Life Cycle Cost Considerations in Seismic Design Optimization of Structures - Structural Seismic Design Optimization and Earthquake Engineering - page 8

In the following section a LCC formulation
is developed for optimal seismic design of RC
structures. Fiber-based finite element modeling
is used to construct the structural models. In-
elastic dynamic time history analysis and static
pushover analysis are used to accurately obtain
the earthquake demand and structural capacity, re-
spectively. In addition, randomness due to ground
motion variability (aleatory uncertainty) and errors
in structural modeling and inherent variability in
material properties (epistemic uncertainty) are
taken into account.

used intensity measures in seismic design are the
peak ground acceleration (PGA) and spectral ac-
celeration (S a ) at the fundamental period of the
structure. Example hazard curves for PGA and S a
are shown in Figure 3(a). These curves are site
specific and they can be obtained from PSHA for
seismic design purposes.
For certain types of structural analysis such
as modal response spectrum and time history
analysis, it is required to obtain the design spec-
trum. The design spectrum could be obtained from
the hazard curves depicted in Figure 3(a) at dif-
ferent return periods. The return period of an
earthquake, T R is simply the inverse of the mean
annual frequency of exceedance, v , i.e. T R = 1/v .
As examples, design spectra at different return
periods obtained from the hazard curves shown
in Figure 3(a) are given in Figure 3(b). These are
also referred to as uniform hazard spectra (UHS)
because each spectral ordinate has the same prob-
ability of exceedance. As an alternative, the design
spectra could be obtained from seismic design
codes. Although, most codes provide the necessary
information to draw design spectra for the maxi-
mum considered earthquake only (e.g. ICC, 2006)
and it is not possible obtain the hazard at different
return periods which is required for LCC analysis,
more detailed seismic zonation maps have been
developed in the recent years and design spectra
for earthquake hazards with different probabilities
of occurrence are now available (USGS, 2008).

LIFE-CYCLE COST FORMULATION

Definition of the Seismic Hazard

In order to evaluate the LCC of a structure due to
repair in future earthquakes, one has to evaluate
the probability of demand exceeding capacity for
the whole-life time of the structure. There is more
than one way of defining the seismic hazard that a
structure could experience throughout its life time.
The most commonly used methods to evaluate
the earthquake hazard for a given location with
known historical seismicity are the deterministic
and probabilistic seismic hazard analysis (DSHA
and PSHA). The main difference between the two
approaches is that PSHA incorporates the ele-
ment of time in hazard assessment. It is beyond
the scope of this chapter to assess and validate
the two approaches for seismic hazard analysis;
however, PSHA has well made its way into the
seismic design codes. As examples, UBC (ICBO,
1997) and FEMA 450 (FEMA, 2003) represented
design response spectrum based on probabilistic
zonation maps. Therefore, here PSHA is used to
characterize the seismic hazard for a selected site,
while DSHA is also a valid option.
In PSHA mean annual frequency of exceedance
(or probability of exceedance) is calculated for a
range of a selected intensity measure that repre-
sents the earthquake hazard. The most commonly

Evaluation of the Structural
Capacity and Earthquake Demand

As described in the previous section, the continu-
ous probabilistic hazard at a selected site, Figure
3(a), is discretized at certain probabilities, repre-
sented with return period in Figure 3(b). The objec-
tive behind this manipulation is to have a means
of evaluating the probability of failure; that is the
probability of earthquake demand exceeding the
structural capacity. By having the design spectra
at different return periods, the hazard levels are