Environmental Engineering Reference
In-Depth Information
analysis. The characteristic shape of a bimodal or gap-
graded soil is the double hump observed from experimental
data. The hump indicates that the particle-size distributions
are concentrated around two separate particle sizes. A
gap-graded soil can be viewed as a combination of two or
more soils (Durner, 1994). The gap-graded soil can be best
fit by “stacking” two (or more) unimodal equations:
A total of nine parameters must be computed when fitting
the bimodal equation to experimental data (i.e., stacking uni-
modal curves). Seven parameters can be determined using
a nonlinear least-squares fitting algorithm while two param-
eters can be fixed, namely,
d
r
and
d
m
. The superposition
method provides a robust method of fitting bimodal data sets.
The bimodal grain-size distribution curve equation has
been found to closely fit essentially any data set. If the data
set tends to be unimodal in character, it is better to best fit
the data using the unimodal grain-size equation. Figure 2.13
shows a bimodal fit for clayey silt sand along with its log-
arithmic PDF.
⎨
⎬
⎡
⎣
⎤
⎦
k
1
P
p
(d)
=
w
i
ln
exp
(
1
)
a
gr
d
n
gr
m
gr
⎩
⎭
i
=
1
+
⎧
⎨
7
⎫
⎬
ln
1
⎡
⎣
⎤
⎦
d
r
d
2.2.2.3 Application of Grain-Size Distribution Function
A mathematical equation for grain-size distribution data is
of considerable value in geotechnical engineering. First, the
unimodal and bimodal grain-size equations provide a con-
tinuous mathematical function for the grain-size distribution
curve. Second, grain-size distributions can be identified by
equations that are best fit to the experimental data. The grain-
size information can be stored in a database and used for iden-
tification purposes. Third, grain-size distribution equations
provide a consistent method for determining physical indices
such as percent clay, percent sand, percent silt, and particle
diameter variables such as
d
10
,
d
20
,
d
30
,
d
50
, and
d
60
.
Grain-size distribution curves have also been used for esti-
mating SWCCs (Gupta and Larson, 1979a; Arya and Paris,
1981; Haverkamp and Parlange, 1986; Ranjitkar and Sunder,
1989; M.D. Fredlund et al., 2000a). An accurate represen-
tation of the grain-size distribution curve forms a reliable
basis for the estimation of the SWCC (i.e., pedo-transfer
estimation functions).
The percentages of each range of particle sizes can be cal-
culated from the grain-size distribution equation. The grain-
size distribution equations are in the form of percent passing
a particular particle size,
P
p
(d)
, where
d
is particle diameter
(mm). The percent clay, percent silt, and percent sand can
be read directly from the curve by designating appropriate
particle diameters. The input diameters used depend upon the
criteria associated with the various soil classification systems.
For example, the USDA classification designates clay-sized
particles as being less than 0.002 mm. Silt-sized particles
are in the range between 0.002 and 0.05 mm and sand-sized
particles are between 0.05 and 2.0 mm. The Unified Soil
Classification System (USCS) uses a boundary of 0.005 mm
for the start of clay size particles. Silt-sized particles are
between 0.005 and 0.075 mm, and sand-sized particles are
between 0.075 and 4.75 mm. The divisions can be deter-
mined for any classification method by substituting the appro-
priate particle sizes into the grain-size equation as shown
in Fig. 2.14.
Diameter variables can also be read from the grain-size
distribution curve equation in an inverse manner. Particle-
size diameters answer to the question “What particle diame-
ter has 10 percent of the total mass smaller than this size?” A
“half-length” algorithm can be used to read diameters from
+
ln
1
×
1
−
(2.5)
⎩
d
r
d
m
⎭
+
where:
P
p
(d)
=
percentage by mass of particles passing a par-
ticular size,
k
=
number of “subsystems” for the total particle-
size distribution, and
w
i
=
weighting
factors
for
each
of
the
subcurves
adhering
to
the
condition;
0
<
w
i
<
1
and
w
i
=
1.
For a bimodal curve,
k
=
2 and the number of parame-
ters to be determined is 4
k
1
)
(i.e., 9). A unimodal
equation is used as the basis for the prediction of the bimodal
equation. The final bimodal curve equation in its extended
form can be written as
+
(k
−
w
1
P
p
(d)
=
ln[exp
(
1
)
(a
bi
/d)
n
b
]
m
b
+
(
1
−
w
)
+
1
×
ln[exp
(
1
)
(j
bi
/d)
k
bi
]
l
bi
+
1
7
ln
(
1
+
d
r
/d)
×
−
(2.6)
ln
(
1
+
d
r
/d
m
)
where:
a
bi
=
parameter related to the first breaking point along
the coarse-grained portion of the grain-size distri-
bution curve,
n
bi
=
parameter related to the first steep slope along the
coarse-grained part of the grain-size distribution
curve,
m
bi
=
parameter related to the first portion of the grain-
size distribution curve,
j
bi
=
parameter related to the second breaking point of
the curve,
k
bi
=
parameter related to the second steep slope along
the curve,
l
bi
=
parameter related to the second shape along the
curve, and
d
r
=
parameter related to the particle size in the fines
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