3477
Technical Committee CFMS /
Comité technique CFMS
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
are shown in Figure 5. In these cases the horizontal and vertical
correlation lengths of both top and bottom layers were kept
constant as
. As shown in the figure, the average
bearing capacity factor for all of the stochastic cases
(represented by
) is less than the deterministic case. This is
consistent with the reports of Nobahar and Popescu (2000),
Griffiths et al. (2002) and Cassidy et al. (2012). Further, when
the COV of both layers is increased from 0.1 to 0.3 the average
bearing capacity reduces from 0.98 to 0.93 and the normalised
standard deviation increases from 0.01 to 0.02. This is as
expected and is shown in the two extremity curves of Figure 5.
Comparing the cases where the COV of only the top layer (case
3:
COV
b
=0.1,
COV
t
=0.3) and only the bottom layer (case 4:
COV
b
=0.3,
COV
t
=0.1) provides more insight into the
mechanisms of failure. We can see from Figure 5 that the COV
of the top layer has a more significant effect with case 3
trending towards case 1 where both layers are 0.3. Moreover,
the similarity of the shapes of case 2 and case 4 as well as case
3 and case 1 implies that the top layer COV determines the
variation (standard deviation) of the curves.
Figure 4 Histogram of
for case 2
Figure 5 Cumulative probability curves for varying COV of
cases 1 to 4
The output samples of stochastic bearing capacity factor
normalized by the deterministic value were analysed with the
aim of estimating the frequentist probability of exceedence of
unity, i.e., the probability that the stochastic bearing capacity
factor exceeds the deterministic bearing capacity factor. This
assessment is important in the context of engineering design, as
it provides a measure of the unconservatism in using
deterministic bearing capacity factors, i.e., in neglecting
uncertainty and spatial variability. In only 6 cases out of the 12
analyzed, output samples resulted to be lognormal at the 95%
confidence level using the Anderson-Darling test. Hence,
estimating the probability of exceedence of unity from
cumulative values of fitted lognormal samples would not allow
confident assessement for all cases. Empirical cumulative
distribution functions were calculated for each sample. The
empirical probability
P
e
of exceedence of unity for each case is
noted in the rightmost column in Table 1.
The failure mechanisms of three selected realizations of case
1 (
) are shown in Figure 6. These
represent the minimum, median and maximum
cases and
are shown alongside the deterministic failure mechanism
(uniform and mean parameter values). In all three cases the
existence of the random field results in a non-symmetric failure
mechanism, with the minimum bearing capacity case most
unsymmetrical. With the increasing of bearing capacity, the
failure mechanism tends to resemble the deterministic case. The
importance of spatial variability in the top layer can be
observed, with the majority of the failure mechanism residing in
that layer. Further, with higher variability and potential for
weaker zones the mechanism for lower bearing capacity is both
more unsymmetric and shorter (pulling it further into the top
layer).
(a) (b)
(c) (d)
Figure 6. Failure mechanisms from finite element analysis for
(a) lowest, (b) median and (c) highest bearing capacity, and (d)
deterministic uniform case (for clarity only a section of 3
B
width and depth 2
B
show)
3.3
Stochastic soil cases: variation of correlation length in
top layer
With the top layer determined to play a more significant role
in the problem configuration of this paper further concentration
on the effect of top layer correlation length is discussed. The
results for correlation length varying from 0.1B to 10B are
presented as cumulative probability curves in Figure 7. These
represent cases 5, 6, 1, 7 and 8 in Table 1. As for the
cases (cases 5, 6 and 1), the largest bearing capacity
corresponds to the largest horizontal correlation length
(case 5) while the minimum corresponds to
(case 6). This is consistent with the observation of Griffiths et
al. (2002) for the single layer case. In general, a large
correlation length results in greater standard deviation of the
bearing capacity, i.e. the foundation becomes more “non
-
