# Pseudo-static Slope Stability Analysis for Cohesive-Frictional Soil by Using Variational Method

• Sourav Sarkar
• Manash Chakraborty
Conference paper
Part of the Lecture Notes in Civil Engineering book series (LNCE, volume 55)

## Abstract

In this research work, the factor of safety of a rectilinear slope consisting of cohesive-frictional soil subjected to seismic load is determined on the basis of the calculus of variation theory and pseudo-static analysis. Unlike the conventional limit equilibrium method, variational calculus method neither requires kinematical assumption (i.e. the shape of the critical slip surface) nor any static assumption (i.e. distribution of normal stress along the slip surface). The factor of safety (F) is defined as a functional of normal stress and slip surface. The functional is minimized using Euler-Lagrangian equation. The critical slip surface and consequently critical factor of safety, Fs is being obtained by employing the (i) transversality conditions and the boundary conditions at the intersection of slip surface and the slope surface and (ii) continuity conditions and natural boundary conditions at the intermediate point of the slip surface. The value of Fs is obtained for different combinations of soil friction angle, and slope angle, β corresponding to varying horizontal (kh) and vertical (kv) seismic coefficients. The results suggest that the seismic coefficients, especially the value of kh have significant impact on the stability of the slope. The design charts are prepared for different combinations of , β, kh and kv. For a certain , the factor of safety decreases with the increase of kh, kv and β. The available solutions compare quite well with the available solution for the pseudo-static slope stability analysis.

## Keywords

Variational calculus Slope stability Seismic coefficient Limit equilibrium method

## Notes

### Acknowledgements

The corresponding author acknowledges the support of ‘Department of Science and Technology (DST), Government of India’ under grant number DST/INSPIRE/04/2016/001692.

## References

1. 1.
Ausilio E, Conte E, Dente G (2000) Seismic stability analysis of reinforced slopes. Soil Dyn Earthq Eng 19(3):159–172
2. 2.
Baker R, Shukha R, Operstein V, Frydman S (2006) Stability charts for pseudo-static slope stability analysis. Soil Dyn Earthq Eng 26(9):813–823
3. 3.
Baker R (2003) Sufficient conditions for existence of physically significant solutions in limiting equilibrium slope stability analysis. Int J Solids Struct 40:3717–3735
4. 4.
Baker R, Garber M (1978) Theoretical analysis of the stability of slopes. Geotechnique 28(4):395–411
5. 5.
Bishop AW (1955) The use of the slip circle in the stability analysis of slopes. Geotechnique 5(1):7–17
6. 6.
Chen J, Yang Z, Hu R, Zhang H (2016) Study on the seismic active earth pressure by variational limit equilibrium method. Shock Vib 1–11. Hindawi Publishing CorporationGoogle Scholar
7. 7.
Chen ZY, Morgenstern NR (1983) Extensions to the generalized method of slices for stability analysis. Can Geotech J 20(1):104–119
8. 8.
Choudhury D, Basu S, Bray J (2007) Behaviour of slopes under static and seismic conditions by limit equilibrium method. J Embankments Dams Slopes 1–10Google Scholar
9. 9.
Fellenius W (1936) Calculation of stability of earth dam. In: Transactions. 2nd Congress Large Dams, Washington, DC, vol 4, pp 445–462Google Scholar
10. 10.
Goodman RE, Seed HB (1966) Earthquake-induced displacements in sand embankments. J Soil Mech Found Eng ASCE 92:125–146Google Scholar
11. 11.
Janbu N (1954) Application of composite slip surface for stability analysis. Proc Eur Conf Stab Earth Slopes Sweden 3:43–49Google Scholar
12. 12.
Kopacsy J (1961) Distribution des constraintes a' 1a rupture, forme de 1a surface de G1issement et hauteur Theorique des Talus. In: Proceedings 5th international conference on soil mechanics and foundation engineering, vol II. ParisGoogle Scholar
13. 13.
Kramer SL, Smith MW (1997) Modified Newmark model for seismic displacements of compliant slopes. J Geotech Geoenviron Eng ASCE 123(7):635–644
14. 14.
Leshchinsky D, San KC (1994) Pseudostatic seismic stability of slopes: design charts. J Geotech Eng ASCE 120:1514–1532
15. 15.
Ling HI, Leshchinsky D, Mohri Y (1997) Soil slopes under combined horizontal and vertical seismic accelerations. Earthquake Eng Struct Dyn 26(12):1231–1241
16. 16.
Ling HI, Leshchinsky D (1995) Seismic performance of slopes. Soils Foundation 35:85–94
17. 17.
Loukidis D, Bandini P, Salgado R (2003) Stability of seismically loaded slopes using limit analysis. Geotechnique 53(5):463–479
18. 18.
Morgenstern NU, Price VE (1965) The analysis of the stability of general slip surfaces. Geotechnique 15(1):79–93
19. 19.
Narayan CGP (1975) Variational methods in stability analysis of slopes. PhD dissertation submitted to Indian Institute of Technology, Delhi, IndiaGoogle Scholar
20. 20.
Newmark N (1965) Effects of earthquake on dams and embankments. Geotechnique 15(2):139–160
21. 21.
Rathje EM, Bray JD (2000) Nonlinear coupled seismic sliding analysis of earth structures. J Geotech Geoenviron Eng ASCE 126(11):1002–1014
22. 22.
Revilla J, Castillo E (1977) The calculus of variations applied to stability of slopes. Geotechnique 27(1):1–11
23. 23.
Sarma SK (1973) Stability analysis of embankments and slopes. Geotechnique 23:423–433
24. 24.
Sarma SK (1975) Seismic stability of earth dams and embankments. Geotechnique 25:743–761
25. 25.
Sarma SK (1999) Seismic bearing capacity of shallow strip footings adjacent to a slope. In: Proceedings of 2nd international conference on earthquake geotechnical engineering, Lisbon, Balkema, pp 309–313Google Scholar
26. 26.
Spencer E (1967) A method of analysis of the stability of embankments assuming parallel inter-slice forces. Geotechnique 17(1):11–26
27. 27.
Taylor DW (1937) Stability of earth slopes. J Boston Soc Civ Engineers 24(3):197–247Google Scholar
28. 28.
Wartman J, Bray JD, Seed RB (2003) Inclined plane studies of the Newmark sliding block procedure. J Geotech Geoenviron Eng ASCE 129(8):673–684