Managing risk and achieving reliable geotechnical designs using Eurocode 7 - Risk and Reliability in Geotechnical Engineering - page 406

Ching and Phoon (2013) analyzing the mobilized strength in spatially variable soils have
shown that there is a clear distinction between the spatially averaged strength along a par-
ticular failure surface and the spatial variability of the parameters within a soil mass. They
found that the physical properties of a particular design situation play an important role
in determining the trajectory of the failure surface so that the idea that the failure surface
should seek the weakest zones within the spatially variable soil may not lead to what is actu-
ally the dominant mechanism. The selection of variance reduction factors and the effects of
spatially variability is an emerging area of research.
Schneider and Schneider (2013) adopted a simplification for the variance reduction fac-
tor, which is that ΓΓΓΓ
S

2 2 2 = , where Γ 2 and Γ 2 are the variance reduction factors in the
two horizontal directions, x and y, and Γ 2 is the variance reduction factor in the vertical
direction, z. They have also adopted the following two equations from Vanmarcke (1983)
as a simplification of Vanmarcke's (1977) Equation 10.16 for Γ i 2 in a particular direction, i:

x yz

( )

δ δ 2

L i > δ

Γ i =

i

1 −

i

if (10.19)

L 3 L

i

i i

and

L

2

Γ i =− 1

i

if L i ≤ δ

(10.20)

3 δ

i

i

where δ i and L i are the SOF and the extent of the failure mechanism in the direction i,
respectively. Comparing the Γ i 2 values obtained using Equations 10.19 and 10.20 with those
obtained using Equation 10.16 it is found that they are similar, but more cautious, particu-
larly for Li i values close to δ i .
If the measurement of the soil parameters is carried out using accurate equipment and
strictly in accordance with the relevant testing standards, the measurement errors should be
small so that COV meas 2 ≅ . Similarly, if a well-established model is used to transform the
measured test results into the required parameter, then COV trans 2 ≅ , and if the parameters
required to describe the statistical distribution are known with reasonable accuracy from
experience with similar soils, then COV stat 2 ≅ . Hence, assuming a normal distribution
and taking into account the above comments regarding the COV values, substituting the
COV total from Equation 10.10 for COV in Equation 10.8 , gives:

2

(10.21)

XX = ( 11645 − . COV Γ )

k mean inher S

whereas for a log-normal distribution, the characteristic value is given by

ln ( 1 + Γ

2

COV

2

)

S inher

XX = 02 . (10.22)

k mean

1 + Γ

2

COV

2

S inher

Schneider and Schneider (2013) gave ranges of typical values for COV inher and recom-
mended the values shown in Table 10.4 . When COV inher < 0.3 for a parameter, for example,
tan ϕ′, a normal distribution may be assumed and, hence, Equation 10.21 may be used to
determine the characteristic value. However, Schneider and Schneider (2013) stated that
when COV ≥ 0.3, for example, for c′ and c u , a log-normal distribution should be assumed