analysis of machine foundation vibrations state of the art by g. gazetas (1983)

8/6/2019 Analysis of Machine Foundation Vibrations State of the Art by G. Gazetas (1983)

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A n a l y s i s o f m a c h i n e f o u n d a t i o n v i b r a t io n s : s t a te o ft h e a r tG E O R G E G A Z E T A SRen sse la er Po l y t ec h n i c l n s t i t u t e , T ro y , Ne w Y o rk , U S ATh e p ap er r ev i ews th e s t a t e -o f - th e -a r t o f an a ly s in g th e d y n am ic r esp o n se o f fo u n d a t io n s su b jec t edto m ach in e- ty p e l o ading s . Fo l lo win g a b r i e f o u t l i n e o f t h e h i s to r i ca l d ev e lo p m en t s in t h e f i e ld , t h eco n cep t s as so c i a t ed wi th t h e d ef in i t i o n , p h y s i ca l i n t e rp re t a t i o n an d u se o f t h e d y n am ic im p ed an cefu n c t io n s o f fo u n d a t io n s a re e lu c id a t ed an d th e av a il ab le an a ly t i ca l / n u m e r i ca l m e th o d s fo r t h e i rev a lu a t io n a re d i scu ssed . Gro u p s o f c ru c i a l d im en s io n less p ro b lem p aram ete r s r e l a t ed t o t h e so ilp ro t~ il e an d th e fo u n d a t io n g e o m et ry a re i d en t i f i ed an d th e i r e f f ec t s o n t h e r esp o n se a re s tu d i ed .Resu l t s a r e p resen t ed i n t h e fo rm o f s im p le fo rm u lae an d d im en s io n less g rap h s fo r b o th t h e s t a t i can d d y n am ic p ar t s o f im p ed an ces , p e r t a in in g to su r f ace an d em b e d d ed fo u n d a t io n s h av in g c i r cu la r ,s t r ip , r ec t an g u la r o r a rb i t r a ry p l an sh ap e an d s u p p o r t ed b y th ree t y p e s o f i d ea l i zed so i l p ro fd es : t h eh a l f sp ace , t h e s t r a tu m -o v er -b e d ro ck an d th e l ay er -o v er -h a l f p ace . Co n s id era t i o n is g iv en to t h e e f f ec t so f i n h o m o g en e i ty , an i so t ro p y an d n o n - l i n ear i t y o f so i l . Th e v ar io u s r esu l t s a r e sy n th es i zed in a cases tu d y r e fe r r in g t o t h e r esp o n se o f tw o r ig id m assive fo u n d a t io n s , an d p rac t i ca l r eco m m e n d a t io n s a rem ad e o n h o w to i n ex p en s iv e ly p red i c t t h e r esp o n se o f fo u n d a t io n s su p p o r t ed b y ac tu a l so il d ep o s i t s .

I N T R O D U C T I O NTh e b as i c g o a l i n t h e d es ig n o f a m ach in e fo u n d a t io n i s t ol im i t i t s m o t io n to am p l i t u d es wh ich wi l l n e i t h e r en d an g erth e sa t i s f ac to ry o p era t i o n o f t h e m ach in e n o r wi l l t h eyd i s tu rb t h e p eo p le wo rk in g in t h e im m ed ia t e v i c in it y . Th u s ,a k ey in g red i en t t o a su ccess fu l m ach in e fo u n d a t io n d es ig ni s t h e ca re fu l en g in eer in g an a ly si s o f th e fo u n d a t io n r esp o n set o t h e d y n a m i c l o a d s f r o m t h e a n t i c ip a t e d o p e r a t i o n o f th em a c h i n e . F u r t h e r m o r e , w h e n e x c e ss iv e m o t i o n s o f a nex i s t in g fo u n d a t io n o b s t ru c t t h e o p era t i o n o f t h e su p -p o r t ed m ach in ery , an a ly s i s i s n ecessa ry i n o rd er t o u n d er -s t an d th e cau ses o f t h e p ro b lem an d h en ce t o g u id eap p ro p r i a t e r em e d ia l ac t io n .Th e th eo ry o f an a ly s in g th e fo rced v ib ra t i o n s o f sh a llo wan d d eep fo u n d a t io n s has ad v an ced r em ark a b ly in t h e l as t1 5 y ear s an d h as cu r r en t ly r each ed a m atu re s t a t e o fd e v e l o p m e n t . A n u m b e r o f f o r m u l a t i o n s a n d c o m p u t e rp ro g ram s h av e b een d ev e lo p ed to d e t e rm in e i n a r a t i o n a lway th e d y n am ic r esp o n se i n each sp ec i f i c case . Nu m ero u ss tu d i es h av e b een p u b l i sh ed ex p lo r in g th e n a tu re o f as so c i-a t ed p h en o m en a an d sh ed d in g l i g h t o n t h e ro l e o f sev era lk ey p aram e te r s i n f lu en c in g th e r esp o n se . So lu t io n s a re a lsop resen t ly av a i l abl e i n t h e fo rm o f d im en s io n less g rap h s an ds im p le m ath em at i ca l ex p ress io n s f ro m wh ich o n e canread i ly es t im ate t h e r esp o n se o f su rf ace , em b ed d e d an d p i l efo u n d a t io n s o f v a r io u s sh ap es an d r i g id i t i e s , su p p o r t ed b ydeep or shal low layered so i l deposi ts . Clear ly , the curren ts t a t e -o f - th e .a r t o f an a ly s in g m ach in e fo u n d a t io n v ib ra t i o n sh as p ro g ressed su b s t an t i a l l y b ey o n d th e s t a t e o f t h e a r t o fth e l a t e 1 9 6 0 s wh ich h ad b een r ev i ewed b y Wh i tm an an dRic har t in 19671 and b y McNei l in 1969 . 2

In ad d i t i o n t o t h e se l ec t i o n an d ap p l i ca t i o n o f an aly s isp ro ced u res t o p red i c t t h e r esp o n se , th e d es ig n o f a m ach in efo u n d a t io n in v o lv es (1 ) t h e es t ab l i sh m en t o f p e r fo rm an cecr i t e r i a , (2 ) t h e d e t e rm in a t io n o f d y n am ic l o ad s , an d (3 )

* Presented at the International Conference on Soil Dynamics andEarthquake Engineering, he ld at the University of Southampton,England, 13-15 July 1982.

th e es t ab l i sh m en t o f t h e so i l p ro f i l e an d ev a lu a t io n o fc r i t ica l so i l p ro p er t i es . G rea t p ro g ress h as a l so b een m ad e inc u r r e n t y e a r s i n d e v e l o p i n g / n s i tu an d l ab o ra to ry t es t i n gp ro ced u res t o o b t a in r ep resen t a t i v e v a lu es o f d y n am ic so i lp a ram ete r s ; a co m p reh en s iv e r ev i ew o f t h e av a il ab le ex p er i -m e n t a l m e t h o d s h a s b e e n p r e s e n t e d b y W o o d s , 3 w h i leO z a y d i n et a l . , 4 Wo o d s s an d Rich a r t 6 h av e su m m ar i zedth e p resen t k n o wled g e o n th e f ac to r s i n f lu en c in g th ed y n am ic so i l p a ram ete r s . Th ese d ev e lo p m en t s i n d e t e rm in -in g m ate r i a l p ro p er t i es co m p lem en t t h e ad v an ces i nan a ly s in g fo u n d a t io n v ib ra t i o n s , an d p ro v id e co n s id erab l eju s t i f i ca t i o n fo r t h e u se o f so p h i s t i ca t ed n u m er i ca l fo rm u la -t i o n s i n t h e d es ig n o f m ach in e fo u n d a t io n s .On th e o th e r h an d , l i t t l e i f an y p ro g ress h as b een m ad ein r e l i ab ly es t im at in g d y n am ic m ach in e l o ad s an d im p ro v in g( th ro u g h ca l ib ra t i o n wi th f i e ld d a t a ) t h e av a i l ab l e p er fo rm -ance cr i ter ia . The s tate-of- the-ar t in these two areas hasrem ain ed essen t i a l l y u n ch an g ed d u r in g th e l as t d ecad e ;r e fe ren ce i s m ad e to McNeil 2 an d Rich ar t , W o o d s an d Hal l 7fo r co m p reh en s iv e r ev i ews o f t h ese su b jec ts .An ad d i t i o n a l an d o f t e n o v er lo o k ed s t ep i n m ach in efo u n d a t io n d es ig n i s t h e p o s t - co n s t ru c t io n o b serv a t io n o ft h e f o u n d a t i o n p e r f o r m a n c e a n d i t s c o m p a r i s o n w i t h t h ep red i c t ed fo u n d a t io n b eh av io r . Su ch co m p ar i so n s a ren eed e d to ca l i b ra te n ew an a ly s i s p ro ced u res - a n essen t ia lt ask i n v iew o f t h e s im p l i fy in g assu m p t io n s o n w h ich ev enso p h i s t i ca t ed fo rm u la t io n s a re b ased .In t h e f i n a l an a ly s i s , co n f id en ce i n t h e ad v an tag es p ro -v id ed b y th e u se o f ad v an ced m eth o d s o f an a ly si s can o n lyb e g a in ed i f t h ese a re sh o wn to h av e t h e cap ab i l i t y t o p re -d i c t t h e f i e ld p er fo rm an c e o f ac tu a l m ach in e fo u n d a t io n s .Un fo r tu n a t e ly , o n ly a l im i t ed n u m b er o f case h i s to r i es h asso f a r b een p u b l i sh ed ev a lu a tin g s ta t e -o f - th e -a r t m e th o d s o fan a ly si s t h ro u g h d e t a i l ed f i e ld o b serv a t io n s .Th e o b j ec t iv e o f t h i s p ap er i s t o r ev i ew th e p resen t s t a t e -o f - th e -a r t o f d e t e rm in in g th e d y n am ic r esp o n se o f fo u n d a-t i o n s su b jec t ed t o m ach in e- ty p e l o ad in g s. Th e o u t l in e o fth e p ap er fo l l o ws th e ch ro n o lo g y o f h i s to r ica l d ev e lo p -m en t s : f ro m th e d y n am ics o f c i r cu l a r fo o t in g s r es t i n g o nth e su r f ace o f an e l as ti c h a l f sp ace t o t h e b eh av io r o f c ir -

0261-7277/83/010002-41 $2.002 S o i l Dyn a mics a n d Ea r th q u a ke En g in eerin g , 1 9 8 3 , Vo l. 2 , No . 1 1983 CM L Publications


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Analysis of machine foundation vibrations: state o f the art: G. Gazetas

cu l a r an d n o n -c i r cu l a r fo u n d a t i o n s emb ed d ed i n a l ay eredso il d ep o s i t an d , f i na l l y, t o t h e r esp o n se o f p il e s . P a r t i cu l a remp h as i s i s acco rd e d t o t h e e f f ec t s o f d imen s io n l ess g ro u p so f g e o m e t r i c a n d m a t e r i a l p a r a m e t e r s o n t h e d y n a m i cs t i ffn ess fu n c t i o n s an d o n t h e r esp o n se o f mass iv e fo u n d a-t i o n s . No rm al i zed g rap h s an d s imp le fo rmu las a r e p resen t edfo r a v a r i e ty o f id ea l i zed soi l p ro f i l e s an d fo u n d a t i o n g eo -met r i es . Th e u se o f su ch d a t a t o es t imat e t o t r an s l a t i o n a la n d r o t a t io n a l m o t i o n s o f a c t u a l f o u n d a t i o n s in p r a c t ic ei s c l ea r l y d emo n s t r a t ed an d t h e v a r i o u s r esu l t s a r e sy n -th es i zed b y mean s o f a ease s t u d y . P rac t i ca l r eco m me n d a-t i o n s a r e t h e n m a d e o n h o w t o a p p r o x i m a t e l y o b t a i nd y n a mic s t i f fn ess an d d amp in g c o ef f i c i en t s fo r ac tu a lfo u n d a t i o n s , acco u n t i n g o n ly fo r t h e mo s t c ru c i a l p a ra -m e t e r s o f t h e p r o b l e m .S in ce t h e l imi ti n g mo t io n fo r sa t i s f ac to ry p e r fo rm an ceo f a mach in e fo u n d a t i o n u su a l l y i n v o lv es d i sp l acemen ta m p l i t ud e s o f a f e w t h o u s a n d t h s o r e v e n te n - t h o u s a n d t h s o fan inch , so i l deformat ions are quasi -elas t ic , invo lv ing neg l i -g ib le n o n l i n ear i t y an d n o p erm an e n t d e fo rmat io n s . Th u s ,mo s t o f t h e so lu t i o n s r ep o r t ed h e re in as su me l in ea r i so -t ro p i c v i sco e l as t i c so f t b eh av io r , w i th a h y s t e r e t i c so i ld amp in g t o mo d e l en erg y l o s ses a t t h o se smal l s t r a i namp l i t u d es . Ho wev er , so me co n s id e ra t i o n i s a l so g iv en t oth e e f f ec t s o f so f t n o n l i n ea r i t y o n t h e v ib ra t i o n o f s t r i pfo o t i n g s u n d er s t ro n g h o r i zo n t a l an d ro ck in g ex c i t a t i o n .M o r e o v e r , t h e i m p o r t a n c e o f s o il a n i s o t r o p y a n d s o ili n h o m o g e n e i t y a r e a l s o c o n s i de r e d .OLD ER METHOD S OF A N A LY SISIn t h e p as t , mach in e fo u n d a t i o n s were f r eq u e n t l y d es ig n edb y r u l e s - o f - t h u m b w i t h o u t a n y a n a l y s i s o f t h e e x p e c t e dv ib ra t i o n amp l i t u d es . F o r i n s t an ce , o n e su ch d es ig n ru l eca l l ed fo r a m assive co n cre t e fo u n d a t i o n o f a t o t a l we ig h teq u a l t o a t l eas t t h r ee t o f i v e t imes t h e w e ig h t o f t h e su p -p o r t e d m a c h i n e ( s ) . A l t h o u g h s u c h a p r o p o s i t i o n m a y a tf i r s t g l an ce seem lo g i ca l , i t i s i n f ac t an o b so l e t e o n e s i n cei t i g n o res t h e e f f ec t o n t h e m o t io n o f a ll t h e o th e r v a r i ab l eso f t h e p r o b l e m ( e .g . t y p e o f e x c i t a ti o n , n a t u r e o f s u p p o r t-i n g so i l , an d so o n ) . F o r o n e t h in g , i n c reas in g t h e mass o fa f o u n d a t i o n d e c r e a s e s t h e r e s o n a n t f r e q u e n c y o f th esy s t em an d , p e rh ap s m o re imp o r t an t l y , r ed u ces i t s e f f ec t i v ed amp in g . 7 Ob v io u s ly , t h is i s n o t wh a t t h o se ap p ly in g t h eru l e h ad i n min d .

F o l l o win g t h e p io n eer in g ex p er imen ta l s t u d i es ca r r i edo u t b y t h e G e r m a n D e g e b o in t h e e a r ly 1 9 3 0 s , a n u m b e r o femp i r i ca l an a ly s i s p ro ced u res were d ev e lo p ed an d u sedex t en s iv e ly a t l eas t u n t i l t h e 1 9 5 0 s . Th ese meth o d s fo cu sedo n d e t e r m i n i n g o n l y t h e ' n a t u r a l f r e q u e n c y ' o f a f o u n d a -t i o n . To t h i s en d , t h e co n cep t s o f ' i n -p h ase mass ' an d' r e d u c e d n a t u r a l f r e q u e n c y ' w e r e d e v e l o p e d . T h e f o r m e rassu mes t h a t a ce r t a i n mass o f so il imm ed ia t e ly b e lo w th efo o t i n g m o v es as a r i g id b o d y , i n -p h ase wi th t h e fo u n d a t i o n .Th e l a t t e r p o s tu l a t es t h a t t h e ' n a tu ra l f r eq u en cy ' i s so l e lya fu n c t i o n o f t h e co n t a c t a r ea , th e so i l b ea r i n g p res su re an dt h e t y p e o f s o il .

P h y s i ca l r ea l i ty co n t r a d i c t s t h e c o n ce p t o f an ' i n -p h asemass ' . No so i l mass mo v es as a r i g id b o d y wi th t h e fo u n d a-t i o n . In s t ead , sh ear an d d i l a t i o n a l wav es eman a t e f ro m th efo o t i n g - so i l i n t e r f ace i n to t h e so i l , cau s in g o sc i l l a t i n gd e f o r m a t i o n s a t t h e s u r f a c e a n d c a r r y i n g a w a y s o m e o f t h ein p u t en erg y . Th e f ac to r s t h a t h av e an i n f l u en ce o n t h esep h e n o m e n a c a n n o t b e p o s s i b l y a c c o m m o d a t e d t h r o u g hsu ch an a r t i f i c i a l co n ce p t . I n d ee d , t h e ea r l y a t t em p t s t oo b t a in sp ec i f i c v a lu es o f t h e ' i n -p h ase mass ' were f ru s t r a t ed

b y t h e sen s i t i v i ty o f t h i s 'mass ' t o t h e fo u n d a t i o n we ig h t ,m o d e o f v i b r at i o n, t y p e o f e x c i ti n g f or c e , c o n t a c t a r e a , a n dn a tu re o f t h e u n d er ly in g so i l. Ap p a ren t l y , t h e re is ab so lu t e lyn o v a lu e in t h i s co n cep t an d i t s u se i n p rac t ice may v erywel l mislead the designer .T s c h e b o t a r i o f f ' s ' r e d u c e d n a t u r a l f r e q u e n c y ' m e t h o d ,b ased o n t h e r esu l t s o f a f ew case h i s t o r i es , wen t a s t epb e y o n d t h e o r ig i na l ' in - p h a se m a s s ' m e t h o d s ) T h e ' r e d u c e dn a t u r a l f r e q u e n c y ' w a s d e f i n e d as t h e ' n a t u r a l f r e q u e n c y 'm u l t i p li e d b y t h e s q u a r e . r o o t o f t h e a v e ra g e v e rt i c a l c o n t a c tp res su re an d was g iv en g rap h i ca l ly as a fu n c t i o n o f t h e t y p eo f so ft an d o f t h e co n t ac t a r ea . A l th o u g h t h i s meth o d w asn o t w i t h o u t m e r i t , i t w a s o f t e n i n t e r p r e t e d t o m e a n t h a t' t h e s in gle m o s t i m p o r t a n t f a c t o r in m a c h i n e - f o u n d a t i o ndesign w as the soft bea r ing press ure ' . 2 Thus , in more thano n e o ccas io n , t h e d es ign was b ased o n so i l b ea r i n g cap ac i t yv a lu es t ak en f ro m lo ca l b u il d in g co d es !In ad d i t i o n t o t h e a fo remen t io n ed d rawb ack s , t h ese o ldr u l e s w e r e o n l y c o n c e r n e d w i t h t h e r e s o n a n t f r e q u e n c y ,p r o v i d i n g n o i n f o r m a t i o n a b o u t v i b r a t i o n a m p l i t u d e s t h a ta r e p r imar i l y n eed ed fo r d es ig n p u rp o ses . As a co n seq u en ce ,su ch ru l es a r e n o w o b so l e t e an d wi l l n o t b e fu r t h e rad d ressed i n t h i s p ap er . Refe ren ce is ma d e t o R ich ar t et al. 7fo r mo re d e t a i ls o n t h e su b j ec t .Dynamic Winkler model

T h i s m o d e l w a s i n t r o d u c e d a s a n e x t e n s i o n o f th e w e l lk n o wn 'Win k l e r ' o r ' e l a s t ic su b g rad e r eac t i o n ' h y p o th es i s ,wh ich i s s t i l l r a t h e r su ccess fu l l y emp lo y ed i n so me s t a t i cso i l - fo u n d a t i o n i n t e r ac t i o n p ro b l ems . 9 In o rd er t o s imu la t eth e s t i f fn ess ch arac t e r i s t i c s o f t h e ac tu a l sy s t em, t h e mo d e lr ep l aces t h e s u p p o r t i n g so i l b y a b ed o f i n d ep en d en t e l as t i cspr ings res t ing on a r ig id base. P late bear ing tes ts , con-d u c t e d i n t h e f i e l d , f o rm th e b asi s fo r ev a lu a ti n g t h e sp r in gco n s t an t s (o f t en ca l l ed ' co e f f i c i en t s o f su b g rad e r eac t i o n ' ) .On t h e b as is o f f i e ld mea su rem en t s i n t h e US S R, Bark an 1 h as p resen t ed t ab l es an d emp i r i ca l f o rmu lae wi th w h ich o n ecan r ead i l y es t imat e d es ig n v a lu es o f t h e c o ef f i c i en t fo rsev era l t y p es o f so f t , f o r each p o ss ib l e mo d e o f v ib ra t i o n( t r an s l a t i o n a l o r ro t a t i o n a l ) . He h as a lso sh o wn th a t , i n ea chcase , t h e d y n am ic co ef f i c i en t is ap p ro x im ate ly eq u a l t o t h era t i o o f ap p l i ed p res su re i n c rem en t t o t h e r esu l ti n g d isp lace-men t d u r in g s t a t i c r ep ea t ed l o ad in g t es t s . I n t h ese t e s t ss t a t i c l o ad s ' simi l a r ' t o t h e co m b in ed d e ad an d l iv e l o ad o ft h e a c t u a l f o u n d a t i o n a r e f i r s t i m p o s e d , f o l l o w e d b yrep ea t ed s l o w lo ad in g , a t f r eq u en c i es o f t h e o rd er o f 0 .0 0 1cp s , i. e . mu ch s l o wer t h an t h o se ex p e c t ed i n r ea l i ty .I t i s ev id en t t h a t t h i s mo d e l can a t l eas t g iv e so mereaso n ab l e i n fo rma t io n o n t h e l o w- f r eq u e n cy (n ear - s t a t i c )r esp o n se o f a fo u n d a t i o n . Bu t s i nce n o r ad i a t i o n d amp in g isi n c l ud e d , t h e a m p l i t u d e o f m o t i o n a t f r e q u e n c ie s n e a rr eso n an ce c an n o t b e r ea l i s t ica l l y es t imat ed . I t h as b eenarg u ed t h a t b y n eg l ec t i n g d amp in g o n e o b t a in s co n serv a t i v ee s t im a t e s o f t h e r e s p o n s e a n d v e r y g o o d e s t im a t e s o fn a tu ra l f r eq u en c i es . I n f ac t , t h i s i s t h e p ro ced u re cu r r en t l yi n c o r p o r a t e d i n t o t h e 1 9 7 0 ' I n d i a n S t a n d a r d C o d e o f Pr ac -t i ce fo r Des ig n o f M ach in e F o u n d a t i o n s ' . H Th ere i s l i tt l eme r i t i n t h is a rg u m en t , h o wev er . F o r i n s t an ce , t h e h ig hd amp in g v a lu es p resen t i n t h e t r an s l a t i o n a l mo d e s o f vibra-t i o n ( o f t h e o r d e r o f 5 0 % o f c r i t ic a l ) d o a f f e c t th e' r eso n an t ' f r eq u en c i es , i n ad d i t i o n t o d ras t i ca l l y r ed u c in gamp l i t u d es . M o reo v er , av o id in g ' r e so n an ce ' ( b y a sa f e tyfac to r o f 2 ) i n su c h cases i s an u n fo r tu n a t e d es ig n r eco m-men d a t i o n wh ich may l ead t o an o v er ly co n serv a t i v e so lu -t i o n . In o th e r eases , e sp ec i a l ly wh e n t h e ro t a t i o n a l mo d esare o f main c o n cern , an u n sa fe d es ig n i s q u i te p o ss ib le s i n ce

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Ana l y s i s o f m ac h i ne f ounda t i on v i b r a ti ons : s ta t e o f t h e a r t: G . G az e t as

t h e ac tu a l fo u n d a t i o n s t i f fn ess a t h ig h f r eq u en c i es ma y v erywel l b e ap p rec i ab ly smal l e r t h an t h e s t a t i c s t i f fn ess u sed inthe analysis (see, fo r example, F ig . 5 ) .A n i m p r o v e d v e r s io n o f t h e d y n a m i c W i n kl e r m o d e l( ca l led 'Win k l e r -V o ig t ' mo d e l ) p laces a se t o f i n d ep e n d en tv i sco u s d amp er s i n p a ra l l e l w i th t h e i n d ep en d en t e l as t i csp r in g s t o p ro v id e t h e ' d y n am ic su b g rad e r eac t i o n ' . Acco rd -ing to Barke n an d I ly ichev , 12 th is m ode l fo rm s the basis o ft h e 1 9 71 U S S R ma ch in e- fo u n d a t i o n co d e . Ag a in , h o wev e r ,t h e mo d e l i t s e l f p ro v id es n o i n fo rma t io n o n i t s sp ri n g an dd ash p o t co ef f i c i en t s . Th ese a r e i n s t ead b ack f ig u red f ro md y n am ic p l a t e - lo ad t es t s co n d u c t ed i n t h e f i e l d . Bo th t h eo b serv ed am p l i t u d e an d f r eq u en cy a t re so n an ce a r e u t i l izedto b ack f ig u re t h e two co ef f i c i en t s . An a ly z in g t h e r esu l ts o fn u m ero u s f i e ld t e s t s , Bark an an d h is co -wo rk er s fo u n d ad i sc rep an cy b e tween t h e sp r i n g co n s t an t s b ack f ig u red f ro mreso n an ce p l a t e t e s t s an d f ro m s t a t ic r ep ea t ed l o ad in g t es t s(d esc r i b ed p rev io u s ly ) . Th ey , t h u s , r e so r t ed t o t h e ' i n -p h aseso i l mass ' co n cep t t o es sen t i a l l y match t h e mo d e l co n s t an t so b t a in ed f ro m th e tw o t y p e s o f t e s t s. Th i s ad d e d soi l massw a s f o u n d t o d e p e n d o n t h e s iz e a n d e m b e d m e n t o f t h ef o u n d a t i o n a n d o n t h e n a t u r e a n d p r o p e r t i e s o f t h e so i ld ep o s i t , f o r a g iv en m o d e o f v ib ra t i o n .I t t h e r e fo re ap p ear s t h a t t h e 'W in k l e r -Vo ig t ' mo d e l isa p u re ly emp i r i ca l o n e , r eq u i r i n g f i e l d s t a t i c an d d y n am icp l a t e - lo ad t es t s fo r e ach p ar t i cu l a r s i tu a t i o n . S u ch t es t sa r e n o t o n ly v e ry ex p en s iv e an d d i f f i cu l t t o su ccess fu ll yc o n d u c t , b u t , m o r e o v e r , t h e y y i e ld r e su l ts w h i c h c a n n o t b er e a d il y i n t e r p r e t e d a n d e x t r a p o l a t e d t o p r o t o t y p e c o n d i -t ions . I f I ma y s l igh t ly rephrase G ibson: 13

'Th e m o d e l co n sp i cu o u s ly l ack s wh a t a l l mo d e l ss h o u ld p o s s e s s - p r e d ic t iv e p o w e r . 'Th e o n ly p o ss ib l e ex p l an a t i o n fo r t h e p resen t -d ay u se o fd y n amic Win k l e r mo d e l s i n mach in e- fo u n d a t i o n an a ly s i s i st h e a c c u m u l a t i o n i n s o m e c o u n t r i e s o f a w e a l t h o f p e r t i n e n tf ield data . Such d ata , o f te n available in the fo rm of tab les , ~2can b e d i r ec t l y u t i l i zed i n p rac t i ce , t h u s av o id in g t h eb u rd e n o f p e r fo rm in g p la t e - l o ad t es ts . Ag a in , o n e sh o u ld b ev ery c a re fu l i n p i ck in g u p v a lu es fo r t h e co ef f i c i en t s f ro mpubl ished f ield data . Fo r i t i s p ract ica l ly impos sib le toen su re a s imi l a r i ty i n a l l t h e c ru c i a l p h y s i ca l an d g eo m et r i cr e s p on s e p a r a m e t e r s o f t h e n e w p r o t o t y p e a n d o f t h e o ldm o d e l f o u n d a t i o n s c h e m e s.

F U N D A M E N T A L S O F C U R R E N T M E T H O D S O FV I B R A T I O N A N A L Y S I SH i s t or ic a l pe r s pe c t i v e

M o d ern meth o d s o f an a ly s i s o f fo u n d a t i o n o sc i l l a t i o n sa t t e m p t t o r a t i o n a l l y a c c o u n t f o r t h e d y n a m i c i n t e r a c t i o nb e tw een t h e fo u n d a t i o n an d t h e su p p o r t i n g so i l d ep o s i t .C o r n e r s t o n e o f t h e d e v e l o p e d m e t h o d s i s t h e t h e o r y o fwave propagat ion in an elas t ic o r v iscoelas t ic so l id (con-t i n u u m) . Th i s t h eo ry h as seen a r emark ab l e g ro wth s i n ce1 9 0 4 , wh en Lamb p u b l i sh ed h i s s t u d y o n t h e v ib ra t i o n o fan elas t ic semi- in f in i te so l id (hal f -space) caused by ac o n c e n t r a t e d l o a d ( ' d y n a m i c B o u s s in e s q ' p r o b l e m ) . N u m e r -o u s ap p l i ca ti o n s , p r imar i l y i n t h e f i e l d s o f se ismo lo g y an dap p l i ed mech an i cs , h av e g iv en a g rea t imp e tu s i n t h ed ev e lo p men t o f t h e ' e l a s t o d y n amic ' t h eo ry . Re i s sn er i n1 9 3 6 1 4 a t t emp ted wh a t i s co n s id e red t o b e t h e f i r s t en g in -ee r i n g ap p l i ca t i o n ; h i s p u b l i ca t i o n o n t h e r esp o n se o f aver t ical ly loaded cy l indr ical d isk on an elas t ic hal fspacemark ed t h e b eg in n in g o f mo d ern so i l d y n amics . Th e so lu -t i o n was o n ly an a p p ro x im ate o n e s in ce a u n i fo rm d i s t il -

b u t i o n o f co n t ac t s t res ses was assu med fo r math em at i ca ls imp l i f i ca ti o n . N o n e th e l es s , Re i s sn er 's t h eo ry o f f e r ed amajo r co n t r i b u t i o n b y r ev ea l i n g t h e ex i s t en ce o f r ad i a t i o nd a m p i n g - a p h e n o m e n o n p r e vi o us ly u n s us p e c te d b u tt o d a y c l e a rl y u n d e r s t o o d . E v e r y t i m e a f o u n d a t i o n m o v e sagainst the so i l , s t ress waves o r ig inate at the con tact surfacea n d p r o p a g a te o u t w a r d i n t h e f o r m o f b o d y a n d s u r fa c ewav es . Th ese w av es ca r ry aw ay so m e o f t h e en erg y t r an s -m i t t e d b y t h e f o u n d a t i o n o n t o t h e so i l, a p h e n o m e n o nr e m i n i s c e n t o f t h e a b s o r p t i o n o f e n e r g y b y a v i s c ou sd a m p e r ( h e n c e t h e n a m e ) .F o r man y mass iv e fo u n d a t i o n s t h e as su mp t io n o f au n i fo rm co n t ac t s t r ess d i s t r i b u t i o n i s an u n rea l i s ti c o n e , fo ri t y i e ld s a n o n -u n i fo rm p a t t e rn o f d i sp laceme n t s a t t h e so il -fo o t i n g i n t e r f ace . To c lo se r ap p ro x imate t h e r i g id b o d ym o t i o n o f s u c h f o u n d a t io n s , a n u m b e r o f a u t h o r s in t h emid d l e 1 9 5 0 s as su med co n t ac t s t r es s d i s t r i b u t i o n s wh ichp ro d u ce u n i fo rm o r l i n ea r d i sp l acemen t s a t t h e i n t e r f ace ,u n d er s t a t i ca l l y ap p l i ed fo rce o r mo men t l o ad in g s , r e sp ec-t ively . Th us, Sung Is and Quin lan ~6 prese n ted resu l t s fo rv er t i ca l l y o sc i l l a t i n g c i r cu l a r an d r ec t an g u l a r fo u n d a t i o n swh i l e Arn o ld et al . 17 a n d B y c r o f taa s t u d i ed b o th h o r i zo n t a lan d mo men t l o ad in g o f a c i r cu l a r fo u n d a t i o n . Th ese so lu -t i o n s a r e o n ly ap p ro x im ate : i n rea l i t y th e p res su re d i s tr i b u -t i o n s r eq u i r ed t o main t a in u n i fo rm o r l i n ea r d i sp l acemen t sa r e n o t c o n s t a n t b u t v a r y w i t h t h e f r e q u e n c y o f v i b ra t i on .Th e f i r s t ' r i g o ro u s ' so lu t i o n s ap p eared ab o u t t en y ear sl a t e r wh e n t h e v ib ra t i n g so i l - fo u n d a t i o n sy s t em wasan a ly sed as a mix ed b o u n d ary -v a lu e p ro b l em, wi th p re -sc r i b ed p a t t e rn s o f d i sp l acemen t s u n d er t h e r i gid fo o t i n gan d v an i sh in g s t res ses o v er t h e r emain in g p o r t i o n o f t h esu r f ace . I n t ro d u c in g so me s imp l i fy in g as su mp t io n s r eg ard -in g t h e seco n d ary co n t ac t s t r es ses ( ' r e l ax ed ' b o u n d ary ) ,A w o j o b i et al. 19 s tu d i ed a l l p o ss ib l e mo d e s o f o sc i l l a ti o n o fr i gid c i r cu l a r an d s t r i p fo o t i n g s o n a h a l f sp ace , b y r eco u r seto i n t eg ra l t r an s fo rm t ech n iq u es . On t h e o th e r h an d ,L y s m e r2 o b t a in ed a so lu t i o n fo r t h e v e r t i ca l ax i sy m me t r i cv ib ra t i o n b y d i sc r e t i z i n g t h e c o n t ac t su r f ace i n to co n c en t r i cr i ng s o f u n i fo rm b u t f r eq u en cy -d ep e n d en t v e r t i ca l s t r es sesc o n s i s t e n t w i t h t h e b o u n d a r y c o n d i t i o n s . A c o n c e p t u a l l ys imi la r a p p ro ach w as fo n o w ed b y E lo rd u y et al. 21 fo r ver-t i ca l ly l o ad ed r ec t an g u l a r fo u n d a t i o n s .P e r h a p s e q u a l l y i m p o r t a n t w i t h t h e a f o r e m e n t i o n e dth eo re t i ca l d ev e lo p men t s o f t h i s p e r i o d was t h e d i sco v eryb y Hs i eh 22 an d b y Ly sm er2 t h a t t h e d y n a m i c b e h a v i or o f av er t i ca l ly l o ad ed mass iv e fo u n d a t i o n can b e r ep resen t ed b ya s ing le-degree-o f- f reedom 'mass-spr ing-d ashpot ' osci l la to rwi th f r eq u en c y -d ep e n d en t s t i ffn ess an d d amp in g co ef f i -c i en t s . Ly smer2 we nt a s tep far th er b y suggesting the u seo f t h e f o l lo w i n g f r e q u e n c y - i n d e p e n d e n t c o e f f i c i e n ts t oa p p r o x i m a t e t h e r e s p on s e i n th e l o w a n d m e d i u m f r e q u e n c yrange:

4 G R 3 . 4 R 2Kv = ; Cv = ~ X / ~ ( 1)1 - - v 1 - - ~i n wh ich : K v = spr ing consta n t (s t i f fness) , C v = d a s h p o tco n s t an t (d amp in g ) , R = r ad iu s o f t h e c i r cu l a r r ig id l o ad in garea , G an d v = sh ear mo d u lu s an d P o i s so n 's r a t i o o f t h eh o m o g en eo u s h a l f sp ace ( so i l ) , an d p = mass d en s i t y o f so il .N o t e t h a t t h e e x p r e s s i o n f o r K v i n eq u a t i o n (1 ) i s id en t i ca lwi th t h e ex p ress io n fo r t h e s t a t i c s t i f fn ess o f a v e r t i ca l l ylo ad ed r i gid c i r cu l a r d isk o n a h a l f sp ace .

Th e su ccess o f Ly s rn er ' s ap p ro x imat io n (o f t en ca l l ed' L y s m e r ' s A n a l o g ') i n r e p r o d u c i ng w i t h v e r y g o o d a c c u r a c yt h e a c t u a l r e s po n s e o f t h e s y s t e m h a d a p r o f o u n d e f f e c t o nth e fu r t h e r d ev e lo p m en t an d en g in eer in g ap p l i ca t i o n s o f th e

4 S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e e r i n g , 1 9 8 3 , V ol . 2 , N o . 1


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5/41

An a ly s i s o f ma c h in e fo u n d a t io n v ib r a t io n s : s ta te o f th e a r t : G . Ga z e ta si n h o mo g en eo u s o r l ay ered so f t d ep o s i t s , an d t h e f i r s ta t t em p t s h av e a l r ead y b een m ad e t o o b t a in so lu t i o n s fo rd y n am ic l o ad ed p i le g ro u p s . F o r co mp reh en s iv e li st s o fr e l a t ed r e f e r en ces , s ee Do b ry et at. , s7 Kagawa et al. , ss a n dNovak . s9I m p e d a n c e a n d c o m p l i a n c e f u n c t i o n s : d e f i n i ti o n a n dp h y s ic a l in te r p r e ta t io n

A n i m p o r t a n t s t e p i n c u r r e n t m e t h o d s o f d y n a m i canalysis o f r ig id massive machine foundat ions i s the deter-min a t i o n (u s in g an a ly t i ca l o r n u mer i ca l me th o d s ) o f th ed y n a m i c i m p e d a n c e f u n c t i o n s , K ( ~ ) , * o f a n ' a ss o c i a te d 'r ig id b u t massl es s fo u n d a t i o n , a s a fu n c t i o n o f t h e ex c i ta -t ion f requ enc y , ~o . As sh own in F ig. 1 the 'asso ciated 'fo u n d a t i o n - so f t sy s t em i s i d en t i ca l ( i n b o th mate r i a l p ro p -e r t i e s an d g eo me t ry ) wi th t h e ac tu a l sy s t em, ex cep t t h a tt h e fo u n d a t i o n mass i s t ak en eq u a l t o ze ro . I t w i l l b eex p l a in ed i n t h e fo l l o win g sec t i o n h o w, o n ce th e h a rmo n icresp o n se o f su ch a mass l es s fo u n d a t i o n h as b een d e t e r -min ed , t h e s t ead y - s t a t e r esp o n se o f t h e mass iv e fo u n d a t i o n ,o r o f a n y s t r u c t u r e s u p p o r t e d o n i t , m a y b e e v a l u a t e du s ing s tan d ard p ro ced u res . I n ad d i t i o n , t h e t r an s i en tr esp o n se t o n o n -h arm o n ic m ach in e fo rces can a lso b eev a lu a t ed b y r eco u r se t o F o u r i e r an a ly s i s an d sy n th es i st ech n iq u es .F o r e a c h p a r t i c u la r h a r m o n i c e x c i t a t i o n w i th f r e q u e n c y~o , t h e d y n amic imp ed an ce i s d e f i n ed as t h e r a t i o b e tweenth e s t ead y - s t a t e fo rce (o r mo men t ) an d t h e r esu l t i n g d i s -p l acem en t (o r ro t a t i o n ) a t t h e b ase o f t h e mass les s fo u n d a-t i o n . F o r ex am p le , t h e v e r ti ca l imp ed an ce o f a fo u n d a t i o nw h o s e p l a n h a s a c e n t e r o f s y m m e t r y is d e f i n e d b y : *

R ~ ( t )x o = - - ( 3 )V( t )

i n wh ich R ~ ( t ) = R v ex p ( i6 o t) i s t h e h a rm o n ic v e r t i ca l f o r ceap p l i ed a t t h e b ase o f t h e d i sk , an d v ( t ) = v ex p (i~ot) is theu n i fo rm h arm o n ic se t t l em en t o f t h e so i l - fo u n d a t i o n i n t e r -face. I t i s ev iden t that R~ is the to tal so f t rea ct ion againstt h e fo u n d a t i o n ; i t i s mad e u p o f t h e n o rmal s t res ses ag a in stt h e h asem at p lu s , in case o f em b ed d e d fo u n d a t i o n s , t h eshear s t resses along the ve r t ical s ide walls , as i l lus t rated inFig. 1.S imi l a r l y o n e may d efme t h e t o r s i o n a l imp ed an ce , K t ,f r o m t h e t o r s io n a l m o m e n t a n d r o t a t i o n ; t h e h o ri z o n t a limp ed an ces , K a , f ro m th e h o r i zo n t a l f o r ces an d d isp lace-men t s a l o n g t h e p r i n c ip a l ax es o f t h e b ase ; an d t h e ro ck in gi m p e d a n c e s, K r , f r o m t h e m o m e n t s a n d r o t a t i o n s a r o u n dthe same h or iz on ta l p r incipal axes . Ho weve r , s ince hor i -zo n t a l f o r ces a lo n g t h e p r i n c ip a l ax es p ro d u ce ro t a t i o n s i nad d i t i o n t o h o r i zo n t a l d i sp l acemen t s , c ro ss -h or i zo n t a l-ro t a t i o n a l imp ed an ces K rh may a l so b e d e f i n ed ; t h ey a r e

D 0 0 0 ~ O 0 0 0B

rigid, masslessfoundationL. . . . . . . o . . -

0 0 0 0 0 ~ 0 g B 0

Fig u r e 1. Ma c h in e fo u n d a t io n a n d th e a s s o c ia te d r ig idma s s le ss fo u n d a t io n* Bold letters are use d in the tex t for impedanc es, comp liances andsome stiffness and dam ping coefficients (eq uatio n (17)); in thefigures, calligraphic characters are us ed fo r these quantities.

usual ly neg l ig ib ly smal l in case o f surface and very shal lowf o u n d a t i o n s , b u t t h e i r e f f e c t m a y b e c o m e a p p r e c ia b l e fo rg r e a t e r d e p t h s o f e m b e d m e n t .Refe r r i n g t o e q u a t i o n (3 ) , i t i s i n t e r es t i n g t o n o t e t h a td y n a mic fo rce an d d i sp l acemen t a r e g en era l l y o u t o f p h ase .I n f a c t , a n y d y n a m i c d i s p l a c e m e n t c a n b e r e so l ve d in t o t w oc o m p o n e n t s : o n e i n p h a s e a n d o n e 9 0 o u t o f p ha s e w i t hth e imp o sed h armo n ic l o ad . I t i s co n v en i en t t h en t o i n t ro -d u ce co m p lex n o t a t i o n t o r ep rese n t fo rces an d d isp lace-men t s . As a co n seq u en ce , imp ed an ces may a l so b e wr i t t enin t h e fo rm :*

K a ( w ) = Ka ]( ~o ) + i K a 2 ( w ) ( 4 )a = v , h , r , h r , t ; i = x / = l

T h e r e a l a n d i m a g i n a r y c o m p o n e n t s a r e b o t h f u n c t i o n so f t h e v i b r at i o na l f r e q u e n c y t o . T h e r e a l c o m p o n e n t r e f l e c t st h e s t if fn ess an d i n e r t i a o f t h e su p p o r t i n g so il ; i t s d ep en -d en ce o n f r e q u en cy is a t t r i b u t ed so l e ly t o t h e in f l u en cewh ich f r eq u en c y h as o n i n e r t ia , s i nce so il p ro p er t i e s a reessen t i a l l y f r eq u en cy i n d ep en d en t . Th e imag in ary co m-p o n e n t r e f l ec t s t h e r ad i a t i o n an d mate r i a l d amp in g o f t h esy s t em. T h e fo rmer , b e in g t h e r esu l t o f en erg y d i ss i p a t io nb y w av es p ro p ag a t i n g aw ay f ro m th e fo u n d a t i o n , i s f r e -q u en cy d ep en d en t ; t h e l a t t e r , a r i s i n g ch i e f l y f ro m th eh y s t e r e t i c cy c l i c b eh av io r o f so i l, i s p r ac t i ca l l y f r eq u en cyi n d e p e n d e n t .

A v ery i n s t ru c t iv e an a lo g y b e twe en t h e d y n am ic r esp o n seo f a s imp le 1 -d o f o sc i l l a t o r an d o f a t h r ee -d imen s io n a lmassl es s fo u n d a t i o n . so f t sy s t em h as b een d rawn b y Ro esse t .6A s s u m i n g a h a r m o n i c e x c i t a t i o n P ( t ) = P o e x p ( i w t ) , t h es t ead y - s t a t e r esp o n se x ( t ) = Xo e x p ( i ~ t ) o f t h e 1 - d o f o sc il-l a t o r m a y b e o b t a i n e d b y s u b s t i tu t i o n i n t o e q u a t i o n ( 2 ) ;P ( t )( K - - m ~ : ) + i C ~ = ( 5)x ( t )

Co n t ras t i n g eq u a t i o n s (5 ) an d (3 ) p ro mp t s t h e d e f i n i t i o no f a d y n a m i c i m p e d a n c e f u n c t i o n f o r t h e 1 - d o f m a ss -sp r in g -d ash p o t sy s t em:K = ( K - - m ~ 2) + iCco (6)

a n d , b y c o m p a r i s o n w i t h e q u a t i o n ( 4 ) :K 1 = K - - m w 2 (7 )K2 = C~o (8)

I n o t h e r w o r d s , t h e d y n a m i c i m p e d a n c e o f o u r fa m i l ia r1 -d o f o sc i l l a t o r i s i n d eed a co mp lex n u mb er wi th a f r e -q u en cy d ep en d e n t r ea l p a r t r ep resen t i n g t h e s t i ffn ess an din er t i a ch arac t e r i s t i c s o f t h e sy s t em, an d a f r eq u en cyd ep e n d en t imag in ary p a r t ex p ress in g t h e en erg y l o s s in t h esy s t em. Th e re fo re , i t is q u i t e n a tu ra l t o e x p ress t h e d y n a micimp ed a n ce o f so f t -fo o t in g sy s t ems i n a co mp lex fo rm, asd o n e i n eq u a t i o n (4 ) .Hav in g , t h u s , e s t ab l i sh ed t h e an a lo g y b e twe en 1 -d o f an dmassl es s fo o t i n g -so f t sy s t ems , l e t eq u a t i o n (6 ) fo r t h e1- d o f b e r e w r i t te n a s :

o r

K = K . {(1 - - ~---2]+ i2~ } (9a)

K = K . { k + i w c s } ( 9 b )in wh ich t h e c r i t ica l v i scou s d amp in g r a t i o i s:

C C. . . . ( 1 0 )Cc r 2 K/~o n

6 S o i l Dy n a m ic s a n d Ea r th q u a k e En g in e e r in g , 1 9 8 3 , Vo l. 2 , No . 1


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An a lys i s o f ma ch in e f o u n d a t i o n v ib rat i on s : s t a t e o f t h e a r t : G . Ga ze ta s

! !

~I ~ ~ ~/~n----.~-I i I Ili ~ I 0 , 5 1

Fig u re 2 . Dy n a m ic s ti f f n ess a n d d a mp in g co e f f i c i en t s o fa I - d o f s i m p l e o s c il la t or

t h e n a t u r a l f re q u e n c y ~ n = (K /m ) 1 /2 ,k = ( 1 - ~ 2 / co n ) a n dc s = C/K. E q u a t i o n ( 9 b ) i m p l ie s t h a t t h e d y n a m i c i m p e d-an ce o f a 1 -d o f s imp le o sc i l l a t o r may b e ex p ressed as ap r o d u c t o f t h e s p ri ng c o n s t a n t K , w h i c h h a p p e n s t o b e th es t a ti c s t i ff n e ss o f t h e s y s t e m , t im e s a c o m p l e x n u m b e rk + i o c s , w h ich e n co mp a sses t h e d y n am ic ch arac t e r i s t ic so f t h e sy s t em ( i n e r t i a an d v i sco u s d amp in g ) an d i s h e re -a f t e r ca l l ed ' d y n a m i c p a r t ' o f t h e i m p e d a n c e . A t z e r of r e q u e n c y t h e d y n a m i c p a r t b e c o m e s a r e a l n u m b e r , e q u a lt o 1 , an d t h e imp ed an ce co in c id es wi th t h e s t a t i c s t i f fn essK o f t h e s imp le sy s t em, k an d cs a r e n amed r esp ec t i v e lys t i ffn ess an d d am p in g co ef f i c i en t s an d t h e i r v a r i a t io n wi thf r eq u en cy fo r t h e 1 -d o f ' s i s p lo t t ed i n F ig . 2 . No t i ce t h a tk d ecreases as a seco n d d eg ree p a rab o l a wi th i n c reas in gt~, w h ereas c s r em ain s c o n s t an t .I t sh o u ld n o t su rp r ise t h e r ead er t h a t t h e ac tu a l v a ri a-t i o n w i th t~ o f t h e s t i ffn ess an d d am p in g co ef f i c i en t s , k van d csv , o f a v e r t i ca l l y v ib ra t i n g c i r cu la r d i sk o n an e l as t ich a l f sp ace i s i n d eed v ery s imi l a r t o t h e v a r i a t i o n o f th ek an d c s o f t h e 1 -d o f sy s t em! ( To see t h i s s imi l a r i t y j u s tcom pare F ig . 2 to F ig . 5 (a) . ) Ho we ver , in general , k and c so f a fo u n d a t i o n - so i l sy s t em may v ary i n a r a t h e r co mp l i -c a t e d m a n n e r w i t h c o, d e p e n d i n g p ri m a r i ly o n t h e m o d eo f v i b ra t i on , t h e g e o m e t r y , r i gi d it y a n d e m b e d m e n t o f t h efo u n d a t i o n , an d , t 'mal ly , t h e p ro f i l e an d p ro p e r t i e s o f t h esu p p o r t i n g so i l d ep o s i t . F ig u res 5 , 8 , 9 , 1 0 an d 2 0 may b ep rev i ewed t o co n f 'L rm th i s s t a t em en t . No n e th e l es s , i n a l lcases , t h e d y n a mic im p ed an ce fu n c t i o n s can b e ex p ressedas p ro d u c t s o f a s t a t i c an d a d y n amic p a r t , a s d esc r i b edb y eq u a t i o n (9 b ) . A l t e rn a t i v e ly , a d imen s io n l es s f r eq u en cyf a c t o r is o f t e n i n t r o d u c e d :

a o = - - ( 1 1 )Vsi n w h ich : B = a c r i t i ca l f o u n d a t i o n d imen s io n l i k e , e.g. ,t h e r ad ius o f a c i r cu l a r fo u n d a t i o n o r h a l f t h e wid th o f as t r i p o r a r ec t an g u l a r fo u n d a t i o n ; an d Vs = a ch arac t e r i s t i csh ear wav e v e lo c i t y o f t h e so i l . Co mb in in g eq u a t i o n s (9 b )an d (1 1 ) a l l o ws t h e imp ed an ce t o b e case i n t h e fo rm:

K = K ( k + i ao c) (1 2 )w i t h

Vs- - ( 1 3 )C=Cs BS in ce b o th ao an d c a r e d imen s io n l es s q u an t i t i e s , eq u a t i o n(1 2 ) i s s t ro n g ly p re fe r r ed t o eq u a t i o n (9 b ) i n p resen t i n g t h eresu l ts o f d y n a mic an a ly ses .L e t i t n o w b e a s s u m e d t h a t a ' h y s t e r e t i c d a m p e r ' i sad d ed i n -p ara l l e l w i th t h e sp r i n g an d t h e ' v i sco u s d amp er '

t o su p p o r t t h e mass Of t h e s imp le o sc i l l a to r . Th i s d am p eri s d esc r i b ed t h ro u g h a h y s t e r e t i c d amp in g r a t i o , ~ . Du r in ge a c h c y c l e o f m o t i o n i t d is s ip a te s a n a m o u n t o f e n e rg yp ro p o r t i o n a l t o t h e m ax im u m s t r a in en erg y , I, o f th esy s t em:A Wh = 4 n ~W (1 4 )

in wh ich W = ()Kx ~ . O n t h e o th e r h an d , d u r in g a cy c l eo f m o t i o n t h e v i s c o u s d a m p e r h a s c o n s u m e d a n a m o u n to f e n e r g y e q u a l t o :

AW~ = ~C~,~o{ D

= 4 ~ 1 3 - - W ( 1 5 )6 0 n

so t h a t t h e t o t a l d i s s ip a t ed en erg y , AW = AWh + AWv, asa f u n c t i o n o f W i s:'A "W 4" t r ( '8 ~ + ~ ( 1 6 )

Th i s ex p re ss io n su g ges ts t h a t t h e s imp le ad d i t i o n ru l e ,+ ~% o /wn , m ay b e u sed t o o b t a in t h e ' e f f ec t i v e ' d amp in grat io o f a syste m possessing bo th v iscous, 13, and hy ster et ic ,~ , dam ping . A w~orating found at ion-o n-so i l i s one suc hsy s t em, wi th i t s r ad i a t i o n d am p in g b e in g o f a v isco u s n a tu rewh i l e t h e mate r i a l d am p in g is o f t h e h y s t e r e t i c t y p e .T h e p r e s e n c e o f m a t e r i a l d a m p i n g i n t h e s o il a f f e c t s b o t hth e s t i ffn ess an d d am p in g co ef f i c i en t s , k an d c . I n ana t t e m p t t o i s o la t e t h e e f f e c t s o f h y s t e r e ti c m a t e r ia l d a m p -in g , an a l t e rn a t i v e ex p res s io n to eq u a t i o n (1 2 ) i s o f t enu s e d f o r th e d y n a m i c i m p e d a n c e :

K = K ( k + i a o c ) . ( 1 + 2 i ~ ) ( 1 7 )Reca l l in g t h e so -ca ll ed ' co r r esp o n d en c e p r i n c ip l e , 6~ o n em ay an t i c i p a t e t h a t t h e n ew co ef f i c i en t s , k an d c , a r ein d ep en d e n t o f mat e r i a l d amp in g . I f t h is were t ru e , itwo u ld t h en b e su f f i c i en t t o o b t a in so lu t i o n s fo r a p u re lye l as ti c so i l an d t h en ex t r ap o l a t e t h e r esu l t s t o so i ls w i tha n y h y s t e r e t i c d a m p i n g r a t i o b y m u l t i p l y in g th e u n d a m p e dimp ed an c es b y 1 + 2 i~ . I n d ee d , fo r v e ry d eep so i l d ep o s i t swh ich can b e mo d e l ed as a h a l f sp ace t h e ab o v e ' p r i n c ip l e 'i s r easo n ab ly accu ra t e an d h as b een r ep ea t ed ly u ti l i zed t oobta in so lu t io ns fo r dam ped so il s . 29 '62 ,6a How ever , in the

case o f a sh a l lo w s t r a tu m o n r i gid ro ck b o th k an d c a ref a i r l y sen s i t i v e t o t h e as su med mate r i a l d amp in g r a t i o ( seeFig . 9 , fo r example) ; th is d iscred i t s to a large ex ten t the' co r r esp o n d en c e p r i n c ip l e ' , a s K au se l 3a h ad f i r s t n o t i ced .No n e t h e l es s , i t is co n v e n i en t t o ex p ress t h e imp ed an cefu n c t i o n s i n t h e fo rm o f eq u a t i o n (1 7 ) , an d t h i s p r ac t i ce i sf r eq u en t l y fo l l o w ed i n t h e seq u e l . A l t e rn a t i v e ly , h o we v er ,eq u a t i o n (1 2 ) i s a l so u sed i n so me cases .D y n a m i c c o m p l i a n c e f u n c t i o n s

Also g iv en t h e n ames d y n amic ' d i sp l acemen t ' f u n c t i o n san d d y n amic ' f l ex ib i l i t y ' f u n c t i o n s , t h ey a r e es sen t i a l l y t h er a t io s b e t w e e n d y n a m i c d i s p l a c e m e n t s ( o r r o t a t i o n s ) an dt h e d y n a m i c r e a c t iv e fo r c e s ( o r m o m e n t s ) a t t h e b a s e o f afo u n d a t i o n . Th ey were f i r s t i n t ro d u ce d b y R e i s sner . ~4F o l lo win g t h e p rev io u s d i scu ss ion , i t i s co n v en i en t t oe x p r e s s e a c h c o m p l i a n c e u si n g c o m p l e x n o t a t i o n :F a = Fal(o + iFa2(co ) ( l g )

a = v , h , r , h r , tTh e r ea l an d im ag in ary p a r t s r ep resen t t h e d i sp l acemen tco m p o n e n t s wh ich a r e i n -p h ase an d 9 0 -o u t .o f -p h ase wi th

S o i l Dyn a mics a n d Ea r th q u a ke En g in eer ing , 1 9 8 3 , Vol. 2 , No . 1 7


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A n a l y s i s o f m a c h i n e f o u n d a t i o n v i b ra t i on s : s t a t e o f t h e a r t: G . G a z e t a s

t h e r eac t iv e fo rce , r esp ec t iv e ly , an d t h e y b o th a re fu n c t i o n so f f r eq u en cy , as d i scu ssed i n d e t a i l p r ev io u s ly . Fo r af o u n d a t i o n w h i c h i n p l a n ha s a c e n t e r o f s y m m e t r y , t h ev er t i ca l an d t o r s io n a l co m p l i an ces a r e s im p ly t h e i n v er seo f t h e v er t i ca l an d t o r s io n a l im p ed an ces :1F b = - - ; b = v , t ( 1 9 a )K t,

Ho wev er , d u e t o t h e co u p l in g b e tween ro ck in g an dsway in g m o t io n s , t h e co r r esp o n d in g co m p l i an ces sh o u ldb e o b t a i n e d b y i n v e r ti n g th e m a t r i x o f i m p e d a n c es :

Th e fo l l o win g a l t e rn a t i v e fo rm to eq u a t io n (1 8 ) i s a l sof r eq u e n t ly u sed in p resen t in g co m p l i an ce fu n c t i o n s :1Fo = ~ [ f . l (~o) + i f .2(~o)] (20)

w h e r e K a is t h e co r r esp o n d in g s t a t ic s t if fn ess.C o m p u t a t i o n a l p r o c e d u r e s f o r d e t e r m i n i n gi m p e d a n c e / u n c t i o n s

Sev era l a l t e rn a t i v e co m p u ta t i o n a l p ro ced u res a r e p re -sen t ly av ai lab le t o o b t a in d y n a m ic im p ed an ce fu n c t io n s fo reach sp ec i f ic m ac h in e- fo u n d a t io n p ro b l em . T h e ch o i ceam o n g th ese m e th o d s d e p en d s t o a l arge ex t en t o n th ereq u i r ed accu racy , wh ich i n t u rn i s p r im ar i l y d i c t a t ed b yth e s ize an d im p o r t a n ce o f t h e p ar t i cu l a r p ro j ec t . Fu r th er -m o r e , t h e m e t h o d t o b e s e l e ct e d m u s t r e f l e c t t h e k e ych arac t e r i s t i c s o f t h e fo u n d a t io n an d t h e su p p o r t i n g so i l .Sp ec i f i ca l l y , o n e m ay b ro ad ly c l as s i fy so i l - fo u n d a t io nsy s t em s acco rd in g t o t h e fo l l o win g m ate r i a l an d g eo m e t rycharacter is t ics :

1 . Th e sh ap e o f t h e fo u n d a t io n ( c i r cu la r , s t r ip , r ec t -angular , arb i t rary) .2 . T h e t y p e o f so il p ro f il e (d eep u n i fo rm d ep o s i t , d eepl ay ered d ep o s i t , sh a ll o w l ay ered s t r a tu m o n ro ck ) .3 . T h e a m o u n t o f e m b e d m e n t ( s u rf a c e f o u n d a t i o n ,e m b e d d e d f o u n d a t i o n , d e e p f o u n d a t i o n ) .4 . Th e f l ex u ra l r ig id i ty o f t h e fo u n d a t io n ( r ig id fo u n d a-t i o n , f l ex ib l e fo u n d a t io n ) .

T w o c o m p u t a t i o n a l l y d i f f e r e n t a p p r o a c h e s h a v e b e e nfo l l o wed o v er t h e y ear s t o o b t a in t h e d y n am ic im p ed an ceso f fo u n d a t io n s wi th v ar io u s ch arac t e r i s t i c s : a ' co n t in u u m 'ap p ro ach , wh ich l ed t o t h e d e v e lo p m en t o f an a ly t i ca l an dsem i .an a ly t i ca l fo rm u la t i o n s , an d a ' d i sc re t e ' ap p ro ach ,wh ich r esu l t ed i n t h e d ev e lo p m e n t o f f i n i t e -d i f f e ren ce an d ,p r im ar i l y , f i n i t e - e l em en t m o d e l s . In t h e p as t (m id -1 9 7 0 s ) ,co n s id erab l e co n t ro v er sy was h e ld ab o u t t h e r e l a t i v em er i t s an d d ef i c i en c i es o f each ap p ro ach an d so m e ex t r em ean d u n ju s t i f ied p o s i t i o n s were ad v o ca t ed . T o d ay , it is q u i tec l e a r t h a t b o t h p r o c e d u r e s , i f c o r r e c t l y u n d e r s t o o d a n dim p lem en ted , a r e v e ry u se fu l t o o l s i n an a ly s in g t h e b eh av io ro f d y n am ica l l y l o ad ed fo u n d a t io n s . Mo reo v er , t h ey y i e ldvery s imi lar resu l ts i f they are appropr iately used to so lveth e sam e p ro b l em . Had j ian e t a l . 64 an d Jak u b e t a l . 6s h av ep resen t ed ex ce l l en t d i scu ss io n s an d co m p ara t i v e s t u d i es o nth i s su b j ec t . Th e fo l l o win g p arag rap h s i n t en d t o r a th erb r i e f l y i n t ro d u ce t h e m o s t im p o r t a n t an a ly t i ca l , s em i-an a ly t i ca l an d n u m er i ca l p ro ced u res wh ich a re cu r r en t l yavai lab le to the machine-foundat ion analyst . The l i s t i s byn o m ean s ex h au s t i v e , an d t h e em p h as i s i s o n d i scu ss in g t h es t r on g a n d w e a k p o i n ts o f e a c h m e t h o d .

' C o n t i n u u m ' m e t h o d s . Star t ing p oin t o f al l the devel -o p ed fo rm u la t i o n s is t h e an a ly t i ca l so lu t i o n o f t h e p er t i n en twav e eq u a t io n s g o v ern in g t h e im p o sed d efo rm at io n s i n eachu n i fo rm so il l ay er o r h a lf sp ace . Ho wev er , th e b o u n d aryco n d i t i o n s a t t h e so i l -fo o t in g i n t e r f ace a re h an d led d i f f e r -e n t l y b y t h e v a r io u s m e t h o d s . I n t h a t r e s p e c t , o n e m a y v e r yb ro ad ly c l as s ify t h e av a il ab le co n t in u u m fo rm u la t i o n s i n toan a ly t i ca l an d sem i -an a ly t ica l so lu t i o n s .T h e k n o w n a n a l y t i c a l so lu t i o n s s im p l i fy t h e m ech an ica lb eh av io r o f t h e so i l - fo o t in g co n t a c t su r f ace b y assu m in g a' r e l ax e d ' b o u n d ary . Th a t is , n o f r i c t i o n a l sh ear t r ac t i o n s cand ev e lo p d u r in g v er t i ca l an d ro ck in g v ib ra t i o n s , wh i l e fo rh o r i zo n t a l v ib ra t io n s t h e n o rm al t r ac t i o n s a t t h e i n t e r f aceare as su m ed to b e ze ro . Th i s as su m p t io n h as b een n ec essa ryt o a v o id t h e m o r e c o m p l e x m i x e d b o u n d a r y c o n d i ti o n sresu l ti n g f ro m th e co n s id era t i o n e i t h e r o f a p e r f ec t a t tach -m e n t b e t w e e n f o u n d a t i o n a n d s o i l ( ' r o u g h ' f o u n d a t i o n ) o ro f a c o n t a c t o b e y i n g C o u l o m b ' s f r i c ti o n l a w (a n e v e n m o r ereal i s t ic ideal izat ion) .By r eco u r se t o i n t eg ra l t r an s fo rm t ech n iq u es ( i n v o lv -i n g H a n k e l o r F o u r i e r t r a n s f o r m s f o r a x i s y m m e t r i c o rp l an e- s tr a in g eo m et r i es , r esp ec t i v e ly ) t h e r e l ax ed b o u n d aryco n d i t i o n s y i e ld se t s o f d u a l i n t eg ra l eq u a t io n s fo r eachm o d e o f v ib ra t io n . Each se t is t h en r ed u c ed t o a F red h o lmin t eg ra l eq u a t io n wh ich i s f i n a l l y so lv ed n u m er i ca l ly .Su ch an a ly t i ca l so lu t i o n s h av e so f a r b een p u b l i sh ed fo rsu r f ace c i r cu l a r an d s t r i p fo u n d a t io n s o f i n fin i t e f l ex u l a rr i g id it y su p p o r t ed b y an e l as t ic o r v i sco e l as ti c h a l fspa ce ;26-29fo r c i r cu l a r fo u n d a t io n s o n a l ay ered e l as t i c o r v i sco e l as t i cso i l d ep o s i t ; 3 s '~ fo r c i r cu l a r fo u n d a t io n s o f f i n i t e f l ex u ra tr i g id i t y su p p o r t ed o n a h a l f sp ace ;49 fo r c i r cu l a r fo u n d a t io n so n a c ro ss -an i so t ro p i c h a l fsp ace ;67 and even f or ve r t ical lylo ad ed r i gid r ec t an g u la r fo u n d a t io n s o n a h a l f sp ace . 4 s

T h e s e m i - a n a l y t i c a l t y p e so lu t i o n s a re b ased o n t h ed e t e r m i n a t i o n o f t h e d i s p la c e m e n t s a t a n y p o i n t w i t h in t h efo o t in g - so i l i n t e r f ace , cau sed b y a u n i t n o rm al o r sh eart i m e - h a r m o n i c f o r c e a p p l ie d a t a n o t h e r p o i n t o f t h e s a m ein t e r f ace . Th en , b y p ro p er ly d i sc re ti z in g t h e co n t ac t su r-f a c e , th e m a t r i x o f d y n a m i c i n f l u e n c e o r G r e e n ' s f u n c t i o n si s as sem b led an d t h e p ro b l em i s so lv ed a f t e r im p o s in g t h er ig id -b o d y m o t io n b o u n d ary co n d i t i o n s . Sev era l d i f f e r en tt ech n iq u es ( i n es sen ce d i f f e ren t i n t eg ra t i o n p ro ced u res )h a v e b e e n f o r m u l a t e d t o c a r r y o u t t h e s e s t e p s o f t h ean a ly s i s . Fo r ex am p le , E lo rd u y et al . 21 an d Wh i t t ak ere t a l . s u t i l i zed Lam b ' s so lu t i o n fo r a p o in t l o ad ed h a l f -sp ace ; Lu co e t a l f l 7 o b ta in ed p a i r s o f Cau ch y ty p e i n t eg ra leq u a t io n s wh ich t h ey n u m er i ca l l y so lv ed a f t e r r ed u c in g t oco u p led F red h o lm eq u a t io n s ; Gaze t as 36 an d G aze t as e t a l . 3aut i l ized a fas t F our ier t r ans form algor i thm ; Wong 68 andW o n g e t a l . 44 u sed t h e so lu t i o n fo r a u n i fo rm ly l o ad edrec t an g l e ; an d so o n .Fo r t h e p u rp o se o f t h i s d i scu ss io n , o n e m ay l is t a s asem i -an a ly t ica l so lu t io n t h e fo rm u la t i o n o f Do m in g u ez an dRo esse t , a7 wh o ap p l i ed t h e so -ca ll ed ' b o u n d ary i n t eg ra le q u a t i o n ' o r , m o r e s i m p ly , ' b o u n d a r y e l e m e n t ' m e t h o dt o o b t a i n d y n a m i c i m p e d a n c e f u n c t i o n s o f r e c ta n g u l a rfo u n d a t io n s a t t h e su r f ace o f , o r em b ed d ed in a h a l f sp ace .To th i s en d , t h e y u t i l ized t h e c lo sed - fo rm so lu ti o n t o th e'd y n am ic K elv in ' p ro b l em o f a co n cen t r a t ed l o ad i n anin f in i t e m ed iu m , 69 an d d i sc re t ized e i t h e r o n ly t h e co n t a c tsu r f ace , i n t h e case o f su r f ace fo o t in g s wi th ' r e l ax ed 'b o u n d ar i es , o r b o th t h e co n t ac t an d t h e su r ro u n d in g so i lsu r f aces , i n t h e cases o f em b e d d ed fo o t in g s an d o f su r f acefo o t in g s ' ad h es iv e ly ' a t t ach ed t o t h e so il .So f a r r i g o ro u s sem i -an a ly t i ca l so lu t i o n s h av e b een p u b -l i sh ed fo r r ig id s t r i p fo u n d a t io n s o n t h e su r f ace o f a l ay ered

8 S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e er i n g , 1 9 8 3 , V o l. 2 , N o . 1


8/41

An a lys i s o f ma ch in e f o u n d a t i o n v ib ra ti o n s : s t a t e o f t h e a r t : G . Ga ze ta shalfspace or s t ratum -on-ro ck; as ' 38, ~9, 7o for r lg id re ctang ularfo un da tio ns on a hal fspa ce; 21' 36' 44' 46-4a' ~,6s,6a, 71 for rec t-an g u la r fo u n d a t io n s o f f i n i t e f l ex u ra l r i g id i t y ; s ' sl fo r rigidrec t an g u la r fo u n d a t io n s em b ed d ed in a h a l f sp ace ;47 and,f 'mal ly , for r igid f oun da t ion s o f arb i t rary shape. 44N o t e t h a t a p p ro x ima te semi -a n a l y t i ca l p r o c e d u r e s h a v ea l r e a d y b e e n d e v e l o p e d t o o b t a i n t h e i m p e d a n c e s o f c y l in -dr ical e mb edd ed fou nda t ion s a nd ci rcu lar p i les , a-43 ,sT 'TzT h e s e p r o c e d u r e s a s s u m e t h a t o n l y h o r i z o n t a l l y p r o p a -g a t in g wav es g en era t e a t t h e v er t i ca l fo u n d a t io n - so i l i n t e r -f a c e , a n d t h e y n e g l e c t t h e c o u p l i n g b e t w e e n f o r c e s a n dd i sp l acem en t s a t v a r io u s p o in t s . In s t ead , t h ey o n ly co m p u teth e d i sp l acem en t a t t h e p o in t o f ap p l i ca t i o n o f t h e l o ad .Th u s , i n e f f ec t , t h e so il is m o d e l ed as a Win ld er m ed iu m ,th e sp r in g an d d ash p o t ch arac t e r i s t i c s o f wh ich a re es t i -m ated f ro m rea l i s t i c , a l b e i t s im p l i f i ed , wav e p ro p ag a t io nanalyses .Final ly , several s imi lar a p p ro x ima te a n a l y t i ca l fo rm u la -t i o n s h av e b een d ev e lo p ed , ag a in fo r d eep ly em b ed d e dcy l in d r i ca l fo u n d a t io n s an d en d -b ear in g p i l es i n so i ls t r a t a . 7 3-7s Th ese p ro ced u res a t t em p t t o an a ly t i ca l l y solv eth e g o v ern in g wav e eq u a t io n s fo r t h e s t r a tu m , b y n eg l ec t i n gt h e s e c o n d a r y c o m p o n e n t o f d i s p l a c e m e n t (i .e . t h e v e r ti c a lco m p o n en t fo r l a t e r a l v ib ra t i o n s o r t h e r ad i a l o n e fo rv er t i ca l v ib ra t i o n s ) . Th e b o u n d ary co n d i t i o n s a t t h e so i l -p i le i n t e r f ace a re an a ly t i ca l l y en fo rc ed b y ex p an d in g t h eco n ta c t p ressu re d i s t r i b u t i o n t o a n i n f in i t e se ri es i n t e rm s o fth e n a tu ra l m o d es o f v ib ra t i o n o f t h e so i l l ay er .'D i scre te ' m o d e l s . Dy n am ic f i n i t e d i f f e r en ce an d f i n i t ee l e m e n t m o d e l s h a v e b e e n d e v e l o p e d f o r p r o b l e m s o fc o m p l i c a t e d g e o m e t r y w h i c h a r e n o t e a s i l y a m e n a b l e t oan a ly s i s wi th co n t in u u m ty p e , an a ly t i ca l o r sem i -an a ly t i ca lfo rm u la t i o n s . T o d ay , f i n i t e d i f f e r en ce fo rm u la t i o n s su cha s t ho s e p ro p o s e d b y A n g e t al., 79 Ag ab e in et a l . , s K r i z e ket a l . , s l an d Tsen g et a l . , s2 f r ed v ery l i t t l e i f an y ap p l i ca t i o nin so lv in g fo u n d a t io n v ib ra t i o n p ro b l em s , an d , t h e re fo re ,wi l l n o t b e fu r th er ad d ressed i n t h i s p ap er . On th e o th erh an d , sev era l f 'm it e e l em en t fo rm u la t i o n s an d co m p u te rp ro g ram s a re p resen t ly wid e ly av a i l ab l e an d f r eq u en t lyu sed i n an a ly s in g fo u n d a t io n o sc i l l a ti o n s .T h e u s e o f f i n i te e l e m e n t s i n d y n a m i c f o u n d a t i o n pr o b -l em s i s d i f f e r en t f ro m o th er ap p l i ca t i o n s o f f i n i t e e l em en t sin s t a t i cs an d d y n a m ics i n t h a t so i l s t r a t a o f i n f i n i t e ex t e n ti n t h e h o r i zo n t a l an d ev en i n t h e v er t i ca l d i r ec t i o n m u s t b erep resen t ed b y a m o d e l o f a f i n i t e s i ze . Su ch a f i n i t e m o d e lc r e a t e s a f i c ti t io u s ' b o x ' e f f e c t , t ra p p i n g t h e e n e r g y o f t h esy s t em an d d i s to r t i n g i t s d y n am ic ch arac t e r i s t i c s . To av o idth i s p ro b l em , wav e ab so rb in g l a t e r a l b o u n d ar i es a r e i n t ro -d u c e d t o a c c o u n t f o r t h e r a d i a t io n o f e n e r g y i n t o t h e o u t e rr e g io n n o t i n c l u d e d i n t h e m o d e l . T w o m a i n t y p e s o f s uc hb o u n d ar i es a r e av a i lab l e . Th e a p p ro x im ate ' v i sco u s ' b o u n -d a r y p r o p o s e d b y L y s m e r e ta l . s3 a n d e x t e n d e d b y V a l l ia p p a net a l . ~ m u s t b e p l a c e d a t s om e d i s ta n c e f r o m t h e f o u n d a -t i o n . T h e a l te r n a ti v e ' c o n s i s t e n t ' b o u n d a r y d e v e l o p e d b yWaas a t an d ex t e n d ed b y Kau se1 33 i s v e ry e f f ec t i v e i n accu r -a t e ly r ep ro d u c in g t h e p h y s i ca l b eh av io r o f t h e sy s t em , an di t a l so r esu l t s i n co n s id erab l e eco n o m y b y b e in g p l acedd i r e c t l y a t t h e e d g e o f t h e f o u n d a t i o n . T h i s ' c o n s i s t e n t 'b o u n d a r y p r o v i d e s a d y n a m i c s t i f f n e s s m a t r i x f o r t h em ed iu m su r ro u n d in g t h e p l an e o r cy l i n d r i ca l v e r t i ca l cav i tywh ich i s as su m ed to o cc u p y th e ce n t r a l r eg io n u n d er th es t r i p o r c i r cu la r fo u n d a t io n . Th i s m a t r i x co r r esp o n d se x a c t l y t o t h e b o u n d a r y s t if f ne s s m a t r i x t h a t w o u l d b eo b t a i n e d f r o m a c o n t i n u u m t y p e f o r m u l a t i o n .Un fo r tu n a t e ly , ' co n s i s t en t ' b o u n d ar i es h av e b ee n dev el -o p ed o n ly fo r p l an e- s t r a in an d ax i sy m m et r i c ( cy l i n d r i ca l )g eo m et r i es . No su ch b o u n d ary i s av a i l ab l e fo r t ru ly t h ree -

d im en s io n a l (3 D ) g eo m et r i es , i n ca r t es ian co o rd in a t es .Th u s , t o so lv e 3 D p ro b l em s a fm i t e -e l em en t m o d e l m u s tr eso r t t o ' v i sco u s ' o r e l em en ta ry b o u n d ar i es p l aced f a rawa y f ro m th e l o ad ed a rea. In t h i s way th e f i c t i ti o u s lyre f l ec t ed wav es a re d i s s ip a t ed t h ro u g h h y s t e res i s an d f r i c -t i o n (m ate r i a l d am p in g ) i n t h e so i l b e fo re t h ey r e tu rn t oth e fo u n d a t io n r eg io n . Ho wev er , t h e co s t o f su ch an a ly sesi s p ro h ib i t iv e an d t ru ly 3 D so lu t i o n s a re v ery r a re ly u sedi n p r a c t ic e . A n a t t e m p t h a s b e e n m a d e t o m o d i f y a 2 Dc o m p u t e r p r o g r a m b y a d d i n g v i s c o u s d a s h p o t s t o t h el a t e ra l f aces o f i t s p l an e- s t r a in e l em en t s , i n o rd er t o s im u -l a t e t h e r ad i a t i o n d am p in g o f 3 D s i t u a ti o n s , s s No tw i th -s t a n d in g t h e p o p u l a r i t y e n j o y e d b y t h is p s e u d o - 3 D m o d e l ,i ts o n l y d i f f e r e n c e f r o m t h e 2 D m o d e l is t h a t i t i n t r o d u c e san a r t i f i c i a l i n c rease i n d am p in g , wh ich can n o t p o ss ib lyrep ro d u ce a l l a sp ec ts o f t h e t ru e 3 D b e h av io r . In f ac t , i nso m e cases th e ac tu a l 3 D rad i a t i o n d am p in g i n ro ck in g i so v e r - e s ti m a t e d r a t h e r t h a n u n d e r - e s ti m a t e d b y a 2 D m o d e l ; ~th u s b y ad d in g v isco u s d ash p o t s t h e s i t u a t i o n m ay w o rseninstea d of improv ing , s6 , es

C o n s e q u e n t ly , t o d a y , t w o t y p e s o f f i n i te - e l e m e n t m o d e l sare prac t ical ly avai lab le: p lane-st rain 2D mod els approp r iatefor s t r ip fo o t ings o r elong ated re ctang ular s t ructur es; 34,s4,s7a n d 3 D a x i s y m m e t r i c . g e o m e t r y m o d e l s a p p r o p r i a te f o rcy l indr ic al fo und at io ns and near ly squa re s t ructure s . 31' 33 , ss

I t is n o t e d t h a t e m b e d d e d f o u n d a t i o n s a n d l a y e r e d s oi ls t r a t a can b e ro u t in e ly h an d led w i th al l t h e f 'm i t e -e lem en tf o r m u l a t io n s . O n t h e o t h e r h a n d , t h e p r e s e n c e o f a f ix e db o t t o m b o u n d a r y i s r e q u i r e d b y m o s t o f th e a v ai la bleco d es . T h i s i s h a rd ly a d raw b ack i f a s t i f f, r o ck - l ik e s t r a tu md o es ex i s t a t a r e l a t i v e ly sh a l l o w d ep th . Oth erwi se , w h enth e su p p o r t i n g so i l d ep o s i t i s v e ry d eep , t h e co s t o f ar ea l is t ic f i n i t e - e l em en t an a ly s i s m a y b eco m e su b s t an t i a lCo n c lu s io n . With t h e av a i lab l e an a ly t i ca l, s em i -an a ly t i ca lan d f 'm i t e -e l em en t co m p u te r p ro g ram s th e fo u n d a t io n v ib ra -t i o n a n a l y s t m a y o b t a i n s o l u t i o ns f o r f o u n d a t i o n s o f v a r io u ss h a pe s , s u r fa c e o r e m b e d d e d , s u p p o r t e d b y d e e p o r s h a ll o wso f t d ep o s i t s . I n se l ec t i n g t h e m o s t ap p ro p r i a t e co d e fo reach sp ec i f i c s i t u a t i o n , a t t en t i o n sh o u ld f i r s t fo cu s o n t h ed e p t h o f e m b e d m e n t a n d t h e n a t u r e o f th e u n d e r l y in g s of t.Wh en d ea l i n g wi th v e ry sh a l l ow fo o t in g s o n d eep d ep o s i t sw h i c h c a n b e w e l l r e p r o d u c e d b y a s m a ll n u m b e r o f l a y e rsw i t h d i f f e r e n t p r o p e r ti e s , c o n t i n u u m t y p e a n a l y ti c a l o rsem i -an a ly t ica l fo rm u la t i o n s a r e c l ea r ly m o re ad v an tag eo u s ;t h e c h o i c e o f t h e m o s t a p p r o p r i a t e a m o n g t h e m w i l l b em ain ly d i c t a t ed b y t h e sh ap e o f t h e fo o t in g ( s t ri p , c i r cu l a r,r ec t an g u la r , a rb i t r a ry ) an d t h e d es i r ed d eg ree o f accu racy .O n t h e o t h e r h a n d , f o r e m b e d d e d f o u n d a t i o n s in a s h a ll o ws t r a t u m o r w h e n e v e r a l ar ge n u m b e r o f l a y e rs w i t h s h a rp l yd i f f e ren t p ro p er t i es ex i s t s b e lo w th e fo o t in g , f i n i t e e l em en tm o d el s a r e p ar t i cu l a r ly ap p ro p r i a t e .F u r t h e r m o r e , a t t e n t i o n s h o u l d b e a c c o r d e d t o t h e o p e r a-t i o n a l f r eq u en c i es o f t h e m ach in e an d t h e i n e r t i a ch arac t e r -i s t i c s o f t h e fo u n d a t io n . At v ery h ig h f r eq u en c i es o f v ib ra -t i o n , f , d i s c re t e m o d e l s m a y b e c o m e v e r y c o s t l y ; b e c a u s e ,i n o rd er t o t r an sm i t h ig h f r eq u en c i es , a la rg e n u m b er o fsu f f i c i en t l y sm al l, s i zed e l em en t s m u s t b e u sed . F o r i n s t an ce ,i t i s us u a ll y r e c o m m e n d e d t h a t t h e m a x i m u m d i m e n s i o n o fan e l em e n t sh o u ld n o t ex cee d X/8 , wh ere ~, = V / f i s th ewav e l en g th i n a p ar t i cu l a r so il l ay er h av in g sh ear wav ev e lo c i t y V. Th ere fo re , wi th h ig h f r eq u en c i es , an a ly t i ca lm o d e l s m a y b e c o m e a d v a n t a g eo u s . N o t i c e , th o u g h , t h a t t h ec o m p u t e r c o s t s o f s e m i -a n a l yt ic a l fo r m u l a t i o n s m a y a l s o b ead v er se ly a f f ec t ed b y a l a rg e i n c rease i n t h e o p era t i o n a lf r eq u en cy , s i n ce t h ey , t o o , d i sc re t i ze t h e co n t ac t a r ea o rt h e w h o l e u p p e r m o s t s u r f a c e .Reg ard in g t h e i n e r t i a ch arac t e r i s t ics o f t h e fo u n d a t io n ,

S o i l Dy n a mic s a n d Ea r th q u a ke En # n eer in g , 1 9 8 3 , Vo l. 2 , No . 1 9


9/41

A n a l y s i s o f m a c h i n e f o u n d a t i o n v i b ra t io n s : s t a t e o f t h e a r t: G. G a z e t a s

t h e au th o r an d Ro esse t 39 h av e d emo n s t r a t ed t h a t fo r h eav yfoundat ions ( i .e . wi th h igh mass rat ios) smal l er rors inmo d e l i n g t h e d i f f e r en t so i l l ay er s a r e u n imp o r t an t an d o n ecan sa fely base the design on availab le h al fspace so lu t ionso r o n t h e r esu l t s o f an a ly ti ca l t y p e co m p u te r p ro g rams .On th e o th e r h an d , r e l a t i v e ly l i gh t fo u n d a t i o n s a r e q u i tesens it iv e t o t h e ex i s t en ce o f co m p ete n t ro ck a t a sh a l l o wd ep th an d o f d i f f e r en t so i l l ay er s b en ea th t h e fo o t i n g , t h u sreq u i r in g a g o o d so i l ex p lo ra t i o n fo l l o wed b y f i n i t e - e l eme n tan a ly ses . Th ese co n c lu s io n s a r e fu r t h e r i l l u s t r a t ed an dg en era l ized i n a l a t e r s e c t i o n o f t h i s p ap er .I n a d d i t io n t o t h e e x i s ti n g c o m p u t e r p r o g r am s n u m e r o u sso lu t i o n s h av e b een p u b l i sh ed i n t h e l i t e r a tu re i n t h e fo rmo f d imen s io n l es s g rap h s , t ab l es an d s imp le fo rmu lae fo rimp ed an ce an d co m p l i an ce fu n c t i o n s o f fo u n d a t i o n s wi ths e v e r a l d i f f e r e n t g e o m e t r i e s , d e p t h s o f e m b e d m e n t a n dst i f fness character i s t ics , suppor ted by var ious ideal ized so i lp ro f i l e s (h a l f sp ace , s t r a t u m, e t c . ) . Th ese so lu t i o n s can g iv ev ery sa t i s f ac to ry r esu l t s i n man y p rac t i ca l cases an d a r eesp ec i a l ly v a lu ab l e i n co n d u c t i n g p re l imin ary an a ly ses an dp aram ete r sen s i t i v i ty s t u d i es . On e o f t h e g o a l s o f t h is s t a t e -of- the-ar t paper i s to p resen t and d iscuss the most s ign i -f ican t o f these avai lab le so lu t ions . Before do ing th is ,h o wev er , i t i s ex p ed i en t t o i l l u s t r a t e h o w th e imp ed an cefu n c t i o n s m ay b e u t i l ized t o o b t a in t h e d y n a mic r esp o n seof r ig id massive foun dat io ns .

U s e o f i m p e d a n c e f u n c t i o n s : r e sp o n s e o f m a s s i ve m a c h i n ef o u n d a n ' o n s

The f i rs t s tep in analysing the response of a massivemach in e fo u n d a t i o n i s t o ev a lu a t e t h e p e r t i n en t d y n amicimp ed an ces a t t h e an t i c i p a t ed f r eq u en cy , o r r an g e o f f r e -q u en c i es , o f t h e m ach in e . Th i s i s d o n e e i t h e r b y u t i l iz i n gex i s t i n g d i sc r e t e o r co n t i n u u m ty p e fo rmu la t i o n s , o r b yresor t ing to publ ished so lu t ions avai lab le in the so i l dyn-amics l i t e r a tu re . Th e u se o f d y n am ic imp ed an ce t o o b t a inthe response i s i l lus t rated herein .

F igure 3 po r t ra ys a massive, r igid found at io n hav ing equa ld ep th o f em b ed m en t a l o n g a ll t h e s id es an d p o ssess ing twoo r th o g o n a l v e r t i ca l p l an es o f sy mmet ry , t h e i n t e r sec t i o n o fwh ich d e f i n es a v e r t i ca l ax i s o f sy m me t ry . Th e fo u n d a t i o np l an , h av in g two ax es o f sy mm et ry , m ay b e o f an y ax i-sy mmet r i c o r o r t h o g o n a l sh ap e , i n c lu d in g t h e i n f i n i t e l ylo n g s t r i p (2 D g eo met ry ) . F o r su ch fo u n d a t i o n s , v e r t i ca lan d t o r s i o n a l o sc i l l a t i o n s a r e u n co u p l ed , wh i l e h o r i zo n t a lfo rces an d mo men t s a l o n g an d a ro u n d t h e p r i n c ip a l ax esp ro d u ce d i sp l acemen t s an d ro t a t i o n s o n ly a lo n g an d a ro u n dth e same ax es . Th u s , w i th t h e n o t a t i o n o f F ig . 3, t h e eq u a-t i o n s o f mo t io n i n v e r t i ca l t r an s l a t i o n v ( t ) , t o r s i o n a l ro t a -t i o n O(t ) , an d co u p l ed h o r i zo n t a l t r an s l a t i o n h ( t ) an dro ck in g r ( t ) , a ll r e f e r r ed t o t h e cen t e r o f g r av i t y o f t h ema ch in e- fo u n d a t i o n sy s t em, a r e r esp ec ti v e ly :

m . ~ )( t) + R , ( t ) = Q ~ ( t ) ( 2 1 )I z " O ( t) + T z ( t ) = M z ( t ) (2 2 )

m . h ( t ) + R n ( t ) = Qh ( t ) ( 2 3 )I o x . E ( t ) + T r ( t ) - - R h ( t ) . z e = M r ( t ) ( 2 4 )

in wh ich : m = to t a l f o u n d a t i o n m ass; I o x = m a s s m o m e n to f i n e r t i a ab o u t a p r i n c ip a l h o r i zo n t a l ax i s p ass in g t h ro u g hth e c en t e r o f g r av i t y ; I z = m a ss m o m e n t o f i n e r ti a a r o u n dt h e v e r ti c a l ax is o f s y m m e t r y ; R n , Tz, R n a nd T r = ver t ical ,t o r s i o n a l , h o r i zo n t a l an d ro c k in g r eac t i o n s o f t h e so i l ac t i n ga t t h e c e n t e r o f t h e f o u n d a t i o n b a s e ( r e m e m b e r F ig . l b ) ;Qn, Mz , Qh and M r = ver t ical , to rs iona l , h or iz on ta l and

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ro ck in g ex c i t i n g fo rces an d mo men t s , ac t i n g a t t h e cen t e ro f g r av i t y an d r esu l t i n g f ro m th e o p era t i o n o f th e mach in e .As a l r ead y men t io n ed , o n ly t h e s t ead y - s t a t e r esp o n sed u e t o a h a rm o n ic ex c i t a t i o n i s o f i n te r es t h e re . No t o n lyb ecau se m o s t m ach in es u su a ll y p ro d u c e u n b a l an ced fo rceswh ich i n d ee d v ary h a rm o n ica l l y wi th t im e ( ro t a ry o r r ec ip -ro ca t i n g en g in es ) , b u t a l so b ecau se n o n -h armo n ic fo rces( s u c h a s t h o s e , f o r e x a m p l e p r o d u c e d b y p u n c h p r e s se s a n dfo rg in g h amm ers ) can b e d eco mp o sed i n to a l arge n u m b ero f s i n u so id s t h ro u g h F o u r i e r an a ly s is . Th ere fo re , t h e ex c i t a -t i o n s may b e w r i t t en as :

Q a = Q a ex p [ i(~o t + Ca)] a = v , h (25 )M a = M a ex p [ i (w t + Ca)] a = z , r (26)

in wh ich t h e amp l i t u d es Q a an d M a are e i t h e r co n s t an t s o r( m o r e f r e q u e n t l y ) p r o p o r t io n a l t o t h e s q u a r e o f t h e o p e ra -t i o n a l f r eq u en c y ~ = 2 r r f ; ~a a r e t h e p h ase an g les o f t h efo u r ex c i t a t i o n s , v , h , r an d z .Wi th t h e ex c i t a t i o n fo rces d esc r i b ed b y eq u a t i o n s (2 5 ) -( 2 6 ) , t he s t e a d y - s t a te m o t i o n s m a y b e c a s t in t h e f o r m :v ( t ) = v . ex p ( i co t ) ; v = v l + iv 2 (2 7 )

O ( t ) = 0 . exp ( i6o t ) ; 0 = 01 + i02 (28)I t ( t ) = h . ex p ( iw t) ; h = h~ + ih2 (29)r ( t ) = r . e x p ( i ~ t ) ; r = r l + J r2 ( 3 0 )

in w h ich : v , 0, h an d r a re co mp lex , f r eq u en cy -d ep e n d en td i s p l a c e m e n t a n d r o t a t i o n a m p l i t u d e s a t t h e c e n t e r o fg rav i t y . No te t h a t eq u a t i o n s (2 7 ) - (3 0 ) d o n o t b y a n ym e a n s i m p l y t h a t t h e f o u r c o m p o n e n t s o f m o t i o n a r e a l lin phase, nor that the phase-angles between the corre-sp o n d in g ex c i t a t i o n s an d m o t io n s a r e eq u a l t o Ca ( eq u a t i o n s(2 5 ) - (3 0 ) ) . I n s t ead , t h e t ru e p h ase an g l es Ca a r e ' h i d d en 'i n t h e c o m p l e x f o r m o f e a c h d i s p la c e m e n t c o m p o n e n t . F o rin s t an ce , t h e v e r t i ca l m o t io n wi ll ex h ib i t :

Ja = arc tan ( v 2 / v O ( 3 1 )

1 0 S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e e r in g , 1 9 8 3 , V o l. 2 , N o . 1


10/41

A n a l y s i s o f m a c h i n e f o u n d a t i o n v i b ra t i on s : s t a t e o f t h e a r t: G . G a z e t a sin wh ich 7)1 and v2 axe the real an d im agina ry par ts o f v( equa t ion ( 27) ) , w hi le i t s amp l i tude i s o f a magn i tude :

I v l = ( v 2 + v 2 ) ' / 2 ( 3 2 )Also, s ince Qa a n d M a i n equa t ions ( 25) - ( 26 ) a re r ea l quan-t i t i e s , the phase lags be tw een exc i ta t ions and mot ions w i l lbe s imp ly equa l to Ca - - ~ka-U s ing s imi la r a r guments w i th r egar d to the so i l r eac t ions ,one may, w i th out los s of gener a li ty , se t :

R a = R a ex p (loot) a = v, h (33 )T a = T a exp ( i t~t) a = z , r (34)

w h e r e b y t h e c o m p l e x a m p l it u d e s R a a n d T a are related tot h e c o m p l e x d i s p l a c e m e n t a n d r o t a t i o n a m p l i tu d e s t h r o u g ht h e c o r re s p o n d in g d y n a m i c i m p e d a n c es K a , a = v , h , r , h r ,t ( see equa t ions ( 3) - ( 4) ) . Reca l l ing tha t the la t t e r a rer e f e r red to the cen te r o f the f ou nda t ion base, one canp r o m p t l y w r i t e:R v = K o . v ( 35)T z = K t " 0 ( 36)

R h = K h " ( h - - Z e r ) + K h r" r ( 37)T r = K r . r + K h r . ( h - - z e r ) ( 38)

Subs t i tu t ing equa t ions ( 25) - ( 30) an d ( 33) - ( 38) in to thegover n ing equa t ions o f mo t ion ( 21) - ( 24) and so lv ing ther esu l t ing sys tem of f our a lgebr a ic equa t ions y ie lds thef o l low ing complex- va lued d i sp lacement and r o ta t ionampl i tudes a t the cen te r o f g rav i ty :Qv" exp ( i~v)v = ( 39)Xt,(o - m ~ 2

M z . exp ( i~z)0 = ( 40)K t ( c o ) - - I z 6 0 2h = { K ~ . Q h e x p ( i ~ h ) - - K ~ r . M r e x p ( i ~ r ) } . N ( 41)r = ( X t " M r e x p ( i ~ r ) - K ~ r Q h e x p ( i ~ h )} - N ( 4 2 )

in w hich the f o l low ing subs t i tu t ions have been per f or med :X t = X h ( ~ ) - - m ~ z (4 3)K ~ r = K h r ( c o ) - -K h ( C O ) Z c ( 44)K * = K r ( c o ) - - I o x c O 2 + K h ( t ~ ) Z 2c 2Khr(co z c ( 45)

and, f inal ly,N = ( X t K * - - K ~ h 2 ) - ' ( 4 6 )

N o t i c e t h a t , f o r a p a r t ic u l a r fr e q u e n c y w , d e t e r m i n a t i o n o fthe mot ions f r om equa t ions ( 39) - ( 42) i s a s t r a igh t f or w ar do p e r a t io n o n c e t h e d y n a m i c i m p e d a n ce s ar e k n o w n . O fcour se , the com puta t io ns a r e somew h at t ed ious i f pe r-f or med by hand , s ince comp lex number s a r e involved; bu teven w i th smal l mic r ocom pute r s the ca lcu la t ions can bedone r ou t ine ly , a t a m in imal cos tTher e f or e , the au th or pr oposes tha t th i s p r ocedur e( equa t ions ( 39) - ( 42) , in connec t ion w i th an appr opr ia teeva lua t ion of impedances a t the f r equency( ies ) o f in te r es t ,should be used in mac hine f oun da t io n ana lys i s in p lace ofthe cur r en t ly popula r ' equ iva len t lumped f r equency-independent - par amete r ' appr oach .PRE SE NTAT ION OF RE SUL T S FOR SURFACE ANDE MB E DDE D FOUNDAT IONSThe subsequent f o ur s ec t ions of the paper pr esen t a com-pr ehens ive compi la t ion of char ac te r i s t i c numer ica l r esu l t s

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__d2B

F i g u r e 4 . ( a ) T h e t h r e e s o i l p r o f i l e s s t u d i e d ; ( b ) d e f i n i t i o no f g e o m e t r i c p a r a m e t e r s

f or the dyn amic imp edances ( or compl iances ) o f massles sf ounda t ions , pe r ta in ing to a l l poss~ le ( t r ans la t iona l andr o ta t iona l ) mod es o f w ~or at ion . These r esu l t s can be d i r ec t lyused in equa t ions ( 40) - ( 43) to m ake sa t i s f ac tor y and inex-pensive pr ed ic t ions of the dyn amic behavior of machinef oun da t io ns in m any pr ac t ica l cases, w i tho ut the need tor esor t to cos t ly comp ute r p r ogr ams f or eva lua t ing theimpedan ces ; th i s shou ld b e of espec ia l ly gr ea t value inpr e l iminar y des ign ca lcu la t ions .A second, equa l ly impor tan t ob jec t ive of the pr esen ta -t ion i s to as sess the s ign i ficance of va r ious pheno me na andto i l lust r a te the r o le o f key d imens ionless geomet r ic andmate r ia l pa r amete r s on the r esponse . I t i s thus hoped tha tthe r eader can ga in a va luable ins igh t in to the mechanicso f f o u n d a t i o n v i b ra t io n s .Results are presented for three categor ies of ideal izedsoil prof i les (Fig. 4) : the halfspaee, the uniform s tratumon r ig id base and the layer on top of a ha l f space . Thesemode ls r epr esen t a w ide spec t r um of ac tua l ly encounte r edso il p r of i le s and a r e s imple enough f or the i r geo met r y to bedesc r ibed in t e r ms of a s ing le quan t i ty , namely , the th ick .hes s H of the u pper m os t l ayer . ( For the ha l f space H - ~**. )For mos t p r oblems cons ider ed , the f o l low ing gr oups ofd imens ionles s pa r amete r s w hich appr ec iab ly in f luence thed y n a m i c i m p e d a n c e s h av e b e e n i d e n t if i ed :

( a ) the r a t io H / B o f t h e t o p l a y e r th i c k n e s s , / 4 , o v e r ac r i t i ca l f ounda t ion- p lan d imens ion , B; the la t t e rmay be in te r pr e ted as the r ad ius , R , o f a c i r cu la rf o u n d a t i o n o r h a l f t h e w i d t h o f a re c t an g u l a r o r as t ri p f o u n d a t i o n( b ) t h e e m b e d m e n t r a t i o D / B , w her e D i s the depthf r om the sur face to the hor izonta l so i l - f oo t ing in te r-face

S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e e r i n g , 1 9 8. 3, V o L 2 , N o . I 1 1


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An a l y s i s o f ma c h i n e f o u n d a t i o n v i b r a ti o n s : s t a t e o f t h e a r t : G . Ga z e t a s

( c ) t he s hape o f t he f oun da t i on p l an : c i r cu la r , s t r ip ,rec tang ular , c i rcular r ing; in the l as t two cases thep l a n g e o m e t r y m a y b e d e f 'm e d in t e r m s o f t h el eng t h - t o - w i d t h o r ' a s pec t ' r a ti o , L / B , or the i

analysis of machine foundation vibrations state of the art by g. gazetas (1983)

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