- Please check and comment entries here.

## Definition

The issue of slope stability is one of the most important and yet most difficult geotechnical problems. Assessing slope stability is particularly difficult because of the many uncertainties involved in the process. To take these uncertainties into account, probabilistic methods are used, and the reliability approach is adopted. There are many methods for reliability assessment of earth slope stability. However, there is no system that would organize all of these methods in an unambiguous way. In fact, these methods can be classified in different ways: by assignment to a deterministic classification of methods, by description of uncertainties of soil parameters, by level of reliability according to the theory of reliability, etc. The huge number of articles summarizing the research in this field, but in various “disordered” directions, certainly do not facilitate the understanding or ultimately the practical application of the reliability approach by the engineer. We propose a universal classification system of reliability methods for evaluating the stability of earth slopes. This proposal is preceded by a brief literature review of both historical background and contemporary study on reliability analysis of earth slope stability.

## 1. Introduction

This paper proposes a classification system of reliability methods for earth slope stability assessment that integrates deterministic slope stability methods, modelling of uncertain soil parameters and reliability level (commonly used in the structural safety analysis). Also, in the case of the most sophisticated approaches, further divisions related to improvement of computation efficiency are suggested.

## 2. Methods for Evaluating the Stability of Earth Slopes

^{[1]}

^{[2]}

^{[3]}

^{[4]}

^{[5]}

^{[6]}. In the FEM by SRM used in evaluating slope stability, soil strength parameters continuously decrease until the first indications of failure appear. The safety factor is defined as the ratio of the real shear strength of the soil to the reduced shear strength. This method seems to be superior to the LEM because there is no need for the primary guess at determining the critical failure surface. In addition, this method does not require any assumptions about interslice forces. FEM analysis is a more rigorous and universal technique, but often less attractive due to its dependence on mesh density and the available computational capacity. The primary advantage of this method is that the critical slip surface is found automatically from the shear strain, which increases as the shear strength decreases. Unfortunately, other ‘‘slip’’ surfaces (i.e., local minima) are omitted. However, because of the high speed of modern computer systems, analysis by FEM is used today more often than before. The finite element method by strength reduction method was first proposed and applied to slope stability by Zienkiewicz et al.

^{[7]}and then used to assess slope stability in, among others,

^{[8]}

^{[9]}

^{[10]}. Limit analysis models soil as a material that is perfectly plastic and obeys an associated flow rule. This method employs a dichotomy of theorems to provide a solution: either upper bound or lower bound plasticity. The upper bound theorem of limit analysis is predominantly used in solving slope stability problems. Unfortunately, the application of LA is still limited, since most of the research findings are chart-based and prepared for particular cases, and there is no stability chart available to cover a wide range of different slope material properties, geometries, etc. The concept of limit analysis was proposed by Drucker and Prager

^{[11]}and was utilised in slope stability in

^{[12]}

^{[13]}

^{[14]}and others. A review of the three basic deterministic approaches of slope stability, including their shortcomings and possible errors, is discussed in

^{[15]}.

^{[16]}

^{[17]}

^{[18]}

^{[19]}

^{[20]}

^{[21]}

^{[22]}

^{[23]}

^{[24]}

^{[25]}. In the case of a highly nonlinear function of factor of safety, computations of the derivatives are impossible or inconvenient, thus rendering FOSM results inaccurate. In addition, different results can be obtained depending on how the limit state function is formulated. To avoid the main drawback of the FOSM and SOSM methods (in which the results unduly depend on the mean value), as observed in

^{[26]}

^{[27]}, the first-order reliability method (FORM) has begun to be widely used

^{[28]}

^{[29]}

^{[30]}

^{[31]}. It is well known that FORM works only for slopes with a small probability of failure or a high reliability index. Otherwise, this method underestimates the probability of failure of slopes. A summary of research on probabilistic analysis of slope stability was given in the monograph

^{[32]}. A review of the literature on this topic can also be found in

^{[24]}or

^{[33]}. A significant change in approach to probabilistic slope analysis occurred with the use of the finite element method (FEM) for geotechnical problems. Different probabilistic methods related to FEM have been proposed, such as the perturbation method, the Neumann expansion method, the partial differential method, the spectral stochastic finite element method (SSFEM), etc. Along with the development of computers and software, the MC simulation became dominant because of its relative ease of application. This method is a conceptually simple tool for reliability analysis of slope stability regardless of the form of the performance function or the number of scenario failure events. It employs statistical averaging over random samples generated from the probability density function of the parameters to evaluate the probability of failure. It is the easiest to apply; however, its simulations are usually time consuming and computation demanding. The MC method is also robust to various deterministic analysis methods for slope stability analysis, such as LEM or FEM/FDM.

## 3. Classification System

In a structural safety analysis, there are different levels of reliability, depending on the importance of the structure, grouped under four basic levels:

- level 1—partial factor approach — employs only one “characteristic” value of each uncertainty parameter;

- level 2—estimates two values of each uncertainty parameter, usually the mean value, standard deviation, and the correlation between these parameters;

- level 3—best estimate of the probability of failure—knowledge of the join distribution of all uncertain parameters is required;

- level 4—reliability methods appropriate for structures of major economic importance, taking into account the structures’ economic value, including the consequences of their failure.

The level 2 reliability methods are included in limit state design codes. Methods of level 2 include a range of approximate or iterative procedures such as the perturbation method, FOSM, SOSM, FORM, SORM, PEM, etc. Methods of level 3, in the strict sense, require determining the mathematically exact probability of structural failure as a result of integrating the joint probability density function of random variables. In the case of slope stability, they can only be used for simplified, idealised cases. However, in a broader sense, level 3 methods require estimates of all probabilistic measures. The Monte Carlo simulation method has become the dominant procedure here as a result of the rapid development of computer techniques that have taken place in recent years. In order to improve the efficiency of this method while maintaining the accuracy of calculations, various reduction techniques have been developed (e.g., stratified sampling, Latin Hypercube simulation, importance sampling and Russian roulette and splitting). The response surface method and the methods of artificial neural networks have also grown in popularity in the analysis of the reliability of slopes. Both methods allow all probabilistic measures to be estimated and can also be qualified to the level 3 reliability method.

In the deterministic approach to slope stability, three basic approaches—the limit equilibrium method (LEM), the displacement‐based finite/different element method (FEM/DEM), and limit analysis (LA)—have been developed. Probabilistic methods used in geotechnical engineering are commonly divided into two groups, depending on the description of the uncertainties of the soil parameters: the single random variable (SRV) approach and the random field (RF) approach.

The proposal of a classification of reliability methods of slope stability is presented the **Table 1**. It combines deterministic methods of slope stability with random modelling of the soil medium and includes levels of reliability.

**Table 1.** Classification of reliability methods of earth slope stability.

The abbreviations in Table 1 refer to the random soil model, deterministic slope stability method and reliability level. For example: SRVLEM1—soil modelled as a single random variable (SRV), limit equilibrium method (LEM) and Level 1 reliability method; RFFEM3—soil modelled as a random field, finite/different element method and Level 3 reliability method; etc.

Currently, the FEM/DEM is the predominant deterministic method of stability assessment, soil is usually modelled as a random field and level 3 reliability methods are usually applied. Most of the research focus here is on computation efficiency. Thus, the RFFEM3 methods can be divided into two groups depending on how the performance function of slope stability is simplified and on the reduction techniques (**Figure 1**).

**Figure 1.** Division of RFFEM3 methods. RSM—response surface method. Improved RSM methods: KM—Kriging methodology, SM—surrogate models, ANN—artificial neural network, SVM—support vector machine, GA—genetic algorithms, SS—subset simulation, IS—importance sampling, LHS—Latin Hypercube sampling, AS—adaptive sampling, DS—directional simulation.

Slopes can be subjected both to static (gravity) and dynamic (earthquake, waves) loads as well as environmental loads (rainfall, temperature changes). Also, slope failure modelling can be carried out in a two‐dimensional or three‐dimensional analysis. However, these factors should not affect the proposed classification system of reliability methods. Instead, new subdivisions could be introduced on their basis.

## 4. Conclusions

This literature review shows that although reliability methods are extremely advanced and the computational possibilities almost unlimited, in practice, the simplest methods are usually used. This is mainly due to the fact that engineers are not familiar with probabilistic concepts; thus, it is difficult to incorporate them into practice.

The entry is from 10.3390/su13169090

## References

- Fellenius, W. Calculations of the Stability of Earth Dams. In Proceedings of the Second Congress of Large Dams, Washington, DC, USA, 7–12 September 1936; Volume 4, pp. 445–463.
- Bishop, A.W. The use of the slip circle in the stability analysis of slopes. Geotechnique 1955, 5, 7–17.
- Janbu, N. Slope Stability Computation, Embankment- Dam Engineering: Casagrande Volume; John Wiley & Sons: New York, NY, USA, 1973; pp. 47–86.
- Nonveiller, E. The stability analysis of slopes with a slide surface of general shape. In Proceedings of the 6th International Conference on Soil Mechanics and Foundation Engineering, Montreal, QC, Canada, 8–15 September 1965; Volume 2, pp. 522–525.
- Spencer, E. A method of analysis of the stability of embankments assuming parallel inter-slice forces. Geotechnique 1967, 17, 11–26.
- Morgenstern, N.R.; Price, V.E. The Analysis of the Stability of General Slip Surfaces. Geotechnique 1965, 15, 77–93.
- Zienkiewicz, O.C.; Humpheson, C.; Lewis, R.W. Associated and Non-Associated Visco-Plasticity and Plasticity in Soil Mechanics. Geotechnique 1975, 25, 671–689.
- Dawson, E.M.; Roth, W.H.; Drescher, A. Slope Stability Analysis by Strength Reduction. Geotechnique 1999, 49, 835–840.
- Griffiths, D.V.; Lane, P.A. Slope stability analysis by finite elements. Geotechnics 1999, 49, 387–403.
- Cheng, Y.M. Location of critical failure surface and some further studies on slope stability analysis. Comput. Geotech. 2003, 30, 255–267.
- Drucker, D.C.; Prager, W. Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 1952, 10, 157–165.
- Chen, W.F.; Giger, M.W.; Fang, H.Y. On the limit analysis of stability of slopes. Soils Found. 1969, 9, 23–32.
- Donald, I.B.; Chen, Z.Y. Slope stability analysis by the upper bound approach: Fundamentals and methods. Can. Geotech. J. 1997, 34, 853–862.
- Michalowski, R.L. Slope stability analysis: A kinematical approach. Géotechnique 1995, 45, 283–293.
- Przewłócki, J. Some comments on slope stability evaluation. Part I: Deterministic analysis. Inżynieria Morska Geotech. 2004, 2, 141–149.
- Wu, T.H.; Kraft, L.M. Safety Analysis of Slopes. J. Soil Mech. Found. Div. 1970, 96, 609–630.
- Cornell, C.A. First-order uncertainty analysis of soil deformation and stability. In Proceedings of the 1st International Conference Applications of Statistics and Probability in Soil and Structural, Hong Kong, 13–16 September 1971; pp. 129–144. Available online: https://trid.trb.org/view/128568 (accessed on 16 July 1974).
- Alonso, E.E. Risk analysis of slopes and its application to slopes in Canadian sensitive clays. Geotechique 1976, 26, 453–472.
- Tang, W.H.; Yucemen, M.S.; Ang, A.H.S. Probability based short term design of soil slopes. Can. Geotech. J. 1976, 13, 201–215.
- Vanmarcke, E.H. Reliability of earth slopes. J. Geotech. Eng. 1977, 103, 1227–1246.
- Chowdhury, R.N.; Grivas, D. Probabilistic model of progressive failure of slopes. J. Geot. Eng. 1982, 108, 803–917.
- Tobutt, D.C. Monte Carlo simulation methods for slope stability. Comput. Geosci. 1982, 8, 199–209.
- Chowdhury, R.N.; Tang, W.H.; Sidi, I. Reliability model of progressive slope failure. Géotechnique 1987, 37, 467–481.
- El-Ramly, H.; Morgenstern, N.R.; Cruden, D.M. Probabilistic slope stability analysis for practice. Can. Geotech. 2002, 39, 665–683.
- Griffiths, D.V.; Fenton, G.A. Probabilistic Slope Stability Analysis by Finite Elements. J. Geotech. Geoenviron. Eng. 2004, 130, 507–518.
- Christian, J.T.; Ladd, C.C.; Baecher, G.B. Reliability applied to slope stability analysis. J. Geot. Eng. 1994, 120, 2180–2207.
- Hassan, A.M.; Wolff, T.F. Search algorithm for minimum reliability index of earth slopes. J. Geotech. Geoenviron. Eng. 1999, 125, 301–308.
- Low, B.K.; Tang, W.H. Probabilistic Slope Analysis Using Janbu’s Generalized Procedure of Slices. Comput. Geotech. 1997, 21, 121–142.
- Low, B.K.; Gilbert, R.B.; Wright, S.G. Slope reliability analysis using generalized method of slices. J. Geotech. Geoenviron. Eng. 1998, 124, 350–362.
- Cho, S.E. Effects of spatial variability of soil properties on slope stability. Eng. Geol. 2007, 92, 97–109.
- Cho, S.E. First-order reliability analysis of slope considering multiple failure modes. Eng. Geol. 2013, 154, 98–105.
- Knabe, W.; Przewłócki, J. Probabilistic Slope Stability Analysis; Institute of Hydro-Engineering of Polish Academy of Sciences: Gdańsk, Poland, 1990.
- Przewłócki, J. Some comments on slope stability evaluation. Part II: Probabilistic analysis. Inżynieria Morska Geotech. 2004, 3, 141–149.