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b
Y
c
ð
X
2
0
:
1
Þ¼ð
0
:
047 log
X
2
0
:
023
Þb
2
b
Y
c
ð
0
:
1
<
X
2
0
:
25
Þ¼ð
0
:
01 log
X
2
þ
0
:
034
Þb
2
(12)
Y
c
ð
X
2
>
0
:
25
Þ¼ð
0
:
0517 log
X
2
þ
0
:
057
Þb
2
10
3
[
Dg
/(
u
2
b
2
)]
1/3
d
; and
where
X
2
4.67
b
2
(0.265
M
1)
¼
(
M
+ 2)/
(1
2
M
).
Egiazaroff (
1965
) gave yet another derivation for
Y
c
(
R
). He assumed that
the velocity at an elevation of 0.63
d
(above the bottom of particle) equals the fall
velocity
w
ss
of particle. His equation is
þ
1
:
33
Y
c
¼
(13)
C
D
½
a
r
þ
5
:
75 log
ð
0
:
63
Þ
where
a
r
¼
0.4 for large
R
. Both
a
r
and
C
D
increase for low
R
. His results do not correspond with the Shields diagram.
Mantz (
1977
) proposed the extended Shields diagram to obtain the condition
of maximum stability (Fig.
2
). Yalin and Karahan (
1979
) presented a curve of
Y
c
versus
R
, using a large volume of data collected from literature (Fig.
2
). Their
curve is regarded as a superior curve to the commonly used Shields curve.
Cao et al. (
2006
) derived the explicit equation for the curve of Yalin and
Karahan (
1979
). It is:
8.5; and
C
D
¼
drag coefficient
¼
b
R
0
:
23
d
Y
c
ð
R
d
6
:
61
Þ¼
0
:
1414
=
0
:
35
2
:
84
Þ¼
½
1
þð
0
:
0223
R
b
Þ
b
(14)
Y
c
ð
6
:
61
<
R
d
282
:
84
09
R
0
:
68
d
3
:
r
Y
c
ð
R
d
282
:
84
Þ¼
0
:
045
1
Yalin and Karahan
Upper curve of Mantz
0.1
Low curve of Mantz
0.01
0.01
0.1
1
10
100
1000
10000
R
*
Fig. 2 Curves (
Y
c
vs.
R
) of Mantz (1977) and Yalin and Karahan (
1979
)