τ0 = τc

(3.38)

where:

Average bed shear stress, N/m2, lb/ft2

=

τ0

Critical bed shear stress at incipient motion, N/m2, lb/ft2

=

τc

The average bed shear stress applied by the steady uniform flow, derived in Chapter 2, is as

follows:

τ0 = γ R S f

(3.39)

Using y for the hydraulic radius (R) and the Manning equation to determine the slope (Sf)

The average shear stress can be expressed as follows:

ρ gn2 V 2

τ0 = ρ g y S f =

(SI)

(3.40)

1/ 3

y

ρ gn2 V 2

τo =

(English)

(3.41)

(1.49)2 y1/ 3

For noncohesive bed materials, the Shields relation (Vanoni 1975) can be used to determine

the relation between the critical shear stress and bed material size for beginning of bed

material movement. The relation is as follows:

τ c = K s (ρ s - ρ) g D

(3.42)

At the beginning of sediment movement the applied shear stress is equal to the critical shear

stress as given in Equation 3.36 (τ0 = τc) resulting in the following:

ρ gn2 V 2

= K s (ρ s - ρ) gD s

(SI)

(3.43)

1/ 3

y

ρ gn2 V 2

= K s(ρ s - ρ) gD s

(English)

(3.44)

2.22 y1/ 3

where:

y

=

Average depth of flow, m, ft

=

Slope of the energy grade line, m/m, ft/ft

Sf

V

=

Average velocity, m/s, ft/s

=

Diameter of smallest non transportable bed material particle, m, ft

Ds

Unit weight of water (9,800 N/m3, 62.4 lb/ft3)

=

γ

n

=

Manning's roughness coefficient

=

Shield's coefficient

Ks

3.40

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