Environmental Engineering Reference
In-Depth Information
Lytton (1994) presented the following typical values for
the coefficients of lateral earth pressure back calculated from
field observations of heave and shrinkage:
The differential expression of normal strain in terms of
displacement in the
y-
direction (i.e.,
v
)isasfollows:
∂
v
∂y
⎧
⎨
ε
y
=
(14.79)
0 when soil is dry and cracked
0.333 when soil is dry and cracks are opening
0.500 when cracks are closed and suction
is at steady-state condition
0
.
667 when cracks are closed and soil is wetting
1 when soil is wetting and is in hydrostatic
stress condition
2-3 when soil is approaching passive
earth pressure
The governing differential equation for the
K
0
loading
condition can be obtained by substituting the expression for
net normal stress (i.e., Eq. 14.78) into the force equilibrium
equation (i.e., Eq. 14.77). Pore-air pressure is assumed to
remain at atmospheric conditions:
K
0
=
⎩
∂ν
∂y
−
u
w
)
∂
∂y
E(
1
−
μ)
1
+
μ
μ)
(u
a
−
(
1
+
μ)(
1
−
2
μ)
H(
1
−
(14.76)
+
ρg
=
0
(14.80)
The coefficient of earth pressure at rest presented by
Eq. 14.76 can be used to estimate net normal stress.
The elastic moduli
E
and
H
can be calculated from the
swelling indices, initial void ratio, and assumed Poisson's
ratio as follows:
14.8 ONE-DIMENSIONAL SOLUTION USING
INCREMENTAL ELASTICITY FORMULATION
E
=
n
t
(σ
−
u
a
)
ave
(14.81)
n
m
u
a
−
u
w
ave
The example problem shown in Fig. 14.38 was earlier solved
for total heave using longhand calculation with known val-
ues for swelling index
C
s
and swelling pressure
P
s
.The
same example can now be solved with a numerical solution
using an incremental elasticity-type formulation. A compar-
ison of the two methodologies and the calculated total heave
is presented.
H
=
(14.82)
where:
n
t
=
coefficient that relates net normal stress
with the elastic modulus
E
,
n
m
=
coefficient that relates matric suction with
the elastic modulus
H
,
(σ
−
u
a
)
ave
=
average of the initial and final net normal
stress for an incremental change in stress,
and
14.8.1 Heave Analysis Formulation
The force equilibrium equation for
K
0
loading conditions
(i.e., volume change in the
y-
direction only) can be
expressed as follows:
(u
a
−
u
w
)
ave
=
average of the initial and final matric suc-
tions for an incremental suction change.
∂σ
y
∂x
+
ρg
=
0
(14.77)
Equations 14.81 and 14.82 show that the elastic parame-
ters change linearly with either of the stress state variables.
Equations 14.81 and 14.82 are assumed to be linear, an
assumption based on a history of laboratory measurements
that have shown the swelling index of a soil,
C
s
, to approxi-
mate a constant value. Laboratory measurements by Meilani
et al. (2005) showed that the elastic moduli
H
can vary in
a nonlinear manner with respect to stress state. The coeffi-
cients of proportionality
n
t
and
n
m
can be written in terms
of Poisson's ratio and the respective compressive indices:
where:
ρ
=
total density of the soil and
g
=
acceleration due to gravity.
The soil structure constitutive relationship for
K
0
loading
can be written as follows:
ε
ν
−
u
w
)
(14.78)
(
1
−
μ)E
1
+
μ
1
σ
y
−
u
a
=
μ)
(u
a
−
(
1
+
μ)(
1
−
2
μ)
e
0
0
.
4343
C
t
+
(
1
+
μ)(
1
−
2
μ)
H(
1
−
n
t
=
(14.83)
1
−
μ
where:
1
+
μ
e
0
0
.
4343
C
m
1
+
n
m
=
(14.84)
−
μ
1
E
=
elastic modulus for the soil structure with respect to
net normal stress,
where:
H
=
elastic modulus for the soil structure with respect to
matric suction, and
C
t
=
volume change index with respect to net normal
stress,
μ
=
Poisson's ratio.
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