indian geotechnical conference (igc-2010)igs/ldh/files/igc 2015 pune/theme 15...sahithi arukonda, b...

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50th INDIAN GEOTECHNICAL CONFERENCE

17th – 19th DECEMBER 2015, Pune, Maharashtra, India

Venue: College of Engineering (Estd. 1854), Pune, India

RELIABILITY ANALYSIS OF SOIL SLOPES USING ORDINARY AND BISHOP

METHOD OF SLICES

Sahithi Arukonda1, B. Munwar Basha2, K.V.N.S. Raviteja3

ABSTRACT

Slope stability analysis is a classical problem of geotechnical engineering characterized by many sources

of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the

analysis. Current practice of slope stability analysis relies in the deterministic characterization and

assessment of performance of embankments, excavations and Municipal Solid Waste (MSW) landfills.

These slopes have been evaluated in terms of the factor of safety, where the shear strength mobilized along

the failure envelop is compared with the shear stresses generated due to self-weight of the soil mass and

surcharge loading on the slope. The significant uncertainties associated with the shear strength and shear

stresses render deterministic modeling potentially misleading. For example, two slopes with the same factor

of safety can have significantly different probabilities of failure. The traditional engineering approaches

like method of slices used for evaluating the slope stability are frequently questionable because they do not

adequately account for uncertainties included in analytical modeling and natural variability. The present

work builds on probabilistic assessment approaches to develop reliability based design optimization

(RBDO) methodology. Moreover, RBDO quantifies the contribution of uncertainty to engineering analyses

of slope factors of safety and thereby produce a more accurate and informative method in geotechnical

sustainability of slopes. The reliability index or probability of occurrence or probability of failure is directly

influenced by how well the slope mechanism is understood, and how much uncertainty exists with the

performance of sliding limit state. Therefore, a probabilistic slope stability analysis should account for

inherent uncertainty and modeling uncertainty. The mean and standard deviations associated with unit

weight, cohesion and angle of internal friction of the soil are taken into account in the probabilistic

optimization. Reliability analysis of soil slopes is presented using first order reliability method (FORM) i.e.

Hasofer-Lind method. The results of these methods are compared using two recognized methods of slope

stability. These are Ordinary method of slices (OMS) and simplified Bishop's method (BMS). A limit state

function is formulated against sliding failure. Reliability indices against sliding failure using OMS and

BMS have been computed. Then, a procedure is presented for locating the surface of minimum reliability

index for slopes.

1Former M.Tech Student, Dept. of Civil Engg., IIT Hyderabad, India, [email protected] 2Asst. Professor, Dept. of Civil Engg., IIT Hyderabad, India, [email protected] 3Research Scholar, Dept. of Civil Engg., IIT Hyderabad, India, [email protected]

mailto:[email protected]



Sahithi Arukonda, B Munwar Basha, KVNS Raviteja

This paper advances reliability based slope stability approaches and strategies that address such

uncertainties. Probabilistic methods have been successfully applied taking into account two primary

categories of uncertainties: natural variability and modeling uncertainties. The following figure shows the

coordinates of critical centers and reliability indices obtained from RBDO methodology.

Fig. 1 Comparison of deterministic and probabilistic critical centers for ordinary and Bishop method of

slices

Keywords: Reliability, slope stability, uncertainty, variability, FORM, ordinary method, Bishop’s method

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RELIABILITY ANALYSIS OF SOIL SLOPES USING ORDINARY AND BISHOP

METHOD OF SLICES

Sahithi Arukonda, Former PG Student, Dept. of Civil Engineering, IIT Hyderabad, [email protected]

B Munwar Basha, Asst. Professor, Dept. of Civil Engineering, IIT Hyderabad, [email protected]

KVNS Raviteja, Research Scholar, Dept. of Civil Engineering, IIT Hyderabad, [email protected]

ABSTRACT: Soil slopes often tend to fail if designed by neglecting the variability associated with the geotechnical

parameters of the soil involved in slope. The traditional engineering approaches like method of slices used for

evaluating the slope stability are frequently questionable as they do not adequately account for the uncertainties,

variability associated with shear parameters of the soil and modelling errors. Therefore, a probabilistic slope stability

analysis is proposed to account for inherent uncertainty and modelling uncertainty. The present work builds on

probabilistic assessment approach to develop a reliability based design optimization (RBDO) methodology for slope

stability analysis. Reliability analysis of soil slopes is presented using first order reliability method (FORM) by

employing Ordinary method (OMS) and simplified Bishop's method of slices (BMS).

INTRODUCTION

Geotechnical engineers have to deal with materials

and geometries provide by the nature. These

conditions are not predefined and hence must be

inferred via intense observational and experimental

studies which are often costly. Uncertainties arise

when we look for accuracy of these experiments, in

modelling in-situ conditions of the soils perfectly in

a laboratory and in prediction of the resistances that

the materials will be able to mobilize. The

uncertainties in geotechnical engineering are

inductive i.e. there are limited observations to begin

with, the judgment of the engineer isn’t reliable, and

the limited knowledge of geology, and the statistical

reasoning’s that are employed to infer the behaviour

undefined naturally occurring materials.

Decisions have to be made on the basis of

information which is limited or incomplete. For

instance, there might exist spatial variability in the

strength of soil in a slope, the measurement of

parameters might not be perfect and there is a

possibility that the samples collected do not

correctly represent the entirety of the slope material.

Hence there is a considerable uncertainty with

regards to our knowledge of the input parameters.

Geotechnical engineers deal with uncertainties by

recognizing that risk and uncertainty are inevitable

and by applying the observational method [1] to

maintain control over them. However, the

observational method is applicable only when the

design can be changed during construction on the

basis of observed behaviour. The critical behaviour

cannot be observed until too late to make changes,

the designer must rely on a calculated risk. It is,

therefore, desirable to use methods and concepts in

engineering planning and design which facilitate the

evaluation and analysis of uncertainty. Traditional

deterministic methods of analysis, which use the

factor of safety as a measure of safety, must be

supplemented by methods which use the principles

of statistics and probability. These latter methods,

often called probabilistic methods, enable a logical

analysis of uncertainty to be made and provide a

quantitative basis for assessing the reliability of

foundations and retaining structures. Consequently,

these methods provide a sound basis for the

development and exercise of engineering judgment.

Occasionally a range of factor of safety values

ensures the long term stability of slopes where

uncertainties are involved in the soil properties [2].

But different slopes with the same factor of safety

can have significantly different probabilities of

failure [3]. In this study, we are mainly concerned

about the uncertainties involved in slope failure

mechanism and their influence on the overall slope





stability analysed using ordinary and Bishop

methods of slices.

Reliability Analysis of Soil Slopes

Over the years, a more formal way of dealing with

the uncertainties has been developed by applying

the reliability theory to geotechnical engineering.

Reliability studies provide a way of quantifying

those uncertainties and handling them consistently.

As discussed above, the deterministic method do not

take into account the variation or the uncertainty in

the various parameters that are involved in the

calculation of the stability. When uncertainty in the

parameters exists, the factor of safety, which is

dependent on these parameters may not be a

consistent measure of the stability of slopes.

Therefore, slopes with the same factor of safety can

have different levels of probability of failure

depending on the variability of those design

parameters. As deterministic slope models do not

take into consideration the associated variability and

use only average input parameters, may provide

misleading results for slope reliability. Reliability

calculations provide a means of evaluating the

combined effects of uncertainties. Moreover, it also

provide a means of distinguishing between

conditions where uncertainties are particularly high

or low. Reliability is theoretically defined as the

probability of success i.e.

1 fR P (1)

where, fP = probability of failure

Reliability also accounts for the heterogeneity of the

system under consideration. The effect of the

various parameters which vary continuously across

the system are called the random variables.

Literature review on reliability analysis of soil

slopes

Hassan & Wolff [4] proposed an algorithm to search

for the minimum reliability index for soil slopes.

They investigated the similarities and differences of

the surface of minimum factor of safety (FS) and the

surface of minimum reliability index ( ). The first-

order second-moment method (FOSM) and Monte

Carlo simulation (MCS) methods used for

measuring variability associated with the cohesion

of the soil (c), the angle of internal friction (ϕ) and

the unit weight of the soil (γ) for the reliability

analysis of the soil slopes [5]. Low [4] implemented

Spencer’s method of slices for probabilistic

approach to slope stability. First order reliability

method (FORM) can be coupled with the method of

slices for evaluation of factor of safety and

reliability index simultaneously [7]. In the

probabilistic analysis of slope stability, the input

parameters which are essentially the engineering

and shear parameters of the soil, are considered as

random variables. Xue and Gavin [8] reported an

approach to calculate the minimum reliability index

with the variability associated with the soil

properties.

Slope reliability analysis provide a means of

evaluating the combined effects of uncertainties in

the parameters involved in the calculations. The

computational effort that goes into probabilistic

analysis is much more than that required for

deterministic analysis. As stated before, experience

and engineering judgement are required to establish

an order of magnitude of the acceptable failure

probability. The acceptable probability of failure

depends on the importance and service time of a

slope as well as the consequences of failure. Both

deterministic and probabilistic methods should be

performed and these alternative methods can be

considered as complementary to each other.

Reliability Based Design Optimization (RBDO)

As the traditional factor of safety based design does

not include uncertainties of in the soil properties,

there is a need to apply reliability based design

optimization (RBDO). A target reliability based

design optimization method developed by Basha

and Babu [9] is used in this work to obtain a

reliability index considering the variability

associated with cohesion, the unit weight and the

angle of internal friction of the soil. RBDO has been

carried out in order to study the influence of

uncertainties associated with soil properties and

geometry of the slope on the critical slip surfaces.

OBJECTIVE OF THE PRESENT STUDY

The above literature clearly indicating the fact that

there is a high degree of variability and uncertainty

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involved in the shear parameters of soil slope which

have not been incorporated in the calculations of

factor of safety. Though a few studies reported

regarding the probabilistic analysis of soil slopes,

the complete mechanism behind the slope failures

has not been understood completely. This paper

presents the reliability based analysis of soil slopes

by ordinary method of slices (OMS) and Bishop’s

simplified method of slices (BMS). The present

investigation focused on slope reliability analysis to

locate critical failure surfaces and the corresponding

reliability indices by taking into account the

variability associated with the shear strength

parameters and unit weight of the soil.

FORMULATION

Slope stability analysis has been done by using

OMS and BMS. The performance function of the

slope against sliding failure is then formulated.

Ordinary Method of Slices

The OMS is the basic and simplest method for

determining the factor of safety (FS) against slope

failure. It considers the circular slip surface which is

divided into ‘n’ number of vertical slices and uses

moment equilibrium about the centre of slip surface

to calculate the factor of safety. This method ignores

the inter-slice shear and normal forces.

As shown in Fig. 1, the geometry is formulated by

considering the toe as the origin (0, 0) for the entire

slope. The inputs include the center of the circular

slip surface (xo, yo) for which the FS has to be

calculated, the geometric parameters H and α, the

soil properties γ, c and ϕ and the number of slices

(n). The geometry and various parameters used are

shown in Fig. 1. The crest point (p, q) can be

calculated using the Eq. 2 given below:

cotp H , q H (1)

Fig. 1 Slope geometry and ordinates of the section

Geometry of the individual ith slice is shown in Fig.

2. The x co-ordinates of the ith slice on the slip

surface xi can be found using

2ix p ib

(2)

(3.32)

Based on this corresponding yi’s can be computed

such that yi < q (as they should lie within the

geometry of the slope). To determine the heights of

each slice, we use an additional parameter h1i such

that if i0 < x < p , then

1i i

qh x

p (3)

else, 21ih q (4)

Thus heights of slices can be determined as

1i i ih h y (5)

The area of each slice can then be determined

using

1

1( )

2i i iz h h (6)

The weights are given by W =γbzi i (7)


Fig. 2 Schematic representation of coordinates of

the ith slice

The inclinations of each slice θi are calculated using

1 1

1

tan i ii

i i

y y

x x

(8)

The factor of safety against slope failure can be

computed using Eq. 9

( cos tan )

sin

i i i

OMS

i i

cdl WFS

W

(9)

where, c = cohesion of the soil, ϕ = angle of internal

friction of soil, Wi = the weight of ith slice, γ = unit

weight of soil, b = base length of each slice, zi =

average height of ith slice, H = height of the slope, α

= the angle the slope with the horizontal, dli = length

along slip surface of ith slice = b sec θi, θi = the

inclination of the base length of the slice with the

horizontal.

Bishop’s Simplified Method

The Modified (or Simplified) Bishop's Method

proposed by Alan W. Bishop is a most widely used

method for calculating the stability of slopes. It is an

extension of OMS. By making some simplifying

assumptions, the problem becomes statically

determinate. The vertical interslice forces on the

sides of each slice are considered and horizontal

interslice force are neglected.

By using all the parameters given in Eq.’s 1-8, the

factor of safety against slope failure can be given as:

cos tan

sin

i i i

Bishop

i i

cdl W

mFS

W

(10)

where, sin tan

cos ii

Bishop

mFS

(11)

Grid and Radius search technique

In order to find the global critical factor of safety, a

range of areas which will cover several points and

different radii has to be generated. One of the way

to do this is to use the Grid and Radius method. The

idea is to create a grids which covers the possible

positions of the center and radii of the slip surfaces,

divide them into desired number of parts and search

each possibility. The co-ordinates of these grids are

given manually in the initial stages.

Centre Grid

For the grid, let (x1,y1), (x2,y2), (x3,y3) and (x4,y4)

are the top left, bottom left, bottom right and top left

co-ordinates respectively as shown in Fig. 3. It is

assumed that ‘nl’ and ‘nb’ are the number of parts

that the centre grid has to be divided into lengthwise

and breadthwise respectively, then the co-ordinates

of various points on the grid can be given as a

combination of xi and yi as follows:

i 1 lx x ( 1)i m (12)

i 1 b( 1)y y i m (13)

where, i varies from 1 to nl + 1 for xi and 1 to nb+1

for yi. The ‘ml’ and ‘mb’ can be expressed as

follows:

4 1l

l

x xm

n

(14)

4 3b

b

y ym

n

(15)

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Fig. 3 Geometry and coordinates of the center grid

Radius Grid

The radius grid is similarly generated as (X1, Y1),

(X2, Y2), (X3, Y3) and (X4, Y4) and is divided into nr

parts breadthwise as shown in Fig. 4. The sides of

the grid are assumed to be at an angle so as to

provide some diversification in the calculation of

the radius values. The corresponding co-ordinates

along the both breadths are then joined to make

lines.

1 21

r

Y Yny

n

(16)

4 32

r

Y Yny

n

(17)

The radius of the slip circle will be the perpendicular

distance to this line from the center under

consideration.

1 21

1 2

Y Yd

X X

(18)

3 42

3 4

Y Yd

X X

(19)

The width of the radius grid along length wise can

be given as:

11

1

nynx

d (20)

22

2

nynx

d (21)

Fig. 4 Geometry and coordinates of the radius grid

Increments in the coordinates along x & y directions

on the left side can be given as:

1 2 1( 1)ixx X i nx (22)

1 3 1( 1)iyy Y i ny (23)

Increments in the coordinates along x & y directions

on the right side can be given as:

2 3 1( 1)ixx X i nx (24)

2 3 2( 1)iyy Y i ny (25)

Based on the above coordinates, the slope of the line

can be determined as:

2 1

2 1

i ii

i i

yy yym

xx xx

(26)

1 1i i i icc yy m xx (27)

The radius of the slip circle can be given as:

21

o i o i

i

i

abs y m x ccR

m

(28)

A MATLAB program is coded to find FSOMS and

FSBishop. The individual combinations of the points

of the grid and their corresponding radii are


calculated and fed into the MATLAB program to get

the individual values for factor of safety. Once the

factors of safety for a given centre and different radii

combination are calculated, the least of these values

is assigned as a value to the position in a matrix

which constitutes FSoms and FSbishop. This results in

a matrix, FSoms and FSbishop as shown in Figs. 5(a)

and 5(b). The lowest values of all factors of safety

is taken as the critical factor of safety. Then,

corresponding centers and radius are identified as

critical centers (xc, yc) and critical radius (Rc). It can

be noted from Figs. 5(a) and 5(b) that Not Feasible

(NF) solution is written in some places of the matrix.

NF denotes that "no solution" when the iteration

process does not converge to a solution or when a

specified slip surface does not intersect the either the

crest or toe of slope.

6.56 17.54 NF NF NF NF NF NF NF NF NF

3.27 3.71 5.17 9.65 NF NF NF NF NF NF NF

3.08 3.04 3.10 3.37 4.38 6.86 19.56 NF NF NF NF

2.62 2.48 2.34 2.22 2.11 3.14 3.91 5.46 10.24 NF NF

2.92 2.76 2.61 2.46 2.32 2.36 2.20 2.07 2.11 2.33

3.56 3.38 3.21 3.05 2.89 2

2.00

.74 2.60 2.46 2.33 2.21 2.24

4.21 4.07 3.93 3.80 3.69 3.58 3.48 3.41 3.37 3.41 3.66

5.17 5.07 4.98 4.92 4.86 4.84 4.86 4.94 5.12 5.27 5.80

6.48 6.45 6.46 6.50 6.58 6.74 7.01 7.45 8.23 9.77 13.95

10.87 11.03 11.30 11.73 12.39 13.43 15.10 18.06 24.56 44.55 NF

27.63 29.51 32.39 36.98 45.04 61.82 NF NF NF NF NF

Fig. 5a Matrix showing the critical factor of safety

of 2.0 for OMS


3.29 3.73 5.18 9.66 NF NF NF NF NF NF NF

3.10 3.06 3.12 3.40 4.39 6.87 19.57 NF NF NF NF

2.65 2.50 2.37 2.25 2.14 3.17 3.93 5.48 10.26 NF NF

2.96 2.80 2.65 2.51 2.38 2.40 2.25 2.13 2.19 2.43

3.61 3.44 3.28 3.13 2.98 2

2.07

.85 2.72 2.61 2.47 2.38 2.49

4.33 4.21 4.09 3.98 3.88 3.81 3.75 3.73 3.77 3.87 4.29

5.40 5.33 5.27 5.24 5.24 5.28 5.38 5.58 5.95 6.38 7.38

6.88 6.90 6.96 7.07 7.25 7.53 7.97 8.67 9.87 12.23 18.52

11.54 11.80 12.20 12.80 13.70 15.10 17.33 21.30 30.06 57.44 NF

29.33 31.57 34.98 40.41 49.90 63.37 NF NF NF NF NF

Fig. 5b Matrix showing the critical factors of safety

2.07 for BMS

First order reliability method (FORM)

FORM derived by approximating the performance

function by the first order Taylor series. This

method considers the first two moments of the

random variables (for normally distributed random

ariables i.e. mean and variance). The performance

functions for OMS and BMS can be written as

( ) -1OMS OMSg x FS (29a)

( ) -1BMS BMSg x FS (29b)

The basic formulation for FORM can be stated as

follows:

Find , which

Tminimizes

subjected to 0

u u

g u

(30)

This problem is modeled as a nonlinear constrained

optimization problem which can be solved using the

method of Lagrange multipliers and is given by,

2

1

( )n

i

i

L u g u

(31)

where, is the Lagrange multiplier. The stationary

points of L can be found by solving the following

set of equations 0iL u and 0L .

2

1

0j

nj j

i

i

uL g

u uu

(32)

where,

1 1

n ni

i

i ij i j i

xg g g

u x u x

and j = 1,

2…n. (3.109)

0L

g u

(33)

After simplification, Lagrange multiplier ( ) can

be written as:

2

1 1

1

n n

i

j i i

g

x

(34)

where, k = 1, 2…n

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Now the design point in the standard normal space (

ku ) can be expressed as

1

2

1 1

n

i

i kk k

n n

i

j i i

g

xu

g

x

(35)

where, k = 1, 2…n

(3.112)

Rearranging the above equation we get,

2

1 1 1 1

n n n n

k i i

j i j ik k

g gu

x x

(36)

The reliability index against slope failure can be

expressed as:

1 1

2

1 1

n n

k i

j i k

n n

i

j i k

gu

x

g

x

(37)

A MATLAB program is coded to find OMS and

Bishop . Once the reliability indices, OMS and

Bishop are calculated for a given center and

different radii combination, the least of these values

is assigned as a value to the position in a matrix

which constitutes oms and bishop . This results in

a matrix, oms and bishop as shown in Figs. 6(a)

and 6(b). The lowest values of all oms and bishop

is taken as the critical reliability indices. Then,

corresponding centers and radius are identified as

critical centers (xc, yc) and critical radius (Rc) for

reliability analysis. It may be found from Figs. 6(a)

and 6(b) that Not Feasible (NF) solution is written

in some places of the matrix due to convergence

problem of solution or improper intersection of

slope. 9.73 9.92 NF NF NF NF NF NF NF NF NF

9.64 9.54 9.64 9.83 NF NF NF NF NF NF NF

10.28 9.96 9.67 9.50 9.55 9.75 9.93 NF NF NF NF

11.24 10.92 10.55 10.14 9.75 9.47 9.48 9.68 9.85 NF NF

12.07 11.94 11.71 11.38 10.96 10.45 9.91 9.46 9.06 9.42

11.89 12.18 12.

8.99

35 12.40 12.31 12.06 11.63 11.01 10.22 9.55 9.79

14.03 14.53 14.46 11.28 11.99 12.52 12.88 13.02 12.89 12.36 11.52

15.55 16.27 16.90 17.46 17.96 14.39 15.24 15.99 16.66 17.16 19.85

20.64 21.14 21.62 22.07 22.50 22.92 23.32 23.69 23.98 25.88 29.31

24.49 24.90 25.29 25.67 26.03 28.07 28.37 28.61 NF NF NF

29.35 NF NF NF NF NF NF NF NF NF NF

Fig. 6(a) Matrix showing the critical reliability

index for OMS


9.69 9.58 9.66 9.84 NF NF NF NF NF NF NF

10.32 10.01 9.73 9.55 9.59 9.77 9.93 NF NF NF NF

11.28 10.96 10.60 10.21 9.83 9.55 9.71 9.86 NF NF

12.10 11.97 11.75 11.43 11.02 10.53 10.01 9.59 9.58 9.61 9.84

11.90 12.20 1

9.55

2.38 12.43 12.35 12.12 11.72 11.14 10.42 9.84 10.10

14.08 14.60 15.01 11.29 12.01 12.56 12.94 13.12 13.06 12.72 12.52

15.66 16.42 17.08 17.69 18.25 14.53 15.46 19.85 20.47 20.64 22.01

20.97 21.54 22.09 22.63 23.18 23.65 26.34 26.84 25.11 28.40 30.55

27.26 27.67 28.10 26.42 28.73 29.26 30.81 30.42 NF NF NF

NF NF NF NF NF NF NF NF NF NF NF

Fig. 6(b) Matrix showing the critical reliability

index for BMS

RESULTS AND DISCUSSION

Probabilistic analysis of the soil slopes is performed

by taking into account the variability associated

with unit weight, cohesion and friction angle of the

soil in this study using RBDO methodology. A limit

state function is framed for circular sliding failure.

Table 1 Range of parameters considered in the

present study

Variable

Statistics

Mean COV

(%) Distribution

Stability

number,

Hc / 0.1

10-

30 Normal

Friction angle,

20o 5-20

Log-

Normal

21.80o -

63.44o - -


In this study, for reliability indices, the stability

number c/γH is considered for calculations. When

the values of mean and COV of c and γ (COVc and

COVγ) are known, the COV value of c/γH can be

calculated as follows:

COV

(38)

The variance of a fraction, c/γH can be written as 2 2 2 2

2

4

H c c H

H

c

H

(39)

2 2cc H

H

cCOV COV

H

(40)

2 2

c H

c cCOV COV

H H

(41)

2 2

c H

cCOV COV COV

H

(42)

By substituting the values of COVc and COVγ, COV

of c/γH is computed from Eq. (42). The range of

deterministic parameters and random variables

considered in the present study are given in Table

1. In this study, a slope with a slope angle ranges

from 21.80o - 63.44o, /c H = 0.1, friction angle

( ) = 32o, COV of /c H = 30% and COV of

= 15% are considered for the reliability analysis.

Since the surfaces of sliding for many slope failures

have been observed to follow approximately the arc

of a circle, it is assumed that the shape of slip circle

is circular failure arc in the current study. Reliability

indices against sliding failure using OMS and BMS

are computed. The effect of slope angle ( ) on the

critical center coordinates (cx /H ,

cy /H ), critical

radius of circle (cR /H ) and critical reliability

indices ( OMS and Bishop ) are presented in Figs. 7

to 11.

Factor of safety versus Reliability Index

Approach

Fig. 7 shows the formation of the critical slip circles

and shift in their centres with deterministic and

probabilistic approaches. The geometry of the soil

slope and the formation of the critical slip circles on

normalized x & y axis can also be seen in Fig. 7. An

increase in the diameter of the slip circle can be

observed in case of probabilistic approach for both

OMS and Bishop’s methods. This indicates that the

slip circles are moving away from the slope when

variability associated with shear parameters is

considered in the design.

Fig. 8 shows the magnified view of slip circles as

shown in Fig. 7, in which the effect of considering

variability is clearly visible. It can be noted from

Fig. 7 that the radius of the circle (in terms of Rc/H

ratio) has been increased from 2.58 to 4.96, 2.77 to

3.61 respectively for OMS and BMS.

Effect of Slope Angle (α)

To study the effect of slope angle on the formation

of slip circles and slope failure, the slope angle (α)

has been varied from 2.5:1 to 0.5:1 (i.e. 21.80o to

63.44o) as shown in Figs. 9-12. Fig. 9 shows the

effect of change in slope angle on the critical slip

surfaces for ordinary method of slices. A magnified

view of Fig. 9 is presented in Fig.10 which shows

the effect of change in slope angle on the shear

bands for the ordinary method of slices. It can be

observed Fig. 10 that as the value of α increases, the

volume of soil involved in the failure mechanism

(and hence the location of the critical centers) is

decreasing. This relatively reduces the normal force

at the base of the slip surface and consequently

reduces the factor of safety as well as reliability

index ( OMS ). Similar behavior is also observed in

case of Bishop’s method as shown in Figs. 11 to 12.

It can also be noted from Figs. 11 to 12 that for α =

45o, the shear band is shallower than α = 53.13o case.

This behavior can be attributed to the possibility that

a local minima is being chosen instead of a global

minima which causes the anomaly in the behavior.

It may be eliminated by taking a more fine grids.

An observation that can be made from Fig. 2 that the

magnitude of reliability indices ( OMS and Bishop )

reduces significantly from 3.019 to -0.165 and 3.131

to -0.148 respectively for OMS and BMS when

increases from 21.80o - 63.44o.

50

th

IG

C




Fig. 7 Variation in slip circle formation with

deterministic and probabilistic approaches

Fig. 8 Magnified view of slip circles in deterministic

and probabilistic approaches

Fig. 9 The effect of change in slope angle α on the

critical slip surfaces for Ordinary Method of Slices


shear bands for OMS (Magnified view)


critical slip surfaces for BMS


shear bands for BMS (Magnified view)


CONCLUSIONS

A probabilistic analysis of slope stability using

FORM method is discussed in this paper. Slope

stability analysis has been performed by using

ordinary and Bishop’s methods of slices. A

comparative study is presented between the

deterministic and probabilistic approaches by

locating respective critical centers and radius. The

study indicates that the slip circles moves away from

the slope when the variability associated with the

shear strength parameters is considered in the

design. An Important observation that be made from

the present study that the increase in slope angle

value reduces reliability of the slope significantly.

The developed probabilistic, makes use of

MATLAB coding. The underlying procedures of

evaluating factors of safety and reliability indices

are simple and transparent, requiring only

fundamental knowledge of statistics and probability

theory. However, the analysis does not accounts for

the spatial variability of the input variables. The

developed approach is flexible in handling real

slope problems, including various loading

conditions, complex geometry, c– soils, and

circular slip surfaces.

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