Stability of Slopes

General
Slopes can be natural or man-made slopes. Natural slopes (hilly areas) exist in nature and are formed
by natural causes whereas sides of cuttings, the slopes of embankments constructed for roads, railway
lines and canals and slopes of earth dams are examples for man-made slopes. These slopes can be
considered into two types such as Infinite and finite slopes.

1) Infinite slope: the slope is used to designate a constant slope of infinite extent. (mountain)
2) Finite slope: limited slopes (slopes of embankments and earth dams)

Stability of construction of earth dams, embankments and natural slopes are important. The results of
a slope failure can be often catastrophic, involving the loss of considerable property and many lives.
Causes of failure of slopes are

1. Gravitational force
2. Force due to seepage force
3. Erosion of the surface of slopes due to flowing water
4. Sudden lowering of water adjacent to a slope
5. Forces due to earthquakes

General Assumptions

1. Testing of samples to determine the cohesion and angle of internal friction- If the analysis is for
natural slope, it is necessary that the sample is undisturbed.
2. The study of items are known to enter but which cannot be accounted for in the computations-
progressive cracking, the effects of non-homogeneous nature of the typical soil
3. Computation- Assumed that the problem is two-dimensional, which theoretically requires a long
length of slope normal to the section.
Infinite Slope Stability

R  W cos 
T  W sin 
W  H cos 
R
   H cos2 
A
 mob  H cos  sin 
c1   1 tan  1
Fs 
 mob
c  H cos2  tan  1
1
Fs 
H sin  cos 
c1 tan  1
Fs  
H sin  cos  tan 

If it is cohesion less soil, then c1= 0. Then

tan  1
Fs 
tan 

For slope stability  1  

Critical depth Hcr

If Fs=1, then slope stability is critical

c1 tan  1
1 
H cr sin  cos  tan 
c1 tan  1
1 
H cr sin  cos  tan 
c1 1
H cr 
 cos  (tan   tan  1 )
2
Infinite Slope with Steady Seepage

R  u  W cos 
T  W sin 
W  H cos 
R
  H cos2   u
A
 mob  H cos  sin 
c1   1 tan  1
Fs 
 mob
c  (H cos2   u ) tan  1
1
Fs 
H sin  cos 
c1  1 u 
Fs     tan  1
H sin  cos   tan  H sin  cos  
c1  u  tan  1
Fs   1 
H sin  cos   H cos2   tan 

But u   w h p

Pore water pressure (Assume cohesion less soil)

hp 
z  z w  w cos  cos
cos(   )
  z  z w  w cos  cos   tan  1
Fs  1  
 H cos  cos     tan 
2

Above equation shows that the factor of safety depends on slope angle, direction of flow and depth of
slip plane.
When flow is parallel to surface, then β=α
  z   tan  1
Fs  1  (1  w ) w 
 z   tan 
For dry slope
tan  1
zw= z, Fs 
tan 

For water lodged slope

  w  tan  1
zw= 0, Fs  1  
   tan 
w 1
On the other hand, assuming ratio  then
 2

tan  1  zw 
Fs  1  z 
2 tan 
zw
Above equation show that factor of safety is proportional to the ratio of .
z
For horizontal flow
α= 0

  z  z w  w cos  cos  tan  1

Fs  1  
 H cos  cos     tan 
2

 z  w  tan  1
Fs  1  (1  ) 
 z w  cos2   tan 
For vertical flow

 tan  1
zw=0, α= , Fs 
2 tan 

This shows that stability of the slope is not affected by water seepage in the case of vertical downward
flow.

Effect of cohesion on the stability of infinite slopes

c1   z   w cos  tan  1
Fs   1  1  w 
z sin  cos    z   cos  cos    tan 
2c1
If cohesion is considered, then factor of safety is simply increased by an amount .
z sin 2

Undrained analysis of the stability of infinite slopes

For slopes undertaken in clays, the excess pore water pressure generated during construction or a short
time after the end of construction (that is few days or even few weeks) may trigger failure.

Undrained conditions u  0

2c1
FOS 
z sin 2
 zw
F  F 1 (1  )
z
Example 1:

For the infinite slope shown below (consider that there is no seepage through the soil), determine:

a) The factor of safety against sliding along the soil-rock interface

b) The height, H, that will give a factor (Fs) of 2 against sliding alone the soil-interface

c) If there is seepage through soil as shown and the groundwater table coincides with the ground
surface, what is the factor of safety Fs given H=1.16 m and  sat  18 .55 kN / m 3 ?

Example 2:

The frictional soil of a water logged slope, inclined at an angle of β=15o to the horizontal, is
characterized by a critical angle of shearing resistance c1  32 o . In-situ tests undertaken at different

locations indicated the presence of layer of fissured hard rock at a depth z=5m, running parallel to
the surface. Also, piezo meter readings indicated that water is flowing parallel to the slope.
Determine:

(a) factor of safety against sliding under the stated conditions

(b) the depth zw t which the water table must be lowered if a minimum factor of safety F=1.5 were
needed.

Example 3:

The clayey soil of natural slope, inclined at an angle of β=12o to the horizontal is characterized by
a residual angle of shearing resistance c1  14 o . A thorough investigation revealed that the
presence of an old shallow slip plane, running roughly parallel to the slope and situated at a depth
of z=3.0 m. The water table is at a depth of zw=1.5 below ground water level and the flow is
assumed to be parallel to the slope.