RELIABILITY ANALYSIS OF REINFORCED SLOPE BY RANDOM
SET THEORY
Dr. – Ing. Hong Shen, Dipl. Ing. Slamet Widodo
TU Bergakademie Freiberg, Institut für Geotechnik, Freiberg
Prof. Dr. – Ing. Herbert Klapperich & Priv. – Doz. Dr. – Ing. habil. Nándor Tamáskovics
TU Bergakademie Freiberg, Institut für Geotechnik, Freiberg
Abstract: the Random Set Limit Equilibrium Method (RS-LEM) is developed and applied to the stability of a steep slope
reinforced by gabion and geosynthetics in this study. The internal stability of this slope is verified by the utilisation factor
analysis according to DIN 1054 in software GGU-GABION. Furthermore, the uncertainties of the strength parameters of
soil: cohesion and friction angle are considered in the reliability analysis. The case study approved that Random Set-Limit
Equilibrium Method framework is an efficient way to deal with the practical geotechnical problems where engineers have
to face not only imprecise data but also lack of information at the beginning of the project.
Keyword: Reliability analysis; reinforced slope; uncertainty; random set theory
1
INTRODUCTION
In geotechnical engineering, the uncertainties such as
the variability and uncertainty inherent in the geotechnical
property have caught more and more attentions from researchers and engineers. People have found that a single
“Factor of Safety” or “Utilisation Factor” calculated by traditional deterministic analyses methods can not represent
the slope stability exactly. Recently in order to provide a
more rational mathematical framework to incorporate different types of uncertainties in the slope stability estimation, reliability analyses and probabilistic methods, have
been applied widely, like Monte Carlo Simulation, First Order Reliability Method or Point Estimate Method (Christian
and Bächer; 1999; Malkawi, 2000; low, 2007). These
methods are dependent on the assumed or tested probability distribution of input variables.
In many practical cases, the input data are insufficient,
for example number of samples is not enough to determine
the probability distributions of variables. Therefore, people
proposed many imprecise methods recent years to deal
with these problems, such as interval approach (Moore,
1966; Goodman and Nguyen, 1985), evidence theory
(Dempster, 1967; Shafer, 1976), fuzzy set theory (Zadeh,
1965; Dodagoudar, 2000), possibility theory (Zadeh,
1978), imprecise probabilities (walley, 1991), random set
theory (Kendall, 1974), and convex model (Ben-Haim and
Elishakoff, 1990). Among them, random set theory is one
of the most practical and reliable method.
The random set theory was proposed first by Kendall in
1974, and then developed by several authors (e.g.
Matheron 1975, Goodman 1995, and Dubois 1991). It is a
mathematical model which can cope with uncertainty of
system while the exact values of input parameters are not
available, only the interval of these values can be obtained. In the beginning of 21 century, random set theory
has been applied extensively in geotechnical engineering.
In 2000, Tonon et al. used random set theory to deal with
the uncertainty in geomechanical classification of jointed
rock masses and reliability analysis of a tunnel lining.
Peschl (2004) and Schweiger et al. (2007) have developed
random set finite element method (RS-FEM) and investi-
gate the feasibility of this method in Tunneling design.
Shen (2012) developed random set distinct element method (RS-DEM) and applied it in a rock slope stability analysis.
In present paper, a slope reinforced by concrete gabion
and geosynthetics has been adopted to demonstrate the
Random Set theory in practical engineering. The shear
strength parameters of soil layer cohesion and friction are
considered as random sets. The slope stability is estimated in a commercial software GGU-GABION (2012). A selected result of the system response, internal stability, is
depicted in terms of discrete cumulative probability distributions.
2
RANDOM SET THEORY
Random Set Theory provides a general framework for
dealing with set-based information and discrete probability
distributions. It yields the same results as Interval Analysis
when only range information is available and under certain
conditions results are similar to Monte Carlo simulations
(Peschl 2004).
Suppose a system response is F=X1×…×Xp, where ×
indicates Cartesian product, F is the system response results, X1,…Xp are the influence parameters of the system
response. Because of the uncertainties, the influence parameters are not a single value, but some sets with different possibility. Therefore, F is also some sets impacted by
the variation of parameters X.
Because of the imprecision, the set X is also composed
by many subsets A={Aj, j=1,…, M}which are called focal
elements, and m(Aj) is the basic probability assignment of
Aj. So that m(Ф)=0 and
∑𝑀
𝑗 𝑚(𝐴𝑗 ) = 1
(1)
For example, in geotechnical engineering, each set A is
the interval of one parameter values measured for one
sample, and m(A) is the frequency of this kind of sample
occurring in all the samples. The parameter values measured from all samples compose the set X. Alternatively, the
sets Aj could be ranges of a variable obtained from another
source with relative credibility m(Aj).
In probabilistic analysis method, all of the possible values of an input parameter conform to a certain probability
distribution. While in Random Set approach, the parameters are composed of several intervals shown in Fig. 1 as
an example. Only lower and upper limits of these intervals
are considered as input variables in the deterministic model.
geomechanical parameters, geometries, loadings. Second
step is sensitivity analysis. It is to identify the degree of influence of every input parameter on slope stability. The
purpose of this step is to reduce the number of basic variables in RS-LEM analysis. And then input every possible
upper or lower limit of basic variables in limit equilibrium
models to calculate every possible safety factor. At last, we
can get the ranges of most possible safety factors.
3
RELIABILITY ANLYSIS OF A REINFORCED SLOPE
For demonstrate the application of random set theory in
the reliability analysis of reinforced slope, an example is
adopted here as a case study in GGU-GABION.
3.1 Overview of the reinforced slope
Fig. 1 Types of random set visualization: a) random interval b) pbox (after Nasekhian, 2011)
If A1i ,…, A1j are random sets on X1 and X1,…, Xp are
stochastically independent, then the joint basic probability
assignment is given by
𝑚�𝐴1𝑖 × … × 𝐴𝑝𝑗 � = 𝑚(𝐴1𝑖 ) × … × 𝑚(𝐴𝑝𝑗 )
(2)
In this paper, Random Set Theory will be applied with
Limit Equilibrium Method (LEM) in reinforced slope stability
analysis.
The geometry of the slope reinforced by concrete gabion and geogrid is shown in Fig. 3. The height of slope is 6
m, and the slope of the gabion is 85°. The gabion is divided into 10 same concrete elements with a size of 0.5 m ×
0.5 m. In the bottom of the gabion, there is a footing with a
size of 1 m × 1 m. The material of the gabion and footing
are pure concrete, and the unit weight is 18 kN/m3, and
the Young’s module is 2.5e4 kN/m2. The backfill soil behind the gabion is reinforced by geogrid. The geogrids is
4.5 m long and horizontal settled with a spacing 0.6 m. The
first geogird is on the surface of footing. The design
strengths for the used Secugrid® - geogrids are followed
by EBGEO (1997):
Secugrid® 120/40 R6 = 49.9 kN/m (1st and 2nd
geogrid)
Secugrid® 60/60 Q6 = 19.9 kN/m (3st to 10nd geogrid)
The geogrids are banded in the concrete elements frictionally. The design resistance at junction is 80% of design
resistance. The bonding stress τ of geogrid in soil is automatically calculated by the program. Usual practice is to
calculate the bonding stress τ from the effective stress (σ')
and the tangent of the friction angle φ and then to reduce
this value by λ. In this study, λ is settled as 0.7.
In the crown of the slope, there is a surcharge of 10
kPa. The random sets of basic variables can be collected
from different sources, like geotechnical tests, expert experience, similar projects, and published literature. These are
in the form of ranges, and at least two sets are needed in
order to build the probability distributions. In this case, the
basic values are presented in Table 1. The probability assignments of both sources are considered to be 0.5, so
there is no preference and both sources have the same reliability. In Table 1, the reference values are the values
adopted for the previous deterministic analysis. These are
the mean value of the upper and lower limits of two sets.
The visions of the random sets of input parameters are
shown in Fig. 4.
Fig. 2 RS-LEM procedure (modified from Peschl, 2004)
There are some important steps in this process. First is
to define the basic variables and their discrete probability
distributions. Every input parameter in slope stability analysis can be defined as random variable, like
Fig. 3 geometry of reinforced slope (from Geogrid® Handbuch)
Table 1 input parameters of soil
Unit weight Cohesion Friction angle
3
(kN/m )
(kPa)
(°)
15-20
25-30
soil
1900
17-22
27-32
Reference value
1900
18.5
28.5
need to be calculated based on the input of upper and
lower limits of each random set. For there are one upper
and one lower limit of every random set, there are 4 possible combination of each case. As an example, the deterministic input values of such analysis for the case of (c1,
φ1) are presented in Table 2.
﹡
Table 2 inputs variables of case 1 in deterministic LEM calculation
Var. Prob. Set No.
c
0.5
1
φ
0.5
1
Utilisation factor
LL
15
25
0.700
LU
15
30
0.607
UL
20
25
0.632
UU
20
30
0.558
units c, kPa; φ, degree.
﹡ L denotes the lower limit of a random set variable and U
denotes the upper limit.
Fig. 4 Random sets of input parameters: cohesion and friction angle
3.2 Verification of internal stability
GGU-GABION is a very powerful and practical software.
Many system responses can be verified, like bearing capacity, settlements, overturning safety, etc. The present
study focused on the verification of the internal stability of
the reinforced earth system, which is regulated in EBGEO.
Verification uses the methods described in DIN 4084. The
program assumes a failure system consisting of two slip
planes (two-part wedge failure mechanism) (Fig. 5).
From Table 2, the best and the worst combinations are
UU and LL, and the smallest and largest utilisation factors
are 0.558 and 0.700 respectively. It means the cohesion or
the friction angle has positive influence on the slope stability, the cohesion or friction angle is higher, the utilisation
factor will be smaller. These change laws also can be improved by the monotonicity of system response. The influences of these two basic variables on the utilisation factor
can be illustrated in Fig. 5.
Fig. 5 Influences of two basic variables on utilisation factor
Fig. 5 Two-part wedge failure mechanism (from GGU-GABION
Manuel)
In the analysis of ultimate limit state of geosynthetics,
there are two types of failure. First is reinforcement failure
and second is the pull-out or shear failure. In slope design,
the resistance of geosynthetics is the minimum value of
structure resistance and the pull out resistance. And if the
design value of effect is less than the design value of resistance, the ultimate limit state is satisfied.
Therefore, in every case, only two combinations, (1) the
highest cohesion and friction angle and (2) the lowest cohesion and friction angle are necessary to be realized in
order to obtain the lower and upper limits of utilisation factor. This has reduced the computation effect significantly,
and the total number of LEM runs has been decreased
from 16 to 8.
The next step is to determine the probability of the assignment of each realization. In this case, the random variables are considered as independent from each other, the
joint probability of the response focal element obtained
from the distinct element model is the product of the probability assignment m of input focal elements by each other.
For the mass probability of each set equals to 0.5, the first
realization can be obtained as follows:
3.3 Reliability analysis
Since the geometry uncertainty of slope has not been
considered, the geometry model of the slope in random set
approach is the same. The Mohr Coulomb criterion is
adopted in the slope material. The two basic variables: cohesion c and friction angle φ will be changed in every deterministic Limit Equilibrium analysis. For every basic variable, there are two random sets. Therefore, there are 8
different cases, which are given in the following vector:
𝑐 × 𝜑 = {(𝑐1 , 𝜑1 )1 , (𝑐1 , 𝜑1 )2 , (𝑐2 , 𝜑1 )3 , (𝑐2 , 𝜑2 )4 }
(3)
here the index of parameters denotes the relevant set
number and the index of pairs signifies one case of basic
variables.
After that, the best (smallest utilisation factor) and the
worst (largest utilisiation factor) combinations of each case
𝑚�𝑓(𝑐1 , 𝜑1 )� = 𝑚(𝑐1 ). 𝑚(𝜑1 ) = 0.5 × 0.5 = 0.25
(4)
The results of these 8 combinations are shown in Table
3.
Table 3 Results of 8 combinations and their probabilities
No.
cases
Probability
1
2
3
4
c1×φ1
c1×φ2
c2×φ1
c2×φ2
0.25
0.25
0.25
0.25
utilisation factor
Lower limit
Upper limit
0.558
0.700
0.535
0.660
0.540
0.671
0.519
0.634
At last, the random set results of utilisation factor can be
constructed in the form of p-box as Fig. 6. Contrary to
classical probability theory, the probability of failure computed by RS-LEM cannot be interpreted as a frequency of
failure. The utilisation factor ranges with different credibility
are presented in Fig. 6.
The practical purpose of the random set analysis of reinforced slope stability is to get the most likely utilisation
factors when every possible parameters values input in the
LEM model. For simplification, it is assumed that the most
likely results are those values, whose measure of their belief degree are less than 50% and their corresponding
plausible likelihood of occurrence are larger than 50%
(Nasekhian, 2011) as shown in Fig. 6. The design value of
utilisation factor is 0.593, which is calculated by the mean
value of the input parameters.
The mean value of the true system response obtained
by random set bounds is within the following range given
by Tonon et al. (2000a):
𝜇 = �∑𝑛𝑖 𝑚𝑖 ∙ 𝑖𝑛𝑓(𝐴𝑖 ) ; ∑𝑛𝑖 𝑚𝑖 ∙(𝐴𝑖 ) �
0.538
0.666
Interval of most likely
values
lower
upper
0.540
0.671
Fig. 6 Lower and upper bounds on utilisation factors of reinforced
slope
4
ACKNOWLEDGEMENT
This work has been funded by project “Interaction between
geosynthetics and clay soil” of the DFG.
6
Table 4 lower and upper mean values of true system response
Safety
factor
5
(5)
where inf(A) and sup(A) denote lower and upper limits of
focal element A, respectively. There is a good conformity
between the intervals obtained from both the most likely
range definition and those calculated from Eq. 5, and they
have been tabulated in Table 4.
Interval of true mean
values
lower
upper
Geosynthetics have been applied in civil engineering
since Roman days. In recent years as geosynthetics applied worldwide, some practical and comparable international design approaches have been developed, like
“EBGEO”. The benefit of our study is to introduce a rational mathematical framework in reinforcement design to consider the uncertainties. It can be regarded as a very important complement for existed approaches.
CONCLUSION
In this paper, the RS-LEM analysis was applied in a
steep slope reinforced by gabion and geogrids. And the
basic variables soil cohesion and friction angle which have
the most influence on the slope stability for the RS-LEM
analysis were chosen. The RS-LEM results for the slope
internal stability were presented in form of lower and upper
probability bounds.
It was illustrated that the RS-LEM is very practical to
deal with the uncertainty in rock slope stability analysis.
Based on the imprecise probabilities concepts, it can predict the system response within a range in the form of a pbox. RS-LEM provides user-friendly framework that can
apply the advanced constitutive models available in LEM
programs.
The relatively smooth bounds on the system responses
were obtained after small number of simulations. In this
case with 2 basic variables, after the monotonicity analysis, only 8 realizations were needed. The application of
RS-LEM requires much less computational effort as compared to fully probabilistic methods like Monte Carlo Simulation.
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