Environmental Engineering Reference
In-Depth Information
The first and third terms in Eq. 13.40 are used for isotropic
loading whereas the second and fourth terms are used for
uniaxial compression. The sum of the water volume changes
during isotropic and uniaxial compression gives the total
change in volume of water during triaxial loading. Sub-
stituting
du
a
=
du
ai
+
du
au
The inverse of the coefficients in the first
term (i.e.,
E (
1
2
μ)
]) is referred to as the “cons-
trained modulus” (Lambe and Whitman, 1979).
The water phase constitutive relation is obtained in a sim-
ilar manner (i.e., substituting Eqs. 13.42 and 13.43 into
Eq. 13.29):
−
μ) /
[
(
1
+
μ)(
1
−
and
du
w
=
du
wi
+
du
wu
into
Eq. 13.40 gives Eq. 13.38.
dV
w
V
0
=
+
μ
E
w
(
1
1
−
μ)
d(σ
y
−
u
a
)
13.3.7
K
0
Loading
A total stress increment of
dσ
y
is applied in the vertical
direction for
K
0
loading, while the soil is not permitted
to deform laterally (i.e.,
dε
x
=
dε
z
=
1
H
w
−
d(u
a
−
u
w
)
2
(E/H )
E
w
(
1
+
(13.46)
−
μ)
0). The
K
0
loading
condition occurs during one-dimensional compression where
deformation is only allowed in the vertical direction (i.e.,
dε
y
). The total stress increment
dσ
y
is applied in the vertical
direction. The strain conditions during the
K
0
loading can be
applied to Eqs. 13.21 and 13.22. Multiplying Eq. 13.23 by
Poisson's ratio
μ
and adding the result to Eq. 13.21 gives
1
13.3.8 Plane Strain Loading
Many geotechnical problems can be simplified into a two-
dimensional form using the concepts of plane strain and
plane stress loading. If an earth structure is significantly
long in one direction (e.g., the
z-
direction) in comparison to
the other two directions (e.g., the
x-
and
y-
directions) and the
loadings are applied only on the
x-
and
y-
planes, the struc-
ture can be modeled as a plane strain problem. The stability
of slopes, earth pressures on retaining wall, and the bear-
ing capacity of strip footing are examples of problems that
are commonly analyzed assuming plane strain loading con-
ditions. The soil deformation in the
z-
direction is assumed
to be negligible for plane strain conditions (i.e.,
dε
z
=
d(σ
z
−
u
a
)
=
−
μ
2
E
μ
E
(
1
+
μ) d(σ
y
−
u
a
)
1
d(u
a
−
u
w
)
(13.41)
+
μ
H
−
The above equation can be arranged as follows:
0).
Imposing a condition of zero strain in the
z-
direction in
Eq. 13.23 gives
μ
1
d(σ
y
−
u
a
)
d(σ
z
−
u
a
)
=
−
μ
2
u
a
−
(H/E) d
u
a
−
u
w
(13.47)
Volumetric strain during plane strain loading is obtained
by substituting Eq. 13.47 into Eq. 13.25:
d
σ
z
−
u
a
=
μd
σ
x
+
σ
y
−
E
−
−
μ)H
d(u
a
−
u
w
)
(13.42)
(
1
Similarly, Eq. 13.23 can be multiplied by Poisson's ratio
μ
and added to Eq. 13.21 to give
μ
1
d(σ
y
−
u
a
)
d
σ
ave
−
u
a
2
(
1
+
μ)(
1
−
2
μ)
dε
v
=
d(σ
x
−
u
a
)
=
E
−
μ
2
1
d
u
a
−
u
w
+
μ
H
E
+
(13.48)
−
−
μ)H
d(u
a
−
u
w
)
(13.43)
(
1
where:
Substituting Eqs. 13.42 and 13.43 into Eq. 13.22 gives
2
μ
2
1
−
μ
−
σ
ave
=
average total normal stress for two-dimensional
dε
y
=
d(σ
y
−
u
a
)
loading [i.e.,
σ
x
+
σ
y
/
2].
E(
1
−
μ)
1
−
μ
+
2
μ
+
d(u
a
−
u
w
)
(13.44)
H(
1
−
μ)
The above equation can be used as the soil structure con-
stitutive relationship for plane strain loading.
The water phase constitutive equation is obtained from
Volumetric strain
dε
v
is equal to the strain in the vertical
direction,
dε
y
,for
K
0
loading conditions. Equation 13.44
for the soil structure can then be rewritten as follows:
Eq. 13.29 by replacing the
d
σ
z
−
u
a
term with Eq. 13.47:
2
1
d(σ
ave
−
u
a
)
dV
w
V
0
=
+
μ
E
w
(
1
+
μ)(
1
−
2
μ)
dε
v
=
d(σ
y
−
u
a
)
E(
1
−
μ)
1
H
w
−
d(u
a
−
u
w
)
+
μ
H(
1
1
(E/H )
E
w
+
−
μ)
d(u
a
−
u
w
)
(13.45)
+
(13.49)
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