HYDRODYNAMIC EXPERIMENTS WITH RIGID CYLINDRICAL T A N K S S U B J E C T E D TO T R A N S I E N T M O T I O N S * B y LYDIK S. JACOBSEN and ROBEaT S. AYRE ABSTRACT Four tanks, from 6 inches to 4 feet in diameter, have been subjected simultaneously to transient, horizontal "ground motions" of simplified type. The important parameters, in addition to size of tank, were depth of fluid and frequency, duration, and amplitude of ground motion. The envelopes of the gravity-wave profiles have been recorded on a vertical plane of symmetry placed within each tank in a direction parallel to the ground motion. The data include samples of the wave envelopes, photographic studies of the wave formation, maximum wave heights and the locations of these maxima, and the fluid damping coefficients. Equivalent mass and overturning moment due to the fluid have been shown for various degrees of confinement of the upper surface, from complete confinement (owing to use of a rigid cover) to a free surface. The study relates to the effect of earthquakes and other ground motions on oil and water storage tanks. The results can be extrapolated with reasonable certainty to full-scale tanks.
INTRODUCTION Previous work.--The present investigation is related to earlier work carried on in the Vibration Laboratory at Stanford University. The first of the series, published by Leander 5/[. Hoskins and Lydik S. Jacobsen in 1934 (i),I was an experimental and analytical study of the water pressures in a tank of rectangular cross section and finite length which was subjected to horizontal transient ground motions parallel to the length of the t a n k . The second investigation, made b y Brooks T. Morris in 1938, consisted largely of experiments on the gravity-wave formations in cylindrical tanks (2). The third, an analytical investigation (3), deals with the impulsive hydrodynamic pressures and velocities, and the effective mass and mass moment, in a cylindrical tank as well as around a cylindrical pier. Some less closely related investigations are those of H. M. Westergaard on the hydrodynamic pressures in a reservoir of rectangular cross section and infinite length in which one end (the dam) is subjected to a horizontal simple-harmonic oscillation (4), and of P. Wilh. Werner and K. J. Sundquist on the hydrodynamics of fluid containers of a variety of shapes when the containers experience simple-harmonic motions (5). In the experimental work of Arthur C. Ruge on the effects of earthquakes on elevated water tanks (6) the tank is elastically supported, with the result that a coupling exists between the fluid system and the tank-tower system. The existence of this coupling has been demonstrated on a full-scale basis in experiments conducted b y 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 the end of this paper. [ 313 ]
314
BULLETIN OF THE SEISMOLOGICAL SOCIETY OF A1VfERICA
Y
.y (a)
X
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(b)
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EXPERIMENTS
WIT]=[ T A N K S S U B J E C T E D
TO T R A N S I E N T
MOTIONS
315
ship of the Office of N a v a l Research, to whom a more detailed report has already been m a d e (15). Mr. E d w a r d P. Hollis, Miss Gudrun Gytel, and Mr. Richard Warrick assisted in various par~s of the study.
r, 0, z a
h c 7"m
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NOTATION Cylindrical location coSrdinates Radius of tank wall Static depth of fluid Clearance between static surface of fluid and tank cover Elevation of fluid surface with respect to static level, z = h Maximum wave height Maximum wave height at tank wall, r'= a Maximum wave height at center of tank, r = 0 Radial location of ~,~ Ratio of effective hydrodynamic mass to total mass of fluid of static depth h i Subscripts designating natural mode of gravitational oscillation of fluid, class
s-~l Natural periods of fluid system Natural frequencies of fluid system / Frequency of ground motion (free vibration frequency of shaking table) Maximum amplitude of ground motion A Maximum harmonic velocity of ground motion V t Time Duration of impulse interval of ground motion tl Duration of step motion (ground) t2 D Duration of oscillatory motion (ground) Number of cycles in oscillatory motion; N = Df N (Time is expressed in seconds; displacement and length, in inches; frequency, in cycles per second.) Ti
THE PI~OBLEM Figure 1 shows a rigid t a n k of radius a filled with fluid to the level h. I t experiences the horizontal translatory ground motion xu(t) parallel to the x - z plane. The sign of the ground motion is shown positive to the left in order to conform with the experimental setup. The gravity wave surface has the height ~ with respect to the equilibrium plane, z = h. Type of ground motion.--The ground motion consists of an approximately impulsive start from rest, followed b y a sinusoidal motion having a nearly constant rate of decay. The impulse interval tl is short (in relation to the lower natural periods of the fluid system) to the degree t h a t the gravity wave heights remain small during the interval and consequently the impulsive hydrodynamic pressures and velocities can be determined without great difficulty b y analysis (3). Two varieties of the ground motion have been used: 1) the step motion, in which the ground motion is arrested at time t2 when xa = A and ~o = 0; and
316
BULLETIX OF THE SEISMOLOGICAL SOCIETY OF AMERICA
2) the oscillatory motion, in which the "ground" continues to vibrate until it comes to rest, largely as a result of Coulomb damping, after a time duration D. These motions are reasonably characteristic of portions of actual transient ground disturbances. They satisfy the principal purpose of the investigation, namely, to obtain a general understanding of the problem rather than to explore the effects of particular, and often very complicated, disturbances. Plane of symmetry of the tank.--The x-z plane is a plane of symmetry, and the fluid velocities across this plane are zero. The results thus also apply to a half 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 particular interest as a background against which to record the wave profiles. ~ Elliptieity of the tanlc.--Actual tanks are not perfectly circular in cross section, but have some out-of-roundness, or ellipticity. If the major axis of the ellipse is not parallel to the direction of the ground movement, the wave motion will tend to break into two components (2), one parallel to the major axis and one to the minor axis, and after several cycles the resultant motion becomes very complicated in appearance. The variation in tank diameter was no~ more than 0.4 per cent, but the effect on the wave motion was still noticeable. The tanks were, therefore, carefully oriented with the major axis in the x-z plane. EXPERIMENTAL METHODS
Shaking table and tanks.--Figure 2 is a general view of the four tanks and the shaking table. The tank diameters were 6, 12, 23, and 47 inches. 3 The construction and analysis of the shaking table have been described previously (1, 8). Briefly, it consists of a massive platform mounted on 'concentrically ground wheels and connected at one end through springs of adjustable modulus to a fixed pier. At the other end is a bumper spring. The table is set in motion by striking the bumper with a heavy pendulum, the pendulum remaining in contact with the bumper for the time interval 0 < t < tl, which is the impulse interval. The table motion may be arrested at time t2 (figs. 1, b; 3, a; 3, b) by a simple, gravity-operated latch (15). Ground motion.--The natural frequency of the table and anchor spring, i.e., the frequency f of the ground motion, has been varied in small steps from about 0.7 to 7.0 cycles per second. During the impulse interval the entire system, neglecting the action of the fluid, acts as a two-degree-of-freedom system (pendulum, bumper spring, table, anchor spring) the fundamental natural frequency of which is a little less than f, and the second natural frequency, f2, is 6.5 to 7 cycles per second. The impulse interval tl ranges between 0.075 and 0.08 second; both f2 and tl are thus relatively constant. 2 T h e earlier work (2) on cylindrical t a n k s did n o t m a k e use of t h e x-z plane as a recording device, t h e w a v e h e i g h t s being recorded only a t t h e cylindrical b o u n d a r y , r = a. 3 T h e d i a m e t r a l d i s t o r t i o n s a t t h e top~ of t h e two larger t a n k s were m e a s u r e d during t h e experiments a n d found to be less t h a n 0.001 inch.
EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT IA~IOTIONS
317
The ground-motion criteria are: (1) constant maximum harmonic velocity, V, and (2) constant time duration, D (except with respect to the step motion, for which the duration is t2). The choice of these criteria has already been discussed (10); H. M. Westergaard originally suggested the constant V criterion (9). The criteria may be explained in terms of ground energy, and mean (1) that the energy required to set up the ground motion at any frequency f
Fig. 2. General view of s h a k i n g t a b l e and four tanks.
is essentially constant, and (2) that the average rate of dissipation of the ground energy is approximately constant. The maximum harmonic: velocity is given by V = 2~rAf (1) and can be held to within 2 per cent of the desired value. The standard duration was 2.1 seconds, =55 per cent. Measuring the effective mass and overturning moment.--In order to determine the effective mass and overturning moment of the fluid, the 12-inch and the 23-inch tanks have been mounted on dynamometers, as shown in figures 2 and 4, and the same general procedure has been followed as was used in the earlier work on rectangular tanks (1). Two sets of tests are made: 1) with fluid in the tank at various depths h; 2) with the fluid replaced by various solid
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EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS
319
weights which are rigidly clamped to the base of the tank. By comparing the dynamometer responses for these two conditions, during the impulse interval, it is possible to determine the effective mass of the fluid and the overturning moment. Recording the gravity wave formation.--Extensive experiments were made with several methods of recording the envelope of the gravity wave profiles in the x-z plane, the most successful one using a black dye solution and strips of Bristol board. The recording card is attached to a thin metal holder which is hung on the background plane (fig. 4, e). The dynamometers were blocked out of action during the wave-profile studies. NATURAL-~/[ODE GRAVITATIONAL OSCILLATIONS OF THE FLUID
Modes of order s = 1.--Horizontal ground motions excite a class of gravitational modes which are characterized by a single nodal diameter and by onefold symmetry with respect to the x-z plane. They are describable by Bessel functions of the first kind and first order, the J~ functions where s = 1, and are often called the s = 1 modes? The fundamental mode has no nodal circle, but is simply a "swaying" back and forth of the fluid. ~ The second mode has one nodal circle, the third has two, etc. (see fig. 5). If an imaginary cylinder is dipped concentrically into the tank, the fluid surface will draw a cosine curve on it. The motion in a particular natural mode is simple harmonic if friction is neglected. The natural periods 6 are given by T~ =
2~- , , / a ~/gk~a t a n h (k~a. h/a)
(2)
where the subscript refers to the particular natural mode and where the constants characteristic of the natural modes are ki = 0.586 ~r/a, kii = 1.697 ~r/a, k m = 2.717 lr/a, k~ -~ (i - 1/4) v/a with increasing accuracy (see fig. 6). EXPERIMENTAL RESULTS DAMPING
The fundamental mode can easily be set up by pushing vertically against the water surface in a periodic manner near the wall of the tank, ~ and the second 4 The basic theoretical work on cylindrical containers was published by Poisson (1828) and b y Lord Rayleigh (1876). 5 This mode can easily be set up in a glass of water; in fact, many of the phenomena can be d e m o n s t r a t e d in a qualitative way by moving a can of water back and forth on a smooth surface. 6 See L a m b ' s Hydrodynamics (11), Arts. 191 and 257. Some interesting experiments on the natural gravitational modes in b o t h circular and rectangular tanks have been described by Frederick Guthrie (1875) (12). Among his m e t h o d s was the use of a sheet of cardboard on which the w a t e r surface marked out its own profile. However, he apparently 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 TO T R A N S I E N T MOTIONS
321
m o d e can be excited b y p u s h i n g near the point of m a x i m u m displacement, r = 0.35a; b u t the higher modes are difficult to f o r m b y h a n d (15). T h e sample graphs in figure 7 show the viscous n a t u r e of the damping. I n figure 8 the logarithmic decrements for the f u n d a m e n t a l m o d e h a v e been plotted against t a n k diameter, the curves a p p r o x i m a t i n g equilateral hyperbolas, s T h e decrements for the second m o d e were generally higher t h a n those for the fundamental. E F F E C T I V E MASS A N D O V E R T U R N I N G MOMENT~ C O V E R E D T A N K
T h e r e are two limiting cases: 1) If the t a n k is full to the cover, 9 the fluid is completely confined at all boundaries and should have an effective mass and an o v e r t u r n i n g m o m e n t equal to those for a solid of the same density and dimensions, z° 2) If there is distance enough between the fluid surface and the cover, the s y s t e m will act as one w i t h o u t a cover (3). T h e intermediate cases are extremely complicated. T h e r e were four series of tests for effective mass (f = 1 and 4 cps, V = 2 and 3 in/sec.; 2a = 23), each series including at least seven values of clearance, c, between the static fluid level and the cover. T h e results h a v e been s u m m a r i z e d in figure 9, where it is shown t h a t if only a small proportion of the fluid is removed, the tank becomes, effectively, an open one. I t is s o m e w h a t surprising t h a t there was no significant difference in the shapes of the experimental curves for the two conditions of V. One would expect that, as V is increased, t h e a m o u n t of fluid which m u s t be r e m o v e d in order to reduce m l / m f r o m u n i t y to the open t a n k value of 0.68 would also increase. T h e effect of increasing V f r o m 2 to 3 in/see, is a p p a r e n t l y small enough to be lost in the inherent errors 8 The decrements may be compared with values obtained by Carder (7) on elevated tanks having a volume from 25 to 50 times as great as the four-foot model. The comparable full-scale logarithmic decrements are of the order 0.01 to 0.015. These agree reasonably well with an extrapolation of figure 8, if it is recognized that they include the frictional effects due to the behavior of the supporting tower as well as those of the fluid. 9 See figure 6 of reference (3) for experimental results for the tank without cover. 10 Care was taken to insure that the cover was rigid (fig. 4, d) and that there were no air pockets. Fig. 4. Experimental apparatus. a) Side elevation of horizontal-force dynamometer. Tank base is supported at four points by hinges H and blocks B. Wire-resistance strain gauges are attached at eight locations, G. Moment dynamometer is not in action when force dynamometer is in use. b) Side elevation of moment dynamometer. Tank base is supported at three points by hinges H and, through the cantilever dynamometer bar M, by the ball and pedestal P. Strain gauges are attached at G. When the moment dynamometer is in use, the force dynamometer is blocked by shear plates S. c) View of 23-inch glass tank showing horizontal-force dynamometer, electrical terminal block for connecting strain gauges in Wheatstone bridge circuit, and wave-envelope recording card holder E. d) View of 23-inch tank with rigid cover. Moment dynamometer shown in operating condition. e) Dynamometer recording apparatus, showing: C, steady-current and bridge-balance control box; A, a.c. amplifier; O, low-frequency oscillograph. Apparatus for recording ground motion is at R.
i.
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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
323
of the experiment. The amount of fluid which must be removed is so small that a more precise investigation does not seem warranted. An even more spectacular reduction in ml/m can be expected when the depth is shallow. For example, when h/a = 1/4 then ml/m -= 0.16 when the tank is open. The re0.5
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duction in rnl/m to the open-tank value is probably more closely associated with the ratio of clearance to tank radius, c/a, than with the ratio of clearanee to fluid depth, c/h. Two series of tests were made for overturning moment, the experimental data having been compared with the limiting theoretical cases in figure 10. The moments are with respect to the hinge H shown in figure 4, b. The very rapid decrease in overturning moment is due to two causes: 1) reduction in effective mass, and 2) lowering of the centroid of effective mass.
324
BULLETIN 0I: THE SEISMOLOGICAL SOCIETY OF AMERICA
2a =47 FUNDAMENTAL
MODE
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Fig. 7. N a t u r e of t h e fluid damping. W a v e h e i g h t h a s been p l o t t e d on a l o g a r i t h m i c scale. PHOTOGRAPHIC STUDIES OF
Step motion.--Photographs of the
THE GRAVITATIONAL WAVES
intersection of the water surface with the background plane (0 = 0 °) were taken at a known rate, tracings of the intersection lines or profiles having been shown in figure 11, a. The first frame has arbitrarily been taken as zero time. The wave motion consists mainly of the fundamental mode, although m a n y other modes are obviously present. Calculated profiles, assuming a particular start, have been shown to an arbitrary scale in figure ]1, b; qualitatively, the agreement is good. The basis of the calculation is as follows.
EXPERIMENTS
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
325
The u n d a m p e d free oscillation with general starting conditions can be described b y the series n = ~
A ~ J l ( k i r ) cos 0-cos \ ~ (
+ ¢i
)
(3)
where Ti and ki are given b y equation (2). Let us assume a particular start such t h a t ~i = ~i~i = Cv . . . . 0, ¢~i = ~iv = CvI . . . . ~, and, to an arbit r a r y scale, let AI -- 1, AII = 1/3, AIII = 1/5, A i r = 1/7, Av -- 1/9, etc. 11
FUNDAMENTAL MODE
OJO
0.08 E o
'~ 0,06
.3 I 0.04
0,02
i "-""-" --=¢
0 0
1 ~
! I
! 2 TANK DIAMETER.FEET
I 4
Fig. 8. Effect of tank diameter and of static depth on damping. The logarithmic decl/ements have been obtained from curves of the type shown in figure 7. (fig. 12, a). T h e above choice of the ~'s makes the terms all additive at the boundary, which is consistent with the impulsive nature of the start of the ground motion. The stun of the first five terms at zero time, and thereafter at intervals of TI/8, has been shown in figure 12, b, and to different scale in figure 11, b. The lack of more complete agreement with the photographs is due: a) to the limitation of analysis to five terms; b) to the omission of d a m p ing; c) to the arbitrary choice of start; and d) to the fact t h a t the ground motion is not purely an impulsive one. 11An equ~'lly reasonable choice would have been AI = 1, AII = k i / k i i = 0.345, AII~ = 0.216, etc.
]~I/]fIII =
326
BULLETIN OF T H E SEIS~IOLOGICAL SOCIETY OF AMERICA
ml
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.
/COVERED
TANK, EXPERIMENTAL
0.7
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F i g . 9. I n f l u e n c e of a r i g i d c o v e r on t h 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 of t h e fluid. T h e d a s h e d lines 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 of 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 full line is t h e a v e r a g e .
Oscillatory motion.--Photographs were also taken of the wave profiles set up under two conditions of oscillatory motion (fig. 13). Since the wave motions are considerably more complicated than those excited by the step motion, it would be useless to represent them by a free vibration from a simple impulsive start. ENVELOPES OF THE WAVE PROFILES
Step motion.--Representative samples of the wave-profile envelopes taken in the 12-inch tank appear in figure 14. An increase in V results in increased wave 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 change occurs with the large decrease in h/a.
EXPERIMENTS W I T H TANKS SUBJECTED TO TRANSIENT IV[OTIONS
327
Oscillatory motion.--Samples of the envelopes h a v e been shown in figure 15 for h/a = 1, a n d in figure 16 for h/a = 1/4. V is c o n s t a n t while f has been varied f r o m zero I2 to a b o u t 6 cycles per second. T h e greatest difference between the conditions h/a = 1 and h/a = 1//4 appears in the frequency range in which the f u n d a m e n t a l m o d e is of m o s t i m p o r t a n c e (see profiles for f = 1.03 a n d 350
LIMITING
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FLUID REMOVED, PER CENT OF FULL TANK
Fig. 10. Influence of a rigid cover on the effective hydrodynamic overturning moment due to the fluid. The maximum value of the impulsive acceleration was 0.12 g. Full tank h/a = 1.50. f = 1.64). A t g r o u n d frequencies higher t h a n a b o u t 3 cycles per second the differences due to d e p t h are more or less secondary. A direct, left-to-right comparison between the records, based on t a n k diameter, should n o t be m a d e unless it is recognized t h a t the n a t u r a l frequencies are different in the two t a n k s and hence the " t u n i n g " with the g r o u n d f r e q u e n c y is different. Table 1 1, The ground motion for the "zero frequency" case was obtained by removing the anchor springs from the table, the motion consisting of the impulsive start followed by an aperiodic motion in which the table gradually came to rest as a result of Coulomb damping. The starting conditions were adjusted until a maximum velocity of 3 in/see, was indicated by the slope of the xa, t curve (see fig. 20, b).
328
BULLETIN OF TItE SEIS]VIOLOGICALSOCIETY OF A2VIERICA
(a)
EXPERIMENTAL f=l.03 , =0.84 , "V'= G h 5
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EXPERIMENTS WITI~I T A N K S SUBJECTED TO TRANSIENT ~¢IOTIONS
(b)
CALCULATED
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329
330
B U L L E T I : N OF TI=IE SEIS1V[OLOGICAL SOCIETY OF AME,P~ICA
lists t h e n a t u r a l frequencies a n d t h e l o c a t i o n s of t h e c o r r e s p o n d i n g p o i n t s of m a x i m u m elevation. T h e l o c a t i o n s of 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 t e 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 of t h e m a x i m a in t h e r e c o r d e d profiles. F o r e x a m p l e , in figure 15 t a k e t h e case 2a = 23, f = 3.97. A c c o r d i n g to t a b l e 1 t h e g r o u n d freq u e n c y is in n e a r r e s o n a n c e w i t h t h e sixth n a t u r a l f r e q u e n c y , a n d a m a x i m u m w a v e h e i g h t can b e e x p e c t e d in t h e n e i g h b o r h o o d of r/a = 0.10. T h e s a m e TABLE 1 NATURAL FREQUENCIES IN TANKS OF 12-INCH AND 23-INCIt DIAMETER AND LOCATIONS OF POINTS OF MAXIMUM ELEVATION IN THE NATURAL-MODE PROFILES fi cycles per second (s=l) 2a=12 in. h/a = 1
I.
II. III. IV. V. VI. VII. VIII.
1.69 2.94 3.73 4 37 4.91 5.44 5.88 6.30
h/a =
_rfor a maximum*
2a=23 in. 1/4
1.14 2.75 3.67 4.36 4.91 5.44 5.88 6.30
h/a ~ 1
1.22 2.12 2.68 3.16 3.54 3.93 4.23 4.54
h/a ~
1/4
0.83 1.98 2.64 3.14 3.54 3.93 4 23 4.54
1.000 O. 346 0.216 0.156 O, 123 O. 102 0.087 0.076
* Locations of the lesser maxima and of the nodal diameters can be obtained from a table of Bessel's functions.
g r o u n d f r e q u e n c y , w h e n a p p l i e d to t h e 12-inch t a n k , excites b o t h t h e t h i r d a n d t 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 extent. I t is significant t h a t t h e funda± m e n t a l or " s w a y i n g " m o d e is e x c i t e 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 forcing f r e q u e n c y is g r e a t e r t h a n a b o u t 2 cycles p e r second even t h o u g h t h e g r o u n d m o t i o n h a s an i m p u l s i v e s t a r t . Presence of another class of gravitational modes.--The v e r t i c a l fluid m o t i o n s a t t h e c e n t e r of t h e t a n k , w h i c h h a v e a p p e a r e d in s e v e r a l envelopes, a r e n o t c o n s i s t e n t w i t h t h e f o r m of t h e s = 1 modes. I n an e x t r e m e case s h o w n in figure 16 (2a = 12, f = 1.03) t h e g r o u n d f r e q u e n c y was 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 m o d e . L a r g e w a v e h e i g h t s were b u i l t u p a t t h e t a n k wall; in f a c t t h e y were a b o u t t w i c e t h e d e p t h of t h e fluid a t rest. 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 of 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 ever. T h e r e s u l t i n g s h a p e of t h e e n v e l o p e suggests t h a t t h e f u n d a m e n t a l of t h e s = 0 class 13 of m o d e s was also set u p ; if so, this m u s t h a v e b e e n d u e t o a ~ The s = 0 modes are described by Bessel functions of the first kind and zero order, the J0 functions; see Lamb (11), Arts. 191 and 257. They have circular symmetry and are characterized by a system of "annular ridges and furrows." There are no nodal diameters. The fundamental mode of this class has one nodal circle, the second mode has two, etc. The absolute maximum elevation in each mode is at the center.
331
EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS
~
o
:
:
-0.5 ~ ~
"
~ ~ - , _
- ~
~ -
-
Q25
~----~'~-~J~~
o'.~------~ ~
/
~.
t~O~+
,.,
1'5 r~
(b)
S /
~
_._//_____~-,,o
.o" /
I--0.5
5~ Fig. 12. Calculated wave profiles; step motion. (~) Assumed s t a r t i n g conditions. (b) R e s u l t a n t profiles.
332
BULLETIN OF THE SEISlY£OLOGICAL SOCIETY OF A ~ E R I C A
,-.~. ~.~. ,°.,~ (a)
f / f=l,03, "fz=O,S4, ~=0,49
±\
f-2.04, fz-i.66, "~-096
(b)
------lJ~__
t=O
_~
w ~
/
~_
I
I-0.50 - -,
.
~
I -I -Tz- Onl-- ~
t ,I.00 i
~
L
L I
~
~
'
i
I
I t =1.50"~
,
!
~
.~"~-~
/ /
1
-I=2T=-- 1
1=2.00
-D f
t=2.5o--
05
114
~
~ _
j
1/)
Fig. 13. Photographic wave profiles; oscillatory motion. (a) . f / f i = 0.84, fundamental mode prominent. (b) f / f i i = 0.96, second mode prominent. D = approximate end of ground motion. (See fig. 3, c and d.)
EXPERI~IENTS WITH TANKS SUBJECTED TO TRANSIENT IVIOTIOI~S
333
Fig. 14. Samples of wave-profile envelopes; step motion. The equilibrium or static level lines S, and the envelope lines~ have been retraced in order to make them photograph more clearly.
IJ
rm~
0
q~
r~
o
rm~
JI
o
0 o
0
o
b~
336
BULLETIN OF THE SEIS~CIOLOGICAL SOCIETY OF AMERICA
coupling between the two classes of motion, since the s = 0 class cannot be excited directly by a horizontal ground motion. The test was repeated several times with substantially the same results. The presence of the fundamental mode of class s = 0 was observed in all tanks whenever the motions were relatively large, which means that it was most noticeable when the equilibrium depths were small and when the ground frequency was near resonance with the fundamental s - 1 frequency. One can easily visualize that identical waves of fundamental s -- 1 type will be accompanied by much higher fluid velocities in a tank of shallow depth than in a similar tank of large depth. The tendency is for a symmetrical set of velocity components to be generated in directions perpendicular to the x-z plane; such a set is in conformity with the requirements of the s "= 0 modes. Since it was found that these central motions became smaller as the size of the tank was increased, they probably are not of much consequence in full-scale tanks. MAXIMUM WAVE HEIGHTS
About 250 sets of test conditions were used; there were usually two separate tests for each condition, and from two to four tanks were represented in each test. From the resulting envelopes the maximum wave heights and the radial locations of the maxima have been measured. Step motion.--The experiments have been limited to a comparatively small range of ground frequency and to the tank diameters of 12 and 23 inches. Change in fluid depth, h/a: Figure 17 shows the effect of variation in h/a and in f. As the depths are increased the wave heights increase, but by no means linearly; in fact, when h/a is made larger than unity no very appreciable change in ~ occurs. I t is obvious that the effect of depth on wave height
does not follow a simple relationship and that extrapolation to full scale should be on the basis of equivalent h/a. Change in maximum ground velocity, V: The effect of change in V, and therefore in the maximum ground amplitude A, is shown in figure 18. The effect is nearly linear when f = 2.04 cycles per second, but when f -- 1.03 there is linearity only up to V = 4 and beyond that the wave height increases more rapidly. A reSxamination of the corresponding envelopes in figure 14, a, indicates t h a t the nonlinearity is associated with large motions, for example, ~ = 0:33 h when V = 8. The conclusion is t h a t accurate extrapolation is limited to motions which can be assumed to be small. However, there is no general guide to what values of ~ / h can be assumed" small." An extreme case is represented in figure 14, c, in which the variation in w is approximately linear even though at V = 8, ~ = 0.5 h. Change in tank diameter: From figure 19 two comparisons between the tanks can be drawn, one for the depth h/a -- 1, the other for h/a -- 1/2. The reference frequency fi takes four values, each relating to one of the combina-
0
~g
I
I
I
I
I
f
I
I
I
~-~2,'-.. / "--.. W ",~ ~
I
i
I
~
I
04 a
I
~o
z~. /k"y -t I
f
1
I
f
I
I
I
a~_o
o
b2"~
338
BULLETIN
OF TI-IE SEIS!VIOLOGICAL
SOCIETY
OF AMERICA.
tions of tank size and fluid depth. Comparing the wave heights at like values of f/fi, it is found that when h/a = 1 the heights in the 23-inch tank are from 1.37 to 1.42 times the heights in the 12-inch tank. When h/a = 1/2 the factors range from 1.43 to 1.57. Hence, with maximum ground velocity constant, the
wave heights vary approximately as the square root of the tank diameter ( %/~-/12 = 1.39).
I t has already been mentioned (in footnote 12) that some tests were made in which the anchor springs were removed from the table, the resulting ground (a)
v" =3
1.5 I~m
I 1,0
O.5
0,1
I 2
1 ~
I 4,
(b)
0
0,I
02
I 5
T 6
T 7
J
1 0.5
-~-
t
I 1,0
I 1.5
I 20
Fig. 20. Effect of tan]( diameter on wave height; zero ground frequency. Maximum wave heights (a) resulting from ground motion (b).
motion being more nearly of impulsive type. If the maximum wave heights occurring in these tests are plotted as a function of the square root of tank diameter, a nearly linear relationship results (fig. 20). Oscillatory motion.--It is much more difficult to generalize about the effect of the oscillatory motion since it is neither a simple " s t e p motion" nor a steady state. The principal complication is due to the variation in the number of cycles in the ground motion. Even when the system is a simple, singledegree-of-freedom oscillator the response is a complicated function of the number of forcing cycles. 14 The response of an undamped oscillator, which is 14See the paper by It. A. Williams (13) in which he shows the effects of the same ground motion as used here, on a single-degree-of-freedom system with damping.
EXPERII~IE,NTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS
2.0
2o=23 "V'=2
/"x -~-hG =l-~X
339
~
OSCILLATORY GROUND MOTION
1.5
°,,'?,.ff WX >( ~,
rtm
1.0
h I ,.."'~,/ --d=~,,,
~,..~,,
~ . X?'~..~----.';~"~'o - . . . . . . =" ~.-.._~ .."
.,.~=~ . . . . . . -.-z-n,~% ~-"S..~.. ~..
"
0.5
,
0.5
i
-~
"--
,,
I
1.0
--'~--- ~
"*---J-*--~'.S
I'I
,
i,
1.5
2.0 ~
,,--~,,,=~
f il f Z.5
,
~,-0
il
,I
3.5
I~
I~
4.0
,I~
,
4.5
5.0
Fig. 21. Effect of change in static depth and in ground frequency on wave height;" oscillatory motion. The natural frequencies of the three systems have been indicated along the ground-frequency axis. The curve relating to each depth has two branches, one giving the maximum wave height ~ , the other giving the wave height at the tank wall, ha. In the lower frequency range the maximum occurs at the wall, i.e., n~ = w, and the two branches coincide. (It must be kept in mind that the number of cycles in the ground motion is not constant.) 20:23 OSCILLATORY GROUND MOTION
x ~-=1
rl, m .V.= 3
h _._L
~' ~ ' -
2
~rr,'V-=2
;=
0
0,5
ID
1.5
2.0
~ f
=r
2,5
~D
3.5
4.0
4.5
5.0
Fig. 22. Effect of change in maximum ground velocity on wave height; oscillatory motion. The comparison of two sets of tests shows fair agreement with the predicted linear value of 3/2. at resonance with a simple harmonic disturbance varies linearly with the n u m b e r of cycles in 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 m response 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 t h e n u m b e r of cycles. I t can be e x p e c t e d t h a t t h e effect on t h e fluid s y s t e m will be c o n s i d e r a b l y m o r e c o m p l i cated. 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 results in a p r o m i n e n t 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 n a t u r a l m o d e as i n d i c a t e d b y figure 21, b u t t h e r e a r e n o definite " 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 modes. T h e l a c k of s h a r p l y defined, h i g h e r - m o d e p e a k s is p a r t l y d u e to t h e closeness of t h e
340
BULLETIN OF THE SEISMOLOGICAL SOCIETY OF A~VIERICA
)2
)5
Ca)
.;>-" N=Df
f lz
&l
+ I
i
I
i
l
,
I
I
|
I
I
I
I
I
I
0
v.._.~.3 23f"/~-~, 2.0
f'~
.- ~ - /
,
\
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O~OILLATORY GROUNO MOTION %
"
47
,°=j/ ) r \ X
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.; /
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,
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(~) <0"~--~.
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I ,
4,;.=
20=47 / " ~ .
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I
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,,
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;"
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I
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, I
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- .........
I I'6 ~'t:.~.L..=,I':_-.:,: . . . . . . I , ,=;i 45 5.D
.~=,,z ~. 4,0
3.5
"t:--:'-1
xl,=
~
l
, 5,5
-
.
-
-
-
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" - -6.0- -['6, " ~ --7.o-" ' 1
':=
Fig. 23. Effect of tank d i a m e t e r on wave height; oscillatory motion. (a) Amplitude A and number of cycles N of the ground motion. (b) Static depth is large, h/a = 1. (c) Static depth is shallow, h/a = 1/4.
341
EXPEI%I]CIENTS WITH TANKS SUBJECTED TO TI~ANBIENT I~[OTIONS
natural frequencies and to the sharp tuning which consequently is required for resonance, 1~It can be concluded that prominent resonance with modes higher than the fundamental, owing to a transient disturbance of a few cycles, is not to be expected. Furthermore, maximum wave heights Vmoccur at the tank wall only when the ground frequency is less than about 5/4 • fz-
1.5
05
h,I ~o ~
~
0.1 O,
I
2
I 3 ~
,I 4
I 5
I 6
7
Fig. 24. Effect of t a n k d i a m e t e r on absolute maximum wave heights; oscillatory motion: (1) near resonance w i t h t h e f u n d a m e n t a l mode; (2) a t ground frequencies greater t h a n
the 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 the manner 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 to have approximately the same relative effect at any ground frequency, provided that the fluid motions are not excessively large. In figure 22 a comparison has been made between tests on the 23-inch tank when V = 3 with tests when V -- 2; two values of h/a are represented. Change in tank diameter: The results of two series of tests, for which h/a = 1 and 1/4, are shown in figure 23. The two families of curves differ considerably in detail, but as far as general shape of the curves and order of magnitude of wave height are concerned, they are much alike. The variation within each family is reasonably regular, except in the heights of the fundamental-mode resonance peaks for the depth h/a = 1/4. There are two reasons lb T h e higher n a t u r a l frequencies of the s y s t e m v a r y roughly as the square root of t h e n u m b e r of t h e n a t u r a l mode, whereas in m o s t mechanical systems t h e frequencies v a r y more nearly as e i t h e r t h e first or t h e second power of i. T h e effect of t h e same oscillatory ground motion on a mechanical s y s t e m h a v i n g m a n y degrees of freedom will be found in reference (10), where i t is shown t h a t distinct resonances occurred w i t h n a t u r a l modes as high as t h e fourth.
342
BULLETIN OtP THE SEISiIKOLOGICAL SOCIETY OF A]~EI~ICA
for this irregularity; one is the large height Of the waves relative to the static depth, and the other is the presence of inaccuracies in the experimental ground motion at low frequencies. If the absolute maxima of the fundamental mode peaks for the case h / a = 1 be divided b y the number of cycles in the ground motion and plotted against the square root of the tank diameter, an approximately linear relationship is found (fig. 24). Corresponding values for the depth h/a = 1/4 also show a reasonably linear relation, with the exception of the poorly defined value for
Lo'
V-- 2,5 2o=12,.25 OSCILLATOR._~Y GROUN.~DMOTION
Le
,
0.2
o.,
O
-
..........................
'
0,5
"
r ...............
I
1.0
.
,
1,5 ~ f
I
I
, 2.0
~
I
,, 2.5
"- ~
;, ~.0
'
~
"
, 3.5
.
, 4,0
.
.
.
"
, 4.5
f~
F i g . 25. l ~ a d i a l l o c a t i o n s of m ~ x i m u m w ~ 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 r e s o n a n t
frequencies and the locations of the maximum crests for modes I to V have been indicated. the 47-inch tank. A similar relation is found among the absolute maxima associated with the higher natural modes. The following empirical relations for maximum wave height due to an oscillatory motion (V constant) are based on figure 24: ~.~ = C'~/2a V N
(4)
For absolute maximum response in the fundamental mode f = 0.9 f~z h / a = 1, C = 0.07; h / a = 1/4, C = 0.1; where ~ occurs at the tank wall (r = a). For absolute maximum response when f > f~: h/a = 1, C = 0.015; h/a = 1/4, C = 0.02; where nm occurs at r < a/3. I t is evident that for equal N and equal V the maximum wave heights at the tank wall are about five times the value of those which occur at r < a/3. While the relations just given are no more than rough approximations, they do indicate the order of magnitude of ~ . T h e y must be used with discretion when N is significantly larger than the values shown in figure 23, a. In order to apply to large values of N, the relations should include an exponential damping factor, since as N is increased the effect of damping becomes in-
EXPERIMENTS WITI{ TANKS SUBJECTED TO TRANSIENT MOTIONS
343
creasingly important. The most important effect of large N, however, is that the fluid motions m a y become "large" with consequent failure of the linear relationship. This m a y also be true of large V (fig. 18). If the substitution V -- 27r A(f/fi) fI be made in equation (4), w h e r e f i m a y be determined from equation (2), it is apparent that the tank diameter terms % / ~ disappear and that for constant f/f~ and h/a the maximum wave height is given by ~m = CLAN, (4a) which means that, if the ground motion criterion is constant maximum amplitude A, the wave heights are independent of the tank diameter. This is true, however, only for maximum response in the fundamental mode. Radial locations of v~: The locations of the maximum crests are shown in figure 25 for the depth h/a = 1 where rm/a, the ratio of the radius at which the crest occurs, to the wall radius, has been plotted against f/fi. The existence of some asymmetry in the wave motion, as indicated b y the fact that the crest in the 0 = 0° plane does not fall at the same value of r as does the crest in the 0 = 180 ° plane, is due to the transient nature of the ground motion. The experimental points have not been shown. There was much less "scattering" in the location of the crests than there was in their heights. Similar relationships were found for the depths h/a = 1/2 and 1/4. SUMMARY AXD CONCLUSIONS
Effective mass and overturning moment.--[ 1 :] When the tank is not covered, the effective mass and overturning moment due to the behavior of the fluid can be calculated from hydrodynamic theory (3). The agreement between theory and experiment is good. For practical purposes the ratio of effective hydrodynamic mass to actual mass of the fluid can be calculated within about 5 per cent of error from the following simple relations :16 0 < - h < 1, a
ml _ 0 . 5 7 h . m a '
1 < h < 2.5, a
ml _ 0.39 + 0 . 1 8 h m a
(5)
The location of the eentroid of the effective hydrodynamic mass is given to the same degree of approximation b y :17 h
0.36 + 0.027
a .
(6)
[2:] When the tank has a rigid cover, the removal of a small proportion of the fluid makes the tank effectively an open one. The overturning moment is, ~7 C o m p a r e w i t h figures 2 a n d 3 of r e f e r e n c e (3).
344
BULLETIN OF THE SEISMOLOGICALSOCIETYO1~AMERICA
reduces even more rapidly t h a n the effective mass does, since it is being a function not only of the effective mass b u t also of the effective centroidal height. G r a v i t y w a v e s . - - [ 3 : ] The results of the wave studies are applicable to fluids other t h a n water, because: a), as is well known, the gravitational natural modes are independent of the fluid density; b) the viscous damping in the system is inherently small; and c), the surface tension effects are secondary. [4:] The wave heights resulting from the step motion have been calculated for one example with reasonably good agreement with the experiment. The calculation assumes a particular starting condition of the natural-mode oscillations which is consistent with the impulsive nature of the start of the ground motion. [5:] T h e effect of static fluid depth is a complicated one and the comparison of tanks of different diameters should generally be made on the basis of equivalent depth, t h a t is, equal values of h / a . [6:] When the criterion for the ground motion is constant m a x i m u m harmonic velocity, the m a x i m u m wave height can, for m o s t practical purposes, be assumed to v a r y as the square root of the t a n k diameter. [ 7 :] A change in the m a x i m u m harmonic velocity V of the ground motion should result in a directly proportional change in wave height, provided t h a t the fluid motions r e m a i n " small." Since the wave heights tend to v a r y as the square root of the t a n k diameter, it would appear t h a t the question of small motion is of m u c h less importance in full-size tanks t h a n in the models. [8:] The radial location of the m a x i m u m wave height is near the location of the m a x i m u m height of the most greatly excited natural mode. When the forcing frequency is less than about five-fourths of the fundamental natural frequency the m a x i m u m wave height occurs at the wall of the tank, b u t for greater forcing frequencies it occurs away from the wall and consesequently is of less practical importance. [9 :] Well-defined resonance occurs only with the fundamental mode. The higher natural frequencies are closely spaced, with the result t h a t very sharp tuning is required for resonance. NUMERICAL EXAMPLES I. A rigid tank of 120 ft. inside diameter, containing water to a depth of 90 ft., experiences an impulsive horizontal acceleration of 0.20 g. Find the dynamic forces and moments due to the water (3), and compare with those for the same tank when rigidly covered, when there is zero clearance between the fluid and the cover: Compare with shallow tank of the same volume. Estimate the required "open tank" clearance. i. Total weight of water, mg ~ 63.6 X l0 Glbs. it. Effective hydrodynamic mass of water: open tank (h/a = 1.5), m l / m = 0.682 (fig. 2 of ref, 3) or, by equation (5), m l / m = 0.66; covered tank, ml/m--*l.O. (The rigidity of a cover which is 120 ft. in diameter is, of course, questionable.) iii. Maximum horizontal force due to the water: open tank, X1 ~ 63.6 X 106 X 0.682 X 0.20 = 8.67 X 106 lbs., applied at the level ~1 = 0.393 h = 35.4 ft. (fig. 3 of ref. 3), or, by 1.00
equation (6), ~1 -----35 ft.; covered tank, X = ~ level ~ -- 0.5 X h = 45 ft.
X X I ~ 12.7 X 106 lbs., applied at the
EXPERIMENTS WITH TANKS SUBJECTED TO TRANSIENT MOTIONS
345
iv. Maximum overturning m o m e n t due to the water: open tank. [1:] M o m e n t of horizontal force, M1 = X1 ~1 = 307 X l0 s lb. ft. [2:] H y d r o d y n a m i c couple acting on b o t t o m of tank, N1 = 142 X 106 lb. ft. (fig. 4 of ref. 3). [3:] Total overturning moment, M1 -F N1 = 449 X 106 lb. ft. Covered tank. [1:] Moment of horizontal force, M = X~ = 571 X 106 lb. ft. [2:]The hydrodynamic Couple acting on the b o t t o m of the tank is opposed, theoretically, by an equal and opposite couple acting on the cover. [3:] Total overturning moment, M = 571 X 106 lb. ft. v. To the foregoing there must be added the force and overturning moment due to the mass 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., ml/m = 0.305. X1 = 63.6 X 10 s X 0.305 X 0.20 = 8.88 X 108 ibs., where ~ = 0.38 h = 16.5 ft. Compare with calculation in paragraph iii above. N o t e the great reduction in t o t a l hydrodynamic force resulting from the change to a tank of shallower design. vii. Clearance required to reduce the effective mass to the "open t a n k " value. The necessary clearance in the 23-inch tank when the maximum impulsive acceleration was about 0.12 g was c = 0.06 X 11.5 = 0.7 in. (fig. 9). Assuming, on the basis of figure 20, t h a t the clearance must v a r y as the s~ua.re root of the tank diameter, the clearance in the tank of 120-foot d i a m e t e r is c =%//]20 X 12/23 X 0.7 = 6 in. Within the small range of the experiments, no significant difference was found in c as the maximum impulsive acceleration was varied. However, if the conservative assumption is made t h a t c varies directly as the maximum 'impulsive acceleration the clearance required by an acce]cration of 0.20 g becomes c = (0.20/0.12) 6 = 10 inches, which is very small in relation to the dimensions of the tank. II. E s t i m a t e 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. Shallow tank (2a = 173 ft., h/a = 0.5), Tr = 8.90 sec. I t has generally been thought t h a t " s t r o n g " ground motions of a period longer than about 2.0 seconds are rare. If so, t h e r e is little p r o b a b i l i t y of resonance with the fundamental mode in large tanks. (For a tank only 12 feet in diameter and having a relative depth of h/a = 1.5, TI = 2.0 sec.) There is some evidence, however, t h a t significant motions of very long period do occur; Housner has computed in one case a motion of 15.2 sec. period and 0.85 cm. amplitude (14). it. Assuming t h a t the 120-ft. tank experiences the following ground motion, which is near resonance with the fundamental natural frequency, f = 0.9 X fi = 0.9/6.35 = 0.14 cycles per second, V = 1.0 inches per second (A = V/2~rf = 1.1 inches), N = 2.0 cycles
(D = N / f = 14 seconds), the maximum wave height will be ~,~ -- 0.7%/120 X 12 X 1.0 X 2.0 = 5 inches (equation 4). This trivial wave height occurs a t the tank wall. (The coefficient 0.07 was determined at the depth h/a = 1, but the behavior of the system is substantially the same at all values of h/a ~ 1.) iii. Assuming f > fi, such t h a t ~ occurs at r < a/3, and letting V = 3.0 and N = 5.0, the maximum wave height is vm = 0.015%/120 X 12 X 3.0 X 5.0 = 9 inches, which also is trivial. N o c l a i m is 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 of w a v e h e i g h t a n d of
" o p e n t a n k " c l e a r a n 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 e c 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 is e v i d e n t t h a t the question of hydrodynamic force will generally be of much greater importance than that of wave height. W a v e s of s i g n i f i c a n 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 i n
346
BULLETIN OF TtIE SEISMOLOGICALSOCIE~r OF AMEEICA
large t a n k s subjected to earthquakes. On the other hand, large waves m a y occur in smaller t a n k s w h e n subjected: a) to large amplitude motions n e a r the top of a building, or b) to large m o t i o n s arising f r o m elastic s u p p o r t of t h e t a n k on a m o v i n g g r o u n d (elevated storage tanks). REFERENCES (I) "Water Pressure in a Tank Caused by a Simulated Earthquake," by Leander M. Hoskins and Lydik S. Jacobsen, Bull. Seism. Soc. Am., 24:1-32 (1934).
(2) "A Laboratory Model Study of the Behavior of Liquid-Filled Cylindrical Tanks in Earthquakes," by Brooks T. Morris, thesis for Degree of Engineer, Stanford University (June, 1938). (3) "Impulsive Hydrodynamics of Fluid Inside a Cylindrical Tank and of Fluid Surrounding a Cylindrical Pier," by Lydik S. Jacobsen, Bull. Seism. Soc. Am., 39:189204 (1949). (4) "Water Pressure on Dams during Earthquakes," by H. IV[.Westergaard, Proc. Am. Soe. Civil Engineers, Vol. 57, p. 1303 (November, 1931). See also the discussion by John H. A. Brahtz and Carl H. Heilbron in the same publication, Vol. 58, p. 897 (May, 1932). (5) "On Hydrodynamic Earthquake Effects," by P. Wilh. Werner and K. J. Sundquist, Trans. Am. Geophys. Union, Vol. 30, No. 5, pp. 636-657 (October, 1949). (6) "Earthquake Resistance of Elevated Water Tanks," by Arthur C. Ruge, Trans. Am. Soc. Civil Engineers, No. 103, pp. 889-938 (1938). (7) "Observed Vibrations of Steel Water Towers," by D. S. Carder, Bull. Seism. Soc. Am., 26:69-81 (1936). (8) "Vibration Research at Stanford University," by Lydik S, Jacobsen, Bull. Seism. Soc. Am., 19:1-27 (1929), (9) "Measuring Earthquake Intensity in Pounds per Square Foot," by H. M. Westergaard, Engineering News-Record, No. 10, p. 504 (1933). (10) "Experimentally Determined Dynamic Shears in a Sixteen-Story Model," by Lydik S. Jacobsen and Robert S. Ayre, Bull. Seism. Soc. Am., 28:269-311 (1938). (11) Hydrodynamics, by Sir Horace Lamb, 6th ed., Cambridge University Pressj 1932, or Dover Publications, New York, 1945. (12) "On Stationary Liquid Waves," by Frederick Guthrie, Philosophical Magazine, Ser. 4, Vol. 50, pp. 290, 337 (1875). (13) "Dynamic Distortions in Structures Subjected to Sudden Earth Shock," by Harry A. Williams, Trans. Am. Soc. Civil Engineers, No. 102, pp. 838-850 (1937). (14) "Ground Displacement Computed from Strong-Motion Accelerograms," by George W. Housner, Bull. Seism. Soc. Am., 37:299-305 (1947). (15) "Hydrodynamic Experiments with Rigid Cylindrical Tanks Subjected to Transient Motions," Technical Report No. 8, Navy Contract N6-Ori-154~ Task 1, Stanford University, Stanford, California.
