ρf (u - υ) Pi
2
τi Pi = i
(79)
.
8
Substituting for Ffi gives
ρfi (u - υ) Pi
2
t2
t2 x2
t2 x2
∫ Ffidt = ∫ ∫ τi Pidxdt = ∫ ∫
dxdt.
(80)
8
t1
t1 x1
t1 x1
From the assumption that ice jams can be treated as a particulate continuum, the
vertical and horizontal stresses within jams can be related using Rankine stress
theory. Previous researchers, as discussed in the Review of Ice Jam Modeling section,
give the shear stress at the banks τxy as a function of the stress along the jam's
longitudinal axis σx. For a granular material, the normal stress can be expressed
directly as a function of the vertical stress σv, which results from the weight of ice
acting downward to the phreatic surface through the jam, and from the buoyancy
force acting upward in the submerged portion. Vertical stress ascribable to ice weight
acting downward throughout the entire cover thickness is
σvi = ρi gy(1 - p)
(81)
where y is measured from the top of the ice surface. The stress of the water acting
downward in the wetted portion is
(
)
σvw = ρg y - [1 - si ]η p
(82)
where y is measured from the top surface of the ice and y = (1 si)η represents the
phreatic surface. The buoyancy force acting upward in the wetted portion of the
thickness is
(
)
u′ = ρg y - [1 - si ]η .
(83)
From eq 81 to 83, the total or effective vertical stress σv is zero at the upper and
lower surfaces of the jam, and σv reaches a maximum at the phreatic surface. Over
the entire thickness, the average vertical stress is
η
σv = siρg(1 - p)(1 - si ) .
(84)
2
The horizontal or longitudinal stress in a granular material relates to the average
vertical stress as
σx = k1σv
(85)
where k1 is a coefficient of proportionality reflecting the jam's state of stress, i.e.,
passive, neutral, or active. For passive-pressure loading conditions, k1 typically
assumes the value of the passive pressure coefficient Kp, which represents the maxi-
mum or failure stress of the material
φ
Kp = tan2 45 +
(86)
2
where φ is the angle of internal resistance measured in degrees. Equations 81, 82,
39
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