1951-jacobsen-hydrodynamic experiments with rigid cylindrical tanks subjected to transient montions...

7/22/2019 1951-Jacobsen-hydrodynamic Experiments With Rigid Cylindrical Tanks Subjected to Transient Montions (1)

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H Y D R O D Y N A M IC E X P E R IM E N T S W IT H R IG ID C Y L IN D R IC A LTANKS SUBJECTED TO TRANSIENT MOTIONS*

By LYDIK S. JACOBSEN and ROBEaT S. AYREABSTRACT

Four tanks, from 6 inches to 4 feet in diameter, have been subjected simultaneously totransient, horizontal "ground motions" of simplified type. The important parameters, inaddition to size of tank, were depth of fluid and frequency, duration, and amplitude ofground motion. The envelopes of the gravity-wave profiles have been recorded on avertical plane of symmetry placed within each tank in a direction parallel to the groundmotion. The data include samples of the wave envelopes, photographic studies of the waveformation, maximum wave heights and the locations of these maxima, and the fluiddamping coefficients. Equivalent mass and overturning moment due to the fluid have beenshown for various degrees of confinement of the upper surface, from complete confine-ment (owing to use of a rigid cover) to a free surface. The study relates to the effect ofearthquakes and other ground motions on oil and water storage tanks. The results can beextrapolated with reasonable certainty to full-scale tanks.

I N T R O D U C T I O NPrevious work.--The p r e s e n t i n v e s t i g a t i o n i s r e l a t e d t o e a r li e r w o r k c a r r i e d o ni n t h e V i b r a t i o n L a b o r a t o r y a t S t a n f o r d U n i v e r s i t y . T h e f ir s t o f t h e s er ie s,p u b l i s h e d b y L e a n d e r 5/[ . H o s k i n s a n d L y d i k S . J a c o b s e n i n 1 9 3 4 ( i) , w a s a ne x p e r i m e n t a l a n d a n a l y t i c a l s t u d y o f t h e w a t e r p r e s s u r e s i n a t a n k o f r e c t-a n g u l a r cross section and finite length which was subjected to horizontaltransient ground motions parallel to the length of the tan k. The secondinvestigation, made by Brooks T. Morris in 1938, consisted largely of experi-ments on the gr avity- wave formations in cylindrical tanks (2). The third, ananalytical investigation (3), deals with the impulsive hydro dyna mic pressuresand velocities, and the effective mass and mass moment, in a cylindrical tankas well as around a cylindrical pier. Some less closely related investigationsare those of H. M. Wes tergaard on the hy drod ynami c pressures in a reservoirof rectangular cross section and infinite length in which one end (the dam) issubjected to a horizontal simple-harmonic oscillation (4), and of P. Wilh.Werner and K. J. Sundquist on the hydrodynamics of fluid containers of avar iet y of shapes when the containers experience simple-harmonic motion s (5).In the experimental work of Arth ur C. Ruge on the effects of earthquakes onelevated water tanks (6) the tank is elastically supported, with the result thata coupling exists between the fluid system and the tank-tower system. Theexistence of this coupling has been d emon str ated on a full-scale basis in experi-ments con ducted by D. S. Carder on actual tanks (7).A cknowledgment.--The investigation has been carried out under the sponsor-

* Manuscript received for publication May 25, 1950.1 Numbers in parentheses refer to similarly numbered items in the References at theend of this paper.[ 313 ]


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314 BULLET IN OF THE SEISMOLOGICAL SOCIETY OF A1VfERICAY

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E XPE RIM E NT S WIT]=[ T ANKS SUB J E CT E D T O T RANSI E NT MOT IONS 315s h i p o f t h e O f fi ce o f N a v a l R e s e a r c h , t o w h o m a m o r e d e t a i l e d r e p o r t h a sa l r e a d y b e e n m a d e ( 15 ). M r . E d w a r d P . H o ll is , M i s s G u d r u n G y t e l , a n d M r .R i c h a r d W a r r i c k a s s i s te d i n v a r i o u s p a r~ s o f t h e s t u d y .

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N O T A T I O NCyl indr ica l loca t ion coSrd ina tesRad ius o f tank wa l lS ta t ic dep th o f f lu idClea rance be tween s ta t ic su r face o f f lu id and tank coverEleva t ion o f f lu id su r face wi th re spec t to s ta t ic leve l , z = hM a x im u m w a v e h e ig h tM a x im u m w a v e h e ig h t a t t a n k w a ll , r ' = aMaxim um wav e he igh t a t cen te r of tank , r = 0R adia l loc at ion of ~,~Ra t io o f e f fec t ive hydrod ynam ic mass to to ta l m ass o f f lu id o f s ta t ic dep th hSubscrip ts d esignating natura l m ode of grav ita t io nal os ci l la t ion of f lu id, c lasss - ~ lNa tu ra l pe r iods o f f lu id sys temN atu ral f requencies of f lu id syste mFrequency o f ground m ot ion ( free v ib ra t ion f requency o f shak ing tab le )Maxim um amp l i tude o f g round mo t ionMaxim um harmonic ve loc i ty o f g round mot ionT i m eDu ra t ion o f impu lse in te rva l o f g round mo t ionDu ra t ion o f s tep m ot ion (ground)Du ra t ion o f osc i l la to ry mot ion (ground)Nu mb er o f cyc les in osc i l la to ry mot ion ; N = D f

(Time is expressed in seconds; d isplacement and length , in inches; f requency, in cyclesper second.)THE PI~OBLEM

F i g u r e 1 s h o w s a r i g id t a n k o f r a d i u s a f il le d w i t h f l u i d t o t h e l e v e l h . I te x p e r i e n c e s t h e h o r i z o n t a l t r a n s l a t o r y g r o u n d m o t i o n xu( t ) p a r a l l e l t o t h ex - z p l a n e. T h e s i g n o f t h e g r o u n d m o t i o n i s s h o w n p o s i t i v e t o t h e l e f t i n o r d e rt o c o n f o r m w i t h t h e e x p e r i m e n t a l s e tu p . T h e g r a v i t y w a v e s u r fa c e h as t h eh e i g h t ~ w i t h r e s p e c t t o t h e e q u i l i b r i u m p l a n e , z = h .

T y p e o f g r ou n d m o t i o n . - - T h e g r o u n d m o t i o n c o n s i s ts o f a n a p p r o x i m a t e l yi m p u l s i v e s t a r t f r o m r e st , f o ll o w e d b y a s in u s o id a l m o t i o n h a v i n g a n e a r l yc o n s t a n t r a t e o f d e c a y . T h e i m p u l s e i n t e r v a l t l i s s h o r t ( in r e l a t i o n t o t h e l o w e rn a t u r a l p e r i o d s o f t h e f l ui d s y s t e m ) t o t h e d e g r e e t h a t t h e g r a v i t y w a v e h e i g h t sr e m a i n s m a l l d u r i n g t h e i n t e r v a l a n d c o n s e q u e n t l y t h e i m p u l s i v e h y d r o -d y n a m i c p r e s s u r e s a n d v e l o c it i es c a n b e d e t e r m i n e d w i t h o u t g r e a t d i ff i cu l tyb y a n a l y s i s ( 3 ) .

T w o v a r i e ti e s o f t h e g r o u n d m o t i o n h a v e b e e n u s e d : 1) th e s t e p m o t i o n , inw h i c h t h e g r o u n d m o t i o n is a r r e s t e d a t t i m e t2 w h e n xa = A a n d ~o = 0 ; a n d


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316 BULLET IX OF THE SEISMOLOGICAL SOCIETY OF AMERICA2) the oscil latory motion, in which the "ground" continues to vibrate until itcomes to rest, largely as a result of Coulomb damping, after a time duration D.These motions are reasonably characteristic of portions of actual transientground disturbances. They satisfy the principal purpose of the investigation,namely, to obtain a general understanding of the problem rather than toexplore the effects of particular, and often very complicated, disturbances.

P l a n e o f s y m m e t r y o f t h e t a n k . - - T h e x-z plane is a plane of symmetry , andthe fluid velocities across this plane are zero. The results thus also apply to ahalf tank in which the plane side is parallel to the direction of ground motion.Since the greatest wave heights occur in the plane of symmetry, it is of par-ticular interest as a background against which to record the wave profiles. ~

E l l i p t i e i t y o f t h e t a n l c . - - A c t u a l tanks are not perfectly circular in crosssection, but have some out-of-roundness, or ellipticity. If the major axis of theellipse is not parallel to the direction of the ground movement, the wave mo-tion will tend to break into two components (2), one parallel to the major axisand one to the minor axis, and after several cycles the resultant motion be-comes very complicated in appearance. The variation in tank diameter wasno~ more than 0.4 per cent, but the effect on the wave motion was still notice-able. The tanks were, therefore, carefully oriented with the major axis in thex-z plane.

EXPERIMENTAL METHODSS h a k i n g ta ble a n d ta n k s . - - F i g u r e 2 is a general view of the four tanks and theshaking table. The tank diameters were 6, 12, 23, and 47 inches. 3 The con-struction and analysis of the shaking table have been described previously(1, 8). Briefly, it consists of a massive platform mounted on 'concentricallyground wheels and connected at one end through springs of adjustable modulusto a fixed pier. At the other end is a bumper spring. The table is set in motionby striking the bumper with a heavy pendulum, the pendulum remaining incontact with the bumper for the time interval 0 < t < tl, which is the i m p u l s ein terval . The table motion may be arrested at time t2 (figs. 1, b; 3, a; 3, b) bya simple, gravity-operated latch (15).

G r o u nd m o t i o n . - - T h e natural frequency of the table and anchor spring, i.e.,the frequency f of the ground motion, has been varied in small steps fromabout 0.7 to 7.0 cycles per second. During the impulse interval the entiresystem, neglecting the action of the fluid, acts as a two-degree-of-freedomsystem (pendulum, bumper spring, table, anchor spring) the fundamentalnatural frequency of which is a little less than f, and the second naturalfrequency, f2, is 6.5 to 7 cycles per second. The impulse interval tl rangesbetween 0.075 and 0.08 second; both f2 and tl are thus relatively constant.

2 The earlier work (2) on cylindrical ta nks did not ma ke use of the x-z plane as a record-ing device, the wav e heig hts being recorded only at t he cylindrical boun dary, r = a.3 The diametr a l d is tor t ions a t the top~ of the two larger tank s were measured dur ingthe experimen ts and found to be less than 0.001 inch.


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EXPERIMENTS WI TH TANKS SUBJECTED TO TRANSIENT IA~IOTIONS 317The ground-motion criteria are: (1) constant maximum harmonic veloci ty, V,

and (2) constant time duration, D (except with respect to the step motion, forwhich the duration is t2). The choice of these criteria has already been dis-cussed (10); H. M. Westergaard originally suggested the constant V cri-terion (9). The criteria may be explained in terms of ground energy, and mean(1) tha t the energy required to set up the ground motion at any frequency f

Fig. 2. General view of shak ing table and four tanks.

is essentially constant, and (2) that the average rate of dissipation of theground energy is approximately constant. The maximum harmonic: velocityis given by V = 2~rAf (1)and can be held to within 2 per cent of the desired value. The standard dura-tion was 2.1 seconds, =55 per cent.

Measuring the ef fect ive mass and overturning moment.--In order to determinethe effective mass and overturning moment of the fluid, the 12-inch and the23-inch tanks have been mounted on dynamometers, as shown in figures 2and 4, and the same general procedure has been followed as was used in theearlier work on rectangular tanks (1). Two sets of tests are made: 1) withfluid in the t ank at various depths h; 2) with the fluid replaced by various solid


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EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS 319w e i g h t s w h i c h a re r i g id l y c l a m p e d t o t h e b a s e o f t h e t a n k . B y c o m p a r i n g t h ed y n a m o m e t e r r e s p on s e s fo r t h e s e t w o c o n d i t io n s , d u r i n g t h e i m p u l s e i n t e r v a l,i t i s p o s s ib l e t o d e t e r m i n e t h e e f f e c ti v e m a s s o f t h e f l u i d a n d t h e o v e r t u r n i n gm o m e n t .

R e c o r d in g t he g r av it y w a ve f o r m a t i o n . - - E x t e n s i v e e x p e r i m e n t s w e r e m a d ew i t h s e v e r a l m e t h o d s o f r e c o r d i n g t h e e n v e l o p e o f t h e g r a v i t y w a v e p ro f il e s i nt h e x - z p l a n e , t h e m o s t s u c c e s s f u l o n e u s i n g a b l a c k d y e s o l u t i o n a n d s t r i p so f B r i s t o l b o a r d . T h e r e c o r d i n g c a r d is a t t a c h e d t o a t h i n m e t a l h o l d e r w h i c hi s h u n g o n t h e b a c k g r o u n d p l a n e (fig . 4 , e ). T h e d y n a m o m e t e r s w e r e b l o c k e do u t o f a c t io n d u r i n g t h e w a v e - p r o fi l e s t u d i e s.

NATURAL-~/[ODE GRAVITATIONAL OSCILLATIONS OF THE FLUIDM o d e s o f or de r s = 1 . - - H o r i z o n t a l g r o u n d m o t i o n s e x c it e a c la s s o f g r a v i t a -t i o n a l m o d e s w h i c h a r e c h a r a c t e r i z e d b y a s in g le n o d a l d i a m e t e r a n d b y o n e -f o ld s y m m e t r y w i t h r e s p e c t t o t h e x - z p l a ne . T h e y a r e d e sc r ib a b le b y B e s se lf u n c t i o n s o f t h e f i r s t k i n d a n d f i r s t o r d e r , t h e J ~ f u n c t i o n s w h e r e s = 1 , a n da r e o f t e n ca l le d t h e s = 1 m o d e s ? T h e f u n d a m e n t a l m o d e h a s n o n o d a l c ir cl e,b u t i s s i m p l y a " s w a y i n g " b a c k a n d f o r t h o f t h e f l u id .~ T h e s e c on d m o d e h a so n e n o d a l c i r c l e , t h e t h i r d h a s t w o , e t c . ( se e f ig . 5 ) . I f a n i m a g i n a r y c y l i n d e r isd i p p e d c o n c e n t r i c a l l y i n t o t h e t a n k , t h e f l u id s u rf a c e w i ll d r a w a c o s i n e c u r v eo n it . T h e m o t i o n i n a p a r t i c u l a r n a t u r a l m o d e i s s i m p l e h a r m o n i c i f f r i c t i o ni s n e g l e c te d . T h e n a t u r a l p e r i o d s 6 a r e g i v e n b y

T~ = 2~- , , / a ( 2 )~ / g k ~ a t a n h (k~a . h / a )

w h e r e t h e s u b s c r i p t r e fe r s t o t h e p a r t i c u l a r n a t u r a l m o d e a n d w h e r e t h e c o n -s t a n t s c h a r a c t e r i s t ic o f t h e n a t u r a l m o d e s a r e k i = 0 .5 8 6 ~r/a , ki i = 1 .697 ~r/a ,k m = 2 . 7 1 7 l r /a , k~ -~ ( i - 1 / 4 ) v / a w i t h i n c r e a s i n g a c c u r a c y ( se e f ig . 6 ) .

EXPERIMENTAL RESULTSDAMPING

T h e f u n d a m e n t a l m o d e c a n e a s i l y b e s e t u p b y p u s h i n g v e r t ic a l l y a g a in s t t h ew a t e r s u r f a c e in a p e r i o d ic m a n n e r n e a r t h e w a l l of t h e t a n k , ~ a n d t h e s e c o n d

4 The basic theo reti cal wor k on cylindrical containers was published by Poisson (1828)and by Lord Rayleigh (1876).5 This mode can easily be set up in a glass of wat er; in fact, ma ny of the p henom ena canbe demon stra ted in a qualita tive way by movi ng a can of water back and forth on a smoot hsurface.6 See Lamb 's Hydrodynamics(11), Arts. 191 and 257.Some interes ting experiments on the natur al gravit ational mod es in both circular andrect angu lar tanks ha ve been describe d by Fred eri ck Guthr ie (1875) (12). Among hismethods was the use of a sheet of cardboard on which the water surface marked out itsown profile. However, he apparent ly did not use a dye in the water.


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E X P E R I M E N T S W I T H T A N K S S U B J E C T E D T O T R A N S I E N T M O T I O N S 321

m o d e c a n b e e xc i t ed b y p u s h i n g n e a r t h e p o i n t o f m a x i m u m d i sp l ac e m e n t ,r = 0 . 3 5 a ; b u t t h e h i g h e r m o d e s a r e d if f ic u l t t o f o r m b y h a n d ( 1 5) . T h es a m p l e g r a p h s i n fi g u re 7 s h o w t h e v i s c o u s n a t u r e o f t h e d a m p i n g . I n f i g u re 8t h e l o g a r i th m i c d e c r e m e n t s f o r t h e f u n d a m e n t a l m o d e h a v e b e e n p l o t t e da g a i n s t t a n k d i a m e t e r , t h e c u r v e s a p p r o x i m a t i n g e q u i l a t e r a l h y p e r b o l a s , s T h ed e c r e m e n t s f o r th e s e c o n d m o d e w e r e g e n e r a l ly h i g h e r t h a n t h o s e f o r t h ef u n d a m e n t a l .

E F F E C T I V E M A S S A N D O V E R T U R N I N G M O M E N T ~ C O V E R E D T A N KT h e r e a r e t w o l i m i t i n g c a s e s : 1 ) I f t h e t a n k i s f u l l t o t h e c o v e r , 9 t h e f l u id i sc o m p l e t e l y c o n f in e d a t a ll b o u n d a r i e s a n d s h o u l d h a v e a n e f f e ct iv e m a s s a n da n o v e r t u r n i n g m o m e n t e q u a l t o t h o s e fo r a s o li d of t h e s a m e d e n s i t y a n dd i m e n s i o n s , z 2 ) I f t h e r e is d i s t a n c e e n o u g h b e t w e e n t h e f lu i d s u r f a c e a n d t h ec o v e r , t h e s y s t e m w i ll a c t a s o n e w i t h o u t a c o v e r ( 3) . T h e i n t e r m e d i a t e c a s esa r e e x t r e m e l y c o m p l i c a t e d .

T h e r e w e r e f o u r s e r ie s o f t e s t s f o r e f f e c t i v e m a s s ( f = 1 a n d 4 c p s , V = 2 a n d3 i n / s e c . ; 2 a = 2 3 ) , e a c h s e r ie s i n c l u d i n g a t l e a s t s e v e n v a l u e s o f c l e a r a n c e , c,b e t w e e n t h e s t a t i c f lu i d l e v e l a n d t h e c o v e r . T h e r e s u l ts h a v e b e e n s u m m a r i z e di n f i g u r e 9 , w h e r e i t i s s h o w n t h a t i f o n ly a sm a l l p ro p o r t io n o f th e f lu id i s re -moved, the tank becomes, effectively, an open one. I t is s o m e w h a t s u r p r is i n g t h a tt h e r e w a s n o s i g n i f ic a n t d i ff e r en c e i n t h e s h a p e s o f t h e e x p e r i m e n t a l c u r v e sf o r t h e t w o c o n d i t i o n s o f V . O n e w o u l d e x p e c t t h a t , a s V is i n c r e as e d , t h ea m o u n t o f f lu id w h i c h m u s t b e r e m o v e d i n o r d e r t o r e d u c e m l / m f r o m u n i t yt o t h e o p e n t a n k v a l u e o f 0 . 6 8 w o u l d a ls o in c r e a se . T h e e f fe c t o f i n c r e a s i n g Vf r o m 2 t o 3 i n / s e e , i s a p p a r e n t l y s m a l l e n o u g h t o b e l o s t i n t h e i n h e r e n t e r r o r s

8 T h e d e c r e m e n t s m a y b e c o m p a r e d w i t h v a lu e s o b t a i n e d b y C a r d e r ( 7) o n e l e v a t e dt a n k s h a v i n g a v o l u m e f r o m 2 5 t o 5 0 t i m e s a s g r e a t a s t h e f o u r - fo o t m o d e l . T h e c o m p a r a -b l e fu l l- sca le l oga r i t hm i c d ec rem ent s a re o f t he orde r 0 .01 t o 0 .015 . These agree rea son abl ywe l l wi t h an ex t rap ol a t i on of f i gure 8 , if i t i s r e cogni zed t ha t t he y i nc l ude t he f r i c t i ona le f fec t s due t o t he beh avi o r o f t he su ppor t i ng t ow er a s we ll a s t hose of t he f l u i d .9 See fi gure 6 o f r e fe rence (3) fo r expe r iment a l r e su l t s fo r t he t ank w i t hou t cov e r .10 Ca re w as t aken t o i nsure t ha t t he c ove r was r i g i d ( f ig . 4 , d ) and t h a t t he re we re noa i r pock e t s .F i g . 4 . Expe r i ment a l appa ra t us .

a ) S i d e e l e v a ti o n o f h o r i z o n t a l- f o r c e d y n a m o m e t e r . T a n k b a s e i s su p p o r t e d a t f o u rp o i n t s b y h i n g e s H a n d b l o c k s B . W i r e -r e s is t a n ce s t r a i n g a u g e s a r e a t t a c h e d a t e i g h tl o c a ti o n s , G . M o m e n t d y n a m o m e t e r is n o t in a c t i o n w h e n f o rc e d y n a m o m e t e r is i n u s e.b ) S i d e e le v a t i o n o f m o m e n t d y n a m o m e t e r . T a n k b a s e is s u p p o r t e d a t t h r e e p o i n t s b yh i n g es H a n d , t h r o u g h t h e c a n t i le v e r d y n a m o m e t e r b a r M , b y t h e b a l l a n d p e d e s t a l P .S t r a i n g a u g e s a r e a t t a c h e d a t G . W h e n t h e m o m e n t d y n a m o m e t e r is in u s e, t h e f o r c ed y n a m o m e t e r is b l o c k e d b y s h e a r p la t e s S .c ) Vi ew of 23- i nch g l a s s t ank sho wi ng hor i zont a l - force dyn am om et e r , e l ec t r i c a l t e rm i na lb l ock for connec t i ng s t ra i n gauges i n W hea t s t o ne br i dge c i rcu i t, and wave -enve l ope re -c o r d i n g c a r d h o l d e r E .d ) V ie w o f 2 3- in ch t a n k w i t h r i g i d c o v e r . M o m e n t d y n a m o m e t e r s h o w n in o p e r a t i n gcondi t i on .e ) D y n a m o m e t e r r e c o r d i n g a p p a r a tu s , s h o w i n g : C, s te a d y - c u r r e n t a n d b r i d g e - b a l a n c econt ro l box; A, a . c . ampl i f i e r ; O , l ow-f requency osc i l l ograph . Appa ra t us fo r r ecord i ngg r o u n d m o t i o n i s a t R .


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,~ q3


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EXPEEIIVfENTS W I T : L I T A N K S S U B J E C T E D T O T R A N S I E N T 1%lOTIONS 323of the experiment. The amount of fluid which must be removed is so small tha ta more precise investigation does not seem warranted. An even more spec-tacular reduction in m l / m can be expected when the depth is shallow. Forexample, when h /a = 1/4 then ml/m -= 0.16 when the tank is open. The re-

0 . 5

=

0. 4

0. 3

T rW

I I ~ ~ - o , 2 3 s

0.1580,1090.0930.083O.OT5~,.~0.069, 0 6 5

o ~0.5 ID_ _ h

F i g . 6 . N ~ t u r a l p e r i o d s o f g r a v i t a t i o n a l o s c i l l a t io n , types = I . T a n k r = d i u s ( a ) i s [ i n i n c h e s .duction in rnl/m to the open-tank value is probably more closely associatedwith the ratio of clearance to tank radius, c/a, than with the ratio of clear-anee to fluid depth, c/h. Two series of tests were made for overturning moment,the experimental data having been compared with the limiting theoreticalcases in figure 10. The moments are with respect to the hinge H shown infigure 4, b. The very rapid decrease in overturning moment is due to twocauses: 1) reduction in effective mass, and 2) lowering of the centroid of effec-tive mass.


12/34

3 2 4 BULLETIN 0I: THE SEISMOLOGICAL SOCIETY OF AMERICA2 a = 4 7

F U N D A M E N T A L M O D E

0 . 80 . 6

0 . 2

( a )h _~ o ~ , . = o " O "- I

~""~~ ~ ~ . . . . . ~ O , r . . ~ 0 x " ~ * c ' ~ ,, ,~ . ~ 0

I I I f I t t

, . 5 o . ( b )0 . 8 o0 .6 x ~ ' ~ o0. 4

I I I I I x , ~0 ,1 0 2 0 4 0 6 0 8 0 I 0 0 1 2 0 1 4 0N U M B E R O F C Y C L E S

Fig. 7. Nat ure of the fluid damping. Wave heig ht has been plo tted on a logarithm ic scale.PHOTOGRAPHIC STUDIES O F THE GRAVITATIONAL WAVESStep motion.--Photographs o f t h e i n t e r s e c t i o n o f t h e w a t e r s u r f a c e w i t h t h e

b a c k g r o u n d p l a n e (0 = 0 ) w e r e t a k e n a t a k n o w n r a t e , t r a c i n g s o f t h e i n t e r -s e c t i o n l in e s o r p r o fi l e s h a v i n g b e e n s h o w n i n f ig u r e 1 1, a. T h e f i r s t f r a m e h a sa r b i t r a r i l y b e e n t a k e n a s z e ro t im e . T h e w a v e m o t i o n c o n s i st s m a i n l y o f th ef u n d a m e n t a l m o d e , a l t h o u g h m a n y o t h e r m o d e s a re o b v i o u s ly p r e s en t . C a l -c u l a t e d p r o fi le s , a s s u m i n g a p a r t i c u l a r s t a r t , h a v e b e e n s h o w n t o a n a r b i t r a r ys c a le in f i g ur e ] 1 , b ; q u a l i t a t i v e l y , t h e a g r e e m e n t i s g o o d . T h e b a s i s o f t h ec a l c u l a t i o n i s a s f o l lo w s .


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E X P E R I M E N T S W I T H T A N K S S U B J E C T E D TO TP~ANSIENT M O T I O N S 3 2 5

T h e u n d a m p e d f r ee o s ci ll a ti o n w i t h g e n e r a l s t a r t i n g c o n d i t io n s c a n b e d e -s c r i b e d b y t h e s e r ie s )= ~ A ~ J l ( k i r ) c o s 0 - c o s \ ~ ( + i (3 )w h e r e T i a n d k i a r e g i v e n b y e q u a t i o n ( 2) . L e t u s a s s u m e a p a r t i c u l a r s t a r ts u c h t h a t ~ i = ~i ~i = C v . . . . 0 , ~i = ~ i v = C v I . . . . ~ , a n d , t o a n a r b i -t r a r y sc a l e , l e t A I - - 1 , A I I = 1 / 3 , A I I I = 1 / 5 , A i r = 1 / 7 , A v - - 1 / 9 , e t c . 11

FUNDAMENTAL MODE

O JO

0 . 0 8Eo' ~ 0 , 0 6

.3I 0 . 0 4

0 , 0 2 i " - " " - "- - =

0 1 ! ! I0 ~ I 2 4T A N K D I A M E T E R . F E E T

Fig . 8 . Ef fec t o f tank d iam ete r and o f s ta t ic dep th on damping . The log a r i thmic decl /emen ts have been ob ta ined f rom curves o f the typ e shown in figu re 7 .( fig . 12 , a ) . T h e a b o v e c h o i c e o f t h e ~ ' s m a k e s t h e t e r m s a l l a d d i t i v e a t t h eb o u n d a r y , w h i c h i s c o n s i s t en t w i t h t h e i m p u l s i v e n a t u r e o f t h e s t a r t o f t h eg r o u n d m o t i o n . T h e s t un o f t h e f i r st fi v e t e r m s a t z e ro ti m e , a n d t h e r e a f t e r a ti n t e r v a l s o f T I / 8 , h a s b e e n s h o w n i n f i g u r e 1 2 , b , a n d t o d i f f e r e n t s c a l e i nf ig u r e 11, b. T h e l a c k o f m o r e c o m p l e t e a g r e e m e n t w i t h t h e p h o t o g r a p h s i sd u e : a ) t o t h e l i m i t a t i o n o f a n a l y s i s t o f i v e t e r m s ; b ) t o t h e o m i s s i o n o f d a m p -i n g ; c ) t o t h e a r b i t r a r y c h o ic e o f s t a r t ; a n d d ) t o t h e f a c t t h a t t h e g r o u n dm o t i o n i s n o t p u r e l y a n i m p u l s i v e o n e.

11 An equ~'lly reason able choice would ha ve been A I = 1 , A I I = k i / k i i = 0.345, AI I~ =] ~ I / ] f I I I = 0.216, etc.


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326 B U L L E T I N O F T H E S E IS ~ IO L O G IC A L S O C IE T Y O F A M E R I C Aml

0.~

o ,e

0 .7

0 .6rn'IT=0. 5

0 ,4

0.~

.C O V E R E D T A N K , E XP ER IM E NT AL

L O P E N T AN K ~ T HE O R ET IC A L

0 .2

F U L L T AN K 4 = 1 , 5 0

I r I I I I. O I . 0 2 . 0 4 c . 0 6 . 0 8 . 10l

I I o L ~ ~ k ~ ~. -. -- ,, ,, -FLUID RE MO VE D, PER CENT OF FU LL TANK

F i g . 9 . I n f l u e n c e o f a r i g i d c o v e r o n t h e e f f e c t iv e h y d r o d y n a m i c m a s s o f t h e f l u id . T h ed a s h e d l i n e s r e p r e s e n t t h e u p p e r a n d l o w e r b o u n d s o f t h e e x p e r i m e n t a l c u r v e s ; t h e i n t e r -m e d i a t e f u ll li n e i s t h e a v e r a g e .

Oscillatory motion.--Photographs were also taken of the wave profiles setup under two conditions of oscillatory motion (fig. 13). Since the wave motionsare considerably more complicated than those excited by the step motion, itwould be useless to represent t hem b y a free vibration from a simple impulsivestart.

E N V E L O P E S O F T H E W A V E P R O F I L E SStep motion.--Representative samples of the wave-profile envelopes taken

in the 12-inch tank appear in figure 14. An increase in V results in increasedwave height, but not in very marked change in the shape of the envelope,except when V and the resulting motions are large. The most prominent changeoccurs with the large decrease in h/a.


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EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT IV[OTIONS 3 2 7Oscillatory motion.--Samples o f t h e e n v e l o p e s h a v e b e e n s h o w n i n f i g u re 1 5

f o r h/a = 1 , a n d i n f ig u r e 1 6 f o r h/a = 1 / 4 . V i s c o n s t a n t w h i l e f h a s b e e nv a r i e d f r o m z e r o I2 t o a b o u t 6 c y c l e s p e r s e c o n d . T h e g r e a t e s t d i f fe r e n c e b e t w e e nt h e c o n d i t io n s h/a = 1 a n d h/a = 1 //4 a p p e a r s i n t h e f r e q u e n c y r a n g e i n w h i c ht h e f u n d a m e n t a l m o d e is o f m o s t i m p o r t a n c e ( se e p r o fi le s f o r f = 1 .0 3 a n d

3 5 0

i3 0 0

~ 2 5 0dM.

2 0 0o

15 CoZ

IO C, , = ,o >t

L I M I T I N G T H E O R E T I C A L V A L U ECOVERED TANK\~ g

\ :f..~COV ERE D TANK , EXPERIMENTAL- \

O ~I 2 5 4 5 6 7FLUID REMOVE D, PER CENT OF FULL TANK

F i g . 1 0 . I n f lu e n c e o f a r i g i d c o v e r o n t h e e f f e c t iv e h y d r o d y n a m i c o v e r t u r n i n g m o m e n tdue t o t he f l u i d . The max i mu m va l ue of t he i mpul s i ve acce l e ra t i on was 0 .12 g . Fu l lt a n k h/a = 1.50.f = 1 .6 4 ). A t g r o u n d f r e q u e n c i e s h i g h e r t h a n a b o u t 3 c y c l es p e r s e c o n d t h ed i f fe r e n c e s d u e t o d e p t h a r e m o r e o r l es s s e c o n d a r y . A d i r e c t, l e f t - to - r i g h tc o m p a r i s o n b e t w e e n t h e r e c o rd s , b a s e d o n t a n k d i a m e t e r , s h o u l d n o t b e m a d eu n l e s s i t is r e c o g n i z e d t h a t t h e n a t u r a l f r e q u e n c i e s a r e d i f f e r e n t i n t h e t w ot a n k s a n d h e n c e th e " t u n i n g " w i t h t h e g r o u n d f r e q u e n c y is d i ff e re n t. T a b l e 1

1 , T h e g r o u n d m o t i o n f o r t h e " z e r o f r e q u e n c y " c a s e w a s o b t a i n e d b y r e m o v i n g t h ea n c h o r s p r i n g s f ro m t h e t a b le , t h e m o t i o n c o n s i st in g o f t h e i m p u l s iv e s t a r t f o ll o w e d b y a na p e r io d i c m o t i o n in w h i c h t h e t a b l e g r a d u a l l y c a m e t o r e s t a s a r e s u l t o f C o u l o m b d a m p -i ng . Th e s t a r t i n g condi t i ons we re ad j us t ed un t i l a max i mum ve l o c i t y o f 3 i n / see , wasind ica te d by the s lope of the xa , t curve (see f ig. 20, b) .


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328 B U L L E T I N O F T I t E S E I S ] V I O L O G I C A LSOCIETY OF A2VIERICA( a ) E X P E R I M E N T A L

f = l . 0 3 , = 0 . 8 4 , " V '= Gh 5- 5 = ~ - , 2 o = 2 3

t = 0

iII

_ L ' t 2t = O . 2 5 I

I!

t= ~

- - t = - ~

I

, ~ ] - t - - r , = o . e l~ . r , " I " , , , , l , ,I0 O . 5 - r - - i . o

F i g . 1 1. C o m p a r i s o n o f p h o t o g r a p h i c a n d o f c a l c u l a t e d w a v e p r o f i le s ; s t e p m o t i o n .( a ) t~ s h o w s t h e a p p r o x i m a t e e n d o f t h e g r o u n d m o t i o n ( fi g . 3 , a ) . ( b ) C a l c u l a t e d p r o f i l e sb a s e d o n a n a s s u m e d f r ee v i b r a t i o n ( se e f i g. 12 ).


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EXP ERIM ENTS W ITI~ I T A N K S S UBJE CTED TO TRA NS IE NT ~ IOTIONS

(b) CALCULATED329

i

J

fJ I I I j i t i i

o 0.5 r-~ LO


18/34

3 3 0 BU LL ETI :N OF TI=IE SEIS1V[OLOGICAL SOCIETY OF AME,P~ICAl i s ts t h e n a t u r a l f r e q u e n c i e s a n d t h e l o c a t i o n s o f t h e c o r r e s p o n d i n g p o i n t s o fm a x i m u m e l e v a t i o n .T h e l o c a t io n s o f m a x i m u m w a v e h e i g h t in t h e n a t u r a l m o d e s r e l a te a p p r o x i -m a t e l y t o t h e l o c a t i o n s o f t h e m a x i m a i n t h e r e c o r d e d p r o f il e s . F o r e x a m p l e , i nf i g u r e 1 5 t a k e t h e c a s e 2 a = 2 3 , f = 3 .9 7 . A c c o r d i n g t o t a b l e 1 t h e g r o u n d f r e -q u e n c y i s in n e a r r e s o n a n c e w i t h t h e s i x t h n a t u r a l f r e q u en c y , a n d a m a x i m u mw a v e h e i g h t c a n b e e x p e c t e d i n t h e n e i g h b o r h o o d o f r / a = 0 . 1 0 . T h e s a m e

T A B L E 1NATURAL FREQUENCIES IN TANKS OF 12-INCH AND 23-INCIt DIAMETER AND LOCATIONS OF

POINTS OF MAXIMUM ELEVATION IN THE NATURAL-MODE PROFILES

I .I I .I I I .IV .V .V I .V I I .V I I I .

fi cycles per second (s=l )2a=12 in. 2a=23 in.

h/a = 1 h/a = 1/41.692 .9 43 .734 374 .915 .4 45 .886 .3 0

1 .142 .753 .674 .3 64 .915 .445 .886 .30

h/a ~ 1 h/a ~ 1/41 .22 0 .832 .12 1 .982 .68 2 .643 .16 3 .143 .54 3 .543 .93 3 .934.23 4 234 .5 4 4 .5 4

_r fora maximum*

1.000O. 3460 .2160 .156O, 123O. 1020 .0870 .076* Locations of the lesser maxi ma and of the nodal diamet ers can be ob tained f rom a table of Bessel's functions.

g r o u n d f re q u e n c y , w h e n a p p l i e d t o t h e 1 2 - i nc h t a n k , e x c i te s b o t h t h e t h i r d a n dt h e f o u r t h m o d e s t o a c o n s i d e r a b l e e x t e n t . I t i s s i g n i f ic a n t t h a t t h e f u n d am e n t a l o r " s w a y i n g " m o d e i s e x c i te d o n l y t o a l i m i t e d e x t e n t w h e n t h e f o rc i n gf r e q u e n c y i s g r e a t e r t h a n a b o u t 2 c yc l e s p e r s e c o n d ev e n t h o u g h t h e g r o u n dm o t i o n h a s a n i m p u l s i v e s t a r t .

Presence o f ano ther c lass o f gravi ta t iona l m od es . - - Th e v e r t i c a l f l u i d m o t i o n sa t t h e c e n t e r o f t h e t a n k , w h i c h h a v e a p p e a r e d i n s e v e r a l e n v el o p e s , a r e n o tc o n s i s t e n t w i t h t h e f o r m o f t h e s = 1 m o d e s . I n a n e x t r e m e c a s e s h o w n i nf i g u r e 1 6 ( 2 a = 1 2, f = 1 .0 3 ) t h e g r o u n d f r e q u e n c y w a s n e a r r e s o n a n c e w i t ht h e f u n d a m e n t a l s = 1 m o d e . L a r g e w a v e h e i g h t s w e r e b u i l t u p a t t h e t a n kw a l l ; in fa c t t h e y w e r e a b o u t t w i c e th e d e p t h o f t h e f l u id a t re s t . C o n s e q u e n t l y ,t h e a s s u m p t i o n o f s m a l l m o t i o n s c a n n o t b e a p p l i e d w i t h a n y e x a c t n e s s w h a t -e v e r. T h e r e s u l t i n g s h ap e o f t h e e n v e l o p e s u g g e s t s t h a t t h e f u n d a m e n t a l o ft h e s = 0 c la s s 13 o f m o d e s w a s a l s o s e t u p ; i f s o, t h i s m u s t h a v e b e e n d u e t o a

~ T h e s = 0 m o d e s a r e d e s c r i b e d b y B e s s e l f u n c t io n s o f t h e f i r s t k i n d a n d z e r o o r d e r ,t h e J 0 fu n c t i o n s; se e L a m b (1 1), A r t s . 1 91 a n d 2 5 7. T h e y h a v e c i r c u l a r s y m m e t r y a n d a r ec h a r a c t e r i z e d b y a s y s t e m o f " a n n u l a r r i d g e s a n d f u r ro w s . " T h e r e a r e no n o d a l d i a m e t e r s .T h e f u n d a m e n t a l m o d e o f t h i s c l a s s h a s o n e n o d a l c i r c l e , t h e s e c o n d m o d e h a s t w o , e t c .T h e a b s o l u t e m a x i m u m e l e v a t i o n i n e a ch m o d e i s a t t h e c e n t e r.


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EXPERIM ENTS WIT H TA NKS SUBJECTED TO TRANSIENT MOTIONS 331

o : :~ -05~ ~ Q25

" - ~ ~ ~ -- - -~ '~ -~ J~ ~ /~ ~ - , _ - - o ' .~ -- - - - -~ ~ ~ . , .,

t ~ O ~ + 1'5r~

( b) ~ _ . _ / / _ _ _ _ _ ~ - , , o

S / .o" / I--0.5

5~

F i g . 1 2. C a l c u l a t e d w a v e p r o f i le s ; s te p m o t i o n . ( ~) A s s u m e d s t a r t i n g c o n d i t i o n s.( b ) R e s u l t a n t p r o f il e s .


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3 3 2 BU LL ET IN OF THE SEISlYOLOGICAL SOCIETY OF A~ ER IC A

t= O(a )

,-.~. ~.~. ,.,~f /f = l , 0 3 , " fz = O , S 4 , ~ = 0 , 4 9

w ~

I - 0 . 5 0 - - -

t ,I . 0 0 iLI ~ 'i

t = 1 . 5 0 " ~ / /

1 = 2 . 0 0- D

\f - 2 . 0 4 , f z - i .6 6 , " ~ - 0 9 6b )- - - - - - l J ~ _ _ _ ~ ~_

/ I

, . ~I- I - T z - O n l - - ~ ~ ,

L ~

I! ~ . ~ " ~ - ~I1- I = 2 T = - - 1

f 1 1 4 ~ _ j

05 ~ 1/)t = 2 . 5 o - -F i g . 1 3 . P h o t o g r a p h i c w a v e p r o f i le s ; o s c i l l a t o r y m o t i o n . ( a ) . f / f i = 0 .8 4 , f u n d am en ta lm o d e p r o m i n e n t . ( b) f / f i i = 0 .9 6, s e c o n d m o d e p r o m i n e n t . D = a p p r o x i m a t e e n d o f g r o u n dm o t io n . ( See f ig . 3 , c an d d . )


21/34

EXPERI~IENTS WITH TANKS SUBJECTED TO TRANSIENT IVIOTIOI~S 333

Fig. 14. Samples of wave-profile envelopes; step motion. The equilibrium or staticlevel lines S, and the envelope lines~ have been retraced in order to make them photo-graph more clearly.


22/34

IJ

rm~

0

q ~

r~

o

rm~


23/34

JI

o

0o

0

o

b~


24/34

33 6 BUL LE TI N OF TH E SEIS~CIOLOGICAL SOCIETY OF AMER ICAc o u p l i n g b e t w e e n t h e t w o c la s se s o f m o t i o n , s i n c e t h e s = 0 cl as s c a n n o t b ee x c i te d d i r e c t ly b y a h o r i z o n t a l g r o u n d m o t i o n . T h e t e s t w a s r e p e a t e d s e v e ra lt i m e s w i t h s u b s t a n t i a l l y t h e s a m e r e s u l t s .

T h e p r e s e n c e of t h e f u n d a m e n t a l m o d e o f c la s s s = 0 w a s o b s e r v e d i n a llt a n k s w h e n e v e r t h e m o t i o n s w e r e r e l a ti v e l y l ar ge , w h i c h m e a n s t h a t i t w asm o s t n o t i c e a b le w h e n t h e e q u i l i b r iu m d e p t h s w e r e s m a ll a n d w h e n t h e g r o u n df r e q u e n c y w a s n e a r r e s o n a n c e w i t h t h e f u n d a m e n t a l s - 1 f r e q u e n c y . O n e c a ne a s i ly v i s u a l iz e t h a t i d e n t i c a l w a v e s o f f u n d a m e n t a l s - - 1 t y p e w il l b e a c c o m -p a n i e d b y m u c h h i g h e r f lu i d v e l o c i ti e s i n a t a n k o f s h a l lo w d e p t h t h a n i n as i m i l a r t a n k o f l a r g e d e p t h . T h e t e n d e n c y i s f o r a s y m m e t r i c a l s e t o f v e l o c i t yc o m p o n e n t s t o b e g e n e r a t e d i n d ir e c ti o n s p e r p e n d i c u l a r t o t h e x - z p l a n e ; s u c ha s e t is in c o n f o r m i t y w i t h t h e r e q u i r e m e n t s o f t h e s " = 0 m o d e s . S i n c e i t w a sf o u n d t h a t t h e s e c e n t r a l m o t i o n s b e c a m e s m a l le r a s t h e s iz e o f t h e t a n k w a si n c r e a s e d , t h e y p r o b a b l y a r e n o t o f m u c h c o n s e q u e n c e i n f u ll - sc a l e t a n k s .

MAXIMUM WAVE HEIGHTSA b o u t 2 5 0 s e t s o f t e s t c o n d i t i o n s w e r e u s e d ; t h e r e w e r e u s u a l l y t w o s e p a r a t et e s t s f o r e a c h c o n d i t i o n , a n d f r o m t w o t o f o u r t a n k s w e r e r e p r e s e n t e d i n e a c ht e st . F r o m t h e r e s u l ti n g e n v e lo p e s t h e m a x i m u m w a v e h e i g h t s a n d t h e r a d ia ll o c a ti o n s o f t h e m a x i m a h a v e b e e n m e a s u r e d .S t e p m o t i o n . - - T h e e x p e r i m e n t s h a v e b e e n l i m i t e d to a c o m p a r a t i v e l y s m a l lr a n g e o f g r o u n d f r e q u e n c y a n d t o t h e t a n k d i a m e t e r s o f 12 a n d 2 3 i nc h e s.C h a n g e i n f l u i d d e p t h , h / a : F i g u r e 1 7 s h o w s t h e e f f e c t o f v a r i a t i o n i n h / aa n d i n f . A s t h e d e p t h s a r e in c r e a s e d t h e w a v e h e i g h t s i n cr e a se , b u t b y n om e a n s l i n e a r l y ; i n f a c t , w h e n h / a i s m a d e l a r g e r t h a n u n i t y n o v e r y a p p r e -c i a b l e c h a n g e in ~ o c c u r s . I t i s o b v i o u s t h a t the ef fect of depth on wave heightdoes not fol low a s imp le re lat ionship and that ex trapolat ion to fu l l scale should beon the basis of equivalent h/a .C h a n g e i n m a x i m u m g r o u n d v e l o c i ty , V : T h e e f f e ct o f c h a n g e i n V , a n dt h e r e f o r e i n t h e m a x i m u m g r o u n d a m p l i t u d e A , is sh o w n i n f ig u re 1 8. T h ee f f e c t i s n e a r l y l i n e a r w h e n f = 2 . 0 4 c y c l e s p e r s e c o n d , b u t w h e n f - - 1 .0 3t h e r e is l i n e a r i t y o n l y u p t o V = 4 a n d b e y o n d t h a t t h e w a v e h e i g h t i n c re a s e sm o r e r a p i d l y . A r e S x a m i n a t i o n o f t h e c o r r e s p o n d i n g e n v e l o p e s i n fi g u re 1 4, a ,i n d i c a t e s t h a t t h e n o n l i n e a r i t y i s a s s o c i a t e d w i t h l a r g e m o t i o n s , f o r e x a m p l e ,~ = 0 : 3 3 h w h e n V = 8 . T h e c o n c l u s i o n i s t h a t accurate extrapolat ion isl imi ted to mot ions which can be assumed to be smal l . H o w e v e r , t h e r e i s n og e n e r a l g u i d e t o w h a t v a l u e s o f ~ / h c a n b e a s s u m e d " s m a l l. " A n e x t r e m e c a sei s r e p r e s e n t e d i n f ig u r e 14 , c , i n w h i c h t h e v a r i a t i o n i n w is a p p r o x i m a t e l yl i n e a r e v e n t h o u g h a t V = 8 , ~ = 0 . 5 h .C h a n g e i n t a n k d i a m e t e r : F r o m f i g u r e 1 9 t w o c o m p a r i s o n s b e t w e e n t h et a n k s c a n b e d r a w n , o n e f o r t h e d e p t h h / a - - 1 , t h e o t h e r f o r h / a - - 1 / 2 . T h e

r e f e r e n c e f r e q u e n c y f i t a k e s f o u r v a lu e s , e a c h r e l a ti n g t o o n e o f t h e c o m b i n a -


25/34

I I I I I f I I I

0

~g

~ - ~ 2 , ' - . . / " - - . . W ",~

I i I ~ I I

~ 04 a

z~. /k y

I f 1 I f I I

~ o

b2"~-tI oa ~ _ o


26/34

3 38 B U L L E T I N O F T I - IE S E I S ! V I O L O G I C A L S O C I E T Y O F A M E R I C A .t i o n s o f t a n k s iz e a n d f l u i d d e p t h . C o m p a r i n g t h e w a v e h e i g h t s a t l ik e v a l u e so f f / f i , i t is f o u n d t h a t w h e n h / a = 1 t h e h e i g h t s in t h e 2 3 - in c h t a n k a r e f r o m1 .3 7 t o 1 .4 2 t i m e s t h e h e i g h t s i n t h e 1 2 - in c h ta n k . W h e n h / a = 1 / 2 t h e f a c t o r sr a n g e f r o m 1 .4 3 t o 1 . 57 . H e n c e , with maximum ground velocity constant, thewave heights vary app roxim ately as the square root of the tank diameter ( % /~ -/1 2= 1 . 3 9 ) .

I t h a s a l r e a d y b e e n m e n t i o n e d ( in fo o t n o t e 1 2 ) t h a t s o m e t e s ts w e r e m a d ei n w h i c h t h e a n c h o r s p ri n g s w e r e r e m o v e d f r o m t h e t a b l e , t h e r e s u l t i n g g r o u n d

( a ) v " = 31.5

I~ mI1, 0

O .5

0, 1 I 1 I I T T2 ~ 4 , 5 6 7

Jb )

1 I I I0 0 , I 0 2 0 . 5 - ~ - t 1 ,0 1 .5 2 0F i g . 2 0 . E f f e c t o f ta n ] ( d i a m e t e r o n w a v e h e i g h t ; z e r o g ro u n d fr e q u e n c y . M a x i m u m w a v eh e i g h t s ( a ) re s u l t i n g f r o m g r o u n d m o t i o n ( b ) .m o t i o n b e i n g m o r e n e a r l y o f i m p u l s i v e t y p e . I f t h e m a x i m u m w a v e h e i g h t so c c u r r i n g i n th e s e t e s t s a r e p l o t t e d a s a f u n c t i o n o f t h e s q u a r e r o o t o f t a n kd i a m e t e r , a n e a r l y l i n e a r r e l a t i o n s h i p r e s u l t s ( f i g . 2 0 ) .Oscil latory motion.- - I t i s m u c h m o r e d i f f ic u l t t o g e n e r a li z e a b o u t t h e e f f e c to f t h e o s c i l l a t o r y m o t i o n s i nc e it is n e i t h e r a s i m p l e " s t e p m o t i o n " n o r as t e a d y s t a t e . T h e p r i n c i p a l c o m p l i c a ti o n is d u e to t h e v a r i a t i o n i n t h e n u m b e ro f c y c l e s i n t h e g r o u n d m o t i o n . E v e n w h e n t h e s y s t e m is a s im p l e , s i n g l e-d e g r e e - o f - f r e e d o m o s c i ll a to r t h e r e s p o n s e i s a c o m p l i c a t e d f u n c t i o n o f t h en u m b e r o f fo r c i n g c y c le s . 14 T h e r e s p o n s e o f a n u n d a m p e d o s c il la t o r, w h i c h i s

14 See th e pap er by It . A. Williams (13) in which he shows th e e ffects of the same groundm otion as used here, on a single-degree-of-freedom sys tem w ith dam ping.


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EXPERII~IE,NTS W ITH TANKS SUBJECTED TO TRANSIENT MOTIONS 339

2 .0

1.5r t m

1.0

0 .5

2 o = 2 3"V'=2" x-~-hG l-~ X ~ OSC ILLATOR Y GR OU ND MOT ION

ff WX, ,'? , . > ( ~ , ~ , .. ~ ,, . , . ~ = ~ . . . . . . - . - z - n , ~%h I , . . " ' ~ , / ~ . X ? ' ~ . . ~ - - - - . ' ; ~ " ~ ' o . . . . . = " " ~ - " S . . ~ . .- - d = ~ ,, , ~ . - . . _ ~ . . " ~ . .

- ~ " - - - -' ~- -- ~ " * - - - J - * - - ~ ' . S , , - - ~ , , , = ~

, i , , I , i, I'I f il , il , I I ~ I , I ~ ,0 . 5 1 .0 1 .5 2 . 0 ~ f Z . 5 ~ , -0 3 . 5 4 . 0 4 . 5 5 . 0F i g . 2 1 . E f f e c t o f c h a n g e i n s t a t i c d e p t h a n d i n g r o u n d f r e q u e n c y o n w a v e h e i g h t ; "o s c i l l a t o r y m o t i o n . T h e n a t u r a l f re q u e nc i es o f t h e t h r e e s y s t e m s h a v e b e e n i n d i c a t e da l o n g t h e g r o u n d - f r e q u e n c y a xi s . T h e c u r v e r e l a t i n g t o e a c h d e p t h h a s t w o b r a n c h e s , o n eg i v i n g t h e m a x i m u m w a v e h e i g h t ~ , t h e o t h e r g i v i n g t h e w a v e h e i g h t a t t h e t a n k w a l l,h a. I n t h e l o w e r f r e q u e n c y ra n g e t h e m a x i m u m o c c u r s a t t h e w a l l , i .e . , n ~ = w , a n d t h et w o b r a n c h e s c o i n c i d e . ( I t m u s t b e k e p t i n m i n d t h a t t h e n u m b e r o f c y c l e s i n t h e g r o u n dm o t i o n i s n o t c o n s t a n t .)

2 0 : 2 3 x ~ -= 1OSCILLATORY GROUND M OTIONr l , m . V . = 3 h _ . _ L~ ' ~ ' - 2~ r r , ' V - = 2

; = =r

0 0,5 ID 1.5 2.0 2,5 ~ D 3.5 4.0 4.5 5.0~ fF i g . 2 2 . E f f e c t o f c h a n g e i n m a x i m u m g r o u n d v e l o c i t y o n w a v e h e i g h t ; o s c i l l a t o r ym o t i o n . T h e c o m p a r i s o n of t w o s e t s o f te s t s s h o w s f a i r a g r e e m e n t w i t h t h e p r e d i c t e d

l i n e a r v a l u e o f 3 /2 .a t resonance w i t h a s i m p l e h a r m o n i c d i s t u r b a n c e v a r i e s l i n e a r ly w i t h t h en u m b e r o f c y c le s i n t h e d i s t u r b a n c e , b u t a w a y f r o m r e s o n a n c e t h e m a x i m u mr e s p o n s e m a y h a v e l i t t l e a p p a r e n t r e l a t i o n t o th e n u m b e r o f c y c le s . I t c a n b ee x p e c t e d t h a t t h e e f f ec t on t h e f lu i d s y s t e m w i ll b e c o n s i d e r a b l y m o r e c o m p l i -c a t e d .

R e s o n a n c e : T h e o s c i l l a t o r y m o t i o n r e s u l t s i n a p r o m i n e n t r e s o n a n c e w i t ht h e f u n d a m e n t a l n a t u r a l m o d e a s i n d i c a t e d b y f ig u re 21 , b u t t h e r e a r e n od e f in i t e " r e s o n a n c e p e a k s " a s s o c i a t e d w i t h t h e h i g h e r n a t u r a l m o d e s . T h e l a c ko f s h a r p l y d e f in e d , h i g h e r - m o d e p e a k s i s p a r t l y d u e t o t h e c l o se n e s s o f t h e


28/34

3 4 0 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF A~VIERICA

N = D f

C a ). ; > - "

+I i I i , l I I | I I I I I I

v.._.~.32 3 f " / ~ - ~ , f ' ~ . - ~ - / "*1 , O ~ O I L L A T O R Y G R O U N O M O T IO N

2.0 , \ " % 47,=j/ ) r \ X ( b ),o:--."/:.; " % t ......

i - - 4 " " ~ ' " ~ \ ~ " , ' - 2 ) . - * " " . ~ " , , ~ . . . .L . . . t / . - . . . . , \ . . . . ~ , . - - . . . . . ~ . " -. . . . . ~ _ ~ _ .: . ~ .~ . . .

o l , I , I , I I ~ 1 4 ~ , I I , ~ / ~ 1 , I , ~ l ' l , i I , , ~ i , , , ~ , ". . . . . . r 2 1 0 ~ i 4 7 = . . . % 1 ~ ' ' . . . . 0 4 , ; . = . . . . ~ ~ . 0 '= 1 ~ ` " ' , 0

)2

)5

f

l z

&l

0

" , ,L ; " ~ r n| ~ ~ ~q = O SC I LLATO R Y G R O U N D M O TI O N/ / ~ . ~ ~ A 2 0 =4 7 / " ~ ., - 4 / 5 , ~ r ( . , ~ / " x ( ~ )

< 4 ~ - . / ~ , \ , , z f < 0 " ~ - - ~ .

o . , ,' ~ , V " . ; " " ~ " N - ~ - " : " ~ ': ~ - . - . - ~ . ~ ~ . . . . . - . . . . . . . . . " t : - - : ' - 1I . . . . / I ] i47 1~23 I I ' , 'Z' -"-e ; ' ' - ' -~ I I '6 ~' t : .~.L. .=, I ' :_- . : , : . . . . . . ~ l - . - - - ~ ~

o l o ' ! I , , i , ; ~ I , x r ~ i , I . ~ = , , z ~ . I , , = ; i , " - - - - ' , " ~ - - 7 . o - ",5 Io 15 D 2~ 3,O 3.5 4,0 45 5.D 5,5 6.0 [6 '" ' _ _ , ~ 1 , ~ ~ t . x l , = 1 ': =Fig. 23. Effect of tank diame ter on wave height; oscillatory motion. (a) Ampl itude Aa n d n u m b e r o f c y c l e s N o f t h e g r o u n d m o t i o n . ( b ) Static depth i s l a r g e , h / a = 1 . ( c ) Staticd e p t h i s s h a l l o w , h / a = 1 / 4 .


29/34

EXPEI%I]CIENTS W IT H TA NK S SU BJEC TED TO TI~ANBIENT I~[OTIONS 341na tur a l f r e que nc ie s a nd to the s ha r p tun ing whic h c o ns e que nt ly i s r e qu ir e dfor resonance , 1~ I t can be conc lud ed tha t p r om i n e n t r eso n a n c e w i t h m o d es h i g h ert h a n t h e u n d am e n t a l , ow i n g t o a t r a n s i e n t d i s tu r b a n c e o f a ew c y c l es, i s n o t t o b eexpec ted . Fur the r mo r e , ma x imu m wa v e he ig hts Vm o c c ur a t the ta nk wa l l o n lywh e n the g r o und fr e que nc y i s l e s s tha n a bo ut 5 /4 f z-

1 . 5

0 5 h , I~o ~ ~

0.1 I , I I IO , I 2 3 ~ 4 5 6 7F i g . 2 4. E f f e c t of t a n k d i a m e t e r o n a b s o l u t e m a x i m u m w a v e h e i g h t s ; o s c i l l a t o r y m o t i o n :( 1) n e a r r e s o n a n c e w i t h t h e f u n d a m e n t a l m o d e ; ( 2) a t g r o u n d f r e q u en c i e s g r e a t e r t h a nthe fundamental natural frequency.Change in h / a : Figure 21 also shows the effect of changing the fluid depth.

The most pronounced differences occur when f < fii~, owing mainly to themanner in which the natural frequencies are affected by a change in h / a .

Change in V: A change in maximum ground velocity can be expected tohave approximately the same relative effect at any ground frequency, pro-vided t ha t the fluid motions are not excessively large. I n figure 22 a comparisonhas been made between tests on the 23-inch tank when V = 3 with tests whenV -- 2; two values of h / a are represented.Change in tank diameter: The results of two series of tests, for whichh / a = 1 and 1/4, are shown in figure 23. The two families of curves differ con-siderably in detail, but as far as general shape of the curves and order ofmagnitude of wave height are concerned, they are much alike. The variationwithin each family is reasonably regular, except in the heights of the funda-mental-mode resonance peaks for the depth h / a = 1/4. There are two reasons

lb T h e h i g h e r n a t u r a l f r e q u e n c ie s o f t h e s y s t e m v a r y r o u g h l y a s t h e s q u a r e r o o t o f t h en u m b e r o f t h e n a t u r a l m o d e , w h e r e a s i n m o s t m e c h a n i c a l s y s t e m s t h e f re q u e n c ie s v a r ym o r e n e a r l y a s e i t h e r t h e f i r s t o r t h e s e c o n d p o w e r o f i . T h e e f fe c t o f t h e s a m e o s c i l l a to r yg r o u n d m o t i o n o n a m e c h a n i c a l s y s t e m h a v i n g m a n y d e g r e e s o f f r e e d o m w i l l b e f o u n d i nr e f er e n c e ( 10 ), w h e r e i t is s h o w n t h a t d i s t i n c t r e s o n a n c e s o c c u r r e d w i t h n a t u r a l m o d e s a sh i g h a s t h e f o u r t h .


30/34

3 4 2 BU LL ET IN OtP TH E SEISiIKOLOGICAL SOCIETY OF A]~EI~ICAf o r t h i s i r r e g u l a r i t y ; o n e is t h e l a r g e h e i g h t O f t h e w a v e s r e l a t i v e t o t h e s t a t i cd e p t h , a n d t h e o t h e r is t h e p r e s e n c e o f i n a c c u r a c i e s i n t h e e x p e r i m e n t a l g r o u n dm o t i o n a t l o w f r e q u e n c i e s .

I f th e a b s o l u t e m a x i m a o f t h e f u n d a m e n t a l m o d e p e a k s f o r t h e c a s e h / a = 1b e d i v i d e d b y t h e n u m b e r o f c y c le s i n t h e g r o u n d m o t i o n a n d p l o t t e d a g a i n s tt h e s q u a r e r o o t o f t h e t a n k d i a m e t e r , a n a p p r o x i m a t e l y li n e a r re l a t i o n s h i p isf o u n d (fig . 2 4 ). C o r r e s p o n d i n g v a l u e s f o r t h e d e p t h h / a = 1 / 4 a l so s h o w ar e a s o n a b l y l i n e a r r e la t i o n , w i t h t h e e x c e p t i o n o f t h e p o o r l y d e f i n e d v a l u e f o r

Lo' V - - 2 ,5 2o=1 2 , .25L e OSCILLATOR._~YROUN.~D OT IO N

,

0.2 - "

o . , . . .. . .. . .. . .. . .. . .. . . . . .. . r . . .. . .. . .. . .. . . . . . . .I ~ " - ~ ' ~ " "O ' I , I , I , , ; , , , ,0,5 1 .0 1 ,5 2 .0 2 .5 ~ .0 3 .5 4 ,0 4 .5~ f f ~

Fi g . 25. l ~ad i a l l oca t i ons of m~xi m um w~ve he i gh t ; osc i l l a t o ry mot i on . Th e re sonantf requencies and th e locat ions of the maximum cres ts for modes I to V h ave been indicated.t h e 4 7 - i n c h t a n k . A s i m i l a r r e l a t i o n is f o u n d a m o n g t h e a b s o l u t e m a x i m a a s s o -c i a te d w i t h t h e h i g h e r n a t u r a l m o d e s .

T h e f o ll o w in g e m p i ri c al r e la t io n s f o r m a x i m u m w a v e h e i g h t d u e t o a no s c i l la t o r y m o t i o n ( V c o n s t a n t ) a r e b a s e d o n f i g u re 2 4 :

~.~ = C ' ~ / 2 a V N (4 )F o r a b s o lu t e m a x i m u m r e sp o n s e i n t h e f u n d a m e n t a l m o d e f = 0 .9 f~ z h / a =1 , C = 0 .07 ; h / a = 1 / 4 , C = 0 . 1 ; w h e r e ~ o c c u r s a t t h e t a n k w a l l (r = a ) .F o r a b s o l u t e m a x i m u m r e s po n s e w h e n f > f~ : h / a = 1 , C = 0 .0 15 ; h / a =

1 / 4 , C = 0 . 0 2 ; w h e r e nm o c c u r s a t r < a / 3 . I t i s e v i d e n t t h a t f o r e q u a l N a n de q u a l V t h e m a x i m u m w a v e h e i g h t s a t t h e t a n k w a l l a r e a b o u t f iv e t i m e s t h ev a l u e o f t h o s e w h i c h o c c u r a t r < a / 3 .

W h i l e t h e r e la t io n s j u s t g i v e n a re n o m o r e t h a n r o u g h a p p r o x i m a t i o n s , t h e yd o i n d ic a t e th e o r d e r of m a g n i t u d e o f ~ . T h e y m u s t b e u se d w i t h d i s c r e ti o nw h e n N i s s i g n i fi c a n t ly la r g e r t h a n t h e v a l u e s s h o w n i n fi g u re 23 , a. I n o r d e rt o a p p l y t o l a r g e v a l u e s o f N , t h e r e l a t i o n s s h o u l d i n c l u d e a n e x p o n e n t i a ld a m p i n g f a c t o r , s i n ce a s N is i n c r e a s e d t h e e f f e c t o f d a m p i n g b e c o m e s in -


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E X P E R I M E N T S W I T I { T A N K S S U B J E C T E D T O T R A N S I E N T M O T IO N S 3 4 3c r e as i n gl y i m p o r t a n t . T h e m o s t i m p o r t a n t e f fe c t o f l a rg e N , h o w e v e r , i s t h a tt h e f l u id m o t i o n s m a y b e c o m e " l a r g e " w i t h c o n s e q u e n t f a il u re o f t h e l i n e a rr e l a t i o n s h i p . T h i s m a y a l so b e t r u e o f l a r g e V (fig . 1 8 ).

I f t h e s u b s t i t u t i o n V - - 27r A ( f / f i ) f I b e m a d e i n e q u a t i o n ( 4 ), w h e r e f i m a yb e d e t e r m i n e d f r o m e q u a t i o n ( 2) , i t is a p p a r e n t t h a t t h e t a n k d i a m e t e r t e r m s% / ~ d i sa p p e a r a n d t h a t fo r c o n s t a n t f / f~ a n d h / a t h e m a x i m u m w a v e h e ig h ti s g i v e n b y

~m = C L A N , (4a )w h i c h m e a n s t h a t , i f t h e g r o u n d m o t i o n c r i te r i o n is c o n s t a n t m a x i m u m a m p l i -t u d e A , t h e w a v e h e i g h ts a r e i n d e p e n d e n t o f t h e t a n k d i a m e t e r . T h i s is t r u e ,h o w e v e r , o n l y f o r m a x i m u m r e s p on s e i n t h e f u n d a m e n t a l m o d e .R a d i a l l o c a t i o n s o f v ~ : T h e l o c a t i o n s o f t h e m a x i m u m c r e s t s a r e s h o w n i nf ig u r e 25 f o r t h e d e p t h h / a = 1 w h e r e rm/a, t h e r a t i o o f t h e r a d i u s a t w h i c h t h ec r e s t o c c u rs , t o t h e w a l l ra d i u s , h a s b e e n p l o t t e d a g a i n s t f / f i . T h e e x i s t e n c e ofs o m e a s y m m e t r y i n t h e w a v e m o t i o n , a s i n d i c a t e d b y t h e f a c t t h a t t h e c r e s ti n t h e 0 = 0 p l a n e d o e s n o t f a l l a t t h e s a m e v a l u e o f r a s d o e s t h e c r e s t i n t h e0 = 180 p l a n e , is d u e t o t h e t r a n s i e n t n a t u r e o f t h e g r o u n d m o t i o n . T h ee x p e r i m e n t a l p o i n ts h a v e n o t b e e n s h o w n . T h e r e w a s m u c h le ss " s c a t t e r i n g "i n t h e l o c a t i o n o f t h e c r e s t s t h a n t h e r e w a s i n t h e i r h e i g h ts . S i m i l a r r e la t i o n -s h ip s w e r e f o u n d f o r t h e d e p t h s h / a = 1 / 2 a n d 1 / 4 .

SUMMARY AXD CONCLUSIONSEffective mass an d overturning m om en t . - - [ 1 :] W h e n t h e t a n k i s not covered, t h ee f fe c t iv e m a s s a n d o v e r t u r n i n g m o m e n t d u e t o t h e b e h a v i o r o f t h e f l u id c a n b ec a l c u la t e d f r o m h y d r o d y n a m i c t h e o r y ( 3 ). T h e a g r e e m e n t b e t w e e n t h e o r y a n de x p e r i m e n t is g oo d . F o r p r a c t i c a l p u r p o s e s t h e r a t i o o f e f f e c ti v e h y d r o d y n a m i cm a s s t o a c t u a l m a s s o f t h e f l u id c a n b e c a l c u l a t e d w i t h i n a b o u t 5 p e r c e n t o fe r r o r f r o m t h e f o l l o w i n g s im p l e r e l a t i o n s :16

0 < - h < 1 , m l _ 0 . 5 7 h . 1 < h < 2 . 5, m l _ 0 . 3 9 + 0 . 1 8 h (5)a m a ' a m aT h e l o c a t i o n o f t h e e e n t r o i d o f th e e f f e c t i v e h y d r o d y n a m i c m a s s i s g i v e n t ot h e s a m e d e g r e e o f a p p r o x i m a t i o n b y :17

h 0 .36 + 0 .02 7 a . (6 )[ 2 : ] W h e n t h e t a n k h a s a rigid cover, t h e r e m o v a l o f a s m a l l p r o p o r t i o n o f

t h e f l ui d m a k e s t h e t a n k e f f ec t iv e l y a n o p e n o n e. T h e o v e r t u r n i n g m o m e n tis, ~7 C o m p a r e w i t h f i g u r e s 2 a n d 3 o f r e f e r e n c e ( 3 ).


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34 4 BUL LETIN OF THE SEISMOLOGICALSOC IETY O1~ AM ERICAr e d u c e s e v e n m o r e r a p i d l y t h a n t h e e f f ec t iv e m a s s d o e s, s i n ce i t is b e i n g af u n c t i o n n o t o n l y o f t h e e f f e c t i v e m a s s b u t a l s o o f t h e e f f e c ti v e c e n t r o i d a lh e i g h t .

G r a v i t y w a v e s . - - [ 3 : ] T h e r e s u l ts o f t h e w a v e s t u d i e s a r e a p p l i c a b l e t o f l ui dso t h e r t h a n w a t e r , b e c a u s e : a ) , as i s w e l l k n o w n , t h e g r a v i t a t i o n a l n a t u r a lm o d e s a r e i n d e p e n d e n t o f t h e f lu id d e n s i t y ; b) th e v i s c o u s d a m p i n g i n t h es y s t e m i s i n h e r e n t l y s m a l l ; a n d c ), t h e s u r f a c e te n s i o n e f f e c ts a r e s e c o n d a r y .[ 4 :] T h e w a v e h e i g h t s re s u l t in g f r o m t h e s t e p m o t i o n h a v e b e e n c a l c u l a t e df o r on e e x a m p l e w i t h r e a s o n a b l y g o o d a g r e e m e n t w i t h t h e e x p e r im e n t . T h ec a l c u l a t i o n a s s u m e s a p a r t i c u l a r s t a r t i n g c o n d i t io n o f t h e n a t u r a l - m o d e o s ci ll a-t i o n s w h i c h is c o n s is t e n t w i t h t h e i m p u l s i v e n a t u r e o f t h e s t a r t o f t h e g r o u n dm o t i o n . [ 5 :] T h e e f fe c t o f s t a t i c fl ui d d e p t h i s a c o m p l i c a t e d o n e a n d t h e c o m -p a r i s o n o f t a n k s o f d i f fe r e n t d i a m e t e r s s h o u l d g e n e r a l l y b e m a d e o n t h e b a s i so f e q u i v a l e n t d e p t h , t h a t i s, e q u a l v a l u e s o f h / a . [ 6 : ] W h e n t h e c r i t e r i o n f o rt h e g r o u n d m o t i o n is c o n s t a n t m a x i m u m h a r m o n i c v e lo c i t y , th e m a x i m u mw a v e h e i g h t c a n , f o r m o s t p r a c t i c a l p u r p o s e s , b e a s s u m e d t o v a r y a s t h e s q u a r er o o t o f t h e t a n k d i a m e t e r . [ 7 :] A c h a n g e i n t h e m a x i m u m h a r m o n i c v e l o c i t y Vo f t h e g r o u n d m o t i o n s h o u l d r e s u lt i n a d i r e c t l y p r o p o r t i o n a l c h a n g e i n w a v eh e i g h t, p r o v i d e d t h a t t h e f l u id m o t i o n s r e m a i n " s m a l l . " S i n ce t h e w a v e h e i g h t st e n d t o v a r y a s t h e s q u a r e r o o t o f t h e t a n k d i a m e t e r , i t w o u ld a p p e a r t h a tt h e q u e s t i o n o f s m a l l m o t i o n i s o f m u c h l es s i m p o r t a n c e i n f u ll -s iz e t a n k s t h a ni n t h e m o d e l s . [ 8 :] T h e r a d i a l lo c a t i o n of t h e m a x i m u m w a v e h e i g h t i s n e a rt h e l o c a t i o n o f t h e m a x i m u m h e i g h t o f t h e m o s t g r e a t l y e x c i te d n a t u r a l m o d e .W h e n t h e f o r c i n g f r e q u e n c y i s l es s t h a n a b o u t f i v e - f o u r t h s of t h e f u n d a m e n t a ln a t u r a l f r e q u e n c y t h e m a x i m u m w a v e h e i g h t o c c ur s a t th e w a ll o f t h e t a n k ,b u t f o r g r e a t e r f o r c i n g fr e q u e n c ie s i t o c c u r s a w a y f r o m t h e w a l l a n d c o n s e -s e q u e n t l y i s o f l e ss p r a c t i c a l i m p o r t a n c e . [ 9 :] W e l l - d e f in e d r e s o n a n c e o c c u r so n l y w i t h t h e f u n d a m e n t a l m o d e . T h e h i g h e r n a t u r a l f r e q u e n c i es a r e c lo s el ys p a c e d , w i t h t h e r e s u l t t h a t v e r y s h a r p t u n i n g i s r e q u i r e d f o r r e s o n a n ce .

N U M E R I C A L E X A M P L E SI. A r ig id tan k of 120 f t . inside d iam eter , conta ining w ater to a dep th of 90 f t . , ex-pe r iences an im pu ls ive hor izon ta l acce le ra t ion o f 0 .20 g . F ind the dynamic fo rces andm o m e n t s d u e t o t h e w a t e r (3) , and compare wi th those fo r the sam e tank when r ig id lycovered , when the re i s ze ro c lea rance be tween the f lu id and the cover : Com pare wi thsha l low tank o f the sam e vo lume . Es t im a te the requ i red "open tank " c lea rance .i . To t a l w e ig h t o f w a t e r , m g ~ 63 .6 X l0 G bs .i t. Ef fec t ive hydrody nam ic mass o f wa te r : o p e n t a n k ( h / a = 1.5) , m l / m = 0.682 (fig. 2 ofref , 3) or, by equation (5) , m l / m = 0.66; c o v e r e d t a n k , m l / m - - * l . O . (The r ig id i ty o f a coverwhich is 120 f t . in d i am eter is , of course , quest ionable .)i i i. Max im um hor izon ta l fo rce due to the wa te r : o p e n t a n k , X 1 ~ 63.6 X 106 X 0.682 X0.20 = 8.67 X 106 lbs., applie d a t the lev el ~1 = 0.393 h = 35.4 ft. (fig. 3 of ref. 3), or, b y

1 . 00eq ua tio n (6) , ~1 -----35 ft.; c o v e r e d t a n k , X = ~ X X I ~ 12.7 X 106 lbs., app lied a t t helevel ~ -- 0.5 X h = 45 ft.


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EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS 34 5iv. Maximum overturning moment due to the water: open tank . [1:] Moment of hori-

zont al force, M1 = X1 ~1 = 307 X l0 s lb. ft. [2:] H yd ro dy na mi c couple acti ng on bo tt om oftank , N1 = 142 X 106 lb. ft. (fig. 4 of ref. 3). [3:] Tot al ove rt urn ing mom ent , M1 -F N1 =449 X 106 lb. ft. Covered tank. [1:] Moment of horizontal force, M = X~ = 571 X 106 lb. ft.[2:]The h ydrod ynami c Couple acting on the bot tom of the tank is opposed, theoreti cally,by an equal and opposite couple acting on the cover. [3:] Tot al overt urnin g mome nt,M = 571 X 106 lb. ft.

v. To th e foregoing there mu st be added the force and overturning moment due to themass of the tank itself.

vi. Maximum horizontal force due to water in an open, shallow tank of the same volume,for which, h / a = 0.5, 2a = 173 ft ., h = 43.3 ft. , m l / m = 0.305. X1 = 63.6 X 10 s X 0.305 X0.20 = 8.88 X 108 ibs., where ~ = 0.38 h = 16.5 ft. Co mpar e wit h calculati on in par agr aphiii above. Not e the great reduction in total hy drody namic force resulting from the changeto a tank of shallower design.

vii. Clearance required to reduce the effective mass to the "open tank" value. Thenecessary clearance in the 23-inch tank when the maximum impulsive acceleration wasabo ut 0.12 g was c = 0.06 X 11.5 = 0.7 in. (fig. 9). Assuming, on the ba sis of figure 20, t ha tthe clearance must va ry as the s~ua.re root of the t ank diameter, the clearance in the t ankof 120-foot diame ter is c =%//]20 X 12/23 X 0.7 = 6 in. Within t he small range of theexperiments, no significant difference was found in c as the maximum impulsive acceler-ation was varied. However, if the conserva tive assumption is made tha t c varies direct lyas the maximum 'impulsive acceleration the clearance required by an acce]cration of0.20 g bec ome s c = (0.20/0.12) 6 = 10 inches, whi ch is ver y small in re lat ion to th e dim en-sions of the tank.

II. Est ima te the maximum wave heights due to an oscillatory ground motion.i. The fundamental natural periods are, by equation (2) or figure 6: Deep tank (2a= 120 ft ., h /a = 1.5), T~ = 6.35 sec. Shall ow tan k (2a = 173 ft ., h /a = 0.5), Tr = 8.90 sec.

It has generally been thoug ht th at "st ron g" ground motions of a period longer than ab out2.0 seconds are rare. If so, th ere is little pr obabi lit y of resonance with the fundamentalmode in large tanks. (For a tank only 12 feet in diameter and having a rela tive d epth ofh /a = 1.5, TI = 2.0 sec.) There is some evidence, however , th at significant moti ons of verylong period do occur; Housner has computed in one case a motion of 15.2 sec. period and0.85 cm. amplitude (14).

it. Assuming that the 120-ft. tank experiences the following ground motion, which isnear resonance wit h the fundame ntal natura l frequency, f = 0.9 X fi = 0.9/6.35 = 0.14cycles per second, V = 1.0 inches per second (A = V/2~rf = 1.1 inc hes) , N = 2.0 cycl es( D = N / f = 14 second s), th e max imu m wave he ig ht will be ~,~ -- 0.7%/120 X 12 X 1.0X 2.0 = 5 inches (equati on 4). This tri vial wav e height occurs at the tan k wall. (The co-efficient 0.07 was dete rm ine d at the dep th h /a = 1, but the behavior of the sys tem is sub-stant iall y the same at all values of h/a ~ 1 )

iii. Assuming f> fi, such tha t ~ occurs at r < a/3 , and lett in g V = 3.0 and N = 5.0, themax im um wave hei ght is vm = 0.015%/120 X 12 X 3.0 X 5.0 = 9 inches, whi ch also istrivial.

N o c l a i m i s m a d e f o r g r e a t a c c u r a c y i n t h e e s t i m a t i o n o f w a v e h e i g h t a n d o f" o p e n t a n k " c l e a r an c e . H o w e v e r , i t is b e l i e v e d t h a t t h e i r o r d e r s of m a g n i t u d ec a n b e c o r r e c t l y e s t i m a t e d o n t h e b a s i s s h o w n . I t i s e v i d e n t t h a t t h e q u e s t i o no f h y d r o d y n a m i c f o r c e w i l l g e n e r a l l y b e o f m u c h g r ea t er im p o r t a n c e t h a n t h a t o fw a v e h e i g h t . W a v e s o f s i g n i f i c an t h e i g h t a p p a r e n t l y a r e n o t t o b e e x p e c t e d in


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3 4 6 BULLETIN OF Tt IE SEISMOLOGICALS O C I E ~ O F A M E E IC Al ar g e t a n k s s u b j e c t e d t o e a r th q u a k e s . O n t h e o t h e r h a n d , l a r g e w a v e s m a yo c c u r in s m a l le r t a n k s w h e n s u b j e c t e d : a ) t o la r g e a m p l i t u d e m o t i o n s n e a rt h e t o p o f a b u i ld i n g , o r b) t o la r g e m o t i o n s a r i s in g f r o m e l a s ti c s u p p o r t o f t h et a n k o n a m o v i n g g r o u n d ( e le v a t e d s to r a g e ta n k s ) .

R E F E R E N C E S( I ) " W a t e r P r e s s u r e i n a T a n k C a u s e d b y a S i m u l a t e d E a r t h q u a k e , " b y L e a n d e r M .H o s k i n s a n d L y d i k S . J a c o b se n , Bull. Seism. Soc. Am., 24:1-32 (1934).(2 ) " A L a b o r a t o r y M o d e l S t u d y o f th e B e h a v i o r o f L iq u i d - F i ll e d C y l i n d r i c a l T a n k s i nE a r t h q u a k e s , " b y B r o o k s T . M o r r is , t h e s i s f o r D e g r e e o f E n g i n e e r, S t a n f o r d U n i -vers i ty (June , 1938) .( 3) " I m p u l s i v e H y d r o d y n a m i c s o f F l u i d I n s i d e a C y l i n d r i c a l T a n k a n d o f F l u i d S u r -

r o u n d i n g a C y l i n d r i c a l P i e r , " b y L y d i k S . J a c o b s e n , Bull. Seism. Soc. Am., 39:189-204 (1949).(4 ) " W a t e r P r e s s u r e o n D a m s d u r i n g E a r t h q u a k e s , " b y H . IV [.W e s t e r g a a r d , Proc. Am.Soe. Civil Engineers, Vol . 57, p . 1303 (November , 1931) . See a l so the discuss ion byJohn H. A. B rah t z and Ca r l H . He i l b ron i n t he same publ i ca t i on , Vol . 58 , p . 897(May, 1932) .(5 ) "On H y d r o d y n a m i c E a r t h q u a k e E f f e c t s , " b y P . W i lh . W e r n e r a n d K . J . S u n d q u is t,Trans. Am. Geophys. Union, Vol . 30, N o. 5, pp. 636-657 (O ctober , 1949) .(6 ) " E a r t h q u a k e R e s is t an c e o f E l e v a t e d W a t e r T a n k s , " b y A r t h u r C . R u g e , Trans. Am.Soc. Civil Engineers, No. 103, pp. 889-938 (1938).(7 ) "Observed V i b r a t i o n s o f S t e el W a t e r T o w e r s , " b y D . S . C a r d e r, Bull. Seism. Soc.Am., 26:69-81 (1936).( 8) " V i b r a t i o n R e s e a r c h a t S t a n f o r d U n i v e r s i t y , " b y L y d i k S , J a c o b s e n , Bull. Seism.Soc. Am., 19:1-27 (1929),( 9 ) " M e a s u r i n g E a r t h q u a k e I n t e n s i t y i n P o u n d s p e r S q u a r e F o o t , " b y H . M . W e s t e r -gaa rd , Engineering News-Record, N o. 10, p. 504 (1933).( 10 ) " E x p e r i m e n t a l l y D e t e r m i n e d D y n a m i c S h e ar s in a S i x t e e n -S t o r y M o d e l , " b y L y d i kS . J a c o b s e n a n d R o b e r t S . A y r e , Bull. Seism. Soc. Am., 28:269-311 (1938).(11) Hydrodynamics, b y S i r H o r a c e L a m b , 6 t h e d ., C a m b r i d g e U n i v e r s i t y P r e ss j 1 9 3 2, o rD o v e r P u b l i c a ti o n s , N e w Y o r k , 1 9 4 5.(12) "On S t a t i o n a r y L i q u i d W a v e s , " b y F r e d e r i c k G u t h r ie , Philosophical Magazine, Ser .4, Vo l. 50, pp. 290, 337 (1875).( 1 3 ) " D y n a m i c D i s t o r t i o n s i n S t r u c t u r e s S u b j e c t e d t o S u d d e n E a r t h S h o c k , " b y H a r r yA. W i ll iams , Trans. Am. Soc. Civil Engineers, N o. 102, pp . 838-850 (1937).

( 1 4 ) " G r o u n d D i s p l a c e m e n t C o m p u t e d f r o m S t r o n g - M o t i o n A c c e l e r o g r a m s , " b y G e o r g eW . H o u s n e r , Bull. Seism. Soc. Am., 37:299-305 (1947).( 15 ) " H y d r o d y n a m i c E x p e r i m e n t s w i t h R i g i d C y l i n d r i ca l T a n k s S u b j e c t e d t o T r a n s i e n tM o t i o n s , " Technical Report No. 8, N a v y C o n t r a c t N 6 -O r i- 15 4 ~ T a s k 1 , S ta n f o r dUni ve rs i t y , S t anford , Ca l i fo rn i a .

1951-jacobsen-hydrodynamic experiments with rigid cylindrical tanks subjected to transient montions...

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