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CHAPTER 4SLOPE STABILITY ANALYSYS
Introduction: Slope Failures
Types of Slope
Causes of Failures
Types of Failures
Method of Analysis
Slope stabilization
Muhammad Azril Hezmi

Slope Failure
is the movement of mass on slope
(falls, slides, flows)
Landslide: involves an extensive area, mild slope (

TYPES OF SLOPE
Natural Slopes
Long term process
Short process
Manmade Slopes
Excavated Slopes
Slopes of Embankment and Earth Dam

CAUSES OF SLOPE FAILURE
Slope inclination
Additional load or Fill height
Excessive Pore water pressure
Loss of shear strength due to
Weathering
Liquefaction
Water (infiltration and seepage)

TYPES OF FAILURES
Wedge Failure is the soil mass movement dueto external force. This type of failure usually occur on a weak plane or weak joint
Circular Failure or non circular failure, Circular failure are associated with homogeneous soil conditions Noncircular slips are associated with nonhomogeneous conditions
Translational Failures occur where the form of failure is influenced by the presence of weak layer. The failure surface tends to be plane and roughly parallel to the slope surface

TYPES OF FAILURES
Wedge Failure is the soil mass movement dueto external force. This type of failure usually occur on a weak plane or weak joint

TYPES OF FAILURES
Circular Failure or non circular failure, the shape of failure plane maybe circular or noncircular.
In general, circular slips are associated with homogeneous soil conditions while noncircular slips are
associated with nonhomogeneous conditions

TYPES OF FAILURES
Translational Failures occur where the form of failure is influenced by the presence of weak layer. The failure surface tends to be plane and roughly parallel to the slope surface

Principle of Slope Stability
Analysis
Sliding will occur if the shear stress developed
exceeds the corresponding shear resistance of the
soil. In this case, failure is assumes at a certain
plane
W sin Rs
Possible
failure
surface
FS natural slope = 1.25 to 1.4
FS manmade slope > 1.5

METHOD OF ANALYSISLIMIT EQUILIBRIUM METHODS
Factor of safety is the shear strength at the time of failure f compared to the stress acting at that plane m.
If FS = 1, then the slope is in critical condition.
At the time of failure, the shear strength of the soil is fully mobilized along the failure plane. The shear strength is represented by the MohrCoulomb criteria:
= cu (Total stress analysis)
= c + tan (Effective stress analysis)
1
FS
m
f >=

Linear Methods: Relatively simple Infinite slope analysis Linear Failure Plane Analysis for the case of u = 0 (undrained condition) Wedge failure analysis
Non Linear Methods: Method of SlicesNecessary for irregular slope geometry, nonuniform soil condition, and seepagein soil.
METHOD OF ANALYSIS

INFINITE SLOPE ANALYSIS
1
m z cos2
z mzz
W
N or T or
GWTT
Flow net
tan'tan'
sincos2 satsat z
cFS +=

INFINITE SLOPE ANALYSISThe shear strength along the failure plane
The expression for , , and are
= {(1m) + m sat} z cos2 m = {(1m) + m sat} z sin cos = m z w cos2
' tan )  ( c' f +=

Substitute the above expressions to get F
( )m
'tanc'FS
+=
tan
'tan
'
sincos z
cFS
sat2
sat
+=
For special case where c =0, tan
'tan
'FS
sat
=
For the case where water table is far below the failure plane (m = 0)
tan
'tan FS =

Note that when c = 0, then factor of safety is independent of the height of the slope. The slope will be stable as long as slope angle is less than the internal friction angle . If both cohesion and angle of internal friction angle is not zero, then the critical condition (FS = 1) will be achieved when
tancos'c'
zz2cr
==
For a total stress analysis, the shear strength parameters cu and u are used with a zero value of m

FINITE SLOPE WITH LINEAR FAILURE PLANE
H
C B
A
W
L
h
N =W cos T=W sin
Rs
W
WLc
W
RFS s
sin
tancos
sin
+==

From the figure, line AC is the trial failure planeThe weight of soil (ABC) is:
sin
)(sinHL
2
1W
=
The force that will cause the failure is T = W sin
and the resistance to sliding is given by Rs = cd L + W cos tand
The factor of safety will be
sinW
tancosWLc
sinW
RFS s
+==

Critical condition prevails when T = Rs.By substituting FS = 1, then
for critical failure plane = ( + d)/2
Substituting , we get
And solving for H and replacing cd by c, then
Where Hcr is the safe depth of cut and is the slope angle
=
)  ( cos 1
cos sin
4c Hcr
Critical Conditions
( )
=d
dd
cossin
)(sinsinH
2
1c
( )
=d
dd
cossin
cos1
4
Hc

Same principal valid for condition where a slope consists of two layers where the upper layer is assumed to slide along the interface between the two layers
H D
C
B
A
W
h
T = Wsin N= Wcos
Rs L

Circular slope failure

Defining a Failure surface for a toe circle
1 211.3218.4326.5733.79
4560
252525262829
353535353740
Note: there other charts available as guidelines for finding the center of failure circle

zc
R
R B
d
W
La
Pw
yc
b. with tension crack
A
R
R B
d
W
La
a. No tension crack
Hydrostatic pressure in tension crack
SLOPE WITH CIRCULAR FAILURE PLANE(homogeneous cohesive soils, fu = 0)

Slope in Homogeneous Cohesive soils, = 0 analysis
FS
c
FS
uf ==m
au
ams LFS
cLR ==
RLFS
cdW a
u=
dW
RLcFS au=

In the event of tension crack developing, then La is
shortened and hydrostatic force will act normal to the crack
if it is filled with water
cw
au
yPdW
R'LcFS
+=

The use of Charts
Taylors stability number
Janbu stability charts
Bischop and Morgenstein charts for effective stress analysis
Morgensteins graphs for rapid drawdown
Here we discuss the Taylors stability chart only

The Use of Charts, Taylors chart
H nd H

METHOD OF SLICES
In this method, the potential failure surface is assumed to be a circular arc with center O and radius r (see figure).The soil mass (ABCD) above the failure surface (AC) is divided by vertical planes into a series of slices of width b. The base of each slice is assumed to be a straight line. For any slice, the inclination of the base to the horizontal line is i and the height (measured at the centerline) is hi.

forces acting on a slice
Wi
1
87
6
9
54
32
Ei 1
X i1 Xi
METHOD OF SLICES
b
h
x
R

As before,
The factor of safety is defined as the ratio of the available
shear strength to the shear stress acting on the plane
The factor of safety is taken to be the same for each slice,
implying that there must be support between slices
(forces must act between slices)
m
f
FS=

Forces acting on a slice are
The total weight of the slice, W = bh The total normal force on the base: the effective
normal force N = l and the boundary water force U = l. where u is the p.w.p. at the center of the base and l is the length of the base
The shear force on the base, T = m l The total normal forces on the sides, E1 and E2
The shear forces on the sides, X1 and X2
Any external forces must be included in the analysis.

Assumptions must be made regarding the interslice forces E and X
Taking moment about O, the sum of the moments of the shear forces T
on the failure arc AC must be equal to the moment of the weight of
the soil mass ABCD.
= sin/)( WFlf
( )
+=
sin
'tan''
W
NLcF
a
= sinRWTR
=
sinW
lF
f
For analysis in terms of effective stress
( )
+=
sin
'tan''
W
lcF or
Where La is the arc length of AC

The Fellenius (Swedish) MethodFellenius assumed that the resultant of the interslice forces is zero, then
N = W cos ul
Hence the factor of safety in terms of effective stress is given by:
The components W cos and W sin can be determined graphically while angle a can be calculated or measured
For analysis in terms of total stress parameter or u = 0, then
sinWF
= au Lc
( )( )
+=
sin
'tancos'
W
lWlcFm

The Bischop (Routine) MethodBischop assumed that the resultant of the interslice forces are horizontal i.e. X1 X2 = 0, then
)'tanN'l(c'F
1T +=
Resolving forces in the vertical direction:
sintanF
N'sin
F
lc'cosulcosN'W ' +++=
+
=
F
luF
lcW
N
sin'tancos
cossin'
' By replacing l = b secaAnd after some rearrangementWe obtain:

( ){ } ( )
++
=
FaubWbc
aWFS
/'tantan1
sec'tan'
sin
1
By replacing ru = u/h = u/(W/b) then:
( ){ } ( )
++
=
FarWbc
aWFS u
/'tantan1
sec'tan1'
sin
1
The Bischop (Routine) Method (contd)

Since F appear in both sides of the equation, then use trial and error.
To simplify the calculation, the following chart could be used
+=F
' tantan 1 cos m
aa
a
The Bischop (Routine) Method (contd)
( ){ }
+
=a
um
rWbcaW
FS1
'tan1'sin
1
Then

To get FS from the equation,
can use computer program or
graph 1. Assume F right = 1, find m
2. Find F left 3. Take the average of F
right and F left 4. Use this average F,
find m5. Find new F left6. Repeat steps 3 and 4
until the difference between F right and F left is small enough (0.01)reroute to excell program for Bischop

COMMENT ON SLICES METHODS
Due to repetitive nature of the calculations and the need
to select the most critical failure surface, the method
of slices in particularly suitable for solution by
computer. More complex geometry and soil strata can
be introduced.
There are other methods of slices as shown in the following
Table. These methods use different assumption on inter
slices forces.

Slices methods of analysis frequently used in practice.
MethodForce
equilibriumMoment
equilibriumShape of slip surface
Ordinary method of slices (Fellenius, 1927)
Does not satisfy horizontal or vertical forces equilibrium
Yes. Circular
Bishops Modified (Bishop, 1955)
Satisfy vertical force but not horizontal force equilibrium
Yes. Circular only. Non circular may have numerical problems.
Janbus simplified method(Janbu, 1956)
Yes No Any shape. More frequent numerical problems than other methods
Morgenstern and Price (Morgenstern and Price, 1965)
Yes. Permits side forces to be varied
Yes. Any shape.
Spencers Method (Spencer, 1967)
Yes. Side forces are assumed to be parallel
Yes. Any shape.

ASSIGNMENT 1:
SLOPE STABILITY ANALYSIS Pick a problem and the CD + manual
Analyze the problem using SLOPE/W student version (in this case you can use Bischop, Janbu or GLE methods available for Student version).
Find the slip surface that gives the lowest factor of safety (critical failure surface)
Sketch of your slope in graph paper and trace the critical failure surface you obtained from SLOPE/W on your graph
Use method of slices to calculate the factor of safety either using Bischop or Fellenius method (you may make use of Excell for your calculation).

ASSIGNMENT 1:
SLOPE STABILITY ANALYSISDiscuss the results and write a report (Group). The report should include
Introduction (the problem)
Results of SLOPE/W output including contour of FS and the critical failure
surface + analysis of 1 slice
Results of your manual calculation (with the help of Excell program)
Discussion and comparisons