Environmental Engineering Reference
In-Depth Information
Table 14.4 Relationship between Fundamental Elasticity Parameters and Coefficients of Volume Change
Stress
Elasticity
Loading
State Variables
Coefficient of Volume Change Functions
Parameter Functions
a
∂ε
ν
∂(σ
mean
−
0
.
434
1
C
s
σ
mean
−
3
(
1
−
2
μ)
m
1
m
1
=
Three-dimensional (3D)
σ
mean
−
u
a
,
u
a
)
=
E
=
+
e
0
u
a
(u
a
−
u
w
)
3D
∂ε
ν
∂(u
a
−
0
.
434
1
C
m
(u
a
−
3
m
2
m
2
=
u
w
)
3D
=
H
=
+
e
0
u
w
)
3D
∂ε
v
∂(σ
y
−
0
.
434
1
C
s
σ
y
−
(
1
+
μ)(
1
−
2
μ)
m
1
−
1D
=
K
0
loading (1D)
σ
v
−
u
a
,
(u
a
−
u
a
)
=
E
=
μ)m
1
−
1D
+
e
0
u
a
(
1
−
u
w
)
1D
∂ε
v
∂(u
a
−
0
.
434
1
C
m
1
+
μ
m
2
−
1D
=
u
a
−
u
w
1D
u
w
)
1D
=
H
=
μ)m
2
−
1D
+
e
0
(
1
−
ε
y
)
∂(σ
ave
−
∂(ε
x
+
0
.
434
1
C
s
u
ave
−
2
(
1
+
μ)(
1
−
2
μ)
m
1
−
2D
=
Plane strain (2D)
σ
ave
−
u
a
,
u
a
)
=
E
=
m
1
−
2D
+
e
0
u
a
(u
a
−
u
w
)
2D
∂(ε
x
+
ε
y
)
0
.
434
1
C
m
(u
a
−
2
(
1
μ)
m
2
−
2D
+
m
2
−
2D
=
u
w
)
2D
=
H
=
∂(σ
a
−
+
e
0
u
w
)
2D
Source:
After Fredlund and Rahardjo (1993a).
a
Note:
σ
mean
=
(σ
x
+
σ
y
+
σ
z
)/
3;
σ
ave
=
(σ
x
+
σ
y
)/
2.
b
dε
y
=
dε
v
for
K
0
loading;
d(ε
x
+
ε
y
)
=
dε
v
for plane strain.
soil is approximately linear (on the extreme planes) over a
relatively large range of stresses (Ho and Fredlund, 1992a).
The elastic parameter functions
E
and
H
can also be cal-
culated directly from volume change indices
C
s
(on the net
normal stress plane) and
C
m
(on the matric suction plane),
respectively. The elastic parameters are calculated from the
coefficient of volume change, as shown in Table 14.4. The
elastic parameter
E
can be expressed as a function of the
volume change index with respect to net normal stress
C
s
,
initial void ratio, and Poisson's ratio. The elastic parame-
ter
H
can be expressed as a function of the volume change
index with respect to matric suction
C
m
, initial void ratio,
and Poisson's ratio. The equations for these elastic param-
eters can be written for general three-dimensional loading
conditions as follows:
Equations 14.62 and 14.63 can be written for
two-
dimensional plane strain conditions as follows:
4
.
605
(
1
+
μ)(
1
−
2
μ)(
1
+
e
0
)
E
=
(σ
ave
−
u
a
)
(14.64)
C
s
4
.
605
(
1
+
μ)(
1
+
e
0
)
H
=
(u
a
−
u
w
)
(14.65)
C
m
The 4.605 constant arises from the conversion between
the logarithmic and arithmetic scales (i.e., 4
.
605
=
2
/
log
10
2
.
718).
The elastic parameter
E
(or
H
) increases with an increase
in net normal stress (or matric suction) and a decrease of
the volume change index
C
s
(or
C
m
) assuming a constant
value of Poisson's ratio. Figures 14.56 and 14.57 graphically
illustrate the relationship between the elastic parameter
E
and net normal stress for various values of the swelling
index
C
s
and various values of Poisson's ratio, respectively.
Figure 14.58 presents the variation of elastic parameter
H
with matric suction for various values of swelling index
C
m
.
6
.
908
(
1
−
2
μ)(
1
+
e
0
)
E
=
(σ
mean
−
u
a
)
(14.62)
C
s
6
.
908
(
1
+
e
0
)
H
=
(u
a
−
u
w
)
(14.63)
C
m
14.7.2 Determination of Swell Index
The swelling indices can be measured experimentally or
estimated through correlation with the Atterberg limits
The 6.908 constant arises from the conversion between
the logarithmic and arithmetic scales (i.e., 6
.
908
=
3
/
log
10
2
.
718).
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