Using a performance-based earthquake engineering (PBEE) approach to estimate structural performance targets for bridges - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 531

20.2.1 Probabilistic seismic hazard analysis (PSHA) and
ground motion intensity measures

The fi rst part of the problem is the estimation of the annual rate of ex-
ceeding IM. In other words, seismic hazard analysis represents a link be-
tween earthquake scenarios and ground motion intensity in Fig. 20.1.
Traditionally, probabilistic seismic hazard analysis (PSHA) due to a point
source is carried out by evaluating the following equation (Cornell, 1968;
McGuire, 2004):

( ) =

∫ ∫

()( ) ( ) ××

ν IM > z N ⋅ f M f M R P IM , > z M R , d M d R [20.2]

min M R

M R

where R is the distance from the source to site; M is the earthquake mag-
nitude; N min is the annual rate of earthquakes with magnitude greater than
or equal to the minimum magnitude; f M ( M ) and f R ( M , R ) are the probability
density functions for the magnitude and distance, and P (IM> z | M , R ) is the
probability of observing an IM greater than z for a given earthquake mag-
nitude and distance. The IM is the quantifi cation of the characteristics of a
ground motion that are important to the structural response. PSHA requires
the defi nition of seismic sources close to the specifi c site and characteriza-
tion of these seismic sources by appropriate probability density functions
and recurrence models. After defi ning a suite of earthquake scenarios, the
range of ground motions for each earthquake scenario is estimated and the
annual rate of each combination of earthquake scenario and ground motion
is computed. The probability that the IM will exceed z, P (IM> z | M , R ), is
obtained from the ground motion prediction equation (GMPE) and includes
an implicit integration over the ground motion variability. The probability
that the IM will exceed z is given by:

( ) =

∫

() × ( ) ×

PIM zMR > , f ε PIM zMR > , , ε d ε [20.3]

ε

ε

where the epsilon ( ε ) is the number of standard deviations above or below

the median, f ε ( ) is the probability density function for the epsilon (given

by the standard normal distribution) and P (IM> z | M , R ,

ε

ε ) is either 0 or 1.

In this formulation, P (IM> z | M , R , ) identifi es earthquake scenarios and

ground motion combinations that lead to IM greater than z . The hazard
(Equation 20.2) can then be written as:

ε

( )

ν IM > z

∫ ∫ ∫

()( )() ( ) ×××

= N ⋅ f Mf , R f ε P IMz > , R , ε d MR d d ε

min M R ε

M R ε

[20.4]