APPENDIX E
(2) Formulation for frequency of exceedance. The
annual frequency of ground motion exceedance, v(z), is
SITE-SPECIFIC PROBABILISTIC SEISMIC
evaluated using the following expression:
HAZARD ANALYSIS
mi =mU
rj=rmax
∑ ∑ λ (m ) ∑ P (R = r
) (
N
)
E-1. Introduction
ν(z) =
mi P Z > z
mi , r j
n
i
n
j
n=1
rj =0
o
mi =m
a. Purpose. The purpose of this appendix is to
(E-2)
describe details of the methodology used in probabilistic
seismic hazard analysis (PSHA) to develop site-specific
in which
response spectra. More general aspects of the site-specific
approach are presented in Chapter 3. In paragraph E-2, the
λ (mi)
=
the annual frequency of occurrence of
formulation of the basic probabilistic model is described.
n
earthquakes on seismic source n in a
Paragraph E-3 discusses the incorporation of uncertainty in
magnitude interval centered at mi. mi is
PSHA. Paragraph E-4 describes the results of a PSHA and
above a minimum size of engineering
how they can be analyzed to determine the dominant
significance, mo , and below the
contributors to the seismic hazard and sources of
maximum event size, mU .
uncertainty. In paragraph E-5, two examples of
applications of PSHA to develop site-specific response
Pn(R=rj | mi) =
the probability of an earthquake of
spectra are presented.
certain distance rj from the site
E-2. Mathematical Formulation of the Basic Seismic
Hazard Model.
will be exceeded, given an earthquake of
a.
General Formulation.
(1) Formulation for probability of exceedance.
Thus, for a given source, the annual frequency or rate of
The methodology used to conduct PSHA was initially
exceeding a certain ground motion level at the site is
developed by Cornell (1968). The formulation of the basic
obtained by summing over all magnitudes (the second
seismic hazard model is summarized herein. Additional
summation of Equation E-2) and source-to-site distances
discussion and guidance for conducting a PSHA is
(the last summation of Equation E-2) for that source.
described in several publications, including National
Then, the total rate of ground motion exceedance at the
Research Council (1988), Earthquake Engineering
site, v(z), is obtained by adding the rates for all the sources
Research Institute (1989), and Ferritto (1994, 1997). Using
(the first summation of Equation E-2). The components of
a Poisson probability model, the probability of exceedance,
equation E-2 are discussed in paragraphs b, c, and d below.
pz, (z), of a ground motion level, z, in an exposure time or
design time period, t, at a site is related to the annual
b. Frequency of Occurrence of Earthquakes. The
frequency (or rate) of ground motion exceedance at the site,
incremental rate of earthquakes occurrence λ (mi) is
v(z), by:
n
obtained from earthquake recurrence relationships. Two
recurrence models are typically used in PSHA, the
pz (z)=1-e-(v(z) ⋅t)
(E-1)
truncated exponential model and the characteristic
earthquake recurrence model. These two recurrence models
A PSHA is carried out to obtain v(z) and pz (z) can then be
are also discussed in paragraph 3-4e(3)(b) of Chapter 3. For
obtained using Equation E-1. The return period (RP) for
convenience, the subscript n for the source region is
ground motion exceedance at a site is equal to the
eliminated in the following paragraphs.
reciprocal of v(z). The results of a PSHA are, in practice,
expressed in terms of one or more of the parameters, pz(z),
(1) The truncated exponential model of Cornell
v(z), and RP. Note that when (v(z)⋅t) is small
and Vanmarcke (1969) represents the truncation of the
(approximately ≤0.1) pz(z) is approximately equal to
Gutenberg-Richter (1954) earthquake frequency law at a
(v(z)⋅t). For larger values of (v(z)⋅t), pz (z) is less than
(v(z)⋅t).
which expresses the rate of occurrence of earthquakes equal
to or greater than a certain magnitude m, is specified by
E-1
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