Elsevier

Computers and Geotechnics

Volume 37, Issues 1–2, January–March 2010, Pages 132-140
Computers and Geotechnics

Comparison of different probabilistic methods for predicting stability of a slope in spatially variable cφ soil

https://doi.org/10.1016/j.compgeo.2009.08.005Get rights and content

Abstract

Three probabilistic methods of different complexity for slope stability calculations are in the paper evaluated with respect to a well-documented case study of slope failure in Lodalen, Norway. A finite element method considering spatial random fields of uncorrelated parameters c (cohesion) and φ (friction angle) is taken as a reference for comparison with two simpler methods based on Taylor series expansion, known as first-order-second-moment (FOSM) methods. It is shown that the FOSM method enhanced by a reduction of variance of input parameters due to spatial averaging along the potential failure surface (extended FOSM method) leads to a significant improvement in predictions as compared to the basic FOSM method. This method is computationally inexpensive and can be used in combination with any existing finite element code, it is thus a useful approximate probabilistic method for geotechnical practice. Several limitations of the extended FOSM method for calculating probability of a slope failure are identified.

Introduction

The soil mechanical properties obtained from detailed geotechnical site investigations show a marked dispersion, coming from their inherent spatial variability (even in zones which are often regarded as “homogeneous” from the deterministic point of view) and measurement error. Additional uncertainty is introduced by the fact that only limited number of measurements is often available, and from subjective calibration of simple constitutive models, which are often used in geotechnical analyses. These uncertainties are in geotechnical engineering commonly accounted for using deterministic concepts, for example by scaling the uncertain values of material parameters by various factors of safety. This approach, however, discourage clearer understanding of the relative importance of different uncertainties involved in simulations [35], [30], [12]. In this respect, probabilistic approaches are well suited to geotechnical engineering. Their rather limited use in practical applications is mainly caused by the lack of data needed for detailed statistical evaluation of mechanical properties (in fact, only few studies with proper evaluation of geotechnical variability are available, see [2], [39], [15], [22]). Consequently, probabilistic methods are not incorporated in most commercial software tools, and they are thus not used even in projects where their application would be desirable.

From the mentioned uncertainties in soil mechanical properties, we will in this paper focus on inherent spatial variability. A rational means of its quantification is to model the distribution of soil mechanical properties as a random field, in which deviation of a given property from the trend value is characterised using some suitable statistical distribution. Spatial variability is measured by means of the correlation length θ, which describes the distance over which the spatially random values will tend to be significantly correlated [37]. Evaluation of the quantity θ is in geotechnical engineering sometimes difficult due to large amount of data needed. Detailed literature reviews on the values of correlation lengths are presented in Phoon and Kulhawy [30], El-Ramly et al. [9] and Hicks and Samy [24]. It is observed that depending on the geological history and composition of the soil deposit θ in the horizontal direction vary within the range 10–40 m, while θ in the vertical direction ranges from 0.5 to 3 m.

Slope stability analysis is a popular field of geotechnical engineering for application of probabilistic methods. The approaches adopted by different authors vary significantly in the level of complexity and sophistication. The first applications [7], [5] combined classical limit equilibrium methods with approximate analytical probabilistic methods based on Taylor series expansion (see Section 2.1 for details). These methods consider statistical distribution of strength parameters, but they do not incorporate their spatial correlation structure. As demonstrated in terms of slope stability analysis for example by Griffiths et al. [20], this simplification may significantly affect the calculated probability of failure. Though the spatial correlation structure of input variables may be treated within limit equilibrium methods based on the pre-defined shape of the failure surface, as studied by El-Ramly et al. [8], [9], [11], [10], its full potential is exploited by considering random field theories in combination with numerical methods for boundary value problems. In typical applications finite element method in 2D [19], [22], [24] or 3D [23] is used, but they can be combined with other methods, such as discrete element method [25].

In the present paper, finite element method is combined with three different probabilistic methods, starting with a simple Taylor series expansion method (Section 2.1) and ending with more sophisticated methods based on random field theory (Section 2.2). Merits and shortcomings of different approaches are evaluated using data from a well-documented case history, namely slide in Lodalen in Norway [34].

Section snippets

Probabilistic numerical methods

Probabilistic numerical analyses are usually used to evaluate statistical distribution of a performance function Y=g(X1,X2,Xn), based on known statistical characteristics of input variables Xi. In the paper, we will distinguish the following probabilistic methods:

  • Methods which do not consider the random spatial structure of input variables, in other words they assume infinite correlation length θ.

  • Methods based on random field theories, which consider spatial variability of input variables.

Lodalen slide

The slide in the Lodalen marshalling yard near Oslo, Norway [34], was chosen for the purpose of the evaluation of probabilistic numerical methods in this study.

The slide occurred in 1954 at the site where the marshalling yard had been enlarged about 30 years before the slope failed. Mid-section through the slide and the slope geometry in different time periods is shown in Fig. 1a. The inclination of the slope before failure was approximately 1:2. The main part of the slide formed in a

Finite element simulations

All the three methods studied (basic FOSM, extended FOSM and RFEM) were used within the framework of the finite element method. Simulations were performed using commercial finite element package Tochnog Professional [32]. The finite element mesh, including dimensions, is shown in Fig. 3. The inclination of the slope is 1:2. The mesh consists of 840 9-noded isoparametric square elements, which reduce to triangles or irregular quadrilateral elements at the sloping mesh boundary. The geometry

Statistical distribution of the output variable t

Fig. 6a shows the influence of the correlation length on the mean value of the gravity acceleration at failure multiplier t. From the definition, both FOSM methods predict the same value of μ[t] independent of θ. On the other hand, the RFEM method predicts (for the present case and for the range of θ studied) a decrease of the μ[t] value with decreasing θ. Fig. 6b shows the influence of the correlation length on the standard deviation of t. The RFEM method predicts the decrease of σ[t] with

Discussion and concluding remarks

Three probabilistic methods for calculation of slope stability were in the paper compared using a well-documented case study of the slope failure in the Lodalen, Norway. The calculated probability of failure using the RFEM method for θ=10m is 55.7%. The probability of failure is somewhat lower than that obtained by El-Ramly et al. [11] using different approach based on limit equilibrium method (69.4%), but both the methods show that the slope was in meta-stable conditions and the failure was

Acknowledgment

The authors would like to thank to the anonymous journal reviewer for valuable comments on the manuscript. Financial support by The Research Grants GACR 205/08/0732, GAUK 31109 and MSM 0021620855 is greatly appreciated.

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