1070 SEDIMENT TRANSPORT AND EROSION
studies^3 indicate that this sediment discharge, known as the
bed load, q B , is a function of the excess of t o above t c orq B fcn ( t o — t c ). (7)Figure 8 illustrates a typical experimental relationship
between q B and ( t o — t c ).DuBoys in 1879^22 treated the bed material, involved
in the bed load, as it if consisted of sliding layers which
respond to and distribute the applied stress t o. He proposed
the relationq B C s t o ( t o – t c ) (8)Both C s and t c depend on particle size as indicated in Table 3.
Chang,^1 Schoklitsch, 1,16 MacDougall,^1 and Shields^1 have
presented bed load formulae similar to Eq. (8).
The theoretical bed load model developed by Einstein 4,5,8,24
has formed the basis for a number of researches in sediment
transport. 6,15,24,36 Einstein utilized: (1) the statistical nature of
turbulent flow; (2) the fact that in steady uniform flow there
is an equilibrium between the processes of erosion and depo-
sition, that is, (probability of erosion) (the probability of
deposition); (3) the fact that grains near the bed are more in
quick “steps” interrupted by “rest” periods; (4) a separate
hydraulic radius, R ′, associated with grain roughness and
another hydraulic radius, R ′′, associated with the bed form.
Einstein obtained the erosion probability function by
assuming that the lift force, on a grain, consists of an aver-
age component [related to ( U * )^2 ] and a normally distributed
random component. Einstein thus obtained the “bed load”
equationA
A
B tB
o∗∗ o
∗∗ −−−
∗∗∗∗
∫
1 1
11
p chch
(/ )d(/ )
(9)in which
∗iqigdSBBbsg s(^3) () 1
(10)
is Einstein’s bed transport function; A 43.5; B 0.143;
h o 1 2;
cj∗
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
Y
X
S
d
s SR
log
log
.
()
10.6
2
10 6
1
(11)
in the Einstein flow intensity function; i B fraction of q B in the
size range associated with d; d geometric mean of particle
0.011.0
0.1
1.0
10 100 1000
Rs=Unsf
f(Rs)
Laminar
flow of bed
Turbient
flow of bed
FIGURE 7 Shields’ critical shear function (adapted from
Henderson).
0.0 0.001 0.003 0.005 0.007 0.009 0.011 0.013
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.01
T 0 (gm/cm^2 )
SEDIMENT LOAD (gm./sec./cm.)T
0
FIGURE 8 Typical relation of shear and sediment load
(adapted from Leopold, Wolman and Miller).
TABLE 3
Typical values of Cs and tc (after Straub22,23)
d mm 1
8
1
4
1
2
124
Csft^6
162
sec
0.81 0.48 0.29 0.17 0.10 0.06
tclb
ft^2
0.016 0.017 0.022 0.032 0.051 0.09
C019_001_r03.indd 1070C019_001_r03.indd 1070 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM