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ture is obtained using iterative initial stiff-
ness technique.
Despite its robustness, this method suffers from
the considerable number of variables that enter
in the QP sub-problem. Recently Tin Loi has
improved this technique. Among others, a paper
by Tangaramvong and Tin Loi (2011) may be
consulted in this field.
There are also some other techniques that have
been established for inelastic analysis of structures
based on theorems of Structural Variation. Struc-
tural variation theory studies the effect of change
of properties, or even removal, of a member on
the entire structure. It takes advantage of linear
analysis and sensitivity of structure to some self
equilibrating unit loads that are applied at the
end nodes of changing members. This technique
has been applied to analysis of several types
of inelastic skeletal structures including space
trusses, frames, and grids, etc. It has been also
extended to nonlinear finite elements analysis.
Although this method takes advantage of initial
stiffness matrix and does not require a change
in the stiffness matrix of structure during the
analysis process, it is a hierarchical and step by
step method of analysis in which every step uses
information from the previous step and is not
a proper nonlinear analysis that is to be joined
to a performance-based design program. As an
example in this field one may start with a paper
by Saka (1997)
Nonlinear analysis of structures by the math-
ematical programming is another field of research
in this ground. De Donato (1977) presented
fundamentals of this method for both holonomic
(path independent) and nonholonomic material
behaviors. In this method, it is assumed that dis-
placement of nodes of an elasto-plastic structure
comprises two parts namely elastic and plastic
parts. Then, the problem of finding total displace-
ment vector of a structure is formulated in the
form of a quadratic programming (QP) problem
with some complementary yield constraints. These
yield constraints state that individual members
either are stressed within elastic limits and do not
accept plastic deformations or, are stressed up to
yield limit, and as a result, undergo some plastic
deformations. The output of this sub-problem is
linear and nonlinear deformation of the structure.
DESIGN OPTIMIZATION
The traditional design as shown in Figure 4a
consists of a cycle of four components. They
are:
Design, Analysis, Feasibility-check
and
Design improvement
. If the initial design is not
satisfactory or feasible, a revision on the initial
design is made, and the design cycle is repeated.
The process stops when the design is deemed
satisfactory. A design that is obtained in this way
may be satisfactory but not optimum. If, as shown
in Figure 4b, in the design cycle an optimization
component is added to enhance the design intel-
ligently, the outcome of the design cycle may be
both satisfactory and optimum.
Therefore, the design optimization may be
defined as a mathematical means that is used to
evolve a structure from its initial form to a final
form with characteristics of being the optimum.
This process may be employed for a member by
member optimization or whole-structure. To
optimize a design, it should be written in the form
of the standard optimization problem as follows:
Minimize Z x
Subject to
( )
:
g x
( )
≤
0
;
j
=
1 2
,
, ...,
m
j
h x
( )
=
0
.
;
k
=
1 2, ...,
p
,
k
(15)
where
x
is the generalized vector of design vari-
ables including
x
1
to
x
n
.
The design variables
establish the design space.
Z
is the objective func-
tion.
g
and
h
are inequality and equality design
constraints, respectively. Design constraints divide
the design space into feasible and infeasible sec-
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