Efficient Robust Optimization of Structures Subjected to Earthquake Load and Characterized by Uncertain Bounded System Parameters - Structural Seismic Design Optimization and Earthquake Engineering - page 106

INTRODUCTION or nominal value of a performance function disre-

garding its variation due to uncertainty. The RBDO
ensures a target reliability of a design for a specific
limit state. An efficient RBDO was presented for
linear (Jensen, 2005) and nonlinear (Jensen, 2006)
deterministic dynamic systems under earthquake
load. Jensen et al. (2008) have further extended
the approach to include the randomness in the
system parameters. Lagaros et al. (2008) proposed
an RBDO procedure for computationally intensive
system under earthquake load and random system
parameters. Mohsine et al. (2005) presented an
RBDO method where solution has been achieved in
a hybrid design space (HDS) considering probabi-
listic variations of parameters under deterministic
dynamic load. The HDS considers reliability level
in the same design space of the objective func-
tions and constraints. An optimum seismic design
criterion was proposed by Marano et al. (2006)
for elastic structures considering deterministic
system parameters. The optimum design of struc-
ture considering system parameter uncertainty as
discussed above is mainly accomplished in the
framework of RBDO to ensure a target reliability
of structure with respect to desired performance
modes. It may be noted here that the studies on
the optimization of dynamic system considering
system parameter uncertainty primarily apply
the total probability theory concept to obtain the
unconditional response or the failure probability
of the system which is subsequently used as the
performance measure. However, the optimization
has been performed without any consideration to
the possible variation of the performance of the
structure due to system parameter uncertainty. It
may be realized that such a design approach not
necessarily corresponds to an optimum design in
terms of minimum dispersion of the performance
objective of the design. Rather, the system may
be sensitive to the variations of the system pa-
rameters due to uncertainty. In order to obtain a
more viable optimum design, the RDO approach
is more desirable which optimizes a performance

The response of a structural system under en-
vironmental loads such as wind, water wave,
earthquake, etc. is highly uncertain and can be best
modelled as a stochastic process. The optimization
of structure under such loads is normally dealt in
the literature in the form of standard nonlinear
optimization problem. The dynamic responses
to define the stochastic constraints of the related
optimization problem are obtained by random
vibration theory. Subsequently, a standard non-
linear optimization problem is formulated where
the weight of the structure or a desired stochastic
response quantity is minimized. The procedure is
termed as stochastic structural optimization ( SSO ).
The details of the relevant developments can be
found in (Nigam, 1972), Kang et al. (2006). It may
be underlined here that in a typical SSO procedure
the dynamic load is considered to be the only
source of randomness in many cases and all other
system parameters are assumed to be determinis-
tic. But, uncertainty in the system parameters is
inevitable to model a realistic structural system
and incorporation of such uncertainty creates an
interaction between the stochastic descriptions of
the loads and the uncertain parameters (Jensen,
2002). Furthermore, the effect of system parameter
uncertainty is important as the safety of structure
may be endangered due to this (Chaudhuri, &
Chakraborty, 2006) and can affect the final optimal
design significantly (Schuëller, & Jensen, 2008).
Thus, there is a growing interest to consider the
effect of uncertainty in the optimization process
for economic design of structure ensuring neces-
sary safety requirements.
The developments in the optimum design of
structure under uncertainty can be divided into
three broad categories: (i) performance based
design optimization ( PBDO ), (ii) reliability based
design optimization ( RBDO ), and (iii) robust
design optimization ( RDO ). The limitation of the
PBDO is quite obvious as it optimizes the mean