Environmental Engineering Reference
In-Depth Information
The compressibility of an air-water mixture,
C
aw
,
is
calculated by substituting the change in total stress into
Eq. 15.12:
term in Eq. 15.36) multiplied by the pore fluid volume (i.e.,
nV
) and the total stress increment
dσ
y
:
(
1
du
a
dV
a
V
0
=
−
S
+
hS
)n
du
w
dσ
y
(15.42)
(
1
−
S
+
hS
)(
du
a
/dσ
y
)
u
a
u
a
C
aw
=
SC
w
+
(15.36)
The volume change given by the constitutive relation for
the air phase must be equal to the volume change due to
compression of the air:
Combining Eq. 15.35 and Eq. 15.36 results in the follow-
ing relationship for
K
0
-undrained loading:
m
1
k
d
σ
y
−
u
a
+
m
2
d
u
a
−
u
w
m
1
k
d
σ
y
−
u
a
+
m
2
d
u
a
−
u
w
=
(
1
−
S
+
hS
)n
du
a
(15.43)
u
a
du
a
u
a
=
SnC
w
du
w
+
(
1
−
S
+
hS
) n
(15.37)
where:
Rearranging Eq. 15.37 yields an expression for the change
in the pore-water pressure,
du
w
,
in response to a total stress
change,
dσ
y
m
1
k
=
slope of
V
a
/V
0
versus
σ
y
−
u
a
for
K
0
loading.
:
The second equation can be written to give the change in
pore-air pressure,
du
a
,
associated with a total stress incre-
ment
dσ
y
m
2
−
m
1
k
−
du
a
[
(
1
−
S
+
hS
)n/ u
a
]
du
w
=
:
m
2
+
SnC
w
dσ
y
du
w
m
1
k
m
2
+
m
2
+
(15.38)
du
a
=
SnC
w
m
2
−
m
1
k
−
[
(
1
−
S
+
hS
)n/ u
a
]
dσ
y
m
1
k
The compressibility
m
2
can be written as a ratio of the
compressibility with respect to a total stress change,
m
1
k
−
(15.44)
m
2
−
m
1
k
−
[
(
1
−
S
+
hS
)n/ u
a
]
:
The relationship between the modulus values
m
2
and
m
1
k
can be expressed as a ratio,
R
ak
:
m
2
m
1
k
R
sk
=
(15.39)
m
2
m
1
k
Substituting Eq. 15.39 into Eq. 15.38 gives
R
ak
=
(15.45)
R
sk
−
du
a
hS
)n/( u
a
m
1
k
)
]
1
−
[
(
1
−
S
+
du
w
=
SnC
w
/m
1
k
R
sk
+
The above equation can be written in terms of the ratio
R
ak
:
dσ
y
1
+
(15.40)
du
w
SnC
w
/m
1
k
R
sk
+
R
ak
du
a
=
hS
)n/( u
a
m
1
k
)
]
R
ak
−
1
−
[
(
1
−
S
+
Equation 15.40 can be further simplified as follows:
dσ
y
1
−
du
w
=
R
1
k
du
a
+
R
2
k
dσ
y
(15.41)
hS
)n/( u
a
m
1
k
]
R
ak
−
1
−
[
(
1
−
S
+
(15.46)
where:
In a simple form, the change in pore-air pressure can be
written as follows:
hS
)n/( u
a
m
1
k
)
]
R
sk
−
1
−
[
(
1
−
S
+
R
1
k
=
SnC
w
/m
1
k
R
sk
+
1
du
a
=
R
3
k
du
w
−
R
4
k
dσ
y
(15.47)
R
2
k
=
SnC
w
/m
1
k
R
sk
+
where
There are two unknowns (i.e.,
du
w
and
du
a
) in the above
equation. In order to compute the pore-air and pore-water
pressure changes, a second independent equation is
required. The second equation is derived by considering
volume change in the air phase. The change in volume is
described by the constitutive relationship for the air phase.
The volume change due to the compression of air,
dV
a
,is
computed from the compressibility of air (i.e., the second
R
ak
R
3
k
=
hS
)n/( u
a
m
1
k
)
]
R
ak
−
1
−
[
(
1
−
S
+
1
R
4
k
=
hS
)n/( u
a
m
1
k
)
]
R
ak
−
1
−
[
(
1
−
S
+
The pore-air and pore-water pressure parameters for
K
0
-
drained loading can be written as
B
ak
and
B
wk
, respectively.
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