Geoscience Reference
In-Depth Information
2
Q
0
= kU
0
cos(
2
t +
) / cos
with tan
=
(
c
2
/k
) / (1
M
2
/k
)
(12.37b)
Adopting Ehlers' model this becomes in terms of soil properties with (12.36)
w
0
= U
0
cos(
2
t
)
(12.38a)
Q
0
= U
0
(
EA
0
/h
) cos(
2
t +
) / cos
2
/EA
0
)
with tan
=
(
h
2
/a
) / (1
Mh
2
(12.38b)
Soil response to an impact loading
In case of a shock load the eigen frequency
1
of this system will dominate. For
this type of loading the equilibrium of the mass-spring-dashpot system is described
by
2
Mw
0
,tt
+ cw
0
,t
+ kw
0
= Q
00
?
(
t
)
with
?
(
t
) the Dirac delta function
(12.39)
Laplace transformation gives
&
st
= Q
00
/
(
Ms
2
+ cs + k
)
w
e
dt
(12.40)
0
0
Solving with Heaviside's theorem leads to
w
0
= Q
00
e
ct/
2
M
sinh(
2
with
1
=
((
c/
2
M
)
2
k/M
)
½
2
1
t
)
/
2
2
(12.41)
1
This shows that for
D = c/
2(
kM
)
½
> 1
the system shows smooth damping, and
for
D <
1 it is oscillating and damping (resonance). The critical damping is
therefore expressed by
D =
1, which leads to
c =
2(
kM
)
½
EA
0
/a =
2(
EA
0
Mh
)
½
)
½
=
2(
EA
0
M/h
)
½
EA
0
/
(
E/
A
0
h
=
4
M
This gives
A
0
h
=
4
Mg =
4
W =
4
A
0
0
0
=
¼
h
(12.42)
Here,
W
is the weight of the foundation (and structure above it) and
0
the static
contact pressure between the foundation and the soil. For
0
>
¼
h
oscillations
will occur, and for
0
<
¼
h
the system will only show smooth damping. Here, the
choice of
h
plays a role.
Static case determined from dynamics
Solution (12.38) shows for the static case, i.e.
M =
0,
c =
0 and thus
=
0, that
at the surface the displacement and force are described by
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