Analysis of Slope Stability with Imprecise Soil Properties Using Uncertain Sets

Zhou, Xuejun; Li, Jie; Liu, Zhibing; Tang, Yi

doi:https://doi.org/10.1155/2019/1062347

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2019 | Article ID 1062347 | https://doi.org/10.1155/2019/1062347

Analysis of Slope Stability with Imprecise Soil Properties Using Uncertain Sets

Xuejun Zhou,^1,2Jie Li,³Zhibing Liu,¹and Yi Tang⁴

Academic Editor: Giuseppe D'Aniello

Received22 Nov 2018

Revised31 Jan 2019

Accepted20 Feb 2019

Published13 Mar 2019

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.

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

Let be a σ-algebra over a universal set , and the uncertain measure be a set function satisfying the normality, duality, and subadditivity axioms. The triplet is called an uncertainty space, and an uncertain set is a set-valued function ξ on the uncertainty space.

If , , , are uncertain sets on the uncertainty space , and let f be a measurable function. Then is a function of the uncertain sets defined by

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.

Let ξ be an uncertain set whose membership function μ exists. Thenfor any number x.

The value of μ(x) is just the membership degree that x belongs to the uncertain set ξ, and the larger the value of μ(x) is, the more true x belongs to ξ.

On the other hand, the set-valued functionis called the inverse membership function of ξ. For each given α, the set μ⁻¹(α) is also called the α-cut of μ, and it is an interval for each α.

The triangular and trapezoidal membership functions are employed in the present study.

Example 1. An uncertain set ξ is called triangular if it has a membership functiondenoted by (a, b, c) where a, b, and c are real numbers with a < b < c.
The triangular uncertain set ξ = (a, b, c) has an inverse membership functionNote that a and c are the lower and upper x values of the triangle at μ = 0, respectively, and b is the x value that corresponds to μ = 1 (see Figure 1).

Example 2. An uncertain set ξ is called trapezoidal if it has a membership functiondenoted by (a, b, c, d) where a, b, c, and d are real numbers with a < b < c < d.
The trapezoidal uncertain set ξ = (a, b, c, d) has an inverse membership functionNote that a and d are the lower and upper x values of the triangle at μ = 0, respectively, while b and c are the x values that correspond to μ = 1 (see Figure 2). The triangular uncertain sets are a special case of the trapezoidal uncertain sets.

2.3. Operational Law

Let ,,, be independent uncertain sets with inverse membership functions ,,, respectively, and let f be a measurable function. Then has an inverse membership function

Let ξ be a nonempty uncertain set with membership function μ, then its expected valuewhere inf and sup are the infimum and supremum of the α-cut, respectively.

This shows that the triangular uncertain set ξ = (a, b, c) has an expected valueand the trapezoidal uncertain set ξ = (a, b, c, d) has an expected value

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 (F_s) can be expressed as a function of several uncertain and constant variables, i.e.,where γ is the unit weight, c the cohesion, ϕ the angle of internal friction, H the height of slope, and β the inclination of the base.

If the input soil parameters γ, c, and ϕ are 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 that , , and are their membership functions, respectively, and the inverse membership functions are , and , respectively. The safety factor is a continuous function and thus measurable, so its inverse membership function can be written as

According to (9), the expected value of the safety factor F_s can be computed by

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 cohesion c, and the angle of internal friction ϕ are considered as uncertain parameters and denoted by a 3-dimensional uncertain vectorwhere X_i, i=1,2,3, represent the corresponding uncertain parameters γ, c, and ϕ, respectively. Assume that the uncertain vector X has a mean vectorand a standard deviation vector

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 Figure 1, the values of a, b, and c should 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 of a, b, and c in the triangular membership function (4) can be designed aswhere k is 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 values a, b, and c into (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 Figure 2, the values of a, b, c, and d should 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 of a, b, c, and d in the trapezoidal membership function (6) can be designed aswhere k₁ and k₂ 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 relation k₁ > k₂. In the following numerical examples, six cases are considered: (a) Case 1, k₁ = 1.00 and k₂ = 0.50; (b) Case 2, k₁ = 1.50 and k₂ = 0.50; (c) Case 3, k₁ = 1.50 and k₂ = 1.00; (d) Case 4, k₁ = 2.00 and k₂ = 0.50; (e) Case 5, k₁ = 2.00 and k₂ = 1.00; and (f) Case 6, k₁ = 2.00 and k₂ = 1.50.

By substituting the above values a, b, c, and d into (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. Figure 3 illustrates 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:where Wi=γ_iv_i is the weight of the ith slice, v_ithe volume of the ith slice, θi the inclination of the base of the ith slice, ci the cohesion of the ith slice, and li the width of the ith 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 Figure 4. The inclination of the base equals . 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 friction ϕ varies from 0.10 to 0.30, for the cohesion c 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.

4.2.1. Results Analysis

The uncertainties in the variables ϕ and c are 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 circle O (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.

From Figures 5 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 parameter k in (19) also takes the value of 1.00. Table 2 shows the results of the three numerical experiments.

It can be seen from Table 2 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.

From Table 3, as the parameter value of the triangular membership increases, the increase in the variations of E[Fs] is reflected. Therefore, compared to the safety factor of 1.1414 with the original Fellenius method, it can be concluded that the parameter k should 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 Figure 7, 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.

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 circle O (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 parameters k₁ and k₂ in (20) take values of 1.50 and 1.00, respectively. Table 5 shows the results of the three numerical experiments.

From Table 5, 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 and k₂=0.50; k₁=1.50 and k₂=0.50; k₁=1.50 and k₂=1.00; k₁=2.00 and k₂=0.50; k₁=2.00 and k₂=1.00; and k₁=2.00 and k₂=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.

From Table 6, as the parameter value of the triangular membership varies, the change in the variation of E[Fs] is reflected. Therefore, compared to the safety factor of 0.947 with the original Fellenius method, it can be concluded that the parameters k₁ and k₂ 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 parameter k in the triangular membership function should take a best value of between 0.50 and 1.00 for the homogeneous slope, and the parameters k₁ and k₂ in trapezoidal membership function should take a best value of around 2.00 and 0.50, respectively.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The research in this paper is funded by China National Natural Science Foundation under Grant no. 61462096, Humanity and Social Science Foundation of Hubei Provincial Department of Education of China under Grant no. 18Y141, and Huanggang Normal University Natural Science Foundation under Grant no. 04201813103.

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