Proceedings of Indian Geotechnical Conference
December 15-17,2011, Kochi (Paper No. K-250)
PROBABILISTIC ANALYSIS OF A SLOPE STABILITY PROBLEM
Anasua Guha Ray,Research Scholar, Civil Engineering Department, IIT Kharagpur, Email: anasua_guharay@yahoo.co.in
Dilip Kumar Baidya, Professor, Civil Engineering Department, IIT Kharagpur, Email: baidya@civil.iitkgp.ernet.in
ABSTRACT: Probabilistic analysis quantifies the uncertainties present in conventional safety factor approach rationally
and efficiently. This paper presents the reliability analysis of a slope stability problem using Mean First Order Second
Moment Method (MFOSM), Point Estimate Method (PEM) and Monte Carlo Simulation (MCS). Unit weight of soil (Ȗ),
internal angle of friction (ʔ) and cohesion (c) are considered as random variables. The study shows that although
deterministic analysis reveals a safety factor > 1.5, reliability analysis reflects quite high failure probability when COV of
ʔ > 7%. Partial safety factors back-calculated for all the random variables for a target reliability index ȕ in the range of
3-3.2 reveals that a higher safety factor than chosen is required for ʔ, while c and Ȗ do not require a safety factor even as
high as 1.5.
INTRODUCTION which the relevant sources of uncertainty (internal angle of
Slopes may be artificial, i.e. man-made, as in embankments soil friction ʔ, cohesion c and soil unit weight, Ȗ) involved
for highways and railroads, earth dams etc. or natural as in in slope stability analysis can be analyzed. Partial safety
hillside and valleys, coastal and river cliffs etc. Most factors for the random variables are back-calculated by
problems in slope stability are statically indeterminate; target reliability approach method corresponding to a target
hence, some simplifying assumptions are made in order to failure probability of 0.00135 (ȕ=3) as recommended by
determine a unique factor of safety. Uncertainties in any of USACE [9]. Analysis shows that the same partial factor can
the input parameters (e.g. cohesion, angle of internal have different levels of risk depending on the degree of
friction, unit weight, pore pressure parameters etc.) are uncertainty of the mean value of ʔ. These calculated partial
unaddressed in deterministic approach. All these safety factors are recommended in design under static
uncertainties of the variables have been suitably condensed loading. This can be proved to be cost effective and hence
into a single reliability index ȕ in probabilistic approach. the structure can be optimized for specific site conditions.
A number of researchers have contributed to the
development of probabilistic approach for slope stability RELIABILITY EVALUATION METHODS
analysis. Wu and Kraft [1], Tang et al. [2], Venmarcke [3], The probability of a system performing its required
Li and Lumb [4], Christian et al. [5] applied the first-order function adequately for specified period of time under
second-moment method for analyzing slope stability. stated conditions is quantified in terms of an index known
Bhattacharya et al. [6] presents a numerical procedure for as reliability index (ȕ). For uncorrelated normally
locating the surface of minimum reliability index ȕmin for distributed capacity C and demand D, ȕ is calculated using
earth slopes. The procedure uses a formulation similar to the following expression (Baecher & Christian, 2003):
that used to search for the surface of minimum factor of
safety FSmin in a conventional slope stability analysis. The CD
E= (1)
advantage of the formulation lies in enabling a direct search V C2 V D2
for the critical probabilistic surface by utilizing an existing
deterministic slope stability algorithm with the addition of a
USACE (1997) stated that for good performance of a
simple module for the calculation of the reliability index ȕ.
Xue and Gavin [7] used a genetic algorithm approach for geotechnical system, E t 3 .
simultaneously locating the critical slip surface and Furthermore, the probability of failure (Pf) can be estimated
determining the reliability index of slope stability problems. from the reliability index ȕ, using the established equation
Babu and Murthy [8] carried out reliability analysis of Pf = 1-ĭ(ȕ) = ĭ(-ȕ), where ĭ is the cumulative distribution
unsaturated soil slopes. Most of the past researches are function (CDF) of the standard normal variate.
based on supplication of different methodologies to slope
stability problems. Little work has been done on what Mean First Order Second Moment Method (MFOSM)
safety factors should be assigned for design purpose based Here, only the first order terms of a Taylor’s series
on probabilistic approach. expansion of the performance function estimate the mean
and variance of the said function. For uncorrelated
OBJECTIVE OF THE PRESENT STUDY variables, ȕ is calculated by the equation:
This paper presents a probabilistic based approach by
651
Anasua Guha Ray & Dilip Kumar Baidya
P ( xi ) In order to formulate the algorithm to solve for the factor of
E n
§ wg · safety based on the above-mentioned methods, the forces
¦ ¨ ¸V xi (2) acting on a typical slice is considered as shown in Fig. 2.
i 1 © w xi ¹
Generally, the partial derivatives of the factor of safety with
respect to each soil parameter are calculated numerically . b
using the following equation:
wg g g
(3)
w xi 2 mV ( xi ) W
Ei+1
where g+ and g- are values of FS obtained by using
parameter values greater than and less than the mean by an Ei H
increment mV x i , respectively. According to Hassan and Xi+1
Wolff, m = 1.0. Xi
Į T
Point Estimate Method (PEM) N
In this method proposed by Rosenblueth and Harr, discrete
values of the performance function are evaluated at the
mean values of the basic variables, at one standard Fig. 2 Forces acting on a typical slice
deviation above and one standard deviation below the mean
values for uncorrelated variables. If n is the number of Ordinary Method of Slices or Fellenius Method
variables, in general 2n terms are to be added. Fellenius Method [10] assumes that the inter-slice forces
are parallel to the base of each slice, thus they can be
Monte Carlo Simulation (MCS) neglected and the factor of safety is given as follows:
For non-linear functions, such as the factor of safety, F
from most limit equilibrium methods, it is necessary to use c ' L tan I .¦ W cos D
a numerical approach to obtain the statistical moments. F (4)
Among the widely used numerical methods, only the
¦ W sin D
Monte-Carlo Simulation Method (MCS) aims at generating
the probability distribution of F numerically. In this where H, bi, R, LJ and W are defined in Fig 1.
approach, a large number of realizations of the basic L is the length of the slip surface = RLJ
random variables X, i.e. xj, j=1,2,3…N are simulated and Į is the angle the normal acting on the slice makes with
for each of the outcomes xj, it is checked whether or not the vertical.
limit state function taken in xj is positive. All the
simulations for which xj<0 are counted (nf) and after N Bishop Method
simulations the failure probability pf may be estimated In Bishop Method [11], the inter-slice shear forces are
through Pf = n f /N . In fact for N o f ҏ, the estimate of the neglected, and only the normal forces are used to define the
inter-slice forces. The factor of safety is given as follows:
failure probability becomes exact.
ª c 'b W tan I º
RELIABILITY ANALYSIS OF SLOPE ¦« < »
A finite slope having height H=11m and slope 1V:1.5H F ¬ ¼
(5)
with critical slip circle of radius R=17m is considered as ¦ W sin D
shown in Figure 1. Friction angle, cohesion and unit weight
of backfill soil are considered as random variables for the sin D .tan I
where < cos D
analysis. Water table is assumed to be at a great depth from F
the ground surface, thus not affecting the stability of the The factor of safety F is on both sides of the equation. The
slope. The slope is analyzed by Fellenius [10] and Bishop equation has therefore to be solved by successive
[11] methods of analysis. approximation. A trial value is first assumed and the factor
O of safety computed. If the computed value differs from the
LJ assumed value, a second trial is carried out, assuming a new
R=17 h=5.88
h=5 value of F.
To study the relationship between the factor of safety and
1 the variation of soil properties, the factor of safety was
1 H=11
calculated as function of each of soil variables. Each
variable (soil property) allowed varying about its respective
mean value while the other properties are kept fixed at their
Fig.1 Geometry of Slope mean values.
652
Probabilistic Analysis of a Slope Stability Problem
Table 1 provides the statistics of the input parameters
involved for calculating the stability of the slope.
Table 1 Statistics of Input Parameters
Variables Mean μ COV Distri-
bution
Unit Weight DŽ 20kN/m3 3-7% Normal
Friction Angle ʔ 32 degrees 2-30% Normal
Cohesion c 10kN/m2 20-50% Normal
(b)
Reliability index satisfying all the constraints in the form of
performance function is achieved. The termination
tolerance for the convergence of the reliability index is
taken as 10í3 i.e. ȕ above 3 as recommended by USACE.
Expressions for margin of safety M for Fellenius and
Bishop Method are as follows:
Fellenius Method:
M c ' L tan I .¦ W cos D ¦ W sin D
CD
C (c)
Bishop Method: M 1 F 1
D
RESULTS AND DISCUSSIONS
Analysis was carried out both by Point Estimate Method
and Mean First Order Second Moment Method and verified
by Monte Carlo Simulation. In the present paper, for
carrying out MCS, 10,00,000 sample points are generated
by an algorithm coded in commercially available software
MATLAB, which minimizes the effect of variation of the
number of sample point generation. The study reveals that a
sample size of 8,00,000 or greater is a good choice for (d)
Fig.
Fig. 33 Variation
Variation of (d)different
of âȕ for
for different values
values of
of COV
COV of
of ʔ and
and ãȖ
MCS, while for sample size less than 5,00,000 the
reliability index shows a significant variation. for
for (a)
(a) COV
COV of of c=20%
c=20% (b) (b) COV
COV of c=30%
c=30% (c) COV of
. c=40%
c=40%(d) (d)COV
COVofofc=50%
c=50%
The deterministic safety factors calculated from the two
limit equilibrium methods are 1.551 for Fellenius and 1.52 From Figs. 3a-b, it is observed that variation of Ȗ is
for Bishop Method. insignificant on the stability of the slope, especially when
COV of c exceeds 30%. But when COV of ʔ exceeds
The variation of reliability index ȕ with variation of the approximately by 10% and c by 40%, the design has to be
random variables Ȗ, ʔ and c are presented in Fig. 3a-d. modified to bring down ȕ under tolerable limit. So it can be
concluded that ʔ is a very important parameter to be
measured during field investigation. It is also suggested that
a greater partial safety factor may be assigned to ʔ than
either on Ȗ or c, rather than applying an overall safety factor
to the design, thereby economizing the structure.
Comparison of Fellenius and Bishop Methods
Fig. 4 indicates that the calculated ȕ values differ a little
between the results obtained by Fellenius and Bishop
Method which can be attributed to model uncertainty. This
mainly arises due to inappropriate selection of the method
(a) to evaluate the safety factor, which causes some inherent
error in the estimation.
653
Anasua Guha Ray & Dilip Kumar Baidya
the structure and that a reliability based approach is
essential in decision-making process.
It can be concluded that ȕ is more sensitive to the
uncertainty in ʔ than c or DŽ, indicating that ȕ provides more
meaningful information than the deterministic factor of
safety. Hence, greater partial safety factors should be
applied for ʔ, while much smaller value of partial safety
factor is recommended for Ȗ and c, instead of applying a
global factor of safety to the whole structure, thereby
economizing the structure for specific site conditions.
These partial safety factors are recommended for design
Fig. 4 Comparison of ȕ in Fellenius and Bishop Method purpose. So, safety factors should be recommended based
on the site conditions and to what degree of safety is
According to Christian et al. [5], three sources of model required for the particular structure.
uncertainties were defined, these are: three-dimensional
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