Environmental Engineering Reference
In-Depth Information
The movement of a spherical boulder positioned without initial velocity on a
rough inclined plane must comply with the following equations [FAV 47]:
dv d
mmr
ω
⎤
=
=
g
sin
β
−
F
⎥
⎥
dt dt
Nmg
0
=−
cos
β
N mg
=
cos
β
[4.29]
⎥
⎥
⎥
d
ω
J
=−
Fr
Ne
dt
⎦
2
5
2
We can hence deduce, with
J
=
mr
:
mg
e
⎛
⎞
⎠
[4.30]
F
=
2sin
β
+
5
cos
β
⎜
⎟
7
r
⎝
The displacement velocity, parallel to the slope, is equal to:
5
g
e
⎛
⎞
v
=
sin
β
−
cos
β
⎠
t
[4.31]
⎜
⎟
7
r
⎝
The motion is
uniformly accelerated,
except if
e > rtg
β
.
In this case, the boulder
remains in place without rolling, which can only happen for small slopes or for very
soft strata, where
e
would not be very small given the radius of the boulder.
It is different when the shape of the boulder is angular and its section distant
from a circle (see Figure 4.10). The theoretical point of rotation contact coincides
with an edge of the boulder and exercises a reactive stabilizing force as long as the
angle of the slope β does not exceed the half-angle at the center
a
= 180°/
n
(
n
:
number of sides of the polygonal section):
o
- hexagonal stable boulder for
β
≤
30 ;
o
- square, stable boulder for
β
≤
45 ;
without counting the effect of the rolling resistance and apart from the risk of
sliding.
