Geoscience Reference
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be explained with an example that deals with the response of a vertical soil column
to cyclic loading.
' = (1
)
0
cos(
2
t)
=
0
cos(
2
t)
u =
0
cos(
2
t)
sand
clay
z
z
z=H
= 1
= 0
Figure 16.4a Cyclic vertical consolidation
At the surface
z =
0 the conditions
=
0
cos(
2
t
)
and
u =
0
cos(
2
t
) hold.
With Terzaghi's law, this yields for the effective stress at the surface
' =
u =
(1
)
0
cos(
2
t
). Here,
is some constant 0
<
<
1, which expresses a situation
between two extremes: for
=
0 a
loaded shallow submerged footing on a drained layer (Fig 16.4a). For this situation
the consolidation process in the soil is described by
57
=
1 it refers to free waves on a seabed, for
)
-1
c
v
u
,zz
= u
,t
a
,t
with
a =
(1
+n
)
/
(16.3)
Here,
is the compressibility for the soil matrix and
)
for the pore fluid.
Equilibrium at the bottom yields
t
), since at the bottom there is no
relative motion between both phases. Hence, the flow term becomes zero, which
yields, applying equation (16.3),
u
,t
=
0
cos(
2
+ constant
. The constant
can be identified with the effect of volumetric weight (hydrostatic pressure).
Because, here, excess pore pressures are considered, this constant is zero. The
internal stress state at the bottom becomes
a
,t
=
0 or
u = a
u = a
0
cos(
2
t
)
and
' =
u =
(1
a
)
0
cos(
2
t
)
(16.4)
Introduction of the variable
w = a
u
yields for equation (16.3), keeping in
mind that
is only a function of
t
and not of
z
cw
,zz
= w
,t
(16.5)
with boundary conditions
z =
0
, w = a
u =
(
a
)
0
cos(
2
t
)
57
Suffix
,zz
refers to a second partial derivative to the space coordinate
z
and
,t
to a first
partial derivative to the time coordinate
t.
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