Skip to main content
SpringerLink
Log in

Three Dimensional Limit Analysis of Slope Stability with Nonlinear Criterion: Comparison the Effect of Volumetric Strain

  • Original Paper
  • Published:
Indian Geotechnical Journal Aims and scope Submit manuscript

Abstract

In view of the complexity of three-dimensional analysis, traditional slope stability analysis was mainly conducted in 2-dimensional scope, namely in the case of plane strain condition. Based on the 3-dimensional failure mechanism proposed by Michalowski, three-dimensional upper bound analysis of the soil slope was presented in this paper. The two different conditions of both with and without volumetric strain taken into consideration are analyzed. The nonlinear failure criterion was introduced into analysis by virtue of the generalized tangential technique method, and the earthquake action was analyzed on the basis of pseudo-static method. The formulas of external work rate and interior energy dissipation were deduced, and the stability factors of slope with respect to diverse parameters were calculated through optimization. The results show that, with the increase of B/H, the difference between the stability factors of the two conditions decrease. Moreover, the change law of the stability factors of the different conditions is similar.

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

Similar content being viewed by others

Abbreviations

σ 1, σ 3 :

Maximum and minimum principal stresses

M p *, M p and q p :

Parameters related to c and φ, can be determined by triaxial test

C :

Cohesion of soil

φ :

Internal friction angle

γ :

Soil density

c 0 :

Cohesion of soil for nonlinear Mohr–Coulomb failure criterion

m :

Nonlinear coefficient for nonlinear Mohr–Coulomb failure criterion

τ :

Shear strength

σ n :

Normal stress

σ t :

Uniaxial tensile strength

r :

Expression of AC of the failure mechanism in Fig. 2

r′:

Expression of A′C′ of the failure mechanism in Fig. 2

r m :

Generatrix of conic section of the failure mechanism

R :

Radius of a cross section of the failure mechanism

Ω :

Rate of rotation of the moving mass

V :

Linear velocity of a soil weight element

dV :

Volume of a soil weight element

θ :

Rotation angle of the failure mechanism

θ 0, θ B and θ h :

Rotation angles of the failure mechanism (see in Fig. 2)

W soil :

Soil weight work rate of rotating body

W γ-insert :

Soil weight work rate of insert block

\(W_{{k_{h} {\hbox{-}} soil}}\) :

Seismic forces work rate on rotating body

\(W_{{k_{h} {\hbox{-}} insert}}\) :

Seismic forces work rate on insert block

D t :

Internal energy dissipation

S t :

The velocity discontinuity surface

b :

Width of the insert block

B :

Width of the whole failure mechanism

D AB :

Internal energy dissipation of section θ 0 − θ B of rotating body with the consideration of volumetric strain

D BC :

Internal energy dissipation of section θ B  − θ h of rotating body with the consideration of volumetric strain

D insert :

Internal energy dissipation of section on plane insert mechanism with the consideration of volumetric strain

\(D^{\prime}_{AB}\) :

Internal energy dissipation of section θ 0 − θ B of rotating body without the consideration of volumetric strain

\(D^{\prime}_{BC}\) :

Internal energy dissipation of section θ B  − θ h of rotating body without the consideration of volumetric strain

\(D^{\prime}_{insert}\) :

Internal energy dissipation of section on plane insert mechanism without the consideration of volumetric strain

H :

Critical height of slope

N n :

Stability factor of slope

Referneces

  1. Li AJ, Merifield RS, Lyamin AV (2010) Three-dimensional stability charts for slopes based on limit analysis methods. Can Geotech J 47(12):1316–1334

    Article  Google Scholar 

  2. Gens A, Hutchision JN, Cavounidis S (1988) Three dimensional analysis of slides in cohesive soils. Geotechnique 38(1):1–23

    Article  Google Scholar 

  3. Eid HT, Elleboudy AM, Elmarsafawi HG (2006) Stability analysis and charts for slopes susceptible to translational failure. Can Geotech J 43(12):1374–1388

    Article  Google Scholar 

  4. Sagaseta C (1987) Analysis of undrained soil deformation due to ground loss. Geotechnique 37(3):301–320

    Article  Google Scholar 

  5. Farzaneh O, Askari F, Ganjian N (2008) Three dimensional stability analysis of convex slopes in plan view. J Geotech Geoenviron Eng 134(8):1192–1200

    Article  Google Scholar 

  6. Tiwari RC, Bhandary NP, Yatabe R (2014) 3-D elasto-plastic SEM approach for pseudo-static seismic slope stability charts for natural slopes. Indian Geotech J 44(3):305–321

    Article  Google Scholar 

  7. Stark TD, Eid HT (1998) Performance of three-dimensional slope stability methods in practice. J Geotech Geoenviron Eng 124(11):1049–1060

    Article  Google Scholar 

  8. Chowdhury I, Dasgupta SP (2012) Seismic response of piles under earthquake force. Indian Geotech J 42(2):57–74

    Article  Google Scholar 

  9. Michalowski RL (2001) Upper-bound load estimates on square and rectangular footings. Geotechnique 51(9):787–798

    Article  Google Scholar 

  10. Michalowski RL, Drescher A (2009) Three-dimensional stability of slopes and excavations. Geotechnique 59(10):839–850

    Article  Google Scholar 

  11. Michalowski RL, Martel T (2011) Stability charts for 3D failures of steep slopes subjected to seismic excitation. J Geotech Geoenviron Eng 137(2):183–189

    Article  Google Scholar 

  12. Hoek E (1983) Strength of jointed rock masses. Geotechnique 33(3):187–223

    Article  Google Scholar 

  13. Labuz JF, Zang A (2012) Mohr–Coulomb failure criterion. Rock Mech Rock Eng 45(45):975–979

    Article  Google Scholar 

  14. Hobbs D (1966) A study of the behavior of broken rock under trixaial compression and its application to miner roadways. Int J Rock Mech Min Sci 3(1):11–43

    Article  Google Scholar 

  15. Brown ET, Bray JW, Ladanyi B et al (1983) Ground response curves for rock tunnels. J Geotech Eng 109(1):15–39

    Article  Google Scholar 

  16. Bieniawski ZT (1975) The point-load test in geotechnical practice. Eng Geol 9(1):1–11

    Article  Google Scholar 

  17. Zhang XJ, Chen WF (1987) Stability analysis of slopes with general nonlinear failure criterion. Int J Numer Anal Methods Geomech 11(1):33–50

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Traffic and Transportation Engineering, Changsha University of Science & Technology, Changsha, 410004, China

    Feng Liu & Yongqin Rui

  2. School of Civil Engineering and Mechanics, Central South University of Forestry and Technology, Changsha, 410075, China

    Feng Liu

  3. State Key Laboratory of Safety Technology of Metal Mines, Maanshan, 243001, China

    Chun Zhang

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongqin Rui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix

Appendix

$$f_{1} = \frac{1}{{3(1 + 9\tan^{2} \varphi )}}\left[ {(3\tan \varphi \cos \theta_{h} + \sin \theta_{h} )e^{{3(\theta_{h} - \theta_{0} )\tan \varphi }} - 3\tan \varphi \sin \theta_{0} + \sin \theta_{0} } \right]$$
$$f_{2} = \frac{1}{6}\left( {2\cos \theta_{0} - \frac{L}{{r_{0} }}} \right)\frac{{L\sin \theta_{0} }}{{r_{0} }}$$
$$f_{3} = \frac{1}{6}e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} \left[ {\sin \left( {\theta_{h} - \theta_{0} } \right) - \frac{L}{{r_{0} }}\sin \theta_{h} } \right]\left[ {\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \alpha + \cos \theta_{h} e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} } \right]$$
$$f_{4} = \frac{1}{{3(1 + 9\tan^{2} \varphi )}}\left[ {(3\tan \varphi \sin \theta_{h} - \cos \theta_{h} )e^{{3(\theta_{h} - \theta_{0} )\tan \varphi }} - 3\tan \varphi \sin \theta_{0} + \cos \theta_{0} } \right]$$
$$f_{5} = \frac{1}{6}\left( {2\sin \theta_{0} + \frac{L}{{r_{0} }}} \right)\frac{{L\sin \theta_{0} }}{{r_{0} }}$$
$$f_{6} = \frac{1}{6}e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} \frac{H}{{r_{0} }}\frac{{\sin \left( {\theta_{h} + \beta } \right)}}{\sin \beta }\left[ {2\sin \theta_{h} e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} - \frac{H}{{r_{0} }}} \right]$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Rui, Y. & Zhang, C. Three Dimensional Limit Analysis of Slope Stability with Nonlinear Criterion: Comparison the Effect of Volumetric Strain. Indian Geotech J 47, 57–66 (2017). https://doi.org/10.1007/s40098-016-0193-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40098-016-0193-7

Keywords

Search

Navigation