Civil Engineering Reference
In-Depth Information
The “dynamic stiffness matrix” is given by:
4
2
t
>@ >@>@
ªº
K
MC
K
(16.21)
¬¼
2
'
'
t
and
§
4
4
·
^`
>@
^
`
^
` ^
`
FM
u
()
t
u
()
t
u
()
t
¨
¸
2
'
t
©
'
t
¹
(16.22)
§
2
·
>@
^
` ^
` ^
`
Cu
()
t
u
()
t
F
()
t
¨
¸
'
t
©
¹
Since we have already worked out a “dynamic stiffness matrix” of the boundary element
region in Chapter 14 the coupling procedure is now straightforward. For a fully coupled
problem the system of equations is given by
^`
^` ^`
ª
º
ª º
(16.23)
NK
K
u
()
t
F
F
¬
¼
¬ ¼
BE
FE
BE
FE
16.4.1
Example
The example is that of a concrete column embedded in a semi-infinite soil mass. The
description of the problem can be seen in Figure 16.9. The top of the column is subjected
to a suddenly applied load
p(t)
of 1 MN/m
2
. The material properties for the column are:
spec. weight= 2500 kg/m
3
,
E
=30 000 MN/m
2
,
Q
=0.2. For the soil we have: spec.
weight= 2000 kg/m
3
,
E
=100 MN/m
2
,
Q
=0.2.
p t
()
p t
()
t
Figure 16.9
Description of example
Figure 16.10 shows the mesh used for the analysis it consists of a finite element region
that describes the column and a boundary element region that describes the semi-infinite
ground. The mesh has 1500 degrees of freedom. Figure 16.11 shows the time-dependent
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