Environmental Engineering Reference
In-Depth Information
If statistical methods are used to select the characteristic value of a geotechnical parame-
ter, Eurocode 7 states in §2.4.5.2(11) that it should be derived such that the calculated prob-
ability of a worse value governing the occurrence of the limit state under consideration is
not greater than 5%. A note to this clause states that a cautious estimate of the mean value
is a selection of the mean value of the limited set of geotechnical parameter values, with a
confidence level of 95%; where a local failure is concerned, a cautious estimate of the low
value is a 5% fractile. It should be noted that the 5% fractile of test results on small samples
normally gives too conservative of an estimate of the characteristic value.
Considering a geotechnical parameter as aleatory, if the characteristic value, X k of a param-
eter is the 5% fractile of an unlimited series of tests with a normal distribution, it is given by
X k = X mean -1.645σ = X mean (1-1.645 COV)
(10.8)
where X mean is the mean value, σ the standard deviation, COV the coefficient of variation,
and 1.645 the factor defining the 5% fractile for a normal distribution, that is, the Student's
t value. This equation usually gives too cautious a characteristic value of a geotechnical
parameter because, as noted above, the relevant value is the 95% confidence in the mean
value controlling the occurrence of a limit state, rather than the 5% fractile of the test
results. Hence, Schneider (1997) proposed the following equation for selecting the charac-
teristic geotechnical parameter value from a series of test results:
X k = X mean (1- 0.5 COV)
(10.9)
This equation aims to provide a characteristic value that is a cautious estimate of the
mean value with a confidence level of 95% and results in a value that is much less conserva-
tive than the 5% fractile value given by Equation 10.8 .
To account explicitly for the inherent variability and other uncertainties affecting the
characteristic value, the COV in Equation 10.8 can be replaced by the following additive
total COV, COV total proposed by Phoon and Kulhawy (1999):
2
2
2
2
2
COV total
=
Γ S
COV
+
OV
+
COV
+
OV
meas
trans
stat
inher
(10.10)
where
Γ 2 is the variance reduction function (Vanmarcke, 1977) considering the spatial extent
of the governing failure mechanism
COV inher is the coefficient of variation of the soil's inherent variability,
COV meas is the coefficient of variation of the measurement errors,
COV trans is the coefficient of variation of the transformation errors, and
COV stat is the coefficient of variation of the statistical parameters.
For a 1D potential slip line in a 2D problem,
2
2
2
Γ S
=
[
2
L
/
δ
1
+
exp(
2
L
/
δ
)] (
/
2
L
δ
)
(10.11)
v
v
v
v
v
v
where δ v is the SOF along the potential slip line direction, v, and L v is the length of the poten-
tial slip line in the y-z-plane. If the potential slip line is inclined at an angle β to the horizontal,
1
(10.12)
δ
=
v
cos( )
βδ
/
+
sin()
βδ
/
y
z
where δ y and δ z are the SOFs in the horizontal and vertical directions, respectively.
 
 
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