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and offi ce buildings. The aim of the case studies is to highlight the usefulness
of seismic risk curves and PML analysis from the viewpoints of catastrophic
earthquake risk mitigation and risk-based decision-making for asset man-
agement. Finally, conclusions and future trends of the seismic risk analysis
are highlighted.
27.2
Analytical procedure for assessing seismic risk
This section summarizes theoretical background and mathematical formu-
lation of seismic risk analysis. More details of the method can be found in
Yoshikawa (2008), Nakamura and Ugata (2008), and Yoshikawa
et al.
(2012).
27.2.1 Seismic risk curve based on analytical method
A seismic risk curve
G
C
(
c
) is calculated based on two random variables,
earthquake hazard information and seismic vulnerability of a structure. It
quantifi es an adopted seismic risk measure as a function of annual occur-
rence probability. In other words, the
G
C
(
c
) is equivalent to the probability
distribution of the seismic risk measure by taking all relevant uncertainties
in seismic hazard and structural vulnerability into account. Using the seismic
risk curve, other useful quantities and curves, such as expected seismic loss,
NEL, and PML, can be calculated. For the seismic risk measure, damage-
based as well as loss-based indicators can be used (e.g. peak drift of a
structural system or component, and total seismic loss for a building). A
formulation to compute the
G
C
(
c
) from seismic hazard curve (
P
A
(
α
)) and
seismic loss function (
f
C
(
c
|
)) is discussed below.
The seismic loss function
f
C
(
c
|
α
) is characterized by seismic fragility curve
and seismic damage function (Mander 1999; Shinozuka
et al.
2000; Yoshi-
kawa 2008). (Note: detailed descriptions of the fragility curve and damage
function are beyond the scope of this chapter). Typically, the fragility curves
are defi ned as the conditional probability function for the specifi ed damage
states, e.g. minor, major, extensive, and complete damage states in the
HAZUS-Earthquake (FEMA/NIBS 2003), whereas the damage functions
relates the attained damage states and (monetary) consequences. These
curves/functions can be derived from statistical analysis of empirical data
(e.g. observations of damage states and ground shaking from past earth-
quakes; see Shinozuka
et al.
2000) and from analytical methods, such as
nonlinear static pushover analysis and incremental dynamic analysis (Vam-
vatsikos and Cornell 2004).
For a seismic hazard curve
P
A
(
α
is the seismic hazard measure
(i.e. seismic intensity measure); and its derivative) denoted as
p
A
(
α
),
α
α
),
p
A
(
α
)
=
−
d
P
A
(
α
)/d
α
, is gradient of the curve at a particular ground motion
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