Civil Engineering Reference
In-Depth Information
A rigorous, yet practical approach to performance-based earthquake
engineering (PBEE) is presented in this chapter. The approach considers
the seismic hazard, structural response, damage, and repair costs associated
with restoring the building to its original condition using a fully consistent
probabilistic analysis. The approach could be generalized to consider other
performance measures, such as casualties and down-time. The procedure is
organized to be consistent with conventional building design, construction,
and analysis practices so that it can be readily incorporated into a design
approach.
12.2 Performance-based earthquake engineering
(PBEE) framework
Owing to uncertainties in earthquakes, it is best to deal with earthquake-
related problems using a probabilistic approach instead of deterministic
approach. The basic probability theory that is used to derive the PBEE
framework is briefl y summarized herein. Additional references can be
found in probability and statistics textbooks (e.g. Ang and Tang 2007). Eq.
[12.1] shows the notation of conditional complementary cumulative distri-
bution function (CCDF) of a random variable
X
given the value of another
random variable
Y
y
. For example, this function can represent the prob-
ability of the inter-story drift ratio of a building exceeding 1 percent given
the magnitude 7 earthquake has happened. Eq. [12.2] shows total probabil-
ity theorem for the event
A
given the condition that
n
mutually exclusive
and collectivelly exhaustive discrete random variables
E
i
have happened.
For example,
P
(
A
|
E
i
) represents the conditional probability of building
repair cost exceeding 1 million dollars given a magnitude 7 earthquake has
happened, while
P
(
E
i
) represents the probability of a magnitude 7 earth-
quake happening at the building site. Multiplying the conditional probabil-
ity
P
(
A
|
E
i
) by probability
P
(
E
i
) and summing over a range of shaking
intensities (for example, probability of magnitude 6 and 8 earthquakes hap-
pening at the site independently), the product could represent the total
probability that the building has a repair cost exceeding 1 million dollars
under the range of earthquake shaking intensities considered.
=
(
)
=
(
)
Gxy
PX
>
xY
=
y
[12.1]
n
=
∑
1
()
=
(
)
(
)
PA
PAE PE
i
[12.2]
i
i
Eq. [12.2] is further modifi ed to Eq. [12.3] to account for
E
being a con-
tinuous random variable. Similar to Eq. [12.2], Eq. [12.3] shows the total
probability of event
A
>
a
given the event
E
has occurred.
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