Geoscience Reference
In-Depth Information
Av
2
w
0
,
;;
+ k/B w
0
= Q
0
/
B
EIw
0
,
;;;;
+
(12.46)
=
0. The harmonic solution of (12.46) can be found by
trying
w
0
= U
0
e
a
;
, and the corresponding characteristic equation is
The load
Q
0
is acting at
;
EIa
4
+
Av
2
a
2
+ k/B = 0
(12.47)
The roots of this equation are
2
2
2
Av
4
(
Av
)
4
kEI
/
B
a =
4
(12.48)
2
EI
U
0
i
e
a
i
;
. From (12.48) one may
distinguish four independent roots. One may recognise a critical velocity when
(
The general solution is therefore
w
0
=
%
Av
2
)
2
-
4
kEI/B =
0 or
4
kEI
v
c
(12.49)
4
2
2
A
B
Elaboration shows that for
v > v
c
all the roots
a
i
are purely imaginary,
representing undamped vibrations. For
v < v
c
all the roots are complex,
representing attenuating vibrations. Therefore, the critical velocity represents a
characteristic value, essential for the design of foundations, which must sustain
dynamic loads.
The previous analysis is a simplification of the reality; it considers bending
waves in the beam and approximated pressure waves in the subsoil, while material
damping is disregarded. Dynamic response of soils contains vibrations related to
three types of acoustic waves, i.e. pressure waves, shear waves and interface
waves, e.g. Rayleigh wave along a surface, each travelling with a specific celerity,
for pressure waves the celerity is
(
K/
), for shear waves
(
G/
), and for surface
waves slightly less than
). The most disturbing wave is usually the Rayleigh
wave. It carries most of the dynamic energy relatively far. The practical
implementation of dynamics in (rail)way embankments, particularly at specific
construction elements, is rather complex (Degrande, Hölscher, Powri).
(
G/
Soil response including damping
Verruijt introduced a special form of damping: hysteretic damping, which is
based on frictional damping. It may be more realistic in granular soils than viscous
damping. From (12.41) it can be seen that the resonance frequency (eigen
frequency) for the undamped system is
1
=
(
k/M
)
½
2
(12.50)
Here,
k
is the spring constant and
M
the mass. The damping ratio is dependent on
the frequency
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