1. bookVolumen 43 (2021): Edición 4 (December 2021)
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Stress-weighted spatial averaging of random fields in geotechnical risk assessment

Publicado en línea: 22 Dec 2021
Volumen & Edición: Volumen 43 (2021) - Edición 4 (December 2021)
Páginas: 465 - 478
Recibido: 28 Jul 2021
Aceptado: 22 Nov 2021
Detalles de la revista
License
Formato
Revista
eISSN
2083-831X
Primera edición
09 Nov 2012
Calendario de la edición
4 veces al año
Idiomas
Inglés
Introduction

Reliability-based design methods have recently been widely applied to geoengineering, in which the spatial variability of soil parameters (random fields) plays an important role. Early studies from the late 60s focused on random stability of slopes. The first models considered soil strength parameters as spatial random fields; however, simple random variables are more useful in practical applications (EN 1997-1; Low and Phoon 2015, Tietje et al. 2011). Recent risk analyses widely implement random data as spatial random fields of auto- and crosscorrelated parameters or even use complete probabilistic definition with joint probability distributions (Ching et al. 2016, Javankhoshdel and Bathurst 2015, Griffith and Fenton 2001, Jiang et al. 2015, Shen et al. 2021). Data acquisition and operations on such spatial data require sophisticated numerical support, but – alternatively – some techniques of data pre-processed simplifications are also effective; the former direction can be found in some selected papers attached as references (Cho 2007, Deng et al. 2017, Farah et al. 2015, Ji et al. 2018, Jiang et al. 2014, Kim and Sitar 2013, Li et al. 2017, Liu et al. 2017, Liu et al. 2018, Shen 2021, Tietje et al. 2011), while the latter direction is presented hereafter.

Another group of boundary problems analyses random bearing capacity of footings in a random field context focusing on the most important regions, where shear resistance is generated along slip lines or slip surfaces. The undrained conditions are often analysed (frictionless materials, the Tresca model), which do not depend on stresses (Chwała 2019, Griffith and Fenton 2001, Huang et al. 2013, Li et al. 2015, Puła and Chwała 2015, Shen et al. 2021). For frictional or frictional–cohesive materials, the situation is more complex because stress fields cannot be ignored (Fenton and Griffith 2003, Liu et al. 2017).

Modelling of input data as spatial random fields complicates risk analyses; therefore, advanced analytical or numerical methods are widely in use. The fundamental difficulty is that such sophisticated models, often parameter sensitive, should be in a balance with high quality of geoengineering input data and the data look never complete; that is why, for practical design situations, the random models are sometimes reduced from spatial random fields to simple random variables (Vanmarcke 2010, Low and Phoon 2015) by making use of some averaging techniques. Note that the spatial averaging does not necessarily mean that any significant part of information is lost. In a context of interactions between adjacent regions in a subsoil mass, significant up and down fluctuations of random parameters can reduce each other; so, the volume averaging of spatially distributed soil parameters is rationally justified and economical – it reduces point variances of considered random fields (Tietje et al. 2011). In addition, the method of spatial averaging has a strong mathematical background (Vanmarcke 2010) as well as a variety of well-documented applications.

The paper focuses mainly on the subsoil shearing resistance – like τ = qn×tan(φ) for the frictional Coulomb material – and analyses the variance reduction coefficients γa for the homogeneous random field tan(φ(x)), which is reduced by averaging to a corresponding random variable [tan(φ)]a. Fields of stresses in the subsoil mass implement a new element to the averaging procedures because non-uniform (deterministic) normal stresses qn ≠ const(x) generate a kind of subsoil inhomogeneities, in terms of the random shearing resistance τ(x). Indeed, if a normal stress qn along a slip line is locally negligible, then τ = qn×tan(φ) is also locally negligible and such a region does not contribute to total bearing capacity. If locally qn ∼ 0 in shearing resistance, then the friction coefficient tan(φ) is locally not very important and especially its random fluctuations Δtan(φ) can be ignored; also, it is vice versa for a locally concentrated load qn ≫ 0. Vanmarcke's spatial averaging applied to soil strength parameters (or to soil deformation parameters) ignores this mechanism and this is controversial. Clearly, the Vanmarcke averaging of spatial random fields is effective for many ‘stress-neutral’ random parameters like radioactive radiation, pollutant concentration, temperature, etc. For example, the spatial geometrical averaging is very useful in hydrological studies – hydrologists evaluate average rainfalls over a certain geographical region to estimate flood danger in an entire river basin (Vanmarcke 2010); but this model is insufficient for ultimate limit states, which are based on a random field tan(φ(x)) because the Coulomb shearing resistance τ is additionally ‘disturbed’ by spatially distributed normal loads qn(x).

It is assumed that the spatial field of the normal stress qn (x) is deterministic and physically independent of the random field tan(φ(x)). The stress field qn(x) plays the role of a fourth source of inhomogeneity – apart from the inherent one, from deterministic trends and measurement/transformation errors – because the stress field qn(x) can amplify (or suppress) effects of random local fluctuations. A new point of view on the spatial averaging of random fields in geoengineering is presented below, and the paper's objectives focus on simple numerical examples. The paper implements a more general and coherent concept of averaging of random fields and allows a conclusion that the standard (purely geometrical) variance reduction based only on volume integrals can be too optimistic; in a context of geotechnical structural safety, the traditional volume averaging can be dangerous.

Only few papers consider in a distinct way volume averaging in conjunction with fields of stresses, though there is a widespread agreement that the spatial regions being close to loaded places play a pivotal role in the averaging; being ‘close to loaded places’ means not only a short distance, but also local stress components of a maximal intensity. Recently, it was found by Ching and Hu (2017) and Ching et al. (2016) that the traditional spatial averaging model that treats all soil regions equally important cannot satisfactorily represent the effective Young modulus for the footing settlement problem – highly mobilised soil regions close to a footing are more significant than non-mobilised ones; the authors proposed some weight coefficients derived from a random finite element analysis. Similarly, for random shearing resistance of a rectangular sample analysed via finite elements (Ching and Phoon 2013), the authors found that the averaging over the whole sample overestimates the variance reduction, whereas the averaging along slip lines is much more representative; such a conclusion for localised shearing resistance should not be surprising. In the next paper, the authors confirmed this conclusion for a wider class of boundary problems (Ching et al. 2016).

Soil parameters as random variables

Soils display inherent inhomogeneities caused by long-term geological processes, unknown stress history, soil moisture, chemical reactions, etc.; therefore, soil parameters have a random nature, which can be described in terms of random variables or spatial random fields (time effects are not considered hereafter). Such local random fluctuations that happen in situ can be amplified during soil parameter evaluation, that is, by soil testing (usually invasive), measurement errors, calibration procedures, transformation models and others (Phoon and Kulhawy 1999a, 1999b); therefore, in geotechnical risk assessment, the term uncertainty of geotechnical parameters is more adequate than the term inherent inhomogeneity. Reported fluctuations of geotechnical parameters are relatively great, much greater than in cases of man-made construction materials. Considering a subsoil layer as a macro-homogeneous mass, the uncertainty can be expressed using coefficients of variations (c.o.v.) called ν = σ/μ, so a standard deviation σ over an expected value μ > 0. This distribution-free parameter is the minimal information about subsoil uncertainty; it can be sufficient only in simple practical applications.

Numerous authors present the results, literature reviews and discussions on ‘typical’ values of c.o.v. (%), like the values reviewed by Phoon and Kulhawy (1999a). Clearly, the data from many countries, regions, geological units, climate zones and even from different geological companies differ substantially and may be incomparable. This explains inevitable discrepancies in published data. Reported c.o.v. are sometimes based on estimated inherent inhomogeneity, but sometimes they are based on a total uncertainty that includes measurement errors, etc. Moreover, algorithms of de-trending can be different, non-standard testing procedures and individual data transformation methods are sometimes implemented (Phoon and Kulhawy 1999a). For an illustrative character of this paragraph, generalised ‘most typical’ values can be derived based on the review papers by Phoon and Kulhawy (1999a, 1999b): for the internal friction coefficient νtan(φ) ∼ 5%–15% (non-cohesive soils), νtan(φ) ∼ 25% (cohesive soils, even up to 50% is possible; slightly less for the angle φ itself), for the cohesion νc ∼ 20%–30% (for undrained shear strength, even up to 70% is possible) and for the deformation modulus νD ∼ 30%–50%. The presented values refer to the mean values of inherent variability; the intervals for global uncertainty can be still wider, at least 10%–20% due to measurement errors.

The following negative crosscorrelation coefficient is usually reported: −0.75 < rtan(φ);c < 0 and the value rtan(φ);c ∼ −0.3 can be recommended as a conservative estimation. Note that deterministic physical arguments do not always confirm the assumed negative crosscorrelation between φ or tan(φ) and c. Indeed, local fluctuations of soil moisture cause simultaneous decrease (or increase) of both tan(φ) and c; this fact can support suggestion about a positive correlation. On the other hand, local fluctuations of fine particle content result in random changes of tan(φ), which are simultaneously opposite to changes of c; this fact can support the conclusion about a negative correlation. Tests on drained and undrained samples also suggest a negative correlation (cu > c′, φu ∼ 0o < φ′). A review of previous papers and study on the role of negative crosscorrelation can be found in Javankhoshdel and Bathurst (2015).

Reliability-based design methods that use only the random variable format can cause an overdesign of foundations or other geoengineering structures. There are several possible reasons for this: ultimate limit state conditions may be oversimplified (too restrictive), some improvements of soil parameters may happen during construction, the database about geotechnical failures may be still poor – or the values of the c.o.v. may be ‘too large’, even though they are correct as point estimations. Indeed, it is a well-established fact that point variances Var{P(x)} = σ2 = const(x) of a homogeneous random field of soil parameters P(x) are usually not representative (overestimated) in geotechnical safety analyses because they ignore the interactions between adjacent regions in the considered soil mass. Bearing in mind dimensions of such adjacent regions – expressed in metres or dozens of metres – local random fluctuations can compensate each other in a considered soil mass as a whole, if total (summarised) effects are of interest. Formally, these are autocorrelation and crosscorrelation functions (between different parameters), which cause such reduction effects. This observation is a pivot of the monographies by Vanmarcke (2010, 1983) and numerous next papers.

Simple reliability models based on random variables that use only the expected values μ and standard deviations σ – instead of spatial random fields of subsoil parameters – are the most popular tool in safety evaluation, including the Eurocodes (EN 1997-1). In this context, a special attention is paid to averaging techniques called as spatial random homogenisation or volume integrals, which can transform spatially fluctuating random fields to some equivalent random variables. Since for a homogeneous random field, the expected value E{P(x)} = μ = const, any spatial averaging will not change this value, but the averaging yields an ‘effective’ variance Vara = γa×σ2 with a reduction coefficient γa ≤ 1; equivalently, but in terms of standard deviations σa = Γa×σ, therefore, the symbol Γa = √γa is also in use (Vanmarcke 2010, Tietje et al. 2011).

Soil parameters as random fields

Consider a homogeneous random field of a soil parameter (or several soil parameters in a vectorial version) denoted as P(x), where a point x = (x1, …, xn) belongs to a domain DEn in n-dimensional Euclidean space. The assumed homogeneity means that for every two points x, y, the expected values and the variances of two random variables P(x), P(y) are the same, E{P(x)} = E{P(y)} = μ = const, Var{P(x)} = Var{P(y)} = σ2 = const < + ∞; the covariance function equals Cov{P(x);P(y)} = E{[P(x) − μ]×[P(y) − μ]} = σ2×ρ(xy), so the corresponding correlation coefficient ρ depends only on the difference Δx = xy or simply ρ = ρ(Δx). Note that the approach is also effective for some quasi-homogeneous random fields when homogeneous random fluctuations happen around a functional deterministic trend μ + f(x), like for a subsoil stiffness and strength which often increase with depth (Li et al. 2015). In principle, the homogeneity requirement concerns the homogeneous random fluctuations and not so much deterministic trends; so, during the pre-processing, the model should be de-trended (Bagińska et al. 2016, Li et al. 2015). Another technique is used by Shen et al. (2021); the authors recommend a modulating amplitude A(x), which is a deterministic function, but modifies the considered random field of subsoil parameters P(x) = A(xH(x). The randomness is still generated by a homogeneous random field H(x). In particular, subsoil resistance derived from vertical cone penetration tests (CPT) can have the form of a stochastic process qc(z) with increasing both the expected value and the standard deviation (proportionally), but the model is not more complicated.

Joint probability distributions of the random field P(x) are not specified, if the simplest distribution-free second-order model is used. Moreover, it is assumed that the probabilistic averaging (ensemble averaging) and spatial averaging (volume averaging) coincide. This is only a useful hypothesis that cannot be proved in frames of distribution-free theories. In geotechnical practice of soil testing, one cannot carry out any sequence of independent measurements at the same point – the list of fully non-invasive geotechnical tests is very short. Replacing the term ‘at the same point’ by a more liberal ‘at the same place’ can be controversial; this difficulty leads to multiscale models, ambiguous definitions of ‘representative volume elements’ (Ostoja-Starzewski 2006) and nested micro-meso inhomogeneous models (Jaksa et al. 1999, Ching and Phoon 2013), which are not discussed here.

Several types of the decay functions ρ(Δx) became popular; four of them are discussed in detail by Vanmarcke (2010), Bagińska et al. (2016) and Oguz et al. (2019), the squared exponential (Gaussian) in particular (equation 1a). Due to natural sedimentation processes over vast areas, the geological model of the subsoil can reveal a horizontally layered structure like for transversally isotropic continua; hence, the following squared exponential representation of the correlation function ρ(x;y) = ρ(xy) is useful: ρ=exp{(x1y1)2+(x2y2)2(dh)2}exp{(x3y3)2(dv)2} \rho = \exp \left\{ { - {{{{\left( {{x_1} - {y_1}} \right)}^2} + {{\left( {{x_2} - {y_2}} \right)}^2}} \over {{{\left( {{d_{\rm{h}}}} \right)}^2}}}} \right\} \cdot \exp \left\{ { - {{{{\left( {{x_3} - {y_3}} \right)}^2}} \over {{{\left( {{d_{\rm{v}}}} \right)}^2}}}} \right\} where the symbols x3, y3, dv are used for vertical direction. Alternatively, the linear exponential (Markovian) function (equation 1b) is in use: ρ=exp{|x1y1|+|x2y2|dh}exp{|x3y3|dv} \rho = \exp \left\{ { - {{\left| {{x_1} - {y_1}} \right| + \left| {{x_2} - {y_2}} \right|} \over {{d_{{\rm{h'}}}}}}} \right\} \cdot \exp \left\{ { - {{\left| {{x_3} - {y_3}} \right|} \over {{d_{{\rm{v'}}}}}}} \right\} Although the equations (1a) and (1b) apply to general 3D situations, the points of interest x, y are often situated on certain specific subsets like along slide lines (surfaces) in ultimate limit states or limit equilibrium studies.

In data processing, sometimes a negative autocorrelation can happen that is beyond the scope of the exponential functions (1a and 1b); if such negative values are not incidental, and so if they are confirmed by significance tests, then a wider class of cosine exponents like exp{−|Δx|/d′}×cos(|Δx|/d″) can be considered. Such a ‘more chaotic’ behaviour seems to be possible for anthropogenic soils, for example for lignite mine dumping soils (Bagińska et al. 2016) or low-quality sandy backfills (Brząkała, 1981); in the former case, averaged locally negative values up to ρ ∼ −0.2 are reported (but not used, assumed d″ = +∞) and in the latter one, the values are up to ρ ∼ −0.5 (horizontal variability of a subsoil vertical response over a meso-scale laboratory field 4.2 m × 2.0 m, several dozens of testing points). The problem of negative autocorrelation for nearshore sea bottom soils is recently discussed by Oguz et al. (2019). The values derived from CPT tests in stiff, overconsolidated clays can also be negative (up to ρ ∼ −0.2) (Jaksa et al. 1999). In the context of risk assessment and averaging, note that ignoring the negative values of ρ is usually on the safe side (Fig. 1b).

Figure 1

(a) Prototype defined by two random variables tan(φi), i = 1, 2. (b) Variance reduction factors γa(w1), depending on the dimensionless weight coefficient w1.

The decay constants dv, dh (m) in equation (1a) control the speed of convergence to zero for increasing distances and they are in close relation to so-called scales of fluctuations δi (Vanmarcke 2010, Bagińska et al. 2016, Pieczyńska-Kozłowska et al. 2017, Tietje et al. 2011), which are good intuitive measures of random fluctuations. The following definitions of δ and δ′ are in use for unidirectional random variations due to equations (1a) and (1b), respectively: δ=+ρ(r)dr=+exp(r2d2)dr=πd \delta = \int_{ - \infty }^{ + \infty } {\rho (r)dr = \int_{ - \infty }^{ + \infty } {\exp \left( { - {{{r^2}} \over {{d^2}}}} \right)dr = \sqrt \pi d} } δ=+ρ(r)dr=20+exp(rd)dr=2d \delta ' = \int_{ - \infty }^{ + \infty } {\rho (r)dr = 2\int_0^{ + \infty } {\exp \left( { - {r \over {d'}}} \right)dr = 2d'} } Empirical (discrete) correlation function ρe(r) can be applied to approximate a distance ro such that ρe(ro) ≈ 0. In terms of both exponential functions ρ(r), this could be assumed as ρ(r) ≅ 0.05. If so, then ro=√3d and ro=3d‘; thus, d ∼ √3d‘. In this way, the following working hypothesis is derived: δ ∼ 1.5δ‘. A wider statistical analysis of the distance ro and its confidence limits were presented by Jaksa et al. (1999). For d ≫ 0, the model turns to a degenerate spatial random field – just a random variable; for d ∼ 0, the model turns to a very chaotic random field with uncorrelated succeeding values.

There exist several sophisticated methods of evaluation of the δ-value; selected techniques and numerical results were presented by Bagińska et al. (2016), Lloret-Cabot et al. (2014), Pieczyńska-Kozłowska et al. (2017), Phoon and Kulhawy (1999a) and Vanmarcke (2010). Standard CPT testing is most often used – in vertical direction, first of all (Oguz et al. 2019); a very original horizontal CPT testing has been presented by Jaksa et al. (1999). Although the CPT method provides a very large amount of data – continuous realisations of the soil resistance, as expected for a stochastic process qc(z) – the interpretation of results encounters difficulties.

In the common opinion δh ≫ δv, like the ratio δhv = 10 used by Tietje et al. (2014); such proportion δh ≫ δv can be suggested by a regular sedimentation of geological units. Again, for the same reasons as in cases of c.o.v., there is no agreed opinion about representative values of the scales of fluctuations d. Moreover, in published data, the distinction between definitions in equations (2a) and (2b) is not always respected. For example, results of CPT soundings have been presented with the best fitted Gaussian curve (equation 1a) in the paper by Jaksa et al. (1999) – the authors report measured cone tip resistance qc for which δv ∼ 0.17 m (average value) with a c.o.v. on the level of 30%. Independently, another value ro ≅ 0.28 m can be estimated from the authors’ empirical plot of ρe(r), used in place of the least-square fitting; in this way, the following values can be derived: δv = √πdv=√πro/√3 ≅ ro ≅ 0.28 m. Similar estimation δv ∼ 0.8×5.3/20 = 0.21 m can be found from the same data using the Vanmarcke proposal δv ∼ 0.8d, where d denotes the average distance between intersections of the fluctuating resistance qc and its parabolic trend function (Vanmarcke 2010, Jaksa et al. 1999). The horizontal scale of fluctuations δh was also analysed by Jaksa et al. (1999) and its nested structure was documented – the estimated results depend on the scale of sampling: δh ∼ 0.15 m for CPT sampling at 5 mm intervals (so, δh ≅ δv), but δh ∼ 1–2 m for CPT sampling at 0.5 and 1.0 m. The authors’ conclusion says that ‘both the small-scale vertical and lateral correlation distances are a manifestation of the CPT and not the soil itself’ (Jaksa et al. 1999). On the other hand, the review by Phoon and Kulhawy (1999a) is in sharp contrast to the conclusions presented by Jaksa et al. (1999), but the former review is focused on a wider spectrum of testing methods, not only on the CPT sounding. For different geotechnical parameters, as well as for different testing techniques, there is generally δv ∼ 1.0–2.0 m (less than 1.0 m only for CPT tests), but δh is expressed in dozens of metres; so, the ratio δhv ∼ 10 is confirmed. The references presented by Shen et al. (2021) for the offshore soil in the North Sea report δv ∼ 0.48–7.14 m and δh ∼ 24.62–66.48 m.

Conclusion

For the illustrative numerical examples presented hereafter, the range of dv = δv/√π ∼ 0.5–1.5 m and the range of dh = δh/√π ∼ 10–20 m can be assumed as some representative values.

Prototype of the geometric spatial averaging

Standard spatial averaging, called as a geometric one, has a well-documented mathematical background presented by Vanmarcke (2010); also, it has many practical applications (Vanmarcke 2010, Chwała 2019, Tietje et al. 2011, Ching et al. 2016, Puła and Chwała 2015). For a frictional material, where tan(φ) > 0 and c = 0 (kPa), consider two intervals dL extracted from a sliding line and assume the two random variables shown in Fig. 1a: a random variable tan(φ1) on the left part dL and a random variable tan(φ2) on the right one.

Looking for one ‘effective’ parameter, the random vector (tan(φ1);tan(φ2)) is replaced by one averaged random variable tan(φ)ga acting along 2dL. Assume the homogeneity condition: E{tan(φi)} = μ, Var{tan(φi)} = σ2; moreover, Cov{tan(φ1);tan(φ2)} = σ2×ρ1;2. The values dL are geometric weighting coefficients and the intuitively averaged (homogenised) friction coefficient [tan(φ)]ga on 2dL can be calculated as a new random variable: [tan(φ)]ga=[tan(φ1)dL+tan(φ2)dL]/(2dL)=12tan(φ1)+12tan(φ2) \matrix{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{ga}}}} = \left[ {\tan ({\varphi _1}) \cdot dL }\,+ \right.} \cr {\left. {\tan ({\varphi _2}) \cdot dL} \right]/(2dL) = {1 \over 2} \cdot \tan ({\varphi _1}) + {1 \over 2} \cdot \tan ({\varphi _2})} \cr } Therefore, E{[tan(φ)]ga} = μ, Var{[tan(φ)]ga} = σ2×(1 + ρ1;2)/2 < Var{tan(φi) = σ2. The variance reduction factor equals γga = Var{[tan(φ)]ga}/Var{tan(φi) = (1 + ρ1;2)/2 = 1 −(1 − ρ1;2)/2; the lower index g refers to the purely geometric sense of equation (3). Note that potentially, the randomness of [tan(φ)]ga in equation (3) can even disappear, if always both random fluctuations of tan(φ1) and tan(φ2) happen as opposite numbers, thus taking the hypothetic correlation coefficient ρ1;2 = −1.

Generalisations of the geometric averaging are straightforward, if for a longer interval AB, the sum in equation (3) is replaced by an integral over this line AB: [tan(φ)]ga=ABtan(φ(x))dx|AB|=ABtan(φ(x))dxAB1dx {\left[ {\tan (\varphi )} \right]_{{\rm{ga}}}} = {{\int_A^B {\tan \left( {\varphi (x)} \right)dx} } \over {\left| {AB} \right|}} = {{\int_A^B {\tan \left( {\varphi (x)} \right)dx} } \over {\int_A^B {1dx} }} where the interval length equals |AB|; the definition is similar for surfaces in spatial cases (Vanmarcke 2010).

Generalised methodology: prototype of the stress-weighted spatial averaging

Ultimate limit state for random shearing is a start point of this approach due to the context of the random vector (tan(φ1);tan(φ2)). For the frictional material, the two variables tan(φ1), tan(φ2) should be considered in conjunction with two normal deterministic loadings in Fig. 1a, qn1 = const > 0, qn2 = const > 0. The random shearing resistance along AB can be expressed as: 2dT=qn1tan(φ1)dL+qn2tan(φ2)dL 2dT = {q_{n1}} \cdot \tan ({\varphi _1}) \cdot dL + {q_{n2}} \cdot \tan ({\varphi _2}) \cdot dL Looking for one ‘equivalent’ random variable [tan(φ)]sa on the same interval AB: 2dT=qn1[tan(φ)]sadL+qn2[tan(φ)]sadL=[tan(φ)]sa[qn1+qn2]dL \matrix{ {2dT = } \hfill & {{q_{n1}} \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{sa}}}} \cdot dL + {q_{n2}} \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{sa}}}} \cdot } \hfill \cr {} \hfill & { \cdot \;dL = {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{sa}}}} \cdot \left[ {{q_{n1}} + {q_{n2}}} \right] \cdot dL} \hfill \cr } Therefore, on comparing the two equations (5a) and (5b): [tan(φ)]sa=qn1tan(φ1)+qn2tan(φ2)qn1+qn2=w1tan(φ1)+w2tan(φ2) \eqalign{ & {\left[ {\tan (\varphi )} \right]_{{\rm{sa}}}} = {{{q_{n1}} \cdot \tan ({\varphi _1}) + {q_{n2}} \cdot \tan ({\varphi _2})} \over {{q_{n1}} + {q_{n2}}}} = \cr & {w_1} \cdot \tan ({\varphi _1}) + {w_2} \cdot \tan ({\varphi _2}) \cr} where the dimensionless weight coefficients wi > 0 equal w1 = qn1/(qn1 + qn2), w2 = qn2/(qn1 + qn2), w1 + w2 = 1.

In particular, the local contribution of tan(φ1) is negligible if qn1qn2, w1w2, so the randomness is localised, almost not averaged. If a (potentially possible) physical dependence between qn and tan(φ) is ignored, then E{[tan(φ)]sa} = w1×μ + w2×μ = μ and Var{[tan(φ)]sa} = [1 − 2×w1×w2×(1 − ρ1;2)]×σ2.

Therefore, the variance reduction factor equals γsa = Var{[tan(φ)]sa}/σ2 = 1 − 2×w1×(1 − w1)×(1 − ρ1;2); the lower index s refers to the stress-weighted averaging. The variance reduction factor γsa defined above is sensitive to qn gradients and depends on the ratio qn1/qn2, what is not true for the variance reduction factor γga = 1 −(1 − ρ1;2)/2, which is stress-independent. The following inequality γga ≤ γsa is true, but γga = γsa if and only if w1 = w2 = 1/2, like for five dots depicted in Fig. 1b. To conclude, the geometric averaging has only an intuitive character, weakly related to the shearing resistance of frictional materials; the overestimation of the beneficial variance reduction γga ≤ γsa can be dangerous in safety evaluation. Parabolic functions γsa(w1) are depicted in Fig. 1b for five different correlation coefficients ρ = ρ1;2.

Passage to a discrete multiterm sum for n > 2 random variables is straightforward; the same applies for a second-order stochastic process tan(φ(x)) with an autocorrelation function ρ(x;y) along the interval AB, where E{tan(φ(x))} = μ = const, Var{tan(φ(x)} = σ2 = const. The shearing resistance T is a random variable generated by the stationary stochastic process tan(φ(x)) in the formula: T=ABqn(x)tan(φ(x))dx T = \int_A^B {{q_n}(x) \cdot \tan \left( {\varphi (x)} \right)dx} Looking for an equivalent simplified model, the stochastic process tan(φ(x)) is replaced by a random variable [tan(φ)]sa = const(x). If so, the same random variable T has the following form: T=AB[tan(φ)]saqn(x)dx=[tan(φ)]saABqn(x)dx \matrix{ {T = \int_A^B {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} \cdot {q_n}(x)dx = } } \cr {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} \cdot \int_A^B {{q_n}(x)dx} } \cr } Both alternative expressions for the same random variable T formulate the background for the definition of [tan(φ)]sa; if equations (7a) and (7b) are compared, the following definition can be obtained: [tan(φ)]sa=ABqn(x)tan(φ(x))dxABqn(x)dx {\left[ {\tan (\varphi )} \right]_{{\rm{sa}}}} = {{\int_A^B {{q_n}(x) \cdot \tan (\varphi (x))dx} } \over {\int_A^B {{q_n}(x)dx} }} Therefore, E{[tan(φ)]sa}=ABqn(x)E{tan(φ(x))}dxABqn(x)dx=ABqn(x)μdxABqn(x)dx=μ \matrix{ {E\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\} = {{\int_A^B {{q_n}(x) \cdot E\left\{ {\tan (\varphi (x))} \right\}dx} } \over {\int_A^B {{q_n}(x)dx} }} = } \cr {{{\int_A^B {{q_n}(x) \cdot \mu dx} } \over {\int_A^B {{q_n}(x)dx} }} = \mu } \cr } Var{[tan(φ)]sa}=ABABqn(x)qn(y)Cov{tan(φ(x));tan(φ(y))}dxdy[ABqn(x)dx]2 \matrix{ {Var\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\} = } \cr {{{\int_A^B {\int_A^B {{q_n}(x) \cdot {q_n}(y) \cdot {Cov}\left\{ {\tan (\varphi (x));\tan (\varphi (y))} \right\}dxdy} } } \over {{{\left[ {\int_A^B {{q_n}(x)dx} } \right]}^2}}}} \cr } The dimensionless variance reduction factor equals γsa=Var{[tan(φ)]sa}Var{tan(φ(x))}=Var{[tan(φ)]sa}σ2==ABABqn(x)qn(y)ρ(x;y)dxdy[ABqn(x)dx]21 \matrix{ {{\gamma _{{\rm{sa}}}} = {{{Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\}} \over {{Var}\left\{ {\tan (\varphi \left( x \right))} \right\}}} = {{{Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\}} \over {{\sigma ^2}}} = } \cr {{ = }{{\int_{A}^{B} {\int_{A}^{B} {{q_n}(x) \cdot {q_n}(y) \cdot \rho (x;y)dxdy} } } \over {{{\left[ {\int_A^B {{q_n}(x)dx} } \right]}^2}}} \le 1} \cr } where ρ(x;y) = ρ(xy) because of the assumed homogeneity; the expression (10) has only loose ties with the approximate variance reduction factor proposed for practical applications by Vanmarcke (2010): Γ2 = min{1; δ/|AB|}.

Only if qn = const along AB, then the geometric averaging coincides with stress-weighted averaging, but generally the latter one is more representative.

Simple numerical results are presented in Fig. 2; a fixed interval AB of length|AB|=7 m is loaded in a localised way on [0;7]: qn(x) = 0 on [0;S], 0 ≤ S < 7, and qn(x) = q = const > 0 on (S;7], so in the Heaviside form, qn(x) = q×[H(xS) − H(x − 7)]; the Gaussian autocorrelation ρ(x;y) = exp{−[(xy)/d]2}, 0 ≤ x, y ≤ 7 is assumed. Both methods coincide if and only if S = 0 (four dots on the vertical axis), but γga < γsa elsewhere; the geometric spatial averaging is insensitive to the length S, which describes very different stresses along AB.

Figure 2

The dimensionless variance reduction factors γsa(S) depending on the correlation parameter d (m) and the stress parameter S (m).

Example: resistance against horizontal sliding

As a practical example, consider the standard GEO-sliding stability condition (EN 1997-1) for a massive shallow foundation BxLxh, where −B/2 ≤ x1 ≤ +B/2 and −L/2 ≤ x2 ≤ +L/2. Define normal (vertical) contact stresses under the footing as qn(x1,x2) = qn(x) ≥ 0. The foundation base BxL is horizontal and ‘rough’, that is, the stabilising horizontal force can be calculated from the simple frictional model as T = V×tan(φ), if the subsoil is cohesionless; note that this deterministic model does not depend on the eccentricities of the deterministic vertical load V called as eB and eL, respectively.

In the random variable format, the friction coefficient tan(φ) in the expression T = V×tan(φ) has a mean value μ and a variance σ2, so E{T} = Vμ, Var{T} = V2σ2. For a second-order homogeneous isotropic random field tan(φ(x1,x2)) = tan(φ(x)) and the random shearing resistance, T can be expressed following equations (7a) and (7b), that is: T=BxLqn(x_)tan(φ(x_))dx_and,ontheotherhand,T=[tan(φ)]saBxLqn(x_)dx_ \matrix{ {T = \int_{BxL} {{q_n}(\underline x) \cdot \tan \left( {\varphi (\underline x)} \right)d{\underline x}\,\,{\rm{and}},} } \cr {{\rm{on}}\,{\rm{the}}\,{\rm{other}}\,{\rm{hand}},\,\,T = {{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} \cdot \int_{BxL} {{q_n}(\underline x)d{\underline x}} } \cr } Therefore, [tan(φ)]sa=BxLqn(x_)tan(φ(x_))dx_BxLqn(x_)dx_andeventually,γsa=BxLBxLqn(x_)qn(y_)ρ(x_;y_)dx_dy_[BxLqn(x_)dx_]2 \matrix{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} = {{\int_{BxL} {{q_n}({\underline x} ) \cdot \tan \left( {\varphi ({\underline x} )} \right)d{\underline x} } } \over {\int_{BxL} {{q_n}({\underline x} )d{\underline x} } }}{\rm{and}}\,{\rm{eventually,}}} \cr {{\gamma _{{\rm{sa}}}} = {{\int_{BxL} {\int_{BxL} {{q_n}({\underline x} ) \cdot {q_n}({\underline y}) \cdot \rho ({\underline x} ;{\underline y}) \cdot d{\underline x} d{\underline y}} } } \over {{{\left[ {\int_{BxL} {{q_n}({\underline x} )d{\underline x} } } \right]}^2}}}} \cr } Note that the vertical coordinate x3 (or simply z) is not used in this model.

The presented numerical example employs the horizontally isotropic Gaussian autocorrelation function ρ(x;y) = exp{−[(x1y1)/dh]2 − [(x2y2)/dh]2}; following the recommendations of the Eurocode 7 (EN 1997-1), the vertical contact stress qn under the footing can be assumed as a double linear function like qn(x) = ao + a1x1 + a2x2 (kPa). Consider a symmetrical case B = L = 10 m, V =10 MN, q = V/(B×L) = 100 kPa, eB = eL = e (m) ≥ 0.

If both eccentricities 0 ≤ e < 10/12 m, then under the corners: q1 = 100 + 120×e, q2 = q3 = 100, q4 = 100 − 120×e > 0, so qn(x) = 100 + 12×e×(x1 + x2) (kPa).

If both eccentricities e > 10/12, then a gap appears under foundation (zero contact stress, locally); for a maximal value e = 30/12 m, which is the maximal acceptable by the Eurocode 7, there is qn(x) = max{0;60×(x1 + x2)} (kPa).

The values of the variance reduction factor γsa are depicted in Fig. 3 for e ≤ 30/12 m.

Figure 3

The dimensionless variance reduction factors γsa(e) depending on the correlation parameter d = dh (m) and the load eccentricities eB = eL = e (m).

The geometric spatial averaging and the variance reduction factor γga do not depend on the eccentricities e and they overestimate the variance reduction effects (four dots in Fig. 3). Special attention should be paid to the curves for d ∼ 20 m and d ∼ 10 m, which correspond to the most realistic values of the horizontal correlation parameter dh:

for dh = 10 m, the variance reduction γsa is on the level of 74%–87%, whereas γga = 74%;

for dh = 20 m, the variance reduction γsa is only on the level of 92%–97%, whereas γga = 92% and

for dh = 30 m, the variance reduction γsa is only on the level of 96%–99%, whereas γga = 96%.

Conclusion

The results are close to each other and for dh ≥ 20 m, both horizontal variance reductions are not significant; still much greater L or B (for dams, weirs, etc.) would be necessary to observe greater reduction. Both spatial averaging methods do not seem to be necessary for such proportions of the object dimensions B, L and dh, that is, if dh/L > 2–3.

Example: settlement analysis of a shallow foundation

The integral equations (7)–(10) apply not only to shearing resistance, but also to settlement analysis and to any linear operator, in fact. In a simplified linear form suggested by the Eurocode 7 (EN 1997-1), the foundation settlement w (m) can be estimated as follows: w=0zmaxqn(z)D(z)dz w = \int_0^{{z_{\max }}} {{q_n}(z) \cdot D(z)dz} The symbol D(z) > 0 (1/kPa) denotes a random deformation modulus – stationary stochastic process as a function of the depth x3 = z > 0 under foundation.

The following equivalent [D]sa = const(z) as a corresponding random variable can be introduced: w=0zmax[D]saqn(z)dz=[D]sa0zmaxqn(z)dz \matrix{ {w = \int_0^{{z_{\max }}} {{{\left[ D \right]}_{{\rm{sa}}}} \cdot {q_n}(z)dz} = } \cr {{{\left[ D \right]}_{{\rm{sa}}}} \cdot \int_0^{{z_{\max }}} {{q_n}(z)dz} } \cr } Equations (13a) and (13b) define the random variable [D]sa.

Equations (13a) and (13b) are focused on a finite interval 0 ≤ zzmax under a shallow footing; taking into account a founding depth hf under the ground surface, the value of zmax can be estimated from the equation γ×(hf/B + zmax/BB×0.2 = qn(zmax) (EN 1997-1), where g×(hf + z) denotes vertical stresses in situ for a dry soil and γ (kN/m3) is the subsoil unit weight. The average vertical stress qn under a rectangular footing BxL vanishes with depth z; usually, the integrated Steinbrenner formula is in use (a double integral of the Boussinesq solution) or just the following simplified estimation: qn(z) = V/[(B + z)×(L + z)] = q/[(1 + z/B) × (1 + β×z/B)] for β = B/L ≤ 1, q = V/(B×L) = const > 0. The deterministic depth zmax depends on the β coefficient as in Fig. 4. It is assumed that random fluctuations of D(z) do not change the deterministic load function qn(z) and vice versa.

Figure 4

The effective depth zmax under considered foundation and the dimensionless variance reduction factor γsa(β) depending on the correlation parameter d = dv (m) and the footing shape ratio β.

The stochastic parameter D(z) (1/kPa) needs a comment. Clearly, the random deformation modulus D(z) corresponds to 1/E(z) or 1/M(z) by making use of the Young modulus E or the oedometric modulus M of the subsoil; both are assumed as stochastic processes. As far as distribution-free models are considered, the first two moments E{D(z)} = μ = const(z), Var{D(z)} = σ2 = const(z), Cov{D(z1);D(z2)} = σ2×ρ(z1z2) cannot be derived from the corresponding moments E{M(z)}, Cov{M(z1);M(z2)} – the model should be completed by some joint probability distributions of the stochastic process M(z); in particular, correlated lognormal random variables M(z1), M(z2) can be useful. Alternatively, the first-order perturbation is acceptable for a ‘small’ variance of M(z); also, an adequate data pre-processing can be recommended to apply the sample statistics directly to 1/M, not to M, because the stationarity of both 1/M and M has different conditions.

Equations (13a) and (13b) yield the variance of the averaged deformation modulus Var{[D]sa} and then the variance reduction factor equals: γsa=Var{[D]sa}Var{D(z)}=0zmax0zmaxqn(z1)qn(z2)ρ(z1;z2)dz1dz2[0zmaxqn(z)dz]21 {\gamma _{{\rm{sa}}}} = {{{Var}\left\{ {{{\left[ D \right]}_{{\rm{sa}}}}} \right\}} \over {{Var}\left\{ {D(z)} \right\}}} = {{\int_0^{{z_{\max }}} {\int_0^{{z_{\max }}} {{q_n}({z_1}) \cdot {q_n}({z_2}) \cdot \rho ({z_1};{z_2})d{z_1}d{z_2}} } } \over {{{\left[ {\int_0^{{z_{\max }}} {{q_n}(z)dz} } \right]}^2}}} \le 1 Spatial horizontal random fluctuations are much less significant for such rectangular footing BxL; they are not considered here (δhdv).

The numerical example in Fig. 4 uses the Gaussian autocorrelation function ρ(z1;z2) = exp{−[(z1z2)/dv]2}, 0 ≤ z1, z2zmax; the subsoil unit weight γ = 18 kN/m3, q = 250 kPa, hf/B = 2 and the foundation width B = 1 m. For such fixed data, the effective dimensionless zmax/B depends on the dimensionless shape coefficient β = B/L ≤ 1 as in Fig. 4; the shape ratio β = 0 corresponds to an infinite beam and β = 1 corresponds to a square footing.

The geometric variance reduction factor γga in Fig. 5 is also derived from equation (14) for the same zmax, but for qn = const(z). Differences between the factors γsa and γga in Fig. 5 are significant.

Figure 5

The dimensionless variance reduction factors γsa(β) and γga(β) for the correlation parameters dv = d = 1.5 m (solid lines) and dv = d = 0.5 m (dashed lines).

Cohesive soils

For perfectly cohesive materials, the shearing resistance in Fig. 1a equals 2dT = c1×dL + c2×dL and is replaced by 2dT = [c]a×dL + [c]a×dL; hence, [c]a = (c1 + c2)/2. For such a case, the geometric spatial averaging (Vanmarcke 2010) is representative because the model is independent of the normal stress qn > 0. There is only one averaged cohesion: [c]a=[c]sa=[c]ga=Dc(x_)dx_D1dx_ {\left[ c \right]_a} = {\left[ c \right]_{{\rm{sa}}}} = {\left[ c \right]_{{\rm{ga}}}} = {{\int\!\!\!\int_D {c({\underline x} )d{\underline x} } } \over {\int\!\!\!\int_D {1d{\underline x} } }} and only one dimensionless reduction coefficient γa = γsa = γga.

To be precise, there exist some rare exceptions because this is not the soil cohesion (or adhesion) itself which generates the shearing resistance, but the soil cohesion (or adhesion) on a real contact with the foundation. As in section 6, a gap can appear under a rigid foundation for ‘great’ eccentricities e, hence the averaging is limited (localised) only to a soil-foundation direct contact (qn > 0); to a certain extent, for a rigid foundation, the model is not completely independent of qn, so γga can be less than γsa also for perfectly cohesive soils. This aspect is not analysed below.

Although the Tresca model with tan(φ) ≡ 0, c > 0 found certain practical applications to undrained bearing capacity, this is, however, a peculiar case. Generally, the shearing ultimate limit states are stress dependent, not only for the Coulomb material. In a general case of τ(x) = qn(x)×tan(φ(x)) + c(x), two homogeneous random fields are considered for a frictional–cohesive soil. Both autocovariance functions are completed by a crosscovariance function Cov{tan(φ(x));c(y)} = σtan(φ)×σc×rtan(φ);c×ρtan(φ);c(xy). More or less significant distinctions between the decay functions ρtan(φ)(xy), ρc(xy) and ρtan(φ);c(xy) are possible. If qn(x) > 0, then the usual definition of a local shear coefficient tan(ψ(x)) = tan(φ(x)) + c(x)/qn(x) > tan(φ(x)) reduces the situation to an equivalent cohesionless material. The new random field tan(ψ(x)) is not homogeneous if qn(x) ≠ const(x); in particular, E{tan(ψ(x))} = μtan(φ) + μc/qn(x), Var{tan(ψ(x))} ≠ const(x). By making use of the procedure defined by equations (7) and (8) for the frictional soil, T(tan(φ(x_));c(x_))=Dqn(x_)tan(ψ(x_))dx_ T\left( {\tan \left( {\varphi ({\underline x} )} \right);c({\underline x} )} \right) = \int\!\!\!\int_D {{q_n}({\underline x} ) \cdot \tan \left( {\psi ({\underline x} )} \right)d{\underline x} } T([tan(ψ)]sa)=D[tan(ψ)]saqn(x_)dx_=[tan(ψ)]saDqn(x_)dx_ \matrix{ {T\left( {{{\left[ {\tan (\psi )} \right]}_{{\rm{sa}}}}} \right) = \int\!\!\!\int_D {{{\left[ {\tan (\psi )} \right]}_{{\rm{sa}}}} \cdot {q_n}({\underline x} )d{\underline x} = } } \cr {{{\left[ {\tan (\psi )} \right]}_{{\rm{sa}}}} \cdot \int\!\!\!\int_D {{q_n}({\underline x} )d{\underline x} } } \cr } Therefore, [tan(ψ)]sa=Dqn(x_)tan(φ(x_))dx_+Dc(x_)dx_Dqn(x_)dx_=[tan(φ)]sa+[c]aD1dx_Dqn(x_)dx_ \matrix{ {{{\left[ {\tan (\psi )} \right]}_{{\rm{sa}}}} = {{\int\!\!\!\int_D {{q_n}({\underline x} ) \cdot \tan \left( {\varphi ({\underline x} )} \right)d{\underline x} + \int\!\!\!\int_D c ({\underline x} )d{\underline x} } } \over {\int\!\!\!\int_D {{q_n}({\underline x} )d{\underline x} } }} = } \cr {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} + {{\left[ c \right]}_a} \cdot {{\int\!\!\!\int_D {1d{\underline x} } } \over {\int\!\!\!\int_D {{q_n}({\underline x} )d{\underline x} } }}} \cr } Here, the variance reduction factor (equation 10) is less useful because it is a position-dependent function (in the denominator): γsa=Var{[tan(ψ)]sa}Var{tan(ψ(x_))}x_D {\gamma _{{\rm{sa}}}} = {{{Var}\left\{ {{{\left[ {\tan (\psi )} \right]}_{{\rm{sa}}}}} \right\}} \over {{Var}{{\left\{ {\tan (\psi ({\underline x} ))} \right\}}^\prime }}}{\underline x} \in D

Alternatively, the vectorial random field (tan(φ(x));c(x)) can be replaced by the equivalent random vector ([tan(φ)]sa;[c]a) by making use of the previously defined averages. Depending on a stress level of qn(x) – not only on gradients – either the value [tan(φ)]sa or [c]a dominates in the averaging; thus, the interpretation becomes more complex. To overcome this difficulty, the effect of the variance reduction can be studied in terms of the random shearing resistance T, not in terms of its strength parameters. Therefore, γsa=Var{T([tan(φ)]sa;[c]a)}Var{T(tan(φ);c)}=Var{[tan(φ)]saDqn(x_)dx_+[c]aD1dx_}Var{tan(φ)Dqn(x_)dx_+cD1dx_} \matrix{ {{\gamma _{{\rm{sa}}}} = {{{Var}\left\{ {T\left( {{{\left[ {\tan (\varphi )} \right]}_{sa}};{{\left[ c \right]}_a}} \right)} \right\}} \over {{Var}\left\{ {T\left( {\tan (\varphi );c} \right)} \right\}}} = } \cr {{{{Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} \cdot \int\!\!\!\int_D {{q_n}({\underline x} )d{\underline x} + {{\left[ c \right]}_a} \cdot \int\!\!\!\int_D {1d{\underline x} } } } \right\}} \over {{Var}\left\{ {\tan (\varphi ) \cdot \int\!\!\!\int_D {{q_n}({\underline x} )d{\underline x} + c \cdot \int\!\!\!\int_D {1d{\underline x} } } } \right\}}}} \cr } The random solution T(tan(φ);c) in the denominator operates simply on the (correlated) random variables tan(φ), c, yielding from the random fields tan(φ(x)), c(x) when ignoring their spatial effects, that is, for ρ(x;y) ≡ 1.

Example: wedge stability
Data analysis

A simple discretised random field which corresponds to Fig. 1a is depicted in Fig. 6. For a potential sliding mechanism in 2D, four crosscorrelated random variables are considered: X1 = tan(φ1), X2 = tan(φ2), X3 = c1, X4 = c2; the sliding wedge is split into two homogeneous triangles ADB and DCB.

Figure 6

Simplified wedge stability for 2x2 random variables tan(φi), ci, i = 1, 2.

The randomly inhomogeneous material corresponds to a silty-clayey sand. The second-order moments are assumed as follows: E{tan(φi)} = 0.5, σtan(φi) = 0.125, νtan(φi) = 25%, E{ci} = 10.0 kPa, σci = 4.0 kPa, νci = 40%. The covariance matrix is presented in Table 1, and dimensionless correlation coefficients are depicted in Table 2.

Auto- and cross-covariances used in numerical calculations.

Cov{Xi;Xj} tan(φ1) (−) tan(φ2) (−) c1 (kPa) c2 (kPa)
tan(φ1) (−) 0.015625 0.0015625 −0.15 −0.03
tan(φ2) (−) 0.0015625 0.015625 −0.03 −0.15
c1(kPa) −0.15 −0.03 16 1.6
c2(kPa) −0.03 −0.15 1.6 16

Auto- and crosscorrelation coefficients used in numerical calculations.

ρij tan(φ1) tan(φ2) c1 c2
tan(φ1) 1 +0.10 −0.30 −0.06
tan(φ2) +0.10 1 −0.06 −0.30
c1 −0.30 −0.06 1 +0.10
c2 −0.06 −0.30 +0.10 1

The second-order moments of the random variables tan(φi), cj are assumed as the discrete input data; if they are discretised from the continuous spatial moments, like Cov{tan(φ(x));c(y)} in the random field formulation, then they are ‘close’ to mid-point values on AD, DC, but are not exactly the same – particularly, if the distinguished macro-homogeneous regions are not ‘small’. This aspect is not considered here.

Results of the presented parameter-averaging procedures

The obtained averaged values of the friction coefficients are as follows: [tan(φ)]ga=[tan(φ1)×L1+tan(φ2)×L2]/(L1+L2)=0.511×tan(φ1)+0.489×tan(φ2)Var{[tan(φ)]ga}=0.00860=0.550×0.1252,γga=Var{[tan(φ)]ga}/Var{tan(φ)}=0.55[tan(φ)]sa=[tan(φ1)×G1+tan(φ2)×(G2+Q)]×cos(α)/[(G1+G2+Q)×cos(α)]=0.916×tan(φ1)+0.804×tan(φ2)Var{[tan(φ)]sa}=0.0112=0.716×0.1252,γsa=Var{[tan(φ)]sa}/Var{tan(φ)}=0.72>0.55. \matrix{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{ga}}}} = \left[ {\tan ({\varphi _1}) \times {{\rm{L}}_1} + \tan ({\varphi _2}) \times {{\rm{L}}_2}} \right]/({{\rm{L}}_1} + {{\rm{L}}_2}) = } \cr {0.511 \times \tan ({\varphi _1}) + 0.489 \times \tan ({\varphi _2})} \cr {{Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{ga}}}}} \right\} = 0.00860 = 0.550 \times {{0.125}^2},} \cr {{\gamma _{{\rm{ga}}}} = {Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{ga}}}}} \right\}/{Var}\left\{ {\tan (\varphi )} \right\} = 0.55} \cr {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}} = \left[ {\tan ({\varphi _1}) \times {{\rm{G}}_1} + \tan ({\varphi _2}) \times ({{\rm{G}}_2} + {\rm{Q}})} \right] \times \cos (\alpha )/\left[ {({{\rm{G}}_1} + {{\rm{G}}_2}} \right.} \cr {\left. { + {\rm{Q}}) \times \cos (\alpha )} \right] = 0.916 \times \tan ({\varphi _1}) + 0.804 \times \tan ({\varphi _2})} \cr {{Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\} = 0.0112 = 0.716 \times {{0.125}^2},} \cr {{\gamma _{{\rm{sa}}}} = {Var}\left\{ {{{\left[ {\tan (\varphi )} \right]}_{{\rm{sa}}}}} \right\}/{Var}\left\{ {\tan (\varphi )} \right\} = 0.72 > 0.55.} \cr } Moreover, [c]a=[c]ga=[c]sa=(c1×L1+c2×L2)/(L1+L2)=0.511×c1+0.489×c2,Var{[c]a}=8.803=0.550×4.02,γa=γga=γsa=Var{[c]a}/Var{c}=0.55. \matrix{ {{{\left[ c \right]}_{\rm{a}}} = {{\left[ c \right]}_{{\rm{ga}}}} = {{\left[ c \right]}_{{\rm{sa}}}} = \left( {{c_1} \times {{\rm{L}}_1} + {c_2} \times {{\rm{L}}_2}} \right)/({{\rm{L}}_1} + {{\rm{L}}_2}) = } \cr {0.511 \times {c_1} + 0.489 \times {c_2},} \cr {{Var}\left\{ {{{\left[ c \right]}_a}} \right\} = 8.803 = 0.550 \times {{4.0}^2},{\gamma _{\rm{a}}} = {\gamma _{{\rm{ga}}}} = {\gamma _{{\rm{sa}}}} = } \cr {{Var}\left\{ {{{\left[ c \right]}_a}} \right\}/{Var}\left\{ c \right\} = 0.55.} \cr } New ‘macro’ correlation coefficients are negative and they equal:

–0.33 between the random variable [tan(φ)]ga and the random variable [c]a and

–0.28 between the random variable [tan(φ)]sa and the random variable [c]a.

Var{T([tan(φ)]ga;[c]ga} = 1040, Var{T([tan(φ)]sa;[c]sa} = 1310, Var{T(tan(φ;c} = 1975; therefore, the definition in equation (19) results in γsa = 1310/1975 = 0.66 > γga = 1040/1975 = 0.53.

Random factors of safety

Since the wedge stability is analysed, the randomness of the factor of safety (FS) requires a further numerical study. The standard FS is defined using the stabilising frictional–cohesive force Tstab and the destabilising shearing one Tdst: FS=TstabTdst=G1cos(α)tan(φ1)+(G2+Q)cos(α)tan(φ2)+L1c1+L2c2G1sin(α)+(G2+Q)sin(α) \matrix{ {FS = {{{T_{{\rm{stab}}}}} \over {{T_{{\rm{dst}}}}}} = } \cr {{{{G_1} \cdot \cos (\alpha ) \cdot \tan ({\varphi _1}) + ({G_2} + Q) \cdot \cos (\alpha ) \cdot \tan ({\varphi _2}) + {L_1} \cdot {c_1} + {L_2} \cdot {c_2}} \over {{G_1} \cdot \sin (\alpha ) + ({G_2} + Q) \cdot \sin (\alpha )}}} \cr } By analogy, the dimensionless variance reduction factors are expressed as a fraction of Var{FS(tan(φ);c)}, where FS(tan(φ);c) is calculated using the point variances of the random field, that is, σtan(φ) = 0.125 and σc = 4.0 kPa.

1) Reference case No. 1: the model of two random variables, not averaged.

Ignoring the spatial randomness, as in the denominator of equation (19), so for the model of two correlated random variables tan(φ1) = tan(φ2) = tan(φ) and c1 = c2 = c (no spatial decay function, ρ ≡ 1), equation (20a) becomes: FS=TstabTdst=[G1cos(α)+(G2+Q)cos(α)]tan(φ)+(L1+L2)cG1sin(α)+(G2+Q)sin(α)=tan(φ)tan(α)+(L1+L2)cG1sin(α)+(G2+Q)sin(α) \matrix{ {FS = {{{T_{{\rm{stab}}}}} \over {{T_{{\rm{dst}}}}}} = {{\left[ {{G_1} \cdot \cos (\alpha ) \cdot + ({G_2} + Q) \cdot \cos (\alpha )} \right] \cdot \tan (\varphi ) + ({L_1} + {L_2}) \cdot c} \over {{G_1} \cdot \sin (\alpha ) + ({G_2} + Q) \cdot \sin (\alpha )}} = } \cr {{{\tan (\varphi )} \over {\tan (\alpha )}} + {{({L_1} + {L_2}) \cdot c} \over {{G_1} \cdot \sin (\alpha ) + ({G_2} + Q) \cdot \sin (\alpha )}}} \cr } In the considered case FS = FS(tan(φ);c) = 2.25×tan(φ) + 0.0649×c; so, E{FS(tan(φ);c)} = 1.77 and Var{FS(tan(φ);c)} = 0.1030.

2) Reference case No. 2: the exact model with two random fields (two random vectors in discretisation, that is, four random variables), not averaged (Fig. 6).

The explicit equation (20a) is used which has the following form: FS=FS(tan(φ1),tan(φ2);c1,c2)=0.441×tan(φ1)+1.80×tan(φ2)+0.0332×c1+0.0317×c2;so,E{FS(tan(φ1),tan(φ2);c1,c2)}=1.77andVar{FS(tan(φ1),tan(φ2);c1,c2)}=0.0673. \matrix{ {FS = FS(\tan ({\varphi _1}),\tan ({\varphi _2});{c_1},{c_2}) = 0.441 \times \tan ({\varphi _1}) + 1.80 \times \tan ({\varphi _2})} \cr { + 0.0332 \times {c_1} + 0.0317 \times {c_2};{\rm{so}},\,\,E\left\{ {FS(\tan ({\varphi _1}),\tan ({\varphi _2});{c_1},{c_2})} \right\} = } \cr {1.77\,{\rm{and}}\,\,{Var}\left\{ {FS(\tan ({\varphi _1}),\tan ({\varphi _2});{c_1},{c_2})} \right\} = 0.0673.} \cr } The exact variance reduction factor γFS = 0.0673/0.1030 = 0.65.

3) The simplified model of geometrical averaging [.]ga: reduction from two random fields (two random vectors) to two random variables.

Using the geometrical averaging, thus the random variable [tan(φ)]ga which is calculated along AD, DC in section 9.2: FS=TstabTdst=G1cos(α)[tan(φ)]ga+(G2+Q)cos(α)[tan(φ)]ga+(L1+L2)[c]gaG1sin(α)+(G2+Q)sin(α) \matrix{ {FS = {{{T_{{\rm{stab}}}}} \over {{T_{{\rm{dst}}}}}} = } \cr {{{{G_1} \cdot \cos \left( \alpha \right) \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{ga}}}} + \left( {{G_2} + Q} \right) \cdot \cos \left( \alpha \right) \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{ga}}}} + \left( {{L_1} + {L_2}} \right) \cdot {{\left[ c \right]}_{\rm{ga}}}} \over {{G_1} \cdot \sin \left( \alpha \right) + \left( {{G_2} + Q} \right) \cdot \sin \left( \alpha \right)}}} \cr } where [c]ga = [c]a = (L1×c1 + L2×c2)/(L1 + L2).

In the considered case, FS = FS([tan(φ)]ga;[c]ga) = 1.15×tan(φ1) + 1.10×tan(φ2) + 0.0332×c1 + 0.0317×c2; so, E{FS([tan(φ)]ga;[c]ga)} = 1.77 and Var{FS([tan(φ)]ga;[c]ga)} = 0.0543.

The calculated variance reduction factor γFS = 0.0543/0.1030 = 0.53 underestimates the exact result γFS = 0.65.

4) The simplified model of stress-weighted averaging [.]sa: reduction from two random fields (two random vectors) to two random variables.

Using the stress-weighted averaging, thus the random variable [tan(φ)]sa which is calculated along AD, DC in section 9.2: FS=TstabTdst=G1cos(α)[tan(φ)]sa+(G2+Q)cos(α)[tan(φ)]sa+(L1+L2)[c]saG1sin(α)+(G2+Q)sin(α) \matrix{ {FS = {{{T_{{\rm{stab}}}}} \over {{T_{{\rm{dst}}}}}} = } \cr {{{{G_1} \cdot \cos \left( \alpha \right) \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{sa}}}} + \left( {{G_2} + Q} \right) \cdot \cos \left( \alpha \right) \cdot {{\left[ {\tan \left( \varphi \right)} \right]}_{{\rm{sa}}}} + \left( {{L_1} + {L_2}} \right) \cdot {{\left[ c \right]}_{{\rm{sa}}}}} \over {{G_1} \cdot \sin \left( \alpha \right) + \left( {{G_2} + Q} \right) \cdot \sin \left( \alpha \right)}}} \cr } where [c]sa = [c]a = (L1×c1 + L2×c2)/(L1 + L2).

In the considered case, FS = FS([tan(φ)]sa;[c]sa) = 0.441×tan(φ1) + 1.80×tan(φ2) + 0.0332×c1 + 0.0317×c2; so, E{FS([tan(φ)]sa;[c]sa)} = 1.77 and Var{FS([tan(φ)]sa;[c]sa)} = 0.0673.

The obtained variance reduction factor γFS = 0.0673/0.1030 = 0.65 confirms the exact solution from previous reference case No. 2. No probabilistic information is lost during the stress-weighted averaging.

All the approaches are unbiased, that is, E{FS} = 1.77 = const; the presented results reveal that the model of two random variables (ignoring spatial variability) overestimates the exact variance of 53% (0.103/0.0673 = 1.53), the [.]ga model underestimates the exact variance of 20% (0.0543/0.0673 = 0.80) and the [.]sa model is exact (0.0673/0.0673 = 1). For weakly correlated random fields, especially for non-cohesive materials, the differences are more significant.

Note that for n discrete elements, n > 2, and for local angles α1, …, αn, the presented example of the wedge stability has a direct generalisation to the method of slices by Fellenius or Bishop.

Summary and conclusions

There exist three basic sources of subsoil randomness: the inherent randomness in situ (including meso- or macro-inhomogeneity trends), measurement errors and possible (mis)interpretations or data transformation uncertainty; the paper focuses on a fourth factor – the field of stresses in a subsoil, which can amplify (or suppress) some spatial random fluctuations of subsoil parameters.

The basic statistical measures of subsoil parameters’ variability, that is, the coefficient of variation, auto- and crosscorrelation lengths (scales of fluctuations d), are reported in numerous papers, but the results are often divergent; this is caused by mostly unknown geological processes (sedimentation, overloading, 3D consolidation, weathering), different particle morphology, water content, soil testing methodology, etc. Generally, there is a wide margin of uncertainty and only some trends or provisional intervals can be concluded from literature studies – until a local programme of subsoil investigations is implemented.

For practical reasons, in the context of simple design situations, it is recommended to replace the vectorial random field (tan(φ(x));c(x)) by an equivalent random vector consisting of spatially averaged values ([tan(φ)]a;[c]a).

If the strength parameters of the Coulomb material can be analysed as homogeneous random fields (tan(φ(x));c(x)), then the spatial auto- and cross-correlations ρ < 1 reduce the variance of the random shearing resistance T due to interactions between adjacent regions; the reduction effects are significant if the scale of fluctuations δ (m) is short – in vertical direction most frequently.

As far as frictional materials are considered, the method of averaging proposed by E. Vanmarcke is controversial because it has no formalised background and the volume averaging is not representative; it overestimates the variance reduction effects, and hence, it overestimates the structural safety and can be dangerous in risk assessment. The same can be concluded for settlements and serviceability limit states.

The paper presents a rational alternative defined as the stress-weighted averaging, which is a more general approach; only for perfectly cohesive soils or uniform loadings, both methods coincide.

Several simple examples illustrate the proposed averaging procedures and reveal that the averaging is problem dependent; moreover, even for the same design situation, like in the example of wedge stability, the analyses are sensitive to basic deterministic parameters (sliding angle α and others).

The conclusions are true for a much wider class of design situations; for example, the standard Fellenius-type slope stability analysis is a straightforward multidimensional generalisation of the presented numerical example (wedge stability), using the stress-weighted averaging along cylindrical surfaces and local angles αi.

Figure 1

(a) Prototype defined by two random variables tan(φi), i = 1, 2. (b) Variance reduction factors γa(w1), depending on the dimensionless weight coefficient w1.
(a) Prototype defined by two random variables tan(φi), i = 1, 2. (b) Variance reduction factors γa(w1), depending on the dimensionless weight coefficient w1.

Figure 2

The dimensionless variance reduction factors γsa(S) depending on the correlation parameter d (m) and the stress parameter S (m).
The dimensionless variance reduction factors γsa(S) depending on the correlation parameter d (m) and the stress parameter S (m).

Figure 3

The dimensionless variance reduction factors γsa(e) depending on the correlation parameter d = dh (m) and the load eccentricities eB = eL = e (m).
The dimensionless variance reduction factors γsa(e) depending on the correlation parameter d = dh (m) and the load eccentricities eB = eL = e (m).

Figure 4

The effective depth zmax under considered foundation and the dimensionless variance reduction factor γsa(β) depending on the correlation parameter d = dv (m) and the footing shape ratio β.
The effective depth zmax under considered foundation and the dimensionless variance reduction factor γsa(β) depending on the correlation parameter d = dv (m) and the footing shape ratio β.

Figure 5

The dimensionless variance reduction factors γsa(β) and γga(β) for the correlation parameters dv = d = 1.5 m (solid lines) and dv = d = 0.5 m (dashed lines).
The dimensionless variance reduction factors γsa(β) and γga(β) for the correlation parameters dv = d = 1.5 m (solid lines) and dv = d = 0.5 m (dashed lines).

Figure 6

Simplified wedge stability for 2x2 random variables tan(φi), ci, i = 1, 2.
Simplified wedge stability for 2x2 random variables tan(φi), ci, i = 1, 2.

Auto- and crosscorrelation coefficients used in numerical calculations.

ρij tan(φ1) tan(φ2) c1 c2
tan(φ1) 1 +0.10 −0.30 −0.06
tan(φ2) +0.10 1 −0.06 −0.30
c1 −0.30 −0.06 1 +0.10
c2 −0.06 −0.30 +0.10 1

Auto- and cross-covariances used in numerical calculations.

Cov{Xi;Xj} tan(φ1) (−) tan(φ2) (−) c1 (kPa) c2 (kPa)
tan(φ1) (−) 0.015625 0.0015625 −0.15 −0.03
tan(φ2) (−) 0.0015625 0.015625 −0.03 −0.15
c1(kPa) −0.15 −0.03 16 1.6
c2(kPa) −0.03 −0.15 1.6 16

Bagińska, I., Kawa, M. & Janecki, W. (2016). Spatial variability of lignite mine dumping ground soil properties using CPTu results. Studia Geotechnica et Mechanica, 38(1), 3–13. BagińskaI. KawaM. JaneckiW. 2016 Spatial variability of lignite mine dumping ground soil properties using CPTu results Studia Geotechnica et Mechanica 38 1 3 13 10.1515/sgem-2016-0001 Search in Google Scholar

Brząkała, W. (1981). Randomness of subsoil parameters (in Polish). Archiwum Inżynierii Lądowej (Archives of Civil Engineering), XXVII(4), 599–606. BrząkałaW. 1981 Randomness of subsoil parameters (in Polish) Archiwum Inżynierii Lądowej (Archives of Civil Engineering) XXVII 4 599 606 Search in Google Scholar

Ching, J. & Hu, Y.-G. (2017). Effective Young's modulus for a footing on a spatially variable soil mass. Geo-Risk 2017, ASCE Conf. Proceedings, Denver, USA, 360–369. ChingJ. HuY.-G. 2017 Effective Young's modulus for a footing on a spatially variable soil mass Geo-Risk 2017, ASCE Conf. Proceedings Denver, USA 360 369 10.1061/9780784480717.034 Search in Google Scholar

Ching, J., Hu, Y.G. & Phoon, K.K. (2016). On characterizing spatially variable soil shear strength using spatial average. Probabilistic Engineering Mechanics, 45, 31–43. ChingJ. HuY.G. PhoonK.K. 2016 On characterizing spatially variable soil shear strength using spatial average Probabilistic Engineering Mechanics 45 31 43 10.1016/j.probengmech.2016.02.006 Search in Google Scholar

Ching, J. & Phoon, K.K. (2013). Mobilized shear strength of spatially variable soils under simple stress states. Structural Safety, 41, 20–28. ChingJ. PhoonK.K. 2013 Mobilized shear strength of spatially variable soils under simple stress states Structural Safety 41 20 28 10.1016/j.strusafe.2012.10.001 Search in Google Scholar

Cho, S.E. (2007). Effects of spatial variability of soil properties on slope stability. Engineering Geology, 92, 97–109. ChoS.E. 2007 Effects of spatial variability of soil properties on slope stability Engineering Geology 92 97 109 10.1016/j.enggeo.2007.03.006 Search in Google Scholar

Chwała, M. (2019). Undrained bearing capacity of spatially random soil for rectangular footings. Soils and Foundations, 59(5), 1508–1521. ChwałaM. 2019 Undrained bearing capacity of spatially random soil for rectangular footings Soils and Foundations 59 5 1508 1521 10.1016/j.sandf.2019.07.005 Search in Google Scholar

Deng, Z.P., Li, D.Q., Qi, X.H., Cao, Z.J. & Phoon, K.K. (2017). Reliability evaluation of slope considering geological uncertainty and inherent variability of soil parameters. Computers and Geotechnics, 92, 121–131. DengZ.P. LiD.Q. QiX.H. CaoZ.J. PhoonK.K. 2017 Reliability evaluation of slope considering geological uncertainty and inherent variability of soil parameters Computers and Geotechnics 92 121 131 10.1016/j.compgeo.2017.07.020 Search in Google Scholar

EN 1997-1: Eurocode 7. Geotechnical design. Part 1: General rules. EN 1997-1 Eurocode 7. Geotechnical design. Part 1: General rules Search in Google Scholar

Farah, K., Ltifi, M. & Hassis, H. (2015). A Study of probabilistic FEMs for a slope reliability analysis using the stress fields. The Open Civil Engineering Journal, 9(1), 196–206. FarahK. LtifiM. HassisH. 2015 A Study of probabilistic FEMs for a slope reliability analysis using the stress fields The Open Civil Engineering Journal 9 1 196 206 10.2174/1874149501509010196 Search in Google Scholar

Fenton, G.A. & Griffiths, D.V. (2003). Bearing-capacity prediction of spatially random c-φ soils. Canadian Geotechnical Journal, 40(1), 54–65. FentonG.A. GriffithsD.V. 2003 Bearing-capacity prediction of spatially random c-φ soils Canadian Geotechnical Journal 40 1 54 65 10.1139/t02-086 Search in Google Scholar

Griffiths, D.V. & Fenton, G.A. (2001). Bearing capacity of spatially random soil: The undrained clay Prandtl problem revisited. Géotechnique, 51(4), 351–359. GriffithsD.V. FentonG.A. 2001 Bearing capacity of spatially random soil: The undrained clay Prandtl problem revisited Géotechnique 51 4 351 359 10.1680/geot.2001.51.4.351 Search in Google Scholar

Huang, J., Lyamin, A.V., Griffiths, D.V., Sloan, S.W., Krabbenhoft, K. & Fenton, G.A. (2013). Undrained bearing capacity of spatially random clays by finite elements and limit analysis. Proceedings of the XVIII Int. Confer. on SMGE, 731–734. HuangJ. LyaminA.V. GriffithsD.V. SloanS.W. KrabbenhoftK. FentonG.A. 2013 Undrained bearing capacity of spatially random clays by finite elements and limit analysis Proceedings of the XVIII Int. Confer. on SMGE 731 734 Search in Google Scholar

Jaksa, M.B., Kaggawa, W.S. & Brooker, P.I. (1999). Experimental evaluation of the scale of fluctuation of a stiff clay. ICASP-8 Conf. Proceedings, Sydney, Australia. Vol. 1, 415–422. JaksaM.B. KaggawaW.S. BrookerP.I. 1999 Experimental evaluation of the scale of fluctuation of a stiff clay ICASP-8 Conf. Proceedings Sydney, Australia 1 415 422 Search in Google Scholar

Javankhoshdel, S. & Bathurst, R.J. (2015). Influence of cross correlation between soil parameters on probability of failure of simple cohesive and c-φ slopes. Canadian Geotechnical Journal, 2015-0109 (November), JavankhoshdelS. BathurstR.J. 2015 Influence of cross correlation between soil parameters on probability of failure of simple cohesive and c-φ slopes Canadian Geotechnical Journal 2015-0109 (November), Search in Google Scholar

Ji, J., Zhang, C., Gao, Y. & Kodikara, J. (2018). Effect of 2D spatial variability on slope reliability: A simplified FORM analysis. Geoscience Frontiers, 9(6), 1631–1638. JiJ. ZhangC. GaoY. KodikaraJ. 2018 Effect of 2D spatial variability on slope reliability: A simplified FORM analysis Geoscience Frontiers 9 6 1631 1638 10.1016/j.gsf.2017.08.004 Search in Google Scholar

Jiang, S.H., Li, D.Q., Zhang, L.M. & Zhou, C.B. (2014). Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Engineering Geology, 168, 120–128. JiangS.H. LiD.Q. ZhangL.M. ZhouC.B. 2014 Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method Engineering Geology 168 120 128 10.1016/j.enggeo.2013.11.006 Search in Google Scholar

Jiang, S.H., Li, D.Q., Cao, Z.J., Zhou, C.B. & Phoon, K.K. (2015). Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo Simulation. Journal of Geotechnical and Geoenvironmental Engineering, 141(2), 04014096. JiangS.H. LiD.Q. CaoZ.J. ZhouC.B. PhoonK.K. 2015 Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo Simulation Journal of Geotechnical and Geoenvironmental Engineering 141 2 04014096 10.1061/(ASCE)GT.1943-5606.0001227 Search in Google Scholar

Kim, J.M. & Sitar, N. (2013). Reliability approach to slope stability analysis with spatially correlated soil properties. Soils and Foundations, 53(1), 1–10. KimJ.M. SitarN. 2013 Reliability approach to slope stability analysis with spatially correlated soil properties Soils and Foundations 53 1 1 10 10.1016/j.sandf.2012.12.001 Search in Google Scholar

Li, D.Q., Qi, X.-H., Cao, Z.-J., Tang, X.-S., Zhou, W., Phoon, K.-K. & Zhou, C.-B. (2015). Reliability analysis of strip footing considering spatially variable undrained shear strength that linearly increases with depth. Soils and Foundations, 55(4), 866–880. LiD.Q. QiX.-H. CaoZ.-J. TangX.-S. ZhouW. PhoonK.-K. ZhouC.-B. 2015 Reliability analysis of strip footing considering spatially variable undrained shear strength that linearly increases with depth Soils and Foundations 55 4 866 880 10.1016/j.sandf.2015.06.017 Search in Google Scholar

Li, X.Y., Zhang, L.M., Gao, L. & Zhu, H. (2017). Simplified slope reliability analysis considering spatial soil variability. Engineering Geology, 216, 90–97. LiX.Y. ZhangL.M. GaoL. ZhuH. 2017 Simplified slope reliability analysis considering spatial soil variability Engineering Geology 216 90 97 10.1016/j.enggeo.2016.11.013 Search in Google Scholar

Liu, L.L., Cheng, Y.M. & Zhang, S.H. (2017). Conditional random field reliability analysis of a cohesion-frictional slope. Computers and Geotechnics, 82, 173–186. LiuL.L. ChengY.M. ZhangS.H. 2017 Conditional random field reliability analysis of a cohesion-frictional slope Computers and Geotechnics 82 173 186 10.1016/j.compgeo.2016.10.014 Search in Google Scholar

Liu, Y., Zhang, W., Zhang, L., Zhu, Z., Hu, J. & Wei, H. (2018). Probabilistic stability analyses of undrained slopes by 3D random fields and finite element methods. Geoscience Frontiers, 9(6), 1657–1664. LiuY. ZhangW. ZhangL. ZhuZ. HuJ. WeiH. 2018 Probabilistic stability analyses of undrained slopes by 3D random fields and finite element methods Geoscience Frontiers 9 6 1657 1664 10.1016/j.gsf.2017.09.003 Search in Google Scholar

Lloret-Cabot, M., Fenton, G.A. & Hicks, M.A. (2014). On the estimation of scale of fluctuation in geostatistics. Georisk:Assessment and Management of Risk for Engineered Systems and Geohazards, 8(2), 129–140. Lloret-CabotM. FentonG.A. HicksM.A. 2014 On the estimation of scale of fluctuation in geostatistics Georisk:Assessment and Management of Risk for Engineered Systems and Geohazards 8 2 129 140 10.1080/17499518.2013.871189 Search in Google Scholar

Low, B.K. & Phoon, K.K. (2015). Reliability based design and its complementary role to Eurocode 7 design approach. Computers and Geotechnics, 65, 30–44. LowB.K. PhoonK.K. 2015 Reliability based design and its complementary role to Eurocode 7 design approach Computers and Geotechnics 65 30 44 10.1016/j.compgeo.2014.11.011 Search in Google Scholar

Oguz, E.A., Huvaj, N., & Griffiths, D.V. (2019). Vertical spatial correlation length based on standard penetration tests. Marine Georesources & Geotechnology, 37(1), 45–56. OguzE.A. HuvajN. GriffithsD.V. 2019 Vertical spatial correlation length based on standard penetration tests Marine Georesources & Geotechnology 37 1 45 56 10.1080/1064119X.2018.1443180 Search in Google Scholar

Ostoja-Starzewski, M. (2006). Material spatial randomness. From statistical to representative volume element. Probabilistic Engineering Mechanics, 21, 112–132. Ostoja-StarzewskiM. 2006 Material spatial randomness. From statistical to representative volume element Probabilistic Engineering Mechanics 21 112 132 10.1016/j.probengmech.2005.07.007 Search in Google Scholar

Phoon, K.K. & Kulhawy, F.H. (1999a). Characterization of geotechnical variability. Canadian Geotechnical Journal, 36(4), 612–624. PhoonK.K. KulhawyF.H. 1999a Characterization of geotechnical variability Canadian Geotechnical Journal 36 4 612 624 10.1139/t99-038 Search in Google Scholar

Phoon, K.K. & Kulhawy, F.H. (1999b). Evaluation of geotechnical property variability. Canadian Geotechnical Journal, 36(4), 625–639. PhoonK.K. KulhawyF.H. 1999b Evaluation of geotechnical property variability Canadian Geotechnical Journal 36 4 625 639 10.1139/t99-039 Search in Google Scholar

Pieczyńska-Kozłowska, J.M., Puła, W. & Vessia, G. (2017). A collection of fluctuation scale values and autocorrelation functions of fine deposits in Emilia Romagna Palin, Italy. Geo-Risk 2017, ASCE Conf. Proceedings, Denver, USA, 360–369. Pieczyńska-KozłowskaJ.M. PułaW. VessiaG. 2017 A collection of fluctuation scale values and autocorrelation functions of fine deposits in Emilia Romagna Palin, Italy Geo-Risk 2017, ASCE Conf. Proceedings Denver, USA 360 369 Search in Google Scholar

Puła, W. & Chwała, M., (2015). On spatial averaging along random slip lines in the reliability computations of shallow strip foundations. Computers and Geotechnics, 68, 128–136. PułaW. ChwałaM., 2015 On spatial averaging along random slip lines in the reliability computations of shallow strip foundations Computers and Geotechnics 68 128 136 10.1016/j.compgeo.2015.04.001 Search in Google Scholar

Shen, Z., Jin, D., Pan, Q., Yang, H. & Chian, S.Ch. (2021). Effect of soil spatial variability on failure mechanisms and undrained capacities of strip foundations under uniaxial loading. Computers and Geotechnics, 139 (November), 104387. ShenZ. JinD. PanQ. YangH. ChianS.Ch. 2021 Effect of soil spatial variability on failure mechanisms and undrained capacities of strip foundations under uniaxial loading Computers and Geotechnics 139 November 104387 10.1016/j.compgeo.2021.104387 Search in Google Scholar

Tietje, O., Fitze, P. & Schneider, H.R. (2014). Slope stability analysis based on autocorrelated shear strength parameters. Geotechnical and Geological Engineering, 32, 1477–1483. TietjeO. FitzeP. SchneiderH.R. 2014 Slope stability analysis based on autocorrelated shear strength parameters Geotechnical and Geological Engineering 32 1477 1483 10.1007/s10706-013-9693-8 Search in Google Scholar

Vanmarcke, E.H. (2010). Random Fields. Analysis and Synthesis. World Scientific, Singapore. (rev. ed. of MIT Press, 1983). VanmarckeE.H. 2010 Random Fields. Analysis and Synthesis World Scientific Singapore rev. ed. of MIT Press 1983 10.1142/5807 Search in Google Scholar

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