1968 Paper - Engineering Seismic Risk Analysis

8/11/2019 1968 Paper - Engineering Seismic Risk Analysis

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Bullet in of the Seismo logicalS o c ie ty o f A m e r ic a . V o l . 5 8 , N o . 5 , p p . 1 5 8 3 - 1 6 0 6 . O c t o b e r , 1 9 6 8

E N G I N E E R I N G S E I S M I C R I S K A N A L Y S I S

BY C ALLIN CORNELL

ABSTRACT

T h is p a p e r i n t ro d u c e s a m e t h o d f o r t h e e v a l u a t i o n o f t h e s e is m ic r is k a t t h e s i te o f

a n e n g i n e e r i n g p r o j e c t . T h e r e su lts a r e i n t e rm s o f a g r o u n d m o t i o n p a r a m e t e r

s uc h a s p e a k a c c e l e r a t io n ) v e r s u s a v e r a g e r e tu r n p e r i o d . T h e m e t h o d i n c o r p o r a t e s

t h e i n f lu e n c e o f a l l p o t e n t i a l s o u r c e s o f e a r t h q u a k e s a n d t h e a v e r a g e a c t i v i ty r a t e s

a s s i g n e d to t h e m . A r b i t r a r y g e o g r a p h i c a l r e la t io n s h i p s b e t w e e n t h e s it e a n d p o -

t e n t i a l p o i n t , l in e , o r a r e a l s o u r c e s c a n b e m o d e l e d w i t h c o m p u t a t io n a l e a s e . In

t h e r a n g e o f i n te r e s t , t h e d e r i v e d d i s tr ib u t io n s o f m a x i m u m a n n u a l g r o u n d m o t io n s

a r e i n t h e fo r m o f T y p e I o r T y p e II e x t r e m e v a l u e d i s t r ib u t i o n s , i f t h e m o r e c o m -

m o n l y a s s u m e d m a g n i t u d e d i s tr ib u t io n a n d a t t e n u a t io n l a w s a r e u s e d .

I N T R O D U C T I O N

O w i n g t o t h e u n c e r t a i n t y i n t h e n u m b e r , s i z e s , a n d l o c a t i o n s o f f u t u r e e a r t h q u a k e s

i t i s a p p r o p r i a t e t h a t e n g i n e e r s e x p r e s s s e i s m i c r i s k , a s d e s i g n w i n d s o r f l o o d s a r e , i n

t e r m s o f r e t u r n p e r i o d s ( B l u m e , 1 9 6 5 ; N e w m a r k , 1 9 6 7 ; B l u m e , N e w m a r k a n d C o r n i n g ,

1 9 6 1 ; H o u s n e r , 1 9 5 2 ; M u t o , B a i l e y a n d M i t c h e l l , 1 9 6 3 ; G z o v s k y , 1 9 6 2 ) .

T h e e n g i n e e r p r o f e s s i o n a l l y r e s p o n s i b l e f o r t h e a s e i s m i c d e s i g n o f a p r o j e c t m u s t

m a k e a f u n d a m e n t a l t r a d e - o f f b e t w e e n c o s t l y h i g h e r r e s i s t a n c e s a n d h i g h e r r i s k s o f

e c o n o m i c l o s s ( B l u m e , 1 9 6 5 ) . I t r e q u i r e s a s s e s s m e n t o f t h e v a r i o u s l e v e l s o f p e r f o r m -

a n c e a n d e c o n o m i c i m p l i c a t i o n s o f p a r t i c u l a r d e s i g n s s u b i e c t e d t o v a r i o u s l e v e l s o f

i n t e n s i t y o f g r o u n d m o t i o n . T h e e n g i n e e r m u s t c o n s i d e r t h e p e r f o r m a n c e o f t h e s y s t e m

u n d e r m o d e r a t e a s w e l l a s l a r g e m o t i o n s . S o u n d d e s i g n o f t e n s u g g e s t s s o m e e c o n o m i c

l o s s ( e . g . , a r c h i t e c t u r a l d a m a g e i n b u i l d i n g s , a u t o m a t i c s h u t - d o w n c o s t s i n n u c l e a r

p o w e r p l a n t s ) u n d e r t h e s e m o d e r a t e , n o t u n e x p e c t e d e a r t h q u a k e e f f e c t s .

T h i s e n g i n e e r s h o u l d h a v e a v a i l a b l e a l l t h e p e r t i n e n t d a t a a n d p r o f e s s i o n a l j u d g e -

m e n t o f t h o s e t r a i n e d i n s e i s m o l o g y a n d g e o l o g y i n a f o r m m o s t s u i t a b l e f o r m a k i n g

t h i s d e c i s i o n w i s e l y . T h i s i n f o r m a t i o n i s f a r m o r e u s e f u l l y a n d c o m p l e t e l y t r a n s m i t t e d

t h r o u g h a p l o t o f , s a y , M o d i f i e d M e r c a l l i i n t e n s i t y v e r s u s a v e r a g e r e t u r n p e r i o d t h a n

t h r o u g h s u c h i l l - d e f i n e d s i n g l e n u m b e r s a s t h e p r o b a b l e m a x i m u m o r t h e m a x i m u m

c r e d i b l e i n t e n s i t y . E v e n w e l l - d e f i n e d s i n g l e n u m b e r s s u c h a s t h e e x p e c t e d l i f e t i m e

m a x i m u m o r 5 0 - y e a r i n t e n s i t y a r e i n s u f f i c i e n t t o g i v e t h e e n g i n e e r a n u n d e r s t a n d i n g

o f h o w q u i c k l y t h e r i s k d e c r e a s e s a s t h e g r o u n d m o t i o n i n t e n s i t y i n c r e a s e s . S u c h i n f o r -

m a t i o n i s c r u c i a l t o w e l l - b a l a n c e d e n g i n e e r i n g d e s i g n s , w h e t h e r i t i s u s e d i n f o r m a l l y

a n d i n t u i t i v e l y ( N e w m a r k , 1 9 6 7 ) , m o r e s y s t e m a t i c a l l y ( B l u m e , 1 9 6 5 ) , o r d i r e c t l y i n

s t a t i s t i c a l l y - b a s e d o p t i m i z a t i o n s t u d i e s ( S a n d i , 1 9 6 6 ; B e n i a m i n , 1 9 6 7 ; B o r g m a n ,

1 9 6 3 ) .

U n f o r t u n a t e l y i t h a s n o t b e e n a s i m p l e m a t t e r f o r t h e s e i s m o l o g i s t t o a s s e s s a n d e x -

p r e s s t h e r i s k a t a s i t e i n t h e s e t e r m s . I - I e m u s t s y n t h e s i z e h i s t o r i c a l d a t a , g e o l o g i c a l

i n f o r m a t i o n , a n d o t h e r f a c t o r s i n t h i s a s s e s s m e n t . T h e l o c a t i o n s a n d a c t i v i t i e s o f p o -

t e n t i a l s o u r c e s o f t e c t o n i c e a r t h q u a k e s m a y b e m a n y a n d d i f f e r e n t i n k i n d ; t h e y m a y

n o t e v e n b e w e l l k n o w n . I n s o m e r e g i o n s , f o r e x a m p l e , i t i s n o t p o s s i b l e t o c o r r e l a t e

p a s t a c t i v i t y w i t h k n o w n g e o l o g i c a l s t r u c t u r e . I n s u c h c i r c u m s t a n c e s t h e s e i s m o l o g i s t

u n d e r s t a n d a b l y h a s b e e n l e d t o e x p r e s s h i s p r o f e s s i o n a l o p i n i o n i n t e r m s o f o n e o r t w o

s i n g l e n u m b e r s , s e l d o m q u a n t i t a t i v e l y d e f i n e d . I t i s u n d o u b t e d l y d i f f i c u l t , i n t h i s s i t u -

1 5 8 3


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58 4 BULLET IN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

ation, for the seismologist to avoid engineering influences; the seismologist's estimates

will probably be more conservative for more consequential projects. But these de-

eisions are more appropriately those of the design engineer who has at hand more

complete information (such as construction costs, sys tem performance characteristics,

etc.) upon which to determine the optimal balance of cost, performance, and risk.

Seismologists have long recognized this need to provide engineers with their best

estimates of the seismic risk. Numerous regional seismic zoning maps have been de-

velopcd. Familiar examples appear in the Uniform Building Code (1967) and

Richter (1959). Despite reference to probabilities they are seldom clear as to how

the (single) intensity level for each location is to be interpreted. More recently these

values have been associated with specific average return periods (~JIuto, Bailey and

Mitchell, 1963; Kawasumi, 1951; Ipek e t a l 1965). In any case, more information is

needed to define a relationship between a continuous range of average return period

and intensities. Other attempts have been made to provide this more complete in-

formation at regional levels (Ipek, 1965; Milne and Davenport, 1965). These ap-

proaches, which are usually large scale numerical studies based directly on historical

data , have difficulty giving proper weight to the known correlation between geological

structure and most seismic activity. They also are not successful at a fine or local

scale. Lacer (1965) has presented a numerical, Monte Carlo technique designed to

estimate the distribution of the intensity of motion at a particular site given the occur-

rence of an earthquake somewhere in the surrounding region. He is able to account

for geological features, such as faults, but he assumes all the assigned poin t sources

are equal likely to give rise to this earthquake.

In this paper a method is developed to produce for the engineer the desired rela-

tionships between such ground-motion parameters as Modified Mercalli Intensity,

peak-ground velocity, peak-ground acceleration, etc., and their average return period

for his site. The minimum data needed are only the seismologist's best estimates of

the average activity levels of the various potential sources of earthquakes (e.g., a

particular fault's average annual number of earthquakes in excess of some minimum

magnitude of interest, say 4). If, in addition, the seismologist has reason to use other

than average or typical values of the parameters in the function used to describe the

relative frequency of earthquake magnitudes or in the functions of intensity, say,

versus magnitude and distance, he may also supply these parameter values. The tech-

nique to be developed provides the method for integrating the individual influences

of potential earthquake sources, near and far, more active or less, into the probability

distribution of maximum annual intensity (or peak-ground acceleration, etc.). The

average return period follows directly. The results of the development appear in closed

analytical form, requiring no lengthy computation and permitting direct observation

of the sensitivity of the final results to the estimates made.

Unlike the analogous flood or wind problem, in the determination of the distribu-

tion of the maximum annual earthquake intensity at a site, one must consider not

only the distribution of the size (magnitude) of an event, but also its uncertain dis-

tanee from the site and the uncertain number of events in any time period. The presen-

tat ion here will show the mathemat ical development of a simple ease. Results of other

eases of interest will be displayed without complete derivations. An illustration will

demonstrate the application of the method. Finally, the assumptions and limitations

will be discussed more critically. Extensions and advantages of the method will con-

elude the presentation.


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ENGI NEER ING SEISMIC RISK ANALYSIS 585

LINE SOURCE DERIVATION

For illustration of the development of the method of solution, the determination of

the distribution of the annual maximum Modified Mercalli intensity at a site due to

potential earthc uakes along a neighboring fault will be considered. As illustrated in

C

A a Perspective

Site

~/ 2 =

X - -~ ~/2

Si te

b A BD P lane

FIG. 1. Line source.

Figure la, the site is assumed to lie a perpendicular distance, A, from a line on the

surface vertically above the fault at the focal depth, h, along which future earthquake

loci are expected to lie. The length of this fault is l, and the site is located symmetri-

cally with respect to this length.

Concern ~i th focal distances restricts attent ion to the

BD

plane; Figure lb. The

perpendicular slant distance to the source is

d = /~ --F A2 (1)


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1586

B U L L E T I N O F T H E S E I S M O L O G I C L S O C I E T Y O F M E R I C

T h e f o c a l d i s ta n c e , R , t o a n y f u t u r e f o c u s l o c a t e d a d i s ta n c e X f r o m t h e p o i n t B i s

R = v / d ~ + X ~ ( 2 )

S in ce - l / 2 i l R = r]

4)

i n w h i c h P [ A I B ] i s r e a d t h e p r o b a b i l i t y o f A g i v e n B . A s s u m i n g p r o b a b il i s ti c i n d e -

p e n d e n c e o f M a n d R ,

P [ I >= i , R = r] = P [ M >= i + c31n + c l l

= I - - F M [ i + c31nrc2 + c ll (5 )

i n w h i c h F M ( m ) i s t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f e a r t h q u a k e m a g n i t u d e s .

F o r e x a m p l e , R i c h t e r ' s w i d e l y v e r if i ed (1 9 , 2 0 ) r e l a t i o n s h ip b e t w e e n n u m b e r , n , ~ , a n d

m a g n i t u d e , m

loglo n m = a - - b m

imp l i e s

1 - - F ~ ( m ) = e ~ ( ~ - m

m => m0 (6 )

i n w h i c h ~ = b I n 1 0 a n d m o i s s o m e m a g n i t u d e s m a l l e n o u g h , s a y 4 , t h a t e v e n t s o f

l e ss e r m a g n i t u d e m a y b e i g n o r e d b y e n g i n e e r s . T h i s r e s t r ic t i o n t o l a r g e r e v e n t s i m p l ie s

t h a t t h e p r o b a b i li t ie s a b o v e a r e c o n d i t i o n a l o n t h e o c c u r r e n c e o f a n e v e n t o f i n t e r e s t ,

t h a t i s , o n e w h e r e M > m 0 . T h e p a r a m e t e r b i s t y p i c a l l y ( I s a c k s a n d O l iv e r, 1 96 4 )

su ch th a t f~ i s ab o u t 1 .5 to 2 .3 .


5/24

E N G I N E E R I N G S E I S M I C : R I S K A N A L Y S I S

C o m b i n i n g e q u a t i o n s 5 a n d 6 , t h e r e s u l t is

1587

T h e l i m i t o n t h e d e f i n i ti o n o f

F M m ) ,

n a m e l y m ~ m 0 , i m p l ie s t h a t e q u a t i o n 7 h o l d s

f o r

i + c~ lit r + 0 -~

L = i , 0

C2

o i l

i~ c2mo 0 csh l

~

f R r ) ~

ro r

FIG. 2 . Probab i l i ty densi ty fu nct ion of focal d is tance, R.

A t s m a l le r v a l u e s o f t h e a r g u m e n t , i , t h e p r o b a b i l i t y e q u a t i o n 7 ) is u n i t y t h a t I e x c e e d s

i g i v e n t h e o c c u r r e n c e o f a n e v e n t o f m a g n i t u d e g r e a t e r t h a n

m o

a t d i s t a n c e r ) .

I n o r d e r t o c o n s i d e r t h e i n f l u e n c e o f a l l p o s s i b le v a l u e s o f t h e f o c a l d i s ta n c e a n d t h e i r

r e l a t i v e li k e li h o o ds , w e m u s t i n t e g r a t e . W e s e e k t h e c u m u l a t i v e d i s t r i b u t i o n o f I ,

F ~ i ) , g i v e n a n o c c u r r e n c e o f M >= m 0 ,

f d

- - F , i ) =

P [I >= i] = P [I >= i l R

= r ] f ~ r ) d r

9 )

i n w h i c h

f R r )

i s t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f R , t h e u n c e r t a i n f o c al d is t a n c e .

F o r t h e i l l u s t r a t io n h e r e , i t i s a s s u m e d t h a t , g i v e n a n o c c u r r e n c e o f a n e v e n t o f

i n t e r e s t a l o n g t h e f a u l t, i t i s e q u a l l y l i k e ly t o o c c u r a n y w h e r e a l o n g t h e f a u l t. F o r m a l l y ,

t h e l o c a t i o n v a r i a b l e X i s a s s u m e d t o b e u n i f o r m l y d i s t r i b u t e d o n t h e i n t e r v a l

- 1 / 2 , + 1 / 2 ) . T h u s I X I, t h e a b s o l u t e m a g n i t u d e o f X , i s m f i f o r m l y d i s t r i b u t e d o n

t h e i n t e r v a l 0 , 1/2) . T h e c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n , F R r ) , o f R f o l l o w s


6/24

1588 B U L L E T I N O F T H E S E I S M O L O G I C A LS O C I E T Y O F A M E R I C A

i m m e d i a t e l y :

F R r ) = P [ R ~ r] = P [ R 2 ~ r 2]

= P [ X 2 - 4- d 2 ~ r e ] =

P [ I x I ~ ~

- d 21

2 d ~

r - -

1 2

d - < r - < r 0 . ( 10 )

T h e r e f o r e , t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f R i s

d F . r ) d 2 ~ / ~ )

f R r ) - d r - d r

2 r =

d _-< r _-< r0 . (1 1 )

~ / ~ d2

T h i s d e n s i t y f u n c t i o n i s p l o t t e d i n F i g u r e 2 .

S u b s t i t u t i n g e q u a t i o n 1 1 i n t o e q u a t i o n 9 a n d i n t e g r a t i n g i s c o m p l i c a t e d b y t h e

a w k w a r d l i m i t s o f d e f i n i ti o n o f t h e f u n c t i o n s , b u t i n t h e r e g i o n o f g r e a t e s t i n t e r e s t ,

n a m e l y l a r g e r v a lu e s o f t h e i n t e n s i t y t h e r e s u l t is

1 - - F i i ) = P [ >= i ]

l l C G e x p

- ~ i ~ i ( 1 2 )

i n w h i c h i i s t h e l o w e r l im i t o f v a l i d i t y o f t h i s f o r m o f t h e r e s u l t a n d e q u a l s

i t

= c l -4 - c 2 m o

Ca In d (1 3)

a n d i n w h i c h C a n d G a re c o n s t a n t s . T h e f i rs t c o n s t a n t i s r e l a t e d t o p a r a m e t e r s i n t h e

v a r i o u s r e l a t i o n s h ip s u s e d a b o v e :

[ ( C l ) ]

C = ex p ~ ~ -4 - mo . (1 4 )

T h e s e c o n d c o n s t a n t i s r e l a t e d t o t h e g e o m e t r y o f i l l u s t r a t io n :

r o d r

G = 2 r ~ / ~ _ d 2

~ 1 e - l [ r O / d ] (COS U) 5 -1 d u ( 1 5 )

d Y ,-1o

i n w h i c h

= ~ c _ ~ _ 1 . 1 6 )

C2


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E N G I N E E R I N G S E I S M I C R I S K N L Y S I S 5 8 9

T h e i n t e g ra l i n e q u a t i o n 1 5 m u s t b e e v a l u a t e d n u m e r ic a l ly . R e s u l t s a p p e a r i n F i g u r e 3 .

F o r t y p i c a l p a r a m e t e r v a l u e s a n d s u f fi c ie n t ly l o n g f a u l ts i t is c o n s e r v a t i v e a n d r e a s o n -

ab l e t o r ep l ace r0 by in f in i t y . In t h i s case G i s g iven b y

2 ~ r ~ )

G -

i n w h i c h r V ) i s t h e c o m p l e t e g a m m a f u n c t i o n a n d v i s r e s t r ic t e d t o p o s i t iv e v a l u e s.

T h e r e s u l t s a b o v e y i e l d t h e p r o b a b i l i t y t h a t t h e s i t e i n t e n s i t y , I , w i l l e x c e e d a

5 \ \

2

1.0.

0 . 5

ro/d =CO

s e c I r o / d

Q = / ( c o s u ) Y - J d u

0

. m0) will be a special even t is given by eq ua ti on 12.

p ~ = P [ I >= i] = ~ C G e x p [ - ~ i ] .

(19)

Thus the number of times N that the intensity at the site will exceed i in an interval

of length t is

p ~ ( n ) P I N n ] e - ~ t ( P ~ v t) ~

= = - n = 0, 1, 2, . - . . (20)

n

Such probabilities are useful in studying losses due to a succession of moderate inten-

sities or cumulative damage due to two or more major ground motions.

Of par ticular inter est is the probabili ty d istr ibution of I(n[~)x the maximum inte nsit y

over an interval of time t (often one year). Observe that

~ t )

P [ I

.. .. =< i] = P[exactly zero special events in excess of i

occur in the time interval 0 to t]

which from equation (20) is

v(t) - " (21)

[ . . . i'

in which now the ratio ~ = v / 1 appears. This ratio is the average number of occur-

rences per unit length per year.

The conclusion is that for the larger intensities of engineering interest, the annual

maximum intensity has a distribution of the double exponential or Gumbel type. This

distribution is widely used in engineering studies of extreme events. It is important to


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E N G IN E E RIN G SE ISMIC RISK A N A L YSIS 1591

realize that, here, this conclusion is n o t based on the intuitive appeal to the familiar

asymptotic extreme value argument (Gumbel, 1958), which has caused other investi-

gators to seek and find empirical verification of the distribution for maximum magni-

tudes or intensities in a given region (Milne and Davenport, 1965; Nordquist, 1945;

Dick, 1965). The form of the distribution is dependen~ on the functional form of the

various relationships assumed above. Others, too, have found (Dick, 1965; Epstein

and Lomnitz, 1966; Epstein and Brooks, 1948) that the combination of Poisson oc-

currences of events and exponentially distributed sizes of events will invar iably lead

to the conclusion that the largest event has a GumbeLlike distribution (the true

Gumbel distribution is non-zero for negative as well as positive values of the argu-

ment). Any combination of assumptions which leads to the exponential form of the

distribution of I will, in combination with Poisson assumption of event occurrences,

yield this Gumbel distribution. The exponential form of F i ( i ) does not require the

exponential form of

F ~ ( m ) .

If the logarithmic dependence of I on R (equation 3 ) is

retained, for example, even polynomial distributions (Housner, 1952) of magnitude

will lead to the exponential distribution of I.

If the annual probabil ities of exceedance are small enough (say _--_ ~.

(25)

Consider the following typical numerical values of the parameters and site constants,

applicable to a particular site in Turkey, where in one region in 1953 years it was found

(Ipek

e t a l ,

1965) that

log10 n m = a - - b m

= 5.51 -- 0.644m

in which n~ is the number of earthquakes greater than m in magnitude. Assuming these

earthquakes all occur along the 650 km of the maior fault system in the region, the

average number of earthquakes in excess of magnitude 5 (i.e., m0 --- 5) per year per


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1592 B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E T Y O F A M E R I C A

u n i t l e n g t h o f f a u l t i s

n 5

- ( 1 9 5 3 ) ( 6 5 0 ) -

A lso

1.5 X 10 4 ( y e a r ) - 1 ( k i l o m e t e r ) - 1.

= b In 10 = 0 .64 4(2 .3 0) = 1 .48 .

c2 = 1.45

c3 -- 2.46

t h e f o ll o w i ng n u m e r i c a l r e s u lt s a r e o b t a i n e d f o r a s i te l o c a t e d a m i n i m u m s u r f a c e

d i s t a n c e , A , o f 4 0 k m f r o m a l in e so u r c e of e a r t h q u a k e s a t d e p t h h = 2 0 k m :

d = %/'h2 + A s = 4 4 . 6 k m

/~ C3

-- - -- 1 = 1.52

C2

= exp ?

2~ r ~ )

G----~ (2d)~ [ r ( 3 ' _+2) ] 2 = _ _ 7 0 4 X 10-~

( N u m e r i c a l i n t e g r a t i o n g iv e s G = 6 .5 8 X 1 0 - 8 ). T h u s , t h e i n t e n s i t y a t t h i s s it e w i t h

r e t u r n p e r i o d T~ i s

2

i ~ ~- In (~CGT~)

0 .98 In (6 .9T~) .

N o t e t h e l o g a r i t h m i c r e la t i o n s h ip b e t w e e n i a n d T i . T h e r is k t h a t a d e s i gn i n t e n s i t y

w i ll b e e x c e e d e d c a n b e h a l v e d ( T d o u b l e d ) b y i n c r e a s i n g t h e d e s i gn i n t e n s i t y b y a b o u t

0 .7 . T h i s e q u a t i o n i s p l o t t e d i n F i g u r e 4 f o r t h e r a n g e o f v a l i d i t y i > i w h e r e

t

= c l + c 2 m o - c 3 In d = 6 .08.

I f i n t e r e s t e x t e n d s t o s m a l l e r i n t e n si t ie s , i t n e c e s s i t at e s m o r e c u m b e r s o m e i n t e g r a t i o n s

n o t s h o w n h e r e .

PEAK GROUND MOTION RESULTS

T h e p r e v i o u s s e c t i o n d e v e l o p e d t h e d e s i r e d d i s t r i b u t i o n r e s u l t s f o r t h e M o d i f i e d

M e r c a l l i i n t e n s i t y , I , a n d a u n i f o r m l i n e so u rc e , w i t h a p a r t i c u l a r s e t o f a s s u m p t i o n s

o n m a g n i t u d e d i s t r i b u t i o n a n d t h e i n t e n s i t y v e r su s M a n d R r e l a t i o n s h i p . E n g i n e e r s

a r e g e n e r a l l y m o r e d i r e c t l y c on c e r n e d w i t h s u c h g r o u n d m o t i o n p a r a m e t e r s a s p e a k-

cl = 8.16

U s i n g a t t e n u a t i o n c o n s t a n t s f o u n d e m p i r i c a l l y ( E s t e v a a n d R o s e n b l u e t h , 1 9 6 4) f o r

C a l i f o r n i a


11/24

ground acce l e r a t i on , A , p e a k - g r o u n d v e l o c i ty , V , o r p e a k - g r o u n d d i s p l a c e m e n t , D ,

t h a n w i t h i n t e n s i t y i ts e lf .

A n a r g u m e n t p a r a l l e l t o t h a t i n t h e p r e c e d i n g s e c t i o n c a n b e c a r r i e d o u t w i t h a n y

f u n c t i o n a l r e l a ti o n s h ip b e t w e e n t h e s it e g r o u n d - m o t i o n v a r i ab l e , Y , a n d M a n d R .

F o r e x a m p l e , t h e p a r t i c u l a r f o r m

Y = b l e b ~ R - b 3

( 2 6 )

I

h a s b e e n re c o m m e n d e d b y K a n a i ( 1 9 6 1 ) a n d b y E s t e v a a n d R o s e n b l u e t h ( 1 9 64 ) * f o r

p e a k - g r o u n d a c c e l e r a ti o n ( Y = A ) , p e a k - g r o u n d v e l o c i t y ( Y = V ) , a n d p e a k - g r o u n d

d i s p l a c e m e n t ( Y = D ) . T h e l a t t e r a u t h o r s ( E s t e v a a n d R o s e n b l u e t h , 1 9 6 4 ; E s t e v a ,

o 8

7

o

0 )

0

I I ~ I I I I I

E N G I N E E R I N G S E I S M I C R I S K N L Y S I S 1593

I 2 5 I 2 5 T i , y e a r s

0 . 0 1 0 . 0 0 1 - F I m Q x i )

FIG. 4. Nu merical exam ple: Intens ity V ersus return period.

1 96 7 ) ( o n t h e o r e t i c a l a n d e m p i ri c a l g r o u n d s ) s u g g e s t t h a t t h e c o n s t a n t s {bl , b2, b3 } be

{2000 , 0 .8 , 2} , {1 6 , 1 .0 , 1 .7} , and {7 , 1 .2 , 1 .6} fo rA , V , a n d D respec t i ve ly i n sou th e rn

C a l if o rn i a , w i t h A , V , a n d D i n u n i t s o f c e n ti m e t e r s a n d s e c o n d s a n d R i n k i l o m e t e r s .

F o r t h e g e n e r a l r e l a t i o n s h i p i n e q u a t i o n 2 6 , a n a r g u m e n t l i k e t h a t i n t h e p r e v i o u s

s e c t io n y i e ld s fo r t h e a n n u a l m a x i m u m v a l u e o f Y f ro m a u n i f o r m l in e s o u r c e

Fr~.a) = exp [ - -~ CGy ~lb2] y >= y

1 - F r ( . )~ ~ ~ C G y -~/b~ y >= y

1 y ~ i b2

T y ~ n C G

( 2 7 )

28)

*More recently, Es teva (1967), it has been sug gested that the focal depth,

h ,

in kilometers, be

replaced by an em pirically adjusted value, ~ /~ -4- 203, w hich increases the formula s accuracy

shorter focal distances.

( 2 9 )


12/24

1 5 9 4 BULLETIN F THE SEISMOLOGICALSOCIETY OF AMERICA

i n w h i c h

C = e ~ m b l ~ /b 2

3 0 )

a n d G is a s g i v e n i n e q u a t i o n 1 5 o r e q u a t i o n 1 7 ) w i t h

5

v = ~ - - - 1 , 3 1 )

O2

T h e l o w e r l i m i t o f t h e v a l i d i t y o f t h e s e f o r m s o f Fr g 2x i s

y = b l e b ~m o d - b 3 .

3 2 )

F o r d u r a t i o n s , t, o t h e r t h a n o n e y e a r , ~ s h o u l d b e r e p l a c e d b y ~t i n e q u a t i o n s 2 7 a n d 2 8.

N o t i c e t h a t e q u a t i o n 2 7 is o f t h e g e n e ra l fo r m o f t h e T y p e I I a s y m p t o t i c e x tr e m e

v a l u e d i s t r i b u t i o n o f l a r g e s t v a l u e s G u m b e l , 1 9 5 8 ) . T h i s d i s t r i b u t i o n , t o o , i s c o m -

m o n l y u s e d i n t h e d e s c r i p t i o n o f n a t u r a l l o a d in g s o n e n g i n e e r i n g s t r u c t u r e s , t h e m o s t

f a m i l i a r b e i n g m a x i m u m a n n u a l w i n d v e l o c it i es T a s k C o m m i s s i o n o n W i n d F o r c e s ,

h Site

Point Source

FIG. 5. Point source cross section.

1 9 61 ; T h o m , 1 9 6 7 ) . T h e j u s ti f ic a t io n t h e r e is b a s e d o n a s y m p t o t i c l a r g e N ) a r g u -

m e n t s w h i le t h a t h e r e is n o t . T h e r e s u l ts h e r e a r e a c o n s e q u e n c e o f t h e f o r m s o f th e

r e l a t i o n s h i p s a s s u m e d .

U s i n g r e s u l ts s u c h a s t h e s e t h e d e s i g n e r c a n c o m p u t e f o r h is s i te t h e p e a k - g r o u n d

v e l o c i t y , v , a n d p e a k - g r o u n d a c c e l e r a t i o n , a , a s s o c i a t e d w i t h t h e s a m e , s a y t h e 2 0 0 - y e a r ,

r e t u r n p e r i o d . F o r t h e n u m e r i c a l e x a m p l e i n t h e p r e v i o u s s e c t i o n a n d t h e v a l u e s o f

t h e p a r a m e t e r s r e f e r r e d to i n th i s s e c ti o n , t h e s e v a l u e s a r e a p p r o x i m a t e l y

v = 7 .5 c m / s e c = 3 i n / s e c

a = 8 0 c m / s e c 2 = 0 . 0 8 g .

GENERAL SOURCE RESULTS

I n o r d e r t o f a c i l i t a t e r e p r e s e n t i n g t h e g e o m e t r y a n d p o t e n t i a l s o u r c e c o n d i t i o n s a t

a r b i t r a r y s i t e s, i t i s d e s i r a b l e t o h a v e a d d i t i o n a l r e s u l t s f o r p o i n t a n d a r e a s o u r c e s .

I t w i ll b e s h o w n t h a t t h e s e r e s u l ts c a n b e u s e d t o r e p r e s e n t q u i t e g e n e r a l c o n d i ti o n s .

I f a p o t e n t i a l s o u r c e o f e a r t h q u a k e s is c l o se l y c o n c e n t r a t e d i n s p a c e r e l a t i v e t o i t s

d i s t a n c e , d , f r o m t h e s i te , i t s a t i s f a c to r i l y m a y b e a s s u m e d t o b e a p o i n t s o u r c e, F i g u r e

5 . E x a m p l e s m i g h t b e s it es o n e o r t w o h u n d r e d k i l om e t e r s f r o m N e w M a d r i d , M o . o r

C h a r l e s t o n , S . C . ) I n t h i s c a s e t h e r e i s n o u n c e r t a i n t y i n t h e f o c a l d i s t a n c e , d , a n d t h e


13/24

E N G I N E E R I N G S E I S M I C R I S K A N A L Y S I S 1595

p r e v i o u s r e s u l t s ( e . g ., e q u a t i o n s 2 2 , 2 5 , 2 7 , 2 9 ) h o l d w i t h ~ e q u a l t o t h e a v e r a g e n u m b e r

o f e a r t h q u a k e s o f i n t e r e s t ( M ->_ m 0 ) p e r y e a r o r i g i n a t i n g a t t h i s p o i n t a n d w i t h a

g e o m e t r y t e r m ( i n p l a c e o f e q u a t i o n 1 5 ) e q u a l t o

G = d - (~+1) (3 3)

F o r i n t e n s i ti e s 7 i s g i v e n b y e q u a t i o n 16 a n d f o r v a r i a b l e s w i t h r e l a ti o n s h i p s o f t h e

t y p e s h o w n i n e q u a t i o n 2 6 , 7 i s g i v e n b y e q u a t i o n 3 1 . F o r a p o i n t s o u r c e , f o r v a l u e s o f

t h e a r g u m e n t l e ss t h a n i ' o r

y ,

t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n ( e q u a t i o n 2 2 o r 27 )

i s s i m p l y z e r o .

I n s o m e s i t u a t io n s , o w i n g t o a n a p p a r e n t l a c k o f c o r r e l a t i o n b e t w e e n g e o l og ic st r u c -

t u r e a n d s e is m i c a c t i v i t y o r o w i n g t o a n i n a b i l i t y t o o b s e r v e t h i s s t r u c t u r e d u e t o d e e p

o v e r b u r d e n s , i t m a y b e n e c e s s a r y f o r e n g i n e e ri n g p u r p o s e s t o t r e a t a n a r e a s u r r o u n d i n g

t h e s i t e a s if e a r t h q u a k e s w e r e e q u a l l y l ik e l y t o o c c u r a n y w h e r e o v e r t h e a r e a. I t c a n

F I G 6. Annular sources, perspective.

b e s h o w n t h a t f o r a n a n n u l a r a r e a l so u r c e s u r r o u n d i n g t h e s it e, a s p i c t u r e d i n F i g u r e 6 ,

t h e d i s t r i b u t io n s a b o v e ( e q u a t i o n s 2 2 a n d 2 7 ) h o l d w i t h a g e o m e t r y t e r m e q u a l t o

G - ~ : ~ ) d ~ _ 1

w i t h 7 g i v e n b y e q u a t i o n 1 6 o r 3 1. T h e v a l u e o f p s h o u l d n o w b e t h e a v e r a g e n u m b e r

o f e a r t h q u a k e s o f i n t e r e s t ( M >= m 0 ) p e r y e a r p e r u n i t a r e a . I n t e r m s o f ~, t h e a v e r a g e

n u m b e r p e r y e a r o v e r th e e n t i r e a n n u l a r r e g io n , p i s

P

= ( 3 5 )

~ ( l 2 _ ~ 2 )

F o r v a l u e s o f t h e a r g u m e n t le ss t h a n i ' o r

y ,

t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n

( e q u a t i o n 2 2 o r 2 7 ) i s ze r o . N o t e t h a t d w il l n e v e r b e le ss t h a n h . T h u s t h e g e o m e t r y

f a c t o r r e m a i n s fi n it e e v e n w h e n t h e s it e is i m m e r s e d i n t h e a r e a l s o u rc e , i .e ., w h e n

A = 0 , a n d n i l e a r t h q u a k e d i r e c t l y b e l o w t h e s i t e i s a n ( i m p r o b a b l e ) p o s s i b i l i t y .


14/24

1596

B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E T Y O F A M E R I C A

W h e n m o r e c o m p l e x s o u r c e c o n fi g u r a ti o n s e x is t , t h e d i s t r i b u t i o n f u n c t i o n f o r th e

m a x i m u m v a l u e o f s o m e g r o u n d m o t i o n v a r i a b le c a n b e f o u n d b y c o m b i n i n g t h e r e -

s u l ts a b o v e . F o r e x a m p l e , if t h e r e e x i s t i n d e p e n d e n t s o u rc e s ( 1 , 2 , n ) o f t h e v a r i o u s

t y p e s d i s cu s se d a b o v e , th e p r o b a b i l i ty t h a t t h e m a x i m u m v a l u e o f Y , t h e p e a k - g r o u n d

a c c e l e ra t i o n , fo r ex a m p l e , i s l es s t h a n y i s t h e p r o b a b i l i t y t h a t t h e m a x i m u m v a l u e s

f r o m s o u r ce s 1 t h r o u g h n a r e

al l

l e s s t h a n y , o r

~ I I l U X I I I U X . l l l a X 2 I Tt~ X n

= l Y I F r ~ >

= l m a x j

i n w h i c h F r (~ ) i s t h e d i s t r i b u t io n o f t h e m a x i m u m Y ( s a y p e a k a c c e le r a t io n ) f r o m

m a , x j

s o u r c e j , a s g i v e n b y e q u a t i o n 2 7 w i t h t h e a p p r o p r i a t e v a l u e s o f t h e p a r a m e t e r s p j., C j ,

G ~.. N o t e t h a t t h e d i f f e re n t p o s s ib l e f o ca l d e p t h s o n t h e s a m e f a u l t c a n b e a c c o u n t e d

f o r in t h i s m a n n e r .

F o r t h e e x p o n e n t i a l f o r m o f t h e ,. r ~ s) f u n c t io n s ( e q u a t i o n 2 7 )

I _ S j b 2 2

Fr(m~2~ =

e x p

- - ~ ~jCjG~y j y > y '

( 3 6 )

w h e r e y ' is th e l a r g e s t o f t h e Y i F o r y l es s t h a n y , t h e d i s t r i b u t i o n c a n b e f o u n d w i t h

e a s e ( u n l e s s a l in e s o u r c e i s i n v o l v e d ) . I f t h e c o n s t a n t s / ~ , b l , b ~ , ba a r e t h e s a m e f o r a ll

t h e s o u r c e s i n t h e r e g i o n a r o u n d t h e s i te , e q u a t i o n 3 6 b e c o m e s s i m p l y

i n w h i c h

F r ( ~ ) = e x p

[ -C ~ G y -~ /b2] y > y '

m a x

( 3 7 )

p G : ~ p j Gj ( 3 8 )

A s i m i l a r c o n c l u s i o n h o l d s fo r M o d i f i e d M e r c a l l i i n t e n s i t ie s , e q u a t i o n 2 2 .

I n s h o r t t h e d i s t r i b u t i o n s r e t a i n t h e s a m e f o r m s w i t h t h e p r o d u c t , ~ G , e q u a l t o t h e

s u m o f t h e c o r r e s p o n d i n g p r o d u c t s o v e r t h e v a r i o u s s o u rc e s. W i t h r e s p e c t t o t h e s e

p r o d u c t s , t h e n , l i n e a r s u p e r p o s i t i o n a p p l i e s . T h i s c o n c l u s i o n i s a r e f le c t i o n o f t h e f a c t

t h a t t h e s u m o f i n d e p e n d e n t P o i s s o n p ro c e s s i s a P o i s s o n p ro c e s s w i t h a n a v e r a g e a r -

r i v a l r a t e e q u a l t o t h e s u m o f i n d i v i d u a l r a t e s .

T h i s c o n c l u s io n c a n b e u s e d t o d e t e r m i n e g e o m e t r y f a c t o r s f o r u n s y m m e t r i c a l s o u r c e

g e o m e t r i e s . F o r e x a m p l e , f o r t h e c o n d i t i o n i n F i g u r e 7 a , t h e g e o m e t r y f a c t o r, G , m u s t

e q u a l o n e - h a l f o f t h a t f o r t h e s y m m e t r i c a l s i tu a t i o n . T h e g e o m e t r y f a c t o r f o r t h e s i t u a -

t i o n i n F i g u r e 7 b m u s t e q u a l o n e - h a l f of t h a t f o r a s y m m e t r i c a l s o u r c e l e n g t h 2 b m i n u s

o n e - h a l f o f t h a t f o r a s y m m e t r i c a l s o u r c e o f l e n g t h 2 a , o r

G - - [ G ' - G ] ( 3 9 )

i n w h i c h

G '

a n d

G

a r e c a l c u l a te d f r o m e q u a t i o n 1 5 w i t h v a l u e s ro a n d r 0 r e s p e c ti v e l y .

A n e x a m p l e w il l f o ll ow . T h i s r e s u l t a ls o p e r m i t s e a s y t r e a t m e n t o f a f a u l t w i t h a ( s p a -

t i a l l y ) n o n - c o n s t a n t a v e r a g e o c c u r r e n c e r a te , e a c h d i f f e r e n t p o r t i o n o f t h e f a u l t b e i n g

t r e a t e d i n d e p e n d e n t ly .


15/24

,

d

S i t e

r

a ) C a s e I

J

I

b ~.

/

. i

d I r o ro

S i t e

b C a s e 2

S i t e

~ f

h

c

C a s e 5 ,

P e r s p e c t i v e

F I e . 7 . U n s y m m e t r i c a l s o u r c e s .

1 5 9 7


16/24

5 9 8 B U L L E T I N O F T H E S E I SM O L O G I C A L S O C I E T Y O F A M E R I C A

in the same manner the geometry factor for an area such as that shown in Figure 7c

is found to be

_ l

G ~G~

W E

FI ~. 8. Numerical examples: plan

in which G~ is the result for the complete annulus, equation 34. As will be shown in a

numerical example, an areal source of arbitrary shape can be modeled with ease by

approximating it by a number of such shapes.

Note that the approximation to equation 37 for smaller values of the probability

- - F y v )

becomes

m a ~

1 -- F~ I~)~,:_~ C y --~/b~ ~ f,~Gi 41)


17/24

ENGINEERING SEISMIC RISK ANALYSIS 159 9

s u g g e s t in g t h a t t h e s m a l l ) p r o b a b i l i ty t h a t t h e a n r m a l m a x i m u m , Y . . . e x c e e d s y ,

i n a n y y e a r is m a d e u p o f t h e s u m o f t h e p r o b a b i l i t ie s c o a t r i b u t e d b y e a c h o f t h e s o u r c e s .

A l s o , f o r l a rg e r v a l u e s t h e r e t u r n p e r i o d is a p p r o x i m a t e l y

1

y ~ / b 2 4 2 )

T~ ~-- CX~jG~

TAB LE 1, PAR T i

NUMEI~ICAL EXAMPL'm.

Source d ro

G~

V D

Line 1

Ri gh t po rt io n 104 115.3 5.12 X 10 7 1.73 X 10 4 3.66 X 10-a

Lef t po rt io n 104 241 1.06 X 10 6 3.17 X 10 ~ 5.99 X 10- 9

Line 2

To ta l 49 206 3.1 8 X 10 5 1,98 X 10 -3 1.5 5 X 10 ~

Por t io n a l ( - ) 49 57.5 -7 .28 >( 10-0 --0.94 X 10- 3 -1. 08 2 X 10- a

Areal

A n n u l u s

1, a = 2~ 28.3 45 24.9 X 10 4 2.4 4 >( 10 i 1.76

2, ~ = 4.38 45 75.5 7.0 X 10- 4 1.14 X 10 1 1.28

3, ~x = 3.78 75. 5 123. 5 2.1 1 X 10 4 0. 65 X 10 1 0.9 8

4, ~ = 3.44 123.5 252 0.8 0 X 10 4 0.61 X 10 i 1.16

5, a --- ~ 252 ~o 0. 23 X 10 4 0. 60 >( 10 1 10. 49

Poin t

216 4.7 X 10 19 4. 4 X 10- 7 1.1 X 10 6

Assu mpt i o ns : h = X/ 20Y + 202 = 28.3 k m ; ~ = 1.6; m0 = 4

Pe ak ac ce le ra ti on : bl = 2000; b~ = 0. 8; b~ = 2 ; C - 2. 4) < 109

Pe ak ve lo ci ty : bl = 16 ; b2 = 1.0 ; b~ = 1. 7; C = 4.98 X 104

Pe ak dis pl ac em en t: bl = 7 ; b~ = 1.2; b3 = 1. 6; C = 7.8 X 10 s

T A B L E 1 P A R T 2

NUMERIC L EX MPLE

~i i

Source

V D

Line 1

Ri gh t po rt io n 5.12 >( 10 l i 1.73 X 10- g 3.66 )< 10 7

Lef t po rt io n 10.6 )< 10 l l 3.17 X 10 8 5.99 X 10 7

Line 2

To ta l 318 X 10 i ' 19.8 )< 10 8 15.5 X 10 7

Por t io n a i ( - ) -7 2.8 X 10- n -9. 4 X 10 8 --10.82 X 10- 7

Areal

A n n u l u s

1, a = 2~ 249 >( 10-n 24. 4 X 10- s 17.6 X 10 7

2, a = 4.38 70 >( 10 11 11 .4 X 10 - 8 12 .8 )< 10- 7

3, a = 3.78 21.1 X 10 -11 6.5 X 10- 8 9. 8 X 10- 7

4, a = 3.44 8. 0 X 10 11 6.1 X 10 s 11. 6 X 10 7

5, a = 7r 2. 3 X 10 11 6. 0 X 10 .8 104 .9 X 10 7

Poin t

4.3 X 10- ii 4.1 X 10-8 9. 8 X 10 7

Su m 616 )< 10 ll 73.7 X 10 8 182 )< 10 7


18/24

I6 00 BULLETIN OF TH:E S~ISMOLOGICAL SOCIETY OF AMERICA

,, :i :~ : NUMERICAL EXA MPL E

~ For : i l l u s t r a t i on We t r ea t t h~ hypo th e t i ca l s i t ua t ion shown in F igure 8 : Th e s i te i s

loca t ed on a deep a l luv ia l p l ane ( shaded ) such tha t t he geo log ica l s t ruc tu re be low the

s i t e and to t he sou th and eas t i s no t known in de t a i l . Hi s to r i ca l ly , ear thquakes have

occur red th rough ou t t h i s p l ane , bu t no t o f t en enough to de t ermine fau l t pa t t e rns . Th e

eng ineer chooses t o t r ea t t he r eg ion as if t h e nex t ear thq uake were equal ly l ike ly to

occur in any uni t area. T he av erage rate , P = 1 .0 X 10-~ per km 2, was es t imated by

d iv id ing the r eg ion 's t o t a l n um ber o f ear thqu akes (wi th m agn i tudes i n excess o f 4)

by i t s t o t a l a rea .

The except ion, h is tor ical ly , i s a Smal l area, so me 200 km southea st . I t i s al so below

the a l luv ia l p lane . Th e f r equency o f a l l s izes o f ear thquakes there has been re l a t i ve ly

h igh , i nc lud ing severa l o f la rger magn i tudes . Al thou gh the eng ineer can eas i ly accoun t

fo r any suspec t ed loca l d i ff e rence in t he p aram eter ~ ( smal l e r va lues imply h igher

relat ive f requencies of larger ma gni tu des ) , h e chooses to use the same ~ value, 1 .6 ,

for th e ent i re region. In o th er words, he chooses to a t t r ib ute the Small area 's obse rved

larger magn i tudes t o t he same popu la t ion

f , m ) .

The just i f icat ion i s th at the larger

the :average ar r ival rate , the larger i s the nu mb er of observat ions and the more l ikely

i t i s tha t larger ma gni tud es wi l l be inclL~ded am ong the observat ions of a g iven per iod

of t ime. Exa ct ly what a rea (here shown as 30 by 30 k in ) i s used to es t imate t he a rea l

occu rrence rate, ~ = 1.0 10 4 per k m 2, is not cri t ical in this cuse since the area is

smal l enough and fa r enough f rom the s i te t ha t t he en t i r e source wil l be t r ea t ed as a

point w i th rate ~ = 0:09.

F ina l ly , t o t he nor thw es t where the geo log ica l s t ruc tu re i s exposed two fau l t s have

been loca t ed . Nei ther can be assumed inac t ive . Pas t ac t iv i ty on the f i r s t ( and o ther

geological ly s imi lar fau l t s) suggests an avera ge occurrence ra te of P = 1 .0 X 10 4

per k in . No ear thquakes on the second, closer faul t have been recorded, but i t s geo-

logical s imi lar i ty to the f i rs t suggests tha t i t be g iven a s imi lar ac t iv i t y level .

The sectors of annul i used to represent the areal region are shown in Figure 8 . The

geo me try factors , G~, for the var ious sources are shown in Tab le 1 along wi th the

product s ~G~, and the i r sums fo r peak-g rou nd acce l era t ion , A, pe ak-g round ve loc i ty V,

and peak-g rou nd d i sp l acement , D . The conclusion i s t ha t t he ma x im um ground ac-

ce l era tion , ve loc i ty , and d i sp l acement d ur ing an in t e rva l o f t year s ha ve d i s t r ibu t ions

F~(?~)~ = exp [ - 14.7ta-2]

Fv~(~2x = ex p [- 0 .0 3 67 tv -1'6]

F~,~i ~ = exp [-0.142td-1'33].

T h e an n u a l m ax i m a h av e t h e ap p r o x i m a t e d i s tr i bu t io n s

1 -- FA[~2x ~ 14.7 a-2

1 - - F. ~2 X~ 0.0367v -16

1 -- F.(~) __--~0.14 2d -1'33.

m x

I n te rm s of re tu rn pe rio ds T~ ~ 0.068 1a 2, Tv --~ 27.3v 1 6, T~ _~ 7.05d 1 ~3, or a ~ 3.83Ta '5,

v --~ 0.126Tv '62~, d _--~ 0.23 1T d '75.


19/24

E N G IN E E RIN G SE ISM IC RISK A N A L Y SIS 16 1

If a design response spectrum based on a 200-year return period were desired, it

should be based on design ground-motion values of a200 = 55 cm/sec 2= 0.054g,

v200 = 3.5 cm/sec, d2o0 = 12.5 cm. Using the method for constructing a spectrum sug-

gested for design by Newmark 1967), the dynamic response spectrum in Figure 9 is

obtained.

/)

E

0

~10

t -

O

/)

0

n ~

5 C

0

0

Q )

>

o 2 . 0

0

0

u I.O

a .

0 .1 0 . 2 0 . 5 1 .0 2 . 0 5 . 0 I 0

N a t u r a l P e r i 0 d , s e c

Design Response Spectrum; 2 of Cr i tical Dam ping

F IG . 9 . D y n a m i c r e s p o n s e s p e c t r u m .

Notice that, for a proportional increase in all ground-motion factors, the risk

a s

measured by 1 - F or 1/T decreases more rapidly for peak acceleration than for

peak velocity, and more rapidly for velocity than for displacement. The implication is

that shorter-period structures can achieve greater risk reductions for the same per-

centage increase in design level than longer-period structures.


20/24

1 6 2

B U L L E T I N O F T H E S E I S M O L O G I C L S O C I E T Y O F M E R I C

Inspection of the individual contribut ions to Ev~G~ (or, approximately, to the

risks 1 - F) in Table 1 reveals that the closer faults have the predominant influence

on peak acceleration risks, since even relatively small, more frequent magnitudes can

give rise to high accelerations locally.* More distant potential sources contribute sig-

nificantly to the risk of longer-period structures, as evidenced by their contributions

to the E,~G~ factor for velocity and displacement. The slow decay with distance of

peak displacement causes a large contribution even from long distances. This explains

why the displacement of 12.5 cm is considerably larger than those associated with

specific earthquakes records with peak accelerations of the order of 0.05g.

ASSUMPTIONS AND EXTENSIONS

While the assumptions made in the method are considered reasonable for most

purposes of engineering design, a number of them can be relaxed without significantly

altering the basic method. In particular, the distribution of magnitudes and the rela-

tionships used to relate site ground-motion characteristics to the magnitude and focal

distance can be replaced with ease, only the results of certain integrations will change.

In the derivations above the distribution of magnitudes has been assumed to be the

unlimited exponential distribution. For the larger, rarer magnitudes there are in-

sufficient data to substantiate with confidence this or any other assumption (Rosen-

blueth, 1964). The shape as well as the parameters may in fact vary among different

regions for these larger values. The magnitudes of earthquakes may be bounded.

Relatively clean analytical results can be obtained for distribution functions of poly-

nomial form and for the limited exponential distribution. Their influence, which may be

significant for larger return periods, are under investigation.

For different focal distance relationships, the existing results can be used with piece-

wise fits to the other functions. For example, if it is assumed that there is no attenua-

tion of Y with distance for a certain distance, r', from a source (Housner, 1965; Ipek

e t a l 1965) an annular source canbebroken into two regions, one ford -< r -< r' and the

other for r' -- r -- r0. In the first region b3 should be set equal to zero, and the values of

bl and b2 appropriate for near-source conditions adopted. Coupled with a l imited magni-

tude distribution, this process facilitates incorporation of any suspected upper bounds

on maximum ground motions (Housner, 1965).

These functions are, in any case, no better than the parameter estimates used in

them. One primary advantage of an analytical method, as opposed to a numerical one

(Ipek

e t a l

1965; Lacer, 1965) is that the sensitivity ot final conclusions to the ac-

curacy of these parameter estimates can be assessed.

Other of the more basic assumptions in the method can also be relaxed with relative

ease. Specifically, these include the two assumptions (a) that the radiation of effects

can be treated as if the earthquake generating mechanism were concentrated at a

point and (b) that isoseismals are circular. These assumptions are commonly made in

design studies. This is done not so much because it is thought to be t rue, but because

alternative methods and information are seldom available. The vast majority of sta-

tistical data on attenuation and scaling laws, for example, are available in forms

(averages, etc. ) based on these two assumptions. At the expense of added mathematical

complexity alternative assumptions (e.g., finite mechanism length, elliptical isoseis-

mals) can be incorporated into the method above if sufficient data are available to

just ify their inclusion.

Also, of course, the durations are correspondingly short, a factor not explicitly appearing in

the method for construction of response spectra proposed by Newmark.


21/24

E N G I N E E R I N G S E I S M IC R I S K A N A L Y S IS

16 3

The more fundamental assumptions are those of (a) equal likelihood of occurrence

along a line or over an areal source, (b) constant-in-time average occurrence rate of

earthquakes, and (c) Poisson (or memory-less ) behavior of occurrences.

If data or judgement rule against the equM-likelihood assumption and in favor of

other relative values they can be included by simply treating each portion of the

source over which the equally likely assumption is reasonable as an individual source

using the superposition method described above.

If the engineer and the seismologist are prepared to make an assumption about the

time dependence of the average occurrence rate, other than tha t of constant in time, a

minor modification in the method suffices to account for this non-homogeneity in time

(Parzen, 1962; Cornell, 1964). The influence appears, for example, in Equation 21 as

I f 1

L[Iu)m~ = i] = exp -- p~ ~(r) dr (59)

in which v (r ) is the average occurrence rate at time T.

The assumption that the occurrences of earthquakes follow the behavior of the

Poisson process model can be removed only at a grea~er penalty , however. The Poisson

assumption does not reflect earthquake swarms or aftershocks, nor is it physically

consistent with the elastic rebound theory, which implies that a zone of recent past

activity is less likely to be the source of the next earthquake than a previously active

zone which has been relatively quiet for some time. These limitations can, in principle,

be removed by adopting more general renewal process or Markov process models

(Aki, 1956; Vere-Jones, 1966). For engineering purposes the Poisson results are con-

sidered adequate for numerous reasons (Rosenblueth, 1966; Lomnitz, 1966). When

swarms and aftershocks are excluded, data does not clearly reject the Poisson assump-

tion (Lomnitz, 1966; Wanner, 1937; Knopoff, 1964; Niazi, 1964)for the rarer, major

events of engineering interest. Even when more accurate theoretical models become

available, it is not evident tha t sufficient statistical data and other information will be

available in many regions to permit the seismologist to adopt a non-Poisson assumption

or to estimate any more parameters than the average occurrence rates.

The structural engineer is concerned more directly with a design response spectrum.

For random forcing functions such as earthquake ground motions, the duration of

motion also influences the pe~k-response values (Rosenblueth, 1964; Crandall and

Mark, 1963). Given a relationship between durat ion and M and R, and given a func-

tion relating (expected) peak response to duration and to expected peak-ground ac-

celeration or velocity, a simple application of the same method will produce such

response spectra. In addition, inclusion of the randomness of peak response to random

motions with given parameters (Ilosenblueth, 1964; Crandall and Mark, 1963) will

permit the construction of response spectra based on prescribed probabilities of re-

sponses not to be exceeded in a given lifetime. There is strong reason to believe that

this latter influence is negligible (Rosenblueth, 1964; Borges, 1956).

Although developed specifically for the seismic risk analysis of individual sites, the

method systematically applied to a grid of points would yield regional seismic proba-

bility maps. These might t ake a form similar to those used in determining design winds

(Thorn, 1967), namely contours of maximum ground motion of equal return period.

Consistent maps could be produced to as fine a scale as desired. Perhaps the greatest

advantage of this method for this purpose is that it would insure that consistent

assumptions were being used for all portions of the region and among different regions.


22/24

1604 BULLE TIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

A l l a s s u m p t i o n s m a d e b y t h e s e is m o l o g is t s i n v o l v e d w o u l d b e e x p li ci t a n d q u a n t i t a t i v e ,

o p e n t o r e v i e w a n d t o u p - d a t i n g w i t h n e w e v i de n c e. M a j o r d i f fi cu lt ie s w o u l d r e m a i n ,

h o w e v e r , i n t h e j u d g e m e n t o f a c t i v e s o u rc e s , i n t h e e s t i m a t i o n o f t h e i r a v e r a g e a c t i v i t y

r a t e s , a n d i n d e t e r m i n a t i o n o f lo c a l s oi l i n f lu e n c e .

CONCLUSION

A q u a n t i t a t i v e m e t h o d o f e v a l u a t i n g t h e s e is m i c r is k a t a p a r t i c u l a r s i te h a s t h e

a d v a n t a g e t h a t c o n s i s te n t e s t i m a t e s o f t h e s e r is k s c a n b e p r e p a r e d f o r v a r io u s p o t e n -

t i a l s i te s , a ll p e r h a p s i n t h e s a m e g e n e r a l r e g i o n b u t i n s i g n i f ic a n t l y d i f fe r e n t g e o m e t r i -

c a l r e l a t i o n s h i p s w i t h r e s p e c t t o p o t e n t i a l s o u r c e s o f e a r t h q u a k e s .

S u c h a m e t h o d i s n e c e s s a r y to d e t e r m i n e h o w r a p i d l y t h e r i s k d e c a y s a s t h e r e s i s t-

a n c e o f th e s y s t e m s d e s i g n is i n c re a s e d . R e a s o n a b l e e c o n o m i c t r a de - o ff s , b e t h e y w i t h

r e s p e c t t o o p e r a t i n g r e g u l a t i o n s , b e l o w - s t a n d a r d p e r f o r m a n c e , o r s y s t e m m a l f u n c t i o n ,

c a n n o t b e m a d e w i t h o u t s u c h q u a n t i t a t i v e r e la t io n s h i ps .

T h e m e t h o d p r o p o s e d o f fe r s t h e m e a n s b y w h i c h t o m a k e t h e s e e n g in e e r in g a n a l y s e s

c o n s i s t e n t w i t h t h e s e i s m i c it y i n f o r m a t i o n a v a i la b l e . T h i s i n f o r m a t i o n i s t r a n s f e r r e d

f r o m t h e s e i s m o l o g i st i n t h e f o r m o f h i s b e s t e s t i m a t e s o f t h e a v e r a g e r a t e o f s e is m i c

a c t i v i t y o f p o t e n t i a l s o u r c e s o f e a r t h q u a k e s , t h e r e l a t i v e l i k el i h o o d s o f v a r i o u s m a g -

n i t u d e s o f e v e n t s o n th o s e s o u r c e s , a n d t h e r e l a t i o n s h i p s b e t w e e n s i te c h a r a c t e r is t i c s ,

d i s ta n c e , a n d m a g n i t u d e a p p l i c a b le f o r t h e r e gi o n.

T h e c o n c l u s io n s a p p e a r i n a n e a s i l y a p p l i e d , e a s i l y i n t e r p r e t e d f o r m , s u i t a b l e f o r

r e v i e w f o r c o n s i s t e n c y a n d s e n s i t i v i t y t o a s s u m p t i o n s .

F o r t h e m o s t c o m m o n l y a s s u m e d f u n c t i o n a l f o r m s o f t h e r e l at i o n sh i p s u s e d, t h e

u p p e r t a i ls o f t h e p r o b a b i l i t y d i s t ri b u t i o n s o f th e d e s i g n g r o u n d m o t i o n p a r a m e t e r s

a r e f o u n d t h e o r e t i c a l l y to b e o f T y p e I o r T y p e I I e x t r e m e v a l u e t y p e .

ACKNOWLEDGMENTS

Th i s w o r k w a s s u p p o r t e d b y a T . W . La m b e a n d A s s o c i at e s c o ns u l ti n g c o n t r a c t w i t h t h e g o v e r n -

m e n t o f Tu r k e y a n d b y t h e I n t e r - A m e r i c a n P r o g r a m o f t h e C i v il En g in e e ri n g D e p a r t m e n t o f t h e

M a s s a c h u s e t t s I n s t i t u t e o f Te c h n ol o g y . Th i s l a t t e r p r o g r a m s p o n s o r ed , i n p a r t , t h e a u t h o r s

v i s i t ing p ro fesso rsh ip a t the Un ive rs i ty o f Mex ico , where d iscuss ions wi th Lu is Es tev a an d D r .

Em i l io R o s e n b l u e th i n i t i a te d t h e a u t h o r s i n t e r e s t i n t h i s s u b j e c t . S u b s e q u e n t ly I n t . Es t e v a i n -

depen den t ly deve loped a num ber o f the re su l t s p resen ted he re (Es teva , 1967). Th e au thor wishes

t o t h a n k t w o c o -w o r k e rs , O c t a v i o Ra s co n a n d E r i k V a n m a r c k e , w h o c o n t r i b u t ed t o t h i s s t u d y .

REFERENCES

Aki, K. (1956) . Some prob lem s in s ta t is t ic a l se ismo logy, Zisin 8, 205-228.

Al len , C . R. , P . S t . Am and , C. F . Rich te r a nd J . M. No rdqu is t (1965) . Re la t ionsh ip be tw een

se ismic i ty and geo log ic s t ruc tu re in the sou the rn Ca l i fo rn ia reg ion ,

Bull. Seism. Soc. Am. 55

753-797.

Be njam in, J . R. (1967) . Prob abil is t ic m odels for se ismic force design, ASCE National Convention

Sea t t le .

Blume , J . A . (1965) . Ea r thqu ake g round mo t ion and eng inee r ing p rocedures fo r im por ta n t in s ta l -

la t ions nea r a c t ive fau l t s . Proc. Third World Conf. on Eq. Engr. (New Zealand) , IV-53.

Blume, J . A . , N . M. New ma rk and L. H. Corn ing (1961). Design of Multistory Reinforced Concrete

Buildings for Earthquake Motions Po rt la nd Ce me nt Assoc. , Chicago.

Bo rges , J . F . (1956) . S ta t is t ica l es t im ate of se ismic loadings, Preliminary Publ. V. Congress

IABSE

Lisbon .

Borg ma n , L . E . (1963) . Risk c r i t e r ia , Proc. ASCE WW3, 1-35.

Corne l l , C . A. (1964) . S tochas t ic P rocess Mode ls in S t ruc tu ra l Eng inee r ing ,

Dept. of C. E. Tech.

Report 34, S tan fo rd U nive rs i ty , Ca l i f .

Cranda l l , S . H. and W. D. Mark (1963) .

Random Vibration in Mechanical Systems

Academic

P r e s s , N e w Y o r k .


23/24

E N G I N E E R I N G S E I S M I C R I S K A N A L Y S I S 1 6 5

Dick, I. D. 1965). Extreme value theory and earthquakes, Proc. Third World Conf. on Eq. Engr.

New Zealand. III 45-55.

Epste in, B. and tI. Brooks 1948). The theory of extreme values and its implica tions in the study

of the dielectric strength of paper capacitors, J. Appl . Phys. 19, 544-550.

Epste in, B. and C. Lomnitz 1966). A model for the occurrence of large earthquakes, Nature 221,

954-956.

Esteva , L. 1967). Crite ria for the construction of spectra for seismic design, Third Panamerican

Symposium on Structures

Caracas, Venezuela.

Esteva , L. and E. Rosenblueth 1964). Spectra of earthquakes at moderate and large distances,

Soc. Mex. de Ing. Sismica Mexico II, 1-18.

Gumbel, E. J. 1958). Extreme Value Statistics Columbia Univ ers ity Press, New York.

Gzovsky, M. V. 1962). Tectonophysics and earthquake forecasting, Bull. Seism. Soc. Am. 52

485-505.

Housner, G. W. 1952). Spectrum intensi ties of strong motion earthquakes, Proc. Sym. on Earth-

quakes and Blast Effects on Structures Los Angeles.

Housner, G. W. 1965). In tensi ty of earthquake ground shaking near the causative fault, Proc.

Third World Conf. on Eq. Engr. 94-111.

Ipek, M. et al 1965). Earthquake zones of Turkey according to seismological data, Prof. Conf.

Earthquake Resistant Construction Regulations in Turkish) Ankara, Turkey.

Isacks, B. and J. Oliver 1964). Seismic waves with frequencies from 1 to 100 cycles per second

recorded in a deep mine in northern New Jersey, Bull. Seism. Soc. Am. 54, 1941-1979.

Kanai, K. 1961). An empirical formula for the spectrum of strong earthquake motions, Bull. Eq.

Res. Inst. 39, 85-95.

Kawasumi, H. 1951). Measures of earthquake danger and expectancy of maximum int ens ity

throughout Ja pan as inferred from the seismic activity in historical times, Bull Earthquake

Res. Inst. 29.

Knopoff, L. 1964). The statisti cs of earthquakes in south ern California, Bull. Seism. Soc. Am.

54, 1871-1873.

Lacer, D. A. 1965). Simula tion of earthquake amplification spectra for southren California sites,

Proc. Third World Conf. on Eq. Engr. New Zealand III, 151-167.

Lomnitz, C. 1966). Stati stica l predic tion of earthquakes, Reviews of Geophys. 4, 337-393.

Milne, W. G. and A. G. Davenpor t 1965). Stati stica l parameters applied to seismic regionaliza-

tion, PToc. Third World Conf. on Eq. Engr. New Zealand, III, 181-194.

Muto, K., R. W. Bailey and K. J. Mitchell 1963). Special requirements for the design of nuclea r

power stations to withs tand earthquakes, Proc. Instn. Mech. Engr. 117, 155-203.

Newmark, N. M. 1965). Curr ent trends in the seismic analysis and design of high rise struc tures ,

Proc. Syrup. on Earthquake Engr. Vancouver, B.C.

Newmark, N. M. 1967). Design Criteria for Nuclear Reactors Subjected to Earthquake Hazards

Urbana, Ill.

Niazi, M. 1964). Seismicity of nort hern California and western Nevada, Bull. Seism. Soc. Am.

54, 845-850.

Nordqui st, J. M. 1945). Theory of Largest Values Applied to Earthquake Magnitudes, T~ans.

Am. Geo. Union 26, 29-31.

Parzen, E. 1962). Stochastic Processes Holden-Day, San Francisco.

Richter, C. F. 1958). Elementary Seismology W. H. Freeman and Co., San Francisco.

Richter, C. F. 1959). Seismic regionali zation, Bull. Seism. Soc. Am. 49, 123-162.

Rosenb lueth , E. 1964). Probabilisti c design to resist earthquakes, Proc. ASCE 90, 189-219.

Rosenblueth, E. 1966). On seismicity,

Seminar in the Application of Statistics to Structural Me-

chanics Dept. of Civil Engr., Univ. of Penn.

Sandi, H. 1966). Ear thqu ake simulat ion for the estimate of structu ral safety, Proc. Ing. Symp.

R.[.L.E.M. Mexico City.

Task Comm. on Wind Forces 1961). Wind forces on struc tures , Trans. ASCE 126, 1124-1198.

Thorn, H. C S. 1967). New distr ibuti ons of extreme winds in the United States, Conf. Preprint

431 ASCE. Envir. Engr. Conf. Dallas.

Uniform Building Code 1967 Edition 1967). Inter. Conf. of Building Officials, Los Angeles.

Veletsos, A. S., N. M. Newmark and C. V. Chelapati 1965). Deformation spectra for elastic and

elasto-plastic systems subjected to ground shock and earthquake motions, Proc. Third World

Conf. on Eq. Engr. New Zealand, I I, 663-682.


24/24

16 6 BULL ETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

Vere-Jones, D. 1966). A Markov model for aftershock occurrences, Pure and Applied Geophys.

54, 31-42.

Wanner, E. 1937). Statistics of earthquakes I, and II , Ger. Geitr. Geophys. 50, 85-99 and 223-228.

Wiggins, J. H. 1964). Effect of site conditi ons on earthquake inte nsi ty,

Proc. ASCE

90, 279-312.

DEPARTMENT OF CIVIL ENGINEERING

M.I.T.

CAMBRIDGE~ MASSACHUSETTS

Manuscript received January 2, 1968.

1968 Paper - Engineering Seismic Risk Analysis

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