3476
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
B
6
B
4.8
B
B
Top layer
Bottom layer
utilised (version 6.10, Dassault Systèmes 2010). A shallow
foundation with width
B
was founded on the surface of the two-
layered soil, which was modelled by a linear-elastic perfectly-
plastic Tresca constitutive law with an undrained shear strength
(
s
u
). The elastic response wa
s defined by the Young’s modulus
(
E =
500
s
u
) and t
he Poisson’s ratio set as 0.49
. Corresponding
to one of the analysis cases of Merifield et al. (1999), the soil
contained a top layer of 1
B
thickness. For efficiency the infinite
bottom layer of Merifield et al (1999). was shortened to 3.8
B
; a
depth deep enough, however, to ensure no boundary effects.
The analysis width was 6
B
. The lateral soil boundaries were
roller supported and the bottom was pinned. The top surface
was assumed to be free. A fully bonded foundation/soil
interface was used to model the undrained behaviour.
Figure 2. The FE model used
The soil domain was divided into 60 by 48 square zones of
width 0.1
B
, as shown in Figure 2. In each zone the soil
properties were constant and defined by an undrained shear
strength
and Young’s modulus
. However, these
properties changed from zone to zone representing the spatial
variability of the soil. For the majority of the soil domain a zone
was represented by one finite element. However, in a region of
size 3B by 1B close to the strip footing (as bounded by heavy
lines in Figure 2) nine smaller finite elements per zone were
used. These smaller elements, each with the same material
properties, were required to improve the numerical accuracy of
the solution. Therefore, in total there are 5280 finite elements in
the mesh but only 2880 zones of spatially varying soil
properties.
The spatially variable undrained shear strength
of both
top and bottom layer was modelled as a normally distributed
random field with a mean
and standard deviation
. Consistent with the deterministic values of Merifield
et al. (1999), the mean shear strength of the top layer was set as
twice the bottom layer, with values of
and
assumed in this paper. The COV, vertical and
horizontal correlation length
and
for both top and bottom
layer vary systematically. Table 1 details the random variables
assumed for the 12 cases presented.
For each case, 1000 realisations of the random fields of
undrained shear strength
were generated using the Local
Average Subdivision algorithm (Fenton and Vanmarcke 1990;
Fenton 1994). One of the 1000 realisations of the random field
of case 1 (
see Table 1 for details) is illustrated in Figure 3.
3 RESULTS
3.1
Deterministic Case
The modified bearing capacity factor
was
defined in Merifield et al. (1999) as the ultimate bearing
capacity
Q
u
normalised by the footing width
B
and top layer
shear strength
s
ut
. Merifield et al. (1999) reported
as 4.44
(lower bound), 4.82 (upper bound) and 4.63 (average) for the
situation considered in Figure 2. A deterministic case was first
conducted in this paper with uniform undrained strengths of 20
kPa and 10 kPa for the top and bottom layer, respectively. An
of 4.66 was obtained. This good agreement implies that the
FE analyses in this paper are reliable and comparable to the
Merifield et al. (1999) analyses.
Figure 3. Example random field (for case 1)
Table 1. Calculation cases and summary results
Case
Input parameters
Analysis results
Bottom layer
Top layer
(
)
1
0.1 0.1 0.3 0.1 0.1 0.3 0.93
0.02
5.0∙10
-4
2
0.1 0.1 0.1 0.1 0.1 0.1 0.98
0.01
1.3∙10
-3
3
0.1 0.1 0.1 0.1 0.1 0.3 0.95
0.02
5.6∙10
-3
4
0.1 0.1 0.3 0.1 0.1 0.1 0.96
0.01
1.0∙10
-4
5
0.1 0.1 0.3 0.1 10 0.3 0.94
0.07
0.144
6
0.1 0.1 0.3 0.1 1
0.3 0.93
0.05
0.057
7
0.1 0.1 0.3 1
10 0.3 0.93
0.18
0.277
8
0.1 0.1 0.3 1
1
0.3 0.89
0.12
0.133
9
0.1 10
0.3 0.1 0.1 0.3 0.92
0.04
0.022
10
0.1 1
0.3 0.1 0.1 0.3 0.92
0.03
4.0∙10
-4
11
1
10
0.3 0.1 0.1 0.3 0.90
0.09
0.149
12
1
1
0.3 0.1 0.1 0.3 0.90
0.05
0.054
Note:
3.2
Stochastic soil cases: variation of COV, constant
The mean undrained shear strength of the top layer
is
used to defined a modified bearing capacity factor for the
stochastic cases (
) and where
(1)
in which
Q
ur
is the stochastic ultimate bearing capacity. The
values of
of the 1000 realisations random field for each case
were ordered and the sample median value denoted as
. The
standard deviation of the
for the 1000 random field
realisations is calculated as
(
)
.
The values of
and
(
)
evaluated for all the cases presented in this paper are
provided in Table 1. The histogram of
from the 1000
random field realisations for case 2 (
see
Table 1) is depicted in Figure 4, with
= 0.98
and
(
)
. The empirical cumulative distribution
functions for cases 1-4 are shown in Figure 5.
In order to investigate the influence of changing the COV for
both or one of the layers, the cumulative curves for cases 1~4
