Geoscience Reference
In-Depth Information
This renders (9.23) in a vertical storage factor
S
v
=
(
n
)
f
+
)
sf
(
2
-n
)
+
2)
v
(1
-
)
sf
/
)
))
du
(9.25)
For a particular case, involving deep sand and silt/clay layers the Poisson ratio is
estimated to
= 0.30 and the
)
v
becomes
)
v
= (1+
)/(3(1-
))
)
= 0.62
)
. Field data
v
= 3x10
-4
MPa
-1
. Thus
= 4.8x10
-4
MPa
-1
. The fluid-matrix compressibility
show
)
)
sf
= 0.16x10
-4
MPa
-1
. The following values are obtained
for sand layers with
n
= 0.25 and 1% relative grain contact:
(sand/silt) is estimated to
)
2
= 1- (1-
n
)
)
ss
/
)
=
0.75 and
S
v
= (
n
)
f
+ 0.75
)
v
)d
u
. For sand layers with
n
= 0.20 and 5% relative grain
contact:
2
= 1- (1-
n
)
)
ss
/
)
= 0.87 and
S
v
= (
n
)
f
+ 0.88
)
v
)
du
. For silt/clay with
n
=
0.05 and large relative grain contact one finds
2
= 1- (1-
n
)
)
ss
/
)
= 0.97 and
S
v
=
(
n
)
v
)
du
.
In conclusion, the Biot coefficient
f
+ 0.99
)
varies from 0.75 to 0.85 and higher for
sand. For deep clay/silt layers it is practically 1. The previous analysis is based on
elasticity. If plasticity (crushing) is involved the Biot coefficient can be lower.
2
D
A VISCO
-
PLASTIC ANALYTIC ELEMENT METHOD
Many geotechnical numerical models are able to predict mechanical behaviour
until the point of collapse. What happens after the moment of failure cannot be
determined, except for trivial cases where a simple kinematics system is adopted.
This is of particular importance for a proper evaluation of possible consequences of
failure. At present, effort is devoted to improve this shortcoming. It requires an
approach that includes geometric non-linearity, convective terms and rate effects.
The material point method is such an approach.
Finite Element analysis approaches a failure state from the lower bound and it
may underestimate the ultimate limit state sometimes by 10% or more (depending
on mesh size and numerical accuracy criterion). Visco-plastic analysis approaches
a vulnerable situation from the upper bound and it may overestimate sometimes by
10% or more.
Visco-plastic deformation occurs when soil is incapable of supporting deviatoric
stresses. The total strains may become so large that it is acceptable to disregard
elastic strains and consider only viscous strains, treating the material as a viscous
fluid. Though there are numerical methods to solve complicated cases, here, the
focus is on using simple analytical elementary solutions, such as the visco-plastic
wedge and the Rankine wedge, as will be shown. A complex visco-plastic
deformation can be composed of a consistent set of elementary solutions. Next, it
will be applied to validate the effect of dike doweling, compensating for the
likelihood of uplift failure.
Basic concept
During visco-plastic flow any element in a body is continuously changing in
shape (in extension and distortion), translation and rotation. The time variation
defines the rate of change. Plastic strain rate
ij,t
is defined as a tensor, the
components of which are related to the derivatives of deformation velocities
u
i,jt
as
follows (see also Fig 7.2)
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