Civil Engineering Reference
In-Depth Information
The integration schemes used at the ELSA belong to the so-called “
E
-Newmark”
family. For these schemes, when the displacement, speed and acceleration vectors at
time t are known, the values of the same vectors at time t +
'
t are given by
Newmark's formulae:
2
'
t
n
1
n
1
2
n
1
n
1
n
n
n
d
d
'
t
E
a
d
d
'
tv
(1
2
E
)
a
[5.2]
2
(1
n
1
n
1
n
1
n
1
n
n
v
v
'
t
J
a
v
v
'
t
J
)
a
and verify the equilibrium system expressed in equation [5.1], shifted in time thanks
to parameter
D
:
n
1
n
1
n
n
1
n
n
1
n
Ma
(1
D
)
Cv
D
Cv
(1
D
)
r
D
r
(1
D
)
f
D
f
[5.3]
If we choose
D
= 0,
E
= 0 and
J
= ½, we obtain the well-known central finite
difference method. The pattern is explicit, as it is possible to calculate
d
n+1
from
known quantities then impose this displacement, measure the reaction load
n
1
r
,
a
1
v
1
and finally calculate
and
.
This scheme is perfectly fitted to the PSD method. Unfortunately, it is
conditionally stable. The solution may become unstable (i.e. increase indefinitely)
when the value of the reduced frequency
f
is
the highest natural frequency of the dynamic system considered. If
f
is high, the
time step becomes very small. Then we have to impose many displacement
increments on the structure, some of which are likely to be smaller than the accuracy
of the control system, which may bring about very large errors.
' is higher than 2. Here
:
2
S
f
t
0
0
0
The frequency
f
may be high for stiff structures including a small number of
dofs, or for most structures when the number of dofs becomes important. The
excellent accuracy of the displacement checking system used at the ELSA (see
section 5.2.2) allows test stiff structures owing to this explicit scheme without any
real difficulty. The scheme is also used to the exclusion of all others by the
continuous PSD method (see section 5.2). The number of dofs can become very high
in sub-structuring tests (see section 5.2.3), and in these cases, other integration
schemes have to be introduced.
2
>
@
If we choose
E
(1
D
)
4
and
J
(1
2
D
) 2
, with
D
13,0
, we again
n
1
find the
D
method [HIL 77]. This scheme is implicit, because the displacement
d
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