When the ordinates of successive classes are added and plotted against the upper limit of
the size class, the cumulative percent finer distribution diagram is obtained (Figure 3.3). In
this diagram, the abscissa scale (usually logarithmic) represents the intervals of the size
scale and the ordinate scale is the cumulative percent by weight of the sample up to (or
percent finer than) the size in question. From the cumulative percent finer the D35, D50, D85,
Di, etc. sizes can be determined.
In a size frequency distribution curve, it is possible to choose certain particle sizes as
representing significant values, such as particles just larger than one-fourth of the distribution
D25 (the first quartile), and particles just larger than three-fourths of the distribution D75 (the
third quartile). Measures of spread are based on differences or ratios between the two
quartiles. Quartile measures are confined to the central half of the frequency distribution and
the values obtained are not influenced by larger or smaller sizes. Quartile measures are very
readily computed, and most of the data may be obtained directly from the cumulative curve
by graphic means.
In contrast to quartile measures, moment measures are influenced by each individual size
class in the distribution. The first moment of a frequency curve is its center of gravity and is
called the arithmetic mean and is the average size of the sediment. The second moment is a
measure of the average spread of the curve and is expressed as the standard deviation of
the distribution.
Commonly, the size distribution of natural sediments plots as a straight line on log probability
paper. If this is true, then a natural sediment is completely described by the median diameter
(D50 the size of sediment of which 50 percent is finer) and the slope of the cumulative
frequency line on log probability paper. The slope of this line is proportional to the spread of
the size distribution in a sediment sample. It is computed with the expression
1 D 50 D84
G=
+
(3.9)
2 D16 D50
where:
G=
Gradation coefficient
Dx =
Sediment diameter particle of which x percent of sample is finer
The grain roughness used in velocity equations is taken as the D80, D85, or D90.
In studies of scour below culvert outlets, Stevens (1968) was able to consolidate a wide
range of scour data by employing the expression.
1/ 3
10 3
i Di
k = =1
(3.10)
10
for the effective or representative grain size of well-graded materials. Here
3.10
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