element axial compression buckling model of steel CHS members are used to build the dataset required for subsequent machine learning. As shown in Figure 1, the steel CHS member’s numerical model is simulated in Abaqus software (DSS Corporation, 2018). The model is loaded with an axial displacement. The axial degree of freedom U3 is free at the loading end while the other degrees of freedom at the fixed end are constrained. Reference points RP-T and RP-B are established in the model’s ends. The upper and lower end faces of the steel CHS member are coupled to the corresponding reference points while the load and boundary conditions are also attached to the steel CHS member through the reference points. The axial displacement is applied to the reference point at the loading end to simulate the steel CHS member’s axial compressive behavior and large deformation is considered in the simulation process. The material adopts the bilinear constitutive model. The first mode as shown in Figure 2 is selected as the initial imperfection mode. The ratio of the peak of the initial imperfection divided by the length is noted as δ.

Figure 1.

FE model for steel circular tubes: (a) Mesh and reference point; (b) Sectional mesh of tubes.

Figure 2.

First-order eigenvalue buckling mode.

The geometric parameters for finite element modeling of steel CHS members mainly include the thickness (t), diameter (D), and length (L). For a given steel CHE member, its geometric shape can be uniquely determined by one direct geometric parameter along with several relative dimensional ratios like the slenderness ratio and the diameter-to-thickness ratio. Also, in the strength verification of CHS members, instead of relying on direct geometric parameters (t, D, L), designers often prefer to use relative dimensional ratios. To integrate with the specification content and facilitate engineering applications, thickness (t), dimensional parameter diameter-to-thickness ratio (D/t), and slenderness ratio (λ) are selected in this article as design parameters. For steel CHS members, λ can be calculated by equations (1) and (2):

i = 0.354 (D - t)

(1)

λ = \frac{L}{i}

(2)

Figure 3 shows a schematic diagram of the loading device used in the reference literature (Shi et al., 2014). Figure 4 shows the comparison between the axial buckling strength of specimens in the reference literature (Meng and Gardner, 2021; Shi et al., 2014; Yang et al., 2021) and the finite element simulation result. Figure 4(a) shows the axial load-displacement curves of the test specimen marked as Q420-60-6 are in literature (Shi et al., 2014), due to the limitations of FE simulation in accounting for all initial imperfections, the simulation’s peak load tends to be marginally lower than the experimental result. While the experimental value stands at 2282.61 KN, the simulation indicates a peak of 2255.01 KN, reflecting a discrepancy of approximately 1%. To further confirm the effectiveness of the proposed FE method, the axial buckling strength of specimens with different sizes mentioned in the reference literature (Meng and Gardner, 2021; Shi et al., 2014; Yang et al., 2021) was also calculated using the proposed FE method. Figure 4(b) shows the comparison between the FE calculation results and the experimental result. The coefficient of variation (R) between the calculated values and experimental values is 0.989, indicating a relatively strong correlation. The detailed values can be found in Table.1

Figure 3.

Test configuration in reference (Mangalathu et al., 2021).

Figure 4.

(a) Comparison of the Q420-60-6’s axial load-displacement curves in literature (Shi et al., 2014) (b) Comparison between the FE calculation results and the experimental result of specimens in Table 1.

Table 1.

Detailed FE calculation and experimental values.

Specimen ID	Literature	Experimental (KN)	FE Method (KN)	Error (%)
Q420-20-1	(Shi et al., 2014)	2095	1968.010	6.062
Q420-20-2		2173	2080.928	4.237
Q420-20-3		2178	2081.757	4.419
Q420-30-1		2075	2105.993	−1.494
Q420-30-2		2094	1996.748	4.644
Q420-30-3		2042	1967.563	3.645
Q420-60-4		2277	2193.954	3.647
Q420-60-6		2285	2255.013	1.312
1	(Yang et al., 2021)	2476	2554.939	−3.188
2		2492	2462.780	1.173
3		2270	2141.391	5.666
4		2296	2371.191	−3.275
7		2120	2031.362	4.181
8		2248	2253.536	−0.246
CLC1-800	(Meng and Gardner, 2021)	1225.5	1257.116	−2.580
CLC1-1200		1118.3	1179.198	−5.446
CLC1-1600		1041.8	1024.165	1.693
CLC2-800		1523.9	1530.914	−0.460
CLC2-1200		1424.2	1333.565	6.364
CLC2-1600		1286.4	1268.612	1.383

Through the results in Figure 4 and Tables 1, it can be found that the proposed FE simulation method can describe the buckling instability mode of steel CHS members under axial load due to initial imperfections, and can effectively calculate the numerical value of the axial buckling strength. It should be noted that the average convergence time required for each model is approximately 10 minutes.

Database of axial buckling strength of steel CHS member

A comprehensive and high-quality dataset plays a critical role in enhancing the performance of ML models. In this section, the verified numerical modeling methods in Section 2.1 are developed to facilitate parameterized modeling. This section aims to establish a database to predict the axial buckling strength of steel CHS members, subsequently serving as the foundation for training machine learning models.

The axial buckling strength of steel CHS members is determined by material properties, geometric dimensions, and the initial imperfection. Accordingly, five parameters are set as the input, including the thickness (t), diameter-to-thickness ratio (D/t), slenderness ratio (λ), nominal yield stress (f_y), and the initial imperfection (δ). Limit to the time and cost, it is not feasible to cover every existing or potential design scenario with specific parameter values. Therefore, based on Chinese codes (L. Metallurgical Industry Information Standards Research Institute of Pangang Group Chengdu Iron and Steel Co, 2008) and past design cases, several representative values were selected from the range of each parameter. The specific values are outlined in Table 2, and the selection method of values is primarily designed to correspond with the various deviation levels of steel CHS members as specified in Chinese codes (L. Metallurgical Industry Information Standards Research Institute of Pangang Group Chengdu Iron and Steel Co, 2008).

Table 2.

Values of design parameters.

Parameters	t (mm)	$λ$	f_y (MPa)	D/t	δ
Number	5	5	4	3 for each f_y	4
Value	3.5, 6, 8, 10, 13	40, 60, 90, 120, 150	235, 345, 420, 460, 500, 620, 690	16, 34, and 50 for f_y = 235 MPa	0.5%, 0.75%, 1.0%, 1.5%
				11, 22, and 34 for f_y = 345 MPa
				9, 18, and 27 for f_y = 420 MPa
				8, 16, and 25 for f_y = 460 MPa
				16, 31, and 46 for f_y = 500 MPa
				13, 25, and 37 for f_y = 620 MPa
				11, 22, and 33 for f_y = 690 MPa

As shown in Table 2, this research comprises five different values for wall thickness (t), five values for the slenderness ratio (λ), seven values for nominal yield stress (f_y), four values for initial imperfection (δ), and three diameter-to-thickness ratios (D/t) values for each f_y. Consequently, a dataset comprising a total of 2100 simulated samples has been built. It’s noteworthy that this research primarily focuses on the steel CHS member’s global buckling instability. However, under axial compression conditions, if the thickness is too small (i.e., D/t is too large), steel CHS members may transition from global buckling to local buckling, a phenomenon beyond the scope of this investigation. According to relevant Chinese codes (L. Metallurgical Industry Information Standards Research Institute of Pangang Group Chengdu Iron and Steel Co, 2008), the design of steel tubes complies with the regulations to prevent local buckling. Consequently, the values of D/t fall within the prescribed limit range, as determined through the variation of geometric dimensions of steel CHS members. All these parameters have been normalized to ensure that all parameters contribute equally to the model training process, the normalization process can improve the model’s performance and stability.

To effectively evaluate the machine learning models, 2100 samples were calculated through the proposed FE method to construct the database. The whole calculation took approximately nine days to complete. The database was randomly divided into training, validation, and testing sets in a ratio of 7:2:1. Specifically, the training and validation sets are allocated for training to assess the model’s convergence capacity, while the remaining testing set serves to evaluate the model’s generalization capabilities.

Machine learning models for nominal axial buckling strength prediction

Presently, an array of machine learning algorithms has been integrated into the domain of civil engineering, where the algorithms are applied to address both theoretical and practical challenges. The capacity of machine learning algorithms to align with research objectives exerts a profound influence on the outcomes of regression predictions. In pursuit of identifying the most effective ML model for steel CHS members, a rigorous evaluation of performance was undertaken on the dataset built in Section 2.2. This evaluation encompassed ten prevalent machine learning algorithms, thus constituting a comprehensive analysis of predictive modeling alternatives. These ten ML algorithms encompass linear regression, lasso regression, ridge regression, Support Vector Regression (SVR), random forest regression (RF), decision tree regression (DT), K-nearest neighbor regression (KNN), adaptive boosting (AdaBoost), extreme gradient boosting (XGBoost), and artificial neural network (ANN). R² and RMSE are used as evaluation indicators to evaluate the effectiveness of ML models:

R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - \hat{y_{i}})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - \bar{y_{i}})}^{2}}

(3)

R M S E = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - \hat{y_{i}})}^{2}}

(4)

The hyperparameters and information of ten ML models are shown in Table. 3. The selection of initial values for hyperparameters was chiefly informed by the approaches described in Reference (Zhou et al., 2023; Mangalathu et al., 2021; Xu et al., 2022; Hu et al., 2023; Hou and Zhou, 2022). The Python library Scikit-optimize (Holger and Gilles, 2020) is introduced to Optimize hyperparameters.

Table 3.

The Hyperparameters and Information of Ten ML Models.

ML algorithm	Hyperparameters
Linear regression	No hyperparameters
Lasso regression	No hyperparameters
Ridge regression	No hyperparameters
SVR	C = 600; γ = 0.001
KNN	k = 6
DT	30 samples per leaf node
RF	The minimum number of samples in a split leaf node = 2 The number of estimators = 200
XGBoost	Learning rate = 0.3 The maximum depth of the trees = 4
AdaBoost	learning rate = 0.1; The number of estimators = 120
ANN	The number of hidden layers = 3

Ten machine learning models were trained on 2100 samples while R² and RMSE were calculated as evaluation indicators after convergence. The comparison results in Figure 5 indicate that among the ten well-trained machine learning models, ANN has the smallest RMSE and the largest R² showing its best performance in predicting steel CHS members’ axial buckling strength. In addition, the RMSE of ANN is close to 0 and R² is close to 1, proving that the trained machine learning model has good convergence and accuracy. The trained ANN model will be used for nominal axial buckling strength prediction in subsequent chapters.

Figure 5.

Performance of ten trained machine learning models.

Explanation of the developed ANN model

To analyze the input parameter’s contribution, the interpretability of ANN models is gathered from SHAP which has been used widely in past machine learning studies (Feng et al., 2021; Mangalathu et al., 2020; Cakiroglu et al., 2022). The SHAP model can be applied to both global and local interpretations. SHAP is one of the post-model interpretation methods, calculating the marginal contribution of characteristics to the output and explaining the black box model at both the global and local levels are the main concepts of SHAP. The explanation model for ANN based on SHAP is developed from equation (5). Equation (5) shows the output can be calculated by the linear combination of the input. More explication can be found in the reference (Mangalathu et al., 2020; Cakiroglu et al., 2022; Lundberg and Lee, 2017; Zhou et al., 2023).

f (x) = ϕ_{0} + \sum_{i = 1}^{N} ϕ_{k} x_{k}

(5)

in which

ϕ_{0}

means the average value of all prediction results, N means the number of input parameters,

ϕ_{k}

means SHAP value related to the kth input parameters, and

x_{k}

means the simplified input parameter.

The distribution of the generated ANN model’s SHAP value in Section 2.3 is shown in Figure 6. The SHAP value of a variable reflects the marginal effect of the variable. The importance of the variable is positively correlated with the corresponding absolute SHAP value. As shown in Figure 6, t, $λ$ , D/t, and f_y have a great impact while δ shows negligible effects. It should be emphasized that although the initial imperfection (δ) shows the lowest importance based on the SHAP value, this doesn’t mean the initial imperfection (δ) is not important in this research. The SHAP value means the variable’s contribution to the result’s numerical value but the initial imperfection (δ) shows great importance to the results from a mechanistic perspective. A larger initial imperfection (δ) means the steel CHS member is more likely to occur global buckling instead of local buckling. It has been emphasized in Section 2.2 that this research focuses on the global buckling of the steel CHS member, so the chosen parameter values are designed to ensure the occurrence of the global buckling leading to a small SHAP value of δ. The thickness (t) has an absolute SHAP value of more than 1.2 exhibiting its extremely high importance. Average absolute SHAP values are shown in Figure 7. The result confirms the highest impact of t with the average absolute SHAP value of 0.3075 and the lowest impact of δ with the average absolute SHAP value of 0.0277.

Figure 6.

SHAP values of the input parameters.

Figure 7.

Average absolute SHAP values of the input parameters.

What’s more, Figure 6 also shows that with the increase of t, f_y, and $λ$ , there is a smooth increase in SHAP value, this conclusion can also be gathered from the dependence plot in Figure 8. As shown in Figure 8(d), 8(h), 8(l), 8(p), and 8(t), with the decrease of D/t and the increase of t, f_y, and $λ$ , the SHAP value tends to increase. This indicates the positive relationship between the axial buckling strength of the steel CHS member and t, f_y, and $λ$ while D/t has a negative relationship. The above conclusions are compatible with the previous research. In addition, Figure 8 also focuses on the interaction among the parameters. For example, Figure 8(t) shows that a larger f_y leads to a smaller SHAP value of $λ$ indicating that f_y will inhibit the effect of $λ$ .

Figure 8.

SHAP dependence plot.

With the correlation between the design parameters and the global dependence analysis results, sufficient information and references can be provided to engineers in the design and production of the steel tube.

Probabilistic prediction model of axial buckling strength

Although the developed ANN model has a satisfactory performance on the database in Section 2, all the input values are the nominal values. In fact, due to the discreteness of geometries, material, and initial imperfections, the actual axial buckling strength of the steel CHS member is more likely to differ from the nominal value. According to the traditional research method, numerical simulation is used to solve the discreteness of geometries and materials. To investigate the impact of the specific parameter’s discreteness on the outcomes, a substantial number of values should be selected within the distribution range of the parameter and corresponding models are necessary to be constructed to calculate the axial buckling strength. Furthermore, to explore the influence of multiple parameters’ discreteness on the results and their interrelationship, it necessitates the construction and computation of a greater number of models under various combinations of these parameters. Compared with the high cost of the traditional method, the developed ANN model costs almost nothing to calculate the axial buckling strength by inputting f_y, D/t, $λ$ , t, and δ. Thus, ANN is competent in solving the axial buckling strength of the steel CHS member as the proxy model of FE models with the consideration of the discreteness of geometries and material.

To ensure full coverage of sample space, the LHS (McKay et al., 2000) is used to consider the discreteness of geometries and material. The LHS has been proven an effective method to create multivariate samples following specific distribution because the parameters are designed to not overlap in the sampling process. The core method of the LHS is to create n intervals with the same probability by dividing the distributions of the random variable. To cover the distribution of the considered variable, n points are obtained by sampling once in each interval during the sampling process. A two-parameter sampling process following the LHS method is exhibited in Figure 9.

Figure 9.

Schematic diagram of LHS.

According to the investigation by JCSS (Vrouwenvelder, 1997), the geometric dimensions of steel CHS members follow the normal distribution, including the thickness t, the length l, and the diameter D. Based on the research of Shi et al. (Shi et al., 2019), the value of f_y follows the normal distribution. In previous research about the imperfection δ, a larger number of experiments and measurements have been completed, but due to the different production methods (hot rolled and cold bent) and mechanical equipment, there is still no clear conclusion on the distribution of imperfection δ. According to József Szalai (Szalai and Papp, 2009), the imperfection can be assumed to follow the normal distribution. The statistical data for the random variables were meticulously compiled based on a comprehensive review of extensive literature, with a particular emphasis on the statistical data and conclusions drawn from references (Szalai and Papp, 2009) and (Melcher et al., 2004). Additionally, These data were carefully adjusted to meet the requirements of the Chinese code, this adjustment was pivotal in guaranteeing their applicability and ensuring full compliance with the regulatory framework prevalent in China. The distribution functions’ parameters of t, l, f_y, δ, and D are shown in Table 4.

Table 4.

Distribution values of the design parameters.

Parameters	Distribution	Mean	Variation
f _y	Normal	Nominal design value	0.05
D and t	Normal	Nominal design value	0.005
l	Normal	Nominal design value	0.03
δ	Normal	0.008*l	0.00012*l

Considering the discreteness of geometries and materials, the probabilistic axial buckling strength of the steel CHS member prediction model can be developed with the combination of the LHS method and the ANN models. The value of n in the LHS is another parameter that may have an impact on the result of the probabilistic axial buckling strength of the steel CHS member. Four cases were randomly selected to investigate the influence of n. Table 5 shows the specific parameters of four examples. For each example, 9 values of n are considered to calculate the probabilistic axial buckling strength of the steel CHS member, including 40, 80, 120, 160, 200, 240, 280, 320, and 360. In Figure 10, several percentiles of axial buckling strength for four considered cases with different n are shown respectively. From the result, it can be seen that when n < 80, the considered percentiles of the result gradually approach stable values, and when n > 80, the increase of n has a slight influence on the distribution of the axial buckling strength prediction result. Therefore, the above results indicate the LHS method has sufficient sampling accuracy when n ≥ 80. n = 80 was used in this research.

Table 5.

The nominal values of the selected parameters.

Cases	t (mm)	$λ$	δ	f_y (MPa)	D/t
Example 1	3.5	150	0.01*l	460	8
Example 2	8	120	0.015*l	235	34
Example 3	10	90	0.015*l	345	11
Example 4	13	60	0.0075*l	420	9

Figure 10.

The relationship between the value of n and the prediction result.

To investigate the developed model’s efficiency, the FE models of four considered examples were also built to make a comparison. All the calculations were completed on the machine equipped with a 13th Generation Intel® Core™ i7-13700K CPU, an NVIDIA GeForce RTX 4080 GPU, and 32 GB of memory. The FE models of four considered examples took approximately 15 hours to finish the calculation while the developed prediction model just took a few seconds to compute. For further comparison, the corresponding axial buckling strength is also calculated based on Chinses code (B.C.A.B. Press, 2017) and Eurocode 3 (European Committee for Standardization, 2005). Both codes comprehensively introduced a reduction factor to overall consider the reduction caused by all kinds of imperfections.

According to EC3, the axial buckling strength can be calculated from Equation (6) and Equation (7):

Φ = 0.5 [1 + α (\bar{λ} - 0.2) + {\bar{λ}}^{2}]

(6)

χ = \frac{1}{Φ + \sqrt{Φ^{2} - {\bar{λ}}^{2}}} = \frac{1}{0.5 [1 + α (\bar{λ} - 0.2) + {\bar{λ}}^{2}] + \sqrt{{0.5 [1 + α (\bar{λ} - 0.2) + {\bar{λ}}^{2}]}^{2} - {\bar{λ}}^{2}}}

(7)

where

\bar{λ}

is the nondimensional slenderness and χ is the reduction factor. α is an imperfection factor that is based on the class of the cross-sections. α is set as 0.21 when f_y equals to 235Mpa, 345Mpa, and 420Mpa while is set as 0.13 when f_y equals to 460Mpa.

The axial buckling strength calculation formulas in Chinese code are Equation (8) and Equation (9):

when \bar{λ} \leq 0.215 : φ = 1 - α_{1} {\bar{λ}}^{2}

(8)

when \bar{λ} > 0.215 : φ = \frac{1}{2 {\bar{λ}}^{2}} [(α_{2} + α_{3} \bar{λ} + {\bar{λ}}^{2}) - \sqrt{{(α_{2} + α_{3} \bar{λ} + {\bar{λ}}^{2})}^{2} - 4 {\bar{λ}}^{2}}]

(9)

where

α_{1}

α_{2}

, and

α_{3}

are the indicators related to the class of buckling curves which are set as 0.41, 0.986, and 0.152 respectively for the selected examples.

φ

is the reduction factor in Chinese code.

Figure 11 exhibits the result simulated from FE models, calculated by the developed prediction model, and gathered from the codes. In Figure 11(a), 11(b), 11(c), and 11(d), the comparison between the results from the FE model (boxes) and the proposed prediction model (blue curves) indicates that the distribution directly calculated from the prediction model can well match that obtained from the FE models. In Figure 11(e), 11(f), 11(g), and 11(h), the parameters of four specimens from the reference literature (Meng and Gardner, 2021; Shi et al., 2014; Yang et al., 2021) are also taken to test the proposed model. Due to differences in the steel material across various studies, the mean values of the distributions in Figure 11(g) and 11(h) are more closer to the experiment result. Although the imperfections of these specimens differ from each other, it can be seen that the experiment result and the code calculation result are all well located in the distribution of the prediction result.

Figure 11.

(a)-(d) Comparison among the results from FE numerical simulation, EC3, Chinese code, and developed prediction model. (e)-(h) Comparison among the results from experiment, EC3, Chinese code, and developed prediction model.

What’s more, it can be seen that the results from the Chinese Code (purple lines) and EC3 (orange lines) are also included in the distribution from the prediction model. This indicates that the model not only aligns with traditional conservative estimates but also offers a more nuanced understanding of the probabilistic buckling strength. The buckling strength in distribution form allows designers to assess the likelihood of different outcomes rather than relying solely on a single conservative estimate and can describe the buckling strength of steel CHS members under axial load more realistically because the proposed prediction model considers the detailed discreteness of each parameter rather than taking a reduction factor to overall incorporate the discreteness of all parameters. This probabilistic approach could lead to more informed decision-making, particularly in scenarios where there is a need to balance safety and material efficiency. It’s commended to combine the probabilistic results of the proposed prediction model with the deterministic values from design standards. This combined approach can help optimize the design while maintaining the necessary safety margin.

To further evaluate the effectiveness of the prediction model, the KS test is introduced to test whether the result simulated from FE models and that calculated by the developed prediction model follow the same distribution. The result of the test indicates that at the 1% significance level, the axial buckling strength of the steel CHS member calculated from numerical simulations and developed ML models follow the same distribution, which illustrates the reliability and effectiveness of the generated probabilistic axial buckling strength prediction model.

Global sensitive analysis

The relationship between the discreteness of design parameters (f_y, D/t, λ, δ, and t) and the probabilistic axial buckling strength of the steel CHS members is explored in this section. An essential approach for assessing the effect of input’s discreteness on output variables is sensitivity analysis. The sensitivity analysis encompasses two fundamental methods: global sensitivity analysis (GSA) and local sensitivity analysis (LSA). Unlike LSA, GSA offers a more comprehensive assessment by accounting for interdependencies among input variables and considering the discreteness of input parameters. Various techniques have emerged in previous research for GSA, in which Sobol’s method stands out as an efficient and robust approach. It excels in characterizing the complex relationships between input variables and outputs, particularly concerning high-order interactions among input variables. According to Sobol’s method, an n-dimensional hypercube’s output function f(x) can be described as:

f (x) = f_{0} + \sum_{i = 1}^{n} f_{1} (x_{i}) + \sum_{i < j} f_{i, j} (x_{i}, x_{j}) + \dots + f_{1, 2, \dots, n} (x_{1}, x_{2}, \dots, x_{n})

(10)

f_{0} = \int_{n} f (x) d X; X = {x_{1}, x_{2}, \dots, x_{n}}

(11)

The cumulative variance can be computed as follows:

D = {\int [f (x)]}^{2} d X - {f_{0}}^{2}

(12)

The biased variance can be calculated as:

{D_{i_{1},}}_{i_{2}, \dots, i_{s}} = \int {f_{i_{1},}^{2}}_{i_{2}, \dots, i_{s}} d x_{i_{1}} d x_{i_{2}} \dots d x_{i_{s}}

(13)

The first-order sensitivity index is calculated by:

S_{i} = \frac{D_{i}}{D}

(14)

The total-order sensitivity index is calculated by:

S_{i}^{T} = 1 - S_{\sim i} = 1 - \frac{D_{\sim i}}{D}

(15)

S_{i}^{T}

represents the sensitivity primarily impacted by the variable

x_{i}

along with its interactions with all other variables. Additionally,

D_{\sim i}

accounts for the variance originating from the remaining variables except

x_{i}

. The LHS method was also employed to compute

S_{i}^{T}

. A total of 600 cases were subject to analysis, and the parameter n was configured as 100,000 in each LHS process to comprehensively address the discreteness associated with the input parameters (f_y, D/t, λ, t , and δ).

Sobol’s indices (i.e., $S_{i}^{T}$ ) for each considered parameter are shown in Figure 12. It’s important to observe that a larger Sobol’s index signifies that the given design parameter’s discreteness exerts a more substantial impact on the steel CHS member’s probabilistic axial buckling strength. As depicted in Figure 12, Sobol’s indices for t and λ are notably greater than those for the remaining parameters (f_y, D/t, and δ). This observation underscores the substantial sensitivity of the steel CHS member’s axial buckling strength to the discreteness of the thickness (t) and the slenderness ratio (λ). Furthermore, Sobol’s index for δ is nearly equal to zero as illustrated in Figure 12, showing that the initial imperfection’s variations exert minimal influence on the axial buckling strength of the steel CHS member. What’s more, Sobol’s indices for f_y are mostly greater than zero, but the majority of them fall below 0.2, showing that the yielding stress of steel’s discreteness exhibits a discernible yet constrained impact on the axial buckling strength of the steel CHS member. These findings underscore the significance of accounting for the discreteness of materials and geometric dimensions when assessing the axial buckling strength of the steel CHS members.

Figure 12.

Sobol’s index for the parameters.

Figure 13 illustrates the enveloping outer enclosure of 500 axial buckling strength result calculated by the developed model. Within this figure, the red part stands for the probabilistic axial buckling strength of the steel CHS member incorporating all design parameters’ discreteness. Conversely, the blue part signifies the probabilistic axial buckling strength when the specific design parameter is excluded while the discreteness of the remaining design parameters is still considered.

Figure 13.

Comparison between the distribution with and without the consideration of design parameters.

The red part in Figure 13(a) represents probabilistic axial buckling strengths considering all five design parameters’ discreteness, whereas only the discreteness of f_y, D/t, λ, and δ are considered in the blue part. As depicted in Figure 13(b), (c), (d), and (e), it is discernible that the red part is notably bigger than the blue part, particularly in cases where the axial buckling strength is larger than 700 KN. This observation underscores the substantial impact of the discreteness of design parameters on the axial buckling strength of the steel CHS member. Compared with the other parameters, the result without the consideration of the discreteness of t shows a smaller blue region than the others. This reduction can be explained by the selection method of design parameters in Section 2.2. Among five design parameters, the thickness t is the basic absolute geometrical parameter while D/t, and λ are dimensionless proportional parameters. The geometry of the steel CHS member in the database has a strong dependency on the thickness t. Moreover, it can be summarized that the difference between the blue and red regions becomes larger with the increase in the axial buckling strength.

Conclusions

A prediction model using machine learning (ML) methods considering the discreteness of every main imperfection in steel material and geometric dimensions was developed to provide the probabilistic axial buckling strength of steel CHS members in a distribution form in this paper. The nominal axial buckling strength prediction model was developed after the comparison among ten machine learning algorithms where ANN exhibited the best performance. To better interpret the developed prediction model, the feature importance analysis and SHAP methods are introduced. The probabilistic axial buckling strength prediction model with the consideration of the discreteness of steel geometric dimensions and materials was further developed through the LHS method based on the developed ANN model. According to the above works in this research, the main conclusions include:

(1) The ANN model is checked out for the best performance in the nominal axial buckling strength prediction, indicating the ANN algorithm has the best effectiveness.

(2) The axial buckling strength’s probability density distributions gathered from the generated machine learning prediction model exhibit a strong concurrence with those obtained through finite element (FE) simulations across all examined cases and also match the results calculated from the codes. This comparison serves as validation, proving the reliability and trustworthiness of the proposed models.

(3) Compared to a single value from the modern design codes (i.e. Chinese codes and EC3), the proposed prediction model can provide the nominal axial buckling strength prediction result in a distribution form for a more realistic result by considering the detailed discreteness of each parameter.

(4) t, D/t, and λ have significant effects on the nominal axial buckling strength of the steel CHS members. δ shows negligible effect on the axial buckling strength but critical effort on the occurrence of global buckling failure.

(5) The steel CHS member’s nominal axial buckling strength shows a negative influence with δ and D/t while t, f_y, and λ show a positive relationship.

(6) The discreteness of design parameters indeed has a substantial impact on the distribution of the steel CHS member’s axial buckling strength. Due to the value selection method of the parameters, the discreteness of thickness t shows the most significant influence.

It should be noted that the probabilistic buckling stress prediction model for steel circular hollow section (CHS) members is grounded in a series of numerical simulations. The main limitation of this study is the influence of residual stress was not thoroughly considered due to the high correlation between residual stress and input parameters. Their strong coupling makes it challenging to directly incorporate residual stress as a separate parameterized input variable while developing the FE and ML models. As a result, the developed model is applicable to the CHS members that don’t exhibit residual stress or of which residual stress has been eliminated. Although the limitation exists, the research presents opportunities for future research aimed at evaluating the dependability of the established prediction model through the application of empirical test data and the proposed prediction model has the potential for further improvement.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial supports from the Natural Science Foundation of China (NSFC) with Grant Nos. 52078366, 52378182 and National Key Research and Development Program of 14th Five-Year Plan of China with Grant No. 2022YFC3801900 are gratefully acknowledged. This study is also supported by the Top Discipline Plan of Shanghai Universities-Class I with Grant No. 2022-3-YB-18 and Shanghai 2022 Science and Technology Innovation Action Plan Social Development Science and Technology Research Project with Grant No. 22dz1201700.

ORCID iD

Shuling Hu

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