Geology Reference
In-Depth Information
(Oh et al. 2009) as depicted in Figure 12a. In
these installations the device will deform mainly
in shear (Figure 12a). The design domain is the
inner part of the web as shown in Figure 12b. The
two 15 mm strips on the boundaries of the web
are non-designable.
We use the BESO method here and introduce
a simple technique to solve shape optimization
problems using BESO. Restricting the topology
of the design and performing shape optimization
instead of topology optimization is useful when
the fabrication cost is an important factor. We also
address a simple approach to obtain periodic
designs which are produced by repeating a fixed
pattern.
Considering a volume constraint, the energy
absorption maximization problem can be stated as
∂
∂
K
∂
∂
M
z
∂
∂
K
∂
∂
M
∂
∂
C
z
z
T
−
ω
2
−
z
T
−
ω
2
z
−
2
ω
z
C
C
S
S
C
S
x
x
x
x
x
∂
∂
ω
e
e
e
e
e
=
(
)
x
2
z
T
Cz
+
ω
z Mz
T
+
ω
z Mz
T
e
C
S
C
C
S
S
(1.103)
If we normalize the vectors
z
C
and
z
S
with re-
spect to
M
, for sensitivities of
λ
=
ω
2
, we can write
∂
∂
K
∂
∂
M
z
∂
∂
K
∂
M
∂
∂
C
z
z
T
−
ω
2
−
z
T
−
ω
2
z
−
2
ω
z
C
C
S
S
C
S
x
x
x
∂
x
x
∂
∂
λ
∂
∂
ω
e
e
e
e
e
=
2
ω
=
x
x
z
T
Cz
e
e
C
S
+
2
ω
(1.104)
It is easy to verify that in the without damping
this equation reduces to Equation(1.18). Having
the sensitivities, one can solve the optimization
problem using a suitable solution algorithm.
9. MAXIMIZING ENERGY
ABSORPTION
max
Π
p
x x
,
,
,
x
1
2
N
ˆ
such that
r
= − =
p
p
0
N
Apart from controlling natural frequencies and dy-
namic compliance, improving energy absorption
characteristics is also of significant importance
in seismic design of structures. In recent years,
active and passive energy dissipating devices
have been widely studied and utilized to increase
energy absorption of structural systems (Soong and
Spencer 2002). Topology optimization can be used
to maximize energy absorption of these devices.
In this section we consider the problem of maxi-
mizing the energy absorption of passive energy
dissipaters which make use of yield deformation
of metals to mitigate the excitation energy. These
kinds of energy dissipating devices are popular
due to low cost of fabrication and maintenance
and easy installation (Ghabraie et al. 2010).
We consider an energy damping device which
is made of a 100 mm-long cut of a standard struc-
tural wide-flange section with depth, flange width,
web thickness, and flange thickness of 161.8,
152.2, 8, and 11.5 mm respectively. This device
can be installed in braces connections (Chan and
Albermani 2008) or beam-column connections
∑
x v
=
v
e e
e
=
1
0
x
1
,
e
1 2
,
,
,
N
≤ ≤
=
e
(1.105)
where Π
p
is the total plastic dissipation. Because
the problem involves plastic behavior, one needs to
solve a non-linear equilibrium system of the form
r
= − =
p
p
0
(1.106)
ˆ
This requires an iterative solver to eliminate
the residual force vector
r
defined as the differ-
ence between the external and internal force
vectors,
p
and
ˆ
p
respectively.
The internal force vector is defined as
N
N
∑
∑
ˆ
∫
T
T
T
p
=
G B
σ
d
v
=
G q
e
e
e
e
e
e
=
1
e
=
1
v
e
(1.107)
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