Uncertainty and reliability analysis applied to slope stability

ORIGINAL PAPER

Uncertainty and Reliability Analysis Applied to SlopeStability: A Case Study From Sungun Copper Mine

Mohsen Abbaszadeh • Kourosh Shahriar •

Mostafa Sharifzadeh • Mehrdad Heydari

Received: 5 September 2007 / Accepted: 20 April 2011 / Published online: 6 May 2011

� Springer Science+Business Media B.V. 2011

Abstract Nowadays, there are many new methods

for slope stability analysis; including probabilistic

methods assessing geotechnical uncertainties to

develop safety factors. In this paper, a reliability

index analysis for the Sungun copper mine slope

stability is evaluated based on three methods of

uncertainties consisting Taylor series method,

Rosenblueth point estimate method and Monte-Carlo

simulation method. Sungun copper mine will be one

of the Iran’s biggest mines with final pit’s height of

700 meters. For this study two of its main slopes were

assessed, one dipping to the NE (030) and the other to

the SE (140). Probability density function of cohesion

and angle of friction for the slopes were developed

using limit equilibrium methods. These shear

strengths were then used to determine the probability

density function of safety factor and reliability index

using the probabilistic methods. Results of the

probabilistic analysis indicate that with ascending

values of the uncertainties the reliability index

decreases. Furthermore, it was determined that with

the Monte Carlo simulation the seed number used has

little effect on the reliability index of the safety factor

especially with seed numbers in excess of 1200.

Variations in the overall reliability index of safety

factor were observed between the two slopes and this

difference is explained by the differences in com-

plexities of the geology within the cross-section.

Keywords Slope stability � Factor of safety �Reliability index � Monte Carlo simulation �Sungun copper mine

1 Introduction

In geotechnical engineering analysis and design,

various sources of uncertainties are encountered and

well recognized. Several features usually contribute

to such uncertainties, like:(1) those associated with

inherent randomness of natural processes;(2) model

uncertainty reflecting the inability of the simulation

model, design technique or empirical formula to

represent the system’s true physical behavior, such as

calculating the safety factor of slopes using limiting

equilibrium methods of slices;(3) model parameter

uncertainties resulting from inability to quantify

accurately the model input parameters and (4) data

uncertainties including (a) measurements errors,

(b) data inconsistency and non-homogeneity and

(c) data handling. In slope stability analysis, various

sources of uncertainties are encountered, such as

geological details missing in the exploration program,

The paper is Dedicated to Ali Sharifzadeh who lost his life in a

tragic car accident during the preparation of this research work.

M. Abbaszadeh � K. Shahriar � M. Sharifzadeh (&) �M. Heydari

Department of Mining, Metallurgy and Petroleum

Engineering, Amirkabir University of Technology,

Hafez 424, 15875-4413 Tehran, Iran

e-mail: [email protected]

123

Geotech Geol Eng (2011) 29:581–596

DOI 10.1007/s10706-011-9405-1

estimation of rock mass properties that are difficult to

quantify, fluctuation in pore water pressure, testing

errors and many other relevant factors.

In a deterministic analysis, the factor of safety is

defined as the ratio of resisting to driving forces on a

potential sliding surface. The slope is considered safe

only if the calculated safety of factor clearly exceeds

unity. 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:

• Probability of failure (or probability of unsatis-

factory performance)

• Reliability index

• The most probable combination of parameters

leading to failure

• Sensitivity of result to any change in parameters

In his review of slope stability assessment methods,

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.’’

Whereas, in a probabilistic framework the factor of

safety is expressed in terms of its mean value as well

as its variance. Reliability analysis is therefore used

to assess uncertainties in engineering variables such

as factor of safety of slope stability. The reliability

index, b, is often used to express the degree of

uncertainty in the calculated factor of safety. These

uncertainties are assessed by different approaches

such as the Taylor series method, Rosenblueth point

estimate method and Monte Carlo simulation method.

Slope stability analysis is a study within geotech-

nical engineering that is highly amenable to probabi-

listic treatment, and has received considerable

attention in the literature. The earliest papers appeared

in the 1990’s (e.g. Wolff 1985, Christian et al. 1994,

Christian et al. 1994, Lacasse and Nadim (1996),

Hassan and Wolff 2000, Duncan 2000).

Wu and Kraft (1970), Cornell (1971), Alonso (1976),

Tang et al. (1976), Venmarcke (1977), Wolff (1985), Li

and Lumb (1974), and Barabosa et al. (1989), applied a

probabilistic approach to analyze slope stability using

the Taylor series method. Recently, a number of

application of probabilistic slope stability studies using

other numerical approaches such as Monte-Carlo sim-

ulation or Rosenblueth’s point estimate method, have

been reported in the literature (Dai et al. 1993 and

Chowdhury and Xu 1993).

Christian et al. (1994) used the mean-first order

reliability method, which is simplification of the more

general first order reliability method. They found that

the reliability analysis is especially useful in estab-

lishing design values of safety factor representing

consistent risk for different types of failure. Tobutt

(1982) used the Monte-Carlo method as a sensitivity-

testing tool for slope stability and also as a method

for calculating the probability of failure of a given

mine slope.

This paper presents a probabilistic based approach

by which the relevant sources of shear strength (c, u)

uncertainty involved in slope stability analysis can be

modeled and analyzed. Three different methods are

used to quantify the uncertainty in the calculated

safety factor. These are Taylor series method,

Rosenblueth point estimate and Monte Carlo simu-

lation method. Two slopes, were selected to investi-

gate how the probability of failure and the reliability

index may vary among those three methods. The

results were obtained using two well-recognized

methods of slope stability analysis, namely Bishops’

method (1955) and Janbu’s method (1954).

2 Probabilistic Slope Stability

Analysis Procedure

Figure 1 illustrates schematically the methodology

used to evaluate the uncertainty and the reliability of

slope stability factor of safety for the Taylor’s series

method, Rosenblueth’s point estimate method and

Monte Carlo simulation method (before Abdullah

et al. 2000). The first step in the methodology is to

specify the slope geometry and the probability distri-

bution for the rock mass properties. The second step is

to search for the critical slip surface and its associated

factor of safety using limit equilibrium methods

(simplified Bishop’s method and simplified Janbu’s

method). Once the critical slip surface and the

uncertainties of rock mass properties are known (or

assumed), the reliability analysis can be performed. In

the case of the Taylor series method and the Rosen-

blueth method, the partial derivatives of the safety

factor must be evaluated with respect to each of the

rock mass properties. Then the mean and the variance

of the safety factor and the associated probability

582 Geotech Geol Eng (2011) 29:581–596

123

distribution can be determined. Accordingly, the

reliability index (b) and the probability of failure

(Pf) of the slope can be calculated. On the other hand,

for the Monte Carlo simulation method (MCSM)

independent sets of rock mass properties (c, u) are

generated from their assigned probability distribu-

tions. Safety factors for each set are calculated using a

limiting equilibrium method, the mean, the standard

deviation and the associated probability distribution of

the factor of safety are determined. Finally, the

reliability index (b) and the probability of failure

(Pf) are also calculated for the MCSM.

3 Limiting Equilibrium Methods

Most problems in slope stability are unknowns and,

as a result, some simplifying assumptions are made in

order to determine a unique factor of safety. Due to

the differences in assumptions, various methods have

been developed. As already mentioned among the

most popular methods are procedures proposed by

Bishop and Janbu. Simplified Bishop’s method

satisfies overall moment and is applicable to a

circular slip surface, while Janbu’s method satisfies

force equilibrium and is applicable to any shape.

Fig. 1 Schematic

representation of the

methodology used

Geotech Geol Eng (2011) 29:581–596 583

123

Hence, these methods were used to calculate the

factor of safety on a potential sliding surface in

Sungun copper mine slope stability analysis.

The limit equilibrium method was used for slope

stability analysis applying the Slide V.5.0

(Rocscience Inc 2003) computer code. The program

can also perform probabilistic analysis to assess

stability condition. Reliability analysis was per-

formed using the Taylor’s series method and the

Rosenblueth’s point estimate method and the MCSM.

4 Methods of Reliability Evaluation

The main intent of a probabilistic analysis is to

estimate the reliability index, using the mean and

standard deviation of the factor of safety. Alterna-

tively, the probability density function (PDF) of the

factor of safety, F, may be estimated and probability

of failure calculated directly. Limit equilibrium

equations may be used to develop the resultant joint

probability density function with parameters, the

mean and standard deviation due to variations in the

input parameters (cohesion and internal friction

angle). Such predictive models are called perfor-

mance functions and can be represented by an

equation of the form

F ¼ gðX1;X2; . . .;XnÞ þ e ð1Þ

where Xi component input variables, described as random

variables, e model error that accounts for the error

between the response of the adopted model and the real

world behavior (Christian et al. 1994). gð. . .Þ represents

a functional relationship for the factor of safety.

If the reliability index, b, is to be calculated, we

must calculate the mean and standard deviation of the

probability density function of the factor of safety, F,

in response to the input random variables. The

performance function, with continuous random vari-

ables as the input parameters, must be integrated to

generate the joint probability density function for a

closed form solution. Although this may be possible

for simple linear performance functions, the usual

factor of safety, F, equation cannot be integrated

directly, because it is complicated, requires iterations,

and cannot be simplified easily. So, we must resort to

deriving approximate solutions using numerical

methods. Most numerical methods will not provide

us with a probability density function of the factor of

safety, F, (Taylor’s series method and Rosenblueth

point estimate method), but will provide us with the

expected value, E[F], and the variance, Var[F], for

use in calculating the reliability index, b.

As a direct integration of the adopted performance

function for the factor of safety, F, is not feasible, the

expected value, E[F], and the variance, Var[F], of the

factor of safety may be estimated by using (1) Taylor

series method, (2) Rosenblueth’s point estimate

method, (3) Monte Carlo simulation, or (4) Fourier

analysis. The results of a MCSM and the Fourier

analysis will provide an estimate of the probability

density function of the factor of safety, F, as well. Of

these four methods, the first three have been used

extensively, while the Fourier analysis method has

been used primarily for block-slide models to analyze

rock slope stability.

4.1 Taylor Series Method

This method is based on the Taylor series expansion

of the performance function about the expected

values of the random variables (cohesion & internal

friction angle). If only the first-order terms of the

expansion are retained, the expected value of the

performance function for independent random vari-

ables may be approximated by

E½F� ¼ lF � g E½X1�;E½X2�; . . .;E½Xn�ð Þ þ e ð2ÞWhere the function gðX1;X2; . . .Þ is determined

using input parameters that correspond to the mean

values of the component random variables. For the

case where pairs of random variables show some

dependency (that is qXYj ji0), the expected value may

be approximated by

E½F� ¼ lF

� g E½X1�;E½X2�; . . .;E½Xn�ð Þ

þXk

i¼1

Xk

j¼1

o2F

oXioXjCov Xi;Xj

� �þ e

for i \ j

ð3Þ

In the above equation X2;X3; . . .;Xn are the mean

values of the component random variables and

e represents modeling error.

The Taylor series may also be used to estimate the

variance of the response of the performance function

to the input random variables. If all random variables

are independent, all covariance will be zero, and the

584 Geotech Geol Eng (2011) 29:581–596

123

second summation in Eq. 3 reduces to zero. The

assumption of independent (uncorrelated) rock mass

shear strength (c, u) is reported by various research-

ers. For example, Matsuo and Kuroda (1974) and

later on Matsuo (1976), showed that cohesion and

tanu are uncorrelated.

Although this method will give an exact solution

for a linear function, the first-order approximations

introduce error for nonlinear performance functions,

such as the ones typically used for the factor of

safety, F. In considering one of the limit equilibrium

formulations, the partial derivatives cannot be

derived directly and numerical methods must be used

to estimate these derivatives.

The U.S. Army Corps of Engineers recommends

the use of the finite difference approximation of the

derivative and the variance will be given by the

simplified expression (Abramson 2002)

Var½F� ¼ r2F �

1

4

Xk

i¼1

½DFi�2 þ V ½e� ð4Þ

The DF differences are obtained over two standard

deviations. In using this version of the Taylor series

method, the performance function will have to be

evaluated once for the expected value and then

2n times again, for the n random variables for total of

(2n ? 1) evaluations.

4.2 Rosenblueth’s Point Estimate Method

This point estimate procedure was originally pro-

posed by Rosenblueth (1975) where the probability

density functions of the random variables are simu-

lated by ‘‘point’’ masses located at plus or minus one

standard deviation from the mean values. This

method is direct, gives reasonably accurate results

quickly, and has been discussed in the literature

(Baecher and Christian 2003).

For the case of two random variables, X(c) and

Y(u), the performance function for the factor of

safety will take the form

F ¼ g X ; Yð Þ ð5ÞWhere the uncertainty in the random variables is

described by their standard deviation, rX and rY, and

their mean values are taken as lX and lY. For the

general case, we can also presume that the variables

are dependent, with a correlation coefficient of qXY,

and that the probability density functions are distrib-

uted normally.

For two random variables, the first step requires

the evaluation of the factor of safety, F, at four

unique combinations of the random variables, as

shown below

Fþþ ¼ g lX þ rXð Þ; lY þ rYð Þ½ �Fþ� ¼ g lX þ rXð Þ; lY � rYð Þ½ �F�þ ¼ g lX � rXð Þ; lY þ rYð Þ½ �F�� ¼ g lX � rXð Þ; lY � rYð Þ½ �

ð6Þ

In the above set of expressions, the subscripts

indicate the values of X and Y used to evaluate the

performance function. F??, for example, represents

the value of the function g(X,Y), as evaluated for

X = lX ? rX and Y = lY ? rY. Similarly, F?-

represents the value of the function g(X, Y), as

evaluated for X = lX ? rX and Y = lY - rY. Once

these expressions are evaluated, the expected factor

of safety, F, will be given by their sum as:

E½F� ¼ lF

¼ PþþFþþ þ Pþ�Fþ� þ P�þF�þ þ P��F��

ð7Þ

where P, is a coefficient in which defined as follow:

Pþþ ¼ P�� ¼1

41þ qXYð Þ

Pþ� ¼ P�þ ¼1

41� qXYð Þ

ð8Þ

The variance of the factor of safety, F, will be

given by

Var½F� ¼ E½F2� � E½F�ð Þ2 ð9Þ

where E[F] is expected value of the factor of safety,

F, from Eq. 9, and

E½F2� ¼ PþþF2þþ þ Pþ�F2

þ� þ P�þF2�þ þ P��F2

��

ð10ÞCalculating the expected value, E[F], and the

variance, Var[F], using Rosenblueth’s procedure

requires 2n evaluations of the performance function.

For only two variables, this is the less than the

number required for the Taylor series method.

However, for more than two variables, Rosenblueth’s

method will require more evaluations of the perfor-

mance function than with the Taylor series method.

Geotech Geol Eng (2011) 29:581–596 585

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4.3 Monte Carlo Simulation Method

Monte Carlo simulation is used to obtain the prob-

ability distribution of dependent random variables

given the probability distribution of a set of indepen-

dent random variables. Thus, in Monte Carlo simu-

lation studies three steps are usually required:

(a) determining the independent variable (input),

(b) transforming the input as independent variable

(output), and (c) analyzing the output. Monte Carlo

simulation offers a practical approach to reliability

analysis because stochastic nature of the system

response (output) can be probabilistically duplicated.

In this technique, a large number of rock properties

such as shear strength of the rock mass can be

sampled from their known (or assumed) probability

distribution. For this purpose, the probability density

function for each of these rock mass variables must

be specified. Usually, a normal distribution is

assumed for the rock mass properties (Park and West

2001). Then, the corresponding safety factor of each

set is calculated. These values of safety factors are

plotted on a probability paper in order to determine

the distribution of the safety factor. The reliability

index (b) and the probability of failure (Pf) are then

calculated using the safety factor probability distri-

bution. This approach can be applied to any method

of slices, that uses limit equilibrium in the analysis of

slopes. In the present study, uncertainty in slope

stability is quantified by evaluating the reliability

index, which is defines as:

b ¼ E Fð Þ � 1:0

r Fð Þ ð11Þ

where b is the reliability index, E(F) the expected

value of the safety factor, and r(F) is the standard

deviation.

5 Sungun Copper Mine

Sungun Copper Mine (Sungun) in Azerbaijan Prov-

ince of northwest Iran at Lat. 38�38020 north and

Long. 46�45035 east. The mine is an open pit mine

exploiting the Sungun copper porphyry deposit. It is

designed to initially produce 7 Mtpa ore with the

intention to expand capacity to 14 Mtpa. The pit is

roughly semi circular with high walls on three sides

and a low wall to the east in the Sungun valley. The

interim pit which is the subject of this current study

has a maximum height of approximately 675 m. This

is thus considered to be a high to very high slope. The

considerable final slope height and the presence of

faults and intrusive dykes create on of the exceptional

case studies for geotechnical engineering and slope

stability. It is situated in a mountainous area in a

relatively active geological zone, where active faults

are observed.

According to recent studies, due to faults and their

seismic activity, the Sungun area is technically active

and therefore is deformed and heavily jointed (Sun-

gun copper mine geological report 2006).

5.1 Geotechnical Investigation

A specific geotechnical drilling program targeting the

interim and final pit limit conditions was imple-

mented. This used triple tube drilling equipment and

drilling techniques and polymer drilling mud to

minimize disturbance of samples and increase core

recovery. A 15-hole drilling program with inclined

and orientated drill holes drilled on the more critical

geotechnical radial sections through the major slopes

from pit center was undertaken comprising a total of

some 4,600 m of drilling. Figure 2 shows the Sungun

copper mine final pit with layout of geotechnical bore

holes. In the present study, oriented cores were

obtained to assess dip and dip direction of disconti-

nuities for rock mass classification.

Fig. 2 Sungun copper mine final pit with layout of geotech-

nical bore holes

586 Geotech Geol Eng (2011) 29:581–596

123

5.2 Main Lithological Units

The Sungun intrusive complex hosting the Sungun

porphyry copper stock intruded along the Sungun

anticline into Cretaceous limestone, marls and shales.

Pliocene-Quaternary tuff and agglomerate of andes-

itic to trachytic composition followed as late stage

extrusion discordantly overlying the intrusion after a

period of uplift and erosion. The Sungun porphyry

complex developed as a composite stock form an

early monzonite/quartzmonzonite phase to a later

diorite/granodiorite phase which intersects the early

phase as cross cutting dykes.

The Sungun Porphyry (SP) is the host rock of the

copper mineralization forming the main lithological

unit at Sungun. It consists predominantly of quartz

and plagioclase phyric quartz-monzonite. The Sun-

gun Porphyry (SP) is affected by various types of

alteration. Potassic (SP_POT) and propylitic

(SP_PRP) alteration in general do not weaken the

intact rock strength, whereas the phyllic (SP_PHY)

and especially the argillic (SP_ARG) alteration

reduce significantly the intact rock strength.

Dioritic to quartz-dioritic dykes are the most

abundant dykes in the Sungun Porphyry. Based on

alteration type and their chronological order of

intrusion DK1 dykes can be further subdivided into

DK1A, DK1A dykes with DK1A being the older. The

DK1A (DK1A_PHY, DK1A_PRP) is very similar to

Sungun Porphyry and as such is of general poor to

subordinately fair to occasionally good rock mass

quality frequently intersected by very poor rock mass

intervals. DK1B dykes often cross-cut DK1a dykes

and are as such of a later, post-mineralisation stage.

The DK1B rock mass is significantly of better rock

mass quality than the Sungun Porphyry or DK1A

dykes. Medium to wide joint spacing is more frequent

(Sungun Copper Project Mining Geotechnics and

Slope Design Studies, SRK Con., 2008).

5.3 Determination of the Inherent Rock Mass

Properties

Due to significant changes of Sungun copper mine’s

rock mass properties, determination of the mine slope

material engineering properties (i.e. cohesion strength

and internal friction angle uncertainties), effective

parameters on rock mass properties were modelled

using random variables. To aid in these determina-

tions, rock mass classification were done for each run

in the drilling process (Abbaszadeh et al. 2006). In

this study, rock mass classification of Sungun copper

mine slopes were assessed by using RMR and GSI

methods. Confidence level of geotechnical parame-

ters are determined 90% approximately. The rock

mass properties of each lithology-alteration are

shown in Tables 1 and 2 for 30 Azimuth and 140

Azimuth cross sections.

5.4 Probability Density Function Testing

To identify the probability distribution of the safety

factor, two sets of rock mass properties (c, u) for

each rock type were generated from their probability

distributions. Assuming the friction angle and cohe-

sion values are normally distributed random vari-

ables. Factors of safety were determined by limiting

equilibrium methods. For example, the frequency

histogram for the generated factor of safety using

data of 30 Azimuth cross section for the simplified

Bishop’s method is shown in Fig. 3. The individual

values of factor of safety can be plotted on a

histogram; in view of the nature of the input

distributions and complexity of the problem this will

usually take a normal form. It is helpful to process

this histogram further to determine factor of safety

less than some particular value F. In Fig. 3, for

example, 0.5 percent of the values have a factor of

safety less than 1.0 (Probability of failure) and 35

percent are less than 1.15 (Unsatisfactory perfor-

mance). The Chi-square goodness of fit test indicates

that the normal distribution adequately fits the

generated factor of safety for all two sets of random

variables (cohesion and internal friction angle)

(Fig. 4).

6 Factors Affecting Reliability Index

Reliability in slope stability analysis is affected by

various factors such as the uncertainty associated

with rock mass properties and the uncertainty asso-

ciated with analytical methods or models

(Abbaszadeh et al. 2006). The focus of this paper is

to address the reliability of safety factor due to rock

mass shear strength uncertainty.

Geotech Geol Eng (2011) 29:581–596 587

123

6.1 Uncertainty Due to Rock Mass Properties

The randomness and uncertainty in the rock mass

property are the most important factors that may affect

the reliability of safety factor. The uncertainty in rock

mass properties can arise from the inherent spatial

variability in the properties and random testing errors

in their measurements. Also, such uncertainty may

come from systematical errors in the sampling process

and bias in the measurement process itself.

In addition to the randomness in the rock mass

properties, random seed and sequence length may

affect the calculated reliability index (b). Therefore,

the following computer experiments are performed to

see how the reliability index might be affected by

these two factors. Data of 30 Azimuth cross section

was used in theses experiments.

6.1.1 Seed Random Number

In this experiment, the effect of seed random number

generator on the reliability index, b, was investigated.

The first value of the sequence of random number

generation, called the seed, even though many users

of random numbers are tempted to do so. Park and

Miller (1969) and Press et al. (1992) endorse using

75 = 16,807. For this purpose, several computer runs

were conducted by which the seed random number

generator was allowed to vary from 2,000 to 24,000.

For each run, the reliability index was determined.

Figure 5 shows the effect of changing the seed

random number generator on the reliability index

using the two-mentioned slope stability analysis

Table 1 30 Azimuth cross section rock mass properties for each lithology-alteration

No. Lith-Alter C (mean)

kPa

C (Std)

kPa

/ (Mean)

Degree

/ (Mean)

Degree

C.O.V (%)

Cohesion

C.O.V (%)

Cohesion

Unit weight

(KN/m3)

1 DK1A_PHY 226 65 35 4.6 0.2876 0.1314 23

2 DK1A_PRP 97 4.7 22 2.4 0.0484 0.1090 23

3 DK1B_PRP 156.6 19.5 28.3 3 0.1245 0.1060 23.4

4 SP_PHY 171.7 26.3 31.5 2.8 0.1532 0.0888 22.6

5 SP_POT 246 63 37.4 4 0.2561 0.01069 22.6

Table 2 140 Azimuth cross section rock mass properties for each lithology-alteration

No. Lith-

Alter

C (Mean)

kPa

C (SD)

kPa

/ (Mean)

Degree

/ (SD)

Degree

C.O.V (%)

Cohesion

C.O.V (%) Fr.

Angle

Unit weight

(KN/m3)

1 SP_PHY 321 77 42 4.9 0.24 0.117 22.6

2 SP_POT 247 23.6 38.8 1.9 0.0955 0.05 22.6

Fig. 3 Frequency histogram of the factor of safety using

MCSM for 30 Azimuth cross section

Fig. 4 Normal distribution fit for the generated factor of safety

using MCSM for 30 Azimuth cross section

588 Geotech Geol Eng (2011) 29:581–596

123

methods. As can be seen, the reliability index b is not

sensitive to the selected seed random number gener-

ator. Therefore, in this study, the seed of 16807 was

selected. Also in the similar condition there is

significance difference in the calculated reliability

index between Bishop and Janbu methods (Fig. 5).

6.1.2 Sample Size

To perform the reliability analysis, the appropriate

sample size of rock mass properties should be

determined. To test this different sample sizes ranged

from 50 to 2,000 were generated and the associated

reliability index b for each of the two slope stability

methods was calculated. Figure 6 shows the relation-

ship between the sample size (generated number) and

the reliability index (b). It is clearly shown that there

is no significance difference in the calculated b as the

sample size exceeds 1200. But for obtaining proper

random variables distribution (cohesion and internal

friction angle) as well as proper distribution of safety

factor, a higher sample size of 10,000 was chosen

(Abbaszadeh et al. 2006).

6.2 Illustrative Cross Sections Analysis

In this study, two slopes were selected for studying

the reliability and uncertainty of the factors of safety

using the Taylor series, Rosenblueth point estimate,

and Monte Carlo simulation methods. The two

examples used include the stability of a slope with

five rock types (30 Azimuth cross section) and the

second of a slope with two rock types (140 Azimuth

cross section). Two well-known slope stability

methods (Bishop & Janbu) were applied for the

analysis these examples. Table 3 presents the calcu-

lated factor of safety for both slopes.

6.2.1 Section 1: 30 Azimuth Cross Section

This slope includes five lithologies with the geometry

presented in Fig. 7. The rock mass properties of this

slope are shown in Table 1 along with their associated

means, standard deviations, and coefficients of varia-

tions of rock mass properties. As can be seen from

Table 1, the coefficient of variation (COV) for rock

mass cohesion ranges from 0.05 to 0.3, and for rock

mass angle of internal friction it is between 0.05 to

0.15. The uncertainty and reliability analyses of this

example were performed according to the three

mentioned methods of reliability analysis (see Fig. 1).

Three experiments were conducted to illustrate

how the uncertainty in rock mass properties may

affect the calculated reliability index of the safety

factor. These experiments represent three cases of

rock mass uncertainties (low, moderate and high). In

the first experiment, the coefficient of variation

(COV) of the cohesion was allowed to change from

0.1 to 0.4, where the lowest uncertainty for rock mass

angle internal friction was assumed as 0.05. In the

second analysis, a moderate uncertainty was assumed

for the rock mass angle of internal friction (0.1) for

each different lithology. In the third analysis a high

level of uncertainty was assumed for the rock mass

angle of internal friction with the COV 0.2. In these

analyses, the reliability index was estimated using the

Taylor series, Rosenblueth point estimate, and Monte

Carlo simulation methods for the Bishop and Janbu

methods. Figures 8 and 9 show the results of these

experiments for these methods, respectively. It is clear

from Figs. 8 and 9 that, in the case of low uncertain-

ties in rock mass properties high reliability index for

the safety factor was obtained and vice versa.

Fig. 5 Relationship between seed random number generator

and reliability index

Fig. 6 Relationship between sample size and reliability index

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123

In the present research a comparative study

among the Taylor series, Rosenblueth point esti-

mate, and Monte Carlo simulation methods was

done using the rock mass properties of 30 Azimuth

cross section. As Figs. 8 and 9 show the effect of

variation in COV for the angle of internal friction

on the calculated reliability index using the two

methods of slope stability was more than in the

COV for rock mass cohesion. Also values of the

reliability index using simplified Bishop’s method is

more than simplified Janbu’s method in the three

reliability analysis methods for 30 Azimuth cross

section (Figs. 8, 9). The obtained results indicated

that no variations were observed between the Taylor

series and Rosenblueth point estimate approaches

for the Bishop and Janbu slope stability methods.

With increasing of coefficient of variation (COV) of

random variables, value and trend of the variation of

Monte-Carlo simulation method will be the same as

other two methods.

6.2.2 Section 2: 140 Azimuth Cross Section

The geometry of the slope to be analyzed is provided,

including the contacts between each of the litholog-

ical units (Fig. 10). The rock mass properties in each

of the lithological units were input for every section.

These properties are shown in Table 2. It was used to

compare the reliability index calculated using Taylor

series, Rosenblueth point estimate, and Monte-Carlo

simulation methods. Figures 11 and 12 present the

calculated reliability index versus the variation in the

coefficient of variation (COV) of the rock mass

cohesion and angle of internal friction. The results

indicated that no variation was observed between the

Taylor series and Rosenblueth point estimate meth-

ods when the Bishop and Janbu methods were used.

However, a slight variation was observed in case of

Monte-Carlo simulation method. Also, with increas-

ing of coefficient of variation (COV) of random

variables, value and trend of variation of Monte-

Table 3 Values of factor of safety for both slopes

Factor of safety (Mean)

Simplified Bishop’s method Simplified Janbu’s method

Taylor series Point estimate MCSM Taylor series Point estimate MCSM

30 Azimuth 1.179 1.223 1.182 1.14 1.184 1.141

140 Azimuth 1.774 1.788 1.787 1.672 1.684 1.683

Fig. 7 Geometry and types

of lithology-alteration for

30 Azimuth cross section

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Carlo simulation method will be as same as the other

two methods.

7 Discussions

It was shown in this study that the selected seed

random number generator does not affect

Monte-Carlo simulation results. It was also shown

that sample size of 1,200 or greater is sufficient for

Monte-Carlo method to converge in the same

reliability index (b) or probability of failure (Pf).

But to obtain proper random variables distribution

(cohesion and internal friction angle) and also proper

distribution of safety factor, 10,000 iterations were

used. It was also found that the probability

Fig. 8 Variation of factor

of safety reliability index

with rock mass

uncertainties using Bishop

method (30 Azimuth cross

section)

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distribution of the safety factor of such sample size

was normally distributed. Sample of size 10,000 also

confirmed that the distribution of the safety factor

was normal.

In all the tested cases using Bishop and Janbu

methods, it was observed that there was a significant

difference in the calculated reliability index using

Taylor series, Rosenblueth point estimate, and Monte-

Carlo simulation methods. With increasing of the

coefficient of variation (COV) of random variables,

value and trend of variation of Monte-Carlo simulation

will be the same as two mentioned methods.

The main purpose of two slopes (30 Azimuth cross

section with five rock types and 140 Azimuth cross

section with two rock types) consideration was that:

In equal situations of the coefficient of variation



with rock mass

uncertainties using Janbu


section)

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123

(COV) of random variables, the variances of reliabil-

ity index of safety factor decreased when the samples,

diversity and number of rock mass classification (or

rock type) increased (Figs. 8, 9, 11, 12). It was for

utilization of more random variables distribution in

30 Azimuth cross section (five rock types) than 140

Azimuth cross section (two rock types).

8 Conclusions

This study outlines a procedure of probabilistic

analysis of slope stability using Taylor series,

Rosenblueth point estimate, and Monet-Carlo simu-

lation methods. The obtained results of these methods

were compared using two recognized methods of

slope stability (simplified Bishop’s method and

simplified Junbu’s method). For convenience two

illustrative slope examples were used: the first slope

included five lithology changes and the second

included two lithology changes. Our findings warrant

the following conclusions:

1. A sample size of 1200 or greater is a good choice

for Monte-Carlo simulation method, while for

sample size less than 1200 the reliability index

showed a significant variation. Ideally, several

thousand values should be generated for obtaining

proper random variables distribution (cohesion

and internal friction angle) as well as proper

distribution of factor of safety (10000 iterations

were used).

2. For the both 30 and 140 Azimuth cross sections,

a good agreement is observed in the calculated

reliability index for Taylor series and Rosenblu-

eth point estimate methods. For the both slopes

slight variations exist between two mentioned

methods and Monte-Carlo simulation method.

3. With increasing of coefficient of variation (COV)

of random variables, value and trend of varia-

tions of Monte-Carlo simulation will be as same

as Taylor series and Rosenblueth point estimate

methods.

4. In equal situation of the coefficient of variation

(COV) of random variables for both of slopes (30

and 140 Azimuth cross sections), the differences

of reliability index is less where the samples,

diversity and number of rock mass classification

(or rock type) is more. It is for utilization of more

random variables distribution and decreasing

uncertainties.

5. The effect of variation in COV for the angle of

internal friction on the calculated reliability

index of factor of safety using the two methods

of slope stability was more than in the COV for

rock mass cohesion (sensivity analysis).

6. Value of the reliability index using simplified

Bishop’s method is more than simplified Janbu’s

Fig. 10 Geometry and

types of lithology-alteration

for 140 Azimuth cross

section

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123

method in the three reliability analysis methods

for 30 and 140 Azimuth cross sections.

7. It was understood that Taylor series and Rosenblu-

eth point estimate methods require fewer calcula-

tions and computing time compared to

Monte-Carlo simulation method. However, with

the advent of computers in data handling and speed,

Monte-Carlo simulation offers practical tool for

geotechnical engineering problem. The reliability

index of factor of safety was obtained from Monte-

Carlo simulation is more than the values from the

two other methods (Taylor series and Rosenblueth

point estimate methods), this effect is referred to

use much more random variables generated in the

Monte-Carlo simulation. Because of this the

Monte-Carlo simulation proved to be powerful

and effective scheme for more precision reliability

analysis of slope stability.



with rock mass

uncertainties using Bishop


section)

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123

References

Abbaszadeh M, Shahryar K, Sharifzade M, Heydari M (2006)

Classification and determination of Sungun copper mine

rock mass properties uncertainties. J Eng Geol Teach

Train Univ 2(2):377–394

Abdullah I, Waleed F, Fayez A (2000) Uncertainty and reli-

ability analysis applied to slope stability. Struct Saf 22:

161–187

Abramson LW (2002) Slope stability and stabilization meth-

ods. McGraw-Hill, New York

Alonso EE (1976) Risk analysis of slopes and its application

to slopes in Canadian sensitive clays. Geotechnique 26(3):

453–472

Baecher GB, Christian JT (2003) Reliability and statistics in

geotechnical engineering. Wiley, London



with rock mass

uncertainties using Janbu


section)

Geotech Geol Eng (2011) 29:581–596 595

123

Barabosa MR, Morris DV, Sarma SK (1989) Factor of safety

and probability of failure of rock fill embankments.

Geotechnique 39(3):471–483

Bishop AW (1955) The use of slip circle in the stability

analysis of slopes. Geotechnique 5:7–17

Chowdhury RN, Xu DW (1993) Rational polynomial technique

in slope stability analysis. J Geotech Eng Div 119(12):

1910–1928

Christian JT, Ladd CC, Baecher GB (1994) Reliability applied

to slope stability analysis. J Geotech Eng Div 120(12):

2180–2207

Cornell CA (1971) First-order uncertainty analysis of soil

deformation and stability. In: Proceedings of 1st interna-

tional conference on application of probability and sta-

tistics in soil and structural engineering (ICAPI), Hong

Kong, pp 129–144

Dai Y, Fredlund DG, Stolte WJ (1993) A probabilistic slope

stability analysis using deterministic computer soft ware.

In: Proceedings of conference on probabilistic methods in

geotechnical engineering, Canbera, Australia, 10–12

February. pp 267–274

Duncan JM (2000) Factor of safety and reliability in geo-

technical engineering. J Geotechn Geoenviron Eng ASCE

126(4):307–316

Hassan AM, Wolff TF (2000) Effect of deterministic and

probabilistic models on slope reliability index. In: Grif-

fiths DV et al (ed) Slope stability 2000, GSP No. 101

pp 194–208

Janbu N (1954) Application of composite slip surface for sta-

bility analysis. In: Proceedings of European conference on

stability of earth slopes, Stockholm, Sweden, pp 43–49

Lacasse S, Nadim F (1996) Uncertainties in characterizing soil

properties. In Proceedings, Uncertainty ’96. American

Society of Civil Engineers, Reston, Va. Geotechnical

special publication no. 58, vol 1, pp 49–75

Li KS, Lumb P (1974) Probabilistic design of slopes. Can

Geotech J 24:520–535

Matsuo M (1976) Reliability in embankment design. M.I.T.

Department of Civil Engineering Research Report R

76–33. Massachusetts Institute of Technology, Cambridge

Matsuo M, Kuroda K (1974) Probabilistic approach to design

of embankments. Soil Found 14(2):1–17

Park SK, Miller KW (1969) Random number generators: good

ones hard to find. Commun ACM 31:1192–1201

Park H, West TR (2001) Development of a probabilistic

approach for rock wedge failure. J Eng Geol 59:233–251

Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992)

Numerical recipes in Fortran: the art of scientific com-

putation. Cambridge university press, Cambridge

Rocscience Inc (2003) Slide, 2D Limit equilibrium slope sta-

bility for soil and rock slopes, user guide

Rosenblueth E (1975) Point estimates for probability moments.

Proc Natl Acad Sci Math 72(10):3812–3814

Tang WH, Yucemen MS, Ang AHS (1976) Probability-based

short tem design of slopes. Can Geotech J 13(3):201–215

Tobutt DC (1982) Monte Carlo simulation methods for slope

stability. Comput Geosci 8(2):199–208

Venmarcke EH (1977) Reliability of earth slopes. J Geotech

Eng Div ASCE 103(11):1227–1246

Wolff TF (1985) Analysis and design of embankment dam

slopes: a probabilistic approach. Ph.D. thesis, Purdue

University, West Lafayette

Wu TH, Kraft LM (1970) Safety analysis of slopes. J Soil

Mech Found Div ASCE 96(2):609–630

596 Geotech Geol Eng (2011) 29:581–596

123

Uncertainty and reliability analysis applied to slope stability

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