Geology Reference
In-Depth Information
When the structure is subjected to base ac-
celeration
u t
is found for structures loaded by forces excited by
one specific earthquake (El Centro). Moreover, the
objective function is different. Now the maximal
peak values of the bending moment in columns is
the objective function. The aim of this example
and Example 1 is to show how different can be the
optimal solution for various objective functions.
g
( )
, the excitation vector is written
= −
, where
r
is the influence
vector with values 0 or 1. For harmonic external
forces, we have
as
p
t
M r
u t
g
( )
( )
=
λ
, where
U
g
is the amplitude of base acceleration. The displace-
ment response of the structure is given by relation-
ship (30), and
Q
s
(
λ
is determined from:
u t
( )
U
exp(
i t
)
g
g
Example 1: An Eight-Storey Frame
=
Q
λ
H
λ
U
(52)
( )
( )
An eight-storey RC frame with three bays is se-
lected for which the optimized position of VE
dampers and the optimal parameters of dampers
are determined. The frame is designed according
to the requirements of EC8 Part 1 for Class B
(stiff soils). The height of the columns is 3.0 m,
the span of the beams is 5.0 m and Young's
modulus for concrete is 31.0 GPa. The dimensions
of the cross-section of structural elements are
presented in Table 1 while the unit masses of the
frame elements are given in Table 2. The frame
is treated as a planar structure. The axial deforma-
tions and internal damping of the structure are
neglected. However, the internal damping of
structure can be taken into account assuming that
the damping matrix is in the form
C
s
g
where the vector
H
= −
will be called
the vector of frequency transfer functions of
displacements caused by kinematic excitation.
λ
H Mr
λ
( )
( )
NUMERICAL EXAMPLES
Four examples are presented in this section to
illustrate several aspects of the considered optimi-
zation problem. Among others, one aim of all the
examples is to illustrate the possibility of reduction
of vibrations of frame structures with the help of
VE dampers. In the chapter a few structures are
analyzed in order to enlarge the diversity of the
considered problems.
a
0 1
,
where
a
0
and
a
1
are some constant (see Chopra
(2000) for details). In this example, internal damp-
ing of structure is neglected because our aim is
to show the influence of dampers on dynamic
characteristics of structure only. Of course, in real
structures more or less internal damping always
exists and it enlarges the modal damping ratios
of structures.
It is assumed that the dampers could be placed
on all the floors of the structure, in the middle
bay. The dampers are modelled using the general-
ized models with seven parameters. In this ex-
ample we introduce the comparative damper for
which the storage and loss modulus are calcu-
lated from the formulae:
=
M
+
K
Optimal Placement of Dampers
for Frame Structures: Classical
Rheological Models of Dampers
The first example shows the influence of damp-
ers on dynamic characteristics of structure. The
main objective of the second example is to show
that dynamic characteristics of a structure with
dampers, modelled using two different rheological
models of dampers is practically identical if both
models have approximately equal possibilities to
dissipate energy. A frame very similar to the one
analyzed in Example 1 is considered in Example
3. However, the structure is now analyzed in a
time domain and the optimal location of dampers
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