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otherwise a new target point has to be assumed
on capacity spectrum, and the process should be
repeated. Figure 2 illustrates the procedure.
Having the nonlinear demand obtained via any
of aforementioned analyses and subsequent tech-
niques, and comparing it to the specification of
acceptance criteria at desired performance level(s)
identifies the performance level of a structure.
If for a certain degree of freedom at a node,
the internal forces of all members connected to
that node are summed up and equated to the corre-
sponding external load, an equation of equilibrium
for that particular degree of freedom will be gen-
erated. If this is done for all degrees of freedoms,
a set of nonlinear equations can be obtained. The
nonlinear analysis procedure is the art of finding
a displacement vector that satisfies the set of all
nonlinear equations of equilibrium in Eq.(14),
simultaneously. Once the displacement vector
in Eq.(14) is obtained, it remains to calculate the
internal forces in all members. At this stage, the
pre-defined force-deformation relation is used to
obtain internal forces.
Since the performance-based design in its
general form requires nonlinear analysis, design
engineers in this field are recommended to study in
details some useful publications before employing
any commercial software. This will help to under-
stand the merits of different nonlinear schemes
with respect to each other. Crisfield (2000) and
Owen and Hinton (1980) have cited good sum-
maries of classical nonlinear analysis techniques.
A brief description of some of these methods that
are suitable for monotonically increasing curves is
provided hereunder and summarized graphically
in Figure 3.
A BRIEF LOOK INTO
NONLINEAR ANALYSES
Any nonlinear analysis procedure requires the
establishment of nonlinear set of equations of
equilibrium as follows:
[
]{ }
=
{ }
K x
( ,
∆ ∆
P
(14)
This in turn, necessitates the definition of a
force-deformation relation in member level. If
the force-deformation relation is known in its ex-
plicit mathematical form, the assembly of explicit
nonlinear stiffness matrix is theoretically possible,
otherwise, as it often happens, the stiffness matrix
will be a numerically established matrix that is an
implicit function of deformation.
Figure 2. Capacity spectrum method
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