This paper presents a stochastic approach to quantifying reliabilitybased, characteristic values, satisfying the requirements of Eurocode 7. The method for determining these values is demonstrated for a simple slope stability problem. The results show that reliability is a more meaningful measure of stability than conventional factors of safety, for which there is no information regarding probability of failure. They also show that spatial variability is an important factor in stability assessments and that proper account should be taken of its anisotropic nature. However, an accurate knowledge of the degree of anisotropy (of the heterogeneity) may not always be needed.

Recommendations are made for future research in this subject area.

Introduction

In geotechnical engineering, the need for adequate representation of insitu variability has been recognised in recent design codes: in particular, by the European Union's Eurocode 7 (ENV 1997-1 1994), due for release in 2003, which has introduced the concept of characteristic values into the design process.

This is a significant shift from traditional design methods based on global factors of safety. It has also caused much controversy, since the Code gives little guidance as to how such values should be determined: hardly surprising, as there is virtually no experience in this subject area. However, it does suggest statistical methods as an appropriate way forward to achieving the ultimate goal of reliability-based characteristic values.

This paper describes a numerical (stochastic-based) solution to the above problem. It links random field theory, for generating spatially varying property distributions, with finite elements, for analysing structural response. The stability of a structure is then represented by reliability, the probability that failure will not occur, rather than factor of safety, for which there is no information regarding probability of failure. This enables the derivation of reliability-based characteristic values.

Characteristic values and Eurocode 7

Eurocode 7 states that the design value, X d, of the material property, X, is given by where X kis the characteristic value of X and g mis the partial safety factor. The value of the partial factor is defined by the Code. Hence, all engineering judgement is focused on the chosen characteristic value. In terms of the mean property value, X m, it may be represented by where a xis the reduction factor which lies in the range 0 to 1.

The following extracts are taken from Eurocode 7:

Selection of characteristic values of soil and rock properties shall take account of the variabilities of the property values.

The characteristic value of a geotechnical parameter shall be selected as a cautious estimate of the value affecting the occurrence of the limit state.

lf statistical methods are used, the characteristic value should be derived such that the calculated probability of a worse value governing the occurrence of the limit state under consideration is not greater than 5%.

The following observations may be made:

Eurocode 7 has faced up to the issue of natural variability in soils.

It does not specify how the characteristic value should be determined.

This paper demonstrates one approach that may be used for satisfying the requirements of Eurocode 7: that is, by determining reliabilitybased characteristic values for different classes of problem, using stochastic analysis and finite elements. This approach takes account of the fact that soil is a variable material. Importantly, it also recognises that this variability is spatial in nature.

Basis of stochastic approach Figure 1a shows the material property X varying with depth in a socalled uniform layer. In a deterministic analysis, X is represented by the mean or some other characteristic value. In contrast, stochastic analysis makes use of all data, by expressing them in the form of a probability density function, or pdf. These data may then be approximated by some theoretical distribution.

For example, in Figure 1b the data are idealised as a normal distribution, defined by the point statistics of X. These are the mean, m, and the standard deviation about that mean, s. A third statistical parameter, the scale of fluctuation, u, defines the degree of spatial correlation and is illustrated in Figure 1a. Hence, u is a measure of the distance between adjacent 'strong' or 'weak' zones: as u gets larger, so the degree of spatial correlation increases, leading to a more uniform distribution of X.

Based on the derived statistics, m, s and u, it is possible to generate numerical predictions of the spatial distribution of X. These are known as random fields, a term which is something of a misnomer, since the fields are not random at all; rather, they are highly structured and dependent on the input statistics.

Figure 2 shows two such random fields, as generated for a square domain of side length D. In both figures, the darker areas indicate higher values of X, and the lighter areas, lower values of X. In Figure 2a, a small scale of fluctuation relative to the domain size has been used ( u/D=0.1): hence, there is a high degree of spatial variability. Conversely, a larger scale of fluctuation has been used in Figure 2b ( u/D=1.0), leading to greater uniformity.

Note that, in both figures, u is equal in all directions: hence, these are examples of isotropic random fields. Furthermore, for a given set of statistics, there are an infinite number of possible random fields, each yielding a different solution. Stochastic analysis therefore involves repeated realisations as part of a Monte Carlo simulation process and, for stability assessments, this leads to reliability rather than a single factor of safety.

Background in geotechnics In geotechnical engineering, the use of stochastic processes, based on random fields, is still at an early stage. However, applications have often followed the line advocated by Vanmarcke 1983, in which a random field of material properties is approximated by a discrete field of local averages. Also, most applications have been for univariate property distributions: ie involving a single random variable, X.

Three main classes of problem have been investigated: groundwater flow, using a random permeability field (Smith and Freeze 1979a, 1979b, Griffiths and Fenton 1993, Fenton and Griffiths 1996); settlement computations, using a random elastic modulus (Paice et al 1996); and stability assessments for elastic, perfectly plastic materials, using a random shear strength (Paice and Griffiths 1997, Griffiths and Fenton 2000, Hicks and Samy 2002a).

Applications for more complicated material behaviour are rare, and, so far, mainly confined to assessing liquefaction potential. In broad terms, these applications involve more than one, or maybe many, random variables: ie X={X 1X2 X3 . . . . X n}T, in which n is the number of variables. Two solutions are possible:

(a) the multivariate approach is to generate a separate random field for each variable, and to cross-correlate between fields to account for parameter inter-dependency (Fenton and Vanmarcke 1998) - for example, higher friction angles are likely to be associated with an increased tendency for dilation;

(b) the reduced-variate approach is to generate random fields for smaller numbers of parameters (eg relative density), from which other parameters (eg friction angle) may be backfigured - for example, Popescu et al 1997 generated cross-correlated, bi-variate, random fields of cone tip resistance and soil classification index, while Onisiphorou and Hicks 2001 generated univariate fields of Been and Jefferies' 1985 state parameter.

Summary of stochastic process The stochastic procedure may be split into three stages: the pre-analysis stage, involving the preparation of statistical and material data; the analysis stage, comprising repeated realisations of the same problem as part of a Monte Carlo simulation; and the post-analysis stage, in which the results of the realisations are themselves expressed in probabilistic form. The procedure is illustrated schematically in Figure 3 and outlined as follows.

Pre-analysis stage (Figure 3a)

For a given material 'layer', the spatial variation of X is represented as a continuous random field, with the value of X at any point being a function of the statistics of X.

The point statistics may be derived from either insitu or laboratory data. However, insitu data are preferable, since these reduce the possibility of exaggerated estimates for the standard deviation, due, for example, to sampling or testing procedures. Furthermore, insitu data are needed anyway, for determining the scale of fluctuation. The following sequential process may be adopted:

Any depth trend is identified, ie are mand/or sfunctions of the depth, z? (cf the example in Figure 1, in which mand sare both independent of z).

Determine m(z) and s(z), which define the theoretical pdf. Hicks and Samy 2002a argue that a normal distribution is usually adequate for describing strength property variability.

Remove the depth trend from the raw data and determine the vertical scale of fluctuation, u v; for example, by using the method proposed by Wickremesinghe and Campanella 1993.

By comparing closely-spaced property profiles, estimate the horizontal scale of fluctuation, u h. This presents the biggest problem, as, unlike the process for determining u v, it is subjective and difficult to automate. It may also require a high intensity of insitu testing.

For the reduced-variate approach to stochastic analysis, the preanalysis stage also involves calibrating the random variables, X, against material data: that is, so that the related material parameters can be backfigured from X.

For example, Onisiphorou and Hicks 2001 calibrated an advanced soil model against 74 triaxial tests on Erksak sand. The resulting state parameter-dependent calibration meant that soil parameters could be backfigured from univariate random fields of state parameter: these had been generated using CPT-derived state parameter statistics (Onisiphorou 2000, Hicks and Onisiphorou 2000).

Analysis stage (Figure 3b) For a given set of statistics, a series of random property fields is generated and analysed to give a range of solutions.

Each random field is simulated by every element in the finite element mesh being assigned a different value of X. (Note that material properties may instead be assigned to the element integration points. ) This is an approximation, since X is thereby assumed to be constant over each element domain: ie the field is discrete rather than continuous. The aim, therefore, is to generate a discrete random field in which the point statistics are adjusted (to account for the finite size of an element) so that they are equivalent to those of the original continuous field. For this purpose, the authors use local average subdivision (Fenton and Vanmarcke 1990), so-named because of its use of local averaging theory (Vanmarcke 1983).

For each realisation in the Monte Carlo process, the procedure is as follows:

For each random variable, generate a discrete random field of X, based on m, sand u.

For the reduced-variate approach, backfigure the material properties from X.

Carry out finite element analysis.

Post-analysis stage (Figure 3c) The results of the realisations are presented in the form of a 'performance' pdf or cumulative distribution function (cdf). Stability may then be expressed in one of two ways: reliability is the probability of failure not occurring; conversely, risk is the probability that failure will occur. Hence, for a reliability of R=95%, there is an associated risk of 5%. These quantities may be found by proportioning the area under a pdf, or by reading directly from a cdf.

Example computation

The use of stochastics for determining characteristic values is illustrated by the simple example shown in Figures 4 to 9. This considers the stability of a 10m high, 1:2 slope, characterised by a spatially varying undrained shear strength, c u, and has been analysed using the same approach as Hicks and Samy 2002a. Hence, the clay has been modelled using a linear elastic, perfectly plastic, Tresca soil model and the following elastic parameters: Young's modulus, E=10 5kPa; Poisson's ratio, n=0.3.

Figure 4a shows the problem geometry and finite element mesh details. The mesh comprises 1220, 8-node quadrilateral elements, with each element using 2x2 Gaussian integration. This high level of discretisation is needed to accurately model the spatial variability, especially in the vertical direction in which the scale of fluctuation is small relative to the slope height. The boundary conditions are a fixed mesh base and rollers along the left-hand boundary allowing only vertical movement. The insitu stresses are based on a soil unit weight of 20kN/m 3, and have been generated by applying gravitational forces in a single increment, in the manner described by Smith and Griffiths 1998.

The mean undrained shear strength increases linearly with depth, from 10kPa at the horizontal ground surface, to 50kPa at the base of the layer, ie m(z) = 10 + 4z (3) in which m(z) is the mean value of c uat depth z.

The standard deviation also increases linearly with depth, and is defined by a constant coefficient of variation, V, of 0.3, in which V= s/m.

For all analyses, u v=1.0m, while a range of values has been considered for uh. Figure 4b shows a typical undrained shear strength distribution, for a degree of anisotropy of the heterogeneity of j=12, in which j= u h/uv. In this figure, dark and light zones indicate high and low values of c u, respectively.

For a variability of zero, as assumed in a deterministic analysis, the stability of the slope may be quantified by in which F is the deterministic factor of safety, H the slope height, g the soil unit weight, c=c uat depth z, and M and S, the stability coefficient and stability number, respectively, defined by Hunter and Schuster 1968. The stability number is itself a function of M and the slope angle, b.

Therefore, for the present example, H=10m, b=26.6°, g=20kN/m 3and, assuming the undrained shear strength profile given by equation 3, M=0.25, which leads to S=8.0 and F=1.6.

Figure 5a shows the computed deterministic response, by plotting mobilised factor of safety as a function of crest settlement, D, nondimensionalised with respect to H. In this figure, the curve has been produced by conducting a series of separate one-increment analyses.

Specifically, the c udistribution has been changed, from one analysis to the next, by dividing the original c udistribution (of 10kPa-50kPa) by gradually increasing values of F. In each analysis, gravity loading has been applied and the crest settlement recorded, as indicated by the solid symbols in the figure. The limiting value of F is then the factor of safety of the slope for the original c udistribution. This has been defined by an analysis failing to converge in 500 equilibrium iterations (Paice and Griffiths 1997, Hicks and Samy 2002a). The computed result of F=1.62 is in close agreement with the analytical solution (shown in red).

Figure 5b shows the comparable stochastic solution, for u h= `. In this case, there are an infinite number of possible c udistributions. They will all look similar, as each will have been generated using the same set of statistics. However, they will differ with respect to the distribution of strong and weak zones, and each will yield a different solution. Figure 5b shows the results of 30 realisations. It clearly demonstrates that spatial variability leads to a wide range of possible solutions, and that the mean stochastic response is weaker than the deterministic solution based on the mean property value. It is also obvious that traditional factors of safety are inappropriate and that some alternative definition of stability is desirable. This leads to reliability.

Figure 6a expresses the results of 600 realisations, including the 30 shown in Figure 5b, in the form of a pdf of F at failure, and approximates these results using a normal distribution. The value of F corresponding to a given level of reliability, R, may be found as illustrated in Figure 3c: that is, by proportioning the area under the pdf, or by re-plotting the results in the form of a cdf, as in Figure 6b. Hence, for R=95%, F=1.17, or, put another way, there is a 95% probability of the slope having a factor of safety greater than 1.17. Furthermore, the reduction factor defined in equation 2 may be found by dividing the value of F, corresponding to a given level of reliability, by F=1.62, the computed factor of safety based on the mean property value. Therefore, for R=95%, a cu =0.72.

The results shown in Figure 7 have been derived using an alternative, more direct procedure, as described by Paice and Griffiths 1997 and Hicks and Samy 2002a. The first step is to determine the mean c Conclusions and future research The draft Eurocode 7 (ENV 1997-1 1994) has introduced the concept of characteristic values to the geotechnical community. It suggests that these values should be based on a 95% confidence in structural performance, but it does not specify how they may be derived.

This paper has demonstrated a stochastic approach to quantifying characteristic values. Importantly, it takes account of the spatial nature of geotechnical variability and its influence on structural response for different classes of problem. It therefore provides a framework for satisfying the requirements of Eurocode 7.

At present, the application of this method of analysis is still at an early stage. Therefore, research is needed to build up necessary experience. For example, for statistical parameters the following areas may be identified:

Statistical variations of common material properties (Simpson and Driscoll 1998).

The influence of material quality, placement technique and densification method on these statistics.

The required site investigation intensity (number of boreholes, CPTs, etc) to provide reliable characterisation of soil deposits given variability.

Although, for some material properties, there is much published data relating to pointwise variability, there remains some debate (Phoon and Kulhawy 1999): for example, are the statistics reflecting inherent material variability, or some external factor such as sampling technique? Furthermore, there is little information on spatial variability, especially for lateral correlations. Therefore, research is needed to provide:

Guidance on reasonable ranges for statistical properties for those situations in which there are insufficient insitu data.

A database of previous experience against which the quality of new data may be assessed.

Research is also needed to answer the following problem-specific questions:

When is a conventional analysis, based on mean property values, applicable, and when is it significantly in error?

When is an accurate knowledge of the horizontal scale of fluctuation not needed? (This has clear implications for site investigation requirements. ) These questions may be answered through analysing different classes of problem, using similar techniques to those illustrated in Figures 4 to 9. The outcome of such an investigation would be:

The derivation of reliability-based reduction factors, for different classes of problem, and for realistic ranges of statistical parameter values (from which characteristic values could be obtained).

An efficient, PC-friendly method of analysis, for use in non-standard problems.

References

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ENV 1997-1 (1994). Eurocode 7: Geotechnical design - Part 1: General rules. Brussels: CEN, European Committee for Standardisation.

Fenton GA and Griffiths DV (1996). Stochastics of free surface flow through stochastic earth dam. ASCE Journal of Geotechnical Engineering Division, vol 122, pp 427-436.

Fenton GA and Vanmarcke EH (1990). Simulation of random fields via local average subdivision. ASCE Journal of Engineering Mechanics, vol 116, pp 1733-1749.

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