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and the mean value
μ
can be calculated as
The SWCC given by Eq. 5.31 has greater flexibility when
using the beta-type distribution function. For
α
β
,Eq.
5.31 generates a symmetrical S-shaped curve. For
α > β
,the
curve is nonsymmetrical and has a higher air-entry value, a
gentler slope near saturation, and a steeper slope near the
residual water content. For
α<β
, the curve has a smaller
air-entry value, a steeper slope near saturation, and a gentler
slope near the residual water content. When
α
and
β
are
integers, the SWCC defined by Eqs. 5.31 and 5.44 is related
to the binomial probability function as follows (Mendenhall
et al., 1981):
=
θ
s
2
s
μ
=
ψ
aev
+
(5.35)
Let us consider the case where a gamma-type distribu-
tion is used for the function
f(r)
. In this case,
f(h)
takes
the form
h
α
−
1
e
−
h/β
β
α
(α)
f(h)
=
α, β >
00
≤
h<
∞
(5.36)
f(h)
=
0 elsewhere
where
∞
1
h
α
−
1
(
1
h)
β
−
1
B(α, β)
−
h
α
−
1
e
−
h
dh
(α)
=
(5.37)
θ(ψ)
=
θ
s
dh
0
ψ
The SWCC defined by Eq. 5.31 has a smaller air-entry
value, a steeper slope near saturation, and a gentler slope
near the residual water content. In the special case when
α
is an integer the SWCC defined by Eq. 5.31 becomes
α
ψ
i
(
1
α
+
β
−
1
+
β
+
1
ψ)
α
+
β
−
1
−
i
=
θ
s
−
θ
s
−
i
i
=
α
(5.45)
∞
h
α
−
1
e
−
h/β
β
α
(α)
θ(ψ)
=
θ
s
dh
(5.38)
Defining the pore radius
r
over the interval (0, 1) does not
restrict its usage. The beta density function can be applied
to any interval by translation and a change in the scale.
ψ
∞
θ
s
(α)
h
α
−
1
e
−
h
dh
θ(ψ)
=
(5.39)
5.3.3 Proposed Fredlund-Xing (1994) Equation
The pore-size distribution function
f(h)
can also be written
in the form
ψ/β
α
−
1
ψ
i
e
−
ψ/β
i
!
β
i
θ(ψ)
=
θ
s
(5.40)
mnp(pψ)
n
−
1
[1
i
=
0
f(ψ)
=
(5.46)
(pψ)
n
]
m
+
1
+
1, the gamma distribution becomes an expo-
nential type distribution:
When
α
=
Figure 5.21 shows a sample probability distribution for
Eq. 5.46 along with its integration. It can be seen that the
integration drops to zero over a narrow suction range. There-
fore, Eq. 5.21 is not suitable in the high-soil-suction region.
Experimental data show that beyond residual water content
the plot should decrease linearly (on a logarithm scale), to
a value of about 10
6
kPa. To describe the SWCC more
accurately, the following distribution can be used:
1
β
e
−
h/β
f(h)
=
β >
00
≤
h<
∞
f(h)
=
0 elsewhere
(5.41)
The SWCC defined by Eq. 5.31 can then be written as
θ
s
e
−
ψ/β
θ(ψ)
=
(5.42)
Equation 5.42 has the same form as Eq. 5.19, which was the
formused byMcKee and Bumb (1984) to describe the SWCC.
Therefore, Eq. 5.19 gives good results when the pore-size
distribution of the soil reflects a gamma distribution.
Let us consider the case of a beta-type distribution for the
function
f(r)
:
mn(ψ/a)
n
−
1
f(ψ)
=
(5.47)
(ψ/a)
n
]
(ψ/a)
n
]
m
+
1
a
[
e
+
{
log[
e
+
}
Equation 5.47 and its integration formare shown in Fig. 5.22
for the same set of parameters (i.e.,
a
1
/p,n,m
). This dis-
tribution functiondropsmore slowly than shownbyEq. 5.46 as
soil suction increases. Therefore, Eq. 5.47 produces a nonsym-
metrical curve that more closely emulates experimental data.
Integrating Eq. 5.47 using Eq. 5.31 gives the Fredlund and
Xing (1994) relationship between volumetric water content
and soil suction:
=
h
α
−
1
(
1
h)
β
−
1
B(α, β)
−
f(h)
=
α, β >
00
≤
h
≤
1
f(h)
=
0 elsewhere
(5.43)
where
1
m
f
(α)(β)
(α
1
h
α
−
1
(
1
h)
β
−
1
dh
B(α, β)
=
−
=
(5.44)
θ
=
θ
s
(5.48)
+
β)
ln[
e
+
(ψ/a
f
)
nf
]
0
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