Failure of Infinite slopes - Part 01

Considerable evidence is available that slope failure in homogeneous soils usually occur on curved failure surface. In many circumstances ( for  example zoned dames and foundations on weak strata), stability analysis using plane failure of sliding is more appropriate and yields excellent results.

 Failure of infinite slopes, we can analyze using several techniques,

(1). Using planer failure surface (transnational slides)
     - Culman method

(2). Using curved failure surface (rotational slides)
     - Circular failure surface
         1. Circular arc method
         2. Stability number method
         3. Method of slice  (widely use, base the many softwares) 
         4. Morgestern and Bishop method
         5. Cousins charts
    - Non - Circular failure surfaces

(1). Planer failure surface - Culman Method

Example :
A cut is to be made in a soil that has gamma = 16 KN/m^3, C' = 28 KN/m^3 and phi' = 20. the side of the cut slope will make an angle of 45 degrees with the horizontal. what should be the depth of the cut slope that will have a factor of safety Fs (with respect to strength) of 3.5?

Fs = Fc' = Fphi' = 3.5

Fc' = C'/Cd' = 28/3.5
Cd' = 8

Fphi' = tan phi'/tan phid'
tan phid' = tan 20/3.5
phi d ' = 5.94 degrees

m = cd'/gamma H

from the equation cd'/gamma H ; Hcr = 6.29 m 

(2). Circular failure surface

there are basically two types of circular failure modes,

1). Slope failure
   -when the failure circle intercepts the slope at or above the toe, when the failure circle intercepts the slope above the toe, it is referred to as the slope circle failure
   - when the failure circle passes through the toe of the slope it is referred to as the toe circle failure
 
2). Base failure
   - When the failure surface passes some distance below the toe of the slope. the failure circle in the case of base failure is called a midpoint circle failure.  

1). Circular arc method (only valid for phi =0)

- Which gives the FOS lowest is the critical failure path
- Tension crack can also take into account (because phi = 0)
- To find critical failure circle there are softwares or we can do it by using below graph

2). Stability Number method

Stability number method - ( for phi = 0 )

The slope stability issues are solves by Fellunius and Taylor. they expressed the developed cohesion Cd as a non dimensional parameter referred as a stability number "m"

Stability number method - ( for soils with phi and C )

Solution procedure
- Assume the friction anngle, phi
- W - weight of the failure mass is estimated
- The direction of the Cd is parallel to the line AC
- The resultant frictional force is tangential to the friction circle and hence, the direction of the resultant frictional force is known.
- After drawing the force polygon, estimate the developed friction, Cd
- Draw failure circles to obtain the critical circle which gives the maximum developed cohesion.
- Determine the stability number
- Find the m using below graph