The external forces acting on the ice and pore water control volume include:
hydrostatic pressure, a gravity force due to the ice and pore water in x-direction,
shear stress on the underside of the jam from the water flow, and shear stress at the
banks. The strength of the jam also resists ice motion, provided stress acting through
the jam does not exceed the maximum allowable longitudinal stress. That limiting
stress places the jam in the Rankine passive state of stress.
The hydrostatic pressure forces acting at sections x1 and x2 are Fp2′ and Fp2″
respectively. With the level of the phreatic surface above the bottom of the jam
denoted as siη(x), the vertical distance above the bottom of the jam denoted as δ(x),
and the local width as b(δ), then for any section x
s η(x)
Fp2′ = g ∫ ρ[si η(x) - δ] b(x, δ)dδ.
i
(71)
0
For a rectangular channel, b is constant in the vertical and equal to the top width B.
Thus
s η(x)
Fp2′ = g ∫ ρ[si η(x) - δ] Bdδ.
i
(72)
0
Integration of eq 72 yields
(s η) .
2
Fp2′ = ρgB i
(73)
2
Consequently, the time integral of the net pressure Fp2 is
(
)
t2
t2
t2
′
″
∫ Fp2dt = ∫ Fp2 - Fp2 dt = g ∫ (ρI2 )x - (ρI2 )x dt
(74)
t
2
1
t
t
1
1
1
where
(si η)2 .
=B
(75)
I2
2
A gravity force acts on the bottom surface of the ice control volume (the bottom
of the jam) from the weight of the ice and pore water above. The horizontal compo-
nent of this gravity force equals and counterbalances Fg2
x2
x2
[
]
Fg3 = ∫ ρi gAi (1 - p)Sib + ρgAi si pSib dx = ∫ [ρgAisiSib ] dx.
(76)
x1
x1
For the period t1 to t2
t2
t2 x2
d
∫ Fg3dt = ∫ ∫ ρgAisi So - dxdt.
(77)
x
t1
t1 x1
The shear stress on the underside of the jam attributable to water flow is equal
and opposite to the shear force of the cover on the water, i.e.
t2
τi = ∫ Ffidt.
(78)
t1
In the case of the jam, however, only the shear force along the jam underside, or
τiPi, is included. From eq 62 above
38
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