Skip to main content

Probabilistic analysis of the bearing capacity of spatially random Hoek‒Brown rock masses by integrating finite element limit analysis, random field theory, and XGBoost models

  • RESEARCH
  • Published:
Earth Science Informatics Aims and scope Submit manuscript

Abstract

The aim of this study is to investigate the influence of rock variability on the failure mechanism and bearing capacity of strip footings. A probabilistic analysis of the bearing capacity of footings on rock masses is conducted in this paper, where random adaptive finite-element limit analysis (RAFELA) with the Hoek‒Brown yield criterion and the Monte Carlo simulation technique are combined. The stochastic bearing capacity is computed by considering various parameters, such as the mean values of the uniaxial compressive strength of intact rock, Hoek‒Brown strength properties, coefficient of variance, and correlation lengths. In addition to the RAFELA, this study introduces a novel soft-computing approach for potential future applications of bearing capacity prediction by employing a machine learning model called the eXtreme Gradient Boosting (XGBoost) approach. The proposed XGBoost model underwent thorough verification and validation, demonstrating excellent agreement with the numerical results, as evidenced by an impressive R2 value of 99.99%. Furthermore, Shapley's analysis revealed that the specified factor of safety (FoS) has the most significant influence on the probability of failure (PoF), whereas the geological strength index (GSI) has the most significant effect on the random bearing capacity (μNran). These findings could be used to enhance engineering computations for strip footings resting on Hoek‒Brown rock masses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data availability

No datasets were generated or analysed during the current study.

Abbreviations

B :

Width of strip footing

N :

Bearing capacity

N det :

Deterministic bearing capacity

N ran :

Random bearing capacity

Q :

Ultimate vertical force

q u :

Ultimate vertical pressure

\(\sigma\) ci :

Uniaxial compressive strength of intact rock

GSI :

Geological strength index

m i :

Hoek and Brown yield parameter

D :

Disturbance factor

\(\mu\) :

Mean value

\(\sigma\) :

Standard deviation

COV :

Coefficient of variation

\(\mathit\Theta\) :

Spatial correlation length

r s :

Subsample ratio

\(\Omega ({f}_{k})\) :

Regularization term

\({\widehat{y}}_{k}\) :

XGBoost model's prediction

\(\nu\) :

Learning rate

\({Obj}_{(k)}\) :

Loss function

λ :

Regularization factor

R2 :

Coefficient of determination

RMSE:

Root mean squared error

SHAP:

SHapley Additive exPlanations

References

  • Ali A et al (2017a) Probabilistic stability assessment using adaptive limit analysis and random fields. Acta Geotech 12(4):937–948

    Article  Google Scholar 

  • Ali A, Lyamin AV et al (2017b) Undrained stability of an unlined square tunnel in spatially random soil. Geo-Risk 2017 [Preprint]

  • Ali A, Lyamin AV et al (2017c) Undrained stability of a single circular tunnel in spatially variable soil subjected to surcharge loading. Comput Geotech 84:16–27

    Article  Google Scholar 

  • Brahmi N et al (2018) Probabilistic analysis of the bearing capacity of inclined loaded strip footings near cohesive slopes. Int J Geotech Eng 15(6):732–739

    Article  Google Scholar 

  • Breiman L et al (2017) Regression trees. Classification And Regression Trees pp 216–265

  • Bharti JP et al (2021) Slope stability analysis using RF, GBM, CART, BT and Xgboost. Geotech Geol Eng 39(5):3741–3752

    Article  Google Scholar 

  • Chen T, Guestrin C (2016) ‘XGBoost’, Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining [Preprint]

  • Cho SE (2012) Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on Soil Foundation. Eng Geol 133–134:30–39

    Article  Google Scholar 

  • Dessi D et al (2023) Bow slamming detection and classification by Machine Learning Approach. Ocean Eng 287:115646

    Article  Google Scholar 

  • Dong X, Guo M, Wang S (2023) Inclination prediction of a giant open caisson during the sinking process using various machine learning algorithms. Ocean Eng 269:113587

    Article  Google Scholar 

  • Fathipour-Azar H (2021) Data-driven estimation of joint roughness coefficient. J Rock Mech Geotech Eng 13(6):1428–1437

    Article  Google Scholar 

  • Fenton GA, Griffiths DV (2003) Bearing-capacity prediction of spatially random cϕ soils. Can Geotech J 40(1):54–65. https://doi.org/10.1139/t02-086

    Article  Google Scholar 

  • Griffiths DV, Fenton GA (2001) Bearing capacity of spatially random soil: The undrained Clay Prandtl problem revisited. Géotechnique 51(4):351–359

    Article  Google Scholar 

  • Griffiths DV, Fenton GA, Manoharan N (2002) Bearing capacity of rough rigid strip footing on cohesive soil: Probabilistic study. J Geotech Geoenviron Eng 128(9):743–755

    Article  Google Scholar 

  • Halder K, Chakraborty D (2020) Influence of soil spatial variability on the response of strip footing on geocell-reinforced slope. Comput Geotech 122:103533

    Article  Google Scholar 

  • Hamouma D, Messameh AA, Tallah N (2022) Probabilistic analysis of lateral bearing capacity of pile-soil system. Int J Geotech Eng 16(5):525–531

    Article  Google Scholar 

  • Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div 106(9):1013–1035

    Article  Google Scholar 

  • Hoek E, Carranza-Torres C, Corkum B (2002) Hoek–Brown failure criterion—2002 edition. In: Proceedings of the North American rock mechanics society meeting in Toronto, Canada

  • Huang J, Lyamin A, Griffiths D, Sloan S, Krabbenhoft K, Fenton G (2013) Undrained bearing capacity of spatially random clays by finite elements and limit analysis. Proceedings of the 18th ICSMGE: 2013 Paris

  • Jitchaijaroen W, Duong NT, Lai VQ, Sangjinda K, Nguyen TS, Keawsawasvong S, Jamsawang P (2024) Probabilistic analysis of the seismic bearing capacity of strip footings using RAFELA and MARS. Geotechn Geol Eng

  • Kasama K, Whittle AJ, Kitazume M (2019) Effect of spatial variability of block-type cement-treated ground on the bearing capacity of foundation under inclined load. Soils Found 59(6):2125–2143

    Article  Google Scholar 

  • Kasama K, Zen K (2010) The reliability assessment for slope stability considering the spatial variability of soil strength using random field numerical limit analyses. J Soc Mater Sci Jp 59(5):336–341

    Article  Google Scholar 

  • Keawsawasvong S, Thongchom C, Likitlersuang S (2020) Bearing capacity of strip footing on Hoek-brown rock mass subjected to eccentric and inclined loading. Trans Infrastruct Geotechnol 8(2):189–202

    Article  Google Scholar 

  • Krabbenhøft K, Lyamin AV, Sloan SW (2007) Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 44(5):1533–1549

    Article  Google Scholar 

  • Krishnan K, Chakraborty D (2023) Probabilistic seismic passive resistance of hunchback retaining wall considering spatial variability. Comput Geotech 154:105154

    Article  Google Scholar 

  • Lai VQ et al (2023) Coupling FEA with XGBoost model for estimating uplift resistance of circular anchor in Ngi-ADP Soils. Geotech Geol Eng 42(1):767–781

    Article  Google Scholar 

  • Li J et al (2014) Comparative study of bearing capacity of buried footings using random limit analysis and random finite element method. Comput Methods Recent Adv Geomech 1301–1305

  • Liao K et al (2022) Probabilistic risk assessment of earth dams with spatially variable soil properties using random adaptive finite element limit analysis. Eng Comput 39(5):3313–3326

    Article  Google Scholar 

  • Lyamin AV, Sloan SW (2002a) Lower bound limit analysis using non-linear programming. Int J Numer Meth Eng 55(5):573–611

    Article  Google Scholar 

  • Lyamin AV, Sloan SW (2002b) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Meth Geomech 26(2):181–216

    Article  Google Scholar 

  • Lyamin AV et al (2005) Lower bound limit analysis with adaptive remeshing. Int J Numer Meth Eng 63(14):1961–1974

    Article  Google Scholar 

  • Maghous S, de Buhan P, Bekaert A (1998) Failure design of jointed rock structures by means of a homogenization approach. Mech Cohesive-Frictional Mater 3(3):207–228

    Article  Google Scholar 

  • Merifield RS, Lyamin AV, Sloan SW (2006) Limit analysis solutions for the bearing capacity of rock masses using the Generalised Hoek-Brown criterion. Int J Rock Mech Min Sci 43(6):920–937

    Article  Google Scholar 

  • Nguyen HD et al (2023) Seismic fragility analysis of steel moment frames using machine learning models. Eng Appl Artif Intell 126:106976

    Article  Google Scholar 

  • Nguyen TS, Tanapalungkorn W, Keawsawasvong S, Lai VQ, Likitlersuang S (2024) Probabilistic analysis of passive trapdoor in c-φ soil considering multivariate cross-correlated random fields. Geotech Geol Eng 42:1849–1869

    Article  Google Scholar 

  • OptumG2, OptumCE (2020) Copenhagen, Denmark: Optum Computational Engineering. See https://optumce.com/. Accessed 1 Dec 2020

  • Pedregosa F et al (2011) Scikit-learn: Machine learning in Python. J Mach Learn Res 12:2825–2830

    Google Scholar 

  • Pham TA et al (2022) Numerical Analysis of geosynthetic-reinforced and pile-supported embankments considering integrated soil-structure interactions. Geotech Geol Eng 42(1):185–206

    Article  Google Scholar 

  • Phoon K-K, Kulhawy FH (1999) Characterization of geotechnical variability. Can Geotech J 36(4):612–624

    Article  Google Scholar 

  • Saada Z, Maghous S, Garnier D (2008) Bearing capacity of shallow foundations on rocks obeying a modified Hoek-Brown failure criterion. Comput Geotech 35(2):144–154

    Article  Google Scholar 

  • Sadik L (2024) Developing prediction equations for soil resilient modulus using evolutionary machine learning. Trans Infrastruct Geotechnol 11:1598–1620

    Article  Google Scholar 

  • Sadik L, Al-Jeznawi D, Alzabeebee S, Al-Janabi MA, Keawsawasvong, S (2024) An Explicit Model for Soil Resilient Modulus Incorporating Freezing-Thawing Cycles Through Offspring Selection Genetic Algorithm (OSGA). Trans Infrastruct Geotechnol

  • Sangjinda K, Jitchaijaroen W, Nguyen TS, Keawsawasvong S, Jamsawang P (2024) Data-driven modelling of bearing capacity of footings on spatially random anisotropic clays using ANN and Monte Carlo simulations. Int J Geotech Eng

  • Sheridan RP et al (2016) Extreme gradient boosting as a method for quantitative structureactivity relationships. J Chem Inf Model 56(12):2353–2360

    Article  CAS  Google Scholar 

  • Shiau J, Keawsawasvong S (2023) ‘Probabilistic stability design charts for shallow passive trapdoors in spatially variable clays. Int J Geomech 23(6)

  • Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 12(1):61–77

    Article  Google Scholar 

  • Sloan SW (1989) Upper bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 13(3):263–282

    Article  Google Scholar 

  • Sloan SW (2013) Geotechnical Stability Analysis. Géotechnique 63(7):531–571

    Article  Google Scholar 

  • Sloan SW, Kleeman PW (1995) Upper bound limit analysis using discontinuous velocity fields. Comput Methods Appl Mech Eng 127(1–4):293–314

    Article  Google Scholar 

  • Tanapalungkorn W et al (2023) Undrained stability of braced excavations in clay considering the nonstationary random field of undrained shear strength. Sci Rep 13(1)

  • Tran DT et al (2024) An extreme gradient boosting prediction of uplift capacity factors for 3D rectangular anchors in natural clays. Earth Science Informatics [Preprint]

  • Wu G, Zhao H, Zhao M (2021a) Undrained stability analysis of strip footings lying on circular voids with spatially random soil. Comput Geotech 133:104072

    Article  Google Scholar 

  • Wu G et al (2021b) Stochastic analysis of dual tunnels in spatially random soil. Comput Geotech 129:103861

    Article  Google Scholar 

  • Yang X-L, Yin J-H (2005) Upper bound solution for ultimate bearing capacity with a modified Hoek-Brown failure criterion. Int J Rock Mech Min Sci 42(4):550–560

    Article  Google Scholar 

Download references

Funding

This research budget was allocated by the National Science, Research and Innovation Fund (NSRF) and King Mongkut’s University of Technology North Bangkok (Project no. KMUTNB-FF-67-A-05).

Author information

Authors and Affiliations

Authors

Contributions

Thanachon Promwichai: Software, Validation, Data curation, Investigation, Writing–original draft.

Duy Tan Tran: Conceptualization, Investigation, Methodology, Writing–original draft.

Thanh Son Nguyen: Software, Validation, Data curation, Investigation, Writing–original draft.

Suraparb Keawsawasvong: Formal analysis, Investigation, Software, Methodology, Writing–original draft.

Pitthaya Jamsawang: Writing—review and; editing, Supervision, Resource, Project administration.

Corresponding author

Correspondence to Suraparb Keawsawasvong.

Ethics declarations

Conflict of interest

On behalf of all the authors, the corresponding author states that there are no conflicts of interest.

Competing interests

The authors declare no competing interests.

Additional information

Communicated by: Hassan Babaie.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Promwichai, T., Tran, D.T., Nguyen, T.S. et al. Probabilistic analysis of the bearing capacity of spatially random Hoek‒Brown rock masses by integrating finite element limit analysis, random field theory, and XGBoost models. Earth Sci Inform 18, 33 (2025). https://doi.org/10.1007/s12145-024-01634-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12145-024-01634-7

Keywords