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[ Pobierz całość w formacie PDF ]TRADITIONAL AND ADVANCED
PROBABILISTIC SLOPE STABILITY ANALYSIS
D.V. Griffiths,
1
G.A. Fenton,
2
and M.D. Denavit
3
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), β
HL
, 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, P
f
, 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
(
x
1
,...,
x
N
)
≥
0
⎯→
Safe
(1)
g
(
x
,...,
x
)
<
0
⎯→
Failure
1
N
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, β
HL
, 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 β
HL
is an iterative process
and it is defined by
⎧
−
x
µ
⎫
T
⎧
−
x
µ
⎫
[]
β
=
min
⎨
i
i
⎬
R
−
1
⎨
i
i
⎬
i
= 1 ,…, N
(2)
HL
g
=
0
σ
σ
i
i
where {(
x
i
–
µ
i
)/
σ
i
} 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, x
1
and x
2
, are normally distributed and have the
following parameters:
µ
1
=
6
σ
1
=
1
µ
2
=
7
σ
2
=
0
.
75
ρ
1
,
x
2
=
−
0
35
(3)
Failure of the system is given by the limit state function:
g
( )
x
,
x
=
−
0
03
x
3
1
−
0
25
x
2
+
29
.
16
(4)
1
2
2
The probability density function governing two normal random variables correlated
by ρ can be written as (e.g., Fenton and Griffiths 2006):
1
−
β
( )
2
x
,
x
2
1
2
( )
f
x
,
x
=
e
(5)
x
x
1
2
1
2
2
πσ
σ
1
−
ρ
2
x
x
1
2
where
⎧
−
x
µ
⎫
T
⎧
−
x
µ
⎫
⎨
⎬
⎨
⎬
β
( )
x
1
,
x
=
i
x
i
[]
R
−
1
i
x
i
i
= 1, 2
(6)
2
⎩
σ
⎭
⎩
σ
⎭
x
i
x
i
β , given that the limit state function is zero,
is the Hasofer-Lind reliability index, β
HL
.
x
1
,
x
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 β(x
1
,x
2
) (i.e. β(x
1
,x
2
) = 1, 2, 3, 4…), nevertheless, each contour
represents a constant value of the probability density function.
3
Note that the minimum value of
( )
2
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 β = β
HL
, 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 β
HL
= 2.40, corresponding to a probability of failure
of p
f
= 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 x
1
= 8.15 and x
2
= 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(x
1
, x
2
) < 0 gave the probability of failure 0.964%, relatively 16%
higher than given by FORM.
5
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