The document discusses slope stability analysis methods. It describes common features such as calculating a safety factor based on shear strength and mobilized shear resistance. It also discusses total stress analysis, which assumes undrained strength, and the effect of tension cracks and submerged slopes on stability analysis. Methods for analyzing rotational and translational failures as well as granular and cohesive soils are presented.

2. La Conchita, California-a small
seaside community along
Highway 101 north of Santa
Barbara. This landslide and
debris flow occurred in the
spring of 1995. Many people
were evacuated because of
the slide and the houses
nearest the slide were
completely destroyed.
Fortunately, no one was killed
or injured. Photograph by R.L.
Schuster, U.S. Geological
Survey.

3. Victoria, Australia: An aerial view of a coal mine that collapsed, washing
away a road and railway lines
http://images.google.com.my/imgres?imgurl=http://stormwater.files.wordpress.com/2007/

5. COMMON FEATURES OF SLOPE STABILITY ANALYSIS METHODS
 Safety Factor: F = S/Sm where S = shear strength and Sm =
mobilized shear resistance. F = 1: failure, F > 1: safety
 Shape and location of failure is not known, a priori** but
assumed (trial and error to find minimum F)
 Static equilibrium (equilibrium of forces and moments on a
sliding mass)
 Two-dimensional analysis
**relating to what can be known through an understanding of how certain things work rather than by observation; relating to or
derived by reasoning from self-evident propositions ; presupposed by experience ; being without examination or analysis

6. Granular materials
Sand and gravel exhibit friction component of strength
Most soil exhibit cohesive and frictional strength.
N
W
T
Force parallel to slope = W sin
Force perpendicular to slope = W cos
For stability
or
safetyfact
gforce
restrainin
ce
slidingfor =
F
W
W



tan
cos
sin =


tan
tan
=
 F
• Weight of material does not
affect the stability of slope
• Safe angle for slope is same
whether the soil is submerge
or dry.
• Embankment can be at any
height
 = angle of shearing resistance


7. Seepage forces in a granular slope subjected to rapid drawdown
W.L.
G. W.L.
b
l
z
The river level dropped suddenly due to tidal effect.
In soil, permeability effect → water cannot flow fast→ seepage occurs
Water flow from high level to low level of slope
Flow net can be drawn

8. Seepage forces in a granular slope subjected to rapid drawdown
b
l
z
Assume potential failure plane parallel to slope
occurs at a depth of z and weight is W
Excess pore water pressure induced by seepage,
u at the mid point of the base of the element.
Normal reaction N = W cos
Normal stress  =








=
=
cosβ
b
1
b
β
Wcos
l
Wcosβ 2
where
Normal effective stress ’= u
β
γzcos
u
b
β
Wcos 2
2
−
=
−
Tangential shear stress,  = γzsinβcosβ
=
β
W sin 
l

9. Ultimate shear strength of soil = ’ tan = F


 tan
tan
)
cos
1
(
2
u
r
F −
=
Where
z
u
ru 
=
Pore pressure ratio = the pore water pressure to the weight of
material acting on unit area above it.

10. Flow parallel to the surface and at the surface
b
l
z

Equipotential line
A
B
hz
Excess water head = Hw
AB = z cos
Hw=z cos2 
Excess pore water pressure = wzcos2













tan
tan
tan
tan
)
(
tan
tan
)
1
(
'
sat
w
w
F =
−
=
−
=

11. A granular soil has a saturated unit weight of 18.0kN/m3 and angle of
shearing resistance of 300. A slope is to be made of this material. If the
factor of safety is to be 1.25, determine the safe angle of the slope
1. When the slope is dry or submerge
2. If seepage occurs at and parallel to the surface of slope.
1. When dry or submerge


tan
tan
=
F
0
25
462
.
0
25
.
1
30
tan
tan
=
=
=


Example

12. 2. seepage occurs at and parallel to the surface of slope




tan
tan
'
sat
F =
0
5
.
11
205
.
0
18
25
.
1
30
tan
8
tan
=
=
=


x
x
18
x
25
.
1
30
tan
)
0
.
10
0
.
18
(
tan
−
=


13. Soils with two strength components
Tends to be rotational, the actual slip surface approximating to be arc of a circle
crack
Slip surface
Heave of
material at toe
Method of investigating slope stability
•Assuming a slip surface and a center about which it rotates
•Study the equilibrium of the forces acting on this surface
•Repeat the process until the worst slip surface is found

14. Total stress analysis
Often called the  = 0 analysis
Intended to give the stability of embankment
immediately after its construction.
Assumed that the soil in embankment has no time to drain.
Strength parameter used – undrain strength of soil
- unconfined compression test
- undrain triaxial test.
r

A
B
W
We
r
c
moment
disturbing
gmoment
restrainin
F

2
=
=
G
e
O

 2
.
.
.
.
.
.
. cr
r
r
c
r
l
c
e
W =
=
=
At equilibirium,
As ϕ=0°, Ƭmax=σ tan ϕ+C = 0, e=eccentricity

15. Effect of tension crack
r
’
A
B
hc

c
hc
2
=
In pure cohesive soil, depth of crack
Tension crack depth
Fig: Tension crack in a cohesive soil
O
B’

16. Example
Figure below gives details of an embankment of made cohesive soil with
 = 0 and C = 20 kN/m3. The unit weight of the soil is 19 kN/m3. For the trial
circle shown, determine the factor of safety against sliding. The weight of
sliding sector is 346 kN acting at the eccentricity of 5m from the center of
rotation. What would be if the shaded portion of the embankment were
removed? In both cases assume that no tension crack develops.
700
R=9m
W
1.5m
3m
3m
1.1
1
e=5m

17. Solution
Disturbing moment = 346 x 5 = 1730 kNm
Restraining moment =
14
.
1
1730
1980
1980
7
22
180
70
9
20 2
2
=
=
=
=
F
kNm
x
x
x
cr 
Area of portion removed = 1.5 x 3 =4.5m2
Weight of portion removed = 4.5 x 19 = 85.5 kN
Eccentricity from 0 = 3.3 + (3.3+1.5)/2 = 5.7 m
Relief of disturbing moment = 5.7 x 85.5 = 490 kNm
6
.
1
490
1730
1980
=
−
=
F

18. INTRODUCTION TO SLOPE INSTABILITY:-
1. Rotational:
This type of failure surface may be either circular or non-circular. Consists of a
movement of rock or debris, about an axis that is parallel to the slope contours,
involving shear displacement along a concave upward – curving failure
2. Translational : Consists of a non – circular failure which involves motion on a
near – planar slip surface
3.Compound Slip

23. Causes of Slope Failure 1

24. Causes of Slope Failure 2

25. To Be Considered in Assessment of Stability
✓ Ground Investigation
✓ Most Critical Slip Surface
✓ Tension Cracks
✓ Submerged Slopes
✓ Factor of Safety
✓ Long and Short-Term Stability
✓ Effective or Total Stress Parameters?
✓ Progressive Failure
✓ Pre-Existing Failure Surfaces
✓ Assumptions in Limit Equilibrium
Methods of Analysis for Circular Failure extra reading articles : 05 Circular failure.pdf
1. Granular Soils: (the simplest method)
1.1 The C'=0 Method
2. Cohesive Soils:
2.1 Circular Failure Surface
2.1.1 The Basic Idea
2.1.2 Method of Slices
2.1.3 Bishop's Method
2.2 Non-Circular Failure Surface
2.2.1 Janbu's Method
2.2.2 Infinite Slope Method
2.2.3 Stability Charts

26. 1. GRANULAR SOILS
1.1 The C'=0 method
Assumptions:- infinite, dry slope.

27. 2.0 COHESIVE SOIL
2.1 Circular Failure Surface
2.1.2 The Basic Idea
Assumptions:- circular arc, radius R, centre O.

30. 2.1.2 Method of Slices
Assumptions:- circular arc, radius R, centre O
The soil mass above a trial failure surface is divided into slices by vertical planes.
Each slice is taken as having a straight line base.
The Factor of Safety of each slice is assumed to be the same, implying mutual
support between the slices, ie. there must be forces acting between the slices.

32. 2.1.3 Bishop's Method
Another method which follows from the basic "Method of Slices".

34. 2.2 NON-CIRCULAR FAILURE SURFACE
2.2.1 Janbu's Method
The difficulty in analysing a non-circular failure surface is that it is difficult to find a
single point through which many of the force components act.
So, the moment equilibrium method used for circular surfaces is no longer the
most appropriate. Janbu chose instead to use the force equilibrium method in
the analysis which follows.

38. 2.2.3 Stability Charts
2.2.3.1 Taylor’s Chart
2.2.3.2 Spencer’s Chart
It is not practical or a good use of an engineer's time to carry out long stability calculations on each
modification of a slope. The following charts show a range of stability characteristics for slopes with various
soil properties and shapes.
2.2.3.2 Taylor’s Chart

40. Example of Taylor’s Chart

41. 2.2.3.2 Spencer's Charts
These include the effects of porewater pressure. The charts are broadly
similar to Taylor's Charts, but introduce the parameter ru, the porewater
pressure ratio.
Each of the charts is calculated for a single ru value. In a slope, ru will
vary at different points but we use a mean value for simplification. As
the variance isn't large, this is not an over-simplification
Spencer chose to use ru = 0, 0.25, 0.50.
We estimate a FoS for the slope in question using each individual chart,
and then interpolate for the actual ru value for the slope.

42. GROUND INVESTIGATION:
Before any further examination of an existing slope, or the ground onto
which a slope is to be built, essential borehole information must be obtained.
This information will give details of the strata, moisture content and the
standing water level. Also, the presence of any particular plastic layer along
which shear could more easily take place, will be noted.
Piezometer tubes are installed into the ground to measure changes in water
level over a period of time.
Ground investigations also include:-
•in-situ and laboratory tests,
•aerial photographs,
•study of geological maps and memoirs to indicate probable soil
conditions,
•visiting and observing the slope.

43. In homogeneous soils relatively unaffected by faults or bedding, deep
seated shear failure surfaces tend to form in a circular, rotational manner.
We aim to find the most dangerous, ie.the most critical surface, and using
the assumption above, we can find this surface using "trial circles".
The method is as follows:-
Consider a series of slip circles of different radii but the same centre of
rotation.
Plot for each of these circles against radius, and find the minimum FoS. the
Factor of Safety (FoS)
MOST CRITICAL FAILURE SURFACE:

45. This should be repeated for several circles, each investigated from an array
of centres.
The simplest way to do this is to form a rectangular grid from the centres:-
Each centre will have a minimum FoS, and the overall lowest FoS from all
the centres shows that FoS for the whole slope. This assumes that enough
circles, with a large spread of radii , and a large grid of centres have been
investigated.

46. We then have an overall failure, surface, with smaller individual ones which
should not be ignored.

47. TENSION CRACKS:
A tension crack at the head of a slide suggests strongly that instability is
imminent. Tension cracks are sometimes used in slope stability calculations,
and sometimes they are considered to be full of water. If this is the case,
then hydrostatic forces develop as shown below:-
Tension cracks are not usually important in stability analysis, but can
become so in some special cases. We should therefore assume the cracks
don't occur, but take account of them in analysing a slope which has

48. SUBMERGED SLOPES:
When an external water load is applied to a slope, the pressure it exerts
tends to have a stabilising effect on the slope.
The vertical and horizontal forces due to the water must be taken into
account in our analysis of the slope.

50. What is the common cause of slope failures shown
above?
• Failure is along discontinuities
• Orientation of discontinuities controls failure

51. Kinematic Analysis
• Potential for global failure or releasing rockfalls
• Dependent on discontinuity orientations
Types of Discontinuities
 Bedding
 Foliation
 Shear zones
 Fault planes

52. Kinematic Analysis
Planar failure where the discontinuity
intersects the slope face
Two discontinuities intersect and also
intersect the slope face
Steeply dipping discontinuities cause
slabs and columns to separate from
face
Circular failure in soil, waste or heavily
fractured rock with no definable
structural pattern
Planar Wedge
Topple Circular

53. Plane Failure
• Discontinuity should be parallel to slope
face
• Discontinuity should dip at a gentler angle
than slope face
• Discontinuity dip should be greater than
friction angle

54. Wedge Failure
• Line of intersection should intersect the
slope face
• Line of intersection should plunge at a
gentler angle than slope face
• Line of intersection should be greater than
friction angle

55. Toppling Failure Criteria
The kinematic requirement for toppling failures according to Goodman (1989) is: “If
layers have an angle of friction Φj, slip will occur only if the direction of the applied
compression makes an angle greater than the friction angle with the normal to the
layers. Thus, toppling failure with a slope inclined α degrees with the horizontal and
discontinuities dipping at σ can occur if (90 - σ) + Φj < α”.

56. TOPPLING
OF ROCK
SLOPES
FAILURE
Toppling of columnar basalt recorded by Alexander von. Humboldt during his
exploration of the Rio Magdelana in Columbia in 1797

57. • Whenever the resultant weight of a block, W, projects
beyond the downslope outside corner of a rectangular
shaped blocks, toppling can result, as sketched here.
• Toppling usually occurs when low friction
discontinuities dip between 50 and 70 degrees from
horizontal. These can be joints, bedding, or foliation
planes
W

58. Flexural toppling
• Flexural toppling occurs when the length-to-width ratio of adjacent
blocks causes the center of gravity to fall beyond the lower hinge point,
or corner of the block.

59. Slope creep is often confused with toppling and vice versa. This shows classic slope creep in
the Ozarks of Missouri, along iron stained joints in a precambrian rhyolite, leading up into the
bedrock creep zone and overlying residuum.

60. Toppling is often easy to recognize by the textural contrast with the country rock. This shows the Clear
Creek Toppling Complex developed in the Vishnu Schist, in the Granite Gorge of the Grand Canyon,
near River Mile 84.

61. Rock topple along a forest road near Coos Bay, Oregon which killed two and
injured another. Rain had preceded the event.

62. Blocky, jointed rock masses can be modeled using programs like UDEC, shown
here. This shows an example of a toppling failure

63. Topple Failure
Previous failure – sliding of a rock or soil mass along an existing or induced sliding surface
Toppling failure – Involves rotation of columns or blocks of rock about a fixed base
Condition – the tension crack is wider at the top than at the base

64. Types of toppling failure
Individual columns are formed by a set discontinuities
dipping steeply into the face
2nd set, widely spaced orthogonal joints – column
height
Pseudo-continuous flexure along long columns, divided by
numerous cross joints
Accumulated displacements on the cross joints – toppling
Continuous columns, well developed, steeply dipping
discontinuities; bend forward
Thinly bedded shale and slate, orthogonal jointing is
not well developed
Toe sliding, excavation and erosion – start the toppling
process
Lower part – disordered fallen blocks
Block toppling
Block-flexure toppling
Flexure toppling

65. Limit Equilibrium analysis of toppling failure
Involves limit equilibrium analysis
Analysis = dimensions and forces acting on the blocks are calculated; starting the
uppermost block
Each block = quantified as stable, toppling and sliding
Overall slope is unstable = the lowermost block is either sliding or toppling
Basic requirement:
Friction angle, at base of the slope > than the dip angle of the base – no sliding occur
Rectangular block;
width = ∆x; height = yn
Dip of the base of the blocks = Ψp
Dip of the orthogonal planes forming the
faces of the blocks = Ψd (Ψd = 90 – Ψp)
Slope height = H
Angle of excavation = Ψf
Upper slope above the crest = Ψs
Step 1: Dimension calculation

66. Limit Equilibrium analysis of toppling failure
Step 2: Stability calculation
An intermediate set of toppling
The center of gravity lies outside
the base
A set of blocks in the toe region
Pushed by the toppling blocks
above
Stable, topple or slide, depends
on slope and block geometric
Stable blocks in the upper part
Friction angle of the base > dip of
the plane
Height is limited, the center of
gravity lies inside the base
Define the dimensions of the blocks, position and direction of all the forces acting on the blocks

67. Probabilistic analysis
• Design parameters – assumed to be the average and best estimate
values
• In reality – each parameter has range of values; natural variability,
changes over time, the degree of uncertainty
• FS = realistically expressed as a probability distribution, rather a single
value
• Method:
• The margin of safety method
• Monte Carlo method

68. Stereonet-Based Kinematic Analysis

69. Stereonet-Based Kinematic Analysis
Discontinuity Orientation Measurements
Strike and dip of the beds.
1-Strike
2-Dip vector
3- Dip direction
4-Angle of dip

70. Great Circles
Stereonet-Based Kinematic Analysis
Discontinuity Orientation Measurements

71. Pole
Dip Vector
Stereonet-Based Kinematic Analysis
Plotting Discontinuity Orientation

72. Great Circles
Poles
Dip Vectors
Stereonet-Based Kinematic Analysis
Plotting Discontinuity Orientation

73. Poles Contoured Poles
Stereonet-Based Kinematic Analysis
Plotting Discontinuity Orientation

74. Stereonet-Based Kinematic Analysis
Identifying representative great circles using Dips software

75. Stereonet-Based Kinematic Analysis
Plane failure criteria
Pole
Dip Vector

76. Stereonet-Based Kinematic Analysis
Wedge failure criteria

77. Stereonet-Based Kinematic Analysis
Wedge failure criteria - based on lines of intersections

78. Stereonet-Based Kinematic Analysis
Wedge failure criteria - based on lines of intersections

79. Outcrop
No.
Lith-
ology
Form-
ation
Type of
disc.
Disc.
Set no.
Dip Dip Dir. Aperture Continuity
Water
flow
Spacing
Discontinuity Data – Data
collection format