# Analysis of Slope Stability with Imprecise Soil Properties Using Uncertain Sets

Authors: Xuejun Zhou; Jie Li; Zhibing Liu; Yi Tang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

There are many uncertainties with respect to the assessment of slope stability, and those associated with soil properties should be given particular attention. The uncertainty theory provides an alternative to treat these uncertainties using parochial cognitive sources. A novel methodology is proposed to evaluate the stability of slopes based on an uncertain set. The soil properties involved in the deterministic methods, i.e., shear strength parameters and unit weight, are expressed as uncertain sets, and their membership functions can be assumed to be triangular or trapezoidal for a homogeneous or two-layered slope, respectively. The parameter values of membership functions are designed according to the means and variations of the soil properties, and then the expected safety factor can be calculated through the operational laws. Two numerical examples including a homogeneous slope and a two-layered slope illustrate the suitability of the proposed methodology. The relationship between the variation in the safety factor and the changes in the soil properties is investigated; moreover, the determination of the parameter values of membership is also discussed.


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## Body

## 1. Introduction

In the assessment of slope stability, engineers and researchers have reached consensus considering the uncertainties of the soil properties involved. In fact, the value of almost any measured quantity in civil engineering is, to some degree, uncertain due to the intrinsic uncertainty (i.e., it cannot be allocated a fixed value with certainty). Such uncertainties should be considered in the design or evaluation of slope stability in order to ensure the best safety levels.In the past decades, two main methodologies have been proposed to deal with the uncertainties in soil properties in the assessment problems of slope stability, i.e., the probabilistic methods [1–6] and fuzzy methods [7–13]. In the probabilistic methods, the soil properties affecting the stability of a slope are considered as random variables that have certain probability distributions. In the fuzzy methods, the uncertainties associated with the soil parameters involved include fuzziness, and the membership functions are used instead of the eigenfunctions in the probabilistic methods to describe the soil properties with fuzziness. Although the probabilistic and fuzzy methods can provide alternative approaches to cope with these uncertainties, not all uncertainties are random or fuzzy. For example, such uncertainties based on incomplete information because of cognitive sources cannot be handled satisfactorily in the probability or fuzzy theory.Fortunately, the uncertainty theory developed by Liu [14] in 2007 has provided a useful tool to deal with these uncertainties. Accordingly, Zhou et al. [15] built a model of stability assessment by considering the soil properties as the uncertain variables. In the present study, the uncertain set is used to assess the stability of a slope. The uncertain set is an alternative to the uncertain variable for describing the imprecision of data, and its theory has been well studied [6, 12, 16–20] and widely applied [21–23]. Both the uncertain variable and uncertain set belong to the same broad category of uncertain concepts, but they have different foci concerning the description of uncertainties. The uncertain variable concerns the uncertainty of whether a parameter is equal to a value, while the uncertain set focuses on the uncertainty of whether a parameter belongs to an interval [24]. Therefore, the uncertain set is suitable for depicting the uncertainties of soil properties, and can more objectively describe uncertain soil parameters expressed usually as an interval involving an estimate of the lower and upper bounds due to the available means and variations of the parameters involved from the limited information.The major focus of this paper is to propose a methodology for assessing slope stability based on the uncertain set. The soil properties, i.e., unit weight, angle of internal friction, and cohesion, are described as the uncertain sets, and their membership functions are assumed to be triangular or trapezoidal for a homogeneous slope or a two-layered slope, respectively. The safety factor of slope stability is designed based on the deterministic methods, and its expected value is calculated using the operational laws of the uncertain set. Finally, two numerical examples, including a homogeneous slope and a two-layered slope, are used to verify the proposed methodology with the Fellenius method [25].


## 2. Basic Concepts of the Uncertain Sets Theory

The uncertain set attempts to model “unsharp concepts” that are essentially sets but whose boundaries are not sharply described because of the ambiguity of human language [24]. The section introduces some concepts related to uncertain sets, such as membership function, and excepted value, and also introduces the operational laws for the uncertain set via the inverse membership function.


### 2.1. Uncertain Set and the Function of Uncertain Sets

LetL be aσ-algebra over a universal set Γ, and the uncertain measure M be a set function satisfying the normality, duality, and subadditivity axioms. The triplet Γ,L,M is called an uncertainty space, and an uncertain set is a set-valued functionξon the uncertainty spaceΓ,L,M.Ifξ1, ξ2, ⋯, ξn are uncertain sets on the uncertainty space Γ,L,M, and letf be a measurable function. Then ξ=fξ1,ξ2,⋯,ξn is a function of the uncertain sets defined by(1)ξγ=fξ1γ,ξ2γ,⋯,ξnγ,∀γ∈Γ.In the present study, the soil properties are considered as uncertain sets, and the safety factor for slope stability is, therefore, a function of the uncertain sets with respect to the soil properties.


### 2.2. Membership Function and Inverse Membership Function

A membership function is used to describe an uncertain set.Letξbe an uncertain set whose membership functionμexists. Then(2)μx=Mx∈ξ,for any numberx.The value ofμ(x) is just the membership degree thatxbelongs to the uncertain setξ, and the larger the value ofμ(x) is, the more truexbelongs toξ.On the other hand, the set-valued function(3)μ-1α=x∈Rμx≥α,∀α∈0,1,is called the inverse membership function ofξ. For each given α, the setμ−1(α) is also called the α-cut ofμ, and it is an interval for each α.The triangular and trapezoidal membership functions are employed in the present study.Example 1.
An uncertain setξ is called triangular if it has a membership function(4)μx=x-ab-a,a≤x<bx-cb-c,b≤x≤c,denoted by (a,b,c) wherea,b, andc are real numbers witha < b < c.
The triangular uncertain setξ = (a,b,c) has an inverse membership function(5)μ-1α=1-αa+αb,αb+1-αc.
Note thataandcare the lower and upperxvalues of the triangle atμ= 0, respectively, andbis thexvalue that corresponds toμ= 1 (see Figure 1).Figure 1
Triangular membership function.Example 2.
An uncertain setξis called trapezoidal if it has a membership function(6)μx=x-ab-a,a≤x<b1,b≤x<cx-dc-d,c≤x≤d,denoted by (a, b, c, d) wherea, b, c,anddare real numbers witha < b < c < d.
The trapezoidal uncertain setξ= (a, b, c, d) has an inverse membership function(7)μ-1α=1-αa+αb,αc+1-αd.
Note thataanddare the lower and upperxvalues of the triangle atμ= 0, respectively, whilebandc are thexvalues that correspond toμ= 1 (see Figure 2). The triangular uncertain sets are a special case of the trapezoidal uncertain sets.Figure 2
Trapezoidal membership function.


### 2.3. Operational Law

Letξ1,ξ2,⋯,ξn be independent uncertain sets with inverse membership functions μ1-1,μ2-1,⋯,μn-1, respectively, and letf be a measurable function. Then ξ=fξ1,ξ2,⋯,ξn has an inverse membership function(8)μ-1α=fμ1-1α,μ2-1α,⋯,μn-1α.Letξbe a nonempty uncertain set with membership functionμ, then its expected value(9)Eξ=12∫01inf⁡μ-1α+sup⁡μ-1αdα,where inf μ-1α and sup μ-1α are the infimum and supremum of the α-cut, respectively.This shows that the triangular uncertain setξ= (a, b, c) has an expected value(10)Eξ=a+2b+c4,and the trapezoidal uncertain setξ= (a, b, c, d) has an expected value(11)Eξ=a+b+c+d4,


## 2.1. Uncertain Set and the Function of Uncertain Sets

LetL be aσ-algebra over a universal set Γ, and the uncertain measure M be a set function satisfying the normality, duality, and subadditivity axioms. The triplet Γ,L,M is called an uncertainty space, and an uncertain set is a set-valued functionξon the uncertainty spaceΓ,L,M.Ifξ1, ξ2, ⋯, ξn are uncertain sets on the uncertainty space Γ,L,M, and letf be a measurable function. Then ξ=fξ1,ξ2,⋯,ξn is a function of the uncertain sets defined by(1)ξγ=fξ1γ,ξ2γ,⋯,ξnγ,∀γ∈Γ.In the present study, the soil properties are considered as uncertain sets, and the safety factor for slope stability is, therefore, a function of the uncertain sets with respect to the soil properties.


## 2.2. Membership Function and Inverse Membership Function

A membership function is used to describe an uncertain set.Letξbe an uncertain set whose membership functionμexists. Then(2)μx=Mx∈ξ,for any numberx.The value ofμ(x) is just the membership degree thatxbelongs to the uncertain setξ, and the larger the value ofμ(x) is, the more truexbelongs toξ.On the other hand, the set-valued function(3)μ-1α=x∈Rμx≥α,∀α∈0,1,is called the inverse membership function ofξ. For each given α, the setμ−1(α) is also called the α-cut ofμ, and it is an interval for each α.The triangular and trapezoidal membership functions are employed in the present study.Example 1.
An uncertain setξ is called triangular if it has a membership function(4)μx=x-ab-a,a≤x<bx-cb-c,b≤x≤c,denoted by (a,b,c) wherea,b, andc are real numbers witha < b < c.
The triangular uncertain setξ = (a,b,c) has an inverse membership function(5)μ-1α=1-αa+αb,αb+1-αc.
Note thataandcare the lower and upperxvalues of the triangle atμ= 0, respectively, andbis thexvalue that corresponds toμ= 1 (see Figure 1).Figure 1
Triangular membership function.Example 2.
An uncertain setξis called trapezoidal if it has a membership function(6)μx=x-ab-a,a≤x<b1,b≤x<cx-dc-d,c≤x≤d,denoted by (a, b, c, d) wherea, b, c,anddare real numbers witha < b < c < d.
The trapezoidal uncertain setξ= (a, b, c, d) has an inverse membership function(7)μ-1α=1-αa+αb,αc+1-αd.
Note thataanddare the lower and upperxvalues of the triangle atμ= 0, respectively, whilebandc are thexvalues that correspond toμ= 1 (see Figure 2). The triangular uncertain sets are a special case of the trapezoidal uncertain sets.Figure 2
Trapezoidal membership function.


## 2.3. Operational Law

Letξ1,ξ2,⋯,ξn be independent uncertain sets with inverse membership functions μ1-1,μ2-1,⋯,μn-1, respectively, and letf be a measurable function. Then ξ=fξ1,ξ2,⋯,ξn has an inverse membership function(8)μ-1α=fμ1-1α,μ2-1α,⋯,μn-1α.Letξbe a nonempty uncertain set with membership functionμ, then its expected value(9)Eξ=12∫01inf⁡μ-1α+sup⁡μ-1αdα,where inf μ-1α and sup μ-1α are the infimum and supremum of the α-cut, respectively.This shows that the triangular uncertain setξ= (a, b, c) has an expected value(10)Eξ=a+2b+c4,and the trapezoidal uncertain setξ= (a, b, c, d) has an expected value(11)Eξ=a+b+c+d4,


## 3. Methodology of Slope Stability Analysis Using Uncertain Sets

### 3.1. Modelling Uncertainties Using Uncertain Sets

Generally, during the stability analysis of slopes, four major types of uncertainty associated with soil modelling may be encountered and identified(i)
The uncertainty arises from in situ variability because of the variation in the mineral composition and stress history of the soil mass.(ii)
The uncertainty originates from the measurement errors due to sampling disturbance, test imperfection, and human factors.(iii)
The uncertainty derives from imprecise information owing to the limited number of tests and samples.(iv)
The uncertainty results from the difference between the actual behaviour of the geotechnical system and that of the mechanical model.The probabilistic methods and fuzzy methods have long been regarded as effective tools for treating uncertainties, but not all uncertainties are random or fuzzy. In particular, some uncertainties are often both random and fuzzy, such as the uncertainties based on incomplete information because of parochial cognitive sources [7, 24, 26]. Those uncertainties originating from cognitive sources may not be handled satisfactorily based on the probability theory or fuzzy theory, but the uncertain theory is more appropriate. The uncertain set is a different concept to the random set [27], fuzzy set [28], and rough set [29].The present study focuses on using the uncertain sets to describe the preceding third uncertainty of soil properties involved. Compared to the uncertain variable [15], the uncertain set has a different focus concerning the description of uncertainty. The uncertain variable concerns the uncertainty of whether a parameter is equal to a value, while the uncertain set focuses on the uncertainty of whether a parameter belongs to an interval [24]. In many cases, uncertain soil parameters may be expressed as an interval involving an estimate of the lower and upper bounds due to the available means and variations of the parameters involved from the limited information. To this end, these uncertain soil parameters may be expressed by the uncertain sets.


### 3.2. Propagation of Uncertainties through the Deterministic Analysis

In the deterministic analysis, the safety factor is an indicator of assessing the stability of a slope. Generally, the safety factor (Fs) can be expressed as a function of several uncertain and constant variables, i.e.,(12)FS=fγ,c,ϕ,H,β,where γ is the unit weight,cthe cohesion,ϕ the angle of internal friction,H the height of slope, and β the inclination of the base.If the input soil parametersγ,c,andϕare uncertain and can be described as independent uncertain sets, the output of the deterministic model will also be an uncertain set. In this case, uncertainty is propagated through the solution processes.Assume thatμγ, μc, and μϕ are their membership functions, respectively, and the inverse membership functions are μγ-1, μc-1 and μϕ-1, respectively. The safety factor Fs is a continuous function and thus measurable, so its inverse membership function can be written as(13)λ-1α=fμγ-1α,μc-1α,μϕ-1α.According to (9), the expected value of the safety factorFs can be computed by(14)EFS=12∫01inf⁡λ-1α+sup⁡λ-1αdα.


### 3.3. Handling Uncertainties

In calculating the safety factor by the deterministic methods, the greatest uncertainties arise in the selection of such properties as shear strength parameters and unit weight [7, 15]. In the proposed method, the uncertainties associated with these parameters can be described by uncertain sets, and different membership functions (triangular or trapezoidal) are used to represent the uncertain parameters on the basis of the slopes with different soil layers.In (12), the unit weight γ, the cohesionc,and the angle of internal frictionϕare considered as uncertain parameters and denoted by a 3-dimensional uncertain vector(15)X=X1,X2,X3,whereXi,i=1,2,3, represent the corresponding uncertain parameters γ,c, andϕ, respectively. Assume that the uncertain vectorX has a mean vector(16)EX=EX1,EX2,EX3,and a standard deviation vector(17)σX=σX1,σX2,σX3.The uncertain parameters are described by their mean values and variances, and the uncertainty is propagated through the proposed process based on the deterministic analysis for the calculation of the safety factor of a slope.The following procedure is to build the uncertain sets for the above-mentioned uncertain parameters. The construction of an uncertain set should deal with two questions, i.e., how to determine an acceptable universe of discourse and how to substantiate an appropriate membership function?


### 3.4. Determination of Membership Functions

#### 3.4.1. Homogeneous Slope

For a homogeneous slope, if there is no other sample information except for the mean value and variance of each soil parameter from the experiments, then the triangular membership function can be employed to describe the uncertain parameter involved. The advantage of such a treatment is that it can improve the reliability of the safety factor calculated using the deterministic methods because of the inaccuracy of the means of the parameters. Of course, if there is more available information of soil parameters from the experiments, the others membership functions, such as trapezoidal membership function, could be introduced into this case. As shown in Figure1, the values ofa,b,andcshould be determined. For this purpose, each uncertain parameter must be expressed as an interval, which includes an estimate of a lower bound, an upper bound, and a mode (the most likely value). The values ofa,b,andcin the triangular membership function (4) can be designed as(18)a=EX-kσX,b=EX,c=EX+kσX,wherekis the number of sigma units, which takes a value of 0.50 to 2.00 depending on the data available and accuracy of the results desired. In the following numerical examples, four cases are considered: (a) Case 1,k= 0.50; (b) Case 2,k= 1.00; (c) Case 3,k= 1.50; and (d) Case 4,k= 2.00.By substituting the above valuesa,b,andcinto (4) and (5), the (inverse) triangular membership functions of the corresponding uncertain sets can be obtained.


#### 3.4.2. Two-Layered Slope

For a nonhomogeneous slope, the above approach can also be employed to build triangular membership functions for each soil layer; however, the other membership functions could be employed more naturally and directly. Normally, the (mean) values of parameters of each soil layer are required in the deterministic methods. Based on this limited information, especially for a two-layered slope, the trapezoidal membership function is used to describe an uncertain parameter involved. As shown in Figure2, the values ofa,b,c,anddshould be designed. To this end, each uncertain parameter should also be expressed as an interval, which includes an estimate of two lower bounds and two upper bounds for the two soil layers. The values ofa,b,c,anddin the trapezoidal membership function (6) can be designed as(19)a=EX-k1σX,b=EX-k2σX,c=EX+k2σX,d=EX+k1σXwherek1 andk2 are the number of sigma units, which also take the values of 0.50 to 2.00 depending on the data available and accuracy of the results desired, and satisfies the relationk1> k2. In the following numerical examples, six cases are considered: (a) Case 1,k1 = 1.00 andk2 = 0.50; (b) Case 2,k1 = 1.50 andk2 = 0.50; (c) Case 3,k1 = 1.50 andk2 = 1.00; (d) Case 4,k1 = 2.00 andk2 = 0.50; (e) Case 5,k1 = 2.00 andk2 = 1.00; and (f) Case 6,k1 = 2.00 andk2 = 1.50.By substituting the above valuesa,b,c,anddinto (6) and (7), the (inverse) trapezoidal membership functions of the corresponding uncertain sets can be obtained.The essence of this methodology is to express the uncertain soil parameters as uncertain sets. By describing uncertain sets with different membership functions, the analysis of uncertain sets leads to a string of interval analyses. Triangular or trapezoidal uncertain sets are employed to describe the uncertainties due to the limited data available. Figure3 illustrates the methodology of the stability analysis using uncertain sets.Figure 3
Schematic representation for the methodology of the stability analysis using uncertain sets.


## 3.1. Modelling Uncertainties Using Uncertain Sets

Generally, during the stability analysis of slopes, four major types of uncertainty associated with soil modelling may be encountered and identified(i)
The uncertainty arises from in situ variability because of the variation in the mineral composition and stress history of the soil mass.(ii)
The uncertainty originates from the measurement errors due to sampling disturbance, test imperfection, and human factors.(iii)
The uncertainty derives from imprecise information owing to the limited number of tests and samples.(iv)
The uncertainty results from the difference between the actual behaviour of the geotechnical system and that of the mechanical model.The probabilistic methods and fuzzy methods have long been regarded as effective tools for treating uncertainties, but not all uncertainties are random or fuzzy. In particular, some uncertainties are often both random and fuzzy, such as the uncertainties based on incomplete information because of parochial cognitive sources [7, 24, 26]. Those uncertainties originating from cognitive sources may not be handled satisfactorily based on the probability theory or fuzzy theory, but the uncertain theory is more appropriate. The uncertain set is a different concept to the random set [27], fuzzy set [28], and rough set [29].The present study focuses on using the uncertain sets to describe the preceding third uncertainty of soil properties involved. Compared to the uncertain variable [15], the uncertain set has a different focus concerning the description of uncertainty. The uncertain variable concerns the uncertainty of whether a parameter is equal to a value, while the uncertain set focuses on the uncertainty of whether a parameter belongs to an interval [24]. In many cases, uncertain soil parameters may be expressed as an interval involving an estimate of the lower and upper bounds due to the available means and variations of the parameters involved from the limited information. To this end, these uncertain soil parameters may be expressed by the uncertain sets.


## 3.2. Propagation of Uncertainties through the Deterministic Analysis

In the deterministic analysis, the safety factor is an indicator of assessing the stability of a slope. Generally, the safety factor (Fs) can be expressed as a function of several uncertain and constant variables, i.e.,(12)FS=fγ,c,ϕ,H,β,where γ is the unit weight,cthe cohesion,ϕ the angle of internal friction,H the height of slope, and β the inclination of the base.If the input soil parametersγ,c,andϕare uncertain and can be described as independent uncertain sets, the output of the deterministic model will also be an uncertain set. In this case, uncertainty is propagated through the solution processes.Assume thatμγ, μc, and μϕ are their membership functions, respectively, and the inverse membership functions are μγ-1, μc-1 and μϕ-1, respectively. The safety factor Fs is a continuous function and thus measurable, so its inverse membership function can be written as(13)λ-1α=fμγ-1α,μc-1α,μϕ-1α.According to (9), the expected value of the safety factorFs can be computed by(14)EFS=12∫01inf⁡λ-1α+sup⁡λ-1αdα.


## 3.3. Handling Uncertainties

In calculating the safety factor by the deterministic methods, the greatest uncertainties arise in the selection of such properties as shear strength parameters and unit weight [7, 15]. In the proposed method, the uncertainties associated with these parameters can be described by uncertain sets, and different membership functions (triangular or trapezoidal) are used to represent the uncertain parameters on the basis of the slopes with different soil layers.In (12), the unit weight γ, the cohesionc,and the angle of internal frictionϕare considered as uncertain parameters and denoted by a 3-dimensional uncertain vector(15)X=X1,X2,X3,whereXi,i=1,2,3, represent the corresponding uncertain parameters γ,c, andϕ, respectively. Assume that the uncertain vectorX has a mean vector(16)EX=EX1,EX2,EX3,and a standard deviation vector(17)σX=σX1,σX2,σX3.The uncertain parameters are described by their mean values and variances, and the uncertainty is propagated through the proposed process based on the deterministic analysis for the calculation of the safety factor of a slope.The following procedure is to build the uncertain sets for the above-mentioned uncertain parameters. The construction of an uncertain set should deal with two questions, i.e., how to determine an acceptable universe of discourse and how to substantiate an appropriate membership function?


## 3.4. Determination of Membership Functions

### 3.4.1. Homogeneous Slope

For a homogeneous slope, if there is no other sample information except for the mean value and variance of each soil parameter from the experiments, then the triangular membership function can be employed to describe the uncertain parameter involved. The advantage of such a treatment is that it can improve the reliability of the safety factor calculated using the deterministic methods because of the inaccuracy of the means of the parameters. Of course, if there is more available information of soil parameters from the experiments, the others membership functions, such as trapezoidal membership function, could be introduced into this case. As shown in Figure1, the values ofa,b,andcshould be determined. For this purpose, each uncertain parameter must be expressed as an interval, which includes an estimate of a lower bound, an upper bound, and a mode (the most likely value). The values ofa,b,andcin the triangular membership function (4) can be designed as(18)a=EX-kσX,b=EX,c=EX+kσX,wherekis the number of sigma units, which takes a value of 0.50 to 2.00 depending on the data available and accuracy of the results desired. In the following numerical examples, four cases are considered: (a) Case 1,k= 0.50; (b) Case 2,k= 1.00; (c) Case 3,k= 1.50; and (d) Case 4,k= 2.00.By substituting the above valuesa,b,andcinto (4) and (5), the (inverse) triangular membership functions of the corresponding uncertain sets can be obtained.


### 3.4.2. Two-Layered Slope

For a nonhomogeneous slope, the above approach can also be employed to build triangular membership functions for each soil layer; however, the other membership functions could be employed more naturally and directly. Normally, the (mean) values of parameters of each soil layer are required in the deterministic methods. Based on this limited information, especially for a two-layered slope, the trapezoidal membership function is used to describe an uncertain parameter involved. As shown in Figure2, the values ofa,b,c,anddshould be designed. To this end, each uncertain parameter should also be expressed as an interval, which includes an estimate of two lower bounds and two upper bounds for the two soil layers. The values ofa,b,c,anddin the trapezoidal membership function (6) can be designed as(19)a=EX-k1σX,b=EX-k2σX,c=EX+k2σX,d=EX+k1σXwherek1 andk2 are the number of sigma units, which also take the values of 0.50 to 2.00 depending on the data available and accuracy of the results desired, and satisfies the relationk1> k2. In the following numerical examples, six cases are considered: (a) Case 1,k1 = 1.00 andk2 = 0.50; (b) Case 2,k1 = 1.50 andk2 = 0.50; (c) Case 3,k1 = 1.50 andk2 = 1.00; (d) Case 4,k1 = 2.00 andk2 = 0.50; (e) Case 5,k1 = 2.00 andk2 = 1.00; and (f) Case 6,k1 = 2.00 andk2 = 1.50.By substituting the above valuesa,b,c,anddinto (6) and (7), the (inverse) trapezoidal membership functions of the corresponding uncertain sets can be obtained.The essence of this methodology is to express the uncertain soil parameters as uncertain sets. By describing uncertain sets with different membership functions, the analysis of uncertain sets leads to a string of interval analyses. Triangular or trapezoidal uncertain sets are employed to describe the uncertainties due to the limited data available. Figure3 illustrates the methodology of the stability analysis using uncertain sets.Figure 3
Schematic representation for the methodology of the stability analysis using uncertain sets.


## 3.4.1. Homogeneous Slope

For a homogeneous slope, if there is no other sample information except for the mean value and variance of each soil parameter from the experiments, then the triangular membership function can be employed to describe the uncertain parameter involved. The advantage of such a treatment is that it can improve the reliability of the safety factor calculated using the deterministic methods because of the inaccuracy of the means of the parameters. Of course, if there is more available information of soil parameters from the experiments, the others membership functions, such as trapezoidal membership function, could be introduced into this case. As shown in Figure1, the values ofa,b,andcshould be determined. For this purpose, each uncertain parameter must be expressed as an interval, which includes an estimate of a lower bound, an upper bound, and a mode (the most likely value). The values ofa,b,andcin the triangular membership function (4) can be designed as(18)a=EX-kσX,b=EX,c=EX+kσX,wherekis the number of sigma units, which takes a value of 0.50 to 2.00 depending on the data available and accuracy of the results desired. In the following numerical examples, four cases are considered: (a) Case 1,k= 0.50; (b) Case 2,k= 1.00; (c) Case 3,k= 1.50; and (d) Case 4,k= 2.00.By substituting the above valuesa,b,andcinto (4) and (5), the (inverse) triangular membership functions of the corresponding uncertain sets can be obtained.


## 3.4.2. Two-Layered Slope

For a nonhomogeneous slope, the above approach can also be employed to build triangular membership functions for each soil layer; however, the other membership functions could be employed more naturally and directly. Normally, the (mean) values of parameters of each soil layer are required in the deterministic methods. Based on this limited information, especially for a two-layered slope, the trapezoidal membership function is used to describe an uncertain parameter involved. As shown in Figure2, the values ofa,b,c,anddshould be designed. To this end, each uncertain parameter should also be expressed as an interval, which includes an estimate of two lower bounds and two upper bounds for the two soil layers. The values ofa,b,c,anddin the trapezoidal membership function (6) can be designed as(19)a=EX-k1σX,b=EX-k2σX,c=EX+k2σX,d=EX+k1σXwherek1 andk2 are the number of sigma units, which also take the values of 0.50 to 2.00 depending on the data available and accuracy of the results desired, and satisfies the relationk1> k2. In the following numerical examples, six cases are considered: (a) Case 1,k1 = 1.00 andk2 = 0.50; (b) Case 2,k1 = 1.50 andk2 = 0.50; (c) Case 3,k1 = 1.50 andk2 = 1.00; (d) Case 4,k1 = 2.00 andk2 = 0.50; (e) Case 5,k1 = 2.00 andk2 = 1.00; and (f) Case 6,k1 = 2.00 andk2 = 1.50.By substituting the above valuesa,b,c,anddinto (6) and (7), the (inverse) trapezoidal membership functions of the corresponding uncertain sets can be obtained.The essence of this methodology is to express the uncertain soil parameters as uncertain sets. By describing uncertain sets with different membership functions, the analysis of uncertain sets leads to a string of interval analyses. Triangular or trapezoidal uncertain sets are employed to describe the uncertainties due to the limited data available. Figure3 illustrates the methodology of the stability analysis using uncertain sets.Figure 3
Schematic representation for the methodology of the stability analysis using uncertain sets.


## 4. Numerical Experiments

In this section, the proposed model is used to study the stability of slopes using two numerical examples. The first example illustrates the stability of a homogeneous slope, and the second one illustrates the stability of a two-layered slope.


### 4.1. Fellenius Method

The majority of slope stability problems are statically indeterminate, and, as a result, some simplifying assumptions are made for the sake of determining these problems. Various deterministic methods are proposed based on different assumptions, and among the most popular methods, the Fellenius method [25] is considered the simplest method of slices. The Fellenius method assumes that the interslice forces are parallel to the base of each slice, which is applicable to a circular slip surface. In the Fellenius method, the safety factor without considering the pore water pressure can be directly obtained as follows:(20)FS=∑i=1NWicos⁡θitan⁡ϕi+cili∑i=1NWisin⁡θi,whereWi=γivi is the weight of theith slice,vithe volume of theith slice,θithe inclination of the base of theith slice,cithe cohesion of theith slice, andlithe width of theith slice basis.In the present numerical analysis, the Fellenius method is employed as a specific case of (12) to analyse the stability of slopes. Actually, any other method may also be employed with equal ease. The uncertainties in the soil parameters are incorporated in the analysis by representing them as uncertain sets.


### 4.2. Homogeneous Slope

This example is a homogeneous slope with the geometry presented in Figure4. The inclination of the base equals 55°. The soil data used for this example are shown in Table 1. In our study, the means of the soil properties with their coefficients of variation (CV) are assumed according to their reported values in the literature [30], and different values for the coefficient of variations are assumed on the basis of their proposed ranges. For instance, the coefficient of variation for the angle of internal frictionϕvaries from 0.10 to 0.30, for the cohesionc from 0.20 to 0.40, and from 0.01 to 0.03 for the unit weight γ. Since the variability of the unit weight γ is very small, it can be considered as a fixed value.Table 1
Soil data for the homogeneous slope.
Soil parameter Mean CV1 CV2 CV3 CV4 CV5 Unit weight(kN/m3) 18.60 0.00 0.00 0.00 0.00 0.00 Angle of internal friction(°) 12.00 0.10 0.15 0.20 0.25 0.30 Cohesion(kN/m2) 16.70 0.20 0.25 0.30 0.35 0.40Figure 4
Cross-section of the homogeneous slope.


#### 4.2.1. Results Analysis

The uncertainties in the variablesϕandcare considered in the analysis. The safety factor of the slope using mean parameters with the Fellenius method is 1.1414. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(4.12, 13.30), and the radius 8.35m(see Figure 4).In order to study the relationship between the changes in the expected safety factor and the variation in soil parameters, the expected value of the safety factor can be calculated by allowing each parameter to vary independently about its respective mean value while the other parameter is kept fixed at their mean values. In addition, the parameter values of the triangular membership function (18) for Case 2, i.e.,k= 1.00, are considered in this round of discussion. Figures 5 and 6 show the relationships between the expected safety factor and the angle of internal friction and cohesion, respectively.Figure 5
Variation in the safety factor with the angle of internal friction for the homogeneous slope.Figure 6
Variation in the safety factor with cohesion for the homogeneous slope.From Figures5 and 6, it can be found that the variations in the expected safety factor clearly increase with the changes in the angle of internal friction. Nevertheless, the expected safety factor almost keeps unchanged until the coefficient of variations of cohesion reaches 40%. The results demonstrate that the expected safety factor changes sensitively with the fluctuations in the angle of internal friction, and only changes insensitively with the fluctuation in the cohesion when its fluctuation is of a high degree.Three numerical experiments are designed to study how the uncertainty in soil properties may affect the expected safety factor. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parameterkin (19) also takes the value of 1.00. Table 2 shows the results of the three numerical experiments.Table 2
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1407 2 Moderate 0.20 0.30 1.1394 3 High 0.30 0.40 1.1370It can be seen from Table2 that the expected safety factors do not change much in the three experiments, and the stability of the proposed model is verified by this fact. In addition, it can be seen that the values of the expected safety factor become smaller and smaller with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the low degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the triangular membership function. All four cases are considered in the evaluation of the expected safety factors, i.e.,k=0.50, 1.00, 1.50, and 2.00. The expected safety factors estimated by the proposed model considering the moderate uncertainty in soil properties, i.e., the CVs of the angle of internal friction and the cohesion, are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 3.Table 3
The numerical results of the parameter analysis.
k 0.50 1.00 1.50 2.00 E[Fs] 1.1407 1.1394 1.1368 1.1330From Table3, as the parameter value of the triangular membership increases, the increase in the variations ofE[Fs] is reflected. Therefore, compared to the safety factor of 1.1414 with the original Fellenius method, it can be concluded that the parameterkshould take the best value between 0.50 and 1.00 in the case of a homogeneous slope.


### 4.3. Two-Layered Slope

A two-layered slope is presented in Figure7, and the soil properties with their coefficients of variation for both layers are shown in Table 4. The unit weights of the two layers are also considered as fixed values. This example is used to apply the proposed model with a trapezoidal membership function for the stability analysis of a two-layered slope.Table 4
Soil data for the two-layered slope.
Soil parameter Mean CV1 CV2 CV3 CV4 CV5 Soil layer 1 Unit weight (kN/m3) 17.64 0.00 0.00 0.00 0.00 0.00 Angle of internal friction (°) 10.00 0.10 0.15 0.20 0.25 0.30 Cohesion (kN/m2) 10.00 0.20 0.25 0.30 0.35 0.40 Soil layer 2 Unit weight (kN/m3) 18.00 0.00 0.00 0.00 0.00 0.00 Angle of internal friction (°) 5.00 0.10 0.15 0.20 0.25 0.30 Cohesion (kN/m2) 8.00 0.20 0.25 0.30 0.35 0.40Figure 7
Cross-section of the two-layered slope.


#### 4.3.1. Results Analysis

The safety factor of the slope using mean parameters with the Fellenius method is 0.947. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(9.22, 11.98) and the radius 9.38m.Three numerical experiments are also designed to study how the uncertainty in soil properties may affect the expected safety factor in this example. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parametersk1 andk2 in (20) take values of 1.50 and 1.00, respectively. Table 5 shows the results of the three numerical experiments.Table 5
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1043 2 Moderate 0.20 0.30 1.1126 3 High 0.30 0.40 1.1162From Table5, the fact that the expected safety factors do not change much in the three experiments also indicates the stability of the proposed model. In addition, it can be found that the values of the expected safety factor become larger and larger with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the high degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the trapezoidal membership function. All six cases are considered in the evaluation of the expected safety factors, i.e.,k1=1.00 andk2=0.50;k1=1.50 andk2=0.50;k1=1.50 andk2=1.00;k1=2.00 andk2=0.50;k1=2.00 andk2=1.00; andk1=2.00 andk2=1.50. The expected safety factor is estimated by the proposed model considering the moderate uncertainty in soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 6.Table 6
The numerical results of the parameter analysis.
k 1 1.00 1.50 1.50 2.00 2.00 2.00 k 2 0.50 0.50 1.00 0.50 1.00 1.50 E[Fs] 1.1015 1.1071 1.1125 0.9670 1.1163 1.1201From Table6, as the parameter value of the triangular membership varies, the change in the variation ofE[Fs] is reflected. Therefore, compared to the safety factor of 0.947 with the original Fellenius method, it can be concluded that the parametersk1 andk2 should take the best value of around 2.00 and 0.50 in the case of a two-layered slope, respectively.Although only two simplified slope models are discussed in this section, they are still valuable for the study of the stability of real slopes. In fact, for more complicated slopes in practical engineering, other shapes of the membership functions for the uncertain sets can also be incorporated in the approach. Furthermore, the complicated slopes can be considered as a number of homogeneous slopes or two-layered slopes [11], and then the method discussed in this section can be developed to evaluate their stability.


## 4.1. Fellenius Method

The majority of slope stability problems are statically indeterminate, and, as a result, some simplifying assumptions are made for the sake of determining these problems. Various deterministic methods are proposed based on different assumptions, and among the most popular methods, the Fellenius method [25] is considered the simplest method of slices. The Fellenius method assumes that the interslice forces are parallel to the base of each slice, which is applicable to a circular slip surface. In the Fellenius method, the safety factor without considering the pore water pressure can be directly obtained as follows:(20)FS=∑i=1NWicos⁡θitan⁡ϕi+cili∑i=1NWisin⁡θi,whereWi=γivi is the weight of theith slice,vithe volume of theith slice,θithe inclination of the base of theith slice,cithe cohesion of theith slice, andlithe width of theith slice basis.In the present numerical analysis, the Fellenius method is employed as a specific case of (12) to analyse the stability of slopes. Actually, any other method may also be employed with equal ease. The uncertainties in the soil parameters are incorporated in the analysis by representing them as uncertain sets.


## 4.2. Homogeneous Slope

This example is a homogeneous slope with the geometry presented in Figure4. The inclination of the base equals 55°. The soil data used for this example are shown in Table 1. In our study, the means of the soil properties with their coefficients of variation (CV) are assumed according to their reported values in the literature [30], and different values for the coefficient of variations are assumed on the basis of their proposed ranges. For instance, the coefficient of variation for the angle of internal frictionϕvaries from 0.10 to 0.30, for the cohesionc from 0.20 to 0.40, and from 0.01 to 0.03 for the unit weight γ. Since the variability of the unit weight γ is very small, it can be considered as a fixed value.Table 1
Soil data for the homogeneous slope.
Soil parameter Mean CV1 CV2 CV3 CV4 CV5 Unit weight(kN/m3) 18.60 0.00 0.00 0.00 0.00 0.00 Angle of internal friction(°) 12.00 0.10 0.15 0.20 0.25 0.30 Cohesion(kN/m2) 16.70 0.20 0.25 0.30 0.35 0.40Figure 4
Cross-section of the homogeneous slope.


### 4.2.1. Results Analysis

The uncertainties in the variablesϕandcare considered in the analysis. The safety factor of the slope using mean parameters with the Fellenius method is 1.1414. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(4.12, 13.30), and the radius 8.35m(see Figure 4).In order to study the relationship between the changes in the expected safety factor and the variation in soil parameters, the expected value of the safety factor can be calculated by allowing each parameter to vary independently about its respective mean value while the other parameter is kept fixed at their mean values. In addition, the parameter values of the triangular membership function (18) for Case 2, i.e.,k= 1.00, are considered in this round of discussion. Figures 5 and 6 show the relationships between the expected safety factor and the angle of internal friction and cohesion, respectively.Figure 5
Variation in the safety factor with the angle of internal friction for the homogeneous slope.Figure 6
Variation in the safety factor with cohesion for the homogeneous slope.From Figures5 and 6, it can be found that the variations in the expected safety factor clearly increase with the changes in the angle of internal friction. Nevertheless, the expected safety factor almost keeps unchanged until the coefficient of variations of cohesion reaches 40%. The results demonstrate that the expected safety factor changes sensitively with the fluctuations in the angle of internal friction, and only changes insensitively with the fluctuation in the cohesion when its fluctuation is of a high degree.Three numerical experiments are designed to study how the uncertainty in soil properties may affect the expected safety factor. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parameterkin (19) also takes the value of 1.00. Table 2 shows the results of the three numerical experiments.Table 2
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1407 2 Moderate 0.20 0.30 1.1394 3 High 0.30 0.40 1.1370It can be seen from Table2 that the expected safety factors do not change much in the three experiments, and the stability of the proposed model is verified by this fact. In addition, it can be seen that the values of the expected safety factor become smaller and smaller with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the low degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the triangular membership function. All four cases are considered in the evaluation of the expected safety factors, i.e.,k=0.50, 1.00, 1.50, and 2.00. The expected safety factors estimated by the proposed model considering the moderate uncertainty in soil properties, i.e., the CVs of the angle of internal friction and the cohesion, are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 3.Table 3
The numerical results of the parameter analysis.
k 0.50 1.00 1.50 2.00 E[Fs] 1.1407 1.1394 1.1368 1.1330From Table3, as the parameter value of the triangular membership increases, the increase in the variations ofE[Fs] is reflected. Therefore, compared to the safety factor of 1.1414 with the original Fellenius method, it can be concluded that the parameterkshould take the best value between 0.50 and 1.00 in the case of a homogeneous slope.


## 4.2.1. Results Analysis

The uncertainties in the variablesϕandcare considered in the analysis. The safety factor of the slope using mean parameters with the Fellenius method is 1.1414. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(4.12, 13.30), and the radius 8.35m(see Figure 4).In order to study the relationship between the changes in the expected safety factor and the variation in soil parameters, the expected value of the safety factor can be calculated by allowing each parameter to vary independently about its respective mean value while the other parameter is kept fixed at their mean values. In addition, the parameter values of the triangular membership function (18) for Case 2, i.e.,k= 1.00, are considered in this round of discussion. Figures 5 and 6 show the relationships between the expected safety factor and the angle of internal friction and cohesion, respectively.Figure 5
Variation in the safety factor with the angle of internal friction for the homogeneous slope.Figure 6
Variation in the safety factor with cohesion for the homogeneous slope.From Figures5 and 6, it can be found that the variations in the expected safety factor clearly increase with the changes in the angle of internal friction. Nevertheless, the expected safety factor almost keeps unchanged until the coefficient of variations of cohesion reaches 40%. The results demonstrate that the expected safety factor changes sensitively with the fluctuations in the angle of internal friction, and only changes insensitively with the fluctuation in the cohesion when its fluctuation is of a high degree.Three numerical experiments are designed to study how the uncertainty in soil properties may affect the expected safety factor. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for the soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parameterkin (19) also takes the value of 1.00. Table 2 shows the results of the three numerical experiments.Table 2
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1407 2 Moderate 0.20 0.30 1.1394 3 High 0.30 0.40 1.1370It can be seen from Table2 that the expected safety factors do not change much in the three experiments, and the stability of the proposed model is verified by this fact. In addition, it can be seen that the values of the expected safety factor become smaller and smaller with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the low degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the triangular membership function. All four cases are considered in the evaluation of the expected safety factors, i.e.,k=0.50, 1.00, 1.50, and 2.00. The expected safety factors estimated by the proposed model considering the moderate uncertainty in soil properties, i.e., the CVs of the angle of internal friction and the cohesion, are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 3.Table 3
The numerical results of the parameter analysis.
k 0.50 1.00 1.50 2.00 E[Fs] 1.1407 1.1394 1.1368 1.1330From Table3, as the parameter value of the triangular membership increases, the increase in the variations ofE[Fs] is reflected. Therefore, compared to the safety factor of 1.1414 with the original Fellenius method, it can be concluded that the parameterkshould take the best value between 0.50 and 1.00 in the case of a homogeneous slope.


## 4.3. Two-Layered Slope

A two-layered slope is presented in Figure7, and the soil properties with their coefficients of variation for both layers are shown in Table 4. The unit weights of the two layers are also considered as fixed values. This example is used to apply the proposed model with a trapezoidal membership function for the stability analysis of a two-layered slope.Table 4
Soil data for the two-layered slope.
Soil parameter Mean CV1 CV2 CV3 CV4 CV5 Soil layer 1 Unit weight (kN/m3) 17.64 0.00 0.00 0.00 0.00 0.00 Angle of internal friction (°) 10.00 0.10 0.15 0.20 0.25 0.30 Cohesion (kN/m2) 10.00 0.20 0.25 0.30 0.35 0.40 Soil layer 2 Unit weight (kN/m3) 18.00 0.00 0.00 0.00 0.00 0.00 Angle of internal friction (°) 5.00 0.10 0.15 0.20 0.25 0.30 Cohesion (kN/m2) 8.00 0.20 0.25 0.30 0.35 0.40Figure 7
Cross-section of the two-layered slope.


### 4.3.1. Results Analysis

The safety factor of the slope using mean parameters with the Fellenius method is 0.947. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(9.22, 11.98) and the radius 9.38m.Three numerical experiments are also designed to study how the uncertainty in soil properties may affect the expected safety factor in this example. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parametersk1 andk2 in (20) take values of 1.50 and 1.00, respectively. Table 5 shows the results of the three numerical experiments.Table 5
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1043 2 Moderate 0.20 0.30 1.1126 3 High 0.30 0.40 1.1162From Table5, the fact that the expected safety factors do not change much in the three experiments also indicates the stability of the proposed model. In addition, it can be found that the values of the expected safety factor become larger and larger with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the high degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the trapezoidal membership function. All six cases are considered in the evaluation of the expected safety factors, i.e.,k1=1.00 andk2=0.50;k1=1.50 andk2=0.50;k1=1.50 andk2=1.00;k1=2.00 andk2=0.50;k1=2.00 andk2=1.00; andk1=2.00 andk2=1.50. The expected safety factor is estimated by the proposed model considering the moderate uncertainty in soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 6.Table 6
The numerical results of the parameter analysis.
k 1 1.00 1.50 1.50 2.00 2.00 2.00 k 2 0.50 0.50 1.00 0.50 1.00 1.50 E[Fs] 1.1015 1.1071 1.1125 0.9670 1.1163 1.1201From Table6, as the parameter value of the triangular membership varies, the change in the variation ofE[Fs] is reflected. Therefore, compared to the safety factor of 0.947 with the original Fellenius method, it can be concluded that the parametersk1 andk2 should take the best value of around 2.00 and 0.50 in the case of a two-layered slope, respectively.Although only two simplified slope models are discussed in this section, they are still valuable for the study of the stability of real slopes. In fact, for more complicated slopes in practical engineering, other shapes of the membership functions for the uncertain sets can also be incorporated in the approach. Furthermore, the complicated slopes can be considered as a number of homogeneous slopes or two-layered slopes [11], and then the method discussed in this section can be developed to evaluate their stability.


## 4.3.1. Results Analysis

The safety factor of the slope using mean parameters with the Fellenius method is 0.947. The geometric parameters of the slip circle with the mean values of the variables are the centre of the slip circleO(9.22, 11.98) and the radius 9.38m.Three numerical experiments are also designed to study how the uncertainty in soil properties may affect the expected safety factor in this example. These experiments stand for three cases of soil uncertainty (low, moderate, and high). In the first experiment, low uncertainties are assumed for the soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.10 and 0.20, respectively. In the second experiment, moderate uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. In the third experiment high uncertainties are assumed for soil properties, i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.30 and 0.40, respectively. Herein, the parametersk1 andk2 in (20) take values of 1.50 and 1.00, respectively. Table 5 shows the results of the three numerical experiments.Table 5
The results of the three numerical experiments.
Exp. Uncertainty CV forϕ CV forc E[Fs] 1 Low 0.10 0.20 1.1043 2 Moderate 0.20 0.30 1.1126 3 High 0.30 0.40 1.1162From Table5, the fact that the expected safety factors do not change much in the three experiments also indicates the stability of the proposed model. In addition, it can be found that the values of the expected safety factor become larger and larger with the increase in the level of uncertainty for the soil properties involved, which means the slope is considered more secure under the high degree of uncertainty.In aiming to determine the expected safety factor one has to consider the possible parameter values of the trapezoidal membership function. All six cases are considered in the evaluation of the expected safety factors, i.e.,k1=1.00 andk2=0.50;k1=1.50 andk2=0.50;k1=1.50 andk2=1.00;k1=2.00 andk2=0.50;k1=2.00 andk2=1.00; andk1=2.00 andk2=1.50. The expected safety factor is estimated by the proposed model considering the moderate uncertainty in soil properties; i.e., the CVs of the angle of internal friction and the cohesion are allowed to be 0.20 and 0.30, respectively. The numerical results of the parameter analysis are listed in Table 6.Table 6
The numerical results of the parameter analysis.
k 1 1.00 1.50 1.50 2.00 2.00 2.00 k 2 0.50 0.50 1.00 0.50 1.00 1.50 E[Fs] 1.1015 1.1071 1.1125 0.9670 1.1163 1.1201From Table6, as the parameter value of the triangular membership varies, the change in the variation ofE[Fs] is reflected. Therefore, compared to the safety factor of 0.947 with the original Fellenius method, it can be concluded that the parametersk1 andk2 should take the best value of around 2.00 and 0.50 in the case of a two-layered slope, respectively.Although only two simplified slope models are discussed in this section, they are still valuable for the study of the stability of real slopes. In fact, for more complicated slopes in practical engineering, other shapes of the membership functions for the uncertain sets can also be incorporated in the approach. Furthermore, the complicated slopes can be considered as a number of homogeneous slopes or two-layered slopes [11], and then the method discussed in this section can be developed to evaluate their stability.


## 5. Conclusions

In the deterministic analysis, it is difficult to obtain a reliable safety factor due to the uncertainties associated with the soil properties involved. The present study has been oriented to provide a novel methodology based on the uncertain set to analyse slope stability.Considering the deterministic method as a foundational model, the uncertainties in the involved soil properties have been incorporated into the stability analysis. The soil properties, i.e., the unit weight, the angle of internal friction, and the cohesion, are expressed as uncertain sets in the proposed methodology. The triangular and trapezoidal memberships are used to describe their uncertain sets for a homogeneous and two-layered slope, respectively. Moreover, the parameter values in the membership functions are designed on the basis of the means and variations of the soil properties. The expected safety factor can be calculated through the operational laws under such an uncertain setting. Two illustrative examples are used in this paper: the first example is for a homogenous slope and the second is for a slope in layered soil. A series of numerical experiments are designed to investigate the relationship between the variation in the safety factor and the changes in the soil properties, the determination of the parameter values of the membership, and the influence of different levels of uncertainty on the expected safety factor. The findings of this research warrant the following conclusions:(1)
The variations in the expected safety factor increase clearly with the changes in the angle of internal friction, while the expected safety factor almost remains unchanged until the coefficient of variation of cohesion reaches 40%.(2)
The expected safety factors increase slightly with the promotion of the level of uncertainty of the soil properties, and, as a whole, this fact can indicate the stability of the proposed methodology.(3)
The parameterkin the triangular membership function should take a best value of between 0.50 and 1.00 for the homogeneous slope, and the parametersk1 andk2 in trapezoidal membership function should take a best value of around 2.00 and 0.50, respectively.



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*Source: 1062347-2019-03-13.xml*