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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