TRADITIONAL AND ADVANCED
PROBABILISTIC SLOPE STABILITY ANALYSIS
D.V. Griffiths,1 G.A. Fenton,2 and M.D. Denavit3
1
F. ASCE, Geomechanics Research Center, Division of Engineering, Colorado
School of Mines, Golden, Colorado 80401-1887, USA; email: vgriffit@mines.edu
2
M. ASCE, Department of Engineering Mathematics, Dalhousie University, Halifax,
Nova Scotia, Canada B3J 2X4; email: Gordon.Fenton@dal.ca
3
S.M. ASCE, Department of Civil and Environmental Engineering, University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801; email: denavit2@uiuc.edu
ABSTRACT
The paper contrasts results obtained by the traditional First Order Reliability Method
(FORM) and a more advanced Random Finite Element Method (RFEM) in a
benchmark problem of slope stability analysis with random shear strength
parameters. The key difference between the methods is that RFEM takes into account
spatial correlation in a rigorous way allowing slope failure to occur naturally along
the path of least resistance. Both methods lead to predictions of the "probability of
slope failure" as opposed to the more traditional "factor of safety" measure of slope
safety, however they give significant different results depending on the value of the
correlation length. For small correlation lengths FORM is generally conservative,
however it is shown that there is a worst case correlation length for which FORM
leads to unconservative predictions of slope reliability.
INTRODUCTION
Slope stability analysis is one of the main areas of interest to geotechnical designers,
and also seems a natural application for probabilistic approaches since the analysis
leads to a probability of failure as opposed to the more customary factor of
safety . This paper will review a traditional approach to probabilistic slope stability
analysis, the first order reliability method (FORM) and then go on to discuss the more
advanced random finite element method (RFEM). The methods will be compared on
a benchmark slope and conclusions will be drawn regarding the limitations of FORM,
in particular, the effect of the spatial correlation length which can be rigorously
modeled by RFEM.
1
FIRST ORDER RELIABILITY METHOD
Theory
The first order reliability method (FORM) is a process which can be used to
determine the probability of a failure given the distribution data and limit state
function. The method is based on the Hasofer-Lind reliability index (Hasofer and
Lind 1964), ²HL, which can be described as the distance, in standard deviation units,
between the most probable set of values and the most probable set of values that
causes a failure. Calculation of this value is an iterative process, finding the minimum
value of a matrix calculation subject to the constraint that the values result in a
system failure. However, common solver routines found in several software packages
(e.g. Excel and Mathematica) can easily arrive at the solution. Once the reliability
index has been determined, the probability of failure, Pf, is a simple calculation.
Limit State Function
Each reliability analysis requires a limit state function, which defines failure or safe
performance. Limit states could relate to strength failure, serviceability failure, or
anything else that describes unsatisfactory performance. The limit state function, g, is
defined
g(x1,..., xN ) e" 0 Å»# Safe
Å»#
(1)
g(x1,..., xN ) < 0 Å»# Failure
Å»#
where N is the number of random variables. Often it is sufficient for the limit state
function to be the resistance minus the load. Another common form of the limit state
function is the factor of safety minus one or the log of the factor of safety.
The limit state function can be determined from analytical theory for simple
systems. For more complex systems, it may need to be approximated numerically
with curve fitting.
Hasofer-Lind Reliability Index
The reliability index, ²HL, is the distance in standard deviation units between the
most probable set of random variables (the means), and the most probable set of
random variables that causes a failure. Determination of ²HL is an iterative process
and it is defined by
T
ż# - µi -1 xi - µi
«# ż# «#
xi
²HL = min [R] i = 1 ,& , N (2)
¨# Ź# ¨# Ź#
g=0
Ãi Ãi
©# # ©# #
where {(xi µi)/Ãi} is the vector of the random variable values reduced to standard
normal space and [R] is the correlation matrix of the variables.
2
Visualization
To better understand and visualize this method, consider the following arbitrary
problem. Two random variables, x1 and x2, are normally distributed and have the
following parameters:
µx = 6.0 Ã =1.0
x1
1
(3)
µx = 7.0 Ã = 0.75 Áx ,x2 = -0.35
x2
2 1
Failure of the system is given by the limit state function:
g(x1, x2)= -0.03x13 - 0.25x22 + 29.16 (4)
The probability density function governing two normal random variables correlated
by Á can be written as (e.g., Fenton and Griffiths 2006):
-² (x1,x2 )2
1
2
fx x2 (x1, x2 )= e (5)
1
2
2Ä„Ã Ã 1- Á
x1 x2
where
T
ż# - µx ª# -1ª# xi - µx ª#
«# ż# «#
xi
ª#
i i
²(x1, x2)= [R] i = 1, 2 (6)
¨# Ź# ¨# Ź#
à Ã
ª# ª# ª# ª#
xi xi
©# # ©# #
Note that the minimum value of ²(x1, x2), given that the limit state function is zero,
is the Hasofer-Lind reliability index, ²HL.
Plotting the probability density function in three dimensions would result in a
surface in the shape of a bell. By definition, the volume under the surface is unity.
The limit state function divides the volume into a failure region and a safe region.
The probability of failure is defined as the volume under the probability density
function in the failure region. FORM uses a first order approximation of the limit
state function and therefore the calculated probability of failure is also approximate.
Numerical integration of the probability distribution function in the failure region
leads to more accurate results and is discussed later.
In plan view, the probability density function can be visualized as a contour plot
involving a series of ellipses, and the limit state function can be seen as a line
separating the failure and safe regions, see Figure 1. The contours in Figure 1 are
actually contours of ²(x1,x2) (i.e. ²(x1,x2) = 1, 2, 3, 4& ), nevertheless, each contour
represents a constant value of the probability density function.
3
Figure 1. Plan View of the Probability Density Function.
The solid curved line represents the actual limit state function. The smallest ellipse
that the limit state function touches is the contour of ² = ²HL, represented above by
the darker ellipse. The point where they meet represents the most probable failure
point. The dashed straight line that also passes through that point is the first order
approximation of the limit state function.
The first order approximation assumed in FORM could lead to an underestimate of
the probability of failure if the actual limit state function curves towards the mean
values as seen in Figure 1. A more accurate, yet more time consuming, method to
determine the probability is to numerically integrate the probability distribution
function in the region of failure. A relatively simple algorithm involving the repeated
mid-point rule (e.g., Griffiths and Smith 2006) can be devised to accomplish this task.
FORM software
Excel
The limit state function and properties described in equation 3 and 4 have been run
through an Excel spreadsheet using the solver add-in (e.g., Low and Tang 1997,
Denavit 2006) in which the FORM algorithm has been implemented. The Hasofer-
4
Lind reliability index is given as ²HL = 2.40, corresponding to a probability of failure
of pf = 0.814%
Mathematica
Using Mathematica, the same calculations can be performed. The following shows
the lines which must be executed:
Again, the probability of failure is 0.814%, with a reliability index of 2.40,
corresponding to a most probable failure point of x1 = 8.15 and x2 = 7.19. Both the
reliability index and the most probable failure point can be graphically checked using
Figure 1.
As discussed earlier, numerical integration can determine the probability of failure
directly but more slowly. Below is a set of commands which will perform the
numerical integration:
Numerical integration of the volume of the probability density function
corresponding to g(x1, x2) < 0 gave the probability of failure 0.964%, relatively 16%
higher than given by FORM.
5
PROBABILISTIC SLOPE STABILITY ANALYSIS
For slope stability, no analytical equation exists which can serve as a limit state
function. In this case, a numerical approximation will need to be formulated to use as
the limit state function. This can be accomplished by fitting a curve to the results
from several finite element analyses using the strength reduction method (e.g.,
Griffiths and Lane 1999). This method involves applying gravity loads to the finite
element mesh and systematically weakening the soil until a sufficient number of
element have yielded to allow the formation of a failure mechanism.
2 2
For example, with two (N=2) random variables (c , tanĆ ) , a quadratic surface
without cross-terms with five (2N +1 = 5) constants of the form
2 2 2 2 2 2 2
FS(c , tanĆ ) = a1 + a2c + a3 tanĆ + a4c + a5 tan2 Ć(7)
could be used to approximate the factor of safety function.
Figure 2 shows the dimensions and properties of a hypothetical sample slope which
was analyzed using this method.
20 20 20
All dimensions in meters
2
1
H=10
µc2 = 5.0 kPa Ãc2 = 1.5 kPa
µtanĆ2 = 0.364 ÃtanĆ2 = 0.109
5 Ác2 -tanĆ2 = 0.5
Figure 2. Slope Dimensions and Properties.
The following parameters were taken as deterministic: unit weight (Å‚ = 20 kN/m3),
modulus of elasticity (E = 100,000 kPa), Poisson s ratio (½ = 0.3), and dilation angle
(È = 0). It was assumed that there was no correlation between the variables and that
the variables were normally distributed. The limit state function will then be the
factor of safety function minus one, thus
2 2 2 2
g(c , tanĆ ) = FS(c , tanĆ ) -1 (8)
In order to find the constants in equation (7), five finite element analyses were run
with the following input and results.
6
Table 1. Sample Points for the Approximate Limit State Function.
Sample Value of Value of Factor of
c2 (kPa)
tanĆ2
Point c2 Safety
tanĆ2
1 5.00 0.364 1.09
µc2 µtanĆ2
2 6.50 0.364 1.17
µc2 + Ãc2 µtanĆ2
3 3.50 0.364 1.00
µc2 Ãc2 µtanĆ2
4 5.00 0.473 1.33
µc2 µtanĆ2 + ÃtanĆ2
5 5.00 0.255 0.84
µc2 µtanĆ2 ÃtanĆ2
Solving for the five constants yields the limit state function:
2 2 2 2 2 2 2
g(c , tanĆ ) =-0.123 + 0.079c + 2.554 tanĆ - 0.002c - 0.421tan2 Ć -1 (10)
Implementing this function along with the soil properties into Excel or Mathematica
as described earlier will yield a Hasofer-Lind reliability index, ²HL, of 0.301,
corresponding to a probability of failure, pf , of 38.2%.
Random Finite Element Method
The random finite element method (RFEM) is an entirely separate method for
determining the probability of failure. This method involves a Monte Carlo
simulation with many different realizations of the soil properties. Each realization of
the soil properties involves overlaying a random field onto a finite element mesh,
essentially resulting in each element being a random variable. In doing this, a new
parameter becomes evident, the spatial correlation length. This parameter describes
the tendency of elements spatially near each other to be correlated. For slope stability
problems, it is described in the dimensionless form, Åš , which is the correlation
length divided by the height of the slope, H . This parameter can be clearly seen in
the following two figures where the darker elements represent higher strength.
Figure 3. Slope with Low Correlation Length.
7
Figure 4. Slope with High Correlation Length.
Once the properties are assigned, a finite element analysis determines whether or not
failure occurs, and the process is repeated. The nature of RFEM can lead to quite
time-consuming calculations compared with FORM, however the latter method does
not explicitly incorporate the correlation length.
Since RFEM is based on Monte Carlo simulation, it is important to ensure that the
number of realizations is sufficient to provide accurate and repeatable results. To
check this, the slope was analyzed multiple times with increasing numbers of
realizations. The results of two such runs are shown in Figure 5. Note that the
dimensionless correlation length, Åš, was set at 0.1 for this analysis. In both cases the
probability of failure converges to the same constant value as the number of
realizations increases. It can be seen that 1000 realizations yields sufficiently
repeatable results and this value has been used in the subsequent analyses.
25%
Åš = 0.1
20%
15%
10%
5%
0%
10 100 1000 10000
Number of Realizations
Figure 5. Probability of Failure versus Number of Realizations.
To examine the effect of the correlation length on the probability of failure, a
parametric study was performed. The results of RFEM compared with FORM for the
slope shown in Figure 2 are shown in the Figure 6.
8
f
Probability of Failure, P
50%
40%
30%
20%
10%
RFEM
FORM
0%
0.01 0.10 1.00 10.00 100.00
Dimensionless Correlation Length, Åš
Figure 6. Probability of Failure versus Correlation Length.
For small correlation lengths, the probability of failure is essentially zero, for
intermediate correlation lengths, the probability of failure increases rapidly, and for
large correlation lengths, the probability of failure is essentially constant and similar
to that found by FORM.
Consider the limits of zero and infinity for correlation length. As the correlation
length approaches zero, the soil will vary rapidly between any two points and become
essentially homogeneous, with the soil properties tending to their mean values.
Assuming the mean values provide a safe design (FS > 1), the probability of failure
will always be zero. As the correlation length approaches infinity, however, the soil
across the slope is highly correlated and will not vary. It becomes essentially
homogeneous within each realization although different from one realization to the
next. Use of the FORM is therefore equivalent to a system with a correlation length
tending to infinity. For intermediate values of correlation length, the slope is not
homogeneous and anomalies, such as locations of weak areas, control the probability
of failure since the finite element analysis is able to seek out the weakest path
through the slope.
CONCLUSION
The first order reliability method is a powerful tool in probabilistic geotechnical
analysis; however, it fails to explicitly account for the spatial correlation length. In
the example considered, when the correlation length was small, results produced by
9
f
Probability of Failure, P
FORM were inaccurate and conservative. At high correlation lengths, the FORM
results tended to agree with RFEM. While traditional methods like FORM or FOSM
can account for spatial correlation indirectly by including variance reduction, this is
inevitably subjective, since the local averaging zone cannot be known a priori. A
number of investigators have attempted to include the effects of spatial correlation by
locally averaging the random properties over the circular failure surface that would be
predicted by a classical slope stability method (e.g. Bishop). The RFEM studies
described in this paper are more realistic and conservative, in that they allow the
critical mechanism to seek out the weakest path through the soil without any a
priori assumption about the shape or location of the critical failure surface.
Of particular interest for designers is the case when FORM gave unconservative
predictions. This is due to observation of a worst case correlation length (ÅšH" 1)
which gave higher probabilities of failure than FORM. At this intermediate
correlation length, the failure mechanism is able to seek out the optimal path
through the weaker zones of soil and is a phenomenon that has been documented by
the authors in other geotechnical failure analyses by RFEM (e.g. Griffiths and Fenton
2001, Fenton and Griffiths 2003)
REFERENCES
Denavit, M.D. (2006). Probabilistic Geotechnics, Independent Study Report,
Colorado School of Mines, Golden, Colorado.
Fenton, G.A. and Griffiths, D.V. (2003) Bearing capacity prediction of spatially
2 2
random c -Ć soils. Can Geotech J, vol. 40, no.1, pp.54-65.
Fenton, G.A., and Griffiths, D.V. (2006). Risk assessment in geotechnical
engineering, Textbook to appear.
Griffiths, D.V., and Lane, P.A. (1999). Slope stability analysis by finite elements.
Géotechnique, 49, no.3, pp.387-403.
Griffiths, D.V. and Fenton, G.A. (2001). Bearing capacity of spatially random soil:
the undrained clay Prandtl problem revisited. Géotechnique 51, no.4, pp.351-
359.
Griffiths, D.V., and Smith, I.M. (2006). Numerical Methods for Engineers 2nd ed.,
CRC Press, Boca Raton, FL
Hasofer, A.M., and Lind, N.C. (1974). Exact and invariant second-moment code
format. Journal of Engineering Mechanics, ASCE, 100, 111-121.
Low, B.K., and Tang, W.H. (1997). Efficient reliability evaluation using
spreadsheet. Journal of Engineering Mechanics, ASCE, 123(7), 749-752.
10
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