Abstract
One of the most basic topics in rock mechanic is the shear strength criteria for rock joints. Thus, it is of high importance to accurately predict the shear strength of rock joints. In this study, the abilities for accuracy and agreement of Kriging model-based nonlinear interpolation strategy are investigated in terms of predicting the shear strength of rock joints. Totally 84 datasets were used to construct the Kriging models; the datasets were divided into two main parts: training and testing. The prepared database was applied to the training phase in the Kriging model; this way, several nonlinear basic functions were introduced to enhance the predictions of the Kriging model. The examined functions in this paper were linear, 2-order, 3-order, exponential, logarithmic, logistic, hyperbolic tangent, and hyperbolic sine. The sigmoid forms of the basic functions, including logistic and hyperbolic tangent, provide the superior predictions compared to other mathematical functions, while the 2-order and 3-order forms provide the worst performances than the linear, exponential, and logarithmic functions. According to the obtained results, the logistic-based model with coefficient of determination (R2) of 0.916 was found the optimal model that can be successfully applied to estimating the shear strength of rock joints.
Similar content being viewed by others
References
Babanouri N, Fattahi H (2020) An ANFIS–TLBO criterion for shear failure of rock joints. Soft Comput 24(7):4759–4773. https://doi.org/10.1007/s00500-019-04230-w
Patton FD (1966) Multiple modes of shear failure in rock. In: 1st ISRM congress, 1966. International Society for Rock Mechanics and Rock Engineering
Barton N (1973) Review of a new shear-strength criterion for rock joints. Eng Geol 7(4):287–332
Maksimović M (1992) New description of the shear strength for rock joints. Rock Mech Rock Eng 25(4):275–284
Tang Z-C, Liu Q-S, Huang J-H (2014) New criterion for rock joints based on three-dimensional roughness parameters. J Cent S Univ 21(12):4653–4659
Hossaini KA, Babanouri N, Nasab SK (2014) The influence of asperity deformability on the mechanical behavior of rock joints. Int J Rock Mech Min Sci 70:154–161
Wei Y, Fu W, Nie D (2015) Nonlinearity of the rock joint shear strength. Strength Mater 47(1):205–212
Zhang X, Jiang Q, Chen N, Wei W, Feng X (2016) Laboratory investigation on shear behavior of rock joints and a new peak shear strength criterion. Rock Mech Rock Eng 49(9):3495–3512
Sarfarazi V, Haeri H, Shemirani AB, Zhu Z (2017) Shear behavior of non-persistent joint under high normal load. Strength Mater 49(2):320–334
Gentier SS, Hopkins DL (1997) Mapping fracture aperture as a function of normal stress using a combination of casting, image analysis and modeling techniques. Int J Rock Mech Min Sci 34(3–4):132-e1
Grasselli G, Egger P (2003) Constitutive law for the shear strength of rock joints based on three-dimensional surface parameters. Int J Rock Mech Min Sci 40(1):25–40
Li K-h, Cao P, Zhang K, Zhong Y-f (2015) Macro and meso characteristics evolution on shear behavior of rock joints. J Cent S Univ 22(8):3087–3096
Babanouri N, Nasab SK, Baghbanan A, Mohamadi HR (2011) Over-consolidation effect on shear behavior of rock joints. Int J Rock Mech Min Sci 48(8):1283–1291
Babanouri N, Nasab SK (2017) Proposing triangulation-based measures for rock fracture roughness. Rock Mech Rock Eng 50(4):1055–1061
Chen X, Fu J, Yao J, Gan J (2018) Prediction of shear strength for squat RC walls using a hybrid ANN–PSO model. Eng Comput 34(2):367–383
Sarkar K, Tiwary A, Singh TN (2010) Estimation of strength parameters of rock using artificial neural networks. Bull Eng Geol Environ 69(4):599–606. https://doi.org/10.1007/s10064-010-0301-3
Dantas Neto SA, Indraratna B, Oliveira DAF, de Assis AP (2017) Modelling the shear behaviour of clean rock discontinuities using artificial neural networks. Rock Mech Rock Eng 50(7):1817–1831. https://doi.org/10.1007/s00603-017-1197-z
Khandelwal M, Armaghani DJ (2016) Prediction of drillability of rocks with strength properties using a hybrid GA-ANN technique. Geotech Geol Eng 34(2):605–620. https://doi.org/10.1007/s10706-015-9970-9
Murlidhar BR, Ahmed M, Mavaluru D, Siddiqi AF, Mohamad ET (2019) Prediction of rock interlocking by developing two hybrid models based on GA and fuzzy system. Eng Comput 35(4):1419–1430. https://doi.org/10.1007/s00366-018-0672-9
Xia C, Huang M, Qian X, Hong C, Luo Z, Du S (2019) Novel intelligent approach for peak shear strength assessment of rock joints on the basis of the relevance vector machine. Math Probl Eng 2019:3182736. https://doi.org/10.1155/2019/3182736
Zhou J, Li E, Wei H, Li C, Qiao Q, Armaghani DJ (2019) Random forests and cubist algorithms for predicting shear strengths of rockfill materials. Appl Sci 9(8):1621
Krige DG (1952) A statistical approach to some basic mine valuation problems on the Witwatersrand. J South Afr Inst Min Metall 52(9):201–203
Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246–1266
Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423. https://doi.org/10.1214/ss/1177012413
Heddam S, Keshtegar B, Kisi O (2019) Predicting total dissolved gas concentration on a daily scale using Kriging interpolation, response surface method and artificial neural network: case study of Columbia River Basin Dams, USA. Natl Resour Res. https://doi.org/10.1007/s11053-019-09524-2
Keshtegar B, Mert C, Kisi O (2018) Comparison of four heuristic regression techniques in solar radiation modeling: Kriging method vs RSM, MARS and M5 model tree. Renew Sustain Energy Rev 81:330–341
Sakata S, Ashida F, Zako M (2003) Structural optimization using Kriging approximation. Comput Methods Appl Mech Eng 192(7):923–939. https://doi.org/10.1016/S0045-7825(02)00617-5
Huang D, Allen TT, Notz WI, Miller RA (2006) Sequential Kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 32(5):369–382. https://doi.org/10.1007/s00158-005-0587-0
Zhang J, Xiao M, Gao L, Qiu H, Yang Z (2018) An improved two-stage framework of evidence-based design optimization. Struct Multidiscip Optim 58(4):1673–1693
Xiao M, Zhang J, Gao L (2020) A system active learning Kriging method for system reliability-based design optimization with a multiple response model. Reliab Eng Syst Saf 199:106935. https://doi.org/10.1016/j.ress.2020.106935
Zhang J, Xiao M, Gao L, Chu S (2019) A combined projection-outline-based active learning Kriging and adaptive importance sampling method for hybrid reliability analysis with small failure probabilities. Comput Methods Appl Mech Eng 344:13–33
Xiao N-C, Yuan K, Zhou C (2020) Adaptive Kriging-based efficient reliability method for structural systems with multiple failure modes and mixed variables. Comput Methods Appl Mech Eng 359:112649
Li H, Liu T, Wang M, Zhao D, Qiao A, Wang X, Gu J, Li Z, Zhu B (2017) Design optimization of stent and its dilatation balloon using Kriging surrogate model. BioMed Eng OnLine 16(1):13. https://doi.org/10.1186/s12938-016-0307-6
Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241. https://doi.org/10.2514/2.1234
Lu C, Feng Y-W, Liem RP, Fei C-W (2018) Improved Kriging with extremum response surface method for structural dynamic reliability and sensitivity analyses. Aerosp Sci Technol 76:164–175. https://doi.org/10.1016/j.ast.2018.02.012
Keshtegar B, Meng D, Ben Seghier MEA, Xiao M, Trung N-T, Bui DT (2020) A hybrid sufficient performance measure approach to improve robustness and efficiency of reliability-based design optimization. Eng Comput. https://doi.org/10.1007/s00366-019-00907-w
Zhang J, Xiao M, Gao L, Fu J (2018) A novel projection outline based active learning method and its combination with Kriging metamodel for hybrid reliability analysis with random and interval variables. Comput Methods Appl Mech Eng 341:32–52
Zhang Y, Gao L, Xiao M (2020) Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization. Comput Struct 230:106197
Sun Z, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152–165. https://doi.org/10.1016/j.ress.2016.09.003
Xiao M, Zhang J, Gao L, Lee S, Eshghi AT (2019) An efficient Kriging-based subset simulation method for hybrid reliability analysis under random and interval variables with small failure probability. Struct Multidiscip Optim 59(6):2077–2092
Keshtegar B, MeAB Seghier (2018) Modified response surface method basis harmony search to predict the burst pressure of corroded pipelines. Eng Fail Anal 89:177–199
Keshtegar B, Heddam S (2018) Modeling daily dissolved oxygen concentration using modified response surface method and artificial neural network: a comparative study. Neural Comput Appl 30(10):2995–3006
Keshtegar B, Kisi O (2017) Modified response-surface method: new approach for modeling pan evaporation. J Hydrol Eng 22(10):04017045
Coleman JN (2004) Method and apparatus for determining the approximate valve of a logarithmic function. Google Patents
Jordan MI (1995) Why the logistic function? A tutorial discussion on probabilities and neural networks. Computational cognitive science technical report
Mathias AC, Rech PC (2012) Hopfield neural network: the hyperbolic tangent and the piecewise-linear activation functions. Neural Netw 34:42–45
Karlik B, Olgac AV (2011) Performance analysis of various activation functions in generalized MLP architectures of neural networks. Int J Artif Intell Expert Syst 1(4):111–122
Keshtegar B, Ozbakkaloglu T, Gholampour A (2017) Modeling the behavior of FRP-confined concrete using dynamic harmony search algorithm. Eng Comput 33(3):415–430
Gao L, Xiao M, Shao X, Jiang P, Nie L, Qiu H (2012) Analysis of gene expression programming for approximation in engineering design. Struct Multidiscip Optim 46(3):399–413. https://doi.org/10.1007/s00158-012-0767-7
Keshtegar B, Bagheri M, Yaseen ZM (2019) Shear strength of steel fiber-unconfined reinforced concrete beam simulation: application of novel intelligent model. Compos Struct 212:230–242
Chiu SL (1994) Fuzzy model identification based on cluster estimation. J Intell Fuzzy Syst 2(3):267–278
Bezdek JC, Ehrlich R, Full W (1984) FCM: the fuzzy c-means clustering algorithm. Comput Geosci 10(2):191–203. https://doi.org/10.1016/0098-3004(84)90020-7
Qiu B-Z, Li X-L, Shen J-Y (2007) Grid-based clustering algorithm based on intersecting partition and density estimation. In: Washio T, Zhou ZH, Huang JZ, Hu XT, Li J, Xie C, He J, Zou D, Li KC, Freire MM (eds) Emerging technologies in knowledge discovery and data mining. Springer, Berlin, pp 368–377
Kowsar R, Keshtegar B, Marey MA, Miyamoto A (2017) An autoregressive logistic model to predict the reciprocal effects of oviductal fluid components on in vitro spermophagy by neutrophils in cattle. Sci Rep 7(1):4482
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hasanipanah, M., Meng, D., Keshtegar, B. et al. Nonlinear models based on enhanced Kriging interpolation for prediction of rock joint shear strength. Neural Comput & Applic 33, 4205–4215 (2021). https://doi.org/10.1007/s00521-020-05252-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-020-05252-4