Geology Reference
In-Depth Information
2. SINGLE OBJECTIVE DESIGN
OPTIMIZATION PROBLEM
volve realistic modelling of loading and resistance
uncertainty, initial construction cost, damage cost,
failure consequence cost, maintenance cost and
discount cost of distant future failure. Cheng and
Chang (1988), Cheng and Li (1997), Li (1998)
and Zou et al (2007b) studied the application of
minimum expected life-cycle cost. Recently, Liu
et al. (2003) presented a multi-objective genetic
algorithm for optimal seismic design of steel
frames based on life cycle cost considerations.
All these research efforts indicate the importance
of consideration of a life-cycle cost on making
rational design decisions.
This chapter presents an effective single- or
multi-objective optimization technique for the
elastic and inelastic seismic drift performance
design of reinforced concrete buildings under
response spectrum loading and pushover loading.
Using the principle of virtual work, the modal drift
response can be explicitly formulated in terms of
element sizing variables and the peak drift values
can be estimated by modal combination methods.
With careful tracking of the location and extent of
plastic hinge occurrence, the inelastic pushover
drift can also be explicitly expressed in terms of
the sizing variables using the same principle of
virtual work and the Taylor series approximation.
The total life-cycle cost optimization is formulated
as a multi-objective optimization problem subject
to seismic inelastic inter-story drift responses
under pushover loading. Once the multi-objective
function (including both the initial material cost
and the predicted damage loss) and the design
performance constraints are explicitly formulated,
the ε-constraint method is then applied to produce
a Pareto optimal set, from which the decision of the
best compromise solution for the multi-objective
design problem can be achieved. The optimization
methodology for each Pareto optimal solution is
basically established based on a rigorously derived
Optimality Criteria (OC) approach. Finally, one
RC building frame example is presented to il-
lustrate the effectiveness and applicability of the
proposed automated optimization approaches.
In seismic design, it is commonly assumed that
a building behaves linearly elastic under minor
earthquakes and may work nonlinearly inelastic
when subjected to moderate and severe earth-
quakes. Under such an assumption, the entire
design optimization process can therefore be de-
composed into two phases (Zou and Chan 2001;
Zou 2002). The first phase is an elastic design
optimization in which the structural concrete cost
can be minimized subject to elastic spectral drift
responses under minor earthquake loading; and
concrete member sizes are taken as design vari-
ables since concrete material plays a more domi-
nant role in improving elastic drift performance
of a building. In this phase, all concrete sections
are assumed to be uncracked and to behave linear-
elastically. Once the optimal structural member
sizes are determined at the end of the first phase
of the optimization, the steel reinforcement quan-
tities can then be considered as design variables
in the second phase. In controlling the inelastic
drift responses, steel reinforcement is the most
effective element that provides the ductility of
RC building structure beyond first yielding (Zou
2002). In this second design phase or the so-called
inelastic design optimization, the member sizes
are kept unchanged and the cost of steel reinforce-
ment is minimized subject to inelastic inter-story
drift performance constraints based on nonlinear
pushover analysis.
2.1 First Phase: Elastic Design
Optimization Problem
Consider a multi-story concrete framework hav-
ing i= 1, 2,…, Ni member (or member fabrication
groups). Assuming that the concrete elements are
uncracked and have rectangular cross sections
such that the width (
B
i
) and depth (
D
i
) are
taken as design variables, the design objective of
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