Geoscience Reference
In-Depth Information
w
0
= U
0
cos(
2
t
)
(12.31a)
Q
0
= U
0
(
EA
0
/h
) cos(
2
t +
) / cos
with tan
=
2
h/a
(12.31b)
Next consider the Kelvin-Voigt model: a spring-dashpot model (Fig 12.8b)
without mass. Harmonic displacement
w = U
0
e
i
2
t
for a weightless system (mass
M
=
0) is described by the following equilibrium
cw
0,
t
+ kw
0
= Q
0
= Q
00
e
i
2
t
or
(i
2
c + k)U
0
= Q
00
(12.32)
Here,
c
is the dashpot constant and
k
the spring constant. Hence
Q
0
= Q
00
e
i
2
t
= U
0
(
i
2
c + k
) (cos(
2
t
)
+ i
sin(
2
t
))
=
U
0
(
k
cos(
2
t
)
c
2
sin(
2
t
)
+ ik
sin(
2
t
)
+ ic
2
cos(
2
t
))
(12.33)
For the real motion, the real part of (12.33) must be considered
t
))
= U
0
(
k
2
+
(
c
)
2
)
½
cos(
Q
0
= U
0
(
k
cos(
2
t
)
c
2
sin(
2
2
2
t +
?
)
/k
)
2
)
½
cos(
= kU
0
(1 + (
c
2
2
t +
?
)
=
= kU
0
cos(
2
t +
?
) / cos
?
with tan
?
= c
2
/k
(12.34)
Thus, the system's motion of a weightless spring-dashpot system is described by
w
0
= U
0
cos(
2
t
)
(12.35a)
Q
0
= kU
0
cos(
2
t +
?
) / cos
?
with
tan
?
= c
2
/k
(12.35b)
Perfect similarity with Ehlers' model (12.31) is obtained by putting
E
)
½
k = EA
0
/h
and
c = EA
0
/a = A
0
(
(12.36)
The conclusion is that for dynamic oscillations the subsoil, with uniform stress
within a cone, can be represented by a simple spring-dashpot system, the
coefficients of which can be related to intrinsic soil properties, according to
(12.36). Here, the geometric factor
h
is related to the apex of the cone, which is a
choice: the apex
=
90
o
, so
h = r
0
.
The dashpot constant
c
of (12.36) is equivalent to expressions in
literature, e.g. Verruijt:
c
=
is incorporated through
h = r
0
tan(
/2). A common choice is
G
)
½
/2
m
with
m
2
=
(1
).
Next, mass is incorporated. Following the mass-spring-dashpot theory, a
harmonically oscillating circular disk with mass
M
, representing the mass of a
foundation (and structure on it), will induce a response at the interface between soil
and foundation, according to
A
0
(
2
)/(2(1
w
0
= U
0
cos(
2
t
)
(12.37a)
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