(ητxy ) - τi cos si
η
(ησx ) -
- siρgη sin(θ + α) = 0
(15)
x
x
y
where
σx =
normal stress in the streamwise direction
τxy =
shear stress at the banks
τi =
shear on the underside of the cover
θ =
slope of the bed
(θ + α)=
slope of the water surface.
Uzuner and Kennedy expressed σx and τxy as functions of the average vertical
stress σ z within the cover
η
siρg(1 - si )(1 - p) cos(θ + α) = γ e η
σz =
(16)
2
where p is jam porosity and γe is the equivalent unit weight of the jam. The Rankine
and Mohr-Coulomb stress theories for granular materials give
σx = Kp σz
(17)
and
τxy = Co σz + Ci
(18)
where
Kp = passive pressure coefficient
Co = shear stress coefficient
Ci = assumed cohesive intercept.
Substitution of eq 16 through 18 into 15, integration of the modified equation, eq
15, then normalization using xo = x/hn and ηo = η/hn, yields
ηo
= k1 + k2ηo + k3ηo2
ηo
(19)
xo
where
τi
k1 =
(20)
2Kpγ ehn
Ci
siρgSo - 2
B
k2 =
(21)
2Kpγ e
Cohn
k3 = -
(22)
KpB
1
f q
23
hn = b n
(23)
.
8gSo
Also
So = bed slope (sin θ)
fb = Darcy-Weisbach resistance factor of the bed
qn = unit discharge at a location upstream where ice does not affect the flow.
16
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