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**CHAPTER 2 ****PROBABILITY/RELIABILITY METHODS ****FOR SLOPE STABILITY ANALYSIS**

** INTRODUCTION TO SLOPE STABILITY ANALYSIS**

The evolution of slope stability analyses in geotechnical engineering has followed closely the developments in soil and rock mechanics as a whole. Slopes either occur naturally or are engineered by humans. The increasing demand for engineered cut and fill slopes on construction projects has only increased the need to understand analytical methods, investigative tools, and stabilization methods to solve stability problems. Slope stabilization methods involve specialty construction techniques that must be understood and modeled in realistic ways.

An understanding of geology, hydrology, and soil properties is central to applying slope stability principles properly. Analyses must be based upon a model that accurately represents site subsurface conditions, ground behavior, and applied loads. Judgments regarding acceptable risk or safety factors must be made to assess the result of analyses (Whitman, 1984).

**LIMIT EQUILIBRIUM METHOD**

The geotechnical engineer frequently uses limit equilibrium methods of analysis when studying slope stability problems. The method can be approached with single free-body procedures or method of slices. Duncan and Wright (2005) include infinite slope, logarithmic spiral, and Swedish slip circle methods as single free-body procedures.

The methods of slices have become the most common methods due to their ability to accommodate complex geometries and variable soil and water pressure conditions. Some methods of slices accommodate circular slip surfaces, while others accommodate noncircular slip surfaces. There are several commonly used methods, including:

- Ordinary Method of Slices (Fellenius, 1927)
- Janbu’s Simplified Method (Janbu, 1954)
- Janbu’s Generalized Method (Janbu, 1954)
- Bishop’s Simplified Method (Bishop, 1955)
- Bishop’s Rigorous Method (Bishop, 1955)
- Lowe and Karafiath’s Method (Lowe and Karafiath, 1960)
- Morgenstern-Price Method (Morgenstern and Price, 1965)
- Spencer’s Method (Spencer, 1967)
- Sarma’s Method (Sarma, 1973)
- Corps of Engineer’s Method (1982)

Duncan (1992) showed stability methods that satisfy all conditions of equilibrium (horizontal and vertical force equilibrium and moment equilibrium) result in a factor of safety with an accuracy of ±5 percent. As a result, stability methods that satisfy all conditions of equilibrium can be considered to yield an accurate estimate of the factor of safety.

**HARA ET AL. SIMPLIFIED METHOD (1990)**

Ishihara et al. (1990a) and Ishihara et al. (1990b) proposed an infinite slope stability analysis particularly suited to slopes which failed by flow liquefaction. This method was utilized to analyze four failure case histories in Japan: Mochikoshi Tailings Dam (1978), Hokkaido Tailings Dam (1968), Metoki Road Embankment (1968), and Chonan Sand Fills (1987) which failed following an earthquake. All of the case histories were flow failure cases. The method was also utilized to analyze case histories of flow slide in the collapsible loess deposit in Soviet Tajik following 1989 earthquake.

The Ishihara et al. simplified method (1990) is appropriate when the failure mass is very large in lateral extent as compared the depth to the failure surface. The simplified method is slightly modified to account for the depth to the water table. Figure 2.1 shows the assumed free body diagram of the failed soil mass; it is assumed that the side forces are equal in magnitude and opposite in direction and the ground surface is parallel to the failure surface. The water table is assumed to be parallel to the ground/failure surface.

Factor of safety, *FS*, is given by Equation 2.1. Expressions used to compute the liquefied shear strength with the simplified method, are shown in Equation 2.2 and Equation 2.3. Equation 2.1 and Equation 2.2 are derived through equilibrium of forces analysis along the failure surface.

Where *S** _{u-LIQ}* is the liquefied shear strength,

*FS*is factor of safety, α is the angle of the ground/failure surface and

*W*‘ is the effective weight of the soil above the failure surface, defined as:

*W*‘ =

*H*

_{d}*γ*

_{d}*+*

*H*

_{w}*γ*

*−*

_{sat}*H*

_{w}*γ*

*cos*

_{w}^{2}

*α*

*H*

*is the depth to the water table,*

_{d}*H*

*is the height of the water table above the failure surface, γ*

_{w}*is the dry unit weight of the soil above the water table, γ*

_{d}*is the saturated unit weight of the soil above the failure surface, and γ*

_{sat}*is the unit weight of water.*

_{w}**INTRODUCTION TO RELIABILITY METHODS FOR SLOPE STABILITY ANALYSIS**

Geotechnical engineers deal with uncertainties by recognizing that risk and uncertainty are inevitable and by applying the observational method. However, the observational method is applicable only when the design can be changed during construction on the basis of observed behavior. In those cases in which the critical behavior cannot be observed until too late to make changes, the designer must rely on a calculated risk. Recently, several studies of risk analysis for slopes and embankment have been conducted.

Slope stability analysis involves several uncertainties, including computational accuracy, soil unit weight, slope geometry, pore-water pressures, and soil shear strength. These uncertainties are the reasons for a lack of confidence in deterministic analyses (Alonso, 1976). Extensive research has been conducted to evaluate the computational accuracy of two-dimensional limit equilibrium slope stability methods.

Because the uncertainty in computational accuracy is small for methods that satisfy all conditions of equilibrium, it is necessary to consider possible uncertainties in the other previously mentioned parameters. Soil unit weight can be readily measured in the laboratory, slope geometry can be ascertained via elevation surveys and subsurface techniques, and pore-water pressures can be estimated from boreholes and or/ piezometer installations. However, a large source of uncertainty can be introduced during the selection of the soil shear strength parameters.

**VARIABILITY OF SOIL PARAMETERS**

As a natural material, soil contains a large source of uncertainty of soil parameters. Soil parameters are usually measured in the laboratory using special apparatus. Accuracy of the apparatus and human error during the measurement add more uncertainty to measure soil parameters. Figure 2.2 demonstrates one concept of the source of uncertainty in soil parameters (Christian et al., 1994). Systematic error can result from statistical error in mean values arising from limited numbers of measurements, as well as from bias in measurement procedures. Data scatter, on the other hand, includes nonsystematic (random) testing errors and actual spatial variation in the soil profile.

Several researchers have collected coefficient of variations of commonly used soil parameters (Chowdhury, 1984; Harr, 1987; Kulhawy, 1992; Lacasse and Nadim, 1997; Duncan, 2000). Table 2.1 presents data of coefficient of variation of soil parameters as summarized by Duncan (2000). Table 2.2 and Table 2.3 present data of coefficient of variation of dry, moist, and buoyant unit weight of soils from recent study (Gutierrez et. al., 2003).

Where *COV* is coefficient of variation, *μ* is mean value, and *σ* is standard deviation respectively.

** METHOD OF PROBABILISTIC ANALYSIS**

There are several techniques of probabilistic method that are commonly used in Geotechnical Engineering. First Order Reliability Method (FORM) and Monte Carlo Simulation are utilizing in this study. The following paragraphs describe each of these methods in more detail.

** First Order Reliability Method (FORM)**

The First Order Reliability Method (FORM) was introduced for the first time by Hasofer and Lind (1974). This approach is based on independent, normal variable. For other situation, it will not give correct information on the probability of failure. Several study corrected and included information on the probability of the random variables (e.g., Rackwitz and Fiessler, 1978; Chen and Lind, 1983).

The First Order Reliability Method (FORM) is not commonly used for slope stability analyses. Low and Tang (1997a) were pioneering the use of FORM for slope stability analyses. They were conducting various studies utilizing the FORM (Low and Tang, 1997a, 1997b; Low, 2003).

In this study, the reliability of each case history is measured with the Hasofer-Lind reliability index, *β* (Hasofer and Lind, 1974). This study adopts a convenient spreadsheet approach proposed by Low and Tang (2004), where object-oriented constrained optimization is used to obtain a solution for reliability index, *β**,* with Equation 2.5:

where * x* is a vector representing the random variables,

*F*is the failure domain,

*is the correlation matrix, and*

__R__*μ*

_{i}*N*and σ

_{i}*N*are the mean and standard deviation computed from Rackwitz-Fiessler (1978) two-parameter equivalent normal transformations. By using the equivalent normal parameters, model properties with different probability distributions (i.e. normal, lognormal, uniform, beta, etc.) can easily be used in the same analysis.

**Monte Carlo Simulation**

The name « Monte Carlo » was coined by Stanislaw Ulam and Nicholas Metropolis (Metropolis, 1987) during the Manhattan Project of World War II. Monte Carlo simulation encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. The simulation is useful for obtaining numerical solutions to problems which are too complicated to solve analytically.

ABSTRACT

ACKNOWLEDGEMENTS

CHAPTER 1 INTRODUCTION

1.1 INTRODUCTION TO THE PROBLEMS

1.2 LIQUEFACTION

1.3 OBJECTIVES AND METHODOLOGY OF THE STUDY

1.4 ORGANIZATION AND SCOPE

CHAPTER 2 PROBABILITY/RELIABILITY METHODS FOR SLOPE STABILITY ANALYSIS

2.1 INTRODUCTION TO SLOPE STABILITY ANALYSIS

2.2 LIMIT EQUILIBRIUM METHOD

2.3 THE ISHIHARA ET AL. (1990) SIMPLIFIED METHOD

2.4 INTRODUCTION TO RELIABILITY METHODS FOR SLOPE STABILITY ANALYSIS

2.5 VARIABILITY OF SOIL PARAMETERS

2.6 METHOD OF PROBABILISTIC ANALYSIS

CHAPTER 3 THE 1999 KOCAELI/DUZCE, TURKEY EARTHQUAKE

3.1 INTRODUCTION

3.2 GEOLOGY OF TURKEY

3.3 GEOTECHNICAL EFFECT

CHAPTER 4 CASE HISTORIES BACK ANALYSIS

4.1 INTRODUCTION

4.2 HOTEL SAPANCA LIQUEFACTION FAILURE

4.3 OTHER CASE HISTORIES

4.4 COMPARISON OF DATA FROM THIS STUDY AND PREVIOUS DATA FROM SEED AND HARDER (1990)

CHAPTER 5 CONCLUSIONS

5.1 CONCLUSIONS

REFERENCES

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Probabilistic Post-Liquefaction Residual Shear Strength Analyses of Cohesionless Soil Deposits: Application to the Kocaeli (1999) and Duzce (1999) Earthquakes