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be zero. However, the value of
DI
PA
computed
from Eq (5) will be greater than zero. Similarly,
when the system reaches its maximum mono-
tonic deformation, while
DI
PA
should be 1.0,
however, Eq (5) leads to
DI
PA
greater than 1.0.
Chai et al (1995) proposed modification to
DI
PA
to correct for the second drawback only. The study,
also, examined experimentally the implication of
the energy-based linear damage model of
DI
PA
.
Despite the drawbacks of
DI
PA
, it has been exten-
sively used by many researchers, mainly due to
its simplicity and the extensive calibration against
experimentally observed seismic structural dam-
age during earthquakes (mainly for reinforced
concrete structures). Bozorgenia & Bertero (2003)
proposed two improved damage indices that
overcome some of the drawbacks associated with
DI
PA
.
In this chapter, worst earthquakes that maxi-
mize the structure's damage were obtained using
deterministic methods. Critical earthquakes can
be formulated using stochastic processes, random
vibration theory and reliability analysis which
provides a powerful alternative to the methodol-
ogy developed here (see, e.g., Abbas & Manohar,
2005, 2007).
is of substantial importance in deriving critical
earthquake loads for inelastic structures. This is
because damage indices imply that the structure
is damaged by a combination of repeated stress
reversals and high stress excursions. This also
facilitates assessing the safety of the structure by
providing a quantitative measure on the neces-
sary repair.
In this chapter, the worst earthquake load is
derived based on available information using
inverse nonlinear dynamic analysis, optimization
techniques and damage indices. It was seen that if
available information is limited to the energy and
PGA, the resulting earthquake is highly resonant
and produces conservative damage. When extra
information on the Fourier amplitude spectra is
available, more realistic earthquake loads (in
terms of frequency content, amplitude, inelastic
deformations and damage indices produce) are
obtained. The influences of the strain hardening
and damping ratios on the estimated design loads
were studied. Critical damage spectra for the site
were also established. These spectra provide upper
bounds on the structural damage and necessary
repair under possible future earthquakes. The
formulation developed in this chapter was dem-
onstrated for inelastic frame structures modeled
with bilinear and elastic-plastic force-deformation
laws. In other words, non-deteriorating structures
are only considered. Future extension of the
present research requires the use of nonlinear
degradation models that facilitate the development
of plastic hinges in the structure. In this case the
computations will increase considerably due to the
complexity in estimating the structural response.
Finally, it may be emphasized that in the present
work, the structural properties have been kept
unchangeable. It is possible to apply the proposed
methodology for optimal design of the structure
under future earthquakes. Herein, an initial guess
for the dimensions of the structure's members
needs to be assumed and an iterative procedure
has to be carried out leading to the optimal de-
sign of the structure, the system-dependent worst
5. CONCLUDING REMARKS
A methodology for assessing damage in structures
under critical future earthquake loads is developed
in this chapter. The novelty of this research lies
in combining damage indices, nonlinear opti-
mization and nonlinear time-history analysis in
assessing the structural performance under future
earthquakes. Damage descriptors are introduced
in deriving the worst earthquake ground motion.
The structural damage is quantified in terms of
Park and Ang damage indices. As is well known,
damage indices describe the damage state of the
structure and correlate well with the actual damage
displayed during earthquakes. The quantification
of the structure's damage using damage indices
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