Environmental Engineering Reference
In-Depth Information
The horizontal pressure acting on the wall of the tube causes a vertical friction
stress
˄
between the wall and the granular material. A simple balance of the vertical
forces acting on a slab of the granular material gives
dp
z
dz
=
ˀ
r
0
r
0
ˁ
ˀ
g
−
2
ˀ
r
0
˄,
(13)
where
is the bulk density of the granular material, assumed to be independent of
z
.
In the original Janssen's model, the vertical friction stress is assumed to be given by
Coulomb's law,
ˁ
˄
=
μ
w
p
r
, with a constant static wall friction coefficient
μ
w
.The
solution of Eq. (
13
) with the condition
p
z
(
)
=
0
0 is then
p
z
(
z
)
=
P
J
(
z
),
(14)
with
1
exp
z
ʻ
r
0
P
J
(
z
)
=
ˁ
g
ʻ
−
−
and
ʻ
=
μ
w
K
.
(15)
2
Equation (
15
) shows that the vertical pressure increases linearly with
z
,as
p
z
(
z
)
≈
ˁ
is
the characteristic size of the region where the pressure undergoes this transition. In
laboratory columns
gz
,for
z
/ʻ
1 and tends to the limiting value
ʻˁ
g
for
z
/ʻ
1. The length
ʻ
1m, and so the wall of the container supports most of the
weight of the grains when
H
ʻ
≈
0
.
1m.
As it was modified above, the two-parameter model suppresses friction with the
wall in a slice at the top of the column by making
˄
=
0for
z
<
a
and
˄
=
μ
w
p
r
for
z
a
, where
a
, the thickness of the frictionless slice, is the second parameter of
the model. The solution of Eq. (
13
) is then
>
ˁ
gz
for
z
<
a
p
z
(
z
)
=
(16)
+
1
P
J
(
a
ʻ
ˁ
ga
−
z
−
a
)
for
z
>
a
.
˄(
)
causes an additional deformation of the wall of a
tube filled with granular material. The maximum vertical force acting on the wall of
a tube filled to a height
H
is
The vertical shear stress
z
H
0
T
(
H
)
=
2
ˀ
r
0
∫
˄(
z
)
dz
=
T
J
(
H
),
(17)
with
g
H
1
exp
H
ʻ
r
0
ˁ
T
J
(
H
)
=
ˀ
−
ʻ
−
−
.
(18)
The experimental test of the previous equations will be left for future work.
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