engineering seismic risk analysis by cornell

7/28/2019 Engineering Seismic Risk Analysis by Cornell

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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 SBY C . ALLIN CORNELL

ABSTRACTT 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 fa 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 st 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 sa 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 . Int 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 sa 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 um 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 NO 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 nt 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 tm 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 fe 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 fi 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 mu 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 cl 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 rp 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 gt 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 dt 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 nt 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 mc 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 em 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 go 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 ya 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 ns 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 li 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 yn 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 ep 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 tu 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 os 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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15 84 BULLET IN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

ation, for the seismologist to avoid engineering influences; the seismologist's estimateswill probably be more conservative for more consequential projects. But these de-eisions are more appropriately those of the design engineer who has at hand morecomplete 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 bestestimates of the seismic risk. Numerous regional seismic zoning maps have been de-velopcd. Familiar examples appear in the Uniform Building Code (1967) andRichter (1959). Despite reference to probabilities they are seldom clear as to howthe (single) intensity level for each location is to be interpreted. More recently thesevalues have been associated with specific average return periods (~JIuto, Bailey andMitchell, 1963; Kawasumi, 1951; Ipek e t a l , 1965). In any case, more information isneeded to define a relationship between a continuous range of average return periodand 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 historicaldata , have difficulty giving proper weight to the known correlation between geologicalstructure and most seismic activity. They also are not successful at a fine or localscale. Lacer (1965) has presented a numerical, Monte Carlo technique designed toestimate 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 accountfor geological features, such as faults, but he assumes all the assigned "point" sourcesare 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 periodfor his site. The minimum data needed are only the seismologist's best estimates ofthe average activity levels of the various potential sources of earthquakes (e.g., aparticular fault's average annual number of earthquakes in excess of some minimummagnitude of interest, say 4). If, in addition, the seismologist has reason to use otherthan average or typical values of the parameters in the function used to describe therelative 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 influencesof potential earthquake sources, near and far, more active or less, into the probabilitydistribution of maximum annual intensity (or peak-ground acceleration, etc.). Theaverage return period follows directly. The results of the development appear in closedanalytical form, requiring no lengthy computation and permitting direct observationof 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 notonly 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 othereases of interest will be displayed without complete derivations. An illustration willdemonstrate the application of the method. Finally, the assumptions and limitationswill 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 1585

LINE SOURCE DERIVATIONFor illustration of the development of the method of solution, the determination ofthe 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) Persp ective

Site

~/ 2 =- X -- ~ ~/2

Siteb ) ABD P laneFIG. 1. Line source.

Figure la, the site is assumed to lie a perpendicular distance, A, from a line on thesurface vertically above the fault at the focal depth, h, along which future earthquakeloci 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 ABD plane; Figure lb. Theperpendicular 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 A L S O C I E T Y O F A M E R I C AT 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 i n c e - l / 2


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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 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 , th e r e s u l t is1587

T h e l i m i t o n t h e d e f i n it io 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 sf o ri + c~ lit r + 0 -~

L = i , ' 0C2

o i l

i~ c2mo-- 0 - - csh l ~ .

f R ( r ) ~

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

A t s m a l l er 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 si ( 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 rr 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 O- - 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 i s ta 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 is e q u a l l y li k e l y 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 nt 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


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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 Ai 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 ~ ~ 7 - d 212 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 sd 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 th 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 i m 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 ev 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 :

f ro d rG = 2 r ~ / ~ _ d 2"~1 e - l [ rO /d] (COS U ) 5 ' -1 d u ( 1 5 )

d'Y ,-1oi 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 A N A L Y S I S 1 5 8 9

T h e i n t e g r a 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 ig 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 is g iven b y

2 ~ r ( ~ )G -

i n w h i c h r (V ) is 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 ti v e v a lu 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 u0

\- - . 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 intervalof 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 yover an interval of time t (often one year). Observe that

~ ( t )P [ I ... . =< i] = P[e xac tly zero special events in excess of ioccur in the time interval 0 to t]

which from equation (20) isv(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 annualmaximum intensity has a distribution of the double exponential or Gumbel type. Thisdistribution 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 1591realize that, here, this conclusion is n o t based on the intuitive appeal to the familiarasymptotic 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 thevarious relationships assumed above. Others, too, have found (Dick, 1965; Epsteinand Lomnitz, 1966; Epstein and Brooks, 1948) that the combination of Poisson oc-currences of events and exponentially distributed "sizes" of events will invariably leadto the conclusion that the largest event has a GumbeLlike distribution (the trueGumbel 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 thedistribution 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 theexponential form of F ~ ( m ) . If the logarithmic dependence of I on R (equation 3 ) isretained, for example, even polynomial distributions (Housner, 1952) of magnitudewill lead to the exponential distribution of I.

If the annual probabil ities of exceedance are small enough (say _-= i' (24)or, the " T - y e a r " intensity is

i ~ C2 .t_--- ~ In (~C G T~ ) i >-_ ~. (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) thatlog10 n m = a - - b m

= 5.51 -- 0.644min which n~ is the number of earthquakes greater than m in magnitude. Assuming theseearthquakes all occur along the 650 km of the maior fault system in the region, theaverage 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 Au 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 lso1.5 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 -- 2.46t 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 ed 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 = 1.52C2

= exp ? + 2~ r (~ )G----~ (2d)~ [ r ( 3 ' _+2) ] 2 = _ _ 7 " 04 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 hr e t u r n p e r i o d T~ i s

2i ~ ~- 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 yw 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 t0 .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 sn o t s h o w n h e r e .

PEAK GROUND MOTION RESULTST 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 in e s o ur c 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 so 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 sa r e g e n e r a l ly m o r e d i r e c t l y c o n 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 ak -

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 64 ) fo rC a l i f o r n i a


11/24

g r o u n d a c c e l e r a t i o n , A , p e a k - g r o u n d v e l o c i t y , 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 t s e l f .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 yf 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 0

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 rp 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 dd 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" o0 )

0

6

I

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 A N A L Y S I S 1593

I I I I I II 0 0 2 0 0 5 0 0 I 00 0 2 0 0 0 5 0 0 0 Ti , y e a r s0 . 0 1 0 . 0 0 1 I - 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 rnC 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 ss 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~ib2T y ~ n C G

( 2 7 )(28)

*More recently, Este va (1967), it has been sugg ested tha t th e focal depth, h , in kilometers, bereplaced by an em pirically adjusted value, ~ /~ -4- 203, w hich increases the formula's accuracyshorter focal distances.

( 2 9 )


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1 5 9 4 BULLETIN F THE SEISMOLOGICALSOCIETY OF AMERICAi n w h i c h

C = e ~ m b l ~ /b 2 ( 3 0 )a n d G is as 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

53v = ~ - - - 1 , ( 3 1 )O2T 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 ev a l u e d i s t r i b u t i o n o f la 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 tf a m i li a r b e in g m a x i m u m a n n u a l w i n d ve l oc i ti e s ( T a s k C o m m i ss i o n o n W i n d F or c es ,

h Site

Point SourceFIG. 5. Point source, cross section.1 9 61 ; T h o m , 1 9 6 7 ) . T h e j u s t if i c a ti o n t h e r e i s b a s e d o n a s y m p t o t i c ( l a rg 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 er 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 dv 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 ra 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 ft 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 ca = 8 0 c m / s e c 2 = 0 . 0 8 g .

GENERAL SOURCE RESULTSI 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 sd 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 e5 . ( E x a m p l e s m i g h t b e si te s o n e o r t w o h u n d r e d k i l o m e t e rs f ro m N e w M a d r i d , M o . o rC 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 1595p 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 ro 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 ag 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 et 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 ft 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 po 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 gt 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 ro 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 en u m b e r p e r y e a r o v e r t h 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 yf a c t o r r e m a i n s f i 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 nA = 0 , a n d n il 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 li 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 AW 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 ri a b l e c an 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 st 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 da 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 sf r o m s o u r ce s 1 t h r o u g h n a r e a ll 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

= lY I Fr(~>= l m a x ji 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 mm a , x js 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 df 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 / b22Fr(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 he 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 llt 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 hF 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 es 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 ep 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 tt 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 cl u si o 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 to r s f o r u n s y m m e t r i c a l s o u r ceg 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 te 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 so 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 t e 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 gt r e a t e d i n d e p e n d e n t l y .


15/24

%,

d

S i t e

r o

a ) C a s e I

/

J

I - -

b ~ .

, /II . , id I r o roI

S i t eb ) C a s e 2

S i t e~ O fh

c ) C a s e 5 , P e r s p e c t i v eF I e . 7 . U n s y m m e t r i c a l s o u r ce s .

1 5 9 7


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1 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 7cis found to be

_ OlG. ~G~

W E

FI(~. 8. Numerical examples: planin which G~ is the result for the complete annulus, equation 34. As will be shown in anumerical example, an areal source of arbitrary shape can be modeled with ease byapproximating it by a number of such shapes.Note that the approximation to equation 37 for smaller values of the probability1 - - F y ( v ) becomes

ma :~

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


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ENGI NEERI NG SEISMIC RISK ANALYSIS 15 99

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~ "

TABLE 1 , PART iNUMEI~ICAL EXAMPL'm.

Source d ro G~A V D

Line 1Ri gh t po rt io n 104 115.3 5.12 X 10 7 1.73 X 10 4 3.66 X 10-aLef t po rt io n 104 241 1.06 X 10 6 3.17 X 10 ~ 5.99 X 10- 9Line 2To 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- aArealA n n u l u s

1, a = 2~ 28.3 45 24.9 X 10 4 2.4 4 >( 10 i 1.762, ~ = 4.38 45 75.5 7.0 X 10- 4 1.14 X 10 1 1.283, ~x = 3.78 75. 5 123. 5 2.1 1 X 10 4 0. 65 X 10 1 0.9 84, ~ = 3.44 123.5 252 0.8 0 X 10 4 0.61 X 10 i 1.165, 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 6Assu mpt i o ns : h = X/ 20Y + 202 = 28.3 k m ; ~ = 1.6; m0 = 4Pe ak ac ce le ra ti on : bl = 2000; b~ = 0. 8; b~ = 2 ; C - 2. 4) < 109Pe ak ve lo ci ty : bl = 16 ; b2 = 1.0 ; b~ = 1. 7; C = 4.98 X 104Pe 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 2NUMERICAL EXAMPLE

~iGiSource A V D

Line 1Ri gh t po rt io n 5.12 >( 10 l i 1.73 X 10- g 3.66 )< 10 7Lef t po rt io n 10.6 )< 10 l l 3.17 X 10 8 5.99 X 10 7Line 2To ta l 318 X 10 i ' 19.8 )< 10 8 15.5 X 10 7Por t io n a i ( - ) -7 2.8 X 10- n -9. 4 X 10 8 --10.82 X 10- 7ArealA n n u l u s

1, a = 2~ 249 >( 10-n 24. 4 X 10- s 17.6 X 10 72, a = 4.38 70 >( 10 11 11.4 X 10- 8 12. 8 )< 10- 73, a = 3.78 21.1 X 10 -11 6.5 X 10- 8 9. 8 X 10- 74, a = 3.44 8. 0 X 10 11 6.1 X 10 s 11. 6 X 10 75, 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 7Su m 616 )< 10 ll 73.7 X 10 8 182 )< 10 7


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' I 60 0 BULLETIN OF TH:E S~ISMOLOGICAL'SOCIETY OF' AMERICA

',, :i :~ ' : NUMER ICAL EX AM PL 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 show n in F igure 8 : Th e s i te i sloca 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 thes 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 haveoccur 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 eeng 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 tooccur in any uni t area. T he av erage rate , P = 1 .0 X 10-~ per km 2, was est im ated byd 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 belowthe al luvial p lane. The f requency of al l s izes of ear thquakes there has been relat ivelyh 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 tfo 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 igherrelat ive f requencies of larger mag ni tu des ) , he 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 rvedlarger magn i tudes t o t he same popu la t ion f , ( m ) . The jus t i fi ca t ion is t ha t t he l a rgerthe :average ar r ival rate , the larger i s the nu mb er of observat ions and the more l ikelyi 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 iodof 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 loccu rrence rate, ~ = 1.0 10 4 per k m 2, is not cri t ical in this cuse since the area issmal 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 apoint 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 havebeen 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 thergeological ly s imi lar fau l t s) suggests a n a verag e occurrence ra te of P = 1 .0 X 10 4per 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 . Thegeo me try factors , G~, for the var ious sources are shown in Tab le 1 along wi th theproduct 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~(?~)~ = ex p [ - 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 s1 -- FA[~2x ~ 14.7 a-21 - - F .~ 2 X~ 0.0367v -161 -- F .( ~) __--~0.1 42 d -1'33.

m a x

I n te rm s of re tu rn pe rio ds T~ ~ 0.068 1a 2, Tv --~ 27.3v 1"6, T~ _~ 7.05 d 1"~3, or a ~ 3.83Ta '5,v --~ 0.126Tv '62~, d _--~ 0.23 1T d '75.


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1 6 0 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 A L S O C I E T Y O F A M E R I C A

Inspection of the individual "contr ibutions" to Ev~G~ (or, approximately, to therisks 1 - F) in Table 1 reveals that the closer faults have the predominant influenceon peak acceleration risks, since even relatively small, more frequent magnitudes cangive rise to high accelerations locally.* More distant potential sources contribute sig-nificantly to the risk of longer-period structures, as evidenced by their contributionsto the E,~G~ factor for velocity and displacement. The slow decay with distance ofpeak displacement causes a large contr ibut ion even from long distances. This explainswhy the displacement of 12.5 cm is considerably larger than those associated withspecific earthquakes records with peak accelerations of the order of 0.05g.

ASSUMPTIONS AND EXTENSIONSWhile the assumptions made in the method are considered reasonable for mostpurposes 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 focaldistance 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 theunlimited 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 differentregions 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 besignificant 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; Ipeke t a l , 1965) an annular source canbebroken into two regions, one ford -< r -< r' and theother for r' -- r -- r0. In the first region b3 should be set equal to zero, and the values ofbl and b2 appropriate for near-source conditions adopted. Coupled with a l imited magni-tude distribution, this process facilitates incorporation of any suspected upper boundson maximum ground motions (Housner, 1965).These functions are, in any case, no better than the parameter estimates used inthem. 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 relativeease. Specifically, these include the two assumptions (a) that the radiation of effectscan be treated as if the earthquake generating mechanism were concentrated at apoint and (b) that isoseismals are circular. These assumptions are commonly made indesign studies. This is done not so much because it is thought to be t rue, but becausealternative 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 mathematicalcomplexity 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 inthe method for construction of response spectra proposed by Newmark.


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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 1603The 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 ofearthquakes, and (c) Poisson (or "memory-less") behavior of occurrences.

If data or judgement rule against the equM-likelihood assumption and in favor ofother relative values they can be included by simply treating each portion of thesource over which the equally likely assumption is reasonable as an individual sourceusing the superposition method described above.

If the engineer and the seismologist are prepared to make an assumption about thetime dependence of the average occurrence rate, other than that of constant in time, aminor 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 0 1L[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 thePoisson process model can be removed only at a grea~er penalty , however. The Poissonassumption does not reflect earthquake swarms or aftershocks, nor is it physicallyconsistent with the elastic rebound theory, which implies that a zone of recent pastactivity is less likely to be the source of the next earthquake than a previously activezone 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). Whenswarms and aftershocks are excluded, data does not clearly reject the Poisson assump-tion (Lomnitz, 1966; Wanner, 1937; Knopoff, 1964; Niazi, 1964)for the rarer, majorevents of engineering interest. Even when more accurate theoretical models becomeavailable, it is not evident tha t sufficient statistical data and other information will beavailable in many regions to permit the seismologist to adopt a non-Poisson assumptionor 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 ofmotion also influences the pe~k-response values (Rosenblueth, 1964; Crandall andMark, 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 suchresponse spectra. In addition, inclusion of the randomness of peak response to randommotions with given parameters (Ilosenblueth, 1964; Crandall and Mark, 1963) willpermit 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 thatthis latter influence is negligible (Rosenblueth, 1964; Borges, 1956).

Although developed specifically for the seismic risk analysis of individual sites, themethod 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 greatestadvantage of this method for this purpose is that it would insure that consistentassumptions were being used for all portions of the region and among different regions.


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1604 BULLE TIN OF THE SEISMOLOGICAL SOCIETY OF AMERICAA 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 yr 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 .

CONCLUSIONA 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 ig n i s 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 hr 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 l a ti o n s h ip s .

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 sc 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 df 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 ca 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 rr e v i e w f o r c o ns i s te n c y a n d s e n s i ti 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 eu 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 sa 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 .

ACKNOWLEDGMENTSTh 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 eM 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 hn o l 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 re d , i n p a r t , t h e a u t h o r ' sv 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 Ro s e n b l u e t h i n i ti a t e 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 wishest 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 .

REFERENCESAki , K . (1956) . Some p rob lems in s ta t i s t i ca l se i smology , 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 eense 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 ac t ive fau l t s . Proc. Third World Conf. on Eq. Engr. (New Zea land) , IV-53.Blume, J . A . , N . M. New ma rk and L. H. Corn ing (1961). Design of Multistory Reinforced ConcreteBuildings 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 i m ate of se ismic loadings, Preliminary Publ. V. CongressIABSE, Lisbon .Borg ma n , L . E . (1963) . Risk c r i t e r ia , Proc. ASCE, WW 3, 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 , C a l i f.Cranda l l , S . H. and W. D. Mark (1963) . Random Vibration in Mechanical Systems, AcademicP r e s s , N e w Y o r k .


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Dick, I . D. (1965). Extreme value theory and ear thquakes, 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 stu dyof 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). Criteria for the construc tion of spectra for seismic design, Third PanamericanSymposium 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 ear thquake forecasting, Bull. Seism. Soc. Am. 52,485-505.Housner, G. W. (1952). Spectrum intensitie s 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). Ear thquake 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 secondrecorded in a deep mine in nor thern New Jersey, Bull. Seism. Soc. Am. 54, 1941-1979.Kanai, K. (1961). An empirical formula for the spect rum of strong earthquake motions, Bull. Eq.Res. Inst. 39, 85-95.Kawasumi, H. (1951). Measures of earthquake danger and expec tancy of maximum inte nsi tythroughout Ja pan as inferred from the seismic activity in historical times, Bull EarthquakeRes. Inst. 29.Knopoff, L. (1964). The statistics of earthquakes in southern California, Bull. Seism. Soc. Am.54, 1871-1873.Lacer, D. A. (1965). Simulation 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 prediction of earthquakes, Reviews of Geophys. 4, 337-393.Milne, W. G. and A. G. Davenport (1965). Statistical 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 nuclearpower stations to withstand earthquakes, Proc. Instn. Mech. Engr. 117, 155-203.Newmark, N. M. (1965). Curr ent trends in the seismic analysis and design of high rise structures,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 northern Cal ifornia and western Nevada, Bull. Seism. Soc. Am.54, 845-850.Nordquist, 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 regionalization, 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 s imula tion 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 structu res, Trans. ASCE, 126, 1124-1198.Thorn, H. C S. (1967). New dist ribu tions 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 WorldConf. on Eq. Engr. New Zealand, II , 663-682.


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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 I I, Ger. Geitr. Geophys. 50, 85-99 and 223-228.Wiggins, J. H. (1964). Effect of site conditions on earthquake inten sity, Proc. ASCE, 90, 279-312.DEPARTMENT OF CIVIL ENGINEERINGM.I.T.CAMBRIDGE~ MASSACHUSETTS

Manuscript received January 2, 1968.

engineering seismic risk analysis by cornell

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