Environmental Engineering Reference
In-Depth Information
where:
P
p
(d)
The arithmetic particle-size distribution exaggerates the
clay portion of the particle distribution curve and does not
appear to provide meaningful information for engineering
purposes. It is more meaningful to represent the PDF on a
logarithmic scale rather than as an arithmetic PDF. Differ-
entiating the grain-size distribution equation on a logarithm
scale produces a PDF that is more meaningful, as shown in
the equation,
=
percentage by mass of particles passing a par-
ticular particle size,
a
gr
=
parameter designating the inflection point on the
grain-size distribution curve and is related to the
initial breaking point near the coarse particle
sizes,
n
gr
=
parameter related to the steepest slope on the
grain-size distribution curve (i.e., uniformity of
the particle size distribution),
d
P
p
dlog
(d)
=
d
p
p
d
d
p
1
(d)
=
ln
(
10
)d
(2.3)
m
gr
=
parameter related to the shape of the grain-size
curve as it approaches the fine-grained region,
where:
d
r
=
parameter related to the particle size in the fine-
grained region and is referred to as the residual
particle size,
p
1
(d)
=
logarithmic PDF
The peak value in Eq. 2.3 represents the most frequent
particle size. The probability of the logarithmic PDF can be
calculated according to
d
=
diameter of any particle size under considera-
tion, and
diameter of the minimum allowable size particle
(e.g., 0.0001 mm).
Equation 2.1 is unimodal in character and can be used to
fit a wide variety of soils. A quasi-Newton fitting algorithm
can be used to best fit three fitting parameters for a soil
(M.D. Fredlund, 2000). The best-fit particle size distribution
function can be plotted along with the grain-size distribution
data as shown in Fig. 2.6a. The unimodal sigmoidal equation
provides an excellent fit of the grain-size data.
The particle-size distribution equation provides informa-
tion on the amount and distribution of particle sizes. Differ-
entiation of the particle size distribution curve produces a
particle-size probability density function (PDF) (M.D. Fred-
lund, 2000). The differentiated form of the unimodal grain-
size distribution equation is called the arithmetic PDF and
is shown in Eq. 2.2 and Fig. 2.6b:
d
m
=
x
=
log
(d
2
)
probability
(d
1
<d<d
2
)
=
p
1
(x)dx
(2.4)
x
=
log
(d
1
)
Figure 2.6c shows the logarithmic PDF for clayey silt.
The unimodal equation fit for silt sand is shown in Fig. 2.7
along with the logarithmic PDF.
The unimodal equation (i.e., Eq. 2.1) shows characteristics
that are similar to the Fredlund and Xing (1994) equation for
the SWCC (Fig. 2.8). The
a
gr
parameter is related to the initial
break in the grain-size distribution equation (in the coarse
particle size region) and represents the inflection point on the
curve. Its effect on the grain-size distribution curve can be seen
in Fig. 2.8, where
a
gr
is varied from 0.1 to 10 while the other
equation parameters are held constant. The
a
gr
parameter is
related to the largest particle sizes in the soil sample.
Figure 2.9 shows how the parameter
n
gr
influences the
slope of the grain-size distribution. The point of maximum
slope along the grain-size distribution curve (i.e., through
the inflection point) provides an indication of the dominant
particle size in the soil (i.e., on a logarithm scale). Figure 2.9
shows the effect of varying
n
gr
from1to4.
The parameter
m
gr
influences the break onto the finer par-
ticle size region of the soil sample. The effect of varying the
m
gr
parameter from 0.3 to 0.9 can be seen in Fig. 2.10. The
parameter
d
r
affects the shape along the finer particle size
portion of the curve, but its influence on the curve is quite
minimal, as shown in Fig. 2.11. The
d
r
variable can be man-
ually adjusted to improve the fit of the curve to the data when
using a best-fit regression analysis. It has been observed that
a value of 0.001 mm for
d
r
can provide a reasonable fit for
most soils.
⎧
⎨
7
⎫
⎬
ln
1
⎡
⎣
⎤
⎦
d
r
d
+
d
P
p
d
d
1
ln
1
=
ln
exp
(
1
)
a
gr
d
n
gr
m
gr
1
−
⎩
d
r
d
m
⎭
+
+
m
gr
a
gr
d
n
gr
×
n
gr
×
d
exp
(
1
)
a
gr
d
n
gr
ln
exp
(
1
)
a
gr
d
n
gr
+
+
ln
1
6
d
r
d
+
7
+
ln
exp
(
1
)
a
gr
d
n
gr
m
gr
ln
1
7
d
r
d
m
+
+
d
r
d
2
1
×
(2.2)
d
r
d
+
2.2.2.2 Bimodal Equation for Grain-Size Distribution
The unimodal grain-size distribution equation (i.e., Eq. 2.1),
does not provide a satisfactory fit when soils are gap-graded
as shown in Fig. 2.12. In this case it is necessary to use a
bimodal equation when performing the best-fit regression
where:
d
P
p
d
d
=
arithmetic slope of the grain-size distribution
curve in units equal to the inverse of particle
size.
Search WWH ::
Custom Search