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

References

[1] Zavvar E, Rosa-Santos P, Ghafoori E, Taveira-Pinto F. Analysis of tubular joints in marine structures: A comprehensive review.Marine Structures, 2025, 99: 103702

[2] Azari-Dodaran N, Ahmadi H. A numerical study on the ultimate load of offshore two-planar tubular TT-joints reinforced with internal ring stiffeners at fire-induced elevated temperatures.Ocean Engineering, 2021, 230: 108797

[3] Chen B, Zuo Y, Hou Y, Zheng L, Wen Z, Lin J, Chen Q, Rong P, Zheng L, Du S. . Application of concrete-filled steel tubular support in unified strength-bearing ring theory.Journal of Constructional Steel Research, 2024, 223: 109013

[4] Lin G, Zeng J, Li J, Chen G M, Zhou J K. Behavior of hybrid FRP-concrete-steel double-skin tubular member T-joints subjected to brace axial compression.Thin-walled Structures, 2024, 202: 112081

[5] Gao C, Wang J, Wang Y, Wei W. Behaviour of concrete-filled circular steel tubular K-joints in wind turbine towers.Journal of Constructional Steel Research, 2024, 218: 108694

[6] HuangZZhangQYangZZhangYYeZ. Study of model for out-of-plane stability of planar tubular trusses. China Civil Engineering Journal, 2011, 44(5): 49–56 (in Chinese)

[7] HanHZhengTWangX DHuangZ H. Fracture prediction and bearing capacity analysis of weld of tubular circular hollow section X-joints. Building Structure, 2022, 52(13): 99–105 (in Chinese)

[8] Wang C, Wang S, Xie L, Ding H, He W. Characterization of fatigue crack growth behavior in welded tubular T-joint.Marine Structures, 2024, 96: 103625

[9] Rezadoost P, Asgarian B, Nassiraei H. Degree of bending of FRP-reinforced tubular X-joints in offshore jacket-type structures under in-plane bending moment.Ocean Engineering, 2024, 307: 118114

[10] Zhao G Y, Liu W, Zhu L, Li J H, Gao X B. Experimental and numerical study on the stable bearing capacity of steel tubular cross bracing of a transmission tower.Case Studies in Construction Materials, 2022, 17: e01577

[11] Yan Q, Zhang J, Ma Z, Liu F, Song H, Zhao G, Gu C. Experimental and theoretical investigation on residual bearing capacity of CFDST-T joints after lateral impact.Journal of Building Engineering, 2024, 84: 108462

[12] Ahola A, Savolainen J, Brask L, Björk T. Fatigue enhancement of welded thin-walled tubular joints made of lean duplex steel.Journal of Constructional Steel Research, 2024, 218: 108738

[13] Tang Z, Li Z, Wang T. GPR-based prediction and uncertainty quantification for bearing capacity of steel tubular members considering semi-rigid connections in transmission towers.Engineering Failure Analysis, 2022, 142: 106854

[14] Rezadoost P, Nassiraei H. Identification of the most suitable probability distributions for ultimate strength of FRP-strengthened X-shaped tubular joints under axial loads.Ocean Engineering, 2023, 290: 116292

[15] Lesani M, Hosseini A S, Bahaari M R. Load bearing capacity of GFRP-strengthened tubular T-joints: Experimental and numerical study.Structures, 2022, 38: 1151–1164

[16] Wang B, Gao X, Chen A, Yuan L, Wan H, Wang Z. Optimization study of strengthened T-shaped concrete-filled steel tubular (CFST) column-steel beam joint.Journal of Constructional Steel Research, 2024, 223: 109020

[17] Ding F, Zhang S, Pan Z, Lei J, Wang L, Duan L. Seismic performance of square concrete-filled steel tubular column-composite beam single-side bolted joints: An experimental and numerical study.Engineering Structures, 2025, 322: 119035

[18] Hosseini A S, Bahaari M R, Lesani M, Hajikarimi P. Static load-bearing capacity formulation for steel tubular T/Y-joints strengthened with GFRP and CFRP.Composite Structures, 2021, 268: 113950

[19] Choo Y S, Qian X D, Liew J Y R, Wardenier J. Static strength of thick-walled CHS X-joints—Part I.New approach in strength definition. Journal of Constructional Steel Research, 2003, 59(10): 1201–1228

[20] Choo Y S, Qian X D, Liew J Y R, Wardenier J. Static strength of thick-walled CHS X-joints—Part II.Effect of chord stresses. Journal of Constructional Steel Research, 2003, 59(10): 1229–1250

[21] Choo Y S, Qian X D, Wardenier J. Effects of boundary conditions and chord stresses on static strength of thick-walled CHS K-joints.Journal of Constructional Steel Research, 2006, 62(4): 316–328

[22] Wang W, Chen Y Y. Hysteretic behaviour of tubular joints under cyclic loading.Journal of Constructional Steel Research, 2007, 63(10): 1384–1395

[23] Liu T, Qian X, Wang W, Chen Y. Resistance and strain during tearing for tubular joints under reversed axial actions.Journal of Constructional Steel Research, 2024, 213: 108328

[24] Chatziioannou K, Karamanos S A, Huang Y. Ultra low-cycle fatigue performance of S420 and S700 steel welded tubular X-joints.International Journal of Fatigue, 2019, 129: 105221

[25] Yang K, Zhu L, Bai Y, Sun H, Wang M. Strength of external-ring-stiffened tubular X-joints subjected to brace axial compressive loading.Thin-walled Structures, 2018, 133: 17–26

[26] Lan X, Chan T M, Young B. Structural behaviour and design of chord plastification in high strength steel CHS X-joints.Construction & Building Materials, 2018, 191: 1252–1267

[27] Lan X, Chan T M, Young B. Structural behaviour and design of high strength steel CHS T-joints.Thin-walled Structures, 2021, 159: 107215

[28] Li Z, Chang H, Ren T, Meng Z, Yin Y, Liu N, Huang Y, Xia J. Behavior of reinforced CHS T-joints by welding collar plates under load.Thin-walled Structures, 2024, 203: 112187

[29] Guerra M, Pereira D, Neto J˜ B, Nunes G, Sarmanho A, Lima L. Structural behaviour of thin-walled CHS-RHS T-joints: Experimental and numerical assessment.Structures, 2021, 32: 548–561

[30] Zhao B, Li F, Liu C, Wu X, Huang Z, Ke L. Hysteretic behavior of CHS X-joints under in-plane bending moment.Structures, 2022, 43: 1790–1806

[31] van der Vegte G J, Makino Y. Ultimate strength formulation for axially loaded CHS uniplanar T-joints.International Journal of Offshore and Polar Engineering, 2006, 16(4): 305–312

[32] van der Vegte G J, Makino Y. Further research on chord length and boundary conditions of CHS T- and X-joints.Advanced Steel Construction, 2010, 6(3): 879–890

[33] Hu S, Wang W, Lu Y. Explainable machine learning models for probabilistic buckling stress prediction of steel shear panel dampers.Engineering Structures, 2023, 288: 116235

[34] Hu S, Zhu S, Alam M S, Wang W. Machine learning-aided peak and residual displacement-based design method for enhancing seismic performance of steel moment-resisting frames by installing self-centering braces.Engineering Structures, 2022, 271: 114935

[35] Hu S, Wang W, Alam M S, Zhu S, Ke K. Machine learning-aided peak displacement and floor acceleration-based design of hybrid self-centering braced frames.Journal of Building Engineering, 2023, 72: 106429

[36] Hu S, Lei X. Machine learning and genetic algorithm-based framework for the life-cycle cost-based optimal design of self-centering building structures.Journal of Building Engineering, 2023, 78: 107671

[37] Hu S, Zhu S, Wang W. Machine learning-driven probabilistic residual displacement-based design method for improving post-earthquake repairability of steel moment-resisting frames using self-centering braces.Journal of Building Engineering, 2022, 61: 105225

[38] Hu S, Wang W, Lin X. Two-stage machine learning framework for developing probabilistic strength prediction models of structural components: An application for RHS-CHS T-joint.Engineering Structures, 2022, 266: 114548

[39] Gravier J, Vignal V, Bissey-Breton S, Farre J. The use of linear regression methods and Pearson’s correlation matrix to identify mechanical–physical–chemical parameters controlling the micro-electrochemical behaviour of machined copper.Corrosion Science, 2008, 50(10): 2885–2894

[40] Kim S H, Song X, Cho C, Lee C H. Strength prediction of steel CHS X-joints via leveraging finite element method and machine learning solutions.Journal of Constructional Steel Research, 2021, 176: 106394

[41] Cheok E W W, Qian X, Chen C, Quek S T, Si M B I. A local digital twin approach for identifying, locating and sizing cracks in CHS X-joints subjected to brace axial loading.Engineering Structures, 2024, 299: 117085

[42] Chen B, Chen L, Mo R, Wang Z, Zheng L, Zhang C, Chen Y. Strength prediction and uncertainty quantification of welded CHS tubular joints via Gaussian process regression.Engineering Structures, 2025, 332: 120030

[43] Zhang Y, Zheng R. Advanced algorithms to predict time-dependent atmospheric corrosion wastage of low-alloy and high-strength steels based on chemical compositions.Corrosion, 2023, 79(10): 1122–1134

[44] Firouzi N, Żur K K, Amabili M, Rabczuk T. On the time-dependent mechanics of membranes via the nonlinear finite element method.Computer Methods in Applied Mechanics and Engineering, 2023, 407: 115903

[45] Firouzi N, Misra A. New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers.Thin-walled Structures, 2023, 192: 111162

[46] Kim T, Kim K, Hyung J, Park H, Oh Y, Koo J. An interpretable machine learning-based pitting corrosion depth prediction model for steel drinking water pipelines.Process Safety and Environmental Protection, 2024, 190: 571–585

[47] Liu M, Li W. Prediction and analysis of corrosion rate of 3C steel using interpretable machine learning methods.Materials Today. Communications, 2023, 35: 106408

[48] Wang J, Zhang Z, Liu X, Shao Y, Liu X, Wang H. Prediction and interpretation of concrete corrosion induced by carbon dioxide using machine learning.Corrosion Science, 2024, 233: 112100

[49] Chen Y, Hu Z, Guo Y, Wang J, Tong G, Liu Q, Pan Y. Effects of chord pre-load on strength of CHS X-joints stiffened with external ring stiffeners and gusset plates.Engineering Structures, 2019, 195: 125–143

[50] Chen Y, Hu Z, Guo Y, Wang J, Dan H, Liu Q, Pan Y. Ultimate bearing capacity of CHS X-joints stiffened with external ring stiffeners and gusset plates subjected to brace compression.Engineering Structures, 2019, 181: 76–88

[51] YuraJ AZettlemoyerNEdwardsI F. Ultimate capacity equations for tubular joints. In: Proceedings of 12th Offshore Technology Conference. Houston, TX: Offshore Technology Conference, 1980, OTC-3690-MS

[52] Korol R M, Mirza F A. Finite element analysis of RHS T-joints.Journal of the Structural Division, 1982, 108(9): 2081–2098

[53] LuL Hde WinkelG DYuYWardenierJ. Deformation limit for the ultimate strength of hollow section joints. In: Tubular Structures VI. London: Routledge, 1994, 341–347

[54] Cao J J, Packer J A, Kosteski N. Design guidelines for longitudinal plate to HSS connections.Journal of Structural Engineering, 1998, 124(7): 784–791

[55] Izonin I, Tkachenko R, Gregus ml. M, Zub K, Tkachenko P.A GRNN-based approach towards prediction from small datasets in medical application. Procedia Computer Science, 2021, 184: 242–249

[56] Bai X, Liu S, Deng S, Zhang L, Wei M. An optimal control strategy for ASHP units with a novel dual-fan outdoor coil for evener frosting along airflow direction based on GRNN modelling.Energy and Building, 2023, 292: 113136

[57] Zheng X, Yang R, Wang Q, Yan Y, Zhang Y, Fu J, Liu Z. Comparison of GRNN and RF algorithms for predicting heat transfer coefficient in heat exchange channels with bulges.Applied Thermal Engineering, 2022, 217: 119263

[58] Jondhale S R, Deshpande R S. GRNN and KF framework based real time target tracking using PSOC BLE and smartphone.Ad Hoc Networks, 2019, 84: 19–28

[59] Xu F, Zhou C, Liu X, Wang J. GRNN inverse system based decoupling control strategy for active front steering and hydro-pneumatic suspension systems of emergency rescue vehicle.Mechanical Systems and Signal Processing, 2022, 167: 108595

[60] Ghritlahre H K, Prasad R K. Investigation of thermal performance of unidirectional flow porous bed solar air heater using MLP, GRNN, and RBF models of ANN technique.Thermal Science and Engineering Progress, 2018, 6: 226–235

[61] Polat A Ö, Avcı M. Modified GRNN based atomic modeling approach for nanoscale devices and TFET implementation.Materials Today. Communications, 2021, 27: 102294

[62] AengchuanPWiangkhamAKlinkaewNTheinnoiKSukjitE. Prediction of the influence of castor oil–ethanol–diesel blends on single-cylinder diesel engine characteristics using generalized regression neural networks (GRNNs). Energy Reports, 2022, 8(S15): 38–47

[63] Chen Y, Shen L, Li R, Xu X, Hong H, Lin H, Chen J. Quantification of interfacial energies associated with membrane fouling in a membrane bioreactor by using BP and GRNN artificial neural networks.Journal of Colloid and Interface Science, 2020, 565: 1–10

[64] Xie X, Fu G, Xue Y J Y, Zhao Z, Chen P, Lu B, Jiang S. Risk prediction and factors risk analysis based on IFOA-GRNN and apriori algorithms: Application of artificial intelligence in accident prevention.Process Safety and Environmental Protection, 2019, 122: 169–184

[65] Çakir M, Yilmaz M, Oral M A, Kazanci H Ö, Oral O. Accuracy assessment of RFerns, NB, SVM, and kNN machine learning classifiers in aquaculture.Journal of King Saud University. Science, 2023, 35(6): 102754

[66] Santhi T M, Srinivasan K. A duo autoencoder-SVM based approach for secure performance monitoring of industrial conveyor belt system.Computers & Chemical Engineering, 2023, 177: 108359

[67] Liu B, Vu-Bac N, Rabczuk T. A stochastic multiscale method for the prediction of the thermal conductivity of polymer nanocomposites through hybrid machine learning algorithms.Composite Structures, 2021, 273: 114269

[68] Liu B, Vu-Bac N, Zhuang X, Fu X, Rabczuk T. Stochastic full-range multiscale modeling of thermal conductivity of polymeric carbon nanotubes composites: A machine learning approach.Composite Structures, 2022, 289: 115393

[69] Liu B, Vu-Bac N, Zhuang X, Fu X, Rabczuk T. Stochastic integrated machine learning based multiscale approach for the prediction of the thermal conductivity in carbon nanotube reinforced polymeric composites.Composites Science and Technology, 2022, 224: 109425

[70] Liu B, Vu-Bac N, Zhuang X, Lu W, Fu X, Rabczuk T. Al-DeMat: A web-based expert system platform for computationally expensive models in materials design.Advances in Engineering Software, 2023, 176: 103398

[71] Xia Y, Zhang C, Wang C, Liu H, Sang X, Liu R, Zhao P, An G, Fang H, Shi M. . Prediction of bending strength of glass fiber reinforced methacrylate-based pipeline UV-CIPP rehabilitation materials based on machine learning.Tunnelling and Underground Space Technology, 2023, 140: 105319

[72] Liu B, Lu W, Olofsson T, Zhuang X, Rabczuk T. Stochastic interpretable machine learning based multiscale modeling in thermal conductivity of polymeric graphene-enhanced composites.Composite Structures, 2024, 327: 117601

[73] Sheng H, Ren Z, Wang D, Li Q, Li P. Estimation and interpretation of interfacial bond in concrete-filled steel tube by using optimized XGBoost and SHAP.Structures, 2024, 70: 107669

[74] Liu B, Penaka S R, Lu W, Feng K, Rebbling A, Olofsson T. Data-driven quantitative analysis of an integrated open digital ecosystems platform for user-centric energy retrofits: A case study in northern Sweden.Technology in Society, 2023, 75: 102347

[75] Liu B, Vu-Bac N, Zhuang X, Rabczuk T. Stochastic multiscale modeling of heat conductivity of polymeric clay nanocomposites.Mechanics of Materials, 2020, 142: 103280

[76] Liu B, Wang Y, Rabczuk T, Olofsson T, Lu W. Multi-scale modeling in thermal conductivity of polyurethane incorporated with phase change materials using physics-informed neural networks.Renewable Energy, 2024, 220: 119565

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