Mathematical Treatment of Bimodality for the Safety Factor in Riverbank Stability Analysis Using Cusp Catastrophe
Publication: International Journal of Geomechanics
Volume 23, Issue 11
Abstract
The safety factor (SF) in riverbank stability problems for noncohesive soils is treated mathematically in this article to unravel its inherent bimodality, as stipulated by the cusp catastrophe technique in a hyperspace. The developed methodology may be contrasted with traditional approaches delineating the SF space to failure and operational states. The emerging three states are a significant shift from traditional treatments overlooking bimodality. The mathematical treatment presented in the technical note incorporates classic soil equations for noncohesive soils into the cusp catastrophe technique, with clear formulations at multiple levels of a potential function, including the energy level. The integrated mathematical expressions use soil properties to express lower order properties such as catastrophe flags, for example, bifurcation sets, hysteresis, and bimodality delineating sudden and gradual change in the state of a system. These equations show that even the SF of operational states depends on soil properties and gravity, and therefore a safe use of the SF requires a deep knowledge of the SF hyperspace.
Practical Applications
Practical applications of the new mathematical development presented in the technical note may be viewed in three steps. In Step 1, the tacit nature of the cusp catastrophe bimodality of safety factor in riverbank stability problems needs to be tested through experimental data. The laboratory tests and fieldwork can be designed to encompass the full range of cases comprising: (i) the failure region; (ii) the operational region; and (iii) their bimodal zone. In Step 2, the existence of the three cases is verified by various applications to underpin the dependency of the safety factor on soil parameters. In Step 3, this new knowledge is realized by wide applications to gain an insight into behaviors of safety factors in wide-ranging problems such as natural slopes, channels, embankments, riverbanks, and levees.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
No data, models, or code were generated or used during the study.
References
Amirshahi, S. M., A. R. Zarrati, M. K. Tabarestani, and M. Tabesh. 2022. “Effect of turbulence intensity on riprap stability over streambeds.” J. Hydraul. Eng. 148: 06022013. https://doi.org/10.1061/(ASCE)HY.1943-7900.0002008.
Austria, P. M. 1987. “Catastrophe model for the forced hydraulic jump.” J. Hydraul. Res. 25: 269–280. https://doi.org/10.1080/00221688709499270.
BDS (British Dam Society). 1994. Reservoir safety and the environment 1994 Bds conference discussion. Great George Street, London: Institution of Civil Engineer.
Biondi, G., E. Cascone, and M. Maugeri. 2002. “Flow and deformation failure of sandy slopes.” Soil Dyn. Earthquake Eng. 22: 1103–1114. https://doi.org/10.1016/S0267-7261(02)00136-7.
Cobb, L. 1978. “Stochastic catastrophe models and multimodal distributions.” Behav. Sci. 23 (4): 360–374. https://doi.org/10.1002/bs.3830230407.
Dai, W. H., and H. W. Tang. 2010. “A mathematical model of migration and expansion of meander loops.” J. Hydrodyn. 22: 214–220. https://doi.org/10.1016/S1001-6058(09)60047-0.
Das, B. M. 2012. Principles of geotechnical engineering. Stamford, USA: Cengage learning.
Davies, O., M. Rouainia, S. Glendinning, M. Cash, and V. Trento. 2014. “Investigation of a pore pressure driven slope failure using a coupled hydro-mechanical model.” Eng. Geol. 178: 70–81. https://doi.org/10.1016/j.enggeo.2014.05.012.
Delgado, J., F. Vicente, F. García-Tortosa, P. Alfaro, A. Estévez, J. M. Lopez-Sanchez, R. Tomás, and J. J. Mallorquí. 2011. “A deep seated compound rotational rockslide and rock spread in SE Spain: Structural control and DInSAR monitoring.” Geomorphology 129: 252–262. https://doi.org/10.1016/j.geomorph.2011.02.019.
Dong, H., X. Liu, and W. Zhu. 2021. “Experimental and theoretical study of shear instability of rock joints in the direct shear test.” Int. J. Geomech. 21 (3): 04021004. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001943.
Fukuoka, S., and K. Tabata. 2021. “Risk assessment of levee seepage failure based on the levee seepage failure probability Pf and the levee vulnerability index t.” J. Hydraul. Eng. 147: 04020090. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001829.
Ghorbani, M. A., R. Khatibi, B. Sivakumar, and L. Cobb. 2010. “Study of discontinuities in hydrological data using catastrophe theory.” Hydrol. Sci. J. 55: 1137–1151. https://doi.org/10.1080/02626667.2010.513477.
Green, D. 1996. “Latin America: Neoliberal failure and the search for alternatives.” Third World Q. 17: 109–122. https://doi.org/10.1080/01436599650035806.
Hartelman, P. A. I. 1997. Stochastic catastrophe theory. Amsterdam: Faculteit der Psychologie.
Jaeger, J. C., N. G. Cook, and R. Zimmerman. 2009. Fundamentals of rock mechanics. Hoboken, NJ: John Wiley & Sons.
Marandi, S. M., M. Anvar, and M. Bahrami. 2016. “Uncertainty analysis of safety factor of embankment built on stone column improved soft soil using fuzzy logic α-cut technique.” Comput. Geotech. 75: 135–144. https://doi.org/10.1016/j.compgeo.2016.01.014.
Morse, M., and S. S. Cairns. 1969. Critical point theory in global analysis and differential topology – an introduction. New York and London: Academic Press.
Nadiri, A. A., Z. Sedghi, R. Khatibi, and S. Sadeghfam. 2018. “Mapping specific vulnerability of multiple confined and unconfined aquifers by using artificial intelligence to learn from multiple DRASTIC frameworks.” J. Environ. Manage. 227: 415–428. https://doi.org/10.1016/j.jenvman.2018.08.019.
NDRC (National Development and Reform Commission). 2007. Design specification for rolled Earth-Rock Fill Dams. Beijing: China Electric Power Press.
NEAC (National Energy Administration of China). 2011. Design specification for concrete face Rockfill Dams. Beijing: China Electric Power Press.
Ning, B., S. Wu, Y. Tan, X. Xie, J. Yan, Z. Yan, and Y. Geng. 2011. “Coupling effect of seepage flow and river flow on the bank failure.” J. Hydrodyn. Ser. B 23: 834–840. https://doi.org/10.1016/S1001-6058(10)60183-7.
Oyanguren, P. R., C. G. Nicieza, MÁ Fernández, and C. G. Palacio. 2008. “Stability analysis of Llerin Rockfill Dam: An in situ direct shear test.” Eng. Geol. 100: 120–130. https://doi.org/10.1016/j.enggeo.2008.02.009.
Peng, W., M. Zhao, H. Zhao, and C. Yang. 2022. “Seismic stability of the slope containing a laterally loaded pile by finite-element limit analysis.” Int. J. Geomech. 22: 06021033. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002226.
Petterson, K. E. 1955. “The early history of circular sliding surfaces.” Géotechnique 5: 275–296. https://doi.org/10.1680/geot.1955.5.4.275.
Qin, S., J. J. Jiao, S. Wang, and H. Long. 2001a. “A nonlinear catastrophe model of instability of planar-slip slope and chaotic dynamical mechanisms of its evolutionary process.” Int. J. Solids Struct. 38: 8093–8109. https://doi.org/10.1016/S0020-7683(01)00060-9.
Qin, S. Q., J. J. Jiao, and S. Wang. 2001b. “A cusp catastrophe model of instability of slipbuckling slope.” Rock Mech. Rock Eng. 34: 119–134. https://doi.org/10.1007/s006030170018.
Sadeghfam, S., A. Ehsanitabar, R. Khatibi, and R. Daneshfaraz. 2018. “Investigating “risk” of groundwater drought occurrences by using reliability analysis.” Ecol. Indic. 94: 170–184. https://doi.org/10.1016/j.ecolind.2018.06.055.
Sadeghfam, S., Y. Hassanzadeh, A. A. Nadiri, and M. Zarghami. 2016. “Localization of groundwater vulnerability assessment using catastrophe theory.” Water Resour. Manage. 30: 4585–4601. https://doi.org/10.1007/s11269-016-1440-5.
Sadeghfam, S., R. Khatibi, Y. Hassanzadeh, R. Daneshfaraz, and M. A. Ghorbani. 2017. “Forced hydraulic jumps described by classic hydraulic equations reproducing cusp catastrophe features.” Arab. J. Sci. Eng. 42: 4169–4179. https://doi.org/10.1007/s13369-017-2616-x.
Song, Y. S., K. S. Kim, and K. S. Woo. 2012. “Stability of embankments constructed from soil mixed with stone dust in quarry reclamation.” Environ. Earth Sci. 67: 285–292. https://doi.org/10.1007/s12665-011-1507-9.
Stewart, I. 1975. “The seven elementary catastrophes.” New Sci. 68: 447–454.
Tao, Y., J. Cao, J. Hu, and Z. Dai. 2013. “A cusp catastrophe model of mid–long-term landslide evolution over low latitude highlands of China.” Geomorphology 187: 80–85. https://doi.org/10.1016/j.geomorph.2012.12.036.
Thang, N. V., A. Wakai, G. Sato, T. T. Viet, and N. Kitamura. 2022. “Simple method for shallow landslide prediction based on wide-area terrain analysis incorporated with surface and subsurface flows.” Nat. Hazard. Rev. 23: 04022028. https://doi.org/10.1061/(ASCE)NH.1527-6996.0000578.
Thom, R. 1989. Structural stability and morphogenesis. New York: Addison Wesley.
Tung, Y. K., B. C. Yen, and C. S. Melching. 2006. Hydrosystems engineering reliability assessment and risk analysis. New York: McGraw-Hill.
Vandamme, J., and Q. Zou. 2013. “Investigation of slope instability induced by seepage and erosion by a particle method.” Comput. Geotech. 48: 9–20. https://doi.org/10.1016/j.compgeo.2012.09.009.
Wu, Z., C. Chen, X. Lu, L. Pei, and L. Zhang. 2020. “Discussion on the allowable safety factor of slope stability for high rockfill dams in China.” Eng. Geol. 272: 1–5. https://doi.org/10.1016/j.enggeo.2020.105666.
Xia, K., C. Chen, Y. Zhou, X. Liu, Y. Zheng, and Y. Pan. 2019. “Catastrophe instability mechanism of the pillar-roof system in gypsum mines due to the influence of relative humidity.” Int. J. Geomech. 19: 06019004. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001378.
Yan, J., Z. X. Cao, H. H. Liu, and L. Chen. 2009. “Experimental study of landslide dambreak flood over erodible bed in open channels.” J. Hydrodyn. 21: 124–130. https://doi.org/10.1016/S1001-6058(08)60127-4.
Zafir, Z., E. Abraham, and A. Wright. 2022. “Seismic slope stability analysis and mitigation for transmission power lines in Southern California.” In Lifelines, edited by C. Davis, K. Yu, and E. Taciroglu, 228–240. Reston, VA: ASCE.
Zhang, G., R. Wang, J. Qian, J. M. Zhang, and J. Qian. 2012. “Effect study of cracks on behavior of soil slope under rainfall conditions.” Soils Found. 52: 634–643. https://doi.org/10.1016/j.sandf.2012.07.005.
Zhang, X. Y., Y. M. Zhu, and C. H. Fang. 2009. “The role fore air flow in soil slope stability analysis.” J. Hydrodyn. 21: 640–646. https://doi.org/10.1016/S1001-6058(08)60195-X.
Zhou, X. P., X. Wei, C. Liu, and H. Cheng. 2020. “Three-dimensional stability analysis of bank slopes with reservoir drawdown based on rigorous limit equilibrium method.” Int. J. Geomech. 20: 04020229. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001877.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 23 • Issue 11 • November 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Oct 21, 2022
Accepted: May 31, 2023
Published online: Sep 13, 2023
Published in print: Nov 1, 2023
Discussion open until: Feb 13, 2024
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.