1. SLOPE STABILITY ANALYSIS
Tuncer B. Edil
University of Wisconsin-Madison

2. LECTURE OUTLINE Common Features of Slope Stability Analysis Methods
Water Forces on Soil
Infinite Slope Analysis
Finite Slopes: Plane, Circular and Noncircular Failure Surfaces

3. COMMON FEATURES OF SLOPE STABILITY ANALYSIS METHODS
Safety Factor: F = S/Sm where S = shear strength and Sm = mobilized shear resistance. F = 1: failure, F > 1: safety
Shape and location of failure is not known a priori but assumed (trial and error to find minimum F)
Static equilibrium (equilibrium of forces and moments on a sliding mass)
Two-dimensional analysis

4. INFINITE SLOPE ANALYSIS
Translational failures along a single plane failure surface parallel to slope surface
The ratio of depth to failure surface to length of failure zone is relatively small (<10%)
Applies to surface raveling in granular materials or slab slides in cohesive materials
Equilibrium of forces on a slice of the sliding mass along the failure surface is considered

5. INFINITE SLOPE hp d  
sat  c N T hp W’

6. INFINITE SLOPE ANALYSIS
F = f(c’, ’, , , d, u)
F = (c’/ d) seccosec + (tan’/tan)(1-ru sec2) where ru = u/d (different ru for seepage parallel to slope face, seepage emerging, seepage downward, etc)
For Granular Soil: F = (tan’/tan)(1-ru sec2) Dry Granular Soil (ru = 0): F = (tan’/tan)
For Cohesive Soil: F decreases with increasing depth to failure plane; if c is sufficiently large, dc for F = 1 may be large and infinite slope failure may not apply.

7. WATER FORCES ON SOIL
Water fills voids and increase weight which increases driving forces
Water also exerts pore pressures which decrease effective stress and therefore strength
There are mathematically two equivalent ways of taking water forces into account in stability analyses

8. EQUIVALENT METHODS FOR WATER FORCES
1. Boundary water force + total unit weight u = hpw; sat consider soil element (particles and water filled pores) as single solid mass
2. Seepage force + submerged unit weight Fs = i wV; ’ consider soil element as particle skeleton with water external to it

9. BOUNDARY WATER FORCE

10. SEEPAGE FORCE
Hydraulic Gradient, i = sin  ; Seepage Force,Fs = i w Volume
Effective Weight, W’ = ’ Volume; ’ =  - w

11. FINITE SLOPES: PLANE FAILURE SURFACE
Translational Block Slides along single plane of weakness or geological interface
F = c’L + (W cos uL) tan’ / W sin + Fw

12. BLOCK SLIDES

13. BLOCK SLIDES

14. FINITE SLOPES: CIRCULAR FAILURE SURFACE
Rotational Slides - Method of Slices
Applies to slopes containing cohesive soils
Ordinary Method of Slices (Fellenius’ Method)
Bishop’s Simplified Method
Spenser’s Method

16. ORDINARY METHOD OF SLICES
Assumes that resultant of side forces on each slice are collinear and act parallel to failure surface and therefore cancel each other
F = [cn ln + (Wn cosn - un ln) tann] / Wn sinn
Undrained analysis: F = [cn ln] / Wn sinn

17. SIDE FORCES IN ORDINARY METHOD OF SLICES

18. BISHOP’S SIMPLIFIED METHOD
Assumes that resultant of side forces on each slice act in horizontal direction and therefore vertical side force components cancel each other
F = [cn bn + (Wn - un bn) tann](1/m) / Wn sinn
m = cosn + (sinn tann)/F
Undrained analysis: F = [cn ln] / Wn sinn

19. CHART FOR m

20. SIDE FORCES IN BISHOP’S METHOD

21. SPENCER’S METHOD
Assumes that the point of application of resultant of side forces on each slice is at mid-height of each slice but no assumption is made regarding inclination of resultants; inclination is determined as part of the solution
This method is more exact than Bishop’s

22. FINITE SLOPES: NONCIRCULAR FAILURE SURFACE
Wedge Method
Janbu’s Simplified Method
Morgenstern-Price Method

23. WEDGE METHOD
Failure surface consists of two or more planes and applicable to slope containing several planes of interfaces and weak layers
Force equilibrium is satisfied
Assumes that resultant of side forces on each slice either acts horizontally or at varying angles from horizontal (typically up to 15o)

24. WEDGE METHOD

25. WEDGE ANALYSIS
Equilibrium of Forces in each slice is considered to adjust the inter-slice forces and balance them resulting in a correct solution.

26. JANBU’S SIMPLIFIED METHOD
A method of slices applicable to circular and noncircular failure surfaces
F = fo [cn bn + (Wn - un bn) tann](1/ cosnm)} / Wn tann
fo is a correction factor that varies with depth to length ratio of sliding mass and type of soil (c, or c = 0)

28. MORGENSTERN-PRICE METHOD
No assumption is made regarding inclination or point of application of resultants and these are determined as part of the solution
Requires computers for solving the basic equation
Exact but not practical

29. REFERENCES
J.M. Duncan, A.L. Buchignani and M. De Wet (1987), An Engineering Manual for Slope Stability Studies, Virginia Tech Department of Civil Engineering, Blacksburg, Virginia.
L.W. Abramson, T.S. Lee, S. Sharma and G.M. Boyce (1996), Slope Stability and Stabilization Methods, Wiley, N.Y.
Das, B. M., Principles of Geotechnical Engineering, 3rd Ed., PWS Publishing Co., Boston, MA, 1994.
Soil Mechanics Design Manual, NAVFAC DM-7.1, Department of the Navy, 1982.

30. Slide La Conchita, California-a small seaside community along Highway 101 north of Santa Barbara. This landslide and debris flow occurred in the spring of Many people were evacuated because of the slide and the houses nearest the slide were completely destroyed. Fortunately, no one was killed or injured. Photograph by R.L. Schuster, U.S. Geological Survey.