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
� Man-made 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 Non-circular slips are associated with non-homogeneous 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 non-circular.
In general, circular slips are associated with homogeneous soil conditions while non-circular slips are
associated with non-homogeneous 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 sinα Rs
Possible
failure
surface
FS natural slope = 1.25 to 1.4
FS man-made slope > 1.5
METHOD OF ANALYSISLIMIT EQUILIBRIUM METHODS
Factor of safety is the shear strength at the time of failure τf compared to the stress acting at that plane τm.
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 Mohr-Coulomb criteria:
τ = cu (Total stress analysis)
τ = c’ + σ’ tan φ’ (Effective stress analysis)
1τ
τFS
m
f >=
� Linear Methods: Relatively simple• Infinite slope analysis• Linear Failure Plane • Analysis for the case of φu = 0 (undrained condition)• Wedge failure analysis
� Non- Linear Methods: Method of SlicesNecessary for irregular slope geometry, non-uniform soil condition, and seepagein soil.
METHOD OF ANALYSIS
INFINITE SLOPE ANALYSIS
1
m z cos2β β
z mzz
W
N or σ T or τ
µ
GWTT
Flow net
βγφγ
ββγ tan'tan'
sincos2 satsat z
cFS +=
INFINITE SLOPE ANALYSISThe shear strength along the failure plane
The expression for σ, τ, and µ are
σ = {(1-m)γ + m γsat} z cos2 βτm = {(1-m)γ + m γsat} z sin β cosβµ = m z γw cos2β
' φ tan )μ - σ ( c' τf +=
Substitute the above expressions to get F
( )mτ
φ'tanμ-σc'FS
+=
βtan
φ'tan
γ
γ'
βsinβcos zγ
cFS
sat2
sat
+=
For special case where c’ =0, βtan
'φtan
γ
γ'FS
sat
=
For the case where water table is far below the failure plane (m = 0)
tanβ
φ'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 β is less than the internal friction angle ϕ. If both cohesion and angle of internal friction angle is not zero, then the critical condition (FS = 1) will be achieved when
tanφβcosγ'c'
zz2cr
==
For a total stress analysis, the shear strength parameters cu and ϕu are used with a zero value of m
FINITE SLOPE WITH LINEAR FAILURE PLANE
H
β θ
C B
A
W
L
h
N =W cosθ T=W sinθ
Rs
θW
θWLc
θW
RFS s
sin
tancos
sin
φ+==
From the figure, line AC is the trial failure planeThe weight of soil (ABC) is:
βsin
θ)(βsinHLγ
2
1W
−=
The force that will cause the failure is T = W sinθ
and the resistance to sliding is given by Rs = cd L + W cosθ tanϕd
The factor of safety will be
θsinW
φtanθcosWLc
θsinW
RFS s
+==
Critical condition prevails when T = Rs.By substituting FS = 1, then
for critical failure plane θ = (β + φd)/2
Substituting θ, we get
And solving for H and replacing cd by c, then
Where Hcr is the safe depth of cut and β is the slope angle
−=
) φ - β ( cos 1
φ cos βsin γ
4c Hcr
Critical Conditions
( )
−−=d
dd
φcossinβ
)φ(θsinθβsinHγ
2
1c
( )
−−=d
dd
φcosβsin
φβcos1
4
Hγc
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
β
θ
C
B
A
W
h
T = Wsinθ N= Wcosθ
Rs L
Circular slope failure
Defining a Failure surface for a toe circle
β α1 α211.3218.4326.5733.79
4560
252525262829
353535353740
Note: there other charts available as guidelines for finding the center of failure circle
zc
R
R B
d
W
θ
La
Pw
yc
b. with tension crack
A
R
R B
d
W
θ
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, φ = 0 analysis
FS
c
FS
τ uf ==mτ
au
ams LFS
cLτR ==
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
� Taylor’s stability number
� Janbu stability charts
� Bischop and Morgenstein charts for effective stress analysis
� Morgenstein’s graphs for rapid drawdown
Here we discuss the Taylor’s stability chart only
The Use of Charts, Taylor’s chart
H nd H β
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 αi and the height (measured at the centerline) is hi.
α
forces acting on a slice
Wi
1
87
6
9
54
32
α
β
Τι
Νι−µ
EiΕ ι−1
X i-1 Xi
Ο
METHOD OF SLICES
α
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
τ
τFS=
Forces acting on a slice are
� The total weight of the slice, W = γbh� The total normal force on the base: the effective
normal force N’ = σ’l and the boundary water force U = µ 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 = τm 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 inter-slice 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.
∑∑ = αsin/)τ( WFlf
( )∑
Σ+=α
φsin
'tan''
W
NLcF
a
∑∑ = αsinRWTR
∑∑=
ατ
sinW
lF
f
For analysis in terms of effective stress
( )∑
∑ +=α
φsin
'tan'σ'
W
lcF or
Where La is the arc length of AC
The Fellenius (Swedish) MethodFellenius assumed that the resultant of the inter-slice forces is zero, then
N’ = W cos α – ul
Hence the factor of safety in terms of effective stress is given by:
The components W cosα and W sinα can be determined graphically while angle a can be calculated or measured
For analysis in terms of total stress parameter or φu = 0, then
αsinWF
ΣΣ= au Lc
( )( )∑
∑ −+=α
φµβsin
'tancos'
W
lWlcFm
The Bischop (Routine) MethodBischop assumed that the resultant of the inter-slice forces are horizontal i.e. X1 –X2 = 0, then
)φ'tanN'l(c'F
1T +=
Resolving forces in the vertical direction:
αα sintanφF
N'sin
F
lc'cosαulcosαN'W ' +++=
+
−−=
F
luF
lcW
Nαφα
αα
sin'tancos
cossin'
' By replacing l = b secaAnd after some rearrangementWe obtain:
( ){ } ( )∑
+−+
Σ=
FaubWbc
aWFS
/'tantan1
sec'tan'
sin
1
φαφ
By replacing ru = u/γh = u/(W/b) then:
( ){ } ( )∑
+−+
Σ=
FarWbc
aWFS u
/'tantan1
sec'tan1'
sin
1
φαφ
The Bischop (Routine) Method (cont’d)
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
φaa
a
The Bischop (Routine) Method (cont’d)
( ){ }∑
−+Σ
=a
um
rWbcaW
FS1
'tan1'sin
1 φ
Then
To get FS from the equation,
can use computer program or
graph 1. Assume F right = 1, find mα
2. Find F left 3. Take the average of F
right and F left 4. Use this average F,
find mα5. 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.
Janbu’s 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.
Spencer’s 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