Civil Engineering Reference
In-Depth Information
and
{
u
} =
[
D
]
{
}
(6.78)
u
Combining these equations gives,
{
σ
} =
[
D
]
{
}
(6.79)
where
[
D
]
[
D
]
=
+
[
D
]
(6.80)
u
The matrix [
D
] is the familiar elastic stress-strain matrix in terms of effective Young's
modulus
E
v
from (2.77). The matrix [
D
and Poisson's ratio
] contains the apparent bulk
u
modulus of the fluid
K
e
in the following locations:
"
K
e
K
e
0
K
e
!
$
K
e
K
e
K
e
0 000
K
e
K
e
0
[
D
]
=
(6.81)
u
0
K
e
assuming that the third column corresponds to the shear terms in a two-dimensional plane-
strain analysis.
To implement this method in the programs described in this chapter, it is necessary
to form the global stiffness matrix using the total stress-strain matrix [
D
], while effective
stresses for use in the failure function are computed from total strains using the effective
stress-strain matrix [
D
]. Pore pressures are simply computed from:
1
1
0
1
{
u
} =
K
e
(
{
r
} + {
z
} + {
θ
}
)
(6.82)
assuming an axisymmetric analysis.
For relatively large values of
K
e
, the analysis is insensitive to the exact magnitude of
K
e
. For axisymmetric analyses, Griffiths (1985) defined the dimensionless group,
ν
)K
e
E
β
t
=
(
1
−
2
(6.83)
and showed that for an undrained triaxial test in a non-dilative material (
ψ
=
0), consoli-
dated at a cell pressure of
σ
3
, the deviator stress at failure would be given by,
σ
3
(K
p
−
1
)(
3
β
t
+
1
)
D
f
=
(6.84)
(K
p
+
2
)β
t
+
1
tan
2
45
◦
+
φ
/
K
p
=
(
2
)
.
where
