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 = _{n}×tan(φ) for the frictional Coulomb material – and analyses the variance reduction coefficients γ_{a} for the homogeneous random field tan(φ(_{a}. Fields of stresses in the subsoil mass implement a new element to the averaging procedures because non-uniform (deterministic) normal stresses _{n} ≠ const(_{n} along a slip line is locally negligible, then τ = _{n}×tan(φ) is also locally negligible and such a region does not contribute to total bearing capacity. If locally _{n} ∼ 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 _{n} ≫ 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(_{n}(

It is assumed that the spatial field of the normal stress _{n} (_{n}(_{n}(

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).

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

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 ν_{tan(φ)} ∼ 5%–15% (non-cohesive soils), ν_{tan(φ)} ∼ 25% (cohesive soils, even up to 50% is possible; slightly less for the angle φ itself), for the cohesion ν_{c} ∼ 20%–30% (for undrained shear strength, even up to 70% is possible) and for the deformation modulus ν_{D} ∼ 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 < _{tan(φ);c} < 0 and the value _{tan(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 _{u} > c′, φ_{u} ∼ 0^{o} < 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 ^{2} = const(

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 _{a} = γ_{a}×σ^{2} with a reduction coefficient γ_{a} ≤ 1; equivalently, but in terms of standard deviations σ_{a} = Γ_{a}×σ, therefore, the symbol Γ_{a} = √γ_{a} is also in use (Vanmarcke 2010, Tietje et al. 2011).

Consider a homogeneous random field of a soil parameter (or several soil parameters in a vectorial version) denoted as _{1}, …, _{n}) belongs to a domain ^{n} in ^{2} = const < + ∞; the covariance function equals ^{2}×ρ(_{c}(

Joint probability distributions of the random field

Several types of the decay functions ρ(Δ_{3}, _{3}, _{v} are used for vertical direction. Alternatively, the linear exponential (Markovian) function (equation 1b) is in use:

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{−|Δ

The decay constants _{v}, _{h} (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 δ_{i} (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:
_{e}(_{o} such that ρ_{e}(_{o}) ≈ 0. In terms of both exponential functions ρ(_{o}_{o}_{o} and its confidence limits were presented by Jaksa et al. (1999). For

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 _{c}(

In the common opinion δ_{h} ≫ δ_{v}, like the ratio δ_{h}/δ_{v} = 10 used by Tietje et al. (2014); such proportion δ_{h} ≫ δ_{v} 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 _{c} for which δ_{v} ∼ 0.17 m (average value) with a c.o.v. on the level of 30%. Independently, another value _{o} ≅ 0.28 m can be estimated from the authors’ empirical plot of ρ_{e}(_{v} = √π_{v}=√π_{o}_{o} ≅ 0.28 m. Similar estimation δ_{v} ∼ 0.8×5.3/20 = 0.21 m can be found from the same data using the Vanmarcke proposal δ_{v} ∼ 0.8_{c} and its parabolic trend function (Vanmarcke 2010, Jaksa et al. 1999). The horizontal scale of fluctuations δ_{h} was also analysed by Jaksa et al. (1999) and its nested structure was documented – the estimated results depend on the scale of sampling: δ_{h} ∼ 0.15 m for CPT sampling at 5 mm intervals (so, δ_{h} ≅ δ_{v}), but δ_{h} ∼ 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 δ_{v} ∼ 1.0–2.0 m (less than 1.0 m only for CPT tests), but δ_{h} is expressed in dozens of metres; so, the ratio δ_{h}/δ_{v} ∼ 10 is confirmed. The references presented by Shen et al. (2021) for the offshore soil in the North Sea report δ_{v} ∼ 0.48–7.14 m and δ_{h} ∼ 24.62–66.48 m.

For the illustrative numerical examples presented hereafter, the range of _{v} = δ_{v}/√π ∼ 0.5–1.5 m and the range of _{h} = δ_{h}/√π ∼ 10–20 m can be assumed as some representative values.

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 _{1}) on the left part _{2}) on the right one.

Looking for one ‘effective’ parameter, the random vector (tan(φ_{1});tan(φ_{2})) is replaced by one averaged random variable tan(φ)_{ga} acting along 2_{i})} = μ, _{i})} = σ^{2}; moreover, _{1});tan(φ_{2})} = σ^{2}×ρ_{1;2}. The values _{ga} on 2_{ga}} = μ, _{ga}} = σ^{2}×(1 + ρ_{1;2})/2 < _{i}) = σ^{2}. The variance reduction factor equals γ_{ga} = _{ga}}/_{i}) = (1 + ρ_{1;2})/2 = 1 −(1 − ρ_{1;2})/2; the lower index _{ga} in equation (3) can even disappear, if always both random fluctuations of tan(φ_{1}) and tan(φ_{2}) happen as opposite numbers, thus taking the hypothetic correlation coefficient ρ_{1;2} = −1.

Generalisations of the geometric averaging are straightforward, if for a longer interval

Ultimate limit state for random shearing is a start point of this approach due to the context of the random vector (tan(φ_{1});tan(φ_{2})). For the frictional material, the two variables tan(φ_{1}), tan(φ_{2}) should be considered in conjunction with two normal deterministic loadings in Fig. 1a, _{n1} = const > 0, _{n2} = const > 0. The random shearing resistance along _{sa} on the same interval _{i} > 0 equal _{1} = _{n1}/(_{n1} + _{n2}), _{2} = _{n2}/(_{n1} + _{n2}), _{1} + _{2} = 1.

In particular, the local contribution of tan(φ_{1}) is negligible if _{n1} ≪ _{n2}, _{1} ≪ _{2}, so the randomness is localised, almost not averaged. If a (potentially possible) physical dependence between _{n} and tan(φ) is ignored, then _{sa}} = _{1}×μ + _{2}×μ = μ and _{sa}} = [1 − 2×_{1}×_{2}×(1 − ρ_{1;2})]×σ^{2}.

Therefore, the variance reduction factor equals γ_{sa} = _{sa}}/σ^{2} = 1 − 2×_{1}×(1 − _{1})×(1 − ρ_{1;2}); the lower index _{sa} defined above is sensitive to _{n} gradients and depends on the ratio _{n1}/_{n2}, what is not true for the variance reduction factor γ_{ga} = 1 −(1 − ρ_{1;2})/2, which is stress-independent. The following inequality γ_{ga} ≤ γ_{sa} is true, but γ_{ga} = γ_{sa} if and only if _{1} = _{2} = 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 γ_{ga} ≤ γ_{sa} can be dangerous in safety evaluation. Parabolic functions γ_{sa}(_{1}) are depicted in Fig. 1b for five different correlation coefficients ρ = ρ_{1;2}.

Passage to a discrete multiterm sum for ^{2} = const. The shearing resistance _{sa} = const(_{sa}; if equations (7a) and (7b) are compared, the following definition can be obtained:
^{2} = min{1; δ/|

Only if _{n} = const along

Simple numerical results are presented in Fig. 2; a fixed interval _{n}(_{n}(_{n}(^{2}}, 0 ≤ _{ga} < γ_{sa} elsewhere; the geometric spatial averaging is insensitive to the length

As a practical example, consider the standard GEO-sliding stability condition (EN 1997-1) for a massive shallow foundation _{1} ≤ +_{2} ≤ +_{n}(_{1},_{2}) = _{n}(_{B} and _{L}, respectively.

In the random variable format, the friction coefficient tan(φ) in the expression ^{2}, so _{2}σ^{2}. For a second-order homogeneous isotropic random field tan(φ(_{1},_{2})) = tan(φ(_{3} (or simply

The presented numerical example employs the horizontally isotropic Gaussian autocorrelation function ρ(_{1} − _{1})/_{h}]^{2} − [(_{2} − _{2})/_{h}]^{2}}; following the recommendations of the Eurocode 7 (EN 1997-1), the vertical contact stress _{n} under the footing can be assumed as a double linear function like _{n}(_{o} + _{1}_{1} + _{2}_{2} (kPa). Consider a symmetrical case _{B} = _{L} =

If both eccentricities 0 ≤ _{1} = 100 + 120×_{2} = _{3} = 100, _{4} = 100 − 120×_{n}(_{1} + _{2}) (kPa).

If both eccentricities _{n}(_{1} + _{2})} (kPa).

The values of the variance reduction factor γ_{sa} are depicted in Fig. 3 for

The geometric spatial averaging and the variance reduction factor γ_{ga} do not depend on the eccentricities _{h}:

for _{h} = 10 m, the variance reduction γ_{sa} is on the level of 74%–87%, whereas γ_{ga} = 74%;

for _{h} = 20 m, the variance reduction γ_{sa} is only on the level of 92%–97%, whereas γ_{ga} = 92% and

for _{h} = 30 m, the variance reduction γ_{sa} is only on the level of 96%–99%, whereas γ_{ga} = 96%.

The results are close to each other and for _{h} ≥ 20 m, both horizontal variance reductions are not significant; still much greater _{h}, that is, if _{h}/

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 _{3} =

The following equivalent [_{sa} = const(_{sa}.

Equations (13a) and (13b) are focused on a finite interval 0 ≤ _{max} under a shallow footing; taking into account a founding depth _{f} under the ground surface, the value of _{max} can be estimated from the equation γ×(_{f}/_{max}/_{n}(_{max}) (EN 1997-1), where g×(_{f} + ^{3}) is the subsoil unit weight. The average vertical stress _{n} under a rectangular footing _{n}(_{max} depends on the β coefficient as in Fig. 4. It is assumed that random fluctuations of _{n}(

The stochastic parameter ^{2} = const(_{1});_{2})} = σ^{2}×ρ(_{1} − _{2}) cannot be derived from the corresponding moments _{1});_{2})} – the model should be completed by some joint probability distributions of the stochastic process _{1}), _{2}) can be useful. Alternatively, the first-order perturbation is acceptable for a ‘small’ variance of

Equations (13a) and (13b) yield the variance of the averaged deformation modulus _{sa}} and then the variance reduction factor equals:
_{h} ≫ _{v}).

The numerical example in Fig. 4 uses the Gaussian autocorrelation function ρ(_{1};_{2}) = exp{−[(_{1} − _{2})/_{v}]^{2}}, 0 ≤ _{1}, _{2} ≤ _{max}; the subsoil unit weight γ = 18 kN/m^{3}, _{f}/_{max}/

The geometric variance reduction factor γ_{ga} in Fig. 5 is also derived from equation (14) for the same _{max}, but for _{n} = const(_{sa} and γ_{ga} in Fig. 5 are significant.

For perfectly cohesive materials, the shearing resistance in Fig. 1a equals 2_{1}×_{2}×_{a}×_{a}×_{a} = (_{1} + _{2})/2. For such a case, the geometric spatial averaging (Vanmarcke 2010) is representative because the model is independent of the normal stress _{n} > 0. There is only one averaged cohesion:
_{a} = γ_{sa} = γ_{ga}.

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 _{n} > 0); to a certain extent, for a rigid foundation, the model is not completely independent of _{n}, so γ_{ga} can be less than γ_{sa} also for perfectly cohesive soils. This aspect is not analysed below.

Although the Tresca model with tan(φ) ≡ 0, _{n}(_{tan(φ)}×σ_{c}×_{tan(φ);c}×ρ_{tan(φ);c}(_{tan(φ)}(_{c}(_{tan(φ);c}(_{n}(_{n}(_{n}(_{tan(φ)} + μ_{c}/_{n}(

Alternatively, the vectorial random field (tan(φ(_{sa};[_{a}) by making use of the previously defined averages. Depending on a stress level of _{n}(_{sa} or [_{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

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: _{1} = tan(φ_{1}), _{2} = tan(φ_{2})_{,} _{3} = _{1}, _{4} = _{2}; the sliding wedge is split into two homogeneous triangles ADB and DCB.

The randomly inhomogeneous material corresponds to a silty-clayey sand. The second-order moments are assumed as follows: _{i})} = 0.5, σ_{tan(φ}_{i)} = 0.125, ν_{tan(φ}_{i)} = 25%, _{i}} = 10.0 kPa, σ_{ci} = 4.0 kPa, ν_{ci} = 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.

_{i};_{j}} |
_{1}) (−) |
_{2}) (−) |
_{1} (kPa) |
_{2} (kPa) |
---|---|---|---|---|

tan(φ_{1}) (−) |
0.015625 | 0.0015625 | −0.15 | −0.03 |

tan(φ_{2}) (−) |
0.0015625 | 0.015625 | −0.03 | −0.15 |

_{1}(kPa) |
−0.15 | −0.03 | 16 | 1.6 |

_{2}(kPa) |
−0.03 | −0.15 | 1.6 | 16 |

Auto- and crosscorrelation coefficients used in numerical calculations.

_{ij} |
_{1}) |
_{2}) |
_{1} |
_{2} |
---|---|---|---|---|

tan(φ_{1}) |
1 | +0.10 | −0.30 | −0.06 |

tan(φ_{2}) |
+0.10 | 1 | −0.06 | −0.30 |

_{1} |
−0.30 | −0.06 | 1 | +0.10 |

_{2} |
−0.06 | −0.30 | +0.10 | 1 |

The second-order moments of the random variables tan(φ_{i}), _{j} are assumed as the discrete input data; if they are discretised from the continuous spatial moments, like

The obtained averaged values of the friction coefficients are as follows:

–0.33 between the random variable [tan(φ)]_{ga} and the random variable [_{a} and

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

_{ga};[

_{ga}} = 1040,

_{sa};[

_{sa}} = 1310,

_{sa}= 1310/1975 = 0.66 > γ

_{ga}= 1040/1975 = 0.53.

Since the wedge stability is analysed, the randomness of the factor of safety (_{stab} and the destabilising shearing one _{dst}:
_{tan(φ)} = 0.125 and σ_{c} = 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(φ_{1}) = tan(φ_{2}) = tan(φ) and _{1} = _{2} =

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} = 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:
_{ga} = [c]_{a} = (_{1}×_{1} + _{2}×_{2})/(_{1} + _{2}).

In the considered case, _{ga};[_{ga}) = 1.15×tan(φ_{1}) + 1.10×tan(φ_{2}) + 0.0332×_{1} + 0.0317×_{2}; so, _{ga};[_{ga})} = 1.77 and _{ga};[_{ga})} = 0.0543.

The calculated variance reduction factor γ_{FS} = 0.0543/0.1030 = 0.53 underestimates the exact result γ_{FS} = 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:
_{sa} = [_{a} = (_{1}×_{1} + _{2}×_{2})/(_{1} + _{2}).

In the considered case, _{sa};[_{sa}) = 0.441×tan(φ_{1}) + 1.80×tan(φ_{2}) + 0.0332×_{1} + 0.0317×_{2}; so, _{sa};[_{sa})} = 1.77 and _{sa};[_{sa})} = 0.0673.

The obtained variance reduction factor γ_{FS} = 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, _{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 _{1}, …, α_{n}, the presented example of the wedge stability has a direct generalisation to the method of slices by Fellenius or Bishop.

There exist three basic sources of subsoil randomness: the inherent randomness

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(_{a};[_{a}).

If the strength parameters of the Coulomb material can be analysed as homogeneous random fields (tan(φ(

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 α_{i}.

#### Auto- and crosscorrelation coefficients used in numerical calculations.

_{ij} |
_{1}) |
_{2}) |
_{1} |
_{2} |
---|---|---|---|---|

tan(φ_{1}) |
1 | +0.10 | −0.30 | −0.06 |

tan(φ_{2}) |
+0.10 | 1 | −0.06 | −0.30 |

_{1} |
−0.30 | −0.06 | 1 | +0.10 |

_{2} |
−0.06 | −0.30 | +0.10 | 1 |

#### Auto- and cross-covariances used in numerical calculations.

_{i};_{j}} |
_{1}) (−) |
_{2}) (−) |
_{1} (kPa) |
_{2} (kPa) |
---|---|---|---|---|

tan(φ_{1}) (−) |
0.015625 | 0.0015625 | −0.15 | −0.03 |

tan(φ_{2}) (−) |
0.0015625 | 0.015625 | −0.03 | −0.15 |

_{1}(kPa) |
−0.15 | −0.03 | 16 | 1.6 |

_{2}(kPa) |
−0.03 | −0.15 | 1.6 | 16 |

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