Numerical Modeling and Simulation of the Stability of Earth Slopes
A Thesis Submitted to the Department of Nuclear Engineering
DEPARTMENT OF NUCLEAR ENGINEERING
UNIVERSITY OF GHANA
In Partial Fulfilment of the Requirements for the Degree of
MASTER OF PHILOSOPY
July 2016
ii
DECLARATION
…………………………… …………………………..
(Candidate)
I hereby declare that the preparation of this project was supervised in accordance with
………………………… …………………………
Dr. Nii Kwashie Allotey Nana (Prof.) A. Ayensu Gyeabour I
(Principal Supervisor) (Co-Supervisor)
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DEDICATION
I dedicate this work to God Almighty, my family and friends.
iv
ACKNOWLEDGMENT
Firstly I want to thank and praise God for the good health and strength given me during
all this period of schooling.
My sincere and utmost gratitude goes to my Principal Supervisor, Dr. Nii Kwashie
Allotey of the Ghana Atomic Energy Commission (GAEC) and the University of Ghana
for his expertise in the field of research and exemplary guidance towards the progress
of this research. I am also grateful to Prof. (Nana) A. Ayensu Gyeabour my Co-
Supervisor for his creative suggestions and motivation through this research project.
My thanks also goes to Ms. Rita Awura Abena Appiah of GAEC, thank you for pushing
me to make this possible. You helped me improve my programming skills a lot.
To my big family, the Atarigiya, Aguyire, and Allotey families, thank you for the
support. I could not have done it without you.
To my amazing course mate, Linda Sarpong, thank you for your words of
encouragement during out period in school; you are a strong woman! Furthermore, to
the nuclear engineering department (Samiru, Efia, Matilda, Henry…), thanks for the
wonderful time we had together.
Last and not the least, to the woman who stood firmly behind me from day one, from
when this journey began, Mercy Selina Somhayin Namateng, I LOVE YOU.
GOD RICHLY BLESS YOU ALL.
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ABSTRACT
Ghana, as most other countries, has a considerable variation in its topography. In an
attempt to build cheaper, but yet the safe structures (i.e., roads, apartments, etc.), we
are most often times faced with building on hill-sides and in valleys. This then calls
for the need to correctly assess the stability of any adjacent slopes.
In recent times, due to the extensive need for stability analysis in engineering practice,
slope stability analysis programs have been developed. It is noted that these
commercial slope stability programs are used extensively in the industry but are very
expensive and require purchasing yearly licenses. As a result of this, slope stability
analysis is not routinely conducted in local geotechnical engineering practice. The need
for cheaper more accessible options is thus considered needful.
This research initiative uses MATLAB, a commercially available, user-friendly and
easy to access computing platform to develop a slope stability analysis program. The
method used is the General Limit Equilibrium Method (GLE) with the adoption of the
Morgenstern-Price (M-P) factor of safety (FoS) approach to develop a cheap, efficient,
and yet effective model for slope stability analysis and design. The results of the
program are validated by comparing with the results of SLOPE/W, a commercial slope
stability program.
The results show four model outputs from the developed program and SLOPE/W for a
homogeneous material. Two different failure mechanisms are shown (i.e., toe and base
failures). It is noted that the percentage error in the M-P FoS is less than 5%.
It is anticipated that with the availability of this computer code, Ghanaian Engineers
can more readily assess the safety of slopes in routine design works.
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Contents
1.4 Research Goal ................................................................................................. 5
1.5 Research Objectives ........................................................................................ 5
CHAPTER TWO: LITERATURE REVIEW ................................................................ 7
2.1 Factors Causing Instability .............................................................................. 7
2.2 Types of Slip Surfaces..................................................................................... 8
2.4 Slope Stability Analysis Methods ................................................................. 12
2.4.1 Limit Analysis Method .......................................................................... 12
2.4.2 Variational Calculus Method ................................................................. 13
2.4.3 Strength Reduction Method ................................................................... 14
2.4.4 General Discussion on Limit Equilibrium Method ................................ 15
2.5 General Limit Equilibrium Method of Slices (GLE method) ....................... 19
vii
2.7 Simplified Bishop’s Method ......................................................................... 21
2.8 Janbu’s Simplified Method ........................................................................... 22
2.9 Morgenstern-Price (M-P) Method ................................................................. 23
CHAPTER THREE: RESEARCH METHODOLOGY AND SOFTWARE USED ... 27
3.1 Selection of Factor of safety method............................................................. 27
3.2 Morgenstern-Price Method ........................................................................... 28
3.6 Structured Program ....................................................................................... 32
3.7 Numerical Algorithm .................................................................................... 33
3.8.1 SLOPE/W ............................................................................................. 34
3.8.2 MATLAB ............................................................................................... 36
4.1 Introduction ................................................................................................... 37
4.3.1 Case 1 ..................................................................................................... 46
4.3.2 Case 2 ..................................................................................................... 50
CHAPTER FIVE: CONCLUSIONS ........................................................................... 53
Appendix B: MATLAB Code for Solving FoS ....................................................... 63
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LIST OF FIGURES
Figure 1.1: Over 40 m high slope on the Ayi-Mensah-Aburi road 3
Figure 2.1: Types of circular slip failure surface 9
Figure 2.2: Typical non-circular slip surfaces 10
Figure 2.3: Various definitions for FoS 11
Figure 2.4: Swedish Slip Circle Method 16
Figure 2.5: Slice Discretization and Slice Forces in a Sliding Mass 17
Figure 2.6: Forces considered in the Ordinary Method of Slices 21
Figure 2.6: Forces considered in the Ordinary Method of Slices 22
Figure 2.8: Forces considered in the M-P method 24
Figure 2.9: Inter-slice force function types 25
Figure 2.10: Variation of FoS with respect to Fm and Ff vs. λ for the M-P method 26
Figure 3.1: Sketch of a Slope Section 30
Figure 3.2: Forces acting on a Single Slice from a Mass Slope 31
Figure 3.3: Algorithm flowchart for solving for the FoS 33
Figure 3.4: SLOPE/W KeyIn Analyses Page 35
Figure 3.5: SLOPE/W KeyIn Material Page 35
Figure 3.6: SLOPE/W KeyIn Entry and Exit Range Page 36
Figure 4.1: SLOPE/W Output of Toe Failure: Case 1 38
Figure 4.2: MATLAB Output of Toe Failure: Case 1 39
Figure 4.3: SLOPE/W Output for Toe Failure: Case 2 40
x
Figure 4.4: MATLAB Output for Toe Failure: Case 2 41
Figure 4.5: SLOPE/W Output for Base Failure: Case 42
Figure 4.6: MATLAB Output for Base Failure: Case 1 43
Figure 4.7: SLOPE/W Output for Base Failure: Case 2 44
Figure 4.8: MATLAB Output for Base Failure: Case 2 45
Figure 4.9: Homogeneous Slope without Foundation 46
Figure 4.10: Analysis using SLOPE/W - FoS = 1.385 47
Figure.4.11: FoS based on M-P approach for Toe failure - FoS = 1.451 48
Figure 4.12: FoS based on M-P approach for Base Failure: FoS = 1.375 49
Figure 4.13: Slope Model Geometry from Slide 3. 50
Figure 4.14: FoS based on M-P Approach for ACAD Problem 51
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Table 3.1: Brief Comparison of Limit Equilibrium Methods 27
Table 4.1: Slope Dimensions and Material Properties for Toe Failure: Case 1 38
Table 4.2: Slope Dimensions and Material Properties for Toe Failure: Case 2 40
Table 4.3: Slope Dimensions and Material Properties for Base Failure: Case 1 42
Table 4.4: Slope Dimensions and Material Properties for Base Failure: Case 2 44
Table 4.5: Slope dimensions and material properties 46
Table 4.6: Slope Dimensions and Material Properties for ACAD Problem 50
Table 5.1: Summary of FoS Outputs for all Case Studies. 53
xii
dx width of slice
c cohesion of soil
τf shear strength
τ shear stress
angle of internal friction of soil
α inclination from horizontal of the bottom of the slice (degrees)
cal cos(α)
sal sin(α)
tph tan()
N total normal force on the bottom of the slice
S shear force on the bottom of the slice
W weight of the slice
havg = average height of slice
u pore water pressure
E inter-slice force on the downslope of the slice
ns number of slice faces
FoS factor of safety
Xc x coordinate of centre of slip circle
Yc y coordinate of centre of slip circle
R Radius of slip circle
ytop Ground surface
ybot Slope surface
FFm Moment Factor of Safety for Morgentern-Price Method of Slices
FFf Force Factor of Safety for Morgentern-Price Method of Slices
F(x) Interslice force function
1
1.1 Background
Slope stability is the potential for ground slopes to resist movement [1]. Slope
instability has been the subject of continued concern because of the tremendous loss of
life, property and infrastructure caused annually in many places in the world [2]. In the
field of construction, slope instability can occur due to rainfall, increasing the water
table, and the change in stress conditions. Similarly, tracks of land that have been stable
for years may suddenly fail due to changes in the geometry, external forces and loss of
shear strength [3].
Slope failures, also called slides or landslides, whether sudden or gradual, are due to
the increased stress of slope materials or foundations compared with their mobilized
strength [3].
The majority of the slope stability analyses performed in practice still use traditional
limit equilibrium approaches involving the method of slices, and has remained virtually
unchanged for decades.
Analysis of the slope stability is carried out to assess the safety of artificial or natural
slopes (e.g., dams, road cuts, mining open pit excavations and landfills). For human
made slopes, analysis of slope stability is used to evaluate various design options, which
then provides a basis for a form of engineering design with associated costing
comparisons. The efficient engineering of natural and artificial slopes has therefore
become a common challenge faced by both researchers and practitioners.
Slope stability assessment mainly involves the use of the factor of safety (FoS) method
to determine how close a given slope is to the onset of instability, or to what extent the
state of the slope is from failure.
2
When this ratio is well above 1, the resistance to shear failure is generally higher than
the driving shear stress, and the slope is considered stable. When this ratio is near to 1,
the shear strength is almost equal to the shear stress, and the slope is close to failure. If
the FoS is less than 1, the slope is considered to have failed, or considered to be trigger-
point ready [4].
Ghana is not noted to be a frequent serious victim of mass movement (slope failures).
It is, however, noteworthy that Ghana has not been without slope failures. In Ref. [5],
reference is made of a slope failure which occurred in 1968 near Jamasi in the Sekyere
District involving about 1,500 cubic meters of rock, soil and vegetation. The failure
blocked the main Kumasi-Mampong truck road for a total of ten days.
Reference is also made in Refs. [6 & 7] on potential slope failures on the stretch of the
Accra-Aburi road, when rocks began to fall unhindered onto the road in 2014.
Furthermore, Ref. [8] notes that in 2013 after a heavy downpour of rain, loose parts of
the Akoaso Mountain fell and blocked the Kasoa-Weija portions of the road, resulting
in a significant traffic jam for hours.
These recent records of slope instability in the country have served as a wakeup call for
researchers and practicing engineers to take a critical look at this issue.
3
Figure 1.1: Over 40 m high slope on the Ayi-Mensah-Aburi road
1.2 Problem Statement
In an attempt to build cheap and yet the safe structures (i.e., roads, living apartments,
etc.) for man-kind, we are most times faced with building in valleys and on mountains.
Either way, we are faced with the problem of slope instability.
In the past decades, computer software for slope stability analysis and design have been
developed and marketed extensively. These commercial software, which have been
developed over many years, are able to perform rigorous stability calculations, and give
fast and accurate answers to complex slope stability problems. These software have
become widely accepted in industry, and are now part of most large design engineering
offices. These software are, however, expensive and normally require the annual
renewal of licenses. Notwithstanding, their wide acceptance in industry, most
Ghanaian geotechnical engineers still resort to the use of the rule of thumb slope
stability analysis methods, and old charts for their daily slope stability analysis. This is
4
due to the relatively high cost of these commercial software, and the limited financial
capacity of local engineering firms.
This has created a gap between local and international engineers, and has resulted in
major geotechnical engineering projects involving complex slope instability problems
being awarded to international firms, rather than local engineering firms.
1.3 Relevance and Justification of Study
Landslides, rock falls, and mass movement of any kind, are undoubtedly, one of the
oldest natural disasters that have resulted in huge damages, loss of lives, and a great
deal of pain to mankind. Like other mountainous countries, Ghana has large variations
in its topography. The impending threat of landslides in the case of [5 or 6], or rock
falls in the case of [7] is now accepted as life threating, and the need for these slopes to
be properly engineered is critical.
It has been already noted that the available commercial slope stability programs that
are extensive slope stability analysis are very expensive and require purchasing yearly
licenses. This has necessitated the need to develop a simple, yet efficient slope stability
program, that can be easily accessed by local engineers to appraise local slope stability
problems using the most rigorous and accurate methods available.
This research initiative uses MATLAB, (a commercially available, and easily
accessible computing platform with great user-friendly interface) using General Limit
Equilibrium method (GLE) and adopting the Morgenstern-Price FoS to develop a cheap
and efficient, yet effective model for slope stability analysis and design.
5
1.4 Research Goal
The goal of this thesis is to develop and implement a slope stability numerical model
that will aid local geotechnical engineers to readily appraise the stability of slopes in
local practice.
Develop a physical model to represent the problem
Develop mathematical equations to solve the problem
Develop a numerical algorithm and write a code to solve for the FoS of earth
slopes.
Verify and validate the code using Geoslope International’s SLOPE/W
commercial slope stability programme.
1.6 Scope
For the purpose of this study, this thesis is limited to the development of a slope stability
programme for homogenous soil and rock media. In this regard, the goal of the study
is to develop the generic algorithm for slope stability analysis. Furthermore, similar to
the existing commercial programmes, the study is limited to two-dimensional slope
stability problems.
1.7 Format of the Thesis
Following the introduction to slope stability problems in Chapter 1, a detailed literature
review of the methods of slope stability analysis is provided in Chapter 2. Chapter 3
then presents the proposed solution method for calculating the factor of safety, in which
the GLE method is explained, and the solution algorithm developed.
Chapter 4 presents sample results from the developed MATLAB code. It presents
comparisons between the results of the developed code and results from the commercial
slope stability programme, SLOPE/W. Chapter 5 finally presents the conclusions of
the study, and also provides recommendations for further studies.
7
CHAPTER TWO: LITERATURE REVIEW
Analysis of slope stability problem is an important subject area in geotechnical
engineering. The phenomenon of landslides and related slope instability is a problem
in many parts of the world. Slope failure mechanisms and the geological history of a
slope can be very complicated problems and required complex forms of analyses.
2.1 Factors Causing Instability
The failure of the slopes occurs when the downward movements of soil or rock material
because of gravity and other factors, creates shear stresses that exceed the inherent shear
strength of the material. Therefore, factors that tend to increase the shear stress or
decrease the shear strength of a material increases the risk of the failure of a slope.
Various processes can lead to a reduction of the shear strength of a soil/rock mass.
These include: increased pore pressure, cracking, swelling, decomposition of
argillaceous rock fills, creep under sustained loads, leaching, softening, weather and
cyclic loading, among others.
On the other hand, shear stress within a rock/soil mass may increase due to additional
loads on top of the slope, and increase in water pressure due to cracks at the top of the
slope, an increase in the weight of soil due to increasing water content, the excavation
of the base of the slope, and seismic effects. Furthermore, additional factors that
contribute to the failure of a slope include the rock/soil mass properties, slope geometry,
state of stress, temperature, erosion, etc.
The presence of water is the most critical factor that affects the stability of slopes. This
is because it increases both the driving shear stress, and also decreases the soil/rock
mass’ shear strength. The speed of sliding movement in a slope failure can vary from
8
a few millimeters per hour to very fast slides where great changes have occurred within
seconds. Slow slides occur in soils with a plastic stress-strain characteristics, where
there is no loss of strength with increased strain. Fast slides occurs in situations where
there is a sudden loss of strength, as in the liquefaction of sensitive clay or fine sand.
Increase in shear stresses across the soil mass result in movement only when the shear
strength mobilized on given possible failure surface in the ground is less than the
driving shear stresses along that surface.
2.2 Types of Slip Surfaces
To calculate the FoS of a slope, it is always assumed that the slope is failing in some
shape, normally in a circular or non-circular shape. For computational simplicity, the
slide surface is often seen as circular or composed of several straight lines [9]. Different
sliding surfaces are normally assumed with the computation of a corresponding FoS.
The sliding surface with the minimum FoS is then selected as the FoS of the slope in
question.
A circular sliding surface, like that shown in Figure 2.1, is often used because it is
suitable to sum the moments about a centre. The use of a circle also simplifies the
calculations. Wedge-like surfaces have their failure mechanisms defined by three or
more straight line segments defining an active area, central block, and the passive area
as shown in Figure 2.2. This type of sliding surface can be used for analysis of slopes
where the critical potential sliding surface comprises a relatively long linear sector
through low material bounded by a stronger material.
As noted above, the critical slip surface is the surface with the lowest factor of safety.
The critical slip surface for a given problem analysed by a given method, is found by a
9
systematic procedure to generate sliding test surfaces until the one with the minimum
safety factor is found [9].
Figure 2.1: Types of circular slip failure surface [3]
10
2.3 Definition of FoS
FoS is usually defined as the ratio of the ultimate shear strength to the shear stress
mobilized at imminent failure. There are several ways to formulate the FoS, The most
common formulation assumes the safety factor to be constant along the sliding surface,
and it is defined in relation to limit equilibrium, force and moment equilibrium [4].
11
These definitions are given in Figure 2.3 below. As will be developed further in the
limit equilibrium method, the first definition is based on the shear strength which can
be obtained in two ways: a total stress approach (suanalysis) and an effective stress
approach (c’- φ’ −analysis). The type of strength consideration depends on the soil
type, the loading conditions and the time elapsed after excavation. The total stress
strength method is used for short–term conditions in cohesive soils, whereas, the
effective stress method is used in long- term conditions in all soil types, or in short-term
conditions in cohesive soils where the pore pressure is known [3].
Figure 2.3: Various definitions for FoS [3]
12
2.4 Slope Stability Analysis Methods
Slope stability problems deal with the condition of ultimate failure of a soil or rock
mass. Analyses of slope stability, bearing capacity and earth pressure problems, all fall
into this area. The stability of a slope can be analysed by a number of methods, among
others of which are the:
Limit analysis method,
Variational calculus method,
2.4.1 Limit Analysis Method
The limit analysis method theory is based on a rigid-perfectly plastic model material.
Drucker and Prager [10] first formulated and introduced the upper and lower bound
plasticity theorems for soil/rock masses.
The general analysis process includes construction of a statically admissible stress field
for the lower-bound analysis, or a kinematically admissible velocity field for the upper-
bound analysis.
For both upper- and lower-bound analysis, one of the following two conditions has to
be satisfied:
Geometrical compatibility between internal and external displacements or
strains. This is usually concerned with kinetic conditions – velocities must be
compatible to ensure no gain or loss of material at any point.
Stress equilibrium, i.e., the internal stress fields must balance the externally
applied stresses (forces).
13
The basis of limit analysis rests upon two theorems, which can be proved
mathematically. In simple terms, these theorems are:
Lower Bound: any stress system in which the applied forces are just sufficient
to cause yielding.
Upper Bound: Any velocity field that can operate is associated with an upper
bound solution.
The lower- bound approach has been used in 2D slope stability analysis by Zhang [11]
and Kim et al. [12]. The upper-bound approach was first used in 2D slope stability
analysis by Drucker and Prager [10] to determine the critical height of a slope.
Subsequently, Refs. [13, 14 & 15] also applied and extended the upper-bound
approaches in 2D slope analysis. Michalowski [16] proposed an upper-bound approach
based on a translational failure mechanism. The vertical slice techniques, which are
often used in traditional limit equilibrium approaches, were employed to satisfy the
force equilibrium condition for all individual slices. Two extreme kinematical solutions
neglecting the inter-slice strength, or fully utilizing the inter-slice strength of the soil
were then obtained. The traditional limit equilibrium solutions for slices with a proper
implicit assumption of failure mechanism can fall into the range of these two extremes.
Donald and Chen [17] also presented an upper-bound method on the basis of a multi-
wedge failure mechanism, in which the sliding body was divided into a small number
of discrete blocks.
The variational calculus approach does not require assumptions on the inter-slice
forces. It was first used for 2D stability analysis by Baker and Garber [18]. This
14
approach was subsequently employed by Jong [19] for vertical-cut analysis in cohesive
directionless soils. Cheng, et al. [20] also used it in their research where they developed
a numerical algorithm based on the extremum principle by Pan [21]. The formulation
which relies on the use of a modern try and error optimization method and can be
viewed as an equivalent form of the variational method in a discretized form, but is
applicable for a complicated real life problem.
2.4.3 Strength Reduction Method
In recent decades, there have been great developments in the area of the strength
reduction method (SRM) for slope stability analysis. The general procedure of the
SRM analysis is the reduction of the strength parameters by the FoS, while the body
forces (due to the weight of soil and other external loads) are applied until the system
cannot maintain a stable state. This procedure can determine the FoS within a single
framework, for both two and three-dimensional slopes.
The main advantages of the SRM are as follows;
The critical failure surface is automatically determined from the
application of gravity loads and/or the reduction of shear strength,
It requires no assumption about the distribution of the inter-slice shear
forces,
It can give information such as stresses, movements (deformations) and
pore pressures.
One of the main disadvantages of the SRM is the long time required to develop the
computer model and to perform the analysis to arrive at a solution. With the
15
development of computer hardware and software, 2D -SRM can now be done in a
reasonable amount of time suitable for routine analysis and design. This technique is
also adopted in several well-known commercially available geotechnical finite element
or finite difference programs. In strength reduction analysis, the convergence criterion
is the most critical factor in the assessment of the FoS.
Investigation results show that; the FoS obtained and the corresponding slip surface
determined by the SRM, demonstrate good agreement with the results of the Limit
Equilibrium Method (LEM).
2.4.4 General Discussion on Limit Equilibrium Method
2.4.4.1 Swedish Slip Circle Method of Analysis
The Swedish Slip Circle method assumes that the friction angle of the soil or rock is
zero. In other words, when angle of friction is considered to be zero, the effective stress
term tends to zero, which is therefore equivalent to the shear cohesion parameter of the
given soil. The Swedish slip circle method assumes a circular failure interface, and
analyses stress and strength parameters using circular geometry and statics as shown in
Figure 2.4. The moment caused by the internal driving forces of a slope is compared
with the moment caused by resisting forces in the sliding mass. If forces resisting
movement are greater than the forces tending to cause movement, then the slope is
assumed to be stable.
2.4.4.2 Method of Slices
Despite all the above methods, limit equilibrium methods are by far the most used form
of analysis for slope stability studies. They are the oldest best-known numerical
technique in geotechnical engineering. These methods involve cutting the slope into
fine slices so that their base can be comparable with a straight line. The governing
equilibrium equations equilibrium of the forces and/or moments, Figure 2.5 are then
developed. According to the assumptions made on the efforts between the slices and
the equilibrium equations considered, many alternatives have been proposed in Table
2.1. They give, in most cases, quite similar results. The differences between the values
of the FoS obtained with the various methods are generally below 6% [22].
17
Figure 2.5: Slice Discretization and Slice Forces in a Sliding Mass [23.24]
The idea of dividing a potential sliding mass into slices dates back to the early 1900’s.
The first documented use of the method of slices is the analysis of the 1916 failure at
the Stigberg Quay in Gothenburg, Sweden [25].
This Limit Equilibrium method is well known to be a statically indeterminate problem
and assumptions about the inter-slice shear forces are needed to make the problem
statically determinate. On the basis of the assumptions on the internal forces and the
force and/or moment equilibrium, there are more than ten methods developed for
analysis of slope stability problems [26]. Famous methods include those by Fellenius
[27], Bishop [28], Janbu [29, 30], Spencer [31], Morgenstern-Price [32] and the General
Limit Equilibrium (GLE) [25].
Table 2.1 shows the differences between the various methods of stability analysis, on
the basis of forces and moments equilibrium.
18
No Methods Moment
Ordinary Method (1936) Yes No No No Yes No No
2 Bishop Simplified (1955) Yes No Yes No Yes No No
3 Janbu Simplified (1954) No Yes Yes No No Yes No
4 Spencer Method (1967) Yes Yes Yes Yes Yes Yes Constant
5 Morgenstern-Price Method
Constant Half-
Sine
Clipped-Sine
Trapezoidal
Specified
19
All limit equilibrium methods utilise the MohrCoulomb criterion to determine the
shear strength (τf) along the sliding surface. The mobilised shear stress at which a soil
fails in shear is defined as the shear strength of the soil. According to Janbu [29], a state
of limit equilibrium exists when the mobilised shear stress (τ) is expressed as a fraction
of the shear strength. Nash [33] states that, at the moment of failure, the shear strength
is fully mobilised along the failure surface when the critical state conditions are
reached. The shear strength is usually expressed by the Mohr Coulomb linear
relationship as:
where
c is the soil cohesion and is the soil frictional angle.
The strength of the soil available depends partially on the type of soil and the normal
stresses acting on it. The mobilized shear strength on the other hand depends on the
external forces acting on the soil mass. This defines the FoS as a ratio of the τf to τ in a
limit equilibrium analysis [30], as defined in Equation 2.2.
2.5 General Limit Equilibrium Method of Slices (GLE method)
The GLE formulation was developed by Fredlund at the University of Saskatechewan
[34, 35]. This method encompasses the key elements of all the other methods of slices
(the Ordinary, Bishop’s, Janbu and M-P methods). The GLE formulation is based on
two factors of safety equations. One equation gives the factor of safety with respect to
20
moment equilibrium, Fm (Equation 2.3) while the other equation gives the factor of
safety with respect to horizontal force equilibrium, Ff (Equation 2.4).
= ∑ ∗∗∗∗cos[]−μl)∗∗tan[]
∗ (2.3)
= ∑ ∗∗cos[]+(−μ)∗tan[]∗cos[]
∗sin[] (2.4)
Where,
l is the length of the bottom of the slice, R is the radius of the slip circle, W is the weight
of slice, Ni is the total normal force on the bottom of the slice, c is the soil cohesion,
µis the pore water pressure, α is the inclination of the slip surface at the middle of the
slice.
2.6 The Ordinary or Fellenius Method (OMS)
The ordinary method is considered the simplest of the methods of slices since it is the
only procedure that results in a linear factor of safety equation. It is generally stated that
the inter-slice forces can be neglected because they are parallel to the base of each slice
[26]. This notwithstanding, the Newton's principle of 'action equals reaction' is not
satisfied between slices. The change in direction of the resultant inter-slice forces from
one slice to the next results in factor of safety errors that may be as much as 60% [37].
The normal force on the base of each slice is derived either from summation of forces
perpendicular to the base or from the summation of forces in the vertical and horizontal
directions. The forces considered to act in this method are represented in Figure 2.6
below. The FoS is based on moment equilibrium and computed as:
21
Figure 2.6: Forces considered in the Ordinary Method of Slices
= ∗+∗tan()
∗sin() (2.5)
where
µis the pore water pressure,
l is the base length of the slice,
α is the inclination of the slip surface at the middle of the slice.
2.7 Simplified Bishop’s Method
The simplified Bishop method neglects the inter-slice shear forces and thus assumes
that a normal or horizontal force adequately defines the inter-slice forces as shown in
Figure 2.7 below. The normal force on the base of each slice is derived by summing
forces in a vertical direction. The factor of safety is derived from the summation of
moments about a common point as in OMS since the inter-slice forces cancel out.
Therefore, the factor of safety equation is the same as for the ordinary method [28].
22
However, the definition of the normal force is different. The FoS is based on moment
equilibrium and computed as:
Figure 2.7: Forces considered in the Simplified Bishop’s Method
= ∗+∗tan()
∗x (2.7)
∗∗sin[]−∗∗sin[]∗tan[]
]

(2.8)
where ‘x’ is the horizontal distance from the mid-base of the slice to the centre of
rotation.
As can be seen in the equation above, the expression is nonlinear due to the appearance
of Fm in both sides of the equation and will require an iterative procedure to reach a
solution.
2.8 Janbu’s Simplified Method
In Janbu's simplified method, the normal force in each slice is derived from the
summation of vertical forces, with the inter-slice shear forces ignored. The horizontal
force equilibrium equation is used to derive the factor of safety. The sum of the inter-
slice forces must cancel and FoS equation becomes;
23
∗∗sin[]−∗∗sin[]∗tan[]
]

(2.9)
∗sin(α) (2.10)
2.9 Morgenstern-Price (M-P) Method
This has become the most widely used method developed for analysing generalized
failure surfaces. The method was initially described by Morgenstern and Price [36].
The MorgensternPrice method also satisfies both force and moment equilibriums and
the overall problem is made determinate by assuming a functional relationship between
the inter-slice shear force and the inter-slice normal force. According to Morgentern
and price [32], the inter-slice force inclination can vary with an arbitrary function f(x)
as:
where,
f(x) = inter-slice force function that varies continuously along the slip surface and
λ = scale factor of the assumed function.
The method suggests assuming any type of force function, f(x), for example halfsine,
trapezoidal or user defined as shown in Figure 2.9. The relationships for the base normal
force (N) and inter-slice forces (E, X) are the same as given in Janbu’s generalised
method. The forces considered to act in this method are represented in figure 2.8 below.
For a given force function, the inter-slice forces and the Newton-Raphson numerical
technique can be used to solve the moment and force equations for the FoS and until,
Ff is equals to Fm in equations (2.12) and (2.13) [33] below.
24
Figure 2.8: Forces considered in the M-P method [24]
= ∑ ∗∗cos[]+(−μ)∗tan[]∗cos[]
∗sin[] (2.12)
= ∑ ∗∗∗∗cos[]−μl)∗∗tan[]
∗ (2.13)
∗∗sin[]−∗∗sin[]∗tan[]
]

(2.14)
F is Fm or Ff depending on which equilibrium equation is being solved.
and
25
An alternative derivation for the Morgenstern-Price method was proposed by Fredlund
and Krahn in [34]. It presents a complete description of the variation of the factor of
safety with respect to λ. On the first iteration, the vertical shear forces are set to zero.
On subsequent iterations, the horizontal inter-slice forces are first computed and then
the vertical shear forces are computed using an assumed λ value and side force function.
Fm and Ff are solved for a range of λ values and a specified side force function. These
26
FoS are plotted on a graph as shown in Figure 2.10. The FoS vs. λ are fit by a second
order polynomial regression and the point of intersection satisfies both force and
moment equilibrium.
Figure 2.10: Variation of FoS with respect to Fm and Ff vs. λ for the M-P method [34]
27
USED
There are different limit equilibrium methods having varying superiorities over each
other as discussed in Chapter 2. Methods are based on different assumptions on
equilibrium conditions to be satisfied. Duncan and Wright [24] summarized some of
limit equilibrium methods with respect to their limitations, assumptions and
equilibrium conditions to be satisfied as shown in Table 3.1.
Table 3.1: Brief Comparison of Limit Equilibrium Methods [24].
Procedure Use
Ordinary method of slices
Applicable to non-homogeneous slopes soils where slip surface can be
approximated by a circle. Very convenient for hand calculations. In
accurate for effective stress analysis with high pore water pressure.
Simplified Bishop method
Applicable to non-homogeneous slopes soils where slip surface can be
approximated by a circle. More accurate than ordinary method of slices,
especially for analysis with high pore water pressures. Calculations
feasible by hand or spreadsheet.
Spencer's method
An accurate procedure applicable to virtually all slope geometries and soil
profiles. The simplest complete equilibrium procedure for computing
FoS.
method
An accurate procedure applicable to virtually all slope geometries and soil
profiles. Rigorous, well-established complete equilibrium procedure.
Requires solution of nonlinear equations with an iterative procedure.
The analysis and design of failing slopes requires an in-depth understanding of the
failure mechanism in order to choose the right slope stability analysis method.
In this research, as stated above, the slice approach for GLE procedure is used and the
FoS according to the Morgenstern-Price’s procedure, which satisfies all the
28
requirements for static equilibrium is adopted. Regardless of whether equilibrium is
considered for a single free body or a series of individual vertical slices, there are more
unknowns (forces, locations of forces, factor of safety, etc.) than the number of
equilibrium equations; the problem of computing a FoS is thus statically indeterminate.
Therefore, assumptions must be made to achieve a balance of equations and unknowns.
This method allows for analysis of any failure shape (circular, non-circular or
compound). The solution for the (FoS) is derived from the summation of forces
tangential and normal to the base of a slice and the summation of moments about the
centre of the base of base slice. The steps required to provide the input data for
performing the slope stability analysis include [37].
A survey of the elevation of the ground surface on a section perpendicular to
the slope.
Estimation of ground stratigraphy from borehole logs and soil/rock properties
from engineering soil/rock tests.
The determination of ground water level from piezometer readings to estimate
ground water pore-water pressures.
3.2 Morgenstern-Price Method
Morgenstern and Price [32] developed the method for FoS similar to the Spencer
method [31]. The method considers both normal inter slice force and shear forces.
Therefore it satisfies both moment and force equilibrium.
29
3.3 Assumptions
To begin with the generalized formulation, the following assumptions are made for a
body of mass slope:
2. The soil/rock is a homogeneous.
3. The soil/rock is an isotropic material.
4. The failure mass is a rigid body.
5. The base normal force acts at the middle of each slice.
6. The Mohr-Coulomb failure criterion is used.
3.4 Numerical Method Development
In this work, the geotechnical generalized method of slices approach, which is a form
of the Finite Strip Method would be used for the computation of the Factor of Safety
(FoS), which is a measure of the degree of safety of the slope. The Finite Strip Method
is a variant form of the known differential equation numerical discretization
approaches. It discretizes the equilibrium differential equation (based on slices) and
imposes equilibrium boundary conditions at the ends to solve for the required
unknowns.
The method of slices method is a numerical approach used to solve the equilibrium
differential equation at the limiting condition, i.e., at the point of slope failure. The
method consists of dividing the slope into a number of fine slices so that their bases can
be comparable with a straight line. The equilibrium differential equations are then
developed for each slice, and then the global force and moment equilibrium problem
solved numerically to obtain the FOS [24].
30
This study will make available a locally developed slope stability program that would
simulate the behaviour of a simple slope undergoing circular failure. The results from
this work, which will include both factor of safety for moment and force equilibrium,
will be visualized in the command window of MATLAB.
3.5 Derivation of Equations
Figure 3.1. is a sketch of a slope section and Figure 3.2 is a sketch of a slice of mass
from Figure 3.1 with the forces acting on it at the point of failure. By taking moment
about the midpoint of the base of the slice in Figure 3.1;
′ [( − ′) − (−
′ − dy ′ + (
dy
2 ) = 0 (3.1)
After simplification and proceeding to the limit as dx→0, it can be shown readily that;
= (′.
31
Figure 3.2: Forces acting on a Single Slice from a Mass Slope [24].
For equilibrium in the N direction, we have, from Figure 3.2;
dN′ = dWcos[] − dXcos[] − dE′sin[] (3.3)
For equilibrium in the S direction, we have from Figure 3.2;
dS = dE′cos[] − dXsin[] + dWsin[] (3.4)
The Mohr-Coulomb failure criterion in terms of effective stresses may be expressed as;
dS = 1
{′dxsec[] + (dN′)tan[′]} (3.5)
Equation (3.5) defines the factor of safety in terms of shear strength.
Eliminating dS from Equations (3.4) and (3.5);
1
[′dxsec[] + (dN′)sin[′]] = dE′cos[] − dXsin[] + dWsin[] (3.6)
By eliminating dN′ from Equations (3.3) and (3.6) and dividing by dxcos[], we have;
′
dx and equation (3.7) becomes;
32
′
dE′
′ is the effective friction angle,
W is the weight of ith slice,
dN and dS are resultants of the normal and tangential forces on the slice base of length
li
α is the inclination of the slice with respect to the horizontal.
F is the factor of safety.
The governing differential equation is developed for each slice, and summed up to
develop the equilibrium equation for entire mass. Based on the assumption on the inter-
slice forces as provided in Chapter 2, the various expressions for Fm, Ff, N, X & E for
the different slices are computed.
3.6 Structured Program
Below (Appendix A) is an algorithm developed to solve for the force and moment factor
of safety’s (Ff and Fm) using the M-P method.
33
3.7 Numerical Algorithm
Figure 3.3 shows the numerical algorithm used to solve the problem. Equation numbers
referred to here are that from appendix A.
Figure 3.3: Algorithm flowchart for solving for the FoS
No
Yes
10, 7& 6
Output
FSm(i),
FSf(i)
Output
3.8.1 SLOPE/W
SLOPE/W, developed by GeoStudio International Canada, is part of a professional
geotechnical software suite of programmes. SLOPE/W is the programme used for slope
stability analysis. This software is based on the theories and principles of the LE
methods discussed in the previous sections. To check the accuracy of the program
written in MATLAB and its output, SLOPE/W is employed. For this study, a full
licensed version of Geostudio 2007 has been used.
For each model, using the drawing tools, the geometry of the slope is entered. Then by
using the slip surface dialogue box, the slip surface is specified by using a range.
Then by using the “Materials” dialogue box, soil parameters will be entered and the
selected soil will be assigned to the drawing in the software.
After entering all of the input data into the software by hitting the “Start” button under
the “Solve Manager”, the program starts to analyze the slope and find the minimum
factor of safety and its related failure surface. Figures 3.4 to 3.6 show some of the stages
of developing the model using SLOPE/W
35
36
3.8.2 MATLAB
and fourth-generation programming language. A proprietary programming language
developed by MathWorks, MATLAB allows matrix manipulations, plotting of
functions and data, implementation of algorithms, creation of user interfaces, and
interfacing with programs written in other languages, including C, C++, Java, Fortran
and Python [38]. The programme is also easily accessible and is available in most
design engineering offices. These features of MATLAB make it suitable to be used for
this research work.
4.1 Introduction
This chapter presents outputs from the MATLAB code and SLOPE/W programmes and
compares the results of the two. First, the analysis is run for assumed models (the same
geometry, soil properties) of homogeneous soil mass and modelled for two failure
scenarios; i.e., for toe and base failure conditions.
The second set of comparison is done for existing case studies. Case one is the study
from Griffiths and Lane [39] and Rocscience [40] and case two is from the study by
ACADS [41].
4.2 Programme Test Examples
The program was tested with two failure mechanisms, toe and base failures. In slope
stability design and analysis, and for most design projects, the worst case scenario is
always adopted. The worst case when it comes to FoS in slope stability analysis is the
least FoS. This is adopted in choosing the FoS in this MATLAB program in situations
when more than one FoS is outputted.
4.2.1 Toe Failure
Two models were presented for the toe failure mechanism. The slope angles for both
models are 45° or 1H: 1V (H: horizontal distance; V: vertical distance). The rest of the
soil parameters and geometry are presented in Tables 4.1 and 4.2.
38
Case 1
Table 4.1: Slope Dimensions and Material Properties for Toe Failure: Case 1
c´ ’ γ H
(kPa) () (kN/m3) (m)
Figure 4.1: SLOPE/W Output of Toe Failure: Case 1
39
40
Case 2
Table 4.2: Slope Dimensions and Material Properties for Toe Failure: Case 2
c´ ’ γ H
(kPa) () (kN/m3) (m)
Figure 4.3: SLOPE/W Output for Toe Failure: Case 2
41
42
It is noted from the outputs that the MATLAB code give reasonable outputs of FoS
compared to SLOPE/W. In both cases, the percentage error is found to be 4%. The
small error margin is an indication that the developed MATLAB code is working to an
acceptable degree.
4.2.2 Base Failure
Two models were presented for the base failure mechanism. The slope angles for both
models are 1H: 1V. The rest of the soil parameters and geometry are presented in Tables
4.3 and 4.4.
Case 1
Table 4.3: Slope Dimensions and Material Properties for Base Failure: Case 1
c´ ’ γ H
(kPa) () (kN/m3) (m)
Figure 4.5: SLOPE/W Output for Base Failure: Case 1
43
44
Case 2
Table 4.4: Slope Dimensions and Material Properties for Base Failure: Case 2
c´ ’ γ H
(kPa) () (kN/m3) (m)
Figure 4.7: SLOPE/W Output for Base Failure: Case 2
45
46
4.3.1 Case 1
The geometry of a homogeneous slope without foundation is shown in Figure 4.9. This
case follows the analyses performed by [39] and [40], which were used as benchmark
cases to study the applicability of 3-D FE analyses to slope stability. The slope angle is
26.25° or 2H: 1V for this case. According to Griffiths and Lane, the adopted parameters
are based on c/γH = 0.05. The height of the slope is assumed to be 40 m, thus the
corresponding parameters are summarized in Table 4.5.
Figure 4.9: Homogeneous Slope without Foundation
Table 4.5: Slope dimensions and material properties
c´ ’ γ H
(kPa) () (kN/m3) (m)
40 20 20 40
The slope stability analyses using the computer program, SLOPE/W, is shown in Figure
4.10. The FoS based on the GLE and the adopted M-P approach is conducted for both
toe and slope failures. The FoS for toe and slope failures are found to be 1.385 and
1.373 respectively.
47
The same analysis was conducted using the MATLAB code, the FoS for toe and slope
failures are found to be 1.431and 1.375 respectively and shown in Figure 4.10-4.11.
Figure 4.10: Analysis using SLOPE/W - FoS = 1.385
48
Figure.4.11: FoS based on M-P approach for Toe failure - FoS = 1.431
49
Figure 4.12: FoS based on M-P approach for Base Failure: FoS = 1.375
50
4.3.2 Case 2
In 1988 a set of 5 basic slope stability problems, together with 5 variants, was
distributed both in the Australian Geomechanics profession and overseas as part of a
survey sponsored by ACADS [41].
This problem is taken from the verification manual of Slide 3.0 (Verification #1). It was
distributed to the Australian Geomechanics profession and overseas in 1988 with some
other models to verify the efficiency and accuracy of Slide 3.0. The slope model
geometry is presented in Figure 4.13. The slope material properties are shown in Table
4.6.
Table 4.6: Slope Dimensions and Material Properties for ACAD Problem
c´ γ H
(kPa) () (kN/m3) (m)
Figure 4.14: FoS based on M-P Approach for ACAD Problem.
52
The FoS from this case study when modelled with Slide 3.0 and the GLE (M-P)
approach adopted, was found to be 0.986. Upon modelling the same problem with the
MATLAB code, as shown in Figure 4.14 above, the FoS is found to be 1.035. Again,
this can be said to be a reasonably accurate estimate of the FoS by the MATLAB code.
The percentage error is 4.7%.
53
The primary focus of this research was to:
Develop a MATLAB code for solving the FoS of homogeneous earth slopes
Verify and validate the code with SLOPE/W models and also with cases from
literature
Seven cases were studied in this research as summarized below in Table 5.1
Table 5.1: Summary of FoS Outputs for all Case Studies.
Case 1 Case 2
Base
Comparison with Literature
Toe Failure 1.385 1.431 3.2
Base
Failure 1.373 1.375 0.986 1.035 0.1 4.7
From the Table 5.1 above, it is seen that the MATLAB program gives a good estimate
of the FoS when compared with SLOPE/W and some other problem evaluated with
different programs for a homogeneous soil material. Two different failure mechanisms
are shown (i.e., toe and base failures). It is noted that the maximum percentage error
in the M-P FoS is 5%.
It is anticipated that with the availability of this computer code, Ghanaian Engineers
can more readily assess the safety of slopes in routine design works.
54
5.2 Recommendations
The current work could be extended in the future to include the following.
The provision for a more "user friendly" interface and development of the
programme into a stand-alone interactive program.
The direct extension of the model to cater for heterogeneous soil and rock
material
The direct extension of the model to include earthquake effect, surcharge
loading and increase in pore water pressures.
It is hoped that at the end of these improvements, this work that has started during this
research study, would be found in most geotechnical design offices in Ghana and other
less-developed countries, who cannot afford the existing expensive commercial
software available.
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60
APPENDICES
Appendix A: Structured Programme
Below is an algorithm developed to solve for the force and moment factor of safety’s
(Ff and Fm) using the M-P method.
1. By using the equation of a straight line, define the slope surface.
top = ∗ + (1)
2. By using the equation of a circle, define the slip surface.
bot = − √(abs(2 − ( − )2 (2)
3. By using the equation of a straight line, define the piezometric line.
w = ∗ + (3)
4. Divide the slope into n number of slices by vertical lines.
5. For each slice, the width, dx, bottom inclination, α, and average height, havg., are
determined.
dx ) (6)
6. The area of the slice, A, is computed by multiplying the width of the slice (dx)
by the average height, havg.
7. The weight, W, of the slice is computed by multiplying the area of the slice by
the total unit weight of soil:
8. W = γA (7)
9. If piezometric height is above slip surface, continue with step 9, else, skip to
step 17.
61
10. The piezometric height is determined at the downslope boundary, centre and
upslope boundary of each slice. The piezometric height at the downslope and
upslope boundaries of the slice, hb and ht, respectively, are used to compute the
forces from water pressures on the sides of the slice. Here, a triangular
hydrostatic distribution of pressures is assumed on the sides of the slice. The
piezometric height at the centre of the slice, hp, represents the pressure head for
pore water pressures at the base of the slice.
11. Hydrostatic forces from water pressures on the sides of the slice are computed
from,
2 (9)
12. The pore water pressure is computed by multiplying the piezometric head at the
centre of the base of the slice by the unit weight of water: µ = w hp
13. The length of the bottom of the slice, l, is determined; the length can be
computed from the width, dx, and base inclination,
=
cosα (10)
14. A trial FoS (F) is assumed and a λ ranging from 0 to 1 set.
15. Beginning with the first slice at the toe;
= [sin[]−
tan[]cos[]

(11)
16. The inter-slice shear force, X, for each slice is calculated from equation….
= λf() (12)
62
Where λ is the percentage (in decimal form) of the function used and f(x) is
inter-slice force function representing the relative direction of the resultant
inter-slice force given by
− ) ∗ ] (13)
17. The normal force, N, at the base of each slice is calculated from,
= −(X)−[
∗∗sin[]−∗∗sin[]∗tan[]
]

(14)
18. The new factor of safety’s, Ff and Fm , are calculated from
Ff= ∑ ∗∗cos[]+(−μ)∗tan[]∗cos[]
∗sin[] (15)
= ∑ ∗∗∗∗cos[]−μl)∗∗tan[]
∗ (16)
19. Iterate for a number of slip surfaces and determine the slip surface with the least
FoS.
63
Appendix B: MATLAB Code for Solving FoS
%%%this code calculates the factor of safety of a homogeneous soil mass
%%%using the method of slices and the M-P interslice force function
Clear all; close all; clearvars; clc;
%% Slope and soil parameters
H = 5; %inpu('Enter the height of the slope'); % height of
slope
Xc =1; %inpu('Enter the x coordinate of center of slip circle: Xc should be
>0');
% x coordinate of center of circle
Yc = 7; %input ('Enter the y coordinate of center of slip circle: Yc should
be greater than H'); % y coordinate of center of circle
R = 8; %:5: Yc+10 % radius of center of slip circle
fr = 45; % slope angle in degrees
ns = 31; % number of slice faces
G = 20; % unit weigh of soil
c = 40; % cohesion of soil
phi = 20; % frictional angle of soil (degrees)
tph = tand(phi); % tangent of the frictional angle of soil
tfr = tand(fr); % slope angle in gradient
%% Slip surface generation
xmin = Xc - sqrt(abs(R^2 - Yc^2)) % Exit point of slip circle
xmax = Xc + sqrt(abs(R^2 - (Yc-H)^2)) % Entry point of slip circle
x = linspace (xmin,xmax,ns)'; % Positions where the forces
will be analyzed.
circle
ytop = (x>=0).*(x<(H/tfr)).*(x*tfr) + (x>=(H/tfr)).*H; % Slope surface.
equation of a line is used
figure(1)
hold off
%% Other parameters
havg = (hs(1:end-1)+hs(2:end))/2; % 5) Average height of each slice
xavg = abs(((x(1:end-1)+x(2:end))/2)-Xc); % midpoint distance of each
slice.
yb = (diff(ybot));
is clear.
sal = sind(alpha);
cal = cosd(alpha);
s= R*sal; % offset f for non-circular slip surfaces
l = dx. /cal; % 12) Length of the bottom of the slice, assuming
straight border.
u = 0;
downO = sum(W.*sal);
FSom = upO. /downO
OFSf = FSom;
X=E.*lbd.*f;
Nbu = ((W-X) - ((c.*l.*sal) - (u.*l.*tph.*sal)). /OFSm);
Nbd = cal + (tph.*sal). /OFSm;
Nob = Nbu. /Nbd;
upb = sum((c.*l*R) + (Nob.*tph*R) - (u.*l*tph*R));
downb = sum(W.*xavg);
FSm = upb. /downb;
FSmm = ((FSm)) %./(ns-1));
Nju = ((W-X) - ((c.*l.*sal) - (u.*l.*tph.*sal)). /OFSf);
Njd = cal + (tph.*sal). /OFSf;
Noj = Nju. /Njd;
upj = sum( (c.*l.*cal) + (tph*cal.*Noj) - (u.*l.*tph.*cal) );
downJ = sum(Noj.*sal);
FSf = (upj. /downJ);
FSf = (FSf) %./(ns-1));
end
E = (((c.*l-u.*l.*tph).*cal). /OFSm) + (sal.*Nob)-((tph.*cal).*Nob./OFSm);
E = (((c.*l-u.*l.*tph).*cal). /OFSm) + (sal.*Noj)-((tph.*cal).*Noj./OFSm);
OFSm = FSom;
OFSf = FSom;
xlabel('lambda')
ylabel('FoS')
text (xi,yi,strValues,'VerticalAlignment','bottom');