Geoscience Reference
In-Depth Information
pressure distribution is approximated as a uniform pressure
q
over the cone cross-
section. No energy is assumed to be transmitted outside the cone and therefore the
induced soil pressures decrease proportional to depth squared. Also Boussinesq's
solution has this trend, see equation (12.23). Adopting elastic soil, i.e. elasticity
E
and absence of lateral deformation (
=
0), the vertical force acting in the cone
becomes
Q = qA
0
(
z/h
)
2
=
EA
0
(
z/h
)
2
w
,z
(12.25)
Dynamic equilibrium of a thin disk out of the cone at depth
z
, having mass
density
, is expressed in terms of the vertical displacement
w
by
A
0
(
z/h
)
2
w
,tt
=
(
EA
0
(
z/h
)
2
w
,z
)
,z
or
w
,tt
= a
2
(2
w
,z
/z + w
,zz
)
(12.26)
Here,
a =
) is the vertical (compression) wave celerity within the cone.
This equation can be solved for harmonic oscillation, using complex algebra.
The harmonic motion is expressed by
w = Ue
i
2
t
= U
[cos(
(
E/
2
t
) +
i
sin(
2
t
)], where
U
is the amplitude, a function of
z
only. This renders (12.26) into
2
U = a
2
(2
U
,z
/z + U
,zz
)
2
(12.27)
A solution of this equation is sought for
U = Be
i
)
z
/z
. Substitution yields
2
/z =
2
a
2
(
2
/z)
, which is valid for all values of
z
, and gives
)
)
=
4
2
/a
. The solution is
U
= Be
-i
2
z/a
/z
, which satisfies
U = 0
for
z
&
. The value at the surface becomes
U
0
= Be
2
h/a
/h
,
B = U
0
he
i
2
h/a
U = U
0
he
i
2
(
z-h
)
/a
/z
. The displacement becomes
w = Ue
i
2
t
= U
0
he
i
2
(
t
(
z
h
)
/a
)
/z =
=
(
U
0
h/z
) [cos(
2
(
t
(
z
h
)
/a
))
+ i
sin(
2
(
t
(
z
h
)
/a
))]
(12.28)
For the real motion, only the real part of (12.28) should be considered, which is
w = U
0
(
h/z
) cos(
2
(
t
(
z
h
)
/a
))
(12.29)
The corresponding force
Q
can be elaborated from (12.25)
Q = EA
0
(
z/h
)
2
w
,z
=
= EA
0
(
z/h
)
2
U
0
h
[
(1
/z
2
)cos(
2
(
t
(
z
h
)
/a
))
+
(
2
/za
) sin(
2
(
t
(
z
h
)
/a
))]
= EA
0
U
0
/h
[
cos(
2
(
t
(
z
h
)
/a
))
+
(
2
z/a
) sin(
2
(
t
(
z
h
)
/a
))]
=
z/a
)
2
)
½
cos(
= EA
0
U
0
/h
(1 + (
2
2
(
t
(
z-h
)
/a
)
+
)
= U
0
EA
0
cos(
2
(
t
(
z
h
)
/a
)
+
)
/
(
h
cos
)
with tan
=
2
z/a
(12.30)
Finally, at the surface Ehlers' model yields for the displacement and the force
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