Slope stability analysis methods By Zaid Al-Farhan / Geotechnical Engineer Mosul University / Iraq

1. Methods of Stability Analysis of
Slopes
Supervised By Dr.Abdullrahman
Submit By Zaid AL-Farhan
Mosul University

2. Content:
 Introduction
 Objective
 Slopes in Brief
 Type of slip surface
 Method of Analysis
 Conclusion
 References

3. Introduction
 The failure of a slope may lead to loss of life and property. It is, therefore,
essential to check the stability of proposed slopes. With the development of
modern method of testing of soils and stability analysis, a safe and economical
design of slope is possible. The geotechnical engineer should have a thorough
knowledge of the various methods for checking the stability of slopes and
their limitations.
 Various methods are available for slope stability analysis. This is an overview
on various methods of slope stability based on assumptions, Factor of safety
calculation, soil conditions, soil types, applicability of output of the method
with its limitations.

4. Objective:
 To review the methods of slop stability analysis.
 To study principles of limit equilibrium methods and finite element
methods in slope stability analysis.
 To study the suitability of each method for particular soil type and slope
condition with factor of safety.

5. Slopes in Brief:
 Definition: any surface make angle (𝛽) with horizontal. Its found
 in natural (Formation due to geological features of the earth) ,
 man-made (Construction activity like cutting, filling ,Earth dam , embankment,
Road in mountains, etc).
 Types of slopes:
 Causes of Slope failure:
 Definition of Key Terms:
 Types of Slope Failure:
 Types Slip surface:

6.  Types of slopes:
 Infinite slope: the soil
properties for all identical
depths below the surface are
constant is called as infinite
slope. (i.e. They have
dimensions that extend over
great distances).
 Finite slope: if the slope is of
limited extent it is called as
finite slope (i.e. A finite slope is
one with a base and top
surface, the height being
limited.

7. Causes of Slope failure:
 Erosion: The wind and flowing water causes
erosion
 Steady Seepage: The pore water pressure
decrease the shear strength. This condition is
critical for the downstream slope.
 Sudden Drawdown: in this case there is reversal
in the direction flow and results in instability of
side slope. the shearing resistance decreases due
to pore water pressure that does not dissipate
quickly.
 Rainfall: Long periods of rainfall saturate. Water
enters into existing cracks and may weaken
underlying soil layers, leading to failure, for
example, mud slides.
 Earthquakes: They induce dynamic shear forces.
In addition, there is sudden buildup of pore water
pressure that reduces available shear strength.
 External Loading: increases the gravitational
forces that may cause the slope to fail.
 Construction activities: Excavation at the bottom
of the sloping surface will make the slopes steep
and

8.  Definition of Key Terms:
 Types of Slope Failure:
 Translational Slide:
 coarse-grained soils. (infinite slope)
 Rotational Slide:
 Base
 Toe(common one)
 Slope
 Flow Slide:
 internal and external conditions force ,
shallow slopes
 Types Slip surface:
 Plane (Steep slopes)
 Circle (common)
 Non-Circle(soft foundation, stiff dam)

9.  Limit equilibrium Method:
The basic assumption Coulomb's failure
1. Infinite slopes:
• Cohesionless soil:
• Cohesive soil:
2. Finite slopes:
A. Whole free body:
 Culmann’s methods (for planer failure surface).
 Taylor's stability number and stability curve(1948).
 The friction circle method (1948).
 Analysis of Steward, Sivakuga, Shukla, and Das (2011).
A. Vertical slices:
 Ordinary method of slices (Fellenius 1927).(for circular
failure surface).
 Bishop’s method (1955).
 The Bishop and Morgenstern method (1960)
 Other Mthods.
 Finite element method (Stress-Deformation
Analyses):
Method of analysis: Diff
No.
Limit equilibrium method Finite element method
1 In limit equilibrium method
currently most stability analysis
it involves due to most
simplicity and accuracy.
In finite analysis method based
on computer performance has
improved application of FE in
geotechnical analysis.
2 In limit equilibrium method
it must search for critical
surface by using geometry.
In finite element method the
critical surface is automatically
find out by various software’s.
3 The disadvantages of investigating slopes
stability through limit equilibrium method
is lack of prediction of the
deformations occurring over time (This
method assumes that the soil behavior
on failure surface is rigid and
The advantages of finite
element method: In FE method is
to for model slopes with a degree
of very high realism (complex
geometry, sequence of loading,
presence of material for
reinforcement, action of water,
and laws of complex soil
behavior) and also better
visualizes the deformation of soil
in place.
4 It’s required only simple
Mohr-coulomb soil model.
It must have complete stress strain model
for soil.
5 It cannot compute displacement. It can compute displacement.
6 Limit equilibrium method
cannot model progressive
failure.
Finite element method can
model progressive failure.

10. Main assumption:
 Shearing can occur only on the potential
failure surface …. (Plastic failure)
 The available shear strength is assumed
to be mobilized at the same rate at all
points on the potential failure surface.
As a result, the factor of safety is
constant over the entire failure
Because the soil on the potential failure
surface is assumed to be rigid-perfectly
plastic .
 two-dimensional stress. The stresses in
the third direction (perpendicular to the
section of the soil mass) are taken as
zero.
 Depending upon the method of analysis
some additional assumption are made
regarding the magnitude and
distribution of forces along various
planes.
Limit equilibrium Method:
𝐹. 𝑆 =
𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝑚𝑎𝑖𝑛𝑡𝑎𝑖𝑛 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚
=
𝑀 𝑟𝑒𝑠𝑖𝑠𝑡
𝑀 𝑜𝑣𝑒𝑟
=
𝐹 𝑟𝑒𝑠𝑖𝑠𝑡
𝐹𝑜𝑣𝑒𝑟
FACTOR OF SAFETY
In stability analysis, three types of factors of safety are normally used. They are
1. Factor of safety with respect to shearing strength. (𝐹𝑆).
2. Factor of safety with respect to cohesion.(𝐹𝑐).
3. Factor of safety with respect to Friction.(𝐹𝜑).
Generally, the three factors are taken equal, sometime when greater reliance is placed
on ∅ than 𝑐 , 𝐹𝜑 taken as a unity,∅ 𝑑 = ∅
General formula:
𝐹. 𝑆 =
𝜏 𝑓
𝜏 𝑑
𝐹𝑆 =
𝑐 + 𝜎 tan ∅
𝑐 𝑑 + 𝜎 tan ∅ 𝑑
=
𝑐
𝑐 𝑑
=
∅
∅ 𝑑

11. A. Methods that consider the whole
free body:
 Culmann’s methods:
Culmann (1866) considered a simple
failure mechanism of slope of
homogenous soil with plane failure
surface passing through toe of
slope.[2]
Stability analysis of finite slops:
𝐻𝑐 =
4𝑐′
𝑠𝑖𝑛𝛽. 𝑐𝑜𝑠𝜑′
𝛾[1 − cos(𝛽 − 𝜑′)
Allowable
 Culmann’s method is suitable for
very steep slopes.

12.  Culmann’s methods:
 Examples:

13.  Taylor (1937) conceived the idea of
analyzing the stability of a large number of
slopes through a wide range of slope
angles 𝝋′ and angles of internal friction,
and then representing the results by an
abstract number which he called the
"stability number". This number is
designated as 𝑆 𝑛. The expression used is:
𝑆 𝑛 =
𝑐′
𝐹𝑐 × 𝛾 × 𝐻
Let 𝑐 𝑚= mobilsed unit cohesion.
𝐹𝑐 =
𝑐′
𝑐 𝑚
=
𝐻
𝐻𝑐
𝑆 𝑛 =
𝑐 𝑚
𝛾 × 𝐻𝑐
=
𝑐′
𝐹𝑐 × 𝛾 × 𝐻
Taylor's stability number and
stability curve:

14. 𝐷𝑓 =
𝐻+𝐷
𝐻
Taylor's stability number and
stability curve:

15. 𝑆 𝑛 =
𝑐 𝑚
𝛾 × 𝐻𝑐
=
𝑐′
𝐹𝑐 × 𝛾 × 𝐻
Taylor's stability number and
stability curve:

16. 𝑆 𝑛 =
𝑐 𝑚
𝛾 × 𝐻𝑐
=
𝑐′
𝐹𝑐 × 𝛾 × 𝐻
Taylor's stability number and
stability curve:

17.  Homogeneous Clay with Undrained
Condition
 The critical height (i.e., 𝐹𝑠 = 1) of the
slope can be evaluated by substituting
𝐻 = 𝐻𝑐𝑟 and 𝐶 = 𝐶𝑐𝑟 (full
mobilization of the undrained shear
strength) into the preceding
equation. Thus,
For 𝛗 = 𝟎 Condition

18.  For 𝜷 < 𝟓𝟑° , use Figure 15.15 or
Figure 15.16 and Table 15.1 .
 For 𝜷 > 𝟓𝟑° , Use Figure 15.14.
Determine the critical center of failure:

19.  Assumption:
 For c − 𝜑 soil.
 Pore Water = 0
 circular failure (Toe)
 All R intersect at the circle (𝒓. 𝒔𝒊𝒏𝝋)
 Graphical Method:
 Chart Method by Taylor (1937):
The friction circle method (1948):

20. Graphical Method:
 Determine location of Critical
Circle.
 Plot Friction Circle.
 Find 𝑐 𝑚. 𝐿
 Find 𝐹𝑠.

21. Chart Method by Taylor (1937):

22.  Steward et al. (2011) made hundreds of
runs using SLOPE/W to locate the
critical circles of slopes with 𝑐′
− 𝜑′
of
soil. According to this study, the failure
circles are mostly toe circles. However,
in a few cases, they can be midpoint
circles. Based on their study, a design
chart has been developed and is shown
in Figure 15.27. [13]
Analysis of Steward, Sivakuga,
Shukla, and Das (2011):

23. The method of slices is a general method
which is equally applicable to Non
homogeneous soils, fully or partly submerged
soils, non-uniform slopes and to cases when
seepage and pore pressure exist within the
soil.
Methods that divide the free
body into many vertical slices:

24. is found in 1927 by Fellenius.
 Assumption:
 the pore water pressure is assumed to
be zero.
 Slip surface parallel to surface.
 Inter slice forces are neglected.(or
same magnitude and lines of action
coincide).
 trials by changing the center of the
trial circle to find the Min. 𝐹𝑠
Ordinary Method of Slices:
𝐹𝑠 =
𝜏𝑓
𝜏𝑑

25.  Example !
Ordinary Method of Slices:

26.  The Effect of tension Crack:
If a dam is built of cohesive soil, tension
cracks are usually present at the crest.
The depth of such cracks may be
computed from the equation:
𝑧0 =
2𝑐′
𝛾
𝑧0: 𝑑𝑒𝑝𝑡ℎ 𝑜𝑓 𝑐𝑟𝑎𝑐𝑘
Length of Arc failure (𝐿′) = 𝐿 − 𝑧0

27. Bishop (1955) suggests that the accuracy of
the analysis can be improved by taking into
account the forces E and T on the vertical
faces of each slice.
 Assumption:
 Circular surface.
 Internal Shear forces are equal (∆𝑇 =
0).
 Take in account internal slices Force
(𝑃𝑛, 𝑃𝑛+1) and assumed to be collinear.
 Pore water pressure taken in account.
 Use chart illustrated in Figure 15.33 to
find 𝑚 𝛼(𝑛)
Bishop’s method (1955):

28. Bishop’s method (1955):

29. The previous Eq. of Bishop developed by
Bishop and Morgenstern (1960), contains
the term pore pressure 𝒖. they propose
the following equation for the evaluation
of 𝑢.
𝑟𝑢 =
𝑢
𝛾ℎ
where,
𝑢 = pore water pressure at any point on the assumed
failure surface
𝛾= unit weight of the soil
ℎ = the depth of the point in the soil mass below the
ground surface
The factor of safety 𝐹𝑠 is defined as:
𝐹𝑠 = 𝑚 − 𝑛𝑟𝑢
The Bishop and Morgenstern
method (1960):
𝐷𝑓 =
𝐻+𝐷
𝐻

30.  Example !
𝐹𝑠 = 𝑚 − 𝑛𝑟𝑢
The Bishop and Morgenstern
method (1960):

31. Spencer (1967) based on Bishop and
Morgenstern has provided a method to
determine the factor of safety (Fs) by
taking into account the interslice
( 𝑷 𝒏, 𝑻 𝒏, 𝑷 𝒏 + 𝟏, 𝑻 𝒏 + 𝟏, as shown in
Figure (15.32), which does satisfy the
equations of equilibrium with respect to
moment and forces. the final results of
Spencer’s work are summarized in this
section in Figure 15.35.
Spencer (1967)

32. Other Methods of analysis:

33. Applicability of each method:

34. Applicability of each method:
Method Failure Surface
Solution
by
Applicability
Infinit Slope plane By hand grained natural slopes
Swedish Circle Circular By hand
Homogenous clay soil with
𝜑 = 0
Culmann’s methods
Plane By hand Very steep Slopes
Taylor's stability number
and stability curve (1948). Circular By hand
For c − φ soil.
Any slope
The friction circle method
(1948).
Circular
Toe Failure. By hand
For c − φ soil.
Any slope
Analysis of Steward,
Sivakuga, Shukla, and Das
(2011)
Circular
(Toe and
Midpoint)
By hand
For c − φ soil.
Any slope
Ordinary Method of slices
Circular
Slip surface
parallel to surface.
Software
For c − φ
Nonhomogeneous slopes
Inaccurate for high pore
water pressure
Bishops 1955
Circular
Slip surface
parallel to surface.
By hand
or
Software
For c − φ
Nonhomogeneous slopes
More accurate than
Ordeinary for high pore
water pressure
Bishops and Morgenstern
method (1960):
Circular
Slip surface
parallel to surface.
By hand
or
Software
For c − φ
Nonhomogeneous slopes
Accurate for pore water
pressure
Spencer (1967)
Any Shape
By hand
or
Software
For c − φ
Accurate to virtually all
slopes and soil profiles
Sarma (1975)
Any Shape
By hand
or
Software
For c − φ
Accurate to virtually all
slopes and soil profiles
For seismic force affects

35. Finite element method (Stress-Deformation Analyses):
As computer performance has improved, the application of FE in geotechnical
analysis has become increasingly common. These methods have several
advantages: to model slopes with a degree of very high realism (complex
geometry, sequences of loading, presence of material for reinforcement, action
of water, laws for complex soil behavior)
 Strength Reduction Method:
In shear strength reduction method, soil shear strength is gradually decreased, by applying finite
element and finite difference programs as long as the first indications of failure appear. Safety factor is
defined as the ratio of real shear strength of soil to reduced shear strength. [9]
 Monte Carlo simulation:
The mechanism is quite simple. The computer generates a random number between zero and one
from a uniform distribution. By knowing the cumulative distribution of the probability density function
for each variable entering into the design equation, the computer can pick up a unique value for each
variable.[10]

36.  Monte Carlo simulation:
 Geo Stru. software

37. Based on LEM:
 GEO5 : This program is used to
perform slope stability analysis
(embankments, earth cuts, anchored
retaining structures, MSE walls, etc.).
 Slide2: is a powerful, user-friendly, 2D
slope stability analysis program using
limit equilibrium method. Slide2 can
be used for all types of soil and rock
slopes, embankments, earth dams, and
retaining walls.
Softwares for Slope Analysis
The slip surface is considered as circular
(Bishop, Fellenius/Petterson, Janbu,
Morgenstern-Price or Spencer methods) or
polygonal (Sarma, Janbu, Morgenstern-Price
or Spencer methods).
Analyzes the stability of slip surfaces using
vertical slice or non-vertical slice limit
equilibrium methods like (Bishop, Janbu,
Spencer, and Sarma,)

38. Based on FEM:
 RS2 : (Formerly RS2 or Phase2) is a
powerful 2D finite element program
for soil and rock applications. can be
used for a wide range of engineering
projects including excavation design,
slope stability, groundwater
probabilistic analysis, consolidation,
and dynamic analysis capabilities.
 FLAC/Slope : uses the graphical interface
and the automatic factor-of-safety
calculation of FLAC as the core of a user-
friendly code that models slope stability
problems under a wide variety of slope
conditions.
Softwares for Slope Analysis
One of the major features of RS2 is
finite element slope stability
analysis using the shear strength
reduction method. This option is
fully automated and can be used
with various failure criteria,
including Mohr-Coulomb and
Generalized Hoek-Brown.
Uses Shear-Strength Reduction
(SSR) method to calculate FoS.

39. Conclusion:
 Various methods are available for slope stability analysis. It different each to
other by the accuracy of Factor of safety and it depend on the concept of that
method and the condition of the soil with characteristics , and we see there is
some method applicable for specific type or condition of soils but it’s not to
another or give you unreasonable result.
 Generally, we can analysis the slopes by Finite Elements and by Limit
Equilibrium and we mentioned to the difference between them.
 All another methods are developed based on Bishop’s method (1955) which
makes improves to value of factor of safety.
 Finite element is make up the lack of Limit equilibrium method, which is
simulate the reality of soles and the elastic behavior of it .and give indication
about the deformation of slopes.

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