1580
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
principles, but considered the more traditional application of the
soil pressures directly to the soil, assuming therefore flexible
wall.
The loads applied to the soil are: (i) its weight; (ii) a seismic
action represented by an equivalent static horizontal unit force,
directed towards the wall, equal to
, where
is the seismic
horizontal coefficient. No vertical seismic coefficient was
considered. The values of
considered in the calculations were:
0, 0.1, 0.2, 0.3, 0.4 and 0.5.
A horizontal force was applied to the wall, centered in width
b and at 2h/3 below the soil free surface. The wall can not suffer
vertical displacements or slide in the direction of b; it is free to
move otherwise. The collapse load determined is I
as
h
, the
seismic horizontal active force.
The horizontal component of the seismic active earth
pressure coefficient, K
as
h
can be determined using:
2
2
1
hb
I
K
has
has
(2)
The seismic active earth pressure coefficient can be
approximately determined using:
cos
has
as
K
K
(3)
This equation assumes that friction on the soil-to-wall
interface takes place entirely on the vertical direction. In the
three-dimensional calculations there is, however, no imposed
direction for the mobilization of the soil-to-wall friction and the
three-dimensional effect will lead to friction mobilization in the
horizontal direction, particularly for lower b/h ratios. For the
two dimensional calculations, equation (3) is exact.
All calculations were performed on an eight node cluster of
quad-core processor computers, using almost all the available
memory. An example of a three dimensional finite element
mesh used is presented in Figure 2; in fact, each hexahedron
represented in the figure is subdivided into 24 tetrahedral
elements. Additionally, interface elements following
Krabbenhoft et al. (2005) have been introduced between the
rigid wall and the soil, in order to allow considering a friction
angle of
between the two materials. For the case of
’=0, a
very small value of
(0.01º)
was adopted.
In the two-dimensional calculations the soil was modeled
with 3-node triangular finite elements allowing a linear
approximation for the velocity fields. As previously mentioned,
the wall was not explicitly modeled (it was assumed flexible)
and the friction between soil and wall was defined through the
inclination of the stresses applied to the soil.
3 PRESENTATION AND ANALYSIS OF RESULTS
Results of the horizontal seismic active earth pressure
coefficients are presented in Table 1 and are part of the
calculations being performed for a larger range of soil friction
angles and soil-to-wall friction ratios. Some of these results are
also presented in Figure 3, for the friction angles of 30 and 40º,
respectively, and for the two soil-to-wall friction ratios and
three horizontal seismic coefficients.
These results show a significant three-dimensional effect of
the b/h ratio: for small values of this ratio, there is a significant
decrease in the soil seismic horizontal active earth pressure
coefficients, with the greater b/h ratios leading to coefficients
quite close to the two-dimensional case; in fact for b/h>2 there
is small variation in the earth-pressure coefficients, specially for
the lower value of the friction angle.
Table 1. Values of the seismic horizontal active earth pressure
coefficient, K
as
h
, for different b/h and
, for the two values of the soil
riction angle and the two values of the soil-to-wall friction ratio.
f
'
º
'
b/h
1/4
0.1454
0.1621
0.1778 0.1975 0.2191 0.2419
1/2
0.2179
0.2481
0.2826 0.3157 0.3591 0.4005
1
0.2715
0.3086
0.3637 0.4308 0.5121 0.5922
2
0.3005
0.3478
0.4190 0.5115 0.6325 0.7734
5
0.3195
0.3789
0.4601 0.5687 0.7138 0.9184
∞ (2D) 0.3314.
0.3997
0.4897 0.6154 0.7970 1.0860
'
º
'
b/h
1/4
0.1178
0.1333
0.1486 0.1677 0.1891 0.2129
1/2
0.1798
0.2070
0.2393 0.2743 0.3176 0.3642
1
0.2267
0.2652
0.3169 0.3808 0.4608 0.5558
2
0.2534
0.3003
0.3671 0.4553 0.5726 0.7295
5
0.2705
0.3256
0.4005 0.5033 0.6525 0.8743
∞ (2D) 0.2820
0.3405
0.4276 0.5459 0.7212 1.0077
'
º
'
b/h
1/4
0.0730
0.0836 0.0993 0.1020 0.1271 0.1268
1/2
0.1162
0.1438 0.1684 0.1953 0.2244 0.2557
1
0.1643
0.1927 0.2344 0.2834 0.3291 0.3907
2
0.1881
0.2170 0.2781 0.3335 0.4306 0.4964
5
0.2047
0.2511 0.3131 0.3877 0.4965 0.6241
∞ (2D) 0.2170
0.2706
0.3414 0.4364 0.5633 0.7319
'
º
'
b/h
1/4
0.0646
0.0752
0.0860
0.0987 0.1124 0.1272
1/2
0.1035
0.1230
0.1451
0.1707 0.1983 0.2265
1
0.1383
0.1670
0.2046
0.2496 0.2969 0.3572
2
0.1585
0.1941
0.2440
0.3080 0.3882 0.4852
5
0.1710
0.2134
0.2694
0.3473 0.4482 0.5747
∞ (2D) 0.1804
0.2283
0.2924 0.3797 0.5110 0.6613
Figure 4 shows the same results emphasizing the influence
of the seismic horizontal coefficient,
. It can be observed from
these two charts that K
as
h
increases with
and that this increase
is less important for lower b/h ratios.
In this figure, results from the two-dimensional calculations
were also included, as well as the results obtained from
Mononobe-Okabe (M-O) method. Both methods give very
similar results of K
as
h
when
0.2; for
>0.2 the differences
between the two methods increase.
As the active coefficient is being determined and the active
force is the minimum required to ensure stability, the M-O
approximation (also an upper bound solution) is less safe than
the numerical results, because they are smaller.
Examples of the mechanisms obtained automatically from
the program Sublim3D can be inferred from Figure 5, where the
plastic deformation zones in the plan view and in the symmetry
plane are shown for the case b/h=1 and
’=30º, for three values
of
– 0, 0.2 and 0.4 and for the two values of the friction ratio
of 0 and 2/3.
