NigamJennings_CalResponseSpectra1969.pdf

8/20/2019 NigamJennings_CalResponseSpectra1969.pdf

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B u ll etin o f t h e S e i s m o l o g ic a l S o c i e ty o f A m e r i c a . Vo l . 5 9 N o . 2 p p . 9 0 9 - 9 2 2 . A p r i l 1 9 6 9

C A L C U L AT IO N O F R E S PO N S E S P E C T R A F R O MS T R O N G - 5 ~ O T I O N E A R T H Q U A K E R E C O R D S

BY NA VIN C. NIGA~ AND PA UL C. JENNINGSA B S T R A C T

A numerical method fo r computing response spectra from strong-motion earth qua kerecords is de ve lope d based on the exa ct solution to the governing d iffer en tialequation. The method gives a three to fou r-fold saving in computing t ime comp aredto a third ord er Runge-Kutta method of c om par able accuracy. An analysis alsoi s

ma de of the e rrors introduced at various stages in the calculation of spectra so thata l low ab le er rors can b e prescr ibed for the numerica l in tegra t ion. Using the pro-posed method of computing or other methods of com par able accuracy ex am ple

calculations show that the errors introduced b y the numerical proce dure s are muchl s s than the errors inherent in the dig it iza tion of the acce leration record.

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

Since the i r in t rodu ct ion Benioff, 1934; Blo t , 1941; Housn er, 1941) , response spec t rao f s t rong-mot ion ea r thquakes have p roved use fu l and in fo rma t ive in p rob lems o fdes ign and an a lys i s o f s t ruc tu re s su b jec t ed to s t rong ea r thqua ke mot ions . The spec t r a ,ca l cu la t ed f rom the r eco rded g round acce le ra tion , a r e p lo t s o f t he m ax im um responseto the ea r thqu ake o f a s imp le osc i l la to r ov e r a r ange o f va lues o f i t s na tu ra l pe r iod anddamp ing . T hese c u rves p rov ide a desc r ip tion o f t he f r equ ency cha rac t e r i s ti c s o f t he

g r o u n d m o t i o n a n d g i v e t h e m a x i m u m r e sp o n s e o f s im p l e s t ru c t u r e s t o t h e e a r t h q u a k e .B y sup e rpos i t ion o f d if f e ren t modes o f re sponse , spec t rum t echn iques can be app l i ed~o the des ign and ana lys is o f complex s t ruc tu re s such a s bu ild ings and dams . U sed int h is m a n n e r t h e s p e c t r u m t e c h n iq u e r e p r e s e n ts a n a p p r o a c h i n t e r m e d i a te b e t w e e n ades ign based on s t a t i c l oads and a comple t e i n t eg ra t ion o f t he equa t ions o f mo t ion o fthe complex s t ruc tu re .

S t r o n g -m o t i o n e a r t h q u a k e r e c or d s h a v e b e e n o b t a i n e d i n f r e q u e n tl y i n th e p a s t a n dthe r edu c t ion to d ig i ta l fo rm, o r equ iva l en t ana log fo rm, and subse quen t ca l cu la tion o fspec t r a have been pe r fo rmed on more o r l e s s an ind iv idua l bas i s . However, i n r ecen tyea r s t he number o f s t rong-mot ion in s t rumen t s i n t he se i smic r eg ions o f t he wor ld ,

pa r t i cu l a r ly Ca l i fo rn i a , Mex ico and Japan , has i nc reased to the po in t where ama jo r ea r thq uak e in these a reas w il l gene ra t e a l a rge num ber o f r eco rds . The po ten t i a lvo lum e o f t he d a t a a nd the deve lop me n t o f t he t ape - reco rd ing acce l e rog raph ind ica t ec l ea rly tha t r ap id and au tom a ted da t a p rocess ing and sp ec t rum c~ lcu la tion p roceduresare needed. In an effor t to ful fi ll par t of th is need , th is pape r presen ts a rap id , ac cura tem e thod fo r com puta t ion o f r e sponse spec t r a f rom s t rong-mot ion ea r thqu ake r eco rds .

Resp onse spec t r a were f ir s t ob ta ined b y B lo t 1941), u s ing a d i r ec t mechan ica lana log and la t e r by H ousn e r and Me Ca nn 1949) using e l ect ri c ana log t echn iques . Theavai lab i l i ty of d ig i ta l computers and a progress ive increase in the speed of d ig i ta lcomputa t ion in r ecen t yea r s has l ed to an inc reas ing use o f d ig i t a l compute r s i n t he

ca l cu la tion o f spec t r a . The ex tens ive app l ica t ion o f t he r e sponse spec t rum to p rob lem sin ea r thqu ake eng ineer ing has sus t a ined in t e re s t i n me thods o f ca l cu la t ing spe c t r a andhas r a i sed ques t ions r ega rd ing accu racy, r ep roduc ib i li t y and econo my in such ca l cu la -t ions Hu dso n, 1962; Bra dy , 1966; Berg , 1963; Schi ff and Bogd anoff , 1967).

The d ig i ta l com puta t ion o f spec t r a r equ i r es t he r epe a ted num er i cal so lu t ion o f t heO


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9 1 :BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

response of a simple oscillator to a component of recorded ground acceleration. Themotion of the oscillator is described by a second order, linear, inhomogeneous differen-tial equation, and if a digital description of the earthquake record is available, theresponse can be obtained by numerical integration. Several numerical integration tech-

niques have been used for calculation of spectra, for example, the third order Runge-Kutta scheme of integration has been preferred by some investigators because of itsaccuracy, long-range stability, self-starting feature and because it can be adaptedeasily for use when the excitation is not defined at regular intervals Jennings, 1963;Hildebrand, 1956). The trunca tion error in this method is proportional to A0) 4, whereA0 is the normalized interval of integration. Thus by a suitable choice of A0 the calcu-lations can be performed to an acceptable degree of accuracy.

An alternative approach to calculation of spectra is based on obtaining the exactsolution to the governing differential equation for the successive linear segments of theexcitation, then using this solution to compute the response at discrete time intervals

in a purely arithmetical way Hudson, 1962; Iwan, 1960). This method does not intro-duce numerical approximations in the integration other than those inherent in round-off, so in this sense it is an exact method.

In this p~per this method is uss:l to develop a computation technique which leadsto a three-to four-fold s~ving in computing time compared to a third order Runge-Kutta methoJ of comparable accurgcy. If the egrthqu~ke record is digitized at equaltime intervals, the proposed scheme gives spectral values which are exact in the sensementioned above. If the record is digitized at arbitrary time intervals, however, it isnecessary to introduce an approximation into the digitization. An analysis of theerrors introduced at various stages in the preparation of response spectra and the

results of digitization experiments show that the additional numerical approximationsare not detrimental.

The numerical techniques for calculating response spectra have been codified intocomputer programs in Fortran IV; listings of these programs and details not includedherein are available in a recent report Nigam and Jennings, 1968).

FOR MUL ATI ON OF TH E ] \/[ETHOD

Spectra are defined by the maximum response of a simple oscillator subjected tobase acceleration a t ) as shown in Figure 1. The equation of motion of the oscillator is

2 - t- 2 f l ~ 2 + o ~ x = - - a t ) 1)

in which fl = the fraction of critical damping and ~o = the natural frequency ofvibrations of the oscillator.

Assuming that a t ) may be approximated by a segmentally linear function asshown in Figure 2, equat ion 1) may be writ ten as

A a ~ t - - t ~ ) ; t~ _-< t ~ ti+l 2)

with

A t l = ti+l -- tl

A a ~ = a ~ + l - - a . 3 )


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C A L C U LAT IO N O F R E S P O N S E S P E C T R A

Th e solut ion of equ at ion (2) , for t~ < t < t~+~ s g iven by

X = e - f l w ( t - t i ) [ C ' ls i n 50 ~ / 1 - ~ t - - t l) - ~ C2 c o s 50 ~ , / ~ t - - t ,/ ) ]

a s 2 ,3 A a l 1 A a ~ t - - t ~)w 2 - ~ - w 3 A t i w 2 A t i

911

4)

I

m

k k2 2

i - - l - - - - a t )/ / / ' / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

F I G . 1 . A s i m p l e o s c i l l a t o r.

a t )

O i + l

a io i - I

I Ii I

t i - I t i ] i+l

F I G . 2 . I d e a l i z e d b a s e a c c e l e r a t i o n .

in which C1 and C~ are constants of in tegra t ion . Set t ingx = x ~ a n d = ¢ a t t = t ~an d solving for C1 and C2, i t i s found t ha t

1 ( 2/~2-- 1A al /3 / (5 a)

2/3 Aa~ _f a~ (5 b )C2 = x~ ~3 Atl ~

Su bst i tu t ion of these values of C1 and C2 in to equ at ion (4) wi l l show tha t x and 2at t = t~+l are g iven by

X I + I = A ( ~ , 5 0, A t l ) x l ~ - B ( ~ , 5 0, A ~ i ) d i 6 a )


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9 2 B U L L E T IN O F T H E S E IS 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

i n w h i c h

21 ai+~

A _ _ | - a 2f .l l a12 B = I bll b12] .

a.~,._ [.b2 1 b22 .J

T h e e l e m e n t s o f m a t r i c e s A a n d B a r e g i v e n b y

a l l = . V / ~ s i n co %/1 - - 2A t l J r -COS co % /1 - - ,~ At i

e- a,A t iat , - sin co % /1 - 2 Z~h

co % / 1 - , 2

a ,1 = - - % / ~ e ~ s in co % /1 - ,2 At ,:

/e - ~ , o A t i| /1 , 'At , :22 = \ e ° s co - - % / 1 - - , 2

b:: = e - '~ k f (2 -co,At,: ~ ) s i n . c o %/1%'/~ ---, 2-? t i

__ sin co % /1 - ,2 At,:)

( 2 , 1 ) ] 2 #+ ~ + ~ co s co % / i - ~t,: co'~t~

b 2 = - - e - # ~ V ( 2 - - ~ l . ) s h : c o % / 1 - ' 2 A t * 2 f l ]L \ co At~ ~ - ~ i . ~ + ~ cos co / i . zxt~

1 2

--co~ I- o3Ah

L \ c o2 ~t~ + e o s ~ o - z ~ ti % / 1 - . ~

( 5 )~ q_ (co V /1 _ 2 sin co % ./[ -- ,2 A ti q- ,co co s co %./1 -- , ' At,:)Lco2At,:

~ 2 2 = - ~ - ~ ' ~ r o 2 1 2 . _ _L ~ ~ t :

1 (c o s co % /1 - 2At,:X

si n co % /1 - f12 At,:}% / 1 - 2 /

7_ 2 (co % / 1 - - ,~ s i n co % / 1 - - 2 A h q - , w c o s co % / 1 - - 2 A h ) |

co3Ah /

i

~ 2 A t i

( 6 b )

(6c)

6 d )

6 e )


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C A L C U L AT I O N O F R E S P O N S E S P E C T R A 913

Fro m equation 2), it follows th at the absolute acceleration, he, of the mass at timet~ is given by

2i = 2i zr ai = - - 2 f l o ~ 2 i -~ - o ~ x i ) . (7)

Hence, if the displacement and velocity of the oscillator are known at some timeto, the state of the oscillator at all subsequent times t~ can be computed exactlyby a step-by-step application of equations 6) and 7). The computational ad-vantage of this approach lies in the fact that A and B depend only on 5, ~ and Ate.fl and co are constant during the calculat ion of each spectrum value, and if At~ iscons tant also, x~, 2~ and ~ can be evaluated by the execution of only ten multi-plication operations for each step of integration. Matrices A and B, defined by therather complicated expressions equations 6d) and 6e) need to be evaluated only atthe beginning of each spectrum calculation.

If varying time intervals are used, it is necessary, in general, to compute A and Bat each step of integration. However, by rounding the time coordinates of the record,as discussed below, the number of these matrices needed during the calculationcan be reduced to only a few. These, too can be computed at the beginning of thecalculation and called upon when needed, thereby saving computing time.

C O M P U T AT I O N O F S P E C T R A

To construct the response spectra, it is necessary to find the maximum values ofthe displacement, velocity and acceleration during a given excitation. This is doneby computing the response at discrete time intervals through equations 6) and 7) andmonitoring the response parameters to retain the maximum values. Thus, theresponse spectra are given by

S ~ ~ , 5 ) = M a x~(~ , ) ] 8 a )i = I N

(~, ~) = M a x [ 2 ~ ~ , ) ] (8b)I = I N

&(~, ~) = M a x [~(~, )] s o )~ = I N

in which S~, S,and S~ are the spectral values of displacement, velocity, and accelera-tion respectively, for selected values of damping and natural frequency; and N is thetotal number of discrete points at which the response is obtained.

This process of obtaining the maximum response is approximate because the re-sponse is found only at discrete points, whereas the true maxima may occur betweensuch points. This error, called the error of discretization, is inherent in all numericalprocedures, but can be bounded within acceptable limits by a suitable choice of thetime interval. The discretization error operates to give spectrum values lower thanthe true values and the error will be a maximum if the maximum response occurs

exactly midway between two discrete points as shown in Figure 3. An estimate forthe upper bound of this error can be found by noting that at the time of maximumdisplacement or velocity, the response of the oscillator is nearly sinusoidal at a fre-quency near its natural frequency. Under this assumption the error can be related tothe maximum interval of integration Ar),~ and the period of the oscillator as illus-trated in Figure 3.


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9 4 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

Digitization of E arthquake Records.When ea r thquake r eco rds a re d ig i t i zed a t equa lt ime in t e rva l s , t he acce l e ra t ion va lues a re measu red a t r egu la r t ime s t eps and a reassum ed to be con nected by s t ra igh t l ine segm ents . I f the in terv a l o f d ig i tiza t ion , At,i s su ff i c i en t ly sma l l , t h i s p rocedure approx ima tes t he ac tua l ea r thquake r eco rd qu i t e

c losely. The cho ice o f t he in t e rva l o f d ig i ti za tion depend s up on the p e r iod r ange o fi n te r e s t a n d t h e n a t u r e o f t h e e a r t h q u a k e r e c or d ; c o m m o n l y u s e d v a lu e s r an g e f r o m0.01 to 0 .04 seconds . Analy t ica l methods for assess ing the er ror induced by such asamp l ing process a re ava i lab le Schi ff and Bo gdanoff , 1967, B en da t and Pierso l, 1966) .Digi t iza t ion a t equal t im e in terva ls wi l l be a v i r tu a l n ecess i ty for e ff ic ien t process ingof acce l e rog rams r eco rded on magne t i c t ape , and in o the r i n s t ances where ana log tod ig i t a l conve r s ion i s made au toma t i ca l ly.

For r eco rds d ig i t i zed a t unequa l t ime in t e rva l s t he absc i s sae and o rd ina t e s o f t hepo in t s where changes o f s lope a re j udged to occu r a r e measu red and the po in t s a r eaga in a s sumed to be connec ted by s t r a igh t li ne segmen t s F igu re 2 ) . Th i s p rocedu re

l eads to va r i ab le t ime in t e rva l s be tween success ive po in t s . T he accu racy o f th i s ap -

AT ) m M a x i m u m E r r o rp e r c e n t ) - o

X- -< T / 1 0 - -< 4 . 9

T / Z 0 ~ I ° Z

- -< T / 4 0 0 ° 3

j

3-F IG 3 E r r o r d u e t o d i s e r e t i z a t i o n

proach i s d i f f i cu l t t o de t e rmine ana ly t i ca l ly, bu t shou ld be cons ide rab ly g rea t e r t hanwou ld be ach ieved by the sam e num ber o f s amp le po in t s a t equ a l in t e rva l s . Th i sme thod i s t he mos t conven ien t fo r manua l ly con t ro l l ed d ig i t i za t ion t echn iques .

Spectra Com putation for E qu al Time Intervals.E q u a t i o n s 6 ) s h o w t h a t m a t r ic e sA an d B are fun ct ion s of ~0, ¢~ and At+. If At~ is con stan t , th at is, the ea rth qu ak e recordi s d ig it ized a t equa l t ime in t e rva ls , t hese ma t r i ce s need to be co mp uted on ly once fo reach pa i r of co and ~ and the chosen va lue of At , the in terva l o f in tegra t ion . Also , s ince

the m etho d does no t involve t runc at ion er ror, i t i s poss ib le to use la rger in terva ls o fin t eg ra tion than in o the r me thods . The cho ice o f t he in t e rva l o f in t eg ra tion , A t , i scont ro l led b y the in terva l of d ig i t iza t ion A t =< At) and the er ror of d isere tiza t ion .Figure 3 shows tha t the e r ror due to d iscre t iza t ion i s less than 1 .2 per centif Ar ~ T/20 T = Natu ra l Pe r iod ) . Cons ide r ing o the r e r ro r s i n t he computa t ion o fspec t ra , th is choice of in terva l o f in tegra t ion seems reasonable . In fac t , AT =


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CALCULATION OF RESPONSE SPECTRA 915

T h e d a t a i n Ta b l e 1 s h o w t h a t i t i s n e c e s s a ry t o u s e A r


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916 BULLE TIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

appropriate maximum interval of integration, (At)m, and subdividing each timeinterval of the rounded time record into time intervals equal to or less than (At)~,the calculation proeeeds much as before, except that A(~o, 8, Ar~.) and B(¢o, f~, Ar~ )must be calculated for four or five different values of Ar j. This procedure is illustrated

in Table 2.After the time record is rounded and subdivided, the values of the ground aeeelera-

tion at the additional points created by subdivision are computed by linear interpola-tion from the original record. The modified record now consists of a set of points a longthe time axis, spaced at a limited number of known intervals, and the correspondingvalues of the acceleration. The operations necessary to produee the modified recordcan easily be programmed for computer execution.

The number and size of the time intervals in the modified record depend on the wa y

TABLE 2

ROUND-OFF AND SUBDIVISION OF TIME RECORD[(~)~ = 0.04 secs]

Subdivision into IntervalsStep Original Time Rounded Time of Int egra tion Remark s

Ar)m = 0 04

t~ 10.4267 10.43 0.04 Round-off to . 01ti+l 10. 5213 10.52 0.04At~ 0.0946 0.09 0.01

II

0.09a ) b )

t~ 10. 4267 10. 425 0.04 0.04ti , 10. 5213 10. 520 0.04 0.0 4Atl 0.0946 0.095 0.01 0.015

0. 005 0. 095

0.095 0.095

Round-off to .005

in which rounding of the time coordinates is carried out and the choice of ( k r ) ~ . Fo rthe example presented in Table 2, the possibilities are:

I Arj = 0.04, 0.03, 0.02 and 0.01 (4 time- intervals)IIa Arj = 0.04, 0.03, 0.02, 0.01 and 0.005 (5 time-in tervals)IIb Arj = 0.04, 0.035, 0.03, 0.025, 0.02, 0.015, 0.01, 0.005 (8 time-intervals).

Therefore, if the original record is modified as indicated above, the exact method needsonly a limited number of matrices for each pair of o~ and fl, and these can be computedbefore the integration is started. By indexing each possible Arj and the correspondingmatrices, the appropria te matrices can be cMled at each step of integration.

The procedure outlined above requires that the time coordinates of the record berounded to a predetermined decimal fraction. Assuming that the original digital

description of the earthquake record is exact, this is an approximatioI/which mustlead to random errors in the computation of spectra. However, a digital description ofan earthquake at unequal time intervals is obtained by manual, or manually con-trolled, reading of a graphical record and is subject to certain unavoidable errors. Ifthe round-off is carried out in such a way t ha t effects of the approximation so introduced


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C A L C U L AT I O N OF R E S P O N S E S P E C T RA 917

a r e w e l l w i t h i n t h e e r r o r s i n h e r e n t i n t h e p r o c e ss o f d i g i ti z a ti o n , t h e p r o c e ss o u t l i n e da b o v e w i l l b e a c c e p t a b l e .

Digitization ExperimentB e rg 1 9 6 3 ) a n d B r a d y 1 9 6 6 ) h a v e s t u d i e d t h e e r r o rsi n t r o d u c e d d u r i n g t h e p r o c e s s o f d i g i t iz a t i o n a n d t h e i r e f fe c t o n t h e c o m p u t a t i o n o f

s p e c t r a . I n t h e a b s e n c e o f g r o ss p e r s o n a l e r ro r s o r bi as , t h e e r r o rs a r is e p r i m a r i l y f r o mr e s o l u t io n o f t h e s c a li n g d e v i c e a n d f r o m t h e t h i c k n e s s o f t h e t r a c e w h i c h m a k e s t h ec h o i c e o f p o i n t s a t w h i c h a c h a n g e o f s lo p e o cc u r s s o m e w h a t a r b i t r a r y. T o e x a m i n et h e s e e r r o r s a n e x p e r i m e n t o n d ig i t i z a ti o n w a s c o n d u c t e d b y t h r e e p e r s o n s X , Y, Z )o n t h e B e n s o n - L e h n e r D a t a R e d u c e r 0 99 , a t t h e C a l i fo r n ia I n s t i t u t e o f Te c h n o l o g y.T h e f ir s t 5 s e c o n d s o n t h e T a f t , C a l i f o r n ia N 2 1 E r e c o r d w e r e d i g it i z ed i n d e p e n d e n t l yb y X 5 t i m e s ) , Y 2 t i m e s ) a n d Z 2 t i m e s ) , a t t h e h i g h e s t r e s o l u t io n o f t h e D a t aR e d u c e r f o r t h e r e c o r d u s e d 0 . 00 1 5 s ec s a n d 0 . 00 0 1 g ) . I n e a c h c as e p o i n t s w e r e t a k e nw h e n a c h a n g e i n sl o p e w a s j u d g e d t o o c c u r in t h e r e c o r d . Ta k i n g o n e r e c o r d o f e a c hp e r s o n a s a s t a n d a r d , c o r r e s p o n d i n g t i m e c o o r d i n a t e s o n t h e o t h e r r e c o r d s w e r e s u b -

t r a c t e d f r o m i t t o o b t a i n w h a t is c al le d h e r e t h e s e l f- e r ro r o f d i g i ti z a ti o n . T h e m e a n a n d

TA B L ESE LF ER Ro R OF DIGITIZ~4~TION IN T IM E COORDINA.TES

Error in Time X Y ZCoordinates 1 2 3 4 ave rage

M e a n s ee ) -0 . 00 1 0 1 0 . 0 0 0 1 - 0 .0 0 0 21 0 . 0 0 0 6 1 - 0 .0 0 0 4 1 0 . 0 0 0 1 3 0 . 0 00 4 5S t a nd ar d 0 .0 04 32 0 . 0 0 5 0 3 0 . 0 0 4 1 2 0 . 0 0 4 2 6 0 . 0 0 4 4 3 0 . 0 0 3 9 8 0 .0 0 55 6

Devia-

tion sec)

s t a n d a r d d e v i a t i o n fo r e ac h c a se a re s h o w n i n Ta b l e 3 . T h e s t a n d a r d d e v i a ti o n s s h o w ni n Ta b l e 3 i n d i c a t e t h a t a c a r e f u l r e a d e r w i ll b e c o n s i s t e n t w i t h i n 0 . 0 04 s ec a b o u t 7 0p e r c e n t o f t h e t i m e , w i t h a m e a n e r r o r v e r y c l os e t o z e r o.

To c o m p a r e t h e d i g i ti z e d re c o r d s o b t a i n e d b y d i f fe r e n t p e r so n s , o n e o f t h e r e c o r d s o fX w a s t a k e n a s a n o v e r a l l s t a n d a r d a n d t h e r e c o r d s o f Y a n d Z w e r e s u b t r a c t e d f r o mi t t o o b t a i n w h a t i s c a l le d t h e c r o s s -e r r o r o f d i g it i za t io n . T h e m e a n a n d s t a n d a r dd e v i a t i o n o f th e e r r o r a re s h o w n i n Ta b l e 4 . T h e s t a n d a r d d e v i a ti o n s f ro m t h e m e a n

s h o w n i n Ta b l e s 3 a n d 4 i n d i c a t e t h a t t h e c ro s s e rr o r is a b o u t f o u r t i m e s t h e s e lf e r r o r,t h u s s h o w i n g t h e e f fe c t o f p e r so n a l p r e f e r e n c e s i n r e a d in g . T h e c o n s i st e n t , n e g a t i v em e a n s i n d i c a t e a c o n s t a n t s h i f t i n t h e t i m e a x is o f t h e r e c o r d c h o s e n a s a s t a n d a r d a sc o m p a r e d t o t h e o t h e r r e c o r d s . S u c h a s h i ft d o es n o t e f f e ct t h e d e v i a t i o n a b o u t t h em e a n .

A c o m p a r a b l e m e a s u r e o f t h e e r r o r i n t r o d u c e d b y r o u n d i n g - o f f t h e t i m e r e c o r d c a nb e o b t a i n e d b y s u b t r a c t i n g t h e r o u n d e d r e c o rd f r o m t h e o r ig i na l r e c o rd a n d c o m p u t i n gt h e m e a n a n d s t a n d a r d d e v i a ti o n . T h e s e w e r e d o n e f o r o n e o f t h e r e c o r ds d i g it iz e d b yX , a n d t h e r e s u l t s a r e s h o w n i n T a b l e 5 , f o r r o u n d i n g t o 0 .0 1 a n d 0 . 00 5 s ec .

T h e r e s u lt s o f t h e e x p e r i m e n t d e s c ri b e d i n Ta b l e 5 s h o w t h a t t h e e r r o r in t r o d u c e d b y

r o u n d i n g - o f f t h e t i m e c o o r d i n a t e s o f a n e a r t h q u a k e r e c o r d t o 0 . 01 o r 0 . 0 05 s e c s i s w e l lw i t h i n t h e e r r o r i n h e r e n t i n t h e p r o c e s s o f d i g i t i z a t i o n . F o r r o u n d - o ff t o 0 . 01 s e c s, t h es t a n d a r d d e v i a t i o n o f t h e e r r o r d u e t o r o u n d - o f f i s a b o u t h a l f of t h e s t a n d a r d d e v i a t i o no f t h e s e l f- e r ro r o f d i g i t iz a t i o n a n d a b o u t o n e s i x t h o f t h e s t a n d a r d d e v i a t i o n o f t h ec r o s s -e r r o r. F o r r o u n d - o ff t o 0 .0 0 5 s ec s , t h i s m a rg i n i s f u r t h e r i n c r e a s e d b y a f a c t o r o f 2 .


10/14

9 1 8 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

F r o m t h i s i t i s c o n c l u d e d t h a t a r o u n d - o ff t o 0 .0 1 o r 0 . 00 5 se e s i s a n a c c e p t a b l e a p -p r o x i m a t i o n .

Effect of Rounding U po n AccuracyB e rg ( 1 96 3 ) h a s e x a m i n e d e r r o rs i n s p e c t r ac a u s e d b y r a n d o m e r r o r s i n t i m e a n d a c c e l e r a t i o n c o o r d i n a t e s a n d h a s c o n c l u d e d f r o m

a n a p p r o x i m a t e a n a ly s is a n d f r o m c o m p u t e r e x p e r i m e n t a t i o n t h a t a s c a t te r o f t h eo r d e r o f 2 0 p e r c e n t in u n d a m p e d s p e c t r a is t o b e e x p e c t e d f r o m id e n t i c a l c o m p u t i n gp r o c e d u r e s a p p l i e d t o i n d e p e n d e n t l y p r e p a r e d d i g i t i z a t i o n s o f t h e s a m e a c c e l e r o g r a m .H e a t t r i b u t e d t h e s c a t t e r t o r a n d o m e r ro r s i n r e a d in g t h e a c c e le r o g ra m . U s i n g B e r g sa p p r o x i m a t e f o r m u l a e f o r e x p e c t e d e r r o r i n v e l o c i t y s p e c t r a d u e t o r a n d o m e r r o rs i rt

TA B L ECROSS ~RI~OR OF DIGITIZATI ON IN TIME COORDINATES

r r o r in Tim e Y ZCoordinates 1 2 avera ge 1 2 aver age

M ean (see) -0 .02525 -0.02538 -0.0253 2 -0.02212 -0.02257 -0.02235S ta nd ard D e- 0 . 0 1 8 1 9 0 . 0 1 7 9 4 0 . 0 1 8 0 6 0 . 0 1 5 2 0 0 . 0 1 5 2 0 0. 01 5 20

viation (see)

TA B L E 5ERROR DUE TO I~OUND OFF IN TIME COORDINATES

Err or in Ti me Coord inates Roound off to 0.01 sees Round off to 0.005 sees

M ean (see) 0.00003 0.00004Sta nd ard D ev iatio n (see) 0.00285 0.00149Maxim um Er ro r 4-0.005 4-0.0025

t h e t i m e c o o r d i n a t e s o f t h e r e c o r d , a n d a s s u m i n g a = 0 .0 4 5 g, ( S ~ ) a v e ~ 2 0 s i n / s e eand At = 0 .05 sees , the fo l lowing resu l t s a re fou nd :

R o u n d - o ff t o 0 .0 1 s e e s

(a ~ 0 .003 sees)R o u n d - o ff t o 0 . 0 0 5 s e e s

(~ ~ 0 .0015 sees)

E x p e c t e d E r r o r i n S ,

0 . 1 2 ~ J D

E x p e c t e d P e r c e n t a g eE r r o r in S~ ( D =

30 sees)

3.3

0 . 0 6 v / D 1 . 6

H e r e , a = t h e r.m . s , v a l u e o f t h e g r o u n d a c c e l e r a ti o n , ¢ = s t a n d a r d d e v i a t i o n o ft h e e x p e c t e d e r r o r i n t h e t i m e c o o r d i n a t e s a n d D = t h e d u r a t i o n o f t h e e a r t h q u a k e .T h e r e s u lt s i n d i c a t e t h a t t h e e r r o r s i n s p e c t r a l v a l u e s d u e t o r o u n d - o f f t o 0 .0 1 o r 0 .0 0 5s e e s a r e m u c h l e ss t h a n t h e 1 7 p e r c e n t e r r o r B e rg ( 1 9 6 3 ) f o u n d l ik e l y t o o c c u r a s ar e s u l t o f r a n d o m e r r o rs in r e a d i n g t h e t i m e c o o r d i n a t e s d u r i n g d i g i t iz a t io n .

T h e v a l i d i t y o f t h e a b o v e a n a ly s is c a n b e c h e c k e d b y c o m p a r i n g t h e s p e c t r a f o r

u n r o u n d e d a n d r o u n d e d t i m e r e co r d s. T h i s w a s d o n e f o r t h e N 6 5 ° E c o m p o n e n t o f t h es t a t i o n 2 r e c o r d o f t h e P a r k f i e l d e a r t h q u a k e o f J u n e 2 7, 1 96 6 a n d t h e N 8 ° E c o m p o n e n to f t h e L i m a , P e r u e a r t h q u a k e o f O c t o b e r 1 7, 1 96 6. T h e u n d a m p e d v e l o c i t y s p e c tr a f o rt h e o r i g i n a l ( u n r o u n d e d ) r e c o r d a n d f o r t h e r e c o r d s o b t a i n e d a f t e r ro u n d - o f f t h e t i m ec o o r d i n a t e s t o 0 .0 1 s e es a n d t o 0 . 00 5 s e es a r e s h o w n i n F i g u r e s zi a n d 5 . F o r t h e P a r k -


11/14

C A L C U L AT I O N OF R E S P O N S E S P E C T R A 9 9

f ie ld e a r t h q u a k e , t h e t h r e e c u r v e s a r e a l m o s t i n d is t in g u i s h a b le . F o r t h e L i m a , P e r ue a r t h q u a k e t h e a v e r a g e p e r c e n t a g e e r r o r s a r e o f t h e o r d e r e x p e c t e d o n t h e b a s is o fB e r t s a p p r o x i m a t e a n a l ys is .

4O

IM

20 /

lX

r

/ 3 = 0 . 0

0 2 0

[

LIMA-PERUE A RT H Q U A K EO C T I ~ 1 9 6 6

N 8 ° E- - U N R O U N D E D- ~ - R O U N D E D T O . 01 S E C .- ~ - R O U N D E D T O .0 0 5 S E C.

J

~ V

P E R I O D I N S E C S

FIG. 4. E ffect of rounding the t ime co ordinates on velocity spectra.

3 0

80

w

6OZ

z

>

I .

S 4ow>

<~ 2O

~

/0 j

\

B=O.O

PARKFIELD EARTHQUAKESTN. 2J U N E 2 7 1 9 6 6

N 6 5 ° E

i i1.0 2.0

P E R IO D I N S E C S

• I a 5 E f f e c t o f r o u n d i n g t h e t i m e c o o r d i n a t e s o n v e l o c i t y s p e c t r a

5 0

T h e c o m p u t i n g t i m e f o r s p e c t r a b a s e d o n d i g it i ze d a c c e l e ro g r a m s r o u n d e d t o 0 .0 0 5~ e cs w a s f o u n d t o b e o n l y 1 0 t o 1 5 p e r c e n t m o r e t h a n t h e c o m p u t i n g t i m e f o r r e c o r d sr o u n d e d t o 0 .0 1 s ec s. C o n s i d e r i n g t h e e x p e c t e d e r r o r i n t h e s p e c t r a l v a l u e s i n t h e t w o


12/14

92 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

c a s es , i t i s c o n c l u d e d t h a t t o b e c o n s i s t e n t w i t h t h e ch o i c e o f A t ) ~ < T / 2 0 ) , r o u n d -o ff t o 0 .0 05 s e cs w o u l d b e a d o p t e d a s s t a n d a r d . H e r e a g a i n i t m a y b e p o i n t e d o u t t h a tt h e u s e o f AT ) ~ ==_ T / 1 0 a n d r o u n d - o f f t o 0 .0 1 s e cs s h o u l d p r o v e a c c e p t a b l e f o r m o s tp u r p o s e s .

Intervals of Integration.T h e p r o c e d u r e f o r c o m p u t i n g s p e c t r a o u t li n e d a b o v e r e-q u i re s t h e c ho i ce o f a m a x i m u m i n t e r v a l of i n t e g ra t i o n , A r ) ~ . S i n ce t r u n c a t i o ne r r o r is n o t i n v o l v e d , t h e c h o i c e i s g o v e r n e d b y t h e e r r o r i n t r o d u c e d b y d i s c r e t i z a t i o n ,a n d b y c o m p u t i n g t i m e . A s s h o w n i n F i g u r e 3 , i fhr)m < T/20 , t h e e r r o r d u e t o

TA B L E 6UNDAMPED VELOCITY S PECTR A AND REL ATIV E COMPUTING T IME S FOR I~ECORDS DIG ITI ZED AT

U N E Q U A L T IM E I N T ER VA L S

N 65°E Com ponent the Sta t ion 2 Accelerogram of the P arkf ie ld Ea r thqu ake, June 27, 1966,Z~r)~ = 0.05 se es)

S ~ I n I n s / S e cP e r i o d I n S e c s T h i r d O r d e r R u n g e - K u t t a e x a c t

A r ~ T / I O A r


13/14


14/14

9 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

be com puted in the same ope ra t ion a fu r the r s av ings in com put ing t ime cou ld beach ieved by s to r ing the ma t r i ce s u sed in the in t eg ra t ion me thod r a the r t han r ecom-pu t ing th em a t t he beg inn ing o f t he p rocess ing o f each acce l e rog ram.

Becau se o f t he inc reasing in t e re s t i n t he r e sponse o f y ie ld ing s t ruc tu re s t o ea r thquak e

mot ions i t i s app ro pr i a t e t o po in t ou t t ha t t he me thod p re sen ted he re in can be adap tedalso to the ca lcula t ion of spec t ra for b i l inear hys tere t ic an d e las to-p las t ic osc il la tors . Toapp ly th e m e thod i t wou ld be necessa ry to c om pute two se t s o f ma t r i ce s co r respond ingto the two s t iffness coeff ic ien ts of the s t ruc ture .

ACKNOWLEDGMENT

A p p r e c i a t i o n is e x t e n d e d t o t h e N a t i o n a l S c i e nc e F o u n d a t i o n f o r p a r t i a l s u p p o r t o f t h i s r e -s ea rch unde r NSF Gran t 1197X.

R E F E R E N C E S

Al fo rd , J .L , G . W. Housne r, G . W. and R . R . M ar t e l ( 1951 ).Spec t rum A na lys i s o f St rong -Mot ionEar thquakes Ea r thq uak e Eng r. Res . Lab l , Ca l if . I n s t . o f Tech . , Pa saden a . (Rev i sed , Aug .1 9 6 4 )

B e n d a t a n d A . G . P i e r s o l 1 9 6 6 ) .M e a s u r e m e n t a n d A n a l y s i s o f R a n d o m D a t a .J o h n W i l e y a n d

S o n s , N e w Y o r k .

B e n i o f f , H . 1 9 34 ) . C a l c u l a t i o n o f t h e r e s p o n s e o f a n o s c i l l a to r t o a r b i t r a r y g r o u n d m o t i o n ,B u l l .

S e i s m . S o e . o f A m . 3 1 , 3 9 ~ 4 0 3 .

B e r g , G . V . 1 9 63 ) . A s t u d y o f e r r o r s i n r e s p o n s e s p e c t r u m a n a l y s e s ,J o r n a d a s C h i l e n a s d e S e s

mologia e Ingenier ia Ant i s i sm ica1, B1.3, 1-11.Be rg , G . V. , G . W. Hous ne r (1961) . I n t e g ra t ed ve loc i t y and d i sp l a cem en t o f s t rong e a r thq uak e

g round mo t ion ,Bu l l . Se i sm. Soc. o f Am .51,175-189.B lo t , M. A . (1941) . A mecha n ica l ana lyz e r fo r t he p r ed i c t i on o f ea r th quak e s t r e s se s ,Bu l l . Se i sm.

S o c . o f A m .31,151-171.Brady, A. G. (1966) ,S tudies of Response to Ear thquake Ground Mo t ionE a r t h q u a k e E n g r. R e s .

Lab . , Ca l iL In s t . o f Tech . , Pa sad ena .Hi ldebrand , F. B. (1956) .In t roduct ion to Num er ica l Analy s isM c G r a w - H i l l B o o k C o m p a n y I n c . ,

N e w Yo r k .Hou sne r, G . W. (1941). Ca l c u l a t i on o f t he r e sponse o f an o sc i l l a to r t o a rb i t r a r y g round m o t ion ,

Bu l l . Se i sm. Soc. o f Am .32,143-149.I~Iousner, G . W. and G . D . M cCa nn (1949). The a na ly s i s o f s t rong -m ot ion ea r th qua ke r eco rds w i th

e l ec t r i c ana log compu te r,Bu l l . Se i sm . Soc . o f Am :39, 47-56.Hudso n , D . E . ( 1962 ). Some p rob l ems in t he app l i ca t i on o f spec t rum t ec hn iques t o s t ron g -m ot ion

e a r t h q u a k e a n a l y s i s ,Bu l l . Se i sm. Soc. o f Am .52,417-430.Iwan, W. D. (1960) .Dig i ta l Calcula t ion of Response Spec t ra and Four ie r Spec t raU n p u b l i s h e d

Note , Ca l i f . I n s t . o f Tech . , Pa sa dena .Jennings , P. C . (1963) .Veloc i ty Spec t ra of the Mexican Ear thquakes of 11 May and 19 May 1962E a r t h q u a k e E n g r. R e s . L a b . C a l if . I n s t . o f Te c h . , P a s a d e n a .

N igam, N . C . and P. C . J enn ings (1968) .Dig i ta l C alcula t ion of Response Spec t ra f rom St rong-Motion Earthquake RecordsE a r t h q u a k e E n g r. R e s . L a b . , C a l if . I n s t . o f Te c h . , P a s a d e n a .

Sch if f, A . and J . L . Bogdanoff (1967 ). Ana lys i s o f cu r r en t me tho ds o f i n t e rp re t i n g s t rong mo t ionacce log rams ,Bu l l . Se i sm. Soc . o f Am .57,857-874.

DIVISION OF ENGINEERING AND APPLIED SCIENCE

CALIFORNI A INSTITUTE OF TECHNOLOGY

PASADENA CALIFORNIA 91109

M anus c r ip t r ece ived Sep t em ber 12, 1968.

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