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Bulletin of the Seismological Society of America Vol. 53 No. 2 pp. 381-38 7. February 1963

T H E D Y N A M I C B E H A V I O R O F W A T E R T A N K S

BY GEORGE W . HOUSNER

ABSTRACT

During the Chilean earthquakes of May, 1960, a number of large elevated water tanks were

severely damaged whereas others survived without damage. An analysis of the dynamic be-

havior of such tanks must take into account the motion of the water relative to the tank us

well as the motion of the tank relative to the ground. Some simple expressions are given for

the pertinent dynamic properties of tanks with free water surface. A simplified dynamic

analysis is indicated for the response of elevated water tanks to earthquake ground motion.

INTRODUCTION

T h e p e r f o r m a n c e o f e l e v a t e d w a t e r t a n k s d u r i n g e a r t h q u a k e s i s o f m u c h i n t e r e s t

to eng inee rs , no t o n ly because o f the im por tan ce o f the se t anks in con t ro l l ing f ir e s,

bu t a l so because the s imple s t ruc tu re o f an e l eva ted t ank i s r e l a t ive ly ea sy to

a n a l y z e a n d , h e n c e , t h e s t u d y o f t a n k s c a n b e i n f o r m a t i v e a s t o t h e b e h a v i o r o f

s t r u c t u r e s d u r i n g e a r t h q u a k e s . D u r i n g t h e C h i l ea n e a r t h q u a k e s o f M a y , 1 96 0 a

n u m b e r o f e l e v a t e d w a t e r t a n k s w e r e b a d l y d a m a g e d a s d e s c r i b e d i n a c o m p a n i o n

p a p e r b y K . V . S t e i n b r u g g e a n d R . F l o re s . O t h e r e l e v a t e d w a t e r t a n k s s u r v i v e d

w i t h o u t d a m a g e , a s d e s c r i b e d i n a c o m p a n i o n p a p e r b y W . K . C l o u d . A d y n a m i c

a n a l y s i s o f s u c h t a n k s m u s t t a k e i n t o a c c o u n t t h e m o t i o n o f t h e w a t e r r e l a t i v e t o

the t ank a s we l l a s the mot ion o f the t ank re l a t ive to the g round . I f a c lo sed t ank

i s comple te ly fu ll o f wa te r o r com ple te ly emp ty , i t i s e ssen t i a lly a one -mass s t ruc -

ture . I f , as i s usua l , th e t an k has a f ree wa ter s urface there wi l l be s loshing of the

w a t e r d u r i n g a n e a r t h q u a k e a n d t h i s m a k e s t h e t a n k e s s e n t i a l l y a t w o - m a s s s t r u c -

tu re . In th i s c a se , t he dynamic behav io r o f an e l eva ted t ank may be qu i t e d i f fe ren t .

F o r c e r t a in p r o p o r t i o n s o f t h e t a n k a n d t h e s t r u c t u r e t h e s l os h in g o f t h e w a t e r m a y

b e t h e d o m i n a n t f a c t o r , w h e r e as f o r o t h e r p r o p o r t i o n s t h e s l o sh i ng m a y h a v e s m a l l

e f fec t . The re fo re , an unde rs t and ing o f the ea r thquake damage , o r su rv iva l , o f

e l e v a t e d w a t e r t a n k s r e q u i r e s a n u n d e r s t a n d i n g o f t h e d y n a m i c f o r c e s a s s o c i a t e d

wi th the s lo sh ing wa te r .

TANK ON THE GROUND

A wa t e r t a nk o r r e se rvo i r on the g round wi ll have i t s co n ten t s exc i t ed in to s lo sh ing

by an ea r thquake and the ampl i tude o f the s lo sh ing i s ind ica t ive o f the in t ens i ty

o f the g ro und m ot ion . I f a t ank w i th a f ree wa t e r su r face fig. l a ) i s sub jec ted to

h o r i z o n t a l g r o u n d a c c e l e r a t io n a t h e f o r c e s e x e r t e d o n t h e t a n k b y t h e w a t e r a r e o f

two k inds . F i r s t , wh en the wa l l s of the t a nk acce le ra t e back and fo r th a ce r t a in

f rac t ion o f the wa te r i s fo rced to pa r t i c ipa te in th i s mo t ion , wh ich exe r t s a r eac t ive

fo rce on the t ank the same a s wou ld be exe r t ed by a mass M0 tha t i s a t t a ched

r ig id ly to the t ank a t t he p rope r he igh t a s shown in figu re lb . Th e mass M0 i s

a t t a c hed a t a he igh t h0 so th a t the ho r i zon ta l fo rce exe r t ed by i t is co l linea r w i th the

r e s u l t a n t f o r c e e x e r t e d b y t h e e q u i v a l e n t w a t e r . S e c o n d , t h e m o t i o n o f t h e t a n k

wa l l s exc i t e s the wa te r in to osc i l l a t ions wh ich in tu rn exe r t an osc i l l a t ing fo rce on

the t ank . Th i s o sc i ll a t ing fo rce i s the sam e a s wou ld be exe r t ed by a mass M I tha t

381

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38 2 BUL LET IN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

can oscillate horizontally against a restraining spring as shown in figure lb. The

mass M1 corresponds to the fundamental mode of oscillation of the water which is

the mode of importance for most earthquake problems. If the equivalent system

shown in fig. lb is subjected to earthquake ground accelerations a the forces exerted

__

d

R

FIG. la

J~

M

~

h, Mo

FIG. lb

F I G . 1 . E q u i v a l e n t d y n a m i c s y s t e m f o r a w a t e r t a n k .

T he t ank w i th osc i l l a t ing wate r su r f ace i s : shown in l a . T he eq u iva len t s ys tem i s shown in

l b w h e r e M0 a n d M 1 p r o d u c e d y n a m i c f o r c e s e q u i v a l e n t t o t h o s e p r o d u c e d b y t h e w a t e r .

on the tank by M0 and M1 will be the same as would be exerted by the water in the

tank of figure la.

The equivalent system is specified by the following quantities? 2 For cylindrical

tank of radius R and water depth h

tanh 1.7

R h

M0 = M

1.7

R h

1 J a c o b s e n , L . S. , h n 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 ic a l T a n k , B u l l .

S e i s m . S o c . A m .

Vol. 39, 1949.

2 I t o u s n e r , G . W . , D y n a m i c P r e s s u r es o n A c c e l e r a t e d F l u i d C o n t a i n e r s , Bu l l . S e i s m . S o c .

Am., Vol. 47, 1957.

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DYNAMIC BE HAVIOR O F WATER TANKS

8

M ~ = M 0 . 6 )

t a n h 1 . 8

h / R

1.8

h / R

kl = 5.4 M12 gh

M R ~

ho = ~ h { 1 + a I M ~ ~ ) 2 - - 1 7

a = 1 .33

h l = h 1 - 0 . 1 8 5 ~ - 0 . 5 6 ~ - , V \ 3 M ~ h ] 1

= 2.0

T ~ = 2~r - ~ = a p e r io d o f v ib ra t io n

M = t o t a l m a s s o f w a t e r i n t h e t a n k

F o r r e c t a n g u l a r t a n k s o f l e n g t h 2 L a n d w a t e r d e p t h h

M o = M

t a n h 1 . 7

L / h

1.7

L / h

M 1 = M 0 .8 3 ) t a n h

1 . 6 h / L

1.6 h / L

kl = M12 gh

M L 2

ho= ~ h 1 4 a M 1 1

a = 1.33

h i = h ~ l - - -

= 2.0

¢ o

- o . 6 3 L M L y _

3 M1 ~ .28 \M ~ h ] 1

I f t h e h e i g h t s h0 a n d h i ar e t o b e d e t e r m i n e d o n t h e b a s is o f t h e d y n a m i c f l ui d

f o rc e s e x e r t e d o n t h e w a l l s o f t h e t a n k o n l y n o t o n t h e f l o o r) , t h e f o l l o w in g v a l u e s

s h o u l d b e u s e d fo r b o t h c y l i n d r i c a l a n d r e c t a n g u l a r t a n k s : a = 0 , ~ - - 1 .

A g r a p h o f t h e h y p e r b o l i c t a n g e n t t a n h 1 . 6 Z ) i s s h o w n i n fi g u r e 2 .

T h e m a x i m u m o s c i ll a t io n of t h e f l u i d i s r e l a t e d t o t h e o s c i ll a t io n of t h e m a s s

M 1 a s fo l lo w s . I f M 1 i s o s c i l l a t i n g w i t h d i s p l a c e m e n t x = A 1 s i n e t t h e c r e s t o f t h e

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384

:BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

wave at the wall of the tank will oscillate up and down with displacement y = d

sin ~ t and for the cylindrical tank

k,R

0.63

A1 \M1 g]

d =

R~ ~

R

1 0

0 8 f

0 6

Tan h 1 6Z:

0 4

0 2

8

¢

0 2 0 4 ~ 6 a 8

Z

FIG. 2. Graph of tanh 1.6Z.

1 0 1 2 1 4

and for the rectangular tank

0.84A1 kl g~

\M1 g

d =

AI (kl L ~

1 f \ M ~ e~

The foregoing equations give good results for amplitudes of vibrat ion d < 0.2L

and d < 0.2h, although at the larger amplitudes a certain amount of nonl inear ity

is observed in the oscillations.~

As an example of the oscillations that may be induced by earthquake ground

motion, consider a rectangular tank 50 ft long and 10 ft deep L = 25 ft, h = 10

ft) t ha t is resting on the ground. For this tan k there is obtained a period of vibration

of 6 seconds. For relatively strong ground motion with an undamped velocity

spectrum value S~ = 2.0 ft/sec, the displacement of the mass M~ would have an

amplitude

T

A = 2.0~ = 1.gft

3 Conrad, A. , Hydr odyn amic Forces Induced in Fluid Containe rs , Thesis , Cal i fornia

Ins tit ute of Technol ogy, 1956. It is also shown in this thes is that a cylindrical tan k with a

hemispher ical bot to m ma y be t reated as a fia t bot tom t ank of the same volume of water so far

as computing the per iod and the dynamic forces .

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DYNAMIC BEHAVIOR OF WATER TANKS 85

and the ampl i tude o f the s losh ing i s found to be

d = 1 . 5 f t

I t shou ld be no ted tha t the re w i l l a l so be a co r re spond ing osc i l l a t ion o f the f lu id

a t 90 degree s to the fo rego ing mo t ion s ince the ea r thq uak e g round m ot ion wi l l

have a componen t in tha t d i rec t ion a l so .

ELEVATED WATE R TANK

A n e l e v a t e d w a t e r t a n k c a n b e r e p r e s e n t e d b y t h e e q u i v a l e n t s y s t e m s h o w n i n

f igure 3 . Th e spr ing c on s tan t /c repre sents th e e ffec t ive s t i f fness of the s t ru c tu re

i - - x

~ o

Mi

FIG. 3. Eq uivalen t dynam ic system for an elevated water tank .

and M 0 i s the equ iva len t mass o f the s t ruc tu r e p lus M 0. As an i l lu s t ra t ion , a t ank

hav ing R = 20 f t and h = 15 f t ha s a na tu ra l pe r iod o f osc i ll a t ion o f wa t e r g iven by

T - - - 2 7 r ~ - 4 , 0 s e c

where

M1 = 0 .44M = 800,000 lb /g

]~1 = 6 2,000 lb /f t

M0 = 0 .35M = 640,000 lb /g

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386 B U L L E T I N O F T H E S E I S M O L O G I C L S O C I E T Y O F M E R I C

I f t he mass an d s t if fnes s o f the s t r uc tu re a re such th a t

k = 700,000 lb / f t

M0' = (640,000 + 160,000) lb /g = 800,000 lb /g

the two mod es o f v ib ra t ion wi l l hav e the shapes shown in f ig . 4 , and the pe r iods a re

resp ec t iv e ly T1 = 4 .2 sec and T~ = 1 .1 second. I t i s seen th a t the per io d and forces

o f the f i r st mode a re c losely appr ox im a ted by the pe r iod and fo rce o f the f luid w i th

an in f in i t ely r ig id suppo r t ing s t ruc tu re . Th e pe r iod and fo rce s o f the s econd mod e

x

. 0 9 x =

FIG. 4. T he modes of vibration for an elevated water tank .

a re e s sen t i a lly those o f the mass M 0 ' a lone d i s rega rd ing the m ass M 1. Th e ma x im um

shea r fo rce p rodu ced b y M 1 dur ing g ro und m ot ion w i th S~ = 2 .0 is

27

F1 = ~ M1 = 39,000 lb

wi th a pe r iod o f 4 .0 seconds. Th e m ax im um fo rce p rodu ced b y M 0 ' is

F2 = 27r M0 ' = 140,000

lb

a t a pe r iod o f 1 .1 s econds . Comp ar ing w i th the to t a l m as s o f the t a nk the se fo rce s

a re , r e spec t ive ly , equ iv a len t to 2 and 7 g . I f t he t an k were com ple te ly fu l l so

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D Y N M I C B E H V I O R O F W T E R T N K S

387

t ha t t he r e c ou l d be no s l o s h i ng t he pe r i od wou l d be 1 .7 s e e a nd t he f o r c e wou l d be

220,000 pounds .

A c o m p a r i s o n o f t h e m e a s u r e d a n d t h e c a l c u la t e d p e r io d s o f v i b r a t i o n o f a n

e l e v a t e d w a t e r t a n k a r e p r e s e n t e d i n a c o m p a n i o n p a p e r b y W . K . C l o u d.

As a no t h e r e xa m pl e , c ons i de r a go l f - ba l l - on - a -t e e t yp e o f t a n k w i t h r a d i u s R

= 20 f t a nd ha l f f u ll o f wa t e r . Th i s m a y be t r e a t e d a s a c y l i nd r ic a l t a n k o f t he s a m e

r a d i us a nd e qua l vo l u m e o f w a t e P R = 20 f t , h = 13.3 f t ) , a nd t he f o l lowi ng

v a l u e s a r e o b t a i n e d

M = 1 ,000,000 lb /g

M0 = 0 .5 M = 500,000 lb /g

M~ = 0 .42 M = 420,000 lb /g

T = 4.0 sec

I f th i s is c om pa r e d t o t he c a se o f a c om pl e t e l y f u ll t a n k w i t h M 0 = M a nd M 1

= 0 , i t i s s e e n t h a t t h e m a x i m um f o r c e t o wh i c h t he ha l f - f u ll t a n k i s s ub j e c t e d m a y

be s i gn i f i c a n t l y l e s s t ha n ha l f t he f o r c e t o wh i c h t he f u l l t a nk i s s ub j e c t e d i f t he

m a x i m u m f o r c es e x e r t e d b y M 0 a n d M 1 h a p p e n n o t t o c o in c id e . T h e a c t u a l f o r c es

m a y be a s l i t tl e a s 1/d o f t he f o r ce s a n t i c i p a t e d on t he ba s i s o f a c om pl e t e l y f u l l t a nk .

4 Con rad, A.,

loc ci t

C LIFORNI ~NSTITUTE OF TECHNOLOGY

P S DEN ~ C LIFORNI

Manuscript received April 12, 1962.