Probabilistic methods of slope stability analysis. The Case of Wozeka-Gidole Cut Slope

TABLE OF CONTENTS

PROBABILISTIC METHODS OF SLOPE STABILITY ANALYSIS

ABSTRACT

DEDICATION

ACKNOLEDGEMENTS

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF NOTATION

1. INTRODUCTION
1.1. Background of the study
1.2. Description of the slope
1.3. Geology and soils
1.4. Common features of the failure of the cut slope
1.5. Significance implication of the project
1.6. Objective of the research

2. LITERATURE REVIEW
2.1. Introduction
2.1.1. Cut slopes
2.1.2. Mode of failure
2.2. Stability conditions for analysis
2.2.1. End of construction stability
2.2.2. Long term stability
2.3. Methods of Stability analysis
2.3.1. Limiting equilibrium
2.3.2. A General Limit Equilibrium (GLE)
2.3.3. Three Dimensional Analysis
2.3.4. Stress and Deformation Analysis
2.3.5. Finite element and Finite difference method
2.4. Probabilistic Slope Stability Analysis
2.4.1. Uncertainties in Slope Stability Analysis
2.4.2. Types of uncertainties
2.4.3. Coefficient of variation and factor of safety
2.4.4. Probability density function
2.4.5. Cumulative distribution function
2.5.1. Reliability Index, Probability of failure and Reliability
2.5.2. Suggested target values of reliability index and failure probability for slopes
2.5. Slope stabilization techniques
2.5.1. Evaluation of long-term remedial measures

3. MATERIALS AND METHODS
3.1. General
3.2. Data availability
3.2.1. Field data
3.2.2. Ground water
3.2.3. Laboratory tests of the soil
3.2.4. Modulus of elasticity of the soil
3.3. Modeling methodology
3.4. Data analysis
3.5. Limitation and potential problem of the research

4. RESULTS AND DISCUSSIONS
4.1. General
4.2. Results
4.2.1. Laboratory test results
4.2.2. Output of FLAC3D for case-1
4.2.2. Output of FLAC3D for case-2
4.2.3. Output of FLAC3D for case-3
4.3. Probability of failure
4.3.1. Probability of failure based on normal distribution
4.3.1. Probability of failure based on lognormal distribution
4.4. Discussion

5. CONCLUSIONS AND RECOMMENDATIONS
5.1. Conclusions
5.2. Recommendations for further research

REFERENCES

Appendix A

FLAC3D output for the scenarios

Appendix B

Index properties of the soil profile of the slope

Appendix C

Strength parameter of the soil

Appendix D : Standard normal distribution Table

Appendix E

Some of pictures of the research

ABSTRACT

Probabilistic methods of slope stability analysis accompanying conventional analyses provide the means for the reliability and probability of failure of the slope in a quantifying way and provide us an important knowledge for stability condition of the slope. The essential requirement of the soil parameters is determined in the laboratory for the analysis of the slope meeting the insitu conditions. For the location of the phreatic surface three different scenarios were considered for the profile of the water surface to carry out the analysis. Effective strength parameters were considered to assess the long term stability condition.

The spatial variability of the random variables was incorporated into this model and by considering the contributions of the end resistance of 3-D probability of failure. The 3-D aspects of failure were investigated using FLAC3D software incorporating all the parameters of the soil to visualize the failure probability for the scenarios. Both normal and lognormal distribution of FOS were considered in order to compare and contrast the reliability and probabilities of failure in a measureable ways using NORMDIST and NORMSDIST program which is integrated in MS-EXCEL and from the probability density function table. The analysis gave that Probability of failure for normal distribution is higher than that of probability of failure for lognormal distribution. It was also attempted to investigate the best combination of the slope parameters to have the desirable target probability of failure for the scenarios and recommendation were made to provide geosynthetic reinforcement for stabilization and construct appropriate retaining wall with proper drainage as well as provision of sheet pile to achieve the desirable postulated probabilities of failure.

Keywords: Slope stability, Uncertainty, FLAC3D analysis, Factor of safety, Reliability

DEDICATION

To mulubek

ACKNOLEDGEMENTS

I would like to thank all those people who made this thesis possible and an enjoyable experience for me to face challenges and hardship to the great extent as well as make me to be forbearing with the help of the almighty Allah.

I would like to express my deepest sense of gratitude to my advisors Dr.Yoseph Birru Who invigorate me to do my research on slope stability as well as he introduced me the application of robust FLAC3D to model the slope and Dr.R.K.Verma for their patient guidance, support, encouragement, and excellent advice throughout my research project

I take this opportunity to express my profound gratitude to my beloved parents, my sisters and my brothers that have always supported, loved and believed in me throughout every step of my life. Without my family’s encouragement, I would not pursue the research in such a way.

Finally my appreciation and thanks goes to all my beloved friends who have helped me in any form of support and all the lab assistants of Arba Minch University for their devotion to work at the weekend and at the night that are greatly needed for the advancement and accomplishment of this work.

LIST OF FIGURES

Figure 1-1.1: Location of the project site

Figure 1-2: Schematic profile of the slope before failure

Figure 1-3: Failure of the slope (Author’s own work,2012)

Figure 2-1: Aspect ratio of failure mass (Abramson et al., 2002)

Figure 2-2: Typical retrogressive slide (Abramson et al., 2002)

Figure 2-3: 3D column assumption (F.Nadim, 2006)

Figure 2-4: Comparison of two situations with different factors of safety and uncertainty. (F.Nadim, 2006)

Figure 2-5: PDF for Normal and Lognormal

Figure 2-6: A continuous random variable X showing PDF and CDF

Figure 3-1: Cases of water surface profile.

Figure 4-1: water surface profile for different cases.

Figure 4-2: mohr circle for layer 1

Figure 4-3:mode of failure for layer 1

Figure 4-4: direct shear result for layer 2

Figure 4-5: mohr circle for layer 3

Figure 4-6: mode of failure for layer 3

Figure 4-7: direct shear result for layer 4

Figure 4-8: FLAC3D output for case-1

Figure 4-9: FLAC3D output for case-2

Figure 4-10: FLAC3D output for case-3

Figure 4-11: Normal PDF and CDF for the scenarios

LIST OF TABLES

Table 2-1: Geologic Factors Controlling Shape of Potential Failure Surface (Abramson et al., 2002)

Table 2-2: Static equilibrium conditions satisfied by Limit equilibrium methods (Abramson et al., 2002)

Table 2-3: 3D assumption by various methods(Zuyu Chen,2003)

Table 2-4 : coefficient of variation for the soil parameter (Abramson et al., 2002)

Table 2-5: PDF models for different soil parameter (Robin Chowdhury, 2010)

Table 2-6: Suggested target β and for built-up and excavated slopes. (Abramson et al., 2002)

Table 3-1: Index properties of the soil profile.

Table 3-2: selection of strength tests (Abramson et al., 2002)

Table 3-3: Estimated Times to Failures in Triaxial Tests (K.H.Head, 1994)

Table 3-4: Strength parameter of the slope profile

Table 3-5: water surface profile for different cases.

Table 4-1: FOS of the slope for different scenarios

Table 4-2: PF based on Normal distribution

Table 4-3:PF based on lognormal distribution

LIST OF NOTATION

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1. INTRODUCTION

1.1. Background of the study

The Wozeka-Gidole road project is situated in the south west of Arba Minch town The road project crosses a sharp faulted escarpment of Gidole Mountain. The project starts at about 539 km far from Addis Ababa and ends at 573km and the profile of the slope is taken at 5039’00.97’’N and 37022'03.29''E. At the elevation of 2081m amsl and it is part of the Arba Minch-Jinka surface treatment road project which is located in the Southern Nations, Nationalities and Peoples Regional State (SNNPRS) in Arba Minch Zuria and Derashe Woreda . Figure 1.1 shows the location of the project.

According to explanatory note to geological map of Ethiopia (1:2,000,000), the southern segment comprises the Abaya-Chamo rift, the Amaro horst, the Gelana garben and several other features. The Abaya-Chamo is filled with young lacustrine sediments and represents the youngest part of the Ethiopian rift valley. From the preliminary land slide investigation report of (ERA, Preliminary Landslide investigation report, 2010) construction of Wozeka-Gidole road project is part of the Arbaminch-Jinka Surface treatment road project. The first phase of which was constructed by SUR Construction plc. Then the responsibility of taking construction of the current Wozeka-Gidole road project was transferred to own force of ERA following change of the alignment to cross through Gidole town.

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Figure 1-1.1: Location of the project site

1.2. Description of the slope

Wozeka-Gidole road project has a total length of 33.9km and the pavement is Double Bituminous Surface Treatment (DBST) standard. The carriageway is 6.7m and 1.5m shoulder width in both sides in flat to rolling terrain, but only 0.5m shoulder width in both sides in mountainous terrain and also the cut slope of the area is made in successive steps and the whole cut slope has six benches. (ERA, Preliminary Landslide investigation report, 2010)

The geometry of the slope is illustrated in Figure 1.2 which shows a 38 m high slope with five 6m high and one 8m high benches. The individual bench faces are inclined at 45° to the horizontal and the overall slope angle is 34.16o to the horizontal for data input of the software as well as ease of analysis of the problem; moreover there is flowing of stream at the toe of the slope following the excavation. According to the preliminary study of ERA (2009) the slide occurred in the area following of a heavy rainfall in the area.

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Figure 1-2: Schematic profile of the slope before failure

The geometry of the slopes has been modified and trimmed a number of times prior the final geometry presented in Figure 1.2. Moreover due to existence of GW at the foot of the slope it is presented the geometry and assumed water table at the slope, prior to the slide.

From the failures of the slope during construction it indicated that, may be the slide has a short-term stability problem, characterized by the reduction in shear strength due to the induced pore water pressure due to ground water.

From the profile of the soil as shown in Figure 1.2 the soil was determined to be tropical residual soil. The slopes were remarkably consisting of soils of different gradations.

The major factors which initiates landslides were identified and communicated to the resident supervisory staff of TCDSCO & construction crew of ERA own force. It is realized that most of the landslides driving forces can‘t be solved in short terms by the resident supervisory staff and the construction crew as involvement of the local administration and beneficiary community is demanded. However a couple of factors such as distributing the waste material evenly and farther from the route corridor & provision of proper drainage mechanism are believed to be solved at site level. (ERA, Preliminary Landslide investigation report, 2010)

1.3. Geology and soils

Colluviums derived from trachyte-rhyolite rocks are the dominant formations encountered along the route stretch. Highly jointed and fractured dark basalt has formed the Gidole ridge which overlies the completely weathered trachyte-rhyolite.

The cut sections portray layers of reddish brown clayey silty gravel between which is intercalated grayish to very dark grayish clay showing shale like structure.

From observation at the cut slope there were Basalt Boulders having a diameter 3-5 meter are not uncommon along the road cut. The majority of slides are initiated from the intercalated clay. The clay is hard to break between thumbs. However, it disperses easily when in contact with water. In addition to the intercalated continuous thin layers of grayish to very dark grayish clay (black cotton), there are pockets of black cotton soil with in reddish brown clayey silty gravel layers. This may be due to chemical as well as mechanical weathering of the basalt boulders embedded within those layers.

1.4. Common features of the failure of the cut slope

Following the cutting of the slope of the Gidole Mountain in 2009, most instability problems in a slope arise from removal of the material which has supported the upper landmass. As it can be seen in Figure 1.3 there is a relatively deep road cut around the specified stations. It was tried to provide bench to insure stability of the slope despite the unavoidable failure due to combined causes of sliding. Here, it is tried to show how road cut causes instability of slopes combined with other factors such as digging trenches in such delicate areas. (ERA, Preliminary Landslide investigation report, 2010)

The landslide is frequently occurring during wet seasons especially from October to December 2009. The general observation made along and in the vicinity of the road sections shows, slip surfaces are generally shallow. Most of the slip surfaces are associated with the black cotton soil which is inclined towards the RHS of the road to the way to Gidole. The slope stability, especially at 6km from Wozeka town to the way to Gidole town is very critical for the completion or termination of the work due to sever sliding as shown in Figure 1-3. (ERA, Preliminary Landslide investigation report, 2010)

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Figure 1-3: Failure of the slope (Author’s own work,2012)

1.5. Significance implication of the project

The significance and main advantage of this study is immense since, the project route is valuable for transportation and for socioeconomic development of the people of Arba Minch and Gidole as well as those villages around these neighboring towns; moreover the road plays a crucial role in encouraging tourism industry to the great extent as it is found on the major tourist destination of the country.

The project completion time will be extended unreasonably which brings about extra cost or financial loss for the contractor as well as the employer if not resolved timely due to the slope failure. Therefore resolving this problem may have an advantage to the Gidole town, neighboring villages and at large to the country.

1.6. Objective of the research

Because the road has significant value to the individual and to the society at large, resolving the problem is paramount to the region for promoting the socio economic development of the society. Therefore this case study is an indispensable research to know the agent for the cause of failure and the appropriate remedial measurement for the slope failure.

Generally the objectives of this case study are:

- Perform literature review to study the theoretical background of the most widely used slope stability analysis methods.
- Laboratory determination of the soil parameter for the long term analysis of the slope profile.
- Analysis of the slope using 3D model
- Determine the reliability and probability of failure of the slope.
- Searching of appropriate slope stabilization techniques for the slope.
- Give recommendations for further study.

2. LITERATURE REVIEW

2.1. Introduction

2.1.1. Cut slopes

Shallow and deep cuts are important features in railway formations, highway embankments, canal banks, and at many civil engineering projects. A slope failure occurs when the forces causing the failure are greater than the shear strength. Flat cut slopes, which may be stable for an indefinite period of time, are often uneconomical and impractical. On the other hand Slopes that are too steep may remain stable only for a short period of time. So that the failure may pose a danger to life and property at a later date. It could also tremendous inconvenience and the expense of repairs, maintenance, and stabilization measures.

The aim of slope design is to determine a height and inclination that is economical and that will remain stable for a reasonable life span. The design is influenced by the purposes of the cut, geological conditions, in situ material properties, seepage pressures, construction methods, and the potential occurrence of natural phenomena such as heavy precipitation, flooding, erosion, freezing, and earthquakes. Steep cuts often are necessary because of the right-of-way and property line constraints. The design must consider measures that will prevent immediate and sudden failure as well as protect the slope over long term, unless the slope is cut for temporary reasons only. In some situations, cut stability at the end of construction may be a critical design consideration. Conversely, cut slopes, although stable in the short term, can fail many years later without much warning. (Abramson et al., 2002)

Because of the great importance of the consequences of failure (threat to human life, safety as well as threat of economic loss), specific attention must be given to the particular location and function of a slope. For example, an allowable FOS of, say, 1.1 or 1.2 may be considered reasonable for a natural slope, the failure of which is considered unlikely to have adverse consequences. On the other hand, a higher FOS (say 1.5 to 2) may be required for an engineered slope adjacent to a railway line so that deformations are below the level that might cause derailment. (Duncan&Wright, 2005)

2.1.2. Mode of failure

Terzaghi and peck (1967) state, Slides may occur in almost every conceivable manner, slowly or suddenly, and with or without any apparent provocation. These slope failures are usually due to a sudden or gradual loss of strength by the soil or to a change in geometric conditions, for example, the steeping of an existing slope. The types of failure that can be expected to occur in soil slopes take the form of either

- Translational: where there is translational movement of the slope.
- Plane or wedge surface: the failure forms wedge shaped block.
- Circular: the failure is in the form of rotational forms.
- Non circular or a combination of these types: it is a combination of the two or more forms.

The aspect ratio used to differentiate between the translational and rotational surfaces as shown in Figure 2-1

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, has been left to account for the case of a combined rotational and translational failure. (Abramson et al., 2002)

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Figure 2-1: Aspect ratio of failure mass (Abramson et al., 2002)

The planar failure surfaces are usually expected in slopes where a soil layer, or relict jointing, with a relatively low strength strongly influences the shape of the failure surface. The translational type of failure occurs in shallow soils overlying relatively stronger materials and circular failure surface occur in slopes consisting of homogeneous materials. As most soils are generally heterogeneous, non-circular surfaces, consisting of a combination of planar and curved surfaces are most likely. Often retrogressive failures consisting of multiple curved surfaces can occur in layered soils as shown in Figure 2-2. Such failures are typical where the first slip tends to over steepen the slope, which then leads to additional failures. (Abramson et al., 2002) In addition to that the geological factors which influence the shape of the slip surface is given in Table 2-1.

Generally the main items required to evaluate the stability of a slope are the following.

- Shear strength of soils: This is strength parameter of the soil in terms of cohesion and internal friction angle.
- Slope geometry: the height and slope angle of the slope.
- Pore pressures of seepage forces: consideration of the pore water pressure.
- Loading and environmental conditions: influence of environment to the slope.

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Figure 2-2: Typical retrogressive slide (Abramson et al., 2002)

Table 2-1: Geologic Factors Controlling Shape of Potential Failure Surface (Abramson et al., 2002)

2.2. Stability conditions for analysis

Variations of the loads acting on slopes and variations of shear strengths with time, result in changes in the factors of safety of slopes as well as stability condition of the slope. As a consequence, it is often necessary to perform stability analyses corresponding to several different conditions, reflecting different stages in the life of a slope. The stability analysis that we should address are the following.

2.2.1. End of construction stability

Slope stability during and at the end of construction is analyzed using either drained or undrained strengths, depending on the permeability of the soil. Many fine-grained soils are sufficiently impermeable that little drainage occurs during construction. This is particularly true for clays. For these fine-grained soils, undrained shear strengths are used, and the shear strength is characterized using total stresses. For soils that drain freely, drained strengths are used; shear strengths are expressed in terms of effective stresses, and pore water pressures are defined based on either water table information or an appropriate seepage analysis. (Duncan&Wright, 2005)

2.2.2. Long term stability

Over time after construction the soil in slopes may either swell (with increase in water content) or consolidate (with decrease in water content). Long-term stability analyses are performed to reflect the conditions after these changes have occurred. Shear strengths are expressed in terms of effective stresses and the pore water pressures are estimated from the most adverse groundwater and seepage conditions anticipated during the life of the slope. Seepage analyses can be performed using either graphical techniques (flow nets) or numerical analyses (finite element, finite difference), depending on the complexity of the cross section. (Duncan&Wright, 2005)

Long term slope stability is also dependent on seepage forces and, therefore, on the ultimate groundwater level in the slope. After excavation, the free-water surface will usually drop slowly to a stable zone at a variable depth below the new cut surface. This drawdown usually occurs rapidly in cut slopes made in sand but is usually much slower in clay cut slopes. Although typical rates and shapes of groundwater drawdown curves have been proposed for cut slopes, none has proved useful for correctly predicting the time or rate of drawdown of preconsolidated clays. The main obstacle to such prediction comes from the difficulty in correctly modeling the recharge of the area in the vicinity of the cut slope. (Abramson et al., 2002)

2.3. Methods of Stability analysis

Slope stability analysis using computers is an easy task for engineers when the slope configuration and the soil parameters are known. However, the selection of the slope stability analysis method is not an easy task and effort should be made to collect the field conditions and the failure observations in order to understand the failure mechanism, which determines the slope stability method that should be used in the analysis. Therefore, the theoretical background of each slope stability method should be investigated in order to properly analyze the slope failure and assess the reliability of the analysis results.

Two dimensional slope stability methods are the most commonly used methods among engineers due to their simplicity. However, these methods are based on simplifying assumptions to reduce the three-dimensional problem to a two dimensional problem and therefore the accuracy of the analysis results vary between the different analysis methods. (Abramson et al., 2002)

The following methods of analysis have been applied to the solution of slope stability problems

- Limiting equilibrium
- Stress deformation analysis

These are discussed below

2.3.1. Limiting equilibrium

Limiting equilibrium techniques have been developed over many years and are used extensively in slope stability analyses. The most commonly used methods have been developed by Bishop (1995), Janbu (1973), Morgenstern and Price (1955) and Spencer (1967). These methods all make the assumption that the mass of material above a given slip surface is in static equilibrium under gravitational and applied loadings. A factor of safety FOS is defined as the ratio of the material strength at failure (τf) to the mobilized shear stress (τ)

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Since the shear stresses will be balanced by the shear strength ratio a state of limiting equilibrium is said to exist. The three equilibrium equations horizontal, vertical and moment are formulated in terms of the loads. As the slip surface is usually curved the free body above the surface is divided into a number of vertical slices. Assumptions regarding the action of forces within the free body are required to make the problem statically determinate, and the way in which these assumptions are dealt with constitutes the essential difference between the various methods. On formulation the FOS cannot be defined uniquely as it is intrinsic to the normal stress, and therefore the solution procedure is iterative. For some methods not all the equilibrium equations are satisfied, for example Bishop's modified method only satisfies overall moment and vertical equilibrium for circular failure surfaces. Sarma (1973) applied a critical horizontal acceleration to the sliding mass, and the factor of safety is calculated for zero critical acceleration. For failure geometries comprising of two distinct failure planes analysis may be conducted by considering an upper active wedge and the lower passive wedge. Coulthard (1979) developed a two wedge approach based on limiting equilibrium. Dunbavan (1983) proposed a method using the principle of virtual work.

Duncan and Wright (1980) investigated the accuracy of the various methods and Concluded for most practical problems that those which satisfied all equilibrium equations were 'correct' to within ±5%. Also Bishop's modified method was found to be as accurate for circular failure surfaces even though all the equilibrium equations are not satisfied. As all these methods are formulated for static equilibrium they do not account for the stress-strain behavior of the sliding mass. This implies that the same value of shear strength is mobilized over the slip surface independent of deformations.

The static equilibrium conditions satisfied by limit equilibrium methods are shown in Table 2-2.

Table 2-2: Static equilibrium conditions satisfied by Limit equilibrium methods (Abramson et al., 2002)

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2.3.2. A General Limit Equilibrium (GLE)

A general limit equilibrium(GLE) formulation (Fredlund et al. 1981) can be developed to encompass most of the assumption used by various methods and may be used to analyze circular and noncircular failure surfaces. In view of this universal applicability, the GLE formulation has become one of the most popular methods as its generalization offers the ability to model a discrete version of the Morgenstern and Price (1965) procedure via the function used to describe the distribution of the interslice force angles. The method can be used to satisfy both force equilibrium and moment equilibrium, or if required, just the force equilibrium conditions. With this approach, Spencer’s (1973) method can be implemented directly via the use of a constant interslice force functions.

2.3.3. Three Dimensional Analysis

Idealization of real three-dimensional (3D) slope problems as two dimensional (2D) problems is often valid; However, in some cases three-dimensional effects may be significant and, therefore, 3-D analyses are justified. The three dimensional analytical approach has its root in the 1970’s, with papers by (Baligh, 1975) and (Hovland, 1977), Hovland modified the familiar two dimensional methods to the third dimension by substituting column for the more familiar slices.

The general approach used in analyzing slopes for design is to determine the slip surface geometry with the lowest FOS. Methods using dynamic programming techniques, (Li, 1987) , or random search techniques. Several authors have applied a variational calculus approach by expressing the equilibrium equations as a set of integral functional. The solution is obtained by finding the failure geometry which produces an absolute-minimum for the FOS.

In three dimensional analysis assumptions made for the internal shear forces are column. As with other 3D limit equilibrium a method, the failure mass is divided into a number of columns with vertical interfaces (Fig. 2.3). Various literatures on 3D limit equilibrium is given in Table 2-3.

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Figure 2-3: 3D column assumption (F.Nadim, 2006)

Table 2-3: 3D assumption by various methods(Zuyu Chen,2003)

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2.3.4. Stress and Deformation Analysis

In all limiting equilibrium methods assumptions are required to calculate the stress distribution on a particular slip surface; no constitutive law is invoked to determine the deformation within the slope or on the slip surface. A comparison between an excavated and a man-made slope would have identical stress conditions on the failure surface for similar profiles and c-ɸ parameters. The stress distribution and associated deformations within the body of the slope are not considered in these solutions. These shortcomings of limit equilibrium methods have led to an interest in stress deformation analysis in order to further knowledge of model behavior and gain an insight of the stress distribution. Areas of stress concentration within the slope are of particular interest as these may develop into localized failure zones which can be examined in relation to progressive failure due to the strain softening nature of geologic materials. (Robin Chowdhury, 2010)

The relatively complex geometry of a slope is not readily amenable to straight forward closed form analytical solutions even for the case of linear elasticity. Initial studies were conducted using photo elastic models to investigate the normal stress distribution within a slope and these were later verified by linear elastic numerical techniques. On comparison the normal stress distribution from limit equilibrium methods predicts higher stresses at the toe and lower stresses at the crest.

With the evolution of sophisticated numerical techniques based on finite elements, boundary elements and finite difference methods there are a number of very advanced computer codes readily available. These are capable of modeling nonlinear constitutive behavior, plasticity, anisotropy, heterogeneity and shear along discontinuities. Even with the relative cheapness and availability of computer power, these programs require an investment of man-hours to input the slope information and do the calculations. A much more serious drawback to their general acceptance is that the quality of the input data rarely matches the sophistication of the computer model. For an elastoplastic material a yield function needs to be defined and if the material is layered, as is common in geological problems, these parameters must be specified for each layer. The solutions may be very sensitive to input parameters and boundary conditions, which cannot be easily verified in the field or by tests. It is perhaps the paucity or quality of the input data rather than the capabilities of the mathematical models, which have held back the development of these techniques into everyday usage, and at present they are only used in a research environment. (Robin Chowdhury, 2010)

2.3.5. Finite element and Finite difference method

In the classical limit equilibrium and limit analysis methods, the progressive failure phenomenon cannot be estimated except for the method by Pan. (George, 1991) Propose to use the finite element method to overcome some of the basic limitations in the traditional methods of analysis. At present, there are two major applications of the finite element and finite difference in slope stability analysis.

The first approach is to perform an elastic (or elasto-plastic) stress analysis by applying the body force (weight) due to soil to the slope system. Once the stresses are determined, the local factors of safety can be determined easily from the stresses and the Mohr–Coulomb criterion. The global factor of safety can also be defined in a similar way by determining the ultimate shear force and the actual driving force along the failure surface.

The second finite element and finite difference slope stability approach is the strength reduction method (SRM). In the SRM, the gravity load vector for a material with unit weight is determined from

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Where {f} is the equivalent body force vector and [N] is the shape factor matrix. The constitutive model adopted in the non-linear element is usually the Mohr–Coulomb criterion, but other constitutive models are also possible, though seldom adopted in practice. The material parameters c′ and φ′ are reduced according to

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The factor of safety FOS keeps on changing until the ultimate state of the system is attained, and the corresponding factor of safety will be the factor of safety of the slope. The termination criterion is usually based on one of the following:

- The non-linear equation solver cannot achieve convergence after a pre-set maximum number of iteration;
- There is a sudden increase in the rate of change of displacement in the system;
- A failure mechanism has developed.

The location of the critical failure surface is usually determined from the contour of the maximum shear strain or the maximum shear strain rate.

The main advantages of the SRM are as follows:

- The critical failure surface is found automatically from the localized shear strain arising from the application of gravity loads and the reduction of shear strength;
- It requires no assumption on the inter-slice shear force distribution;
- It is applicable to many complex conditions and can give information such as stresses, movements and pore pressures which are not possible with the LEM.

2.4. Probabilistic Slope Stability Analysis

Natural and man-made slopes are expressed by the factor of safety. The factor of safety is defined as the ratio of the characteristic resisting force to the characteristic load (driving force). The conventional approach does not address the uncertainty in load and resistance in a consistent manner. The ambiguous definition of “characteristic” values allows the engineer to implicitly account for uncertainties by choosing conservative values of load (high) and resistance parameters (low). The choice, however, is somewhat arbitrary. Slopes with nominally the same factor of safety could have significantly different safety margins because of the uncertainties and how they are dealt with. Duncan (2000) pointed out that “Through regulation or tradition, the same value of safety factor is often applied to conditions that involve widely varying degrees of uncertainty. This is not logical.”

As shown in Figure 2-4 a low safety factor does not necessarily correspond to a high probability of failure and vice versa. The relationship between the factor of safety and probability of failure depends on the uncertainties in load and resistance. (F.Nadim, 2006)

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Figure 2-4: Comparison of two situations with different factors of safety and uncertainty. (F.Nadim, 2006)

Often the stability situation for slopes where either sliding or large deformations have occurred, and back analyses have been performed to establish design shear strengths, lower factors of safety may be used. In such cases probabilistic analyses may be useful in supporting the use of lower factors of safety for design. Lower factors of safety may also be justified when the consequences of failure are small.

Probability theory and reliability analyses provide a rational framework for dealing with uncertainties and decision making under uncertainty. Depending on the level of sophistication, the analyses provide one or more of the following outputs: (F.Nadim, 2006)

- Reliability index
- Probability of failure (or probability of unsatisfactory performance)
- The most probable combination of parameters leading to failure
- Sensitivity of result to any change in parameters

2.4.1. Uncertainties in Slope Stability Analysis

Neglecting uncertainties in slope analysis is an important limitation of the conventional deterministic approach. In consequence, the conventional ‘factor of safety’ is often not a reliable indicator of slope performance. A probabilistic approach, on the other hand, allows for the systematic analysis of uncertainties and for their inclusion in evaluating slope performance. Important geotechnical parameters such as shear strength parameters and pore water pressures may be regarded as random variables, each with a probability distribution, rather than deterministic values or constants. Consequently, the factor of safety FOS of a slope under specified conditions must also be regarded as a random variable with a probability distribution in terms of ‘Reliability index ‘and ‘probability of failure’ or ‘the probability of inadequate performance’ was first introduced as performance indicators within a probabilistic framework.

In probabilistic analyses the uncertainties associated with the problem are included within the framework of the analysis; therefore the risks associated with the consequences of various strategies can be evaluated in terms of selected design criteria. The word risk replies both the consequence and possibility of an event.

For natural slopes as well as for built-up or engineered slopes such as embankments, excavations and dams, the uncertainties in geotechnical properties are usually the most important. Such uncertainties have both spatial and systematic components. A third source of uncertainty is model error. (Robin Chowdhury, 2010)

The current state of practice relies on using FOS values in the design process to account for uncertainties associated with soil parameters, site stratigraphy, ignorance, and the potential consequences if the slope fails. Under ideal conditions, a FOS of at least 1 should ensure safe design, but because of all uncertainties a higher value of the FOS is desirable for design recommendations.

(Duncan&Wright, 2005) mentions typical minimum values: FOS = 1.5 (long-term), FOS = 1.3 (end-of construction and multi-stage loading) and FOS = 1 to 1.2 (rapid draw-down). Consideration has, of course, to be given to the level of uncertainties and to the consequences of a potential failure.

Unfortunately, the use of such a single value of the design FOS does not effectively relate to the quality of the site characterizations. In other words, if a site is well characterized by a comprehensive geotechnical site investigations, with extensive in situ tests, and high quality laboratory testing, then guidelines should permit a lower FOS for design, because a considerable amount of uncertainty has been eliminated from the analyses .However, this is not the case currently, as the same minimum FOS often is required for all cases covering different levels of uncertainty associated with the characterizations. Similarity, two slopes with the same FOS may have considerably different levels of safety depending on how well the site has been characterized with a view to minimizing uncertainties. To account for such levels of uncertainty, the probabilistic method should be considered for assessing the performance of the slopes.

probabilistic method of analysis can take into account uncertainties associated with the site stratigraphy, soil parameters, and even the method of analysis; of course the final result here will be the probability of failure or alternatively, the probability of unsatisfactory performance. (Robin Chowdhury, 2010)

2.4.2. Types of uncertainties

The uncertainty must address the variance associated with data scatter, which is attributable to

- Spatial variability in the soil profile
- Random testing errors
-ystematic error, which is influenced by
- Statistical modeling of the mean due to the limited amount of sampling and testing in the laboratory or in the field
- Bias in measurement procedures due to features such as sampling methods and the extent of sample disturbance

Our knowledge is limited regarding the exact contribution of these two items of data scatter and systhematic errors to the overall uncertainty. But as more data are gathered within this framework, we hope to separate the influence of each category with more confidence. For most slope stability analyses, the critical parameters includes the following.

I. Shear strength in the form of c and ɸ parameters for various drainage conditions and rates of loading
II. Pore water pressures
III. Unit weight of soils
IV. Locations of failure surface
V. Thickness of a soft, low strength zone
VI. External environmental effects, such as drawdown and surface loads

2.4.3. Coefficient of variation and factor of safety

Slope failure occurs if the total sliding resistance along a potential slip surface is less than the driving force caused by the soil weight and other loads. Hence, in the simplest case, a safety factor can be defined as the ratio of the available resistance, to the applied load

Since the available resistance and available load are each subject to uncertainties, they should be modeled as random variables; thus, FOS, in turn, will also be a random variable.

Reliability and probability of failure can be determined easily once the factor of safety and the coefficient of variation (COV) of the factor of safety have been determined. The value of factor of safety is determined in the usual way, using a computer program, slope stability charts, or spreadsheet calculations. The value of COV can be evaluated using the Taylor series method, which involves these steps:

1) Estimate the standard deviations of the quantities involved in analyzing the stability of the slope: for example, the shear strengths of the soils, the unit weights of the soils, the piezometric levels, the water level outside the slope, and the loads on the slope.
2) Use the Taylor series numerical method (Wolff, 1994; U.S. Army Corps of Engineers, 1998) to estimate the standard deviation and the coefficient of variation of the factor of safety, using these formulas:

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Where ∆F1=(F1+-­­F1-)*F1+ is the factor of safety calculated with the value of the first parameter increased by one standard deviation from its most likely value, and F1- is the factor of safety calculated with the value of the first parameter decreased by one standard deviation.

In calculating F1+ and F1-, the values of all of the other variables are kept at their most likely values. The other values of ∆F2, ∆F3 . . . ∆, FN are calculated by varying the values of the other variables by plus and minus one standard deviation from their most likely values.

Moreover once the shape of the PDF has been selected, other parameters such as the mean and standard deviation will have to be defined for normal and lognormal distributions. In most instances, the standard deviation estimated from published values of the coefficient of variation, . Some values of COV for different soil parameters are given in Table 2-4.

Table 2-4 : coefficient of variation for the soil parameter (Abramson et al., 2002)

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With the availability of such published coefficients and an estimate of the mean value, µx the standard deviation can be calculated using

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If there are insufficient data and published values can’t be obtained, then one must resort to the rule of thumb suggested by Dai and Wang (1992). This rule is based on the premise that most data will be normally distributed, and so 99.73 percent of all values will lie within plus or minus three standard deviations from the mean value. So if one can select a possible range of values, which are symmetric about the mean, the standard deviation can be estimated using the following simple equation.

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Where

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Duncan (2000) believes that there is a tendency to underestimate the range of possible values. In view of this, it would be more realistic to select the standard deviation value on the basis of the equation

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2.4.4. Probability density function

A probability density function can be used to describe the probability distribution of a continuous random variable X. A random variable, which can take on any value, is called a continuous random variable. The probability that such a random variable takes on a specific value is zero. The probability that a continuous random variable, X, is less than a value, x, is given by the probability distribution function. [2.9]

The most common distribution for geotechnical engineers are:

- Normal distribution
- Lognormal distribution

Figure 2.5 shows PDF for normal and lognormal distribution

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Figure 2-5: PDF for Normal and Lognormal

In most situations, however, the choice of the PDF is based on engineering judgment instead of a histogram, because either the sample size of the observations is small, or the engineer believes that the values measured are not representative of the values of the pertinent variable. In a broad sense, the PDF may be used to express the overall feeling of the engineer on the basis of all the evidence that is available. The evidence may include results of various types of tests, geological history, geotechnical performance in similar soils, and the engineer’s intuition. The mean value of the PDF represents the engineer’s best estimate of the random variable without the addition of conservative assumptions, and the standard deviation, σ or COV of the PDF represents the engineer’s assessment of the uncertainty. A convenient probability distribution type (e.g. normal or lognormal) may be selected, and calibrated with those mean values and standard deviations that are consistent with the engineer’s judgment, to yield the judgmentally based PDF of the variable. If an engineer is only confident with the maximum and minimum values of a variable, a uniform distribution over the range may be used as a conservative PDF, whereas a triangular distribution can model approximately the engineer’s perception of the relative likelihood over the given range of values. (Vanmarcke, 1996)

Unfortunately, most geotechnical projects will rarely have sufficient data to define a PDF, so we must select PDFs for the component variables based on past experience with similar projects. (Abramson et al., 2002)

These statistical parameters of FOS are evaluated by an appropriate procedure by combining the respective statistical parameters of all random variables on which FOS, the performance function depends.

Following an extensive review of literature by Lumb (1996), Baechre et al. (1980) and Chowdhury (1984), there is some consensus about selecting the PDF models as shown in Table 2-5.

Table 2-5: PDF models for different soil parameter (Robin Chowdhury, 2010)

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In general, it appears that the normal distribution may be used for most parameters, unless there is sufficient evidence to suggest otherwise. The true normal distribution, which extends from -∞ to +∞, is truncated for most practical purposes here to include values with in ±3σ of the mean. The other significant feature of note is that the parameter tan ɸ is considered to be normally distributed rather than the angle ɸ. For parameters not listed in the above table, the normal probability distribution should be adopted for most cases. If there are concerns that too many unrealistic, erroneous negative random values may overly influence the reliability index, then the lognormal or triangular distribution should be selected as a model of random variable.

It is required to compute the probabilities based on the Normal distribution and the other based on lognormal distribution as in some cases, Normal distribution gave higher value and, in other cases, lognormal distribution gave higher values. Therefore, it is incorrect to generalize that one or the other assumption is conservative in relation to the other. (Duncan&Wright, 2005)

2.4.5. Cumulative distribution function

Cumulative distribution function is the integral of the probability density function between two points. The CDF is extremely useful as we obtain a measure of probability directly, whereas to obtain the probability measure from the PDF, the area under the PDF has to be calculated. For continuous function, the CDF is given as.

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So that the probability that X is between a and b is determined as the integral of f(x) from a to b. Figure 2-6 shows PDF and PDF for continuous variable x.

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Figure 2-6: A continuous random variable X showing PDF and CDF

2.5.1. Reliability Index, Probability of failure and Reliability

Analysis can be performed to calculate the two statistical parameters of the factor of safety, FOS: the mean (expectation) and the standard deviation.

A probability density function (PDF) may be introduced to model the relative likelihood of a random variable. The PDF describes the relative likelihood that the variable will have a certain value within the range of potential values. In a case where the engineer believes that a given set of measured data does not represent a set of realistic sample values of the engineering variable and no other information is available, a PDF can be fitted over the frequency diagram, which is a modified histogram whose ordinate has been scales, so that the area under the histogram is unity. For instance, a normal distribution is a common probability distribution model used to fit a symmetrical bell-shaped histogram. If the engineer adopts a normal distribution to model the undrained shear strength, the parameters of the normal distribution, namely µ and σ, can be estimated by the sample mean and sample standard deviation, respectively.

A performance indicator combining both the mean (expectation) and the standard deviation of FOS can now be introduced as a simple index of reliability or safety.

Reliability RI (often denoted by β ), a ratio of the mean of safety margin, (mean of FOS − 1) and the standard deviation of safety margin (S.D. of FOS), first defined by Cornell (1969) and expressed as in Equation (2.11). This is to complement a deterministic analysis by incorporating uncertainties associated with the performance of the geotechnical structure or facility to be analyzed. (Robin Chowdhury, 2010)

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Where the expected or mean value of FOS is denoted by µ and the standard deviation of FOS is denoted by . Thus, reliability index is the safety margin (FOS−1) standardized in terms of the Standard deviation of FOS.

The reliability index may also be written in terms of the mean of FOS and its coefficient of variation (COV), i.e., which is the ratio of standard deviation and the mean. Thus, standard deviation is the product of µ and COV.

The probability of failure is the probability that FOS is less than 1, expressed as follow

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Thus, reliability or probability of success is given by.

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These may be evaluated by integrating the probability density function (PDF) of FOS between the appropriate limits. Alternatively, it is represented by the area of the PDF curve between the appropriate limits. For a given probability distribution function (PDF) of FOS, the probability of failure is a unique function of the reliability index. Based on the assumption of a Normal or Gaussian distribution for FOS,

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Where Φ is the cumulative distribution function (CDF) of the standard Normal distribution. This is read from table of appendix D or generated from Excel. From such a table or Excel, the area of the PDF corresponding to any value of the Standardized variable (often called standard Normal variate) can be read.

There are several options for calculation of reliability and probability of failure

1. The CDF values of a standard normal distribution are tabulated in appendix D. From such a table, the probability value appropriate to the calculated reliability index may be found.
2. Duncan (2001) has provided a chart from which the probability of failure may be obtained directly given the mean and the coefficient of variation of the factor of safety FOS, assuming that FOS follows a Normal or Gaussian distribution. Earlier, he had provided a similar chart on the assumption that FOS follows a lognormal distribution (Duncan, 2000). Note that, in those charts, the mean or expected value of FOS is denoted as the most likely value (MLV) of FOS.
3. Use of software is another option and, for accurate calculations, this is the best option. For example, one may use the function NORMDIST or NORMSDIST in MS-EXCEL.
4. CDF is also a library function in the mathematical software packages such as MATLAB.

Thus qualitative assessments are made in the first instance, based on visual Observation and guided by past experience and professional judgment. The following five categories of failure likelihood (or probability) are proposed along with the annual probability of failure as shown in parenthesis: (Robin Chowdhury, 2010)

- Very High (>0.2),
- High (0.2 – 0.02),
- Medium (0.02 – 0.002),
- Low (0.002 – 0.0002), and
- Very Low (<0.0002)

Well established reliability methods, such as the first-order second-moment approximation (FOSM), the first-order reliability method (FORM) and Monte Carlo simulation are useful techniques for determining the reliability of the slope and for estimating the probability of failure. The Monte Carlo method has the advantage of being a very general technique and it has been developed by a number of authors for calculating the probability of failure PF (Robin Chowdhury, 2010)

2.5.2. Suggested target values of reliability index and failure probability for slopes

The term ‘target values’ is perhaps preferable to ‘acceptable values’ because of lack of sufficient experience and data. Thus the extent to which calculated values of reliability and failure probability are comparable to actual or observed values is very uncertain. For example, there are few slopes whose performance, at specific stages of their lives, has been compared to the results of associated probabilistic analyses. Moreover, data are even scarcer about the whole of life performance of slopes considered together with associated probabilistic studies.

Based on personal experience of probabilistic analysis and the limited information presented in the literature, suggested target values of reliability index and probability of failure are presented in Table 2-6 for built up and excavated slopes. A range of values corresponds to each range of slope height. With more experience and the publication of more case studies, it might be possible to consider embankment slopes differently from excavated slopes.

Table 2-6: Suggested target β and for built-up and excavated slopes. (Abramson et al., 2002)

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2.5. Slope stabilization techniques

Slope stabilization methods generally reduce driving forces, increase resisting forces, or both. Driving forces can be reduced by excavation of material from the appropriate part of the unstable ground and drainage of water to reduce the hydrostatic pressures acting on the unstable zone

- Drainage that increases the shear strength of the ground.
- Elimination of weak strata or other potential failure zones.
- Building of retaining structures or other supports provision of in situ reinforcement on the ground.
- Chemical treatment(hardening of soils) to increase shear strength of the ground among the various techniques of slope stabilization methods insitu reinforcements should be applied in order to stabilize the slope.

2.5.1. Evaluation of long-term remedial measures

While the short-term threat of instability was considered to be small, the longer-term Stability of the slope was considered to be unacceptable and a study was carried out to evaluate various options for stabilizing the slope. It was agreed that a factor of safety of 1.5 was required to meet long term requirements. The following alternatives are nessary to meet the long term stability of the slope:

- Reducing the height of the slope and the angle of the slope face.
- Drainage of the slope.
- Soil nailing
- Reinforcement of the slope with Geosynthetic reinforcement
- Retaining wall
- Sheet pile etc.

3. MATERIALS AND METHODS

3.1. General

Stability analysis will be done for the individual bench slope and for the overall slope angle of the slope for long term stability analysis of the slope. However; the overall slope is failed, so the analysis was done for the overall slope of the profile. The method applied for the analysis of the slope is using stress and deformation method by FLAC3D. The strength reduction method for determining factor of safety is implemented in FLAC3D through the SOLVE FOS command. This command implements an automatic search for factor of safety using the bracketing approach for the Mohr Coulomb model. In essence, the approach recognizes that field data (such as in-situ stresses, material properties and geological features). It is futile to expect the model to provide design data, such as expected displacements, when there is massive uncertainty in the input data.

The models may be simple, with assumed data that is consistent with known field data and engineering judgment. It is a waste of effort to construct a very large and complicated model that may be just as difficult to understand as the real case. Of course, if extensive field data is available, then this may be incorporated into a comprehensive model that can yield design information directly.

3.2. Data availability

For the analysis of slope stability it is required to have relevant data in order to perform the analysis properly. Generally the geotechnical data which is required for the analysis of the slope are paramount parameters for input data of FLAC3D. Therefore In order to get reliable output from FLAC3D it is essential to input all the parameters that influence the slope. Generally the important data for the slope stability analysis which was gathered are field data and laboratory test results.

3.2.1. Field data

In the field the required data for the input parameter of the slope to the FLAC3D are the height and inclination of the slope profile. The slope angle of the profile before failure is obtained from the AutoCAD drawing of the cut slope. The slope height is obtained using GPS and Tape measure. In addition the depth of the required soil to conduct the test is obtained from Tape measure. Moreover the profile of the slope is modified without a bench having overall slope angle of 34.16[0]

3.2.2. Ground water

An indicator of ground water is found at toe of slope. There is a continuous flow of water from the toe of slope. In addition to that there is existence of saturated soil at some depth of the slope at 15m below the crest of the slope during rainy season. So there is fluctuation of ground water from low level to high level with in the slope.

3.2.3. Laboratory tests of the soil

Evaluation of slope stability analysis require an in depth and reliable estimate of the in situ shear strength of the soil materials. However, the shear strength parameters are wrongly influenced by many complex conditions, including the insitu state of stress, drainage over consolidation ratio, loading rates, and soil composition.

The laboratory tests that were conducted for the progress of the thesis are the following:

- Moisture Content
- Sieve Analysis
- Unit Weight
- Unconfined Compression strength test
- Direct Shear (DS)
- CU triaxial compression test

3.2.3.1. Moisture content

For many soils, the water content may be an extremely important index used for establishing the relationship between the way a soil behaves and its properties. The consistency of a fine grained soil largely depends on its water content. The water content is also used in expressing the phase relationships of air, water, and solids in a given volume of soil. The water content at different depth is given in Table 3-1 and the details are given in appendix B.

3.2.3.2. Sieve Analysis

This test is performed to determine the percentage of different grain sizes contained within a soil. The mechanical or sieve analysis is performed to determine the distribution of the coarser, larger-sized particles, and the hydrometer method is used to determine the distribution of the finer particles. The texture of the soil profile at different depth is given in Table 3-1 and the detail grain size analyses are given in appendix B.

3.2.3.3. Unit weight

Unit weight of the soil is determined as weight of the soil sample to volume of the soil.The unit weight of the soil at each depth of the soil profile is determined using the sample taken from each depth using core cutter. So the unit weights at the respective depth of the soil profile are given in Table 3-1 and the details are given in appendix B.

Table 3-1: Index properties of the soil profile.

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3.2.3.4. Strength parameter

The short term or long term conditions are based on the ability of the soil in the slope to reach equilibrium conditions with respect to volume changes that are a reflection of stress changes affecting the slope. If the pore water pressure is known, the analysis may be performed using effective stress principles with drained strength parameters. For most practical problems, the total stress analysis is used for short-term stability problems and the effective stress is used to assess the long-term stability with the knowledge that any excess pore water pressures generated during the loading have probably dissipated completely. The strength parameters =0 and (undrained strength) are used for total stress analysis assuming the soil behavior is exclusively “cohesive” For effective stress analyses, and , will be required for determining FOS. Table 3-2 shows selection of strength tests

Table 3-2: selection of strength tests (Abramson et al., 2002)

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Based on the foregoing guideline, the types of tests for the slope profile are shown in Table 3-2. The sample for this case study is prepared from compaction of disturbed soil in six layer maintaining in-situ density and moisture content. Soil samples that are unsaturated in the field can either be fully saturated prior to testing, which will give lower bound values for shear strength parameters.

For remolded clay the suggested failure strain for CD and CU in triaxial test is 20-30 % (K.H.Head, 1994).The time to reach failure for CD and CU triaxial test is shown in Table 3-3.

For this case study it was taken a strain of 20% of 140mm or 0.2*140mm=28mm.And also it was taken the shear strain rate by dividing failure strain to estimated times to failure for the appropriate test.

Table 3-3: Estimated Times to Failures in Triaxial Tests (K.H.Head, 1994)

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The shear strength parameter of the soil for all layers is determined by direct shear test and triaxial compression test. Table 3-4 shows the computed shear strength parameter. The detailed strength parameter is given in the appendix C.

3.2.4. Modulus of elasticity of the soil

In addition to unit weight and strength parameter of the soil, one of the input parameter of FLAC3D for the analysis of the slope is modulus of elasticity. The modulus of elasticity of the soil is determined in the laboratory from the stress strain diagram of the soil in the direct shear box and in CU triaxial test. It was conducted both tests to get the required value of modulus of elasticity of the soil. The modulus of elasticity of the soil at the respective depth is given in Table 3-4.

Table 3-4: Strength parameter of the slope profile

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3.3. Modeling methodology

FLAC3D offers an ideal analysis tool for solution of three-dimensional problems in geotechnical engineering. A three-dimensional slope stability analysis was conducted for the overall slope which comprises four different layers which was obtained from the grain size analysis.

The Mohr-Coulomb model is the conventional model used to represent shear failure in soils and rocks. The failure envelope for this model corresponds to a Mohr-Coulomb criterion (shear yield function) with tension cutoff (tension yield function). The position of a stress point on this envelope is controlled by a non-associated flow rule for shear failure, and an associated rule for tension failure.

For computation with FLAC3D we should know the input parameters so the input parameters for the software are outlined as follows.

- Bulk density
- Strength parameter
- Modulus of elasticity of the soil

The bulk density of each layer is measured using samples taken from the site using core cutter at each layer and the shear strength parameter of each layer is determined using direct shear and triaxial compression test. Moreover the modulus of elasticity of the soil is determined from the direct shear box test and triaxial compression test which is one of the input parameters of the software.

It is better to use bulk modulus, K, and shear modulus, G, than Young’s modulus, E, and Poisson’s ratio, ν, for elastic properties in FLAC3D. The pair (K,G) makes sense for all elastic materials that do not violate thermodynamic principles. The pair (E, ν) does not make sense for certain admissible materials. (Itasca Consulting Group, 2005)

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So the input data for FLAC3D were bulk density, Strength parameter, bulk modulus and shear modulus of the each soil. After the relevant data is obtained the slope was modeled considering three different cases due to the fluctuation of the water table of the GW at different season as shown in Figure 3-1 to get FOS.

So considering the fluctuation of the phreatic surface at different season it was taken three different scenarios for the phreatic line. Figure 3-1 shows the cases of the phreatic surface for different occasions.

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Figure 3-1: Cases of water surface profile.

The input parameters of FLAC3D for the profile of the phreatic surface are given in Table 3-5.

Table 3-5: water surface profile for different cases.

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3.4. Data analysis

Once the FOS of the slope was determined using appropriate parameters of the slope for the different scenarios, the slope stability analysis will be carried out probabilistically which incorporates the uncertainties associated with the slope in a quantifiable means from the COV of the soil parameter.

It was determined the reliability and probabilities of failure using NORMDIST and NORMSDIST program which is integrated in MS-EXCEL.

3.5. Limitation and potential problem of the research

The limitation of the research for the analysis of the slope stability probabistically is the difficulty of obtaining the uncertainties due to:

- Random testing errors
- Statistical modeling of the mean due to the limited amount of sampling and testing in the laboratory or in the field
- Bias in measurement procedures due to features such as sampling methods and the extent of sample disturbance
- The strength parameters are obtained from disturbed samples, due to lack of appropriate sampler for the triaxial compression test, it would be better if undisturbed samples were used for triaxial compression test.
- Lack of information concerning different variables clearly limits our capacity to make probabilistic calculations more accurately.

4. RESULTS AND DISCUSSIONS

4.1. General

FLAC3D offers an ideal analysis tool for solution of three-dimensional problems in geotechnical engineering. A three-dimensional slope stability analysis was conducted for the overall slope as shown in Figure 4-1 which comprises four different layers that has different strength parameter on the basis of lab results.

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Figure 4-1: water surface profile for different cases.

4.2. Results

The laboratory results of the shear strength parameters on the basis of the types of the tests conducted resulted the shear strength parameter as shown in Figure 4-2, 4-3, 4-4, 4-5, 4-6, 4-7

The respective output of the model for the scenarios is obtained from FLAC3D software. And some of the outputs of the FLAC3D are indicated in appendix A for the scenarios.

Plots of shear strain-rate contours and velocity vectors, which allow the failure surface to be identified. The results of the contour profile of the X-velocity for each scenario taking consideration of the variation of the water surface at different season are shown in Figure 4-8, 4-9 and 4-10.

4.2.1. Laboratory test results

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Figure 4-2: mohr circle for layer 1

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Figure 4-3:mode of failure for layer 1

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Figure 4-4: direct shear result for layer 2

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Figure 4-5: mohr circle for layer 3

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Figure 4-6: mode of failure for layer 3

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Figure 4-7: direct shear result for layer 4

4.2.2. Output of FLAC3D for case-1

The output of FLA3D for case-1 at which when the water table is at a depth of 20m from the toe level of the slope FLAC3D gave the following results.

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Figure 4-8: FLAC3D output for case-1

4.2.2. Output of FLAC3D for case-2

The output of FLA3D for case-2 at which when the water table is at a depth of 30m from the toe level of the slope FLAC3D gave the following results.

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Figure 4-9: FLAC3D output for case-2

4.2.3. Output of FLAC3D for case-3

The output of FLA3D for case-3 at which when the water table is at a depth of 40m from the toe level of the slope FLAC3D gave the following results.

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Figure 4-10: FLAC3D output for case-3

So the summery of the FOS for the three cases are shown in Table 4-1

Table 4-1: FOS of the slope for different scenarios

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4.3. Probability of failure

The probability of failure may then be calculated from the CDF of a standard normal distribution from tabulated values or from software as mentioned above.

To evaluate the reliability of each scenarios with respect to the sliding-failure mode, due to limited sample data for the soil parameter, The mean safety factor for the slope for the scenarios are taken as the most likely value of the FOS and the standard deviation is taken by thumb rule of Dai and Wang (1992) taking consideration that 99.73 % of all value lie within three plus or minus of standard deviation assuming normal distribution. So we have mean value of FOS of safety 1 for all cases which is the incipient FOS for failure of the slope Which used to measure the performance indicators of the slope.

And the respective standard deviation of the three cases considering thumb rule of Dai and Wang (1992) we have.

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Now for our case we determine the probabilities of failure based on NORMDIST and NORMSDIST program in MS Excel and the respective probabilities of failures for the three cases are outlined below.

4.3.1. Probability of failure based on normal distribution

The PF based on the assumption of normal distribution can be determined from the PDF of normal distribution. So using NORMDIST in MS-EXCEL we can plot the PDF and CDF of the FOS for µ=1and σ=0.166 is shown in Figure 4-11

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Figure 4-11: Normal PDF and CDF for the scenarios

So the PF can be determined from the area of PDF which is the value of CDF of the graph or simply from the NORMDIST program as shown in Table 4-2

Table 4-2: PF based on Normal distribution

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4.3.1. Probability of failure based on lognormal distribution

Similarly for lognormally distributed PDF, the probabilities of failure for the three cases are tabulated for the natural logarithm of the factor of safety in standard normal distribution which has mean 0 and standard deviation of 1. Therefore using NORMSDIST program in MS-EXCEL as shown in Table 4-3

Table 4-3:PF based on lognormal distribution

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4.4. Discussion

The strength parameter determined form lab experiments are essential input parameter served for the modeling of FLAC3D accurately. In the FLAC3D output the velocity magnitude indicates the slip surface of the soil profile and failure will occurs at the slip surface at which the velocity is greater.

The PF based on the assumption of normal distribution is greater than that of PF of lognormal distribution having the same categories of failure. In addition to the foregoing assumption of the phreatic surfaces using three different cases to get FOS for the scenarios attempt was done to have postulated failure probabilities. It was tried to investigate the cases of reducing the angle of the slope and reducing the height of the slope even if there is elementary school at the crest of the slope. However, the results of the FLAC3D are not satisfactory. From the output FLAC3D by changing the geometric elements of the slope we can make the general induction that stability of the slope can’t be achieved by modifying the geometry of the slope and also it indicated that how the effective shear strength parameter of the soil slope is very low.

Uncertainties considered in formal reliability analysis of slopes related to parameters relevant to the basic slope mechanisms and models are essential to model the slope accurately. However, other significant uncertainties are not considered in these analyses. For example, uncertainties in shear strength parameters including systematic and spatial components are included. Considerations of model error and bias may also be included as far as possible. However, there may be significant uncertainties with regard to factors such as:

- construction quality and
- Likelihood of internal erosion and piping within a slope. These uncertainties, which may have a significant influence on slope performance, are generally not included in formal geotechnical analysis.

5. CONCLUSIONS AND RECOMMENDATIONS

5.1. Conclusions

The following conclusions are based on the three-dimensional slope stability analyses of field case histories and a representative slope model. Even if we do the analysis using 3D we draw the following conclusion from the analysis for this case study.

- A three-dimensional slope analysis is beneficial in designing slopes with a complicated topography, shear strength, pore water pressure condition to model the slope. This can be accomplished using a FLAC3D.
- Long-term stability analysis with effective stress parameters would be appropriate in this case study.
- The three dimensional output of FLAC3D is not desirable from the input data. However the factor of safety can be improved significantly by reducing pore water pressure through drainage measures, surface and sub-surface.
- Target probabilities of failure could not achieved with modification of the slope for the three scenarios which results modification of the slope geometry could not mitigate the slope stability problem.
- From the output of FLAC3D it can be concluded that the main reason for the failure of the slope can be because of clay portion is sandwiched between to soil layer of silty sand and gravelly clay which has very low effective shear strength parameter that results long term stability problem.
- The probability failure of the slope for this case study for the three different scenarios using normal and lognormal distribution, it showed that the failure probability based on normal distribution is greater than that of lognormal distribution.
- From the three different scenarios it can be concluded that the probability of failure is more when the water surface profile is higher. It can also be concluded that modifying the slope could not alleviate the slope stability failure, so the only ways for stabilizing the slope is using additional reinforcement and proper designing of retaining wall with proper drainage as well as provision of sheet pile.

5.2. Recommendations for further research

One can also look at this from a cost-benefit point of view. The construction cost associated with an increase in design factor of safety must be compared to the reduction in the expected cost of failure. This is given by a product of the reduction in probability of failure associated with more reliable design multiplied by cost of failure.

I would like to recommend the following to have deep rooted analysis of the slope at this vulnerable area.

- Design retaining wall with facilitating drainage for the slope
- Provision of sheet pile
- Designing of geosynthetic reinforcement for the stability of the slope.

Cost-benefit comparison of alternatives may help in selecting the best approach for remediation of slope to the great extent.

REFERENCES

1. Abramson Lee W., T. S. (2002). Slope Stability and Stabilization methods. New York: John Wiley & Sons, Inc.

2. Baligh, M. A. (1975). End effcts of Stability of Cohesive soils. Journal of the Geotechnical Engineering Division, 1105-1117.

3. Bardet, J.-P. (1997). Experimental Soil Mechanics. London: Prentice Hall, Inc.

4. Bishop. (1995). The use of the slip cirlce in the Stability Analyis of slopes. Geotechnique, 7-17.

5. Budhu, M. (2000). Soil Mechanics and Foundations. New York: John Wiley & Sons, Inc.

6. Craig, R. (2005). Craig’s Soil Mechanics. London: Taylor & Francis Group Inc.

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11. ERA. (2010). Preliminary Landslide investigation report. Addis Ababa: ERA.

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13. Faber, P. D. (2003). Risk and Safety in Civil,Surveying and Environmental engineering. Swiss, Switzerland: Swiss Federal Institute of Technology.

14. Fredlund, D. G. (1981). The Relationship Between Limit Equilibrium slope Stability Methods. ICSMFE, (pp. 409-416). Sweden, Stockholm.

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16. George, J. S. (1991). Probabilistic Methods Applied to Slope Stability Analysis. New Zealand: Auckland University.

17. Hovland, H. J. (1977). Three Dimesnsional slope Stability Methods. Journal of the Geotechnical engineering Division, 971-976.

18. Itasca Consulting Group, I. (2005). FLAC3D user' manual. Minneapolis, Minnesota: Itasca Consulting Group, Inc.

19. K.H.Head, M. (1994). Manual of Soil Laboratory Testing (Vols. 1,2,3). New York: John Wiley & Sons, Inc.

20. Li. (1987). Probabilistic Design of slopes. Canadian Geotechnical Journal, 520-535.

21. Robin Chowdhury, P. F. (2010). Geotechnical Slope Analysis. London: Taylor & Francis Group Inc.

22. Sanjay Kumar Shukla, J.-H. Y. (2006). Fundamentals of Geosynthetic Engineering. London,UK: Taylor & Francis group.

23. T.William Lambe, Robert V.Whitman. (1969). Soil Mechanics. New York: John Wiley & Sons, Inc.

24. Terzaghi, P. (1967). theoretical soil mechanics. NewYork: John Wiley & Sons.

25. Vanmarcke, E. H. (1996). Probabilistic Methods in Geotechnical Engineering. Logan, Utah: ASCE.

26. Y.M. Cheng, C.K. Lau. (2008). Slope Stability Analysis and Stabilization. New York: Taylor & Francis Group Inc.

27. Zuyu Chen, H. M. (2003). A simplified method for 3D slope stability analysis. NRC Canada: NRC Research Press Web site.

Appendix A

FLAC3D output for the scenarios

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Appendix B

Index properties of the soil profile of the slope

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Appendix C

Strength parameter of the soil

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Appendix D : Standard normal distribution Table

Appendix E

Some of pictures of the research

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