characteristics of strong-motion earthquakes

8/3/2019 Characteristics of Strong-motion Earthquakes

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C H A R A C T E R I S T I C S O F S T R O N G - M O T I O N E A R T H Q U A K E S *B y GE ORGE W . HOUS NE R

IN ENGINEERING SEISMOLOGY th e respo nse of str uc tur es to stro ng- mo tionear th quak es is of par t icular in teres t . The a mo unt of d is tor t ion that a s t ructureundergoes dur ing an ear thq uak e and th e magnitude s of the s t resses tha t a redeveloped have an imp or ta n t bear ing on the safe ty o f s t ruc tu res and the i rabi l i ty to survive an ear thquake without suffer ing excessive damage. A s tudy

~F_XNON,CNLI~ MN~,.IO, 1933

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Fig. 1.of this pr ob le m leads to certain rather formidabl e difficulties. ur in g an earth-qu ak e a structure is subjected to vibratory excitation by a grou nd mo ti onwhich is to a h igh degree er rat ic and unpredictable. The inabi l i ty to knowbeforehan d wh at to expect in the w ay of ground m otion is a ser ious handicap.Fur thermore , the average s t ruc tu re , toge ther wi th the g round upon which i ts tands, is an exceedingly complex system f rom the viewpoint of v ibrat iontheory. I t is apparent that the problem divides i tself in to two par ts ; f i r s t , adeterm inat ion of the character is t ics of s t rong-motion ear thqua kes , and second,a determinat ion of the c haracter is t ics of s t ructures w hen subjec ted to ear th-quakes .

A s trong-motion ear thquake record is shown in f igure 1 , f rom which i t isevident that the ground accelerat ion is extremely i r regular . The accelerat ionrecords of ear thquakes dif fer f rom one another ; the chief s imilar i ty betweendif ferent records lies in the ma rked i r regular i ty exhibi ted by all . I t is desirable

* Man uscript received for publicat ion April 20, 1946.[ 1 9 ]


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20 BULLETIN OF THE SEISMOLOGICALSOCIETY OF AMERICAto analyze the records to discover any characteristic that is a common de-scription of all strong-motion earthquakes. The pronounced irregularity ofacceleration records suggests tha t this irregularity may itself be the best com-mon description. Accordingly, the question is raised, What are the character-istics of random earthquakes?--the word " r a n d o m " being here used in thecustomary statistical sense. In an earthquake record, randomness can beintroduced in three ways: by acceleration pulses random in direction, in mag-nitude, or in time. As will be shown later, it is not important, for the questionat hand, just how the randomness occurs. Accordingly, a mathemat ical modelof a random earthquake will be analyzed which will consis t of a series of im-pulses that are random in time.

f "--- ~ cos

Fig. 2.In a st udy of vibrating, undamped structures subjected to a rbitrary ground

motion, certain expressions are obtained for the displacement, velocity, andstress, all of which are made up of one or more terms of the general form:l

A 0 T sin p ( T - - t ) d twhere A is an amplitude factor , is the ground acceleration, p is 27r times thefrequency of vibration, T is the time for which the expression is evaluated,and t is the time variable. The amplitude factor A expresses the influence ofthe physical characteristics of the structure. The rest of the expression intro-duces the effect of the ground motion. An evaluation of the integral (givenabove) over the range of significant frequencies will exhibit pertinent charac-teristics of the ground acceleration. More appropriately, by expanding thetrigonometric term the integral can be written:

j o j o1 See Rayleigh,Theory of Sound, Vol. 1, p. 130.


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22 BULLET IX OF TEE S E I S M O L O G I C A L S O C I E T Y OF A~M:ERICAc a n b e o b t a i n e d b y c 0 m p a r i n g w i t h f i g u r e s 5 - 1 4 . I n c o m p u t i n g t h e c u r v e o ff igure 3 the s cale o f each spec t rum (5 -14) was t ran s fo r me d so a s to have thes a m e a v e r a g e o r d i n a t e .

Al th ough the me an o f R 2 fo r a s e r ie s o f ran do m impulses i s a cons ta n t inde -pen den t o f the pe r iod , the com pu ted va lu es o f R fo r a speci f ic s e t o f impulsesa r e n o t i n d e p e n d e n t o f t h e p e r i od . F l u c t u a t i o n s a b o u t t h e m e a n c a n b e e x-pec ted . T o i llus t ra te these f luc tua t ions , the va lues o f R hav e been com put edf o r a s et o f t w e n t y - f i v e r a n d o m i m p u ls e s. T h e i m p u l se s w e r e d i s t r i b u t e d a tr a n d o m o v e r a n i n t e r v a l o f se v e n s ec o n ds , t h e r a n d o m i z i n g b e i n g o b t a i n e d b yu s e o f a t a b l e o f r a n d o m n u m b e r s . T h e r e s u lt s o f t h e c o m p u t a t i o n s a r e s h o w nin f igure 4 , where i t i s s een th a t the v a lues o f R d o f luc tua te a bo u t th e mean .W h e n t h e v a l u e o f R i s c o m p u t e d f o r a g i v en p e r io d , i t i s u s u a l l y f o u n d t h a t Rp a s s e d t h r o u g h a m a x i m u m v a l u e a t s o m e t i m e p r i o r t o t h e l a s t i m p u l s e . I neng inee r ing s e i smology the ch ie f in te re s t i s in th is ma x im um va lue o f R ra the rtha n in the f ina l va lue . Accord ing ly , a spec t rum of the m ax im um v~Iue o f Ris also sho wn in f igure 4 .

T h e s p e c t r a o f t h e m a x i m u m v a l u e o f R a s c o m p u t e d f r o m t e n d i f f e r e n tea r thquake records a re shown in f igures 5 -14 . Compar ing these wi th f igure 4 ,o n e se es t h a t t h e y e x h i b i t t h e s a m e c h a r a c t e r i s t ic s a s t h e r a n d o m e a r t h q u a k e .The re are the sam e fluctuations abo ut the me an in the case of an actual earth-qu ak e as in the ease of the ra nd om impulses. It is wo rth noting that althoughthe spectra in figures 4-1 4 ha ve be en dr aw n as straight-line segm ents ~ actually,ha d the values of R bee n co mp ut ed for a sufficiently large nu mb er of points,each spectrum would have been a smoo th curve with a very large num ber ofwa v y oscillations. In practice, then, the local characte r of the spe ct ru m curv eis obscured because the co mp ut ed points are separated by s om e distance. Itwill also be noted that so me of the spectra have bee n plotted on the basis of asmaller nu mb er of co mp ut ed points than w as used for the rest.

Th e matl~ematical mod el of a set of ra nd om impulses does not bear an espe-cially close physical re sembl ance to the grou nd acceleration of an earth quake.A closer resem blanc e can be obtained by using pulses of a finite time duration,perh aps half-sine waves, an d th en ran dom izi ng the phasing an d direction.Ho we ve r, in an y case, the sa me significant result is obtaine d; that is, the me anis essentially ind epe nde nt of the period a nd the values of R fluctuate abou tthe me an 2 It does not seem desirable to carry the analogy any further than tosay that the acceleration records of ear thq uak es exhibit the characteristics ofrand omne ss. Th at this should be so is not surprising wh en it is considered thata strong-motion earthqu ake is co mpo sed of a relatively large nu mb er of wav esand that it wo ul d not be expected that a pred eter mine d relationship could befou nd bet wee n the times of arrival of the waves. The se relationships w ou ldlogically dep en d up on the origin of the disturbance an d u po n the character of

3 See Appendix.


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C H A I ~ A C T E I ~ I S T I C S O F S T R O N G - M O T I O N E A I ~ T I c I Q T U A K E S 23

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24 B U L L E T I N O F T I- IE 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

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CHAI=~ACTERISTICS OF STRONG-lYIOTION EART HQUAKE S 27

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28 BUL LETI N OF TI-IE SEISMOLOGICAL SOCIETY OF AMERICA

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CttAI~ACTEI~ISTICS OF STRONG-MOTION EARTYIQUAKES 29the mate rial through which the waves travel. Hence, with respect to these it isonly possible to say th at there is a wide range of possible values, but that thespecific values assumed by a given earthquake depend upon unpredictable con-ditions. The problem therefore is treated properly by statistical methods andthe fact that earthquakes do exhibit characteristics of randomness is notunexpected.

On the basis of random characteristics, the spectra of earthquakes can beexpected to exhibit certain properties. The mean of a sample of earthquakesshould converge toward a straight-line spectrum whose ordinate is a cons tantvalue independent of the period. It is to be understood that this applies onlyover a certain range of frequencies. As is shown in the Appendix, the spectrumdrops off to zero at the high-frequency end. The spectrum for a given earth-quake record should be composed of a series of fluctuations about the mean,and the fluctuations should ha ve relatively sharp pe aks an d valieys. Th e spec-tr um of a specific eart hqua ke ma y sh ow a ten denc y to be high or low in certainregions instead of havi ng rather symm etri cal fluctuations abou t the mea n.This, of course, is consistent with the hypothesis of ran dom nes s. Ther e is,how eve r, a probability distribution associated with these deviations; that is,the freque ncy of occurrence of excessively high or low regions in spectra shouldbe small as com par ed with the frequency of occurrence of moderate ly high orlow regions.

Th e spectra of figures 5-1 4 exhibit certain interesting properties whi ch wer esh ow n to be associated wit h random ness . The re is no indication in thesespectra of a so-called "d om in an t gro und period." In the past, the theory ha sbee n ad va nc ed that a do mi na nt period can be associated with a particularlocality an d that earthqua kes will be especially destructive of structures wh os eperiod of vibration lies in the neig hbo rhoo d of the domi na nt period. On thebasis of the earthq uakes the spectra of whi ch are show n, it appears that thistheory is untenable. An additional point of interest is the fact that the earth-quakes whic h have been analyzed occurred in widely separated locations withext reme diversity of subsoil conditions. Th e spectra, however, do no t reflect toan y m ar ke d degree the variations in gro und conditions. It thus appears thatthe spectra are not dependent to any great degree upon the subsoil conditions,at least within the range covered by figures 5-14.

Since the displacement, velocity, and stress of an undamped structure arelinear functions of the spectrum of the earthquake, the spectra provide aunique method of measuring strong-motion earthquake intensities. For engi-neering seismology it would be more informative to measure the intensity ofan earthquake by the mean value of the spectrum than by an empirical scale.The Mercalli or Modified Merealli scale, even aside from its lack of precision,is unsatisfactory because it attempts to measure the intensity of an earthquakein terms of the damage to buildings. Since buildings differ widely in their


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30 BULLETIN OF T~IE SEISMOLOGICAL SOCIETY OF AMERICAab i l i ty to re s i s t ea r thquakes , i t wou ld be p re fe rab le to measure in tens i t i e si n d e p e n d e n t l y a n d n o t i n t e r m s o f t h e p h y s i c a l c h a r a c t e r is t i c s o f t h e b u i l d -i ng s. S u c h c o m p u t a t i o n s o f e a r t h q u a k e i n t en s i ti e s h a v e b e e n m a d e , a n d a ni n v e s t i g a t i o n o f d i f f er e n t m e t h o d s o f c o m p u t i n g s p e c t r a h a s a ls o b e e n m a d e ?I t w a s f o u n d t h a t t h e s p e c t r a w e r e r a t h e r s e n s i t i v e t o i n a c c u r a c i e s i n t h em e t h o d o f c o m p u t a t i o n . I n o r d e r t o b e s a ti s f a c t o r y m e a s u r e s o f i n t e n s it y , t h es p e c t r a s h o u l d a ll b e c o m p u t e d ir] a u n i f o r m m a n n e r a n d t h e m e t h o d o f c o m p u -t a t i o n s h o u l d h a v e a h i g h d e g r e e o f a c c u r a c y .

T h e s p e c t r a o f t h e e a r t h q u a k e s s h o w n i n fi g ur es 5 - 1 4 w e r e c o m p u t e d a t t h eC a l i f o rn i a I n s t i t u t e o f T e c h n o l o g y i n a p r o g r a m o f e a r t h q u a k e r e s e a rc h u n d e rt h e d i r e c t i o n o f P r o f e s s o r R . R . M a r t e l . T h e r e s e a rc h p r o g r a m w a s s p o n s o r edb y t h e L o s A n g e le s C o u n t y D e p a r t m e n t o f B u i l d i n g a n d S a f e t y .

A P P E N D I XTh e m ean o f R 2 depends in some degree upon the typ e o f pu l se . F or th e ease o fran do m impulses , th e m ean o f R 2, a s was shown, i s a con s tan t ove r the com-p le te range o f f requenc ie s . In t he case o f ran do m pu lses o f ha l f - s ine wave fo rm ,the me an o f R 2 i s no t cons tan t . I t i s app a re n t tha t i f the l en g th o f the pu l se iss m a ll a s c o m p a r e d w i t h t h e p e r i o d o f v i b r a t i o n , t h e e f fe c t b e c o m e s t h a t o f a nimpulse . On the o th e r hand , i f the pe r iod o f v ib ra t ion i s sma l l a s com pare dwi th the l eng th o f the pu l se , the me an Of R 2 approa ches ze ro a s the p e r ioda p p r o a c h e s z er o . F o r i n t e r m e d i a t e c o n d i t io n s i t is n e c e ss a r y t o d e t e r m i n e t h em e a n o f R ~f rom th e fo l lowing expres s ion :

. . . . ~. - cos k t sin p t d tT ~ J o J o ~ - i = a ~ - ~ 2~r($f )], + ~ k_ cos k t co s pt dt dO~d02 dO .+ Jo,- ~ 2

w h e r erk- - c os k t = typ ica l pu l se2

rk cos k t d t = r2

0i = mid po in t o f the i - th pu l seI n p a r t i c u l a r , w h e n n p u l s es o f t h i s t y p e o c c u r a t r a n d o m d u r i n g t h e t i m eT ( T = mul t ip le o f 2~p) and the l eng th o f the pu l se is equa l to one -ha l f the

4 " A n Investigation of the Effects of Earthquakes on Buildings," unpublished doctoraldissertation, California Ins titu te o f Technology, 1941.


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CH~-~AOT~amTICS OF S~rRONG-~OTXONEARtHquAKES 31per iod of v ibra tion , the m ean of R 2 is equal to (~r/~)2nr 2. The correspondingvalue of R is ~r/4 r~v/n = 0 . 7 8 r ~ F o r earthquakes inwhich the length of apulse is 1/12 sec., the mean of R should be approximately const ant for periodsof vibration down to about 0.3 sec., appr oxim atel y 20 per cent smaller at aperiod of 1/6 sec., and zero at a period of zero see. This is indicated by thedotted line in figure 3. The spectrum of the mean of R for a sample of earth-quakes would appear to be fairly well represented by an expression of the formC(1 - e-3~P), where P is the period of vibration. We ma y also note that, inthe spectrum of a given earthquake, as the frequency of vibration approacheszero the ordinate of the spectrum approaches the maximum ground velocity.Hence, a t this end of the frequency range the mean of R ~ does not drop off,but continues at a constant value all the way to zero frequency.

characteristics of strong-motion earthquakes

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