Geology Reference
In-Depth Information
3. STRUCTURAL RESPONSES
UNDER DYNAMIC LOADS
In their approach, Bendsøe and Kikuchi consid-
ered special microstructures as the building cells
of the structure and employed the homogenization
method to find the macro-scale properties of the
cells in terms of their micro-scale dimensions.
By considering the dimensional properties of the
microstructures as design variables, they reduced
the topology optimization of the structure to sizing
optimization of its microstructures.
Using the idea behind the homogenization
method, Bendsøe (1989) introduced a simpler
approach to optimize the topology of structures.
In this new approach instead of using the micro-
structures and homogenization, Bendsøe proposed
an artificial material interpolation scheme relating
the material properties of the elements to their
relative density. After Rozvany et al. (1992), this
approach is referred to as 'Solid Isotropic Micro-
structures with Penalization (SIMP)'. The SIMP
approach is now one of the most established and
popular methods in topology optimization.
A simple FE-based topology optimization
technique was later proposed by Xie and Steven
(1993). Named Evolutionary Structural Optimiza-
tion (ESO), the technique was based on the idea
of evolving the structure towards an optimum
design by progressively removing its inefficient
elements. The Bi-directional ESO (BESO) was
the main successor of the ESO method. Initially
introduced by Querin (1997), Querin et al. (1998)
and Yang et al. (1999a), the BESO algorithm was
capable of adding as well as removing elements.
This method is now a well-known topology op-
timization technique which is widely used due to
its clear topology results and ease of application.
The SIMP and BESO techniques will be de-
tailed and used in later sections of this chapter.
In the next section we investigate the equation of
motion of a structural system to find out which
parameters shape the responses of structures under
dynamic loads.
Consider the equation of motion for a finite ele-
ment discretized linear system
Mu Cu Ku
+
+
=
p
(1.1)
where
M
,
C
and
K
are mass, damping and stiff-
ness matrices respectively and
u
and
p
are time-
dependent vectors of nodal displacement and nodal
force respectively, i.e.,
u
≡
u
(
t
) and
p
≡
p
(
t
). We
assume a classical damping (Chopra 1995), for
example Reyliegh damping of the form
C
=
a
M
+
a
K
(1.2)
M
K
where
a
M
and
a
K
are constants.
We now expand the displacements in terms of
modal contributions
N
d
∑
ϕ
1
u
( )
t
q t
r
( )
=
(1.3)
r
r
=
where
N
d
is the number of degrees of freedom and
1
(1.4)
q t
( )
=
C
cos
ω
t
+
S
sin
ω
t
,
r
=
,
,
N
r
r
r
r
r
d
are harmonic functions and
C
r
and
S
r
are constants
of integration associated with the
r
th degree of
freedom. The natural frequencies
ω
r
and natural
modes
φ
r
are solutions of the following eigenvalue
problem
=
ω
2
K
ϕ
M
ϕ
(1.5)
r
r
r
which represents the free vibration of the un-
damped system. For simplicity, we further require
that the modes are
M
-orthonormal, i.e.,
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