Civil Engineering Reference
In-Depth Information
(
)
=
∫
∫
(
)
(
)
(
)
G dv im
G dv dm
d
G dm edp
d
G edp im
[12.10]
dm
edp
Eq. [12.10] represents the conditional probability of a decision variable
exceeding a value
dv
given a value of intensity measure
IM
im
. Similar
derivation can be made for
G
(
dm
|
im
) and is shown in Eq. [12.11].
=
(
)
=
∫
(
)
(
)
G dm im
G dm edp
d
G edp im
[12.11]
edp
12.2.2 Annual rate of exceeding a threshold value
G
(
dv
|
im
),
G
(
dm
|
im
), and
G
(
edp
|
im
) are the conditional probabilities of the
performance measures
DV
,
DM
, and
EDP
given an intensity measure
IM
=
im
. To translate the conditional probability to a quantity that can be
readily used by engineers and other stakeholders to make a risk manage-
ment decision, these conditional probabilities are multiplied by the absolute
value of the derivative of the annual rate of exceedance of a given value of
i
()
. Because the seismic hazard
d
λ
im
the ground motion intensity measure
curve,
(
im
), is defi ned as the annual rate the earthquake ground motion
intensity measure
IM
exceeds a value
im
, a derivative is used to compute
the annual frequency (rate of occurrence) of the intensity measure
IM
λ
=
im
. An absolute value is used to insure that this rate is a positive number
regardless of the shape of the hazard curve itself.
The product of the annual rate of occurrence and the conditional pro-
bability of the performance measure given an intensity measure gives
the annual rate of the performance measure exceeding a threshold value.
i
()
represents the annual frequency of a random
variable
IM
equaling
im
and
G
(
dv
|
im
) represents the probability that the
random variable
DV
takes values larger than
dv
for shaking intensity equal-
ing
im
. Thus, their product gives the number of occurrences of
DV
d
λ
im
In other words,
>
dv
annually, as shown in Eq. [12.12].
(
)
=
()
λ
DV
>
dv
d
λ
im
im
(
)
Gdvim
[12.12]
dim
d
Integrating Eq. [12.12] for all intensity measures, the annual rate that
DV
exceeds a threshold value
dv
for all intensity measures considered is shown
in Eq. [12.13].
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