Probabilistic Assessment for Slope Using the Generalized Chebyshev Inequalities
Publication: International Journal of Geomechanics
Volume 20, Issue 4
Abstract
This paper aims to estimate the probability of failure () of slope based on Chebyshev inequalities. Traditionally, for a slope reliability analysis, the supposed distribution types of shear strength parameters are employed; this is not accurate because the distribution types of parameters are uncertain. Chebyshev inequalities can estimate the probability even when the distribution types of variables are unknown. The upper bound value of the probability of failure () of a step-shaped slope is derived based on the Chebyshev inequalities. Additionally, the bootstrap method combined with the Akaike information criterion (AIC) are adopted to validate the accuracy and efficiency of the proposed method. It can be concluded from the illustrative example that the of a slope can be estimated by the Chebyshev inequalities. Compared with traditional probabilistic methods, Chebyshev inequalities are a quick and easy way to assess the upper bound value of .
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Nos. 51909087, 51679017), Research Fund of Hunan University of Science and Technology (No. E51975), Open Fund of National Engineering Laboratory of Highway Maintenance Technology (Changsha University of Science & Technology, kfj190107), and Scientific Research Projects of Hunan Education Department (18K064).
References
Adèr, H. J., G. J. Mellenbergh, and D. J. Hand. 2008. Advising on research methods: A consultant’s companion. Huizen, The Netherlands: Johannes van Kessel Publishing.
Akaike, H. 1973. “Information theory and an extension of the maximum likelihood principle.” In Proc., Int. Symp. on Information Theory, 610–624. Budapest, Hungary: Akademiai Kiado.
Alonso, E. E. 1977. “Risk analysis of slopes and its application to slopes in Canadian sensitive clays.” Géotechnique 26 (26): 453–472. https://doi.org/10.1680/geot.1976.26.3.453.
Cho, S. E. 2009. “Probabilistic stability analyses of slopes using the ANN-based response surface.” Comput. Geotech. 36 (5): 787–797. https://doi.org/10.1016/j.compgeo.2009.01.003.
Chowdhury, R. N., and D. W. Xu. 1995. “Geotechnical system reliability of slopes.” Reliab. Eng. Syst. Saf. 47 (3): 141–151. https://doi.org/10.1016/0951-8320(94)00063-T.
Christian, J. T., C. C. Ladd, and G. B. Baecher. 1994. “Reliability applied to slope stability analysis.” J. Geotech. Eng. 120 (12): 2180–2207. https://doi.org/10.1061/(ASCE)0733-9410(1994)120:12(2180).
Dasgupta, A. 2000. “Best constants in Chebyshev inequalities with various applications.” Metrika 51 (3): 185–200. https://doi.org/10.1007/s184-000-8316-9.
Duncan, J. M. 2000. “Factors of safety and reliability in geotechnical engineering.” J. Geotech. Geoenviron. Eng. 126 (4): 307–316. https://doi.org/10.1061/(ASCE)1090-0241(2000)126:4(307).
Efron, B. 1979. “Bootstrap methods: Another look at the jackknife” Ann. Stat. 7 (1): 1–26. https://doi.org/10.1214/aos/1176344552.
Fang, K. T. 1980. “The uniform design: Application of number-theoretic methods in experimental design.” [In Chinese.] Acta Mathematica Applicatae Sin. 3 (4): 363–372.
Fang, K. T., and Y. Zhang. 2000. “Uniform design: Theory and application.” Technometrics 42 (3): 237–248. https://doi.org/10.1080/00401706.2000.10486045.
Huang, X. C., X. P. Zhou, W. Ma, Y. W. Niu, and Y. T. Wang. 2016. “Two-dimensional stability assessment of rock slopes based on random field.” Int. J. Geomech. 17 (7): 04016155. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000858.
Kim, H. G., D. K. Lee, C. Park, Y. Ahn, S. H. Kil, S. Sung, and G. S. Biging. 2018. “Estimating landslide susceptibility areas considering the uncertainty inherent in modeling methods.” Stochastic Environ. Res. Risk Assess. 32 (11): 2987–3019. https://doi.org/10.1007/s00477-018-1609-y.
Li, C., H. Tang, Y. Ge, X. Hu, and L. Wang. 2014. “Application of back-propagation neural network on bank destruction forecasting for accumulative landslides in the Three Gorges Reservoir region, China.” Stochastic Environ. Res. Risk Assess. 28 (6): 1465–1477. https://doi.org/10.1007/s00477-014-0848-9.
Li, D. Q., X. S. Tang, and K. K. Phoon. 2015. “Bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability.” Reliab. Eng. Syst. Saf. 140 (Aug): 99–106. https://doi.org/10.1016/j.ress.2015.03.034.
Liu, Y., L. Q. He, Y. J. Jiang, M. M. Sun, E. J. Chen, and F. H. Lee. 2018. “Effect of in situ water content variation on the spatial variation of strength of deep cement-mixed clay.” Géotechnique 69 (5): 391–405. https://doi.org/10.1680/jgeot.17.P.149.
Liu, Y., Y. J. Jiang, H. Xiao, and F. H. Lee. 2017. “Determination of representative strength of deep cement-mixed clay from core strength data.” Géotechnique 67 (4): 350–364. https://doi.org/10.1680/jgeot.16.P.105.
Luo, Z., S. Atamturktur, and C. H. Juang. 2013. “Bootstrapping for characterizing the effect of uncertainty in sample statistics for braced excavations.” J. Geotech. Geoenviron. Eng. 139 (1): 13–23. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000734.
Miao, F., Y. Wu, Y. Xie, F. Yu, and L. Peng. 2017. “Research on progressive failure process of Baishuihe landslide based on Monte Carlo model.” Stochastic Environ. Res. Risk Assess. 31 (7): 1683–1696. https://doi.org/10.1007/s00477-016-1224-8.
Most, T. 2011. “Efficient structural reliability methods considering incomplete knowledge of random variable distributions.” Probab. Eng. Mech. 26 (2): 380–386. https://doi.org/10.1016/j.probengmech.2010.09.003.
Most, T., and T. Knabe. 2010. “Reliability analysis of the bearing failure problem considering uncertain stochastic parameters.” Comput. Geotech. 37 (3): 299–310. https://doi.org/10.1016/j.compgeo.2009.11.003.
Saw, J. G., and Y. T. C. Mo. 1984. “Chebyshev inequality with estimated mean and variance.” Am. Stat. 38 (2): 130–132. https://doi.org/10.2307/2683249.
Torizin, J. 2016. “Elimination of informational redundancy in the weight of evidence method: An application to landslide susceptibility assessment.” Stochastic Environ. Res. Risk Assess. 30 (2): 635–651. https://doi.org/10.1007/s00477-015-1077-6.
Zhang, J., H. W. Huang, C. H. Juang, and W. W. Su. 2014. “Geotechnical reliability analysis with limited data: Consideration of model selection uncertainty.” Eng. Geol. 181 (Oct): 27–37. https://doi.org/10.1016/j.enggeo.2014.08.002.
Zhou, X. P., and X. C. Huang. 2018. “Reliability analysis of slopes using UD-based response surface methods combined with LASSO.” Eng. Geol. 233 (Jan): 111–123. https://doi.org/10.1016/j.enggeo.2017.12.008.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 20 • Issue 4 • April 2020
Copyright
©2020 American Society of Civil Engineers.
History
Received: May 4, 2019
Accepted: Sep 20, 2019
Published online: Jan 28, 2020
Published in print: Apr 1, 2020
Discussion open until: Jun 28, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.