Rock Slope Stability Analysis: Limit Equilibrium
Method
•Plane failure analysis
•Wedge failure analysis
•Toppling failure analysis
Planar Failure Analysis
The block is considered to undergoes
slippage along the plane for the value of
ratio < 1, else it is stable
A block is rest on a slope having angle θ
Plane failure analysis along a discontinuity
B
Unstable Block blockW
H
W
θ
α
A
Geometry of a slope for plane failure
C
Plane failure analysis along a discontinuity
Planar Failure Analysis
•
the plane on which sliding occurs must strike parallel or nearly parallel
approximately + 200 ) to the slope face
(within
•
the failure must daylight in the slope face. This means that its dip must be
smaller than the dip of the slope face
•
the dip of the failure plane must be greater than the angle of internal friction
angle of this plane
Plane failure analysis along a discontinuity
Block A
R
ShearStren gth
Factor of safety = ShearStres s
Factor of safety =
s
w sin( )
A

w cos( )
A
c
Factor of safety =
w cos
t an
A
w sin 
A
W cosθ
W
c   tan
Normal Stress;  
Shear Stress ,
W sinθ
=
cA  w cos  tan 
w sin 
Water is filled in discontinuities
The effective normal stress due
to present of water in the joint, is
given as
 '   

1
gh 2
4
Tension crack present in the upper slope
surface
Tension crack in upper surface of slope and in the face
b
The depth of critical tension crack, zc and its
B
C
z
location, bc behind the crest can be calculated by the
D
following equations:
W
bc
 (cot cot )  cot
H
plane failure with tension crack
Length of discontinuities; AD 
H  CD
Sin
The weight of the block;
Factor of safety = cA  w cos  tan 
w sin 
Tension crack present in the slope surface
B
z  ( H cot  b)(tan  tan )
C
Length of discontinuities; AD 
D
H  CD
Sin
W
The weight of the block =
Factor of safety = cA  w cos  tan 
w sin 
plane failure with tension crack
c
Compound slope with water in upper slope
angle
Compound slopes have appreciable angle with
the horizontal. High slope formation has in
generally a positive upper slope angle while the
shorter slope has a negative slope angle
Compound slope with a positive upper slope angle
Geometry of slope with tension crack in upper slope angle
Depth of tension crack, Z  H  b tan c  (b  H cot ) tan
Weight of unstable block, W  1 H 2 cot X  bHX  bZ ) 
2
X  (1  tan cot )
or
Area of failure surface, A  ( H cot  b) sec 
Driving water force, V  1  w Z w2
2
Uplift water force, U  1  w Z w A
2
Factor of safety =
cA  ( w cos   U  V sin  ) tan 
W sin   V cos 
Effect of rock bolts
Geometry of slope with tension crack in upper slope and its interaction with rock bolt
FOS =
cA  ( w cos  U  V sin   T cos  ) tan
W sin   V cos  T sin 
Wedge Failure Analysis
Geometric conditions of wedge failure: (a) pictorial view of wedge failure; (b) stereoplot showing
the orientation of the line of intersection
Analysis of wedge failure considering only frictional resistance
Resolution of forces to calculate factor of safety of wedge: (a) view of wedge looking at face showing definition of
angles β and α, and reactions on sliding Plane RA and RB, (b) stereonet showing measurement of angles β and α, (c) crosssection of wedge showing resolution of wedge weight W.
Plane failure analysis along a discontinuity
Analysis of wedge failure with cohesion and
friction angle
Pictorial View of wedge showing the numbering of intersection lines and planes
Analysis of wedge failure with cohesion and
friction angle
FS 


3
(Ca X  CbY )  ( A  w X ) tana  ( B  w Y ) tanb
rH
2 r
2 r
X
sin  24
sin  45 cos na 2
A
cos a cos b cos na.nb
sin i sin 2  na.nb
Y
sin 13
sin 35 cos na1
B
cos b cos a cos na.nb
sin i sin 2  na.nb
Analysis of wedge failure with cohesion and
friction angle
Where, Ca and Cb are the cohesive strength of plane a and b, фa and фb are the angle of
friction along plane a and b, is the unit weight of the rock, and H is the total height of the
wedge. X, Y, A and B are dimensionless factors, which depend upon the geometry of the
wedge, Ψa and Ψb are the dips of planes a and b, whereas, Ψi is the plunge of the line of their
intersection.
Under fully drained slope condition, the water pressure is zero. Therefore, factor of safety of
the wedge against failure is given by:
FS 
3
(Ca X  CbY )  A tana  B tanb
rH
Toppling Failure Analysis
Kinematics of block toppling failure
Case 1:
Case 2:
Case 3:
Case 4:
Inter-layer slip test
If
is the dip of slope face and α is the dip of the planes forming the sides of the blocks, then
the condition for interlayer slip is given by:
(180 −
− α) ≥ (90 − ф)
or
α≥ (90 − ) + ф
Block alignment test
The dip direction of the
planes forming sides of the
blocks, αd is within about 100
of the dip direction of the
slope face αf, i.e.
|(αf− αd)| <10◦
Limit equilibrium analysis for toppling failure
The factor of safety can be calculated as the ratio of resisting
moments to driving moments
Limit equilibrium analysis for toppling failure
Model for limiting equilibrium analysis of toppling on a stepped base (Goodman and Bray, 1976).
Forces acting on the nth column sitting on a stepped base
Figure 17: Limiting equilibrium conditions for toppling and sliding of nth block: (a) forces
acting on nth block; (b) toppling of nth block;