Skip to main content

Advertisement

SpringerLink
Log in

Development of a probabilistic kinematic wedge sliding analysis procedure and application to a rock slope at a hydropower site in China

  • Original Paper
  • Published:
Bulletin of Engineering Geology and the Environment Aims and scope Submit manuscript

Abstract

In performing kinematic wedge stability analysis, in practice, it is usually assumed that the wedge is a “Standard Wedge” (the term is explained in the paper) sliding along the line of intersection of the two discontinuity planes. Intersection of two discontinuities with the lower and upper rock slope surfaces can result in forming Standard Wedges, Overlying Wedges and No Wedges. Vector procedures are developed in the paper to differentiate the aforementioned formation modes for the wedges. Vector procedures are then given to calculate the sliding modes, sliding directions, maximum safe slope angles and factor of safety for the Standard and Overlying wedges. Procedures are also given to incorporate the variability of discontinuity orientations and strength and to calculate the instability cumulative probabilities corresponding to different cut slope dip angles. All these procedures are incorporated in a computer code, named PKWSAC. The developed procedure and the computer program were then validated by applying them to three examples and comparing the results obtained through this procedure with the results obtained through Lucas’ stereographic procedure [Lucas JM (1980) Int J Rock Mech Min Sci Geomech Abstr 17:57–61]. The validated procedure was then applied to a rock slope at a hydropower site in China to perform both deterministic and probabilistic kinematic wedge stability analyses. The paper also shows a comparison between the deterministic and probabilistic analyses results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

CM:

Combination module

CODS:

Combinations of discontinuity sets

COTD:

Combinations of two discontinuities

\( D_{i}\) :

Discontinuity i

FM:

Formation mode: Standard Wedge, Overlying Wedge (D1-overlying-D2 Wedge or D2-overlying-D1 Wedge), No Wedge

F s :

Factor of safety

I 12 :

The vector along the LOI of planes 1 and 2

\( \varvec{I}_{12}^{{\prime }} \) :

The downward vector along the LOI of planes 1 and 2

KWSAM:

Kinematic wedge sliding analysis module

LOI:

Line of intersection

MOF:

Modes of failure: plane sliding, wedge sliding, and toppling

MSSA:

Maximum safe slope angle

n CODS :

Number of combinations of discontinuity sets

n COTD-K :

Number of COTD from CODS K

\({n_i} \) :

Number of discontinuities from set i

\(\varvec{n_i} \) :

Upward unit normal vector of discontinuity or plane i, where i equals to 1, 2, and s for D 1, D 2, and P s

\( \varvec{n}_{i}^{{\prime }} \) :

Unit normal vector of \( D_{i}\) pointing to interior of the formed wedge

n xyi :

Projection of \(\varvec{n_i} \) on the horizontal plane

PAM:

Probabilistic analysis module

PKWSAC:

Probabilistic kinematic wedge sliding analysis code

P s :

Slope face

P(T):

Total cumulative probability obtained by PKWSAC

P(T′):

Total cumulative probability obtained assuming all wedges are Standard Wedges and slide along LOI

P wijm :

The instability probability with respect to WFM = W ij for cut slope dip angle m

m :

Cut slope dip angle

r :

Unit vector of resultant force

S 1, S 2, S 12 :

Sliding along D 1, D 2, and LOI of D 1 and D 2, respectively

s 1 , s 2 , s 12 :

The sliding directions of S 1, S 2, and S 12, respectively

SM:

Sliding mode of a wedge, including single plane (S 1 and S 2) and double plane sliding (S 12)

WFM:

Wedge failure modes

\( x_{i},\, y_{i},\, z_{i}\) :

The components of \(\varvec{n_i} \) in (x, y, z) coordinate system

\( x_{i}^{{\prime }},\, y_{i}^{{\prime }},\, z_{i}^{{\prime }}\) :

The components of \( \varvec{n}_{i}^{{\prime }} \) in (x, y, z) coordinate system

β 12 :

The plunge of the LOI of D 1 and D 2

γ 12l , γ 21k :

The angles of intersection of θ 1 and ψ 2l , and θ 2 and ψ 1k , respectively

γ 12s :

Angle of intersection between the trend of LOI of D 1 and D 2 and the dip direction of the slope face

γ ims , γ ips :

The angles of intersection of θ s and ψ im , and θ s and ψ ip , respectively

δ i :

Dip angle of discontinuity or plane i, where i equals to 1, 2, and s for D 1, D 2, and Ps

γ is :

The angle of intersection between the dip directions of D i and the slope face

ε 12 :

The trend of the LOI of D 1 and D 2

θ i :

Dip direction of discontinuity or plane i, where i equals to 1, 2, and s for D 1, D 2, and P s

φ :

Friction angle

ψ im , ψ ip :

The two directions of the strike of D i

ψ i :

The direction of the strike of D i that intersects the strike of the slope face

References

  • Admassu Y, Shakoor A (2013) DIPANALYST: a computer program for quantitative kinematic analysis of rock slope failures. Comput Geosci 54:196–202

    Article  Google Scholar 

  • Ang AHS, Tang WH (1984) Probability concepts in engineering planning and design, vols I and II. Wiley New York

  • Baecher GB, Christian JT (2003) Reliability and statistics in geotechnical engineering. Wiley, New York

  • Carter BJ, Lajtai EZ (1992) Rock slope stability and distributed joint systems. Can Geotech J 29:53–60

    Article  Google Scholar 

  • Chen Z (2004) A generalized solution for tetrahedral rock wedge stability analysis. Int J Rock Mech Min Sci 41:613–628

    Article  Google Scholar 

  • Cruden DM (1978) Discussion of G Hocking’s paper ‘‘A method for distinguishing between single and double plane sliding of tetrahedral wedges’’. Int J Rock Mech Min Sci Geomech Abstr 15:217

    Article  Google Scholar 

  • Gokceoglu C, Sonmez H, Ercanoglu M (2000) Discontinuity controlled probabilistic slope failure risk maps of the Altindag (settlement) region in Turkey. Eng Geol 55:277–296

    Article  Google Scholar 

  • Goodman RE (1976) Methods of geological engineering in discontinuous rocks. West, San Francisco

    Google Scholar 

  • Goodman RE (1989) Introduction to rock mechanics, 2nd edn. Wiley, New York

    Google Scholar 

  • Goodman RE, Shi GH (1985) Block theory and its application to rock engineering. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Hocking GA (1976) A method for distinguishing between single and double plane sliding of tetrahedral wedges. Int J Rock Mech Min Sci 13:225–226

    Article  Google Scholar 

  • Hoek E, Bray JW (1981) Rock slope engineering, 3rd edn. Institute of Mining and Metallurgy, London

    Google Scholar 

  • Jiang Q, Liu X, Wei W, Zhou C (2013) A new method for analyzing the stability of rock wedges. Int J Rock Mech Min Sci 60:413–422

    Google Scholar 

  • Jimenez-Rodriguez R, Sitar N (2007) Rock wedge stability analysis using system reliability methods. Rock Mech Rock Eng 40(4):419–427

    Article  Google Scholar 

  • Kulatilake PHSW (1985) Fitting of Fisher distribution on discontinuity orientation data. J Geolog Educ 33:266–269

    Article  Google Scholar 

  • Kulatilake PHSW (1988) Minimum rock bolt force and minimum static acceleration in tetrahedral wedge stability: a probabilistic study. Int J Surface Mining 2:19–26

  • Kulatilake PHSW, Fuenkajorn K (1987) Factor of safety of tetrahedral wedges: a probabilistic study. Int J Surface Mining 1:147–154

  • Kulatilake PHSW, Shu B (2015) An approach to predict rock mass deformations in three dimensions for a part of an open pit mine and comparison with field deformation monitoring data. In: Proceedings of the 34th Ground Control Conference, Morgantown, West Virginia, USA, 28–30 July (in press)

  • Kulatilake PHSW, Wu TH (1984) Estimation of mean trace length of discontinuities. Rock Mech Rock Eng 17:215–232

    Article  Google Scholar 

  • Kulatilake PHSW, Wu TH, Wathugala DN (1990) Probabilistic modeling of joint orientation. Int J Numer Anal Methods Geomech 14:325–350

    Article  Google Scholar 

  • Kulatilake PHSW, Um J, Morin B (2003a) Investigation of slope stability for a section of Phelps Dodge Sierrita Open Pit Mine. Tran Soc Min Metall Explor 314:177–182

    Google Scholar 

  • Kulatilake PHSW, Um J, Wang M, Escandon RF, Narvaiz J (2003b) Stochastic fracture geometry modeling in 3-D including validations for a part of Arrowhead East Tunnel site, California, USA. Int J Eng Geol 70(1–2):131–155

    Article  Google Scholar 

  • Kulatilake PHSW, Wang LQ, Tang HM, Liang Y (2011) Evaluation of rock slope stability for Yujian River dam site by kinematic and block theory analyses. Comput Geotech 38:846–860

    Article  Google Scholar 

  • Low BK (1997) Reliability analysis of rock wedges. J Geotech Geoenviron Eng 123:498–505

    Article  Google Scholar 

  • Lucas JM (1980) A general stereographic method for determining possible mode of failure of any tetrahedral rock wedge. Int J Rock Mech Min Sci Geomech Abstr 17:57–61

    Article  Google Scholar 

  • Markland JT (1972) A useful technique for estimating the stability rock slopes when the rigid wedge sliding type failure expected. Imp Coll Rock Mech Res Rep 19:1–10

    Google Scholar 

  • Nilsen B (2000) New trend in rock slope stability analysis. Bull Eng Geol Environ 58:173–178

    Article  Google Scholar 

  • Öcal A, Ozgenoglu A (1997) Determination of sliding mode of tetrahedral wedges in jointed rock slopes. Rock Mech Rock Eng 30(3):161–165

    Article  Google Scholar 

  • Park HJ, West TR (2001) Development of a probabilistic approach for rock wedge failure. Eng Geol 59:233–251

    Article  Google Scholar 

  • Park HJ, West TR, Woo I (2005) Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40, Western North Carolina, USA. Eng Geol 79:230–250

    Article  Google Scholar 

  • Pathak S, Nilsen B (2004) Probabilistic rock slope stability analysis for Himalayan condition. Bull Eng Geol Environ 63:25–32

    Article  Google Scholar 

  • RockWare, ROCKPACKIII (2013). http://www.rockware.com/product/overview.php?id=119. Accessed 21 November 2013

  • Rocscience, DIPS (2013). http://www.rocscience.com/products/1/Dips. Accessed 21 November 2013

  • Sherizadeh T, Kulatilake PHSW (2015) Effect of geological and mining factors on roof stability of a room and pillar mine in US. In: Proceedings of the 34th Ground Control Conference, Morgantown, West Virginia, USA, 28–30 July (in press)

  • Sonmez H, Gokceoglu C (1998) A computer program for the kinematic analysis of rock slopes and its application. Earthsciences 20:75–89

    Google Scholar 

  • Um J, Kulatilake PHSW (2001) Kinematic and block theory analyses for shiplock slopes of the Three Gorges Dam Site in China. Geotech Geol Eng 19:21–42

    Article  Google Scholar 

  • Wang YJ, Yin JH (2002) Wedge stability analysis considering dilatancy of discontinuities. Rock Mech Rock Eng 35(2):127–137

    Article  Google Scholar 

  • Wang YJ, Yin JH, Chen Z, Lee CF (2004) Analysis of wedge stability using different methods. Rock Mech Rock Eng 37(2):127–150

    Article  Google Scholar 

  • Wathugala DN, Kulatilake PHSW, Wathugala GW, Stephansson O (1990) A general procedure to correct sampling bias on joint orientation using a vector approach. Comp Geotech 10:1–31

    Article  Google Scholar 

  • Yoon WS, Jeong UJ, Kim JH (2002) Kinematic analysis for sliding failure of multi-faced rock slopes. Eng Geol 67:51–61

    Article  Google Scholar 

Download references

Acknowledgments

The work was partially funded by the Centers for Disease Control and Prevention (Contract No. 200-2011-39886), National Basic Research Program of China (973 Program; No. 2010CB732005) and the National Natural Science Foundation Project (No. 51079093). Jun Zheng is grateful to the Chinese Scholarship Foundation (No. CSC [2012] 3013) for providing a scholarship to conduct the research described in this paper as a Visiting Research Student at the University of Arizona.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Jun Zheng

  2. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, China

    Jun Zheng, Jianhui Deng & Jinbin Wei

  3. Rock Mass Modeling and Computational Rock Mechanics Laboratories, University of Arizona, Tucson, AZ, 85721, USA

    Jun Zheng & P. H. S. W. Kulatilake

Authors
  1. Jun Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. H. S. W. Kulatilake
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianhui Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinbin Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. H. S. W. Kulatilake.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, J., Kulatilake, P.H.S.W., Deng, J. et al. Development of a probabilistic kinematic wedge sliding analysis procedure and application to a rock slope at a hydropower site in China. Bull Eng Geol Environ 75, 1413–1428 (2016). https://doi.org/10.1007/s10064-015-0764-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10064-015-0764-3

Keywords

Search

Navigation