Worked Example 8.1
A wide open channel has a depth of flow 1.7 m and mean velocity 2.5 m s^1.
Taking (^) s2650 kg m^3 , find the minimum size of bed material needed to
obtain a stable bed.
Solution
Takingc in equation (8.19) as 0.056 (Shields) and using the Man-
ning–Strickler equation (8.6),
dRS/(∆0.056) 11 RS11(Vn)^2 /R1/3(0.04V)^2 11 (d/R)1/3.
Therefore
d1/30.133V/R1/60.1332.5/1.71/60.303, or
d0.028 m28 mm.
Worked Example 8.2
A channel 2.0 m deep, 15 m bed width, with 1:2 (V:H) side slopes is exca-
vated in gravel of d50 mm. What is the maximum permissible channel
slope, and what discharge can the channel carry without disturbing its
stability? Take !37° and the bed critical shear stress as 0.97 gyS.
Solution
For a particle on the bed, bc g∆d0.056 104 1.650.050.056
46.2 N m^2. For a particle on the side, scbc (1 sin^2 /sin^2 !)1/2
30.9 N m^2 (note that tan1/2). Thus
0.75 gyS 1 30.9 N m^2 (side),
0.97 gyS 2 46.2 N m^2 (bed).
ThereforeS 1 0.865S 2 and as S 1 S 2 the side stability is decisive. The
permissible channel slope is S 1 30.9/(10^4 0.752)2.06 10 3. The
permissible discharge is
QAR2/3S1/2/n[38/(0.040.051/6)][38/(15 4 5
)]2/3(2.06 10 3 )1/2
96 m^3 s^1.