Interpretable modelling of deep conical footings on Hoek–Brown rock masses using genetic programming

Abstract

Deep conical footings are used in several civil engineering situations, particularly offshore and in fractured rock terrains, but the study on their bearing behaviour on Hoek–Brown rock masses is limited. In this study, the ultimate resistance of such footings is examined through finite element limit analysis within an axisymmetric setting. The Hoek–Brown criterion is adopted to represent the nonlinearity of jointed rock, and a wide range of geometric and material parameters is explored. The results show how these factors influence the collapse mechanism and the bearing capacity factor, often in ways that differ from conventional soil-based interpretations. A physics-informed data-driven model is developed to strengthen these insights by embedding a genetic programming (GP) algorithm to evaluate the bearing capacity factor. This approach leverages the symbolic-regression capability of GP to automatically construct an explicit nonlinear relationship between the input variables, thereby unifying data-driven modelling with physical constraints. Results demonstrate that, under similar conditions, the proposed model obtained the a coefficient of determination of 0.9721 in training, 0.9705 in testing and 0.9727 in validation phase respectively. Overall, the findings fill a noticeable gap in existing work on conical footings and offer practical guidance for applications where rock conditions dominate the foundation response.

Graphical abstract

Keywords

bearing capacity factor / deep conical footing / finite element limit analysis / GP / Hoek–Brown rock masses

References

[1] Hoek E , Brown E T . Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, 1980, 106(9): 1013–1035

[2] Hoek ECarranza-Torres CCorkum B. Hoek–Brown failure criterion—2002 edition. In: Proceedings of NARMS-TAC Conference. Toronto: University of Toronto Press, 2002, 267–273

[3] Yang X L , Yin J H . Upper bound solution for ultimate bearing capacity with a modified Hoek–Brown failure criterion. International Journal of Rock Mechanics and Mining Sciences, 2005, 42(4): 550–560

[4] Keshavarz A , Kumar J . Bearing capacity of foundations on rock mass using the method of characteristics. International Journal for Numerical and Analytical Methods in Geomechanics, 2018, 42(3): 542–557

[5] Chihi O , Saada Z . Bearing capacity of strip footing on rock under inclined and eccentric load using the generalized Hoek–Brown criterion. European Journal of Environmental and Civil Engineering, 2022, 26(6): 2258–2272

[6] Kumar D R , Samui P , Wipulanusat W , Keawsawasvong S , Sangjinda K , Jitchaijaroen W . Machine learning approaches for prediction of the bearing capacity of ring foundations on rock masses. Earth Science Informatics, 2023, 16(4): 4153–4168

[7] Chakraborty M , Kumar J . Bearing capacity of circular footings over rock mass by using axisymmetric quasi lower bound finite element limit analysis. Computers and Geotechnics, 2015, 70: 138–149

[8] Ukritchon B , Keawsawasvong S . Three-dimensional lower bound finite element limit analysis of Hoek–Brown material using semidefinite programming. Computers and Geotechnics, 2018, 104: 248–270

[9] Houlsby G T , Martin C M . Undrained bearing capacity factors for conical footings on clay. Géotechnique, 2003, 53(5): 513–520

[10] Khatri V N , Kumar J . Bearing capacity factor N_γ for a rough conical footing. Geomechanics and Engineering, 2009, 1(3): 205–218

[11] Chakraborty M , Kumar J . Bearing capacity factors for a conical footing using lower- and upper-bound finite elements limit analysis. Canadian Geotechnical Journal, 2015, 52(12): 2134–2140

[12] Chakraborty M , Kumar J . The size effect of a conical footing on N_γ. Computers and Geotechnics, 2016, 76: 212–221

[13] Liu B K , 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

[14] Liu B K , Vu-Bac N , Zhuang X Y , Fu X L , 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

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

[16] Liu B K , Liu P J , Wang Y Z , Li Z K , Lv H Q , Lu W Z , Olofsson T , Rabczuk T . Explainable machine learning for multiscale thermal conductivity modeling in polymer nanocomposites with uncertainty quantification. Composite Structures, 2025, 370: 119292

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

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

[19] Zhang X , Xia Y Y , Zhang C , Liu B K , Wang C X , Fang H Y , Wang J . A prediction model for dyke-dam piping based on data augmentation and interpretable ensemble learning. Engineering Failure Analysis, 2025, 182: 110174

[20] Liu B K , Lu W Z . Surrogate models in machine learning for computational stochastic multi-scale modelling in composite materials design. International Journal of Hydromechatronics, 2022, 5(4): 336–365

[21] Xia Y Y , Zhang C , Wang C X , Liu H J , Sang X X , Liu R , Zhao P , An G F , Fang H Y , Shi M S . et al. 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

[22] Wang Y ZBai J SLin Z YWang Q MAnitescu CSun JEshaghi M SGu Y TFeng X QZhuang X Y, et al. Artificial intelligence for partial differential equations in computational mechanics: A review. Applied Mechanics Reviews, 2026 (in press)

[23] Wang Y Z , Zhuang X Y , Timon R , Liu Y H . AI for PDEs in solid mechanics: A review. Advances in Mechanics, 2025, 55(2): 231–287

[24] Jin H B , Li Z Q , Xu Q Q , Sang Q Y , Zheng R Y . Model with GA and PSO: Pile bearing capacity prediction and geotechnical validation. Buildings, 2025, 15(21): 3839

[25] Zhu X , Chu J , Wang K D , Wu S F , Yan W , Chiam K . Prediction of rockhead using a hybrid N-XGBoost machine learning framework. Journal of Rock Mechanics and Geotechnical Engineering, 2021, 13(6): 1231–1245

[26] Millán M A , Picardo A , Galindo R . Application of artificial neural networks for predicting the bearing capacity of the tip of a pile embedded in a rock mass. Engineering Applications of Artificial Intelligence, 2023, 124: 106568

[27] Sangjinda K , Keawsawasvong S , Jamsawang P , Senjuntichai A , Senjuntichai T . Optimized ANN-based surrogate models for evaluating the stability of trapdoors in Hoek–Brown rock masses. Earth Science Informatics, 2025, 18(1): 108

[28] Jearsiripongkul T , Keawsawasvong S , Thongchom C , Ngamkhanong C . Prediction of the stability of various tunnel shapes based on Hoek–Brown failure criterion using artificial neural network (ANN). Sustainability, 2022, 14(8): 4533

[29] Keawsawasvong S , Sangjinda K , Jitchaijaroen W , Alzabeebee S , Suksiripattanapong C , Sukkarak R . Soft computing-based models for estimating the ultimate bearing capacity of an annular footing on Hoek–Brown material. Arabian Journal for Science and Engineering, 2024, 49(4): 5989–6006

[30] Khajehzadeh M , Keawsawasvong S . Artificial intelligence for bearing capacity evaluation of shallow foundation: An overview. Geotechnical and Geological Engineering, 2024, 42(7): 5401–5424

[31] Beiki M , Bashari A , Majdi A . Genetic programming approach for estimating the deformation modulus of rock mass using sensitivity analysis by neural network. International Journal of Rock Mechanics and Mining Sciences, 2010, 47(7): 1091–1103

[32] Yong W X , Zhou J , Jahed Armaghani D , Tahir M M , Tarinejad R , Pham B T , Van Huynh V . A new hybrid simulated annealing-based genetic programming technique to predict the ultimate bearing capacity of piles. Engineering with Computers, 2021, 37(3): 2111–2127

[33] Kumar D R , Kumar P , Wipulanusat W , Tran D T , Keawsawasvong S . Seismic stability of unsupported rectangular excavations in cohesive-frictional soils: FELA simulations and comparative analysis using MARS, GP, and GMDH models. Geotechnical and Geological Engineering, 2025, 43(3): 136

[34] Krabbenhoft KLyamin AKrabbenhoft J. OptumG2. Version 2015. Copenhagen, Denmark: Optum Computational Engineering, 2015

[35] Yoonirundorn K , Senjuntichai T , Senjuntichai A , Keawsawasvong S . Predicting bearing capacity factors of multiple shallow foundations using finite element limit analysis and machine learning approaches. Transportation Infrastructure Geotechnology, 2025, 12(3): 105

[36] Ciria H , Peraire J , Bonet J . Mesh adaptive computation of upper and lower bounds in limit analysis. International Journal for Numerical Methods in Engineering, 2008, 75(8): 899–944

[37] Chollet F. Deep Learning with Python. 2nd ed. Shelter Island: Manning Publications, 2021

[38] Kazemi R . Artificial intelligence techniques in advanced concrete technology: A comprehensive survey on 10 years research trend. Engineering Reports, 2023, 5(9): e12676

[39] Kazemi R , Gholampour A . Evaluating the rapid chloride permeability of self-compacting concrete containing fly ash and silica fume exposed to different temperatures: An artificial intelligence framework. Construction and Building Materials, 2023, 409: 133835

[40] Dev K L , Kumar D R , Wipulanusat W . Machine learning prediction of the unconfined compressive strength of controlled low strength material using fly ash and pond ash. Scientific Reports, 2024, 14(1): 27540

[41] Li E M , Zhou J , Shi X Z , Jahed Armaghani D , Yu Z , Chen X , Huang P S . Developing a hybrid model of salp swarm algorithm-based support vector machine to predict the strength of fiber-reinforced cemented paste backfill. Engineering with Computers, 2021, 37(4): 3519–3540

[42] Koza J R. Genetic Programming II (Videotape): The Next Generation. Cambridge: MIT Press, 1994

[43] Keawsawasvong S . Bearing capacity of conical footings on Hoek–Brown rock masses using finite element limit analysis. International Journal of Computational Materials Science and Engineering, 2021, 10(3): 2150015

[44] Liu B K , Liu P J , Lu W Z , Olofsson T . Explainable artificial intelligence (XAI) for material design and engineering applications: A quantitative computational framework. International Journal of Mechanical System Dynamics, 2025, 5(2): 236–265

RIGHTS & PERMISSIONS

Higher Education Press