Probabilistic Stability Evaluation of Oppstadhornet

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O R I G I N A L P A P E R

Probabilistic Stability Evaluation of Oppstadhornet

Rock Slope, Norway

H. S. B. Duzgun R. K. Bhasin

Received: 4 January 2007 / Accepted: 16 April 2008 / Published online: 5 July 2008 Springer-Verlag 2008

Abstract Probabilistic analyses provide rational means to treat the uncertainties

associated with underlying parameters in a systematic manner. The stability of a 734-

m-high jointed rock slope in the west of Norway, the Oppstadhornet rock slope, is

investigated by using a probabilistic method. The first-order reliability method

(FORM) is used for probabilistic modeling of the plane failure problem in the rock

slope. The BartonBandis (BB) shear strength criterion is used for the limit state

equation. The statistical distributions of the BB criterion parameters, for whichcomprehensive data were collected and statistically analyzed, are determined by using

distribution fitting algorithms. The sensitivity of the FORM model for the BB criterion

is also investigated. It is found that the model is most sensitive to the mean value of the

residual friction angle (/r) and least sensitive to the mean value of the slope angle (bf).

It is also found that the standard deviation of joint compressive strength (JCS) causes

the greatest difference in the reliability index, which has the least sensitivity to the

change in the mean and standard deviation of joint roughness coefficient (JRC).

Keywords Rock slope Risk assessment Reliability Probabilistic slope stability

1 Introduction

Rock slope stability problems contain many uncertainties due to inadequate

information about site characteristics and inherent variability and measurement

errors in the geological and geotechnical parameters. Probabilistic modeling of the

H. S. B. Duzgun (&)

Mining Engineering Department, Middle East Technical University, 06531 Ankara, Turkey

e-mail: [email protected]

R. K. Bhasin

Norwegian Geotechnical Institute, P.O. Box 3930, Ullevaal Stadion, 0806 Oslo, Norway

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Rock Mech Rock Eng (2009) 42:729749

DOI 10.1007/s00603-008-0011-3


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rock slope stability problem allows the systematic treatment of these uncertainties.

In probabilistic slope stability analyses the uncertainties are taken into account

through the use of probability distributions, or the moments of the parameters.

Probabilistic rock slope stability analyses are also essential for quantitative risk

assessment. Risk is defined as the probability that a slope failure occurs within agiven period of time, multiplied by the consequences of the slope failure (IUGS

1997). Although there is still some reluctance among engineers to use probabilistic

methods, the need for probabilistic slope stability analysis for quantitative risk

assessment has led to their increased use in recent years.

One difficulty in probabilistic analyses is the lack of acceptable limits, i.e.,

probability of failure (Pf) or reliability index (b), so that the stability/instability

assessment can be made by the comparison of the computed reliability index of the

given slope and acceptable values. Such acceptable values are established for the

safety factor (SF) in deterministic analyses. Usually a SF of 1.3 for temporary and of1.5 for permanent slopes are considered to be acceptable in engineering practice in

deterministic analyses (Hoek and Bray 1981; Hoek 1997; Pine and Roberds 2005).

However, deterministic analyses do not involve treatment of uncertainties, as only

characteristic values of the uncertain parameters are used. This may lead to conditions

where there are different safety margins for slopes with the same factor of safety

(Nadim et al. 2005). Hence, the outputs of deterministic analyses cannot be

incorporated into quantitative risk analyses. A discussion of the pros and cons of the

deterministic and probabilistic safety evaluation of slopes is given by Christian (2004).

The establishment of acceptable values for b and Pf requires probabilisticstability evaluations of existing and failed slopes. For this purpose, the stability of

the 734-m-high Oppstadhornet rock slope in Norway is evaluated in this study using

the first-order reliability method (FORM). The Oppstadhornet rock slope, which has

been investigated by the Geological Survey of Norway (NGU), is located on the

west coast of Norway (Fig. 1) in the Mre and Romsdal County. The investigations

by NGU involved field studies to observe any recent movements in the slope and to

Fig. 1 Location of the Oppstadhornet rock slope on the West Coast of Norway (source: www.ngu.no)

730 H. S. B. Duzgun, R. K. Bhasin

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http://www.ngu.no/http://www.ngu.no/


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identify large-scale instability structures that could potentially collapse into the

fjord and result in generation of high waves or tsunamis (Blikra et al. 2001). NGU

conducted systematic mapping of rock slides and potential rock fall areas in the in

Mre and Romsdal County (west of Norway) to increase the knowledge and

awareness of the risk of such events. In this context, the Oppstadhornet rock slope isreported to be one of the major potential rock slide areas by NGU (Blikra et al.

2001).

Rock slope stability problems are mostly governed by rock discontinuities, which

is the case in Oppstadhornet. For persistent rock discontinuities, the failure is

controlled by the shear strength of the discontinuities. It is widely acknowledged

that shear failure along rock discontinuities exhibits nonlinear behavior (Barton

1976). Hence limit equilibrium functions should be nonlinear. In this paper the

plane failure problem is formulated by considering the BartonBandis shear failure

criterion.

2 Oppstadhornet Rock Slope

The Oppstadhornet rock slope is composed of granitoid gneiss with zones of schist.

The rocks have well-developed foliations striking ENE-WSW and dipping

moderately to steeply to the south. The slope area is described by Robinson et al.

(1997) as an unstable mountainside with a height of about 734 m and width of

several kilometers, stretching from near shore to the peak of the slope. Figure 2illustrates the location of rock blocks where movements have been observed on a

topographic map of the slope.

A considerable amount of large rock avalanches and bedrock failures were

noticed during NGUs investigations on the southern slope of the mountain. A

picture of the steep mountainside towards the southwest slope is given in Fig. 3

and a cross-section of the slope is displayed in Fig. 4. According to the profile

shown in Fig. 4 the major sliding plane is following the foliation in the upper part

of the slope whereas it is cutting through the foliation in the lower part of the

slope. Some of these structural features could be observed from aerial photos, as

shown in Fig. 4, while others were interpreted from field observations. It is also

reported by Blikra et al. (2001) that large blocks of several tens of cubic meters

show internal fracturing and sliding both along the foliation joints and on the

cross-joints (Fig. 4).

The average slope angle between the top of the mountain and the road at the foot

of the slope is about 36, but some sections of the slope are much steeper and some

have a shallower dip. In addition to joints along the foliation, there are also cross-

joints striking NWSE (perpendicular to the plane of the cross-section shown in

Fig. 4). The rock mass is divided into blocks of various sizes that range from several

cubic meters to several tens of cubic meters. At some places block sizes of several

hundred cubic meters exist.

The discontinuities in the rock are mostly characterized as rough and undulating.

Many of the discontinuities are tightly healed with nonsoftening impermeable

filling. A number of the joint walls are slightly altered with only surface stains.

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Some of the foliation joints are persistent in nature and could be followed for

hundreds of meters.

The volume of the entire potentially unstable mountain slope is estimated to be

1020 million m3. This is based on the demarcated area shown in Fig. 2, in which

the thickness of the potentially unstable masses has been estimated to be 5075 m in

the upper part and 25 m in the lower part of the slope. Three possible shear surfaces

(foliation surfaces in Fig. 4) have been indicated in the profile by NGU. This

Fig. 2 Topographic map of the slope with the rock blocks where movements have been observed (Blikra

et al. 2001)


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profile is based on mapping carried out in the field and through aerial photographs

(Braathen et al. 2003; see Fig. 4).

The Geological Survey of Norway concludes that the age of the movements is

less than 11,500 years (i.e., after deglaciation) since the area shows no indication of

being affected by processes related to permafrost conditions. This indicates that

Fig. 3 Photo of the slope (Bhasin and Kaynia 2004)

Fig. 4 Aerial photograph and cross-section of the slope with the foliation surfaces and potential blocks

prone to slide. The cross-section line (AB) is shown on the aerial photograph (Braathen et al. 2003)


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sliding has taken place after the most unstable condition during deglaciation. There

are also some indications of recent movements, but no measurements of possible

movements have been carried out.The identified large-scale instability has the potential to slide into the fjord and

result in generation of high waves or tsunamis (ICG 2004). Since a tsunami caused

by slope failure in Oppstadhornet can have an adverse effect on developed areas and

new development projects along the coastlines of the fjords, slope stability

evaluation is essential to assess the associated risk.

As can be seen from Figs. 2, 3, and 4, there are potentially three layers of rock

which can slide (Fig. 5). The first and the third layers in Fig. 5 are in the form of

single rock blocks (block 1 and block 3, respectively). The second layer is

composed of two blocks, namely block 2.1 and block 2.2 (Fig. 5). The slopegeometry given in Fig. 5 indicates a plane failure mode. Therefore, probabilistic

models are developed for the plane failure case.

3 Probabilistic Analysis of Stability

Stability analysis of a rock slope generally involves four steps: collection and

analyses of data, identification and analysis of failure mechanism, computation of

safety measures [factor of safety (SF) in deterministic analyses, reliability index

(b)/probability of failure (Pf) in probabilistic analyses], and evaluation of

computed safety indices. Data collection and analyses involve obtaining all the

relevant data for computation of resisting and driving forces on the slope.

Identification and analysis of failure mechanisms have two stages, namely

kinematic and kinetic. Kinematic analysis involves determination of potential rock

block instabilities in the rock mass for the considered study region based on the

discontinuity orientation. Usually kinematic analyses are followed by kinetic

analyses, where the potentially unstable rock blocks are investigated by limit

equilibrium or numerical analyses through computation of safety measures related

to each block.

In deterministic analyses, the safety measure is the SF defined by the ratio of the

sum of resisting forces to the sum of driving forces. In probabilistic analyses the

safety measure is either b or Pf. b is the minimum distance from the origin of

normalized basic variables to the limit state function (Fig. 6).

Fig. 5 Slope geometry and

potentially unstable blocks

shown in Fig. 4


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A normalized basic variable has a mean of zero and a standard deviation of one.

Pf is obtained either by Monte Carlo simulation or reliability-based methods. In

Monte Carlo simulation, the SF is evaluated many times; each time a probable valueof the parameters in the SF is sampled from their probability distribution functions.

Then the probability distribution of SF is obtained, which allows one to compute the

probability that SF will be less than 1 or any other specified safety level, i.e., Pf. In

reliability-based methods the computed b is related to Pf under certain satisfied or

imposed conditions.

Finally the computed safety measures are evaluated by comparing them with

some acceptable safety levels.

3.1 Data Collection and Analysis

The data for probabilistic stability analysis of the Oppstadhornet slope were

obtained from previously performed studies on the slope, which are mainly

numerical stability analyses (Bhasin and Kaynia 2004; Bhasin et al. 2004; Dahle

2004). The collected data from field investigations involve geological mapping of

the slope, identification of joint roughness profiles to be used for the BB criterion,

Schmidt hammer testing, and characterization of the exposed joint walls in the

slope. The Emodulus, Poissons ratio (m), and the uniaxial compressive strength (rc)

parameters were obtained by laboratory testing (Table 1). The joint inputparameters to be used in the BartonBandis shear failure criterion and the

mechanical properties of the intact rock used for the numerical modeling studies are

shown in Table 1.

Barton and Choubey (1977) have postulated that the reduction in JCS values due

to the scale effect roughly corresponds to the reduction ofrc with increasing sample

size. Furthermore, they have concluded that there is a significant scale effect on

JRC. The larger the base length considered, the less steep the asperities, which

results in reduced JRC values. The scale correction for in situ block sizes (Ln) is

derived using the following scale correction equations (Barton and Bandis 1990):

JRCn % JRC0Ln

L0

!0:02JRC01

Fig. 6 Graphical representation of the FORM approximation (Nadim et al. 2005)


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JCSn % JCS0Ln

L0

!0:03JRC02

For the present case a JRC0 value of 10 when corrected to full scale (Ln) gives a

JRCn value of 6.3 (Eq. 1). Likewise a JCSo value of 100 MPa when corrected to full

scale gives a JCSn value of 50 MPa (Eq. 2).

In addition, probabilistic modeling of rock slopes requires information about the

probability distributions of the basic variables as well as the moments of the

distributions. There are two main types of basic variables: those related to strength

parameters and those related to geometrical parameters. In this study, the basicvariables related to the strength parameters involve joint roughness coefficient

(JRC), joint wall compressive strength (JCS), and residual friction angle (/r) for the

BartonBandis shear failure criterion. The previous numerical studies for the

Oppstadhornet slope basically contain data for the BartonBandis shear strength

criterion. In this paper the raw data collected for JRC, JCS, and /r (Barton and

Bandis 1990) are statistically analyzed and distribution fitting tests are carried out to

determine appropriate probability distributions. Distribution fitting to the parameters

of the BartonBandis shear failure criterion was carried out using Bestfit software

(version 4.5, Bestfit 2005) and the details of distribution fitting for the strengthrelated basic variables are given in Sect. 3.3. Table 2 lists the descriptive statistical

properties of the considered shear strength parameters.

Geometrical parameters are slope height (H), width (W), and angle (bf) as well as

the discontinuity dip (bs). The discontinuity dip and slope angle are considered to be

Table 1 BartonBandis joint

parameters and intact rock

parameters

Parameters Estimated

mean values

Joint roughness coefficient, JRC0 12

Joint compressive strength, JCS0 (MPa) 89Laboratory scale length L0 (m) 0.1

In situ block size Ln (m) 1.0

Residual friction angle /r () 28

Uniaxial compressive strength rc (MPa) 100

Density (q) kN/m3 27.5

Poissons ratio (m) 0.25

Deformation modulus Ed (GPa) 40

Table 2 Summary statistics for

BartonBandis and Coulomb

shear failure parameters

Variable Min. Max. Mean "x Standarddeviation (s)

Number of

samples (n)

JRC0 6.00 20.00 11.93 4.18 29JRCn 4.55 7.96 6.53 1.04 29

JCS0 (MPa) 16.00 190.00 88.87 50.30 91

JCSn (MPa) 7.02 83.34 38.86 22.05 91

/r () 22.00 34.00 28.00 2.27 26


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basic variables. Since the slope height and width have relatively less uncertainty

they are taken as deterministic variables.

3.2 Analysis of Failure Mechanism

The analysis of the failure mechanism has two basic stages, namely kinematic and

kinetics. Kinematic analysis involves determination of potential rock block

instabilities in the rock mass of the considered study region based on the

discontinuity orientation. Kinematic analyses can be performed either determinis-

tically or probabilistically.

In a typical deterministic approach, the orientation of a particular discontinuity

set is represented by a characteristic orientation value, which is usually the mean.

Then, kinematic tests are performed (e.g., using stereographic projections) in order

to determine potential instabilities as well as possible failure modes. This approachis usually followed for investigating single block instability cases. For kinematic

analysis of multiple blocks, key block theory is widely implemented. Warburton

(1981) and Goodman and Shi (1985) developed the key block theory for movements

in translation, and Mauldon and Goodman (1996) adapted it for movements in

rotation. This theory has later been extended and applied by Mauldon and Ureta

(1996), Fulvio-Tonon (1998), and Sagaseta et al. (2001).

Generally, probabilistic approaches are derived from the stochastic treatment of

deterministic ones. Hence probabilistic kinematic instability approaches use the

same logic as deterministic kinematic methods; the only difference in probabilisticmethods is that they consider the underlying parameters in a stochastic manner.

There are two main types of approaches for assessing the probabilistic kinematic

instability in rock slopes. In the first type, which are also called lumped models, the

discontinuity properties such as spacing, trace length, and orientation are fitted to a

statistical or empirical distribution. Then Monte Carlo simulation is applied by

using the fitted distributions for discontinuity parameters, and kinematic tests are

performed to determine the probability of forming unstable blocks (e.g., McMahon

1971; Carter and Lajtai 1992). The basic shortcoming of the lumped models is that

discontinuity data obtained from different locations in the field are treated as if they

are the same throughout the study region. Hence lumped models do not consider the

spatial distribution and correlation of the discontinuity parameters (Nadim et al.

2005). To overcome this shortcoming, stochastic discontinuity models (e.g., Priest

and Hudson 1983; Einstein and Dershowitz 1996; Ivanova 1998) and geostatistical

methods (e.g., Carosso et al. 1987; Chiles 1988; Young 1993), which fall into the

second group of probabilistic kinematic analyses, are used.

The kinematic analysis is followed by kinetic (mechanical) stability analyses of

the slope. If the probabilistic approaches are used for the kinematic analyses, the

resultant probabilities are multiplied by the Pf values obtained from kinetic

analyses, since Pf is conditioned on the formation of unstable blocks (i.e., Pf given

that an unstable block forms, should be computed). Such conditioned approaches

are mainly used for rock slope design purposes as it is required to predict first the

probable formation of unstable rock blocks (e.g., Einstein et al. 1980; Feng and

Lajtai 1998) for a given slope geometry before combining this with the probability


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of kinetic failure, Pf. For existing or natural slopes the slope geometry is usually

observed and measured in the field, and kinematic analysis is usually unnecessary

prior to kinetic analysis. In the Oppstadhornet rock slide case, as the geometry of

potential rock blocks were observed by NGU, kinematic analyses were not

performed and only the kinetic stability of the blocks is modeled based onprobabilistic approaches. Probabilistic computation of the kinetic instability of the

slope involves the formulation of the limit state equation, which is obtained by

equating the safety margin (subtraction of the driving forces from the resisting

forces) to zero, identification of parameters in the limit state equation to be treated

as random variables (basic variables), assessment of probability distributions with

their moments for each basic variable, and analysis of the uncertainties in the basic

variables by using statistical models for the systematic treatment of various sources

of uncertainties.

The computed value of the probability of slope failure, Pf is basically dependenton the level of uncertainty in the basic variables, which is represented by the

standard deviation or coefficient of variation (c.o.v., a unit-less measure of

uncertainty, which is the ratio of the standard deviation to the mean value of the

basic variable). The higher the c.o.v. or standard deviation, the higher the Pf values.

Hence analyzing uncertainties associated with the basic variables in a systematic

way yields more realistic estimates of the slopes safety. The uncertainties result

from insufficient information and inadequate knowledge about the properties of the

basic variables involved in rock slope stability, such as spatial and temporal

variation in rock properties, limited data collection and laboratory testing,discrepancies between laboratory and in situ conditions, etc. Sources of uncertain-

ties affecting the rock slope stability parameters can be considered to be composed

of three components, namely, inherent variability, and statistical and systematic

uncertainties. The inherent variability is due to the fact that, even in a homogeneous

rock medium, the rock properties exhibit variability by nature. The limited sampling

and laboratory testing cause the statistics (i.e., mean, standard deviation) of a basic

variable to have uncertainty. This type of uncertainty is called statistical, because it

can decrease with increasing number of samples. The systematic uncertainties may

stem from the discrepancies between the laboratory and in situ conditions, due to

factors such as scale, anisotropy, and water saturation. Additional sampling may not

necessarily reduce this type of uncertainty, because the same test conditions are

likely to occur. Duzgun et al. (2002, 2003) presented a comprehensive statistical

model for quantification and analysis of the various uncertainties involved in

estimation of friction angle by aggregating inherent variability, and statistical and

systematic uncertainties within the first-order uncertainty analysis framework.

In the Oppstadhornet case, as there were not enough data to decompose the

uncertainties into inherent, systematic, and statistical uncertainty components, these

analyses cannot be performed. For a more detailed description of uncertainty analysis

on the basic variables see Ang and Tang (1984) and Duzgun et al. (2002, 2003).

The probable failure mode in the Oppstadhornet slope is plane failure (Figs. 2, 3,

4, and 5). Hence formulation of the limit state equation involves analysis of driving

and resisting forces for a plane failure case. The basic mechanism of plane failure is

best expressed by a sliding mass on an inclined plane. The mechanical principles


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state that sliding occurs when resisting forces are smaller than the driving forces that

are parallel to the sliding plane. The mechanical principle of sliding is basically due

to gravitational loading on an inclined plane. Figure 7 illustrates the components oftypical forces acting on a rock block in the plane failure case.

In the analysis of the system of forces the stability of a unit slice of rock in Fig. 7,

is considered. It is convenient to analyze the forces G, U, and V in terms of their

components that lie parallel to the sliding plane, which form the driving forces, and

that are normal to the sliding plane, which contribute to the resisting forces. The

parallel and normal force components are listed in Table 3. Forces that tend to

activate sliding or compress the sliding plane are taken as positive. The details of

this formulation are given by Priest (1993).

The description of the forces listed in Table 3 is as follows:

G: weight of the sliding block

U: water force on the plane of sliding

V: water force in the tension crack

where;

bs: angle of the sliding plane

bc: angle of the tension crack

Fig. 7 Forces on the rock block in the plane failure case (Priest 1993)

Table 3 Forces acting on thesliding block shown in Fig. 7 Force Parallel component Normal component

G GP = G sin bs GN = G cos bs

U UP = 0 UN = -U

V VP Vsin bc bs VN Vcos bc bs


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bf: angle of the slope

bt: angle of the top of the slope

In the plane failure case, the performance function, gx; can be defined as the

difference of the resisting forces, Rf and the driving forces, Df, as given in Eq. 3.gx Rf Df; 3

where x is a vector of basic variables and

Df GP UP VP; 4

where

GP: the component of the weight of the block (G) parallel to the sliding plane AD

in Fig. 7

UP: component of the water force (U) parallel to the sliding plane AD in Fig.7

VP: component of the water force (V) in the tension crack (CD) parallel to the

sliding plane AD in Fig. 7

LAD: length of the sliding plane AD in Fig. 7

GN: component of the weight of the block (G) normal to the sliding plane AD in

Fig. 7

UN: component of the water force (U) normal to the sliding plane AD in Fig. 7

VN: component of the water force (V) in the tension crack (CD) normal to the

sliding plane AD in Fig. 7

Rf for the BartonBandis criterion is defined by

Rf spLAD 5

where

sp: peak shear strength of the foliation joint

The peak shear strength of foliation joint is computed based on the Barton

Bandis nonlinear shear failure criterion as given in Eq. (6).

sp r0n tan JRC log JCSrn

/r

!6

where

r0n: effective normal stress (MPa)

JRC: joint roughness coefficient

JCS: joint wall compressive strength (MPa)

/r: residual friction angle

3.3 Computation of Safety Indices

The computation of safety indices requires the identification of basic variables

(parameters to be considered as random variables). For rock slopes, parameters can

be divided into geometrical parameters (e.g., slope height, width, slope angle,


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existence of joints and cracks and their geometry, discontinuity dip) and mechanical

parameters (e.g., strength, groundwater condition). If the discontinuities are through

going or persistent, mechanical parameters have relatively higher uncertainty than

geometrical parameters, since geometrical parameters can be directly measured

whereas mechanical parameters require prediction based on laboratory testing, insitu investigations, and engineering judgment. However, if the discontinuities are

impersistent, the failure will be governed by both the intact rock failure between the

impersistent discontinuities and the failure along the discontinuities (Einstein et al.

1983), uncertainties in geometrical parameters are higher than those in mechanical

parameters.

In order to determine appropriate probability distribution functions for the basic

variables, data summarized in Table 2 are used for distribution fitting analyses, The

data collected by the Norwegian Geotechnical Institute (NGI) during field surveys

for the parameters of JRC, JCS, and /r, which are the parameters of the BartonBandis shear strength criterion, were fed to distribution fitting procedure of Bestfit

software (version 4.5 2005). The results of distribution fitting analyses are given in

Table 4. As can be seen from Table 4, JRC fits best to triangular, beta, and uniform

distributions, while the best fitting distributions for JCS are beta, triangular,

uniform, Weibull, gamma, and log-normal. The best fitting distributions for the

angles of/r, bs, and bfare found to be triangular and uniform. Except for JCS, all of

the basic variables fit to bounded distributions. This is also consistent with the

nature of the parameters and the recent literature (Low 2007; Jimenez-Rodriguez

and Sitar 2007) as the values of JRC, /r, bs, and bf are defined for a range. JRC isdefined between 0 and 20. The theoretical value for /r is between 0 and 90.

Although theoretically JCS can have values between zero and infinity, it takes on

values within a range for a given rock discontinuity. The triangular, beta, and

uniform distributions are the three best fitting distributions for JRC and JCS

(Table 4). The triangular and uniform distributions are found to be suitable for the

basic variable /r (Table 4).

In the literature Park and West (2001) assumed a normal distribution for friction

angle in probabilistic analysis of a rock slope. However, Park and West (2001) refer

to Hoek (1997) who suggested the use of a truncated normal distribution for friction

angle since using a normal distribution may give unreasonably low or high values.

Muralha and Trunk (1993) used the log-normal distribution in their probabilistic rock

slope stability analyses. Recently, Jimenez-Rodriguez and Sitar (2007) used the beta

distribution for friction angles in analyzing wedge stability, stating that the beta

Table 4 Fitted distribution for the basic variables

Basic variable Range Fitted distributions, decreasing degree of fit

JRC 4.68.0 Triangular, beta, uniformJCS (MPa) 783 Beta, triangular, uniform, Weibull, gamma, log-normal

/r () 2234 Triangular, uniform, beta

bs () 2316 Uniform

bf () 3521 Uniform


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distribution prevents problems of unbounded distributions. Moreover, the uniform

distribution was used for the geometric variables (angles related to wedge geometry)

in the study of Jimenez-Rodriguez and Sitar (2007). Similarly, Low (2007) modeled

a rock slope stability problem in Hong Kong by using the beta distribution for friction

angle, cohesion, and horizontal distance of the tension crack behind the slope crestand indicated that the beta distribution is advantageous over the normal distribution

due to its versatility and flexibility for bounded basic variables.

Consistent with the recent literature (Low 2007; Jimenez-Rodriguez and Sitar

2007), when the available data for the Oppstadhornet case are statistically analyzed

it was found that distributions which are defined for a range are suitable for the

strength parameters of the BartonBandis shear strength criterion. In fact this

finding is consistent with engineering practice as the strength of rock discontinuities

is usually defined for a range of values. In this paper, for ease of computation, the

triangular distribution is used for /r, JRC, and JCS.For the parameters where the data are insufficient to fit a probability distribution,

assumptions are made. As the possible values for the parameters bs and bf are

known inside a range, the uniform distribution is found to be the most suitable.

The computation of Pf and b depends on parameters of the failure criterion (/r),

joint roughness coefficient (JRC), and joint compressive strength (JCS). The results

of the uncertainty analysis are used as input to the probabilistic slope stability

models. Then failure probability (Pf) for the considered rock slope as well as its

reliability index (b) can be evaluated based on Monte Carlo simulation (MCS) or the

first-order reliability method (FORM). In this study FORM is used for theassessment of the probability of slope failure. In FORM, it is easy to perform

sensitivity analyses through the use of direction cosines of the basic variables. The

first step in FORM is the formulation of the performance function, g(X), where X is

the vector of basic variables, which are the stability analysis parameters defined as

random variables. The performance function (Eq. 3) is defined by the safety margin,

which is the difference between resisting forces, Rf (Eq. 5), and the driving forces,

Df (Eq. 4). Then Pf is defined by Eq. 7, if the joint density function of all basic

variables, Fx(X), is known (Nadim et al. 2005).

Pf Z

gX\0

FxXdx 7

Since the analytical solution of the integral in Eq. 7 is generally impossible, Pf is

computed by an approximation. If the basic variables are not normally distributed

and/or correlated with other basic variables, the vector of the basic variables (X) is

transformed to the standard normal space U, where U is the vector of independent

Gaussian variables with mean and standard deviation of 0 and 1, respectively. The

function, g(U) becomes a linear function and takes the following form:

gU Xni1

aiUi b 8

where ai is the direction cosine of the transformed basic variable Ui, b is the

distance between the origin and the hyperplane g(U) = 0, and n is the number of


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basic variables. The graphical representation of reliability index and the perfor-

mance function is given in Fig. 6. Then Pf is given by Eq. 9

Pf PgU\

0 PXni1 a

iUi b\

0" #

1 Ub 9

The basic steps of the FORM approximation algorithm are:

1. Transformation of the vector of basic variables into a standard Gaussian vector

2. Finding the most likely point or design point for the basic variables in the

failure domain

3. Estimation ofPf from Eq. (9)

As it is difficult to predict the design point, an initial guess is made for it, which is

usually the point formed by the mean values of the basic variables. The algorithm isiterated until sufficient convergence is obtained. Because of the iterative nature of

the algorithm, the approximation may not converge, especially for highly nonlinear

performance functions. FORM also allows one to use correlated basic variables. In

this case, the algorithm involves an additional transformation of basic variables into

an uncorrelated space. Examples for the use of FORM models for rock slope

stability for various failure modes are given by Kimmance and Howe (1991), Trunk

(1993), Quek and Leung (1995), Low (1997), Duzgun et al. (2003), and Bafghi and

Verdel (2004).

For the Oppstadhornet rock slope, the probability of slope failure (Pf) and itsreliability index (b) are evaluated based on FORM for the BartonBandis shear

strength criterion, with uncorrelated basic variables. In FORM modeling of

Oppstadhornet rock slope, the basic variables are treated as independent and

distributions related to each basic variable are determined by fitting distribution

functions to the sampled data. The FORM analyses were carried out by using

Comrel software.

The probabilistic analyses for Oppstadhornet are performed by evaluating b for

the likelihood of various block failure scenarios. Then, the sensitivity of b to

changes in the basic variables in FORM is also investigated.

Since there is no information about the existence and measurement of water in

the tension crack and on the sliding plane, the effect of water force on the stability

cannot be investigated. However, any decrease in frictional properties of the

discontinuities, which can be caused by formation of gauge material during

movements, existence of water, weathering, etc., is investigated by calculating Pfand b for the mean values of/r of 28, 25, and 20.

The potentially unstable blocks in Fig. 5 can fail in various ways:

Scenario 1 (AI). All the blocks (blocks 1, 2.1, 2.2, and 3) fail at the same time as

a single block, (1 + 2.1 + 2.2 + 3). Failure is on discontinuity plane 3. Scenario II (AII). Only block 1 fails and the rest are stable (1). Failure is on

discontinuity plane 1.

Scenario III (AIII). Blocks 1, 2.1, and 2.2 fail at the same time as a single block,

block 3 remains stable (1, 2.1, 2.2). Failure is on discontinuity plane 2.


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Scenario IV (AIV). First block 1 fails, then blocks 2.1 and 2.2 fail at the same

time (1, 2.1 + 2.2). First failure is on discontinuity plane 1 and second failure is

on discontinuity plane 2.

Scenario V (AV). First block 1, then sequentially block 2.1, block 2.2, and finally

block 3 fail (1, 2.1, 2.2, 3). The first, second, and third failures are ondiscontinuity planes 1, 2, and 3, respectively.

Since the slope geometry in Fig. 5 for each block on the slope is different,

computations were carried out by using the geometrical properties of each block

illustrated in Fig. 7 (Table 5). The same distribution functions as discussed in the

previous paragraphs are used for the evaluation of each scenario. Moreover, the

distributions and moments of the basic variables (Table 2) related to the strength

parameters are kept the same in the evaluation of each scenario.

Among the five possible slope instability scenarios, Pf for AIAIII is the direct

calculation of the slope failure probability. The scenarios AIV and AV on the other

hand, require the formulation of Pf based on the conditional probabilities, since the

occurrence of these scenarios are conditioned on first the failure of block 1, then the

failures of blocks 2.1 and 2.2 (scenario IV), and finally the failure of block 3

(scenario AV). Hence, Pf for scenarios AIV and AV are given in Eqs. (10) and (11),

respectively;

PfIV Pblock 1 fails Pblock 2:1 and block 2:2 fail togetherjblock 1 failed

10

PfIV Pblock 1 fails Pblock 2:1 failsjblock 1 failed

Pblock 2:2 failsjblock 1 and block 2:1 failed Pblock 3 failsjblock 1; block 2:1 and block 2:2 failed 11

In Table 6, the calculated Pf for the three mean values of/r values for the four

scenarios are given. The highest Pf value is obtained for AII, which involves the

failure of only block 1. AI has the second highest Pf value, indicating the failure of

all the potentially unstable blocks. The Pf value for scenario V is the lowest.

Therefore, it is recommended that in the Oppstadhornet rock slope the potential

failure of block 1 and the failure of the slope as a whole should be considered for theevaluation of landslide risk. As can be seen from Table 6, there is approximately

one order of magnitude difference between the Pf values computed for /r of 28,

25, and 20, respectively. Hence this suggests that a potential decrease in the

frictional properties of the discontinuities should be carefully investigated.

Table 5 Geometrical

parameters of rock blocksInvolved

blocks

Geometrical parameters

Height

H (m)

Width

W (m)

Slope angle

bf ()

Discontinuity

dip bs ()

1 320 171 35 27

2.1 15 13 45 28

2.2 329 140 26 22

3 240 217 21 16


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FORM analyses allow one to investigate the sensitivity of the reliability index b

to the moments (mean and variance/standard deviation) of the basic variables using

the direction cosines. Figures 8 and 9 show the sensitivity ofb to changes in mean

and standard deviation of the basic variables, respectively. When the effects of

mean values of the basic variables are examined, it can be seen that /r has the

strongest influence on b (Fig. 8). The second most influential basic variable on b is

the dip of the discontinuity (sliding) plane (bs), while b is least affected by thechange in the mean value of slope angle (bf) (Fig. 8). In FORM formulated for the

BartonBandis shear failure criterion, a change in the standard deviation of JCS

causes the greatest difference in b (Fig. 9). Figure 9 also illustrates that b has the

second greatest sensitivity to the changes in the standard deviations of/r and bf.

Table 6 Pf values for the five scenarios

Block/scenario Individual block Pf Scenario Pf

/r = 28 /r = 25 /r = 20 /r = 28 /r = 25 /r = 20

1 5 9 10-3 3 9 10-2 2 9 10-1

2.1 4 9 10-4 2 9 10-3 3 9 10-2

2.2 9 9 10-5 8 9 10-4 2 9 10-1

2.1 + 2.2 6 9 10-5 7 9 10-4 2 9 10-2

3 3 9 10-7 4 9 10-6 9 9 10-4

I 1 9 10-3 7 9 10-3 1 9 10-1

II 5 9 10-3 3 9 10-2 2 9 10-1

III 7 9 10-5 8 9 10

-4 2 9 10-2

IV 3 9 10-7 2 9 10-5 4 9 10-3

V 5 9 10-17 2 9 10-13 1 9 10-7

Fig. 8 Sensitivity of the reliability index to changes in the mean value of the basic variables


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variables in the probabilistic models or associating the computed Pf value with the

frequency of failures in the region (Nadim et al. 2003). As such data are not

available for the Oppstadhornet case the annual probability cannot be computed.

However, the approach proposed in this study can form the basis of quantitative

relative hazard assessment and a guide for decision makers investing in detailedworks.

4 Conclusions

Probabilistic analyses treat rationally and explicitly the uncertainties in all the

analysis parameters. The case study of Oppstadhornet illustrates that probabilistic

methods can easily be used for rock slope stability evaluations in engineering

practice.From the sensitivity analyses performed for the Oppstadhornet rock slope, it is

found that the prediction of strength parameters plays a critical role in the safety

evaluation. Hence, more effort should be spent on the data collection for strength

parameters. In addition, comprehensive uncertainty analyses of the strength

parameters could provide more-realistic estimates and increase the reliability of

the probability estimates.

The calibration study for determining acceptable Pf and b values for rock slopes

is needed for improved stability assessments. Although there are some recommen-

dations for acceptable Pf in the literature, they are inconsistent as the conditions forwhich they are computed are not indicated. As the evaluated values of failure

probabilities and reliability indices in this study apply to a real rock slope, they

could serve as a basis for future calibration studies.

Acknowledgments This paper was prepared during the first authors postdoctoral fellowship at the

International Centre for Geohazards (ICG) at the Norwegian Geotechnical Institute (NGI) in Oslo. The

authors would like to thank their colleagues from ICG, NGI, and NGU for their valuable discussions and

contributions. Professor Kaare Heg and Dr. Suzanne Laccase are thanked for their critical review of the

manuscript. The views expressed in this paper are those of the authors and not necessarily of the above-

mentioned organizations.

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Probabilistic Stability Evaluation of Oppstadhornet

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