bulletin crouse

7/26/2019 Bulletin Crouse

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Bul le tin of the Se ismo logica l Soc ie ty of Amer icaVol. 65 No. 1 pp. 13-36. February 1975

S O IL -S T R U CT U R E I N T ER A C T I O N D U R I N G T H E S A N F E R N A N D OE A R T H Q U A K E

BY CHARLES B. CROUSE AND PAUL C. JENNINGS

ABSTRACT

A c c e l e r o g r a m s o b t a i n e d a t t w o s i t e s d u r i n g t h e S a n F e r n a n d o e a r t h q u a k e o f1 9 7 1 w e r e a n a l y z e d t o i n v e s t i g a t e t h e r o le o f s o i l- s t r u c tu r e in t e r a c t i o n u s i n gt e c h n i q u e s d e v e l o p e d b y B i e l a k a n d o t h e r s . A n a l y s i s o f t h e d a t a f r o m t h e s i t e o ft h e H o l l y w o o d S t o r a g e B u i ld i n g f o r w h i ch d a t a f r o m t h e A r v in - Te h a c h a p i e a rt h -q u a k e o f 1 9 5 2 a r e a l s o a v a i l a b l e s h o w e d e v i d e n c e o f s o il - s tr u c t u r e in t e r a c t i o n i nt h e w a y t h e t r a n s f e r f u n c t i o n s b e t w e e n p a r k i n g l o t a n d b a s e m e n t m o t i o n d e c a y e d

w i t h i n c r e a s i n g f r e q u e n c y i n t h e t w o l a t e r a l d i r e c t i o n s . I t i s c o n c l u d e d a l s o t h a ti n t e r a c t io n p r o b a b l y h a d a s m a l l e f fe c t o n t h e r e s p o n s e n e a r t h e E W f u n d a m e n t a lf r e q u e n c y d u r i n g t h e S a n F e r n a n d o e a r t h q u a k e . A l t h o u g h t h e o r e t i c a l a n d e x p e r i-m e n t a l l y d e t e rm i n e d t r a n s fe r f u n c t i o n s a r e b r o a d l y s im i l a r t h e y d o n o t a g r e e i nd e t a i l. T h e l a c k o f g o o d a g r e e m e n t f o r r e a s o n a b l e c h o i c e s o f th e p a r a m e t e r s o f t h et h e o r e t i c a l m o d e l i n d i c a t e s a n e e d f o r s o m e m o d i f i c a t i o n s o f t h e t h e o r y o r i t sa p p l i c a t io n a n d a n e e d f o r m o r e m e a s u r e m e n t s a t t h e s it e .

A s i m i la r a n a l y s i s s h o w e d n o c l ea r e v i d e n c e o f s o i l- s t ru c t u r e i n t e ra c t i o n f o r t h eM i l l i k a n L i b r a r y a n d A t h a n a e u m b u i l d i n g s o n t h e c a m p u s o f t h e C a l i f o r n i aI n s t i t u t e o f T e c h n o l o g y. I f s o i l - st r u c t u r e in t e r a c t io n c a u s e d t h e m a j o r d i f fe r e n ce s

m e a s u r e d i n t h e b a s e m o t i o n s o f t h e s e t w o b u il d in g s i t i s o f a m o r e c o m p l e x f o r mt h a n t h a t c o n s i d e r e d b y p r e s e n t t h e o r i e s .

INTRODUCTION

Theor ies of so i l - s t ruc ture in terac t ion have been developed over the years to predic thow the f lex ib il ity of the so i l beneath a s t ruc ture can mo dify it s dynam ic response . Pr iorto the San Fernando ear thquake , however, i t was d i ff icu l t to eva lua te the theore t ica lresult s because of the lack of exper imenta l da ta : only a t the Ho l lyw oo d Storage Bui ld ingHo usner, 1957) had bo th f ree- fie ld and s t ruc tura l response been m easured . In the present

study , theoret ical techniqu es dev elope d by Bielak and others Bielak, 1971 ; Jennings a ndBie lak , 1973; Parmelee , 1969) a re appl ied to the H ol lyw ood Storage B ui ld ing in LosAngeles and the c lose ly spaced Mil l ikan Library and Athenaeum bui ld ings on thecam pus of the Cal i forn ia Ins t itu te o f Techn ology in Pasadena . The resul ts a re compa redto measu remen t s made du r ing the San Fe rnando ea r thquake . A compar i son wi th theda ta f rom the Arv in -Tehachap i ea r thquake fo r t he Ho l lywood S to rage s it e Housne r,1957; Dukee t a l . 1970) is also included.

The m odel , shown in Figure 1a, consists of a l inear, viscously dam ped , n-story struc turesup po rted on a r igid circular fou nd ation of radius a w hich is bo nd ed to a l inearly elastic,hom ogen eous , i so t ropic ha l f-space . In i ts genera l form the sys tem has n+ 2 degrees of

f reedom , t rans la tion of each s tory mass and the base , and ro ta t ion of the en t i re sys tem.Beginning wi th the sys tem a t res t , the equat ions of mot ion are usual ly der ived for anexci ta tion cons is t ing o f ver t ica l ly inc ident p lanar shear waves , wi th the assum pt ion of nosca t te r ing o f waves f rom the r ig id foundat ion and nearby sur faces. The equat ions are a l soappl icable , how ever, for incom ing wav es a t a rb i t ra ry angles o f inc idence provided tha t thewavelengths a re long wi th respec t to the length o f the foun dat ion .

13


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14 CHARLES B CROUSE AND PAUL C JENNING S

The equa t ions u sed i n t he ana ly s is we re deve loped unde r t he a s su mp t ion t ha t t hesupers t ruc ture possesses c lass ica l nor m al m odes . F or th i s case, a m ode l equiva len t to tha tshow n in F igu re l a c an be cons t ruc t ed (F igu re l b ) . I t consi s ts o f n o sc i ll a to r s a t t a ched t othe r ig id c i rcu la r base , each os c i l l a tor def ined by a na tu ra l f requen cy, ~oj , c r i t ica l dam pin gra t i o , r/j, mod a l mass , M j , m oda l m om en t o f ine r ti a ,ij , a n d m o d a l h e i g h t , / / j .

4

~ ~ ' o ) b )

FIG. 1. M ode ls of Soil-Structure Interaction. (a) M ulti-degree-of-freedom structure, (b) equivalentmo dal representation.

Fo r sma l l d isp l acemen t s t he equa t i ons o f mo t ion fo r t he mode l show n in F igu re l a a r e

M~t+Cb+Kv= 0 ( la )

i mji~+mo Vo+V g)+P t) = 0 ( lb )j = l

i mjhjg/+ltq~+ Q t)= 0. ( lc)j = l

In t he se equa t i ons :v / = to t a l ho r i zon t a l d i sp lacem en t o f t h e f n mass w i th r e spec t t o af ixed vert ical axis , i .e . ,v / = vg+vo+hjtp+vj;I t = s u m o f t h e c e n t ro i d a l m o m e n t s o finer t ia o f the n + 1 masses , andP t) a n d Q t)a re t he ho r i zon t a l r e s to r ing fo r ce andmomen t , r e spec t i ve ly, a t t he i n t e r f ace be tween t he ba se mass and t he ha l f - space . Acomp le t e de sc r ip t i on o f t he sym bo l s is g iven i n t he Append ix .

A r e l a t i onsh ip be tween t he gene ra l i z ed i n t e r ac t i on fo r ce s and gene ra l i z ed base d i s -p l acemen t s (Arno ldet al.,1955; Bycroft , 1956; Koboriet al.,1966) can be w r i t ten as

~a Khh Khm

- ~ J LK h Kmm

a

J

2)

In e qua t i on (2), P ( s ) and Q ( s ) a r e La p l ace t r ans fo rm s o fP t) a n d Q t),respect ively,whi le Khh, Knm = Kmh,a n d Kmr~a re t he d imens ion l e s s , complex impedance func t i ons


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SOIL-STRUCTURE INTERACTION DU RIN G THE SAN FERNANDO EARTHQUAKE 15

which are funct ions of f requen cy and Poisson s ra t io . The effec ts ofKh,nand Kmhare sm allin comp ar ison w i th those o f the d iagonal te rms,Khnand K,, ,, (Lu co and W estm ann , 1971,Veletsos and W ei, 1971 ; Bielak, 1971) an d for th e analysis the y will be set equa l to zero.The funct ionsKhh a n d K m mwere found f rom solu t ions to a re laxed mixed boundary-va lue problem, where in i t i s assum ed tha t the contac t sur face be tween the r ig id base andthe half-space is fr ictionless for vert ical and rockin g m otion s an d is free of no rm al s tressesfor hor izonta l mo t ion . Except for the l imi t ing s ta t ic case, the dyna mic im pedance func-t ions have not ye t been evalua ted for the case of a per fec t ly bonded r ig id p la te on anelas t ic ha l f -space ; however, some exper imenta l ev idence (Bycrof t , 1956; Lysmer andRichar t , 1966; No vak , 1973) indica tes tha t the d i fferences in these two cases a re smal lenoug h no t to a ffec t the p roblem apprec iably for low f requencies . Also , theore tica l resu lt sfor an infini te s tr ip (L uco an d W estma nn, 1972; Oien, 1971) suggest that the effects o f there laxed boundary condi t ions upon the compl iances a re not s igni f icant for the presentp rob lem.

Fo r s teady-s ta te harmon ic exc i ta t ion of the r ig id base ,Khhand K m mcan be expressedas (Bycroft , 1956; Gladwell , 1968)

K h h i ao ) = k h h a o , a ) + i ao c h h a o , a )

K r n m i a o ) = k m m a o , t r) +i a oC m m a o , o r ) , 3 )

w h e r e i = ~ / - 1 a n dao = coal Vs cois the freq uen cy of exci tat ion an d Vs is the ve loci tyof the vert ical ly incident shear waves) ,khh , Chh, k, , ,m,a n d Cramare the real funct ions(Jennings and Bielak, 1973),

8k h h a o, a ) - M a o , ~

2 - - t r

Chh ao, a) = ~h ao, a)k hh ao, or)

8kmr,, ao, a) - f lm ao, or)3 1 - a )

Cm ,, ao, a) = ~m ao, ff )kmm ao,o-). (4)

The func tions f ib, t im, ~h, ~,, , sh ow n in Figure 2, were e valuated from the num ericalresult s presented by L uco an d W estman n (1971) , for va lues of the f requency parameter,

ao , up to 6 and for a Po isson s rat io , a , o f 1 /4 . The funct ions are no t a ffec ted apprec iablyby chang es in a in the range 0 _< a < 1/3.

Since the superstructure is assumed to possess classical normal modes, equations (1)can b e solved for the base displacem ent in terms o f the free-f ield accelerat ion an d transferfunct ions involv ing the m oda l quant i t ies , M j , Hi , l j , the na tura l f requencies , ogj , andthe m odal dam ping ra t ios , r /j , o f the supers t ruc ture . T he result , expressed in te rms ofLap lace transform ation, is (Bielak, 1971)

?vo s) e , t ~ ) A - ~ + , : , t ~ ) 5 )

whereYo = Vo + v~is the to ta l la te ra l d isp lacement o f the base m ass . Taking the b ase massto b e init ially at rest , equ ation (5) becom es (replacing the t ran sform variable s by ico)

~o ~O) -~O~Ao+ZX- 6 ),~ ~o) A


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16 CHARLES B CROUSE AND PAUL C JENNI NGS

Express ions for Ao and A a r e g iven i n t he Append ix . Equ a t ion (6) was u sed fo r c ompar i -sons w i th t he ea r thquak e da t a .

2.~

2.C

1.5

1.Q

0 . 5

= 0 . 2 5

1 2 3 4 5 6:]o

FIG. 2. Value s of1/fland ~ for Poisson s ratio, tr, of 1/4.

RESULTS

Hol lywood Storage Bui ld ingThe Ho l lyw ood S to rage Bu i ld ing i s loca t ed nea r t hec o r n e r o f S a n t a M o n i c a B o u l e v ar d a n d H i g h l a n d Av e n u e i n L o s A n g el e s, a n d w a s a b o u t18 mi le s sou th o f t he cen t e r o f ene rgy r e lea se i n t he San F e rna ndo ea r thqua ke ( J ennings ,1971) . Th e b ui ld ing i s a 14-s tory, re inforced conc re te f ram e s t ruc ture , 150 f t t a l l, w i th abasem en t 9 f t benea th t he g ro und s to ry. The foun da t io n cons is t s o f conc re te p il es 10 t o30 f t long , spaced every 17 f t in each d i rec t ion . L a te ra l d imen s ions o f the bu i ld ing a re51 f t (NS) by 217 f t (EW ). A so i l bor ing to a dep th of 100 f t revea led a sof t , san dy c laymix ture wi th a w eight dens i ty vary in g f ro m 100 lb / f t 3 a t the sur face to ro ughly 130 lb / f t 3(Duk e and Leeds , 1962 ) . F igu re 3 show s the bu i ld ing and a sm a l l shed 112 f t d i r ec t l ywes t o f the sou thwe s t co rne r o f t he ma in bu i ld ing , wh ich con t a ined a S t anda rd s t rong -m ot ion acce l e rog raph . The acce l e rog raph i n t he ba semen t o f the m a in bu i ld ing was a l soa S t anda rd .

T h e m a j o r p o r t i o n s o f th e h o r i z o n t a l c o m p o n e n t s o f t he a c c e l e r og r a m s o b t a i n e d d u r i n gt h e S a n F e r n a n d o e a r t h q u a k e f r o m t h e b a s e m e n t a n d s h e d a r e s h o w n i n F i g u re 4. I n t h epar k ing lo t , the la rger peaks of the acc e le rogram s are gen era l ly 0 .05 g to 0 .1 g grea te rt ha n co r r e spond ing peaks i n t he ba semen t reco rds . I n t he ana ly s i s o f t he Ho l lyw oodS to rage Bu i ld ing , the acce l e rog rams f rom the shed w ere a s sumed r ep re sen t a t ive o f thef ree- fie ld mot ion .

The F our ie r Am pl i tud e Spec t ra (FA S) of se lec ted 20.48-sec por t ion s o f the acee le ro-g r a m s w e r e c a l c u la t e d b y a F a s t F o u r i e r Tr a n s f o r m t e c h n iq u e . C o m p a r i s o n s o f t h e FA S(F igu re 5 ) show c lo se ag reem en t be tween t he ba sem en t an d f r ee - fi e ld mo t ions fo rf r equenc i e s up t o 4 Hz i n bo th l a t e r a l componen t s . Fo r f r equenc i e s g r ea t e r t han 4 Hz ,the f r ee - fi e ld FAS a re s i gni f ic an t ly l a rge r t han t he co r r e spond ing basem en t FA S . T heFAS in F igu re 5 have been smoo thed once w i th a Hann ing spec t r a l w indow we igh t ed


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SOIL STRUCTURE INTERACTION DU RIN G THE SAN FERNANDO EARTHQUAKE 17

vi~ Lookimg Southeast Axrow ~ao~s Locat iom of ~aed in Parkimg

L o t C o n t a i m i n g a ~ A c c e l e ~ g ~ p h

S a n t a M o n i c a _ B l vd

, , 23

i

in b ldg

Fig. 3. Holl ywo od Sto rage Site in Los Angeles.

1/4, 1/2 and 1/4. Th e freque ncy spacing of the calcu lated ordinates is 0.05 H z. A tran sferfunct ion between the basement and free-f ie ld motions in corresponding direct ions wasachieved by dividing the once-sm oothed basem ent FA S by the once -smoothed free-fie ldFA S and smo othing this rat io 10 t imes. The addi t ional sm oothing of the t ransfer funct ionwas do ne to emphasize the overall t rend a nd to el iminate large fluctuat ions. D ata reduc-t ion for the Arvin-Teha chapi accelerograms w as identical to the process just descr ibedfor the San Fernando da ta .

The calculat ion of the necessary parameters for the theoret ical model was based onda ta col lected by Du ke Duk e and Leeds, 1962; Dukee t a l . 1970). A P -wave velocity of2400 f t /sec was mea sured at the s ite a t depths fr om 9 to 60 f t . Experiments mea suringboth the shear-wave and P-wave veloci t ies a t other s i tes indicate that the shear-waveveloci ty is on the average ha lf the P-wave veloci ty Duke and Leeds, 1962). This cor-respon ds to the theoretical result for a = 1/3. Also, there is experim ental evidence tha tthe shear mod ulus of soil is a n onl inear funct ion o f the shearing s t rain. Typical shearingstrains produced by ear thquakes can reduce the shear modulus to approximately one-ha lf the value determ ined by tests at low strain levels Seed and Idriss, 1970). This indi-cates that the shear veloci ty would be reduced by a fa ctor of 1/~/2. Thus, an est imate forthe shear-wave veloci ty during s t rong motion at the s i te would be a bout 800 ft /sec. Theaverage w eight density, p, fro m the soil borings is 115 lb/ft 3. The equiva lent base radius,a for a circle whose area is equal to the cross-sectional area of the main building is59.4 ft . The story masses and equivalent interstory stiffnesses of the building were takenf rom Duke et al . 1970). The natural f requencies and m ode shapes of the superstructurewere fo un d by solving the eigenva lue problem for a 14-degree-of-freedom, linear, s_pring-mass system. The first four calculated frequencies in the NS direction were 1.05, 2.74,4.30 and 5.84 Hz. Vibrat ion tests before the San Fe rna ndo e ar thqua ke Carder, 1964)gave NS re son ant frequencies at 0.83, 2.7 and 4.5 H z. The E W natu ral frequencies, calcu-lated from the eigenvalue problem, were within 5 per cen t of 2, 6 and 10 Hz, the natura l


6/24

] 8 C H A R L E S B C R O U S E A N D PA U L C J E N N 1N G S

5 Z 0 0 - 5 ~ -( 0 1 / O ) ' 7 ~ 3 3 U 0 7 ~ I d 3 3 Y J

~

5 ~ 0 0 5 ~-( 0 W O ) 7 3 3 J U 0 7 3 ] 3 3 3 U ~

S N

5 ~ 0 0 5 ~-( OI / J ) 7 3 2 3 U 1 N 3 H 3 S U 8

S

z

4

5 ~ 0 ' 0 $'~-( O [ I O ) 7 3 3 3 U 1 N 3 N 3 S U ~

M ~

o. . ~

; ~

Q

O


7/24

S O I L -S T R U C T U R E I N T E R A C T I O N D U R I N G T H E S A N F E R N A N D O E A R T H Q U A K E 9

c~

o.

0.

C

0 _ o

g~D

O

I0 2.

L I

I I I I

1 . 2 . 3 . q .

HOLLYWOOD STORROE EW D IRECTION

FREE FIELD

BRSEMENT

"L/ ,,,~J ~ j ~ i - t ,

1 I I I I I I3 . I I . 5 . 6 . 7 . 8 . 9 ,

F f l E O - C P S10 11 12. 13. lq. 15

HOLLYWOOD 5T@RAOE NS DIRECTION

I I

r

/ j. U \ ~ h . . Ili5 . 6 , 7 , 8.

F R E Q - C P

FREE FIELDASEMENT

// ~ /',..~

I I I Iv \ - I / r ' \ d ~ \ ' - - ~9. 10. 11. 12. 13 lq. 15

FIG 5 Four ier Amplitude Spectra of Accelerograms shown in Figure 4


8/24

20 C H A R L E S B . C R O U S E A N D P A U L C . J E N N 1 N G S

f r eq u e n c ie s o f a n a p p r o p r i a t e c a n ti le v e r ed s h e ar b e a m . T h e E W f u n d a m e n t a l f r e q u e n c yf rom v ibra t ion t es t s was 2 .0 Hz (Carder, 1964) .

F igures 6 and 7 com pa re the base m ent to f ree -f ie ld t rans fe r func t ions , ]~o /~ol, de r ivedf r o m t h e FA S o f th e e a r t h q u a k e d a t a a n d t h e t h eo r e ti c a l m o d e l . F o r t h e th e o r e t ic a lt rans fe r fun c t ion the para m ete rs p , Vs, a , and the n a tura l f requenc ies we re f ixed . Di ffe r-e n t c h o i c e s o f t h e b a s e m a s s ,m o m o d a l d a m p i n g r a t io , t/ j, a n d m o d e s h a p e s , X u , w e r em a d e t o i n d i c a t e t h e i r e ff e c t o n t h e t r a n s f e r f u n c t i o n a n d t o s e e k g o o d a g r e e m e n t w i t h t h em e a s u r e d m o t i o n .

7_ . ~

H O L L Y W 0 0 0 S T O R R G E E W D I R E C T I O N

ERRTHOURKE 19 5 2)ERBTHOUFIKE 1971)THEORY

O .

\ \ I / / ~ \ / : A , ,(1)

] ' : , : - , h , ~ ' ,, , /~ , , ,

1 2 3 q 5 6 7 8 9 l O i t 1 2 1 3 l q 1 5FREQ CPS

FIG. 6. Com parison of transfer functions. Values of parameters chosen for the theoretical mo dels are:(1) m o = 5 x 106 lb., It --- 1.21 x 1 0 1 lb-ft, r/--- 0.05 ; mode shapes calculated from Duk ee t a l . (1970).2 ) o = 5 106 lb, It = 1.21 x 101~ lb-ft, r/ = 0.05; mode shap es from Jenningse t a l . (1972). (3)

m o = 3.1 x 107 lb, Ir = 2.2 101 , lb-ft, r/ = 0.05 mo de shapes calculated from Dukeet a l . (1970).

C o m p a r i s o n o f t h e t r a n s fe r f u n c t io n s d e t e r m i n e d f r o m t h e a c c e l e ro g r a m s r e c o r d e d i nt h e S a n F e r n a n d o a n d A r v i n - Te h a c h a p i e a r t h q u a k e s g i ve s a n i n d i c a t io n o f th e a c c u r a c ye x p e c t e d i n c a l c u l a t io n o f a n e x p e r i m e n t a l t r a n s f e r f u n c t i o n . T h e r e is g e n e r a l s i m i l a ri t yin the shapes o f the t rans fe r func t ions , an d i t i s no ted th a t the re i s a d i ffe rence , cons i s te n ti n b o t h e a r t h q u a k e s , i n th e w a y t h e e x p e r i m e n t a l l y d e t e r m i n e d t r a n s f e r fu n c t i o n s d e c a yi n t h e t w o d i r e c t io n s w i t h i n c re a s i n g f r e q u e n c y. T h e f u n c t i o n s d e c a y r a p i d l y a r o u n d 5 H zf o r t h e l o ng , E W d i r e c ti o n , w h e r e a s th e d e c a y is m o r e g r a d u a l f o r t h e s h o r t e r N S d i r e c t i o no f t h e b u i ld i n g . T h e t r a n s f e r f u n c t i o n s d o n o t c o m p a r e s o w e ll i n d e t a il ; t h e l o c a t i o n s o fm a n y o f th e p e a k s m a t c h , b u t m a n y d o n o t . I n a d d i ti o n , t h e t r a n sf e r fu n c t io n s f o u n df r o m t h e w e a k e r m o t i o n s i n th e A r v i n - Te h a c h a p i e a r t h q u a k e a r e c o n s i st e n tl y hi g h e r th a nt h o s e f o u n d f r o m t h e S a n F e r n a n d o r e c o r d s fo r h i g h e r f r eq u e n c ie s .

F i g u r e 6 s h o w s h o w t w o d i f fe r e n t s et s o f m o d e s h a p e s c a n a f f e c t t h e s h a p e o f t h et h e o r e t i c a l t r a n s f e r f u n c t i o n . T h e a m p l i t u d e o f t h e d i s t o r t i o n s i n th e t r a n s f e r f u n c t i o nf r o m h i g h e r m o d e s , t h r o u g h t h e m o d a l q u a n t i ti e s , M j ., i j , H i , d e p e n d s s i g ni f ic a n tl y o nt h e m o d e s h ap e s. S u c h v a r ia t io n s a r e m o r e p r o n o u n c e d f o r th e h i g h e r m o d e s , w h e r e t h e


9/24

S O I L S T R U C T U R E I N T E R A C T I O N D U R I N G T H E S A N F E R N A N D O E A R T H Q U A K E21

HOLLYWOO STORflGE,NS DIRECTIONERRTHOURKE 1952)ERRTHQUAKE 1971)THEORY

i

~ i I I I I I I I I I I I I I I IO . 1 . 2 . 3 . I ~. 5 . 6 . 7 . 6 . 9 . 1 0 . 11 . 1 2 . 1 3 . l t ~ . 1 5 .

F R E O C P S

FiG. 7. Com parison of transfer functions. Values of parameters chosen for the theoretical models are :1) mo = 5 x 104 lb, I t = 6.7 x 109 lb-ft, r /= 0.05; m ode shapes calculated from Du keet aL 1970).2) mo = 3.1 107 lb, It = 1.22 x 101 o lb-ft, q = 0.05; mode shapes calculated from Du keet al. 1970).

d i s t o r t io n s a r e s p r e a d o v e r a w i d e r f r e q u e n c y b a n d t h a n t h e y a r e a t th e f u n d a m e n t a lf r e q u e n c y.

A c o m p a r i s o n o f t h e t h e o r e ti c a l t r a n s f e r fu n c t i o n s i n F i g u r e s 6 a n d 7 s h ow s t h e b a s e -m e n t r e s p o n s e g r e a t l y a t t e n u a t e d a t h i g h f r e q u e n c i e s b y a si x - f o ld i n c r e a s e in t h e b a s em a s s . T h e s m a l l e r v a l u e o fm o 5 x 1 6 l b s, is a n e s t i m a t e o f th e w e i g h t o f th e b a s e m e n ta n d g r o u n d - l e v e l f l o o r s la b s a n d t h e w e i g h t o f t h e c o n c r e t e w a l ls i n th e b a s e m e n t . T h el a rg e r v a l u e ,m o = 3 .1 x 107 lbs , inc ludes a n es t im a te o f the we igh t o f the so il and c lu s te redf o u n d a t i o n p i le s b e tw e e n t h e b a s e m e n t f l o o r a n d t h e r o c k l a y e r o n w h i c h t h e y b ea r. T h i sv a l ue o f th e b a s e m a s s w a s t h o u g h t t o b e a n u p p e r b o u n d f o rm o.

A n e x a m p l e o f h o w t h e m o d a l d a m p i n g , r / j, a ff e ct s t h e t r a n s f e r f u n c ti o n i s n o t i n c lu d e db e c a u s e t h e e f f e ct i s r e l a t iv e l y m i n o r C r o u s e , 1 97 3). L a rg e r v a l u e s o f r/j t e n d t o s m o o t ht h e d i s t o r t i o n s f o r a l l m o d e s ; b u t , a s m i g h t b e e x p e c t e d , m o d e r a t e v a r i a t i o n s i n ~/j d o n o ta l t e r t h e o v e r a l l s h a p e o f t h e t r a n s f e r f u n c t i o n .

T h e p a r t i a l a g r e e m e n t b e t w e e n t h e t h e o re t i c a l a n d a c t u a l t r a n s f e r fu n c t i o n s in t h e E Wd i r e c t i o n i s l i m i t e d t o f r e q u e n c i e s b e t w e e n 0 a n d a b o u t 5 H z . T h e r e i s s o m e e v i d e n c e o fs o i l -s t r u c tu r e in t e r a c t i o n i n t h e f u n d a m e n t a l m o d e f o r t h e S a n F e r n a n d o e a r t h q u a k e .T h e S a n F e r n a n d o t r a n s f e r f u n c t i o n n e a r 1 .5 H z c o n f o r m s i n s h a p e a n d s i ze w i t h t h ed o u b l e t s i n t h e t h e o r e t i c a l t r a n s f e r f u n c t i o n s n e a r t h i s f r e q u e n c y. T h e t h e o r e t i c a l d o u b l e t so c c u r a t s o m e w h a t l o w e r f r eq u e n c ie s t h a n s h o w n b y t h e S a n F e r n a n d o d a t a i n d ic a t in g ,p e r h a p s , t h a t t h e v a l u e o f 8 0 0 f t / s e c u s e d i n t h e a n a l y s i s i s t o o l o w. T h e t r a n s f e r f u n c t i o nf r o m t h e A r v i n - Te h a c h a p i e a r t h q u a k e d o e s n o t c o n f ir m t h is a g r e e m e n t a t t h e f u n d a m e n t a lm o d e , a l t h o u g h t h e r e i s e n o u g h c o r r e s p o n d e n c e o f th e e x p e r i m e n t a l a n d t h e o r e t i c a lt r a n s f e r f u n c t i o n s i n t h e 1 . 2 t o 2 . 5 H z r a n g e t o c o n c l u d e t h a t t h e d a t a a r e a t l e a s t n o tc o n t r a d i c t o r y i n th i s f r e q u e n c y r an g e . B e y o n d 5 H z , i t d o e s n o t a p p e a r t h a t a n y c o m b i n a -t i o n o f p a r a m e t e r s u s e d i n t h e a n a l y si s w il l p r o d u c e g o o d a g r e e m e n t w i t h t h e o b s e r v e d


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22 CHARLES B CROUSE AND PAUL C JENNINGS

behavior. Using the upper bound for m o in the theoretical model will attenuate the ampli-tude of the transfer function enough to give reasonable agreement for frequencies beyond11 Hz for the San Fernando data Figure 6). The amplitude of the Arvin-Tehachapitransfer function shows reasonable agreement with this theoretical curve for frequenciesbeyond 4 Hz. In both cases, however, the choice of the larger m o reduces the agreement inthe low-frequency range.

The theoretical and earthquake transfer functions for the NS direction have about thesame amplitude for frequencies between 0 and 4 Hz if the smaller base mass, m o =5 x 106 lbs, is selected Figure 7). Neither of the earthquake transfer functions, however,gives clear indications of interaction in any of the modes, particularly the fundamental.For frequencies beyond 4 Hz, both earthquake transfer functions attenuate in roughlythe same manner, with the ampli tude levels of the Arvin-Tehachapi slightly larger on theaverage than the San Fernando earthquake. Using the upper limit for m o improves theagreement for larger frequencies but worsens the agreement for frequencies less than4 Hz, as was the case in the EW direction.

Comparisons also were made between the San Fernando free-field accelerogram and atheoretical free-field accelerogram. The theoretical free-field accelerogram was computedby transforming the San Fernando EW basement accelerogram through theoreticaltransfer functions by appropriate Fourier analysis and synthesis. Two theoretical transferfunctions, curves 2 and 3 from Figure 6, were selected because of their dissimilarity.Figure 8 clearly shows that the differences between the computed free-field and recordedbasement motions are relatively minor, although variations do exist in the size and shapeof a few peaks. The similarity of the theoretical free-field motions to each other and tothe measured basement record indicates that the differences of transfer functions 2 and3 from unity Figure 6) in the frequency band that contains most of the energy of thebasement record 0 to 4.5 Hz) are not large enough to be significant. The theoretical free-field motions are quite different in appearance from the assumed free-field motion recordedin the nearby shed Figure 4). The differences are, however, confined to frequenciesgreater than 4.5 Hz, as can be seen in the upper part of Figure 5.

M i l l i k a n L i b r a r y a n d A t h a n a e u m The Millikan Library and the Athenaeum buildingsare located on the main campus of the California Institute of Technology in Pasadena,approximately 21 miles southeast o f the center of energy release during the San Fernandoearthquake. The Athenaeum is 1220 ft due east of the Millikan Library.

The Millikan Library is a nine-story, reinforced concrete building, 144 ft tall with abasement level 14 ft below the ground floor. Figure 9 shows a view of the building andsome foundation details Kuroiwa, 1967).

The Athenaeum is a 2-story, reinforced concrete structure of fairly complex geometryFigure 10). The building is asymmetric and nonuniform in height. The basement floor

area is approximately 127 by 138 ft, with an additional 69 by 32 ft on the north side. Thefoundation consists of conventional spread footings, 7 by 7 x 2 ft thick on the average.Both the Library and the Athenaeum rest on alluvium composed of firm to dense sandmixed with gravel with an average weight density of 115 lb/ft 3. The alluvium extendsabout 900 ft to bedrock. A P-wave velocity of 2200 ft/sec to a depth of 100 ft wasdetermined by experimental tests Duke and Leeds, 1962).

An SMA-1 accelerograph recorded the basement accelerations at the Athenaeum, andan RFT-250 obtained a record in the basement of Millikan Library. Figure 11 comparesthe lateral components of the basement acceleration of each building, while Figure 12shows the FAS of 20.48-sec port ions of the accelerograms. The FAS plots have beensmoothed 10 cycles with the Hanning spectral window. Both the accelerograms and theFAS show that the basement motion was more intense in Millikan than at the Athenaeum,


11/24

SOIL STRUCTURE INTERACTION DURING THE SAN FERNANDO EARTHQUAKE 3

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12/24

4 C H A R L E S B . C R O U S E A N D PA U L C . J E N N I N G S

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SOIL-STRUCTURE INTERACTION DURING THE SAN FERNANDO EARTHQUAKE 25

especial ly in the NS direct ion. This diss imilar i ty in accelerogram s obtain ed so c losetogethe r was widely note d so on af ter the records b ecame avai lable . I t has been suggestedthat the difference may have been cau sed by soi l -s t ructure interact ion (e .g . , Richter,1972).

Th e Ath enae um was mo deled tw o ways because o f i ts re la t ively high s t iffness. Oneme tho d was to consider the ent i re bui lding as a r igid pla te ; the oth er way was to approxi-mate i t as an equivalent one-s tory s t ructure res t ing on a foundat ion base mass . Theweight of the bui lding and fo und at io n determ ined from structural drawings was 18 10 6 lbs . For the s ingle-s tory approximat ion, the tota l weight was divided so that thebase mass, mo weighed 13 10 6 lbs and th e first story, ml, 5 10 6 lbs. Th e height betweenrn o and m l was es t imated to be 20 ft . Th e equivalent base radius was 80 f t . A fun dam entalf requency of 4 Hz was chosen fo r bo th d i rect ions and the mo da l dam ping was assumedto be 5 per cent o f cri t ical .

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Fla. 10. Athenaeum , C al i fornia Inst i tu te of Technology.

The m asses o f each s to ry fo r the Mi l l ikan Libra ry were t aken f rom Kuro iwa (1967).Th e tota l weight of the supers t ructure , 18.7x 10 6 lbs , i s d ivided fa i r ly evenly am ong thenine stories. Values estima ted fo r the weight of the base mass a nd radius were 7 x 106 lbsand 45.9 ft , respectively. Th e fund ame ntal f requencies , based on p re-ear thqu ake am bientand forced-vibrat ion tes ts (Kuroiwa, 1967) were 2 .0 Hz (NS direct ion) and 1.5 Hz (EWdirect ion) . The tests showed the 2nd na tural f reque ncy in the EW direct ion to be 6 .2 Hz.Th e third EW natur al f req uency was arbi t rar i ly chosen as 13.5 Hz. The secon d naturalf reque ncy in the NS direct ion, 10H z was based on the FAS of acce le rograms f rom theroo f and basem ent (Wo od, 1972). Fun dame nta l mod e shapes fo r the superst ruc ture weretaken f rom Kuro iw a ' s fo rced-v ibra t ion t es ts .

To complete the model ing of the supers t ructure , in ters tory s t i ffnesses were calculatedf rom the fundamenta l f requen cy and mod e shape p lus the s to ry masses. The fo rmu la


14/24

26 C H A R L E S B . C R O U S E A N D P A U L C . J E N N I N G S

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F I G . 1 1 . C o m p a r i s o n o f r e c o r d e d a c c e l e r o g r a m s f r o m t h e M i l l i k a n L i b r a r y a n d A t h e n a e u m

used to e s t ima te the in te r s to ry s t i f fnesses (Nie l sen , 1964) was

( - 0 1 2 ~ miX l ik~ = i=1 s = 1 ,2 , n 7 )

. S ' l , s _ _ X l , s _ 1 ' . . . ,

w h e r e 0 91 is t h e fu n d a m e n t a l f r e q u e n c y i n r a d ia n s p e r s e c o n d , a n d m i is t h e m a s s o f t h ei h s t or y. K n o w l e d g e o f i n t e r st o r y s t if f n es s es a n d m a s s o f e a c h s t o r y e n a b l e d t h e c a l c u l a -t i o n o f t h e h ig h e r m b d e s h a p e s b y s o l v i n g t h e e ig e n v a l u e p r o b l e m . T h e h i g h e r n a t u r a lf r e q u e n c i e s o b t a i n e d f r o m t h e s o l u t i o n d i d n o t a g r e e v e ry w e l l w i t h t h e v a l u e s m e n t i o n e dprev ious ly, thus ind ica t ing a d i sc repancy be tween the ca lcu la ted and ac tua l s t i f fnesses .H o w e v e r, t h e e f f e c t o f t h e s e d i f f e r e n c e s o n t h e m o d e s h a p e s a n d , h e n c e , t h e t h e o r e t i c a lt r a n s f e r f u n c t i o n w i l l n o t b e s i g n i f i c a n t w h e n c o m p a r i s o n s a r e m a d e w i t h t h e t r a n s f e rf u n c t i o n c a l c u l a t e d f r o m t h e e a r t h q u a k e d a t a .


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16/24

28 C H A R L E S B C R O U SE A N D P A U L C J E N N IN G S

S o i l p a r a m e t e r s u s e d w e r e t h e s a m e f o r b o t h s o i l -s t r u c tu r e s y st e m s . Va l ue s f o r t h e u n i tweigh t o f the so i l and Po isso n s ra t io were p = 115 lb / f t 3 and a = 0 .25 . Fo r s t ra in leve l sa s s o c i at e d w i t h e a r t h q u a k e r e s p o n s e, a n a p p r o x i m a t e s h e a r- w a v e v e l o c i ty o f 8 0 0 f t /s e cw a s e s t i m a t e d in a m a n n e r s i m i l a r to t h a t d o n e f o r th e H o l l y w o o d S t o r a g e s it e.

To t e st w h e t h e r s o i l- s t ru c t u r e i n t e r a c t io n c o u l d a c c o u n t f o r t h e d i f f e re n c e s n o t e d i n t h eFA S a n d t h e a c c e l e ro g r a m s , t h e f r ee - fi e ld m o t i o n s n e a r e a c h b u i l d i n g w e r e a s su m e d t o b ei d e n ti c a l. F o r v e r t ic a l ly i n c i d e n t w a v e s th i s w o u l d b e a v a l i d a s s u m p t i o n d i s c o u n t i n g a n yl o c a l g eo l o g ic a l ir r e g u la r i ti e s a n d t h e i n f lu e n c e o f o t h e r n e a r b y s t r u c tu r e s . T h e m o d u l u s o fa t r a n s f e r f u n c t i o n , w a s c a l c u l a te d f o r e a c h b u i ld i n g . T h e m o d u l i w e r e t h e nd i v i d e d to g i v e t h e a m p l i t u d e o f a t h e o r e t i c a l tr a n s f e r f u n c t i o n b e t w e e n t h e b a s e m e n t s o feach bui lding, i .e . ,

T h e a m p l i t u d e o f th i s tr a n s f e r f u n c t i o n w a s t h e n c o m p a r e d w i th t h e r a t i o o f th e FA S c a l -c u l a t e d f r o m t h e a c c e l e ro g r a m s .

T h e o r e t i c a l t r a n s f e r f u n c t io n s , a r e s h o w n i n F i g u r e 1 3. M o d e l 2 i n F i g u r e13a represen t s the Athenaeum t rea ted as a r ig id c i rcu la r p la te wi th a weigh t equa l to thet o t a l w e i g h t o f t h e b u i l d i ng . T h e c u r v e i s n e a r l y f la t, i m p l y i n g t h a t t h e b a s e m e n t a n d f r e e-f ie ld m o t i o n s a r e e s s e n ti a ll y id e n t ic a l . N o r e a s o n a b l e i n c r e a s e in b a s e m a s s o r m o m e n t o fi n e rt ia , f o r e x a m p l e b y i n c l u d i n g t h e w e i g h t o f t h e s oi l b e t w e e n t h e b o t t o m o f t h e f o o t in g sa n d b a s e m e n t f l o o r, c o u l d a l te r t h e A t h e n a e u m s t h e o r e t i c a l t r a n s f e r f u n c t i o n e n o u g h s ot h a t t h e t h e o r e t i c a l fr e e- f ie l d m o t i o n w o u l d b e n o t i c e a b l y d i ff e r e n t f r o m t h e b a s e m e n tm o t i o n . B a s e d o n t h e a n a l y s is o f t h e H o l l y w o o d S t o r a g e B u i l d in g , i t c a n a l s o b e c o n -c l u d e d t h a t t h e M i l li k a n L i b r a r y t r a n s f e r f u n c t i o n w o u l d n o t p r o d u c e s i g n if ic a n t d i ff e r-e n c e s b e t w e e n t h e r e c o r d e d b a s e m e n t m o t i o n a n d t h e o r e t i c a l f r ee - f ie l d m o t i o n , b e c a u s et h e t r a n s f e r f u n c t i o n i s f a i rl y cl o s e t o u n i t y o v e r a f r e q u e n c y r a n g e c o n t a i n i n g m o s t o f th ee n e rg y o f t h e s t r o n g m o t i o n . T h u s , t h e t h e o r e t i c a l f r ee - fi e ld a c c e l e r o g r a m s f o r t h eM i l li k a n L i b r a r y a n d t h e A t h e n a e u m a r e e s s en t ia l ly t h e s a m e a s t h e i r r es p e c t iv e b a s e -m e n t a c c e l e r o g ra m s .

F i g u r e 1 4 c o m p a r e s t h e r a t i o o f th e o r e t i c a l t r a n s f e r f u n c t i o n s t o t h e r a t i o o f t h e b a s e -m e n t FA S s m o o t h e d a n a d d i t i o n a l 1 0 c y c le s . T h e a g r e e m e n t b e t w e e n t h e th e o r e t i c a lr a t i o s a n d t h o s e f r o m t h e e a r t h q u a k e r e s p o n s e is p o o r f o r fr e q u e n c i e s b e y o n d 2 H z ,p a r t i c u l a r l y f o r t h e N S d i r e c t i o n . T h e r e d o e s n o t a p p e a r t o b e a n y r e a s o n a b l e w a y t oa d j u s t t h e p a r a m e t e r s o f t h e s o i l - s tr u c t u r e m o d e l o f e i t h e r b u i l d in g t o i m p r o v e t h e a g r e e -m e n t s u b s t a n t i a l l y. T h e a g r e e m e n t c o u l d b e i m p r o v e d s o m e w h a t , m a i n l y i n t h e E Wd i r e c t io n , i f t h e p e a k s i n t h e t h e o r e t i c a l c u r v e s a t 4 H z w e r e r e lo c a t e d a t 3 H z . T h i s w o u l db e t h e c as e if t h e f u n d a m e n t a l N S a n d E W f r eq u e n c ie s o f th e A t h e n a e u m w e r e 3 H z .A m b i e n t m e a s u r e m e n t s o f t h e A t h e n a e u m w e r e m a d e o n J u n e 2 0 , 1 9 74 t o i n v e s ti g a te th i sposs ib i li ty. Th e bu i ld ing i s no t ve ry su i tab le fo r am bien t t e s t ing , b u t the resu l t s o f th i sp r e li m i n a r y t es t in d i c at e d N S a n d E W f u n d a m e n t a l f re q u e n ci e s o f a b o u t 2 .5 H z . A l t h o u g hp o s s ib l e , it s ee m s u n l i k e ly t h a t t h e f u n d a m e n t a l f r e q u e n c ie s a r e n e a r 3 H z f o r e a r t h q u a k eresponse .

D I S C U S S I O N

C o m p a r i s o n s o f d a t a a n d a n al y si s fo r t h e H o l l y w o o d S t o r a g e B u il d in g a r e n o t c o n -c l u s i v e , b u t i t d o e s a p p e a r p r o b a b l e t h a t s o i l - s t r u c t u r e i n t e r a c t i o n o c c u r r e d i n t h e E Wf u n d a m e n t a l m o d e d u r i n g t h e S a n F e r n a n d o e a r t h q u a k e . T h e d a t a f r o m t h e A r v i n -Te h a c h a p i e a r t h q u a k e a r e n o t c o n t r a d i c t o r y t o t h is c o n c l u s io n , b u t d o n o t b y t h e m s e l v essugges t s ign i f i can t in te rac t ion in th i s mode . The theore t i ca l t r ans fe r func t ions ind ica te


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SOIL STRUCTURE INTERACTION DUR ING THE SAN FERNANDO EARTHQUAKE 29

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THEORETICRL TRRNSFER F U N C T I O N E N D I R E C T I O N

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t2. 3 . t l . 5 . 6 . 7 . 8 9 . 1o . 11. 12. 13. l t l . 15.

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THEORETICAL TRRNSFER FUNCT ION N5 DIRECTION

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MILLIKRN

1 2. 3. If . 5 . 6 . ? 8 . 9 . lO . 11 . 12 13 . l i t . 15FflEg - CPS

b)FIG. 13. Com pariso n o f transfer functions. a) EW direction. At hen aeu m treated as rigid plate,

mod el 2. Ath ena eum tre ated as single-degree-of-freedom system, mode l 1. b) NS direction. Athe naeu mtreated as single-degree-bf-freedom system.


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30 C H A R L E S B . C R O U S E A N D PA U L C . J E N N IN G S

R RT I O O FBRSEMENT RES PONSES EN D I R E T I O N

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O . 1 2 3 . tt . 5 6 . 7 8 9 I O . l l . 1 2 1 3 . t q . 1 5F R O CP

b )

FIG. 14 . Compar i son o f t r ans fe r func t ions . ( a ) Theore t i ca l t r ans fe r func t ions ca l cu la t ed f rom F igure13a . (b) Theore t i ca l t r ans fer func t ion ca l cu la t ed f rom F igure 13b .


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SOIL STRUCTURE INTERACTION DURING THE SAN FERNANDO EARTHQUAKE 31

m u c h l e s s i n t e r a c t i o n f o r t h e N S f u n d a m e n t a l m o d e a n d d a t a f r o m b o t h e a r t h q u a k e sshow no i den t i f i ab l e i n t e r ac t i on a t t h i s f r equency. The t heo re t i c a l and expe r imen ta lresu l t s for the NS d i rec t ion agree in the sense tha t the i r overa l l l eve l i s the same in theim por t a n t f r equency band f rom 0 t o 6 Hz , bu t t hey do n o t ag ree i n de ta i l.

I n add i t i on t o pos s ib le e f fec ts a t t he fu ndam en ta l f r equenci e s , the a t t enua t i on o f t heexper im enta l t ransfer fun c t ions a t h igher f requenc ies and the cons is te n t d iffe rences in ther a t e o f a t t enua t i on i n t he NS an d E W expe r imen ta l tr ans f e r f unc t i ons fo r t he two ea r th -quakes a r e p rob ab ly caused by i n t e r ac t ion . U nfo r tuna t e ly, the deg ree o f a ccu racy andcons i s t ency o f t he expe r imen ta l l y de t e rmined t r ans f e r f un c t i ons l im i t s de t a il ed com par i -sons wi th ana ly t ica l resu l ts . As i s seen in F igure s 4 an d 5 , the a t t enu a t io n in the t ransfe rfunc t i ons a r is e s because t he bu i ld ing appa ren t ly f i l te r ed ou t t he h ighe r f r equenc i e s foundin t he f r ee -f ie ld acce l e rog rams . Reaso nab le va r i a t i ons in t he pa ram e te r s o f t he t heo re t i c a lmodel used were unable to pred ic t th i s f i l t e r ing sa t i s fac tor i ly, which occur red for f re -quenc ie s above ab ou t 5 Hz .

The re a r e a nu mb er o f mod i f i ca t i ons t o t he t heo ry t ha t m igh t l e ad t o improved ag ree-me n t be tween expe r ime n ta l and t heo re t ic a l r e su lt s f o r t he h ighe r f r equenc ie s . One obv iousrev is ion to the theory would be to rep lace the c i rcu la r base p la te wi th a rec tangular onewi th t he d imens ions o f t he foun da t ion . Ac co rd ing t o Sa r r az in 097 0 , t he d i ff e rences i nthe com pl i ance func t i ons be tween a 2 b y 1 r ec t angu la r p l a t e and a c i r cu l a r p la t e o f equa la r ea became l a rge fo rao above 2 , wh ich co r r e sponds t o a bou t 4 .3 Hz fo r t he l ong

~ L ~

g.

/ ~ L . 7

incoming \ \w a v e ~ ~ ~ / )

~ - - ~ c F m e c t io n o f p a r t i c l e m o t i o n

FIc. 15. Filtering by Foundation of a plane harmonic wave incident at an angle.

d i r ec t ion o f t he H o l lywo od S to rage Bu i ld ing . I t m ay be pos s ib le a l so t o app rox ima te t h ise ffec t o f a rec tang ular fo und at io n us ing the compl iance s for a c i rcu la r base , by se lec t ingthe equ iva l en t r ad iu s by some p roce du re o the r t han equa t ing t he a r ea s. I f t he equ iva l en tr ad iu s we re t aken a s t he l eng th o f t he found a t ion i n t he d i r ec t i on o f in t e re s t , f o r example ,i t c an be s een f rom F igu re 2 t ha t t he compl i ances i n t he two d i r ec t ions wou ld be d i f fe r en tand t he ag reem en t w i th the da t a migh t pos s ib ly be improved .

An o the r pos s ib il i ty fo r f i lt e r ing h ighe r f r equenc ie s r e su l ts f ro m a s sum ing m os t o f t heea r thquake waves we re i nc iden t upon the founda t ion a t ang l e s o the r t han ve r t i c a l .Un fo r tuna t e ly, t heo r i e s o f so i l - st ruc tu re i n t e r ac t i on have n o t ye t been deve loped t o t r ea tP or V waves a t a rb i t ra ry inc idence , o r sur face waves , The idea o f f i l te r ing for inc ide ntwaves can be seen in F igure 15 , which shows tha t for cer ta in wavelengths , 2 , and anglesof inc idence , c~, the so i l bene a th the bu i ld ing tends to m ove in oppos i te d i rec t ions . Thefou nd at i on w ould tend to f i l t e r ou t such w avelengths because of i t s re la t ive ly la rge r ig id ity.Fo r example , i n the ca se o f t he EW d i r ec t ion o f t he Ho l lywoo d S to rage Bu i ld ing , L


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32 CHARLES B CROUSE AND PAUL C JENNI NGS

200 f t. I f the ve loc i ty, v, and ang le o f inc idence , ~ , o f the incom ing w aves were 800 ft / seca n d 4 5 , r e sp e c t iv e l y, t h e n t h e f r e q u e n c y c o r r e s p o n d i n g t o t h e w a v e l e n g t h s h o w n i nFigu re 15 wo uld be f =v/ Lcos ~) ~- 5.6 Hz . Fo r w aves t rave l ing a long the sur face a tt h is v e lo c i ty, f = 4 H z . F o r w a v e s f r o m t h e N S d i r e c t i o n , f w o u l d b e a b o u t f o u r ti m e sla rger s ince the fou nd a t io n d im ens ion of the bu i ld ing i s on ly 50 f t in th i s d i rec t ion .H o w e v e r, t h e a d j o i n in g o n e - s t o r y s t r u c t u r e a n d t h e s t r u c t u r e 7 8 f t t o t h e n o r t h o f th em a i n b u i l d in g m i g h t i n c r e a s e t h e e f fe c ti v e l e n g t h o f t h e b u i l d in g i n t h e N S d i r e c ti o n .A l t h o u g h t h is a n a ly s is i s a p p r o x i m a t e , f r e q u e n c ie s o f f il te r in g a r e n e a r t h o s e o b s e r v e d t oh e f i l t e r e d b y t h e b u i l d i n g d u r i n g t h e e a r t h q u a k e . S i m i l a r a g r e e m e n t w a s o b t a i n e d b yY a m a h a r a ( 19 70 ) f o r r e c o r d s o b t a i n e d i n a f t e rs h o c k s o f t h e To k a c h i - O k i e a r t h q u a k e .T h e i n a b i l it y o f s i m p l if ie d t h e o r e ti c a l m o d e l s t o i n c o r p o r a t e w a v e s a t a r b i t r a r y i n c i d e n c em a y b e t h e i r m o s t s e r io u s l i m i ta t i o n .

A r e l a t e d m e c h a n i s m f o r f i lt e ri n g h i g h e r f r e q u e n c ie s f r o m t h e b a s e m e n t r e c o r d s ise m b e d m e n t o f t h e s t r u c tu r e , w h i c h m u s t b e c o m e s ig n i fi c an t f o r w a v e l e n g t h s o f m o t i o nc o m p a r a b l e t o t h e d i m e n s i o n s o f t h e f o u n d a t i o n . I n t h e i r s t u d y o f t h e re s p o n s e o f t h eH o l l y w o o d S t o r a g e B u i l d i n g d u r i n g t h e 1 9 5 2 e a r t h q u a k e , H r a d i l e k a n d L u c o ( 1 9 7 0 )m o d e l e d t h e s t r u c t u r e a n d f o u n d a t i o n i n t h e l o n g d i r e c t i o n b y a n i n f i n i t e s h e a r w a l lm o u n t e d o n a h e m i - c y li n d r i ca l f o u n d a t i o n a n d o n a f l a t f o u n d a t i o n . T h e i r r es u lt s,s h o w n i n F i g u r e 1 6, i n d i c a t e m u c h b e t t e r a g r e e m e n t a t h i g h e r f r e q u e n c ie s f o r t h e m o d e lw i t h t h e e m b e d d e d f o u n d a t i o n .

2 ' 0 F r ~ / - E X P E R 'M E N TA L

1.5 ~ ~ I A I ~ - D E E P/ J k 1 1 . / A / L ~ , ] ~ \[FLAT B I

< ~ i n / ~ - ~ . ~ l ~ l l ~ l I I . .'t I I I I H I I I ~ ~ I JJ

0 | I I I I I I I I0 I 0 2 0 3 0 L O 4 .0 5 0 6

FIG. 16. Com parison of transfer functions for m odels of the H ollywood Storage Building, EW direc-tion. The curve labeled FL A T is for an infinite shear wall on a flat foundation; DE EP is for the shearwall on a hem i-cylindrical foundation. Th e experimental resu lt is from the 1952 earthquake. Fro mHradilek and Luco (1970).

T h e d i f f er e n c es i n t h e e x p e r i m e n t a l l y d e t e r m i n e d t r a n s f e r f u n c t i o n s f o r t h e t w o e a r t h -q u a k e s a t t h e s i te o f t h e H o l l y w o o d S t o r a g e B u i l d in g a r e s o m e w h a t l a rg e r t h a n m i g h th a v e b e e n h o p e d . I n s t u d y i n g t h e s e d i ff e re n c e s, i t w a s n o t e d t h a t t h e r e a r e s o m e f e a t u r e sof the expe r ime nta l t r ans fe r func t ions tha t sugges t a non l inear e ffec t in the so i l. I f theso i l had a lowe r e ffect ive m odu lus d ur ing the l a rger s t ra ins exp er ienced in the San F er-n a n d o e a r t h q u a k e , t h e t r a n sf e r fu n c t i o n f o r th e S a n F e r n a n d o e v e n t w o u l d b e l o w e r f o rh i g h e r f r e q u e n c ie s , a n d p e a k s , i n g e n e ra l , w o u l d l ie to t h e l e f t o f c o r r e s p o n d i n g p e a k s i nt h e t r a n s fe r f u n c t io n f o r t h e w e a k e r m o t i o n o f th e A r v i n - Te h a c h a p i e a r th q u a k e . T h e s ee f fe c ts a r e s e e n t o s o m e d e g r e e i n F i g u r e s 6 a n d 7 , b u t a r e n o t c o n v i n c i n g a n d a d d i t i o n a lm e a s u r e m e n t s a r e r e q u i r e d b e f o r e m o r e d e f i ni t e s t a t e m e n t s c a n b e m a d e .

D i f f e re n c e s i n t h e e x p e r i m e n t a l l y d e t e r m i n e d t r a n s f e r f u n c t i o n s m a y a l so a r is e i f t h ep a r k i n g l o t r e c o r d s a r e n o t r e p r e s e n t a ti v e o f t h e t h e o r e t i c a l f r e e- f ie l d m o t i o n f o r h i g h e rf r e q u e n ci e s . T h e r e c o r d i n g s i te is o n l y o n e f o u n d a t i o n s l e n g t h r e m o v e d f r o m t h e b u i l d i n gin the d i rec t ion o f s t ronges t suspec ted in te rac t ion , and i t m ay inc lude , a t h igh frequenc ies ,re f lec tions f ro m the bu i ld ing . Such e ffec t s hav e been n o ted in theo re t i ca l r esu l ts (e .g .,Tr i f u n a c , 1 97 2) a n d c o u l d h a v e b e e n s t u d i ed i f t h e s u r f a c e m o t i o n a t o t h e r p o i n t s i n t h en e i g h b o r h o o d o f t h e b u i l d in g h a d b e e n m e a s u r e d . F o r s u ff ic i en t ly h i g h f r e q u e n c ie s , i t ise x p e c t e d t h a t t h e r e c o r d s f r o m s u c h a s e t o f i n s t r u m e n t s w o u l d s h o w m a j o r d i f f er e n c e s


21/24

S O I L S T R U C T U R E IN T E R A C T I O N D U R I N G T H E S A N F E R N A N D O E A RT H Q U A K E33

a n d n o s in g le r e c o r d w o u l d b e r e p r e s e n t a t i v e o f th e t h e o r e t i c a l fr e e -f i el d m o t i o n . I t s h o u l db e p o s si b le , h o w e v e r, t o d e t e r m i n e f r o m t h e r e c o r d s t h e f r e q u e n c y r a n g e o v e r w h i c h t h ec o n c e p t o f fr e e -f i el d m o t i o n c a n b e a p p l i e d .

S o m e o f t h e o b s e r v e d d i f fe r e n c e s i n t h e a c c e l e r o g r a m s r e c o r d e d i n M i l li k a n L i b r a r ya n d t h e A t h e n a e u m m a y a l s o b e a tt r i b u t a b l e t o t h e e f fe c ts n o t e d a b o v e , a s c a n b e s e enf r o m e x a m i n a t i o n o f F i g u r e 1 2. I t is p os s ib l e , f o r e x a m p l e , t h a t t h e A t h e n a e u m w i t h i tsl a rg e r f o u n d a t i o n i s m o r e e f fe c t iv e a t s u p p r e s si n g w a v e s o f h i g h e r f r e q u e n c y t h a n i s th eL i b r a r y. H o w e v e r, s i n ce b o t h b u i ld i n g s a r e r o u g h l y s q u a r e i n p la n , t h e u s e o f a n e q u i v a -l e n t ci r cl e a s a b a s e s h o u l d n o t b e a s i m p o r t a n t a f a c t o r f o r t h e s e tw o b u i l d in g s a s i t m a yb e f o r t h e H o l l y w o o d S t o r a g e B u i l d in g . I n a d d i t i o n t o d i f fe r e n c es a t h i g h e r f r e q u e n c i e s,t h e r e a r e a l s o s u b s t an t i a l d i f f er e n c e s i n th e i m p o r t a n t f r e q u e n c y b a n d f r o m 0 .3 t o 2 . 5 H z .A l t h o u g h t h e se d i f f e re n c e s a p p e a r i n F i g u r e 1 2, th e y a r e b e s t s e e n in t h e r e s p o n s e s p e c t r as h o w n in F i g u r e 1 7 ( H u d s o n , 1 9 72 ; Tr i f u n a ce t a l . 1973). I t is seen that the p ea k a t 0 .4 seeis a b s e n t i n t h e N S s p e c tr a f r o m t h e A t h e n a e u m , a n d t h a t t h e s p e ct ra o f c o r r e s p o n d i n gcom po ne nts a t the tw o s it es d i ffe r s ign i f ican t ly ove r the en t i re range f ro m 1 to 3 secs.

2

IO/. DAMPED MLRELATIVE VELOCITYRESPONSE SPECTRUM ATH .. . . . . . . .

u SAN FERNANDOz EARTHQUAKE OF FEB. 9, 1971

~c

0 2 3PERIOD- SECONDS

FIG. 17. Response spectra for accelerograms rom the Athenaeum and Millikan ibrary basementsS a n F e r n a n d o e a r t h q u a k e .

T h e d i f f e re n c e s a t l o w f r e q u e n c i e s in t h e r e c o r d s o b t a i n e d i n t h e b a s e m e n t s o f t h eA t h e n a e u m a n d t h e M i l li k a n L i b r a r y c a n n o t b e e x p l a i n e d b y e x i st i ng th e o r i e s o f so i l-s t r u c t u r e i n t e r a c t i o n . S o m e o f t h e d i f fe r e n c e a t h i g h e r f r e q u e n c i e s m a y b e e x p l a i n e d , a tl e a st in p a r t , b y m o d i f y i n g t h e t h e o r y t o i n c o r p o r a t e t h e e ff e ct s o f e m b e d m e n t a n d s c a t te r -ing . I t a l so seems qu i te p ro ba b le tha t the d i s tance be twee n the bu i ld ings and the i r r e la t iveo r i e n t a t i o n w i t h r e s p e c t t o t h e e p i c e n t r a l a r e a , c o u p l e d w i t h t h e p o s s i b i l i t y t h a t m a n yi n c o m i n g w a v e s m a y h a v e b e e n i n c i d e n t a t a n g le s o t h e r t h a n 9 0 , c o u l d a c c o u n t f o r s o m eo f t h e m a j o r d i f fe r e n c e s b y d i f f e r e n t s u p e r p o s i t i o n o f i n c o m i n g w a v e s ( C r o u s e , 1 97 3).

F o r b o t h t h e H o l l y w o o d S t o r a g e a n d c a m p u s s i te s, m o r e r e c o r d s o f e a r t h q u a k e r e s p on s ei n t h e b a s e s o f th e b u i l d in g s a n d o n t h e n e a r b y s o il a re r e q u i r e d t o a n s w e r u n r e s o l v e dq u e s t i o n s. I n p a r t i c u l a r, t h e f r e q u e n c y r a n g e o v e r w h i c h n e a r b y r e c o r d s a r e r e p r e s e n t a -t i v e o f a t h e o r e t i c a l f r ee - fi e ld m o t i o n n e e d s t o b e d e t e r m i n e d b y m e a s u r e m e n t . A l s o ,m o r e d e t a i le d m e a s u r e m e n t s o f f o u n d a t i o n m o t i o n s i n c lu d i ng r o c k i n g a n d t o r si o n , a r er e q u i r e d t o a s se ss t h e i m p o r t a n c e o f e m b e d m e n t a n d o t h e r a d d i t i o n s t o t h e o r e t i c a l re s u lt s .

A C K N O W L E D G M E N T S

Th e results of th is study form a portion of the senior author s doctoral thesis (Crouse, 1973). Th efinancial support of the N ational Science Foundation and the Earthquake Research Affiliates of theCalifornia Institute of Te chnology during the thesis work is gratefully acknowledged.


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4 CHARLES B. CROUSE AND PAUL C. JENNINGS

REFERENCES

Arnold, R. N., G. N. Bycroft, and G. B. Warburton 0955). Forced vibrations of a body on an infiniteelastic solid, J. Appl. Mech. 22, 391-400.

Bielak, J. 1971). Earthquake Response o f Building Foundation Systems, Earthquake EngineeringResearch Laboratory, Report EERL 71-04 California Institute of Technology, Pasadena, California.

Bycroft, R. N. 1956). Forced vibration of a rigid circular plate on a semi-infinite elastic space and on anelastic stratum, Phil. Trans. Roy. Soc. London Ser. A 248, 327-368.

Carder, D. W. Ed.) 1964). Earthquake Investigations in the Western United States U. S. DepartmentCommerce, Coast and Geodetic Survey, Publ. 41-2.

Crouse, C. B. 1973). Engineering studies of the San Fernando Earthquake, Earthquake EngineeringResearch Laboratory, Report EERL 73-04 California Institute of Technology, Pasadena, California.

Duke, C. M. and D. J. Leeds 1962). Site Characteristics of Southern California Strong-Motion Earth-quake Stations, Dept. of Eng., University of California at Los Angeles, Report 62--55.

Duke, C. M., J. E. Luco, A. R. Carriveau, P. J. Hradilek, R. Lastrico, and D. Ostrom 1970). Strongearthquake motion and site conditions: Hollywood, Bull. Seism. Soc. Am. 60, 1271-1289.

Gladwell, G. M. L. 0968). Forced trangential and rotary vibrations of a rigid circular disc on a semi-infinite solid, Intern. J. Eng. Sci. 6, 531-607.

Housner, G. W. 0957). Interaction of building and ground during an earthquake, Bull. Seism. Soc. Am.47, 179-186.

Hradilek, P. J. and J. E. Luco, 1970). Dynamic soil-structure interaction, IDIEM Informa TecnicoNo. 14 Insti tuto de Investigaciones y Ensayes de Materieles, Universidad de Chile, Santiago.

Hudson, D. E. 1972). Local distribution of strong earthquake ground motions, Bull. Seism. Soc. Am.62, 1765-1786.

Jennings, P. C. Ed.) 1971). Engineering Features of the San Fernando Earthquake, Earthquake En-gineering Research Laboratory Report EERL 71-02 California Institute o f Technology, Pasadena,California.

Jennings, P. C., R. B. Matthiesen, and J. B. Hoerner 1972). Forced vibrat ion of a 22-story steel framebuilding, Intern. J. of Earthquake Eng. and Struct. Dyn. 1, 107-132.

Jennings, P. C. and J. Bielak 1973). Dynamics of building-soil interaction, Bull . Seism. Soc. Am. 63,

9-48.Kobori, T., R. Minai, T. Suzuki, and K. Kusakabe 1966). Dynamical ground compliance of rect-

angular foundations, Proc. Japan Natl. Congr. Appl. Mech 16th.Kuroiwa, J. H. 1967). Vibration Test of a Multistory Building, Earthquake Engineering Research

Laboratory Report, California Institute of Technology, Pasadena, California.Luco, J. E. and R. A. Westmann 1971). Dynamic response o f circular footings, J. Eng. Mech. Div.

ASCE 97, EM5, 1381-1395.Luco, J. E. and R. A. Westmann 1972). Dynamic response of a rigid footing bonded to an elastic

half-space, J. Appl. Mech. AS ME39 527-534.Lysmer, J. and F. E. Richart 1966). Dynamic response of footings to vertical loading, J. Soil Mech.

Found. Div. ASCE 92, SM1.Nielsen, N. N. 1964). Dynamic Response of Multistory Buildings, Earthquake Engineering Research

Laboratory Report, California Insti tute of Technology, Pasadena, California.Novak, M. 1973). Vibrations of embedded footings and structures, Meeting Preprint 2029 ASCE Natl.

Struct. Eng. Meeting, San Francisco.Oien, M. A. 1971). Steady motion of a rigid strip bonded to an elastic half-space, J. of Appl. Mech.

ASME 38, 328-334.Parmelee, R. A., D. S. Perelman, and S.-L. Lee 1969). Seismic response of multiple-story structures on

flexible foundations, Bull. Seism. Soc. Am. 59, 1061-1070.Richter, C. F. 1972). Seismic regionalization or zoning: revision, Proc. Int. Conf. on Microzonationfor

Safe Construction--Research and Application Seattle Washington 267-282.Sarrazin, M. A. 1970). Soil-structure interaction in earthquake-resistant design, Research Report R70-59

Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massa-chusetts.

Seed, H. B. and I. M. Idriss 1970). Soil Moduli and Damping Factors for Dynamic Response Analysis,University of California at Berkeley, Report EERC 70-10 Berkeley, California.

Trifunac, M. D. 1972). Interaction of a shear wall with the soil for incident plane waves, Bull. Seism.Soc. Am. 62, 63-83.


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SOIL-STRUCTURE INTERACTION DURING THE SAN FERNANDO EARTHQUAKE 35

Tri funac, M. D. , A. G. Brady, and D. E . Huds on 1973). Analyses of S t rong-M ot ion Ear thquakeAccelerograms, Vol . I I I , P ar t G , Ear thquake Eng. Research Labora tory,Report EERL 73-85Cal i forn ia Ins t i tu te of Technology, Pasadena , Cal i forn ia .

Veletsos, A . S. and Y. T . W ei 1971). Late ral rocking vibrat io n of footings,J. Soil Mech. Found. Div.ASCE97 SM9, 1227-1248.

W ood, J . H. 1972). Analys is of the ear thquake response o f a n ine-s tory s teel frame bui ld ing dur ing theSan Fe rnando Ea r thquake ,Ea r thquake Eng inee r ing Resea rch Labora to ryReport EERL 72~4Califo rnia Inst i tute of Technology, Pasad ena, C alifornia.

DIVISION OF ENGINEERING AND APPLIED SCIENCECALIFORNIA INSTITUTE OF TECHNOLOGYPASADENA CALIFORNIA 91 10 9

Man uscript received July 26, 1974.

L N o t a t i o n

M =

C =

K =

v j =

Vo=

V

Vj t _--

m j =

b =m o .~

I o =

I t

P t ) =

Q ( t ) - ~

p

a

=

X k =

X j k ~

APPENDIX

m a s s m a t r i x o f ri g id b a s e s t r u c tu r ed a m p i n g m a t r i x o f ri g id b a s e s t r u c t u res t if f n es s m a t r i x o f r i g id b a s e s t r u c t u r ef ree - f i e ld acce le ra t ionh o r i z o n t a l d i s p l a c e m e n t o f s u p e r s t ru c t u r e a t f h f l o o r r e l a ti v e t o t h e b a s e

m a s s , e x c l u d i n g r o t a t i o n st r a n s l a t i o n o f b a s e m a s s r e l a t i v e to f r e e -f i el d m o t i o nh e i g h t o f j th s t o r y a b o v e b a s e m a s s( v ) , a c o l u m n v e c t o rr o t a t i o n o f b a s e m a s sv o + v o + h j q ~ + v jm a s s o f j th st o r yc e n t r o id a l m o m e n t o f in e r t ia o f j th m a s sb a s e m a s sc e n t r o i d a l m o m e n t o f i n e rt ia o f b as e m a s s

To I jj = l

h o r i z o n t a l i n t e r a c t i o n f o r c e b e t w e e n b a s e m a s s a n d s o i li n t e r a c t i o n m o m e n t b e t w e e n b a s e m a s s a n d s o i lm a s s d e n s i t y o f so i lP o i s s o n s r a t i o o f s o ils h e a r - w a v e v e l o c i t y o f s o ilr a d i us o f b a s e m a s ss h e a r m o d u l u s o f s o i l ( =V s Z p )k th m o d e s h a p e o f r i g i d b a s e s t r u c t u r e

j t h c o m p o n e n t o f Xk


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36 CHARLES B CROUSE AND PAUL C JENNIN GS

I I . So i l -st ruc tu re in t erac tion t r an sfe r fu nc t ion s.The t r ans fo rm pa rame te r, s , ha s beenrep laced by io9 for ap pl ica t ion in equa t ion 6) .

j , k = lj Tsk

- o 9 2 - o 9 2 1 t+ l ~ a3 K m ) ( _F j (o g) M j )j - tn

_ o92 _ o92mo + ImKhh) E Pj Og)Ij)j = l

.~_f i - - O 9 2 _ ~_ .O k .~ _2qkogk og)[o94moI ,- aa o92moK ,. , , I~ao92 hh lt + 2 a Kh hK ,.m].k = l

A o = _ [ _ o 9 2j , k = lj # k

n+ - o92I, + ImaKm ,.) Z PS Og)MJ)

j = l

-ogZ m o i P i og) iJ )j = l

+ m o ( - m 2 1 + I m a K ,.m ) f i ( - co z + Ogk + i 2 qkOgogk)].k = l

I I I . Th e m oda l quan t i t i es.

~ m i X u ) z

~ m i Yu 2i=1

z j = m j ~ rj =

~ re ,h , X , j ) 2i=1

m i X oi = l

m , h , X , ,i = l i = 1

~ mi.,~fij2i=1

T he fun c t ion s P i O9) an d aejk o9) a re

F og) = o9i2 + i2rlj o9j o9) fi ogk2 _ o9 z + i2qk ogkco)k = lk # j

F jk(og) = (ogj2 + i2qjogjog)(ogk2 + i2~lkogkcO) f i (og t2 -o92 + i2?l tmlog)l = l

l # k , l ~ j

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