Geology Reference
In-Depth Information
As a test example, a 72-bar space steel tower
subjected to the El Centro earthquake is optimized
considering nonlinear time history responses.
The numerical results indicate that the hybrid
methodology is a powerful and efficient tool for
optimal design of structures subjected to earth-
quake loadings.
The problem of structural optimization for
earthquake induced loads can be formulated as
follows:
d
Minimize
f X
(
)
, X;
X
∈
R
;
i
=
1,
,
n
i
(2)
Subject to
g X Z t
j
(
,
( ),
Z t
( ),
Z t
( ),
t
)
≤
0
;
j
=
1
,
,
m
MAIN FOCUS OF THE CHAPTER
(3)
Main focus of the present chapter is to provide
an efficient soft computing based methodology to
achieve optimum design of structures subject to
earthquake. The methodology is a serial integra-
tion of evolutionary algorithms for performing
optimization and neural network for predicting
nonlinear time history responses of structures.
The various issues of the proposed methodology
are described in the following subsections.
L
U
X
≤
X
≤
X i
;
=
1
,
,
n
i
i
i
where
f
,
X
,
g
,
m
and
n
are objective function,
design variables vector, behavioral constraint, the
number of constraints and the number of design
variables, respectively. Also
X
i
L
and
X
i
U
are
lower and upper bounds on the
i
ith design variable.
A given set of discrete values is presented by
R
d
.
In this chapter, for the time-dependent nonlin-
ear optimization problems, only the displacement
constraints are considered in Box 1.where
d
j
and
d
j,all
are the displacement of the
j
th node, and its
allowable value, respectively.
To perform dynamic time history analysis con-
sidering geometrical and material nonlinearities,
ANSYS software (ANSYS, 2006) are employed. It
uses a step-by-step implicit numerical integration
procedure based on Newmark's method to solve
the dynamic equilibrium. In order to consider the
transient nature of earthquake loading a simple
bilinear stress-strain relationship with kinematic
hardening is adopted. As confirmed in (Lagaros
et al
., 2006) this law provides accurate results for
many practical applications.
Optimal Design Problem Formulation
The dynamic equilibrium for a finite element
system subjected to earthquake loading can be
written in the following usual form:
M
Z t
( )
+
C
Z t
( )
+
K
Z t
( )
=
MI
u
( )
t
g
(1)
where
M
,
C
, K, I,
Z ( )
,
Z ( )
,
Z ( )
,
u
g
( )
and t are mass matrix, damping matrix, stiffness
matrix, unit matrix, acceleration vector, velocity
vector, displacement vector, ground acceleration
and the time, respectively.
t
Box 1.
d X Z t
(
,
( ),
Z t
( ),
Z t
( ),
t
)
j
g X Z t
(
,
( ),
Z t
( ),
Z t
( ),
t
)
=
− ≤
1
0
;
j
=
1
, ...,
m
(4)
j
d
j all
,
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