DYNAMICPRESSURESONACCELERATEDFLUI DCONTAINERS ByG.W.I-IovsI~'ER
ABSTRACT An analysis is presented of the hydrodynamic pressures
developed when a fluid container is sub- jectedtohorizontal
accelerations. Simplified formulas aregiven forcontainers having
twofold symmetry, for dams with sloping faces, and for
flexibleretaining walls. The analysis includes both impulsive and
convective fluid pressures. INTRODUCTION T~EDYNAMIC fluid
pressuresdeveloped duringanearthquakeareof importance in the design
of structures such as dams and tanks.The first solution of such
aproblem wast hat
byWestergaard(1933),whodeterminedthepressuresonarectangular,
verticaldamsubjectedtohorizontalacceleration.Jacobsen(1949)solvedthecor-
responding problem for acylindrical tankcontaining fluid andfor
acylindrical pier
surroundedbyfluid.WernerandSundquist(1949)extendedJacobsen' sworkto
includearectangularcontainer,asemicirculartrough,atriangulartrough,anda
hemisphere.GrahamandRodriguez(1952)gaveaverythoroughanalysisofthe
impulsiveandconvective
pressuresinarectangularcontainer.HoskinsandJacob-
sen(1934)determined impulsive
fluidpressuresexperimentally,andJacobsenand
Ayre(1951)gavetheresultsofsimilarmeasurements.Zangar(1953)presented
the pressures on dam faces as measured on an electric analog.
Theforegoing
analyseswereallcarriedoutinthesamefashion,whichrequires finding a
solution of La Place's equation t hatsatisfies the boundary
conditions. With
theseknownsolutionsaschecksonaccuracy,itispossibletoderivesatisfactory
solutions by an approximate method which avoids
partial-differential equations and infinite
seriesandpresentssolutions insimple forms.The approximate methodap-
pealstophysicalintuitionandmakesiteasytovisualizethefluidmotion,andit
thusseemsparticularlysuitableforengineeringapplications.Tointroducethe
method,the problem of the rectangular tankis treatedinsome
detail;applications toother types of containers are treatedmore
concisely. The more exact analyses show thatthe pressurescan be
separated into impulsive andconvective parts.The impulsive
pressuresarethoseassociatedwiththeforces of inertiaproduced by
impulsive movementsof thewallsof thecontainer,andthe
pressuresdevelopedaredirectlyproportionaltotheaccelerationofthecontainer
walls.Theconvective
pressuresarethoseproducedbytheoscillationofthefluid andare
thustheconsequences of the impulsive pressures.Inthe following
analysis the impulsive and convective pressures are examined
separately, the fluid is assumed to be incompressible~and fluid
displacements are assumed to be small. IMPULSIVE PRESSURES
Consideracontainerwithverticalsidewallsandhorizontalbottomthatissym-
metricalwithrespect tothevertical x- yandz-y planes.Letthewallsof
thecon- tainerbegivenanimpulsiveacceleration~0
inthexdirection.Thiswillgenerate Manuscript received for
publication November 17,1955. [15] 16 BULLETI
NOFTHESEISMOLOGICALSOCIETYOFAMERICA fluidaccel er at i ong,b int
hex,ydi rect i onsandma yalsogener at eanaccel er at i on component
@ int hezdi rect i on. For ar ect angul ar t a nk~isobvi ousl
yzero,and Jacobsen(1949)showedt ha t f or acyl i ndri cal t a
nk~isalsozero.I nwhat follows i t willbeassumedt ha t t her at i
oof ~b t o~isei t her exact l yzeroorat l east sosmal lt ha t @ ma
y be negl ect ed. Physi cal l y, t hi sis equi val ent t ohavi agt
he fluid r est r ai ned byt hi n, ver t i cal membr anes,
spaceddzapar t , whi chforcet hefluidmot i ont ot ake - 17" Fi g.
1. ~uu-x~ Fi g. 2.Fi g. 3. pl acei nt hex,ypl aneonl y. I t ist
hensufficientt oconsi dert hei mpul si vepressures gener at edinal
ami naof fluid. Consi deral ami naof fluidof uni t t hi ckness,
figure1,andl et t hewallsbegi vena hor i zont al accel er at i
onit0. The i ni t i al effectoft hi saccel er at i onist oi mpar t
ahori - zont al accel er at i ont ot hefluidandalsoaver t i cal
component ofaccel erat i on. Thi s act i onof t hefluid issi mi l
art ot ha t whi chwoul dresul t if t hehor i zont al component
,u,offluidvel oci t y werei ndependent oft heyeoSr di nat e; t ha t
is,i magi net hefluid t obeconst r ai nedbyt hi n, massless,ver t i
calmembr anes free t omove i nt hexdi rec- t i on, andl et t
hemembr anes beori gi nal l yspacedadi st ancedxapar t . Whent he
wallsof t hecont ai ner aregi venanaccel erat i on, t hemembr anes
willbeaccel er at ed wi t ht hefluid,andfluidwillalsobesqueezedver
t i cal l ywi t hr espect t ot hemem-branes. Asshowninfigure2,t
hefluidconst rai nedbet weent woadj acent mem-DYNAMI
CPRESSURESONACCELERATEDFLUI DCONTAI NERS17
branesisgivenaverticalvelocity du v= (h-y)(1)
Sincethefluidisincompressible,theaccelerationssatisfythesameequation,so
d4(la) =(h-y) d~ The pressure in the fluid is thengiven by
Op=_piJ(2) Oy wherep isthedensityof
thefluid.Thetotalhorizontalforceononemembraneis These equations may
be written d~ b=(h-y)dzp= P= fo h P=pdy(3) fo yd~d~ - -
p(h-y)dxdy=- ph2( y / h-(y/h) 2)dx fo hditd4 - - oh 2( y/ h--(y/ h)
2)dxdy=-Ph3/3dx-- (4)
Theacceleration~isdeterminedfromthehorizontalmotionofthefluidcon-
tainedbetweentwomembranes.Thesliceof fluidshowninfigure2will be
acceler-
atedinthexdirectionifthepressuresonthetwofacesdiffer.Theequationof
motion is dRdx =-phdx dx Usingthevalueof Pfromequation(4)gives
d2~3. u=0(5) dx2h 2 andthesolutionof thisequationis X 6=c1ooshv/
~~+C2 smhV/g ~(6) Equations(4)and(6)determine thefluid
pressures,andthey arestrictly applicable onlywhenthesurfaceof
thefluidis horizonthl,butifconsiderationis restrictedto
smalldisplacementsof fluid theequationsmay be used even when the
surface of the
fluidhasbeenexcitedintomotion,thatis,equations(4)givetheimpulsivefluid
pressures,p( t ) , correspondingtoarbitraryacceleration~0(t). If
thecontaineris slender,havingh>1.5i,somewhat
betterresultsareobtained 18BULLETI
NOFTHESEISMOLOGICALSOCIETYOFAMERICA -[ - - - - -i 'L j -P.Fig. 4.
~e byappl yi ng equat i ons(4)t ot heupperport i on, h t=1.5/,of t
hefluid onl y andcon- sideringt hefluidbelowt hi spoi nt t
omoveasacompl et el yconst rai nedfluidexert- i ngawall
pressurep~=olito(seefig.3).At adept hof1.5l t hemoment exert edon t
hehori zont al pl anebyt hefluidaboveisappr oxi mat el yequal t ot
hemoment( 3 p i l l3)exert edont hesamepl anebyt heconst rai
nedfluidbelowwhi chimplies t ha t t hegenerat i onof fluid vel oci
t y is rest ri ct edessent i al l y t ot hefluid in t he upperpar t
of aslender cont ai ner.CONNECTI VEP R E S S U R E SWhent
hewallsofafluidcont ai neraresubj ect edt oaccelerations, t
hefluiditself isexcitedi nt ooscillations andt hi smot i on
producespressuresont hewalls andfloor oft hecont ai ner. Toexami
net hefirstmodeofvi br at i onoft hefluidconsidercon- st r ai nt st
obeprovi dedbyhori zont al , rigidmembranes, freet orot at e,
asshownin figure4.Let u , v,wbet hex,y, zcomponent soffluidvel oci
t y, anddescribet he const rai nt s on t he flow by t he following
equat i ons:O(ub)=_ b O YOxOy v=x O( 7 )o o ( o uOz --~ +DYNAMI C P
R E S S UR E S ONACCE L E RAT E DF L UI DC ONT AI NE R S 19 where b
and0 are as shown in figure 4. These equations state,respectively,
t hatthe fluid atagiven x, y moves with a uniform u,t hatall the
fluid atagiven x, y moves withthesamev,andt hat continuity of flow
ispreserved.Inamannersimilarto t hat of the preceding section
theappropriateequationsof motion could bewritten
fortheparticularshapeofcontainerunderconsideration.Ageneralsolution,
applicabletoanyshape(twofold symmetry)canbededuced
asfollows.Fromthe precedingequations f ;1O0xbdx u=bOy-R '(8) b ' aO
f f f xbdx~=~ where b~ =db/ dx. The total kinetic energy-is thus: f
o f ? I + { ( o 0 )f : { R O y / )where I z =f A x2d A K=2- . -b-R
2 ( f _ R xbd x ) 2 ( 1 - J r -Z 2 ( ~ - ) 2 ) } d x d y dz + ( o
)The potential energy of the fluid is V=pgOh2 J x 2 d x d zBy
Hamilton' s Principle =pgOh~I~ ft~ a ( T-V) dt =0 tl dt =0 o r),P
OY2/,P ~ Y h "1-" glzOh~Oh dt= 0Thisgives thetwoequations 028Ix0=0
Oy2K o~oo+g~oh Ot2h =0 (lO) 20BULLETINOFTI~IE
SEIS~IOLOGICALSOCIETYOFA2CIERICA Fr
omwhichthereisobtainedbyintegration sinh~ / ~ y 0----0h sinh~ / ~ h
sin~ot (lOa) Thesearet heequationsfort hefreeoscillationandt henat
ural frequencyofthe fundament al
modeofvibration.Foracontainerofspecifiedshape,suchasrect-
angular,circular,elliptical,etc.,itisnecessaryt oeval uat eonlyt
heintegralsI ,and K.Thepressure in t he fluid is given by
Op_pivOp_p~ OzOx p = - p ~ -&+ ~ Q (11) f Q=x b d xR Knowi
ngp,t heforcesandmoment sont hewallsandfloorof t hecontainercanbe
determinedreadily. RECTANGULARCONTAINER Forarectangularcontainerof
uni t wi dt has showninfigure1,t heboundar ycondi- tionsfor t
heimpulsivepressuresare~=~0 at x=4-1, for whichequation(6)gives
Equat i ons(4)t heng i v eX eosh~/ 3 -~0(12) 1 coshv/ ~ p=- o ~ o h
~Y3( y / h -(y/h)=) p= X sinhV~3 h2 - - p G3l cosh~/ ~ X sinh%/3
cosh%//3 / (13) Thewallacceleration,~0,t
husproducesanincreaseofpressureononewallanda decrease of pressure
on t heopposite wall of DYNAMI C P R E S S UR E S ONACCE L E RAT E
DF L UI DC ONT AI NE R S1 =p i t o h ( y / h -( y / h ) 2) v / 3 t
a nhV/ 3 p w 21 (14) andpr oducesapressureont hebot t omof t het a
nk pb=-p~0h~/ ~s i n h %/ 5 x l 2c o s h x / 3The t ot al forceact
i ngononewallis ( 1 5 )h 21 P=p ~ 0 - - t a n h %/3(16) andi t s r
es ul t ant act sat adi st anceabovet hebot t om h0=gh~1.5(17) I t
isseent ha t t heover-al l effectoft hefluidont hewallsoft hecont
ai ner ist he sameas if afract i on, 2 P +2 1 h p N , of t het ot
almassof t he fluid were f ast enedr i gi dl y t ot hewallsof t
hecont ai ner at ahei ght 3/ 8habovet hebot t om. The magni t udeof
t hi sequi val ent mass,Mo , is 1 t a nh~/3 M0=M(18) 1 wher eMist
het ot al massof t hefluid. Thet ot almome nt exer t ed on t hebot
t omof t he t a nkis x p b d x =-p~toh2l11(19) I ncl udi ngthis, t
hecor r ect t ot al moment ont het a nkisgi venwhent heequi val
entmassM0 isat anel evat i onabovet hebot t omof ( ( )) 34~//3 . .
. . t1( 2 0 )h 0 = h l + 5 \ t n h v / 5The accur acyoft hepr ecedi
nganal ysi scanbej udgedbycompar i sonwi t ht he val uescomput
edbyGr a ha mandRodr i guez(1952).Equat i on(18)givesanM0 sl i ght
l yl argert ha nt ha t comput edbyt heseaut hor s wi t hmaxi mumer
r or lesst ha n 2.5per cent , andequat i on(20)givesanh0 sl i ght l
ysmal l ert ha nt hei rswi t hamaxi - mumer r or lesst ha n2per
cent . I t ma y t hus beconcl udedt ha t f or t her ect angul art a
n k t heerrorsi nt r oducedbyt heappr oxi mat i onofequat i
on(1)arenegligibleso f ar asengi neeri
ngpurposesareconcerned.22BULLETI NOFTt-IESE,IS:IVf0LOGICAL
SOCIETYOFAMERI CA I n t h e cas eoff r eeos ci l l at i ons oft h e
fl ui di nt h e f u n d a me n t a l mo d e f or ar e c t -a ngul a
r t a n k of u n i t wi dt h, e q u a t i o n s (9)a r ef+ z2 L=
x~dx= - ~l 2 --Zz t h u sf [ , 2 ( f + x ) 2 415 K2~dxdx=~o l- l=1a
n d e q u a t i o n s (l Oa)a r e1 s i nhi ~ Yl 0=O~si n~ts i n h
~h1 ~2=~ t a n h ~ 2 l (21) T h e v e l o c i t y a t a n y p o i n
t i nt h e fl ui disgi ve nb yl 2- - x2d~ U- -2dy v= t ~ xTh e pr e
s s ur e i nt h e f l ui dis gi venb yOp=_p~t Ox P=- P 5 - 5 d-~
(22) T h e pr e s s ur e e xe r t e dont h e wal l of t h e c ont a
i ne r , (x=l ), is 1 ~20h si ncot(23) pw=p ~h s i n h i i 7Th e f
or cee x e r t e d ononewal l is f 0hI aP=pwdy=p~~20hsi n~t (24) Th
e t o t a l f or ce, 2P, e xe r t e dont h e t a n k b y t h e fl
ui dist h e s a me aswo u l d b e p r o -d u c e d b y a ne q u i v
a l e n t ma s s M1t h a t iss pr i ngmo u n t e d as s howni nf i
gur e5.I f M1 DYNAMICPRESSURESO N ACCELERATEDFLUIDCONTAINERS23
oscillateswi t hdi spl acement Xlt heforce agai nst t het a nkandt
heki net i cener gyof t hemassareas follows: xl=A1 sin ~t F 1 =-M~
A l w ~ sin ~t(25) T=122 ~ M~ A 1 ~sin s ~t Compar i ngt hesewi t
ht hecorrespondi ngequat i onsfort heoscillatingfluidi t is seent
ha th A1=0h t a nh1 Ml = M( l ~ - ~ / t a n h d h ) h( 2 6 )Fig.5.
The el evat i onofM1abovet hebot t omoft het a nkisdet er mi nedsot
ha t i t pro- ducest hesamemome nt ast hefluid.Consi deri ngonl yt
hemoment oft hefluid pressuresont hewalls(negl ect i ngt
hepressuresont hebot t om) , t her eisobt ai ned (27) Whent
hepressuresexer t edont hebot t omarealsot akeni nt oaccount t
hehei ght is hi=h1. . . . . . . . (28) / ~ h. / g~~-smh~
24BULLETINOt~TtIESEISMOLOGICALSOCIETYOFA~[ERICA
ComparingwiththeexactsolutionofGrahamandRodriguez,itisfoundt
hatequation(21) givesavalue for02 t hat isslightly
toolargewithamaximum error
lessthan1percent;equation(26)givesavalueofM~slightly toolargewitha
maximum error less than 2 per cent. As shown in figure 5, the
over-all effect of the fluid upon the container is the same as a
system consisting of the container, a fixed mass M0, and
spring-mounted masses M~,M~;etc.Itwillbenotedt hat
theformulasforthehigherunsymmetrical (n=1,3,5-)modes
arethesameasforthefirst mode if l isreplacedbyl / n.Theresponse of
thesystem shown in Figure 5 when thecontainer is subjected to
arbitrary horizontal acceleration can becomputed readily. Fromthe
motion of M~,
theoscillationofthefluidinthefundamentalmodecanbedeterminedfrom
equation(26), which gives the relation between A1 and0h. Theactual
displacement
ofthewatersurfaceisdeterminedfromequation(22),whichaty=hgives 1)ph=
p ~ x/1- - ~(x/1) 3~20hsin ~t(
29)Thispressureisproducedbytheweightandinertiaforceofthefluidabovethe
plane y=h. The depth d of water above this plane is thus d- ph
p(e-(30) CYLINDRICALCONTAINER Consider acylindrical tank as shown
in figure 6,subjected to ahorizontal accelera-
tion~0andletthefluidbeconstrainedbetweenfixedmembranesparalleltothe
xaxis.Jacobsen(1949) hasshown t hat animpulse~0 does notgenerate
avelocity component ~in thefluid so t hatin thisease themembranes
donotactually intro- duceaconstraint.Eachsliceoffluid
maythusbetreatedasifitwereanarrow rectangular tank and the
equations of the preceding section will apply. The pressure exerted
against the wall of the tank is,from equation(14), pw=- p( t oh( y/
h-(y/h) 2)~/3 tanh( %/ 3hR-cos )(31) Thepressureonthebottomof
thetankis pb=-p0hs i nhx 2 - - ( 32)cos hx/ 3 The preceding
expressions are notconvenient for calculating the total force
exerted bythefluid.Thefollowing modification
givesveryaccuratevaluesforR/ hsmall and somewhat overestimates the
pressure when R/ his not small. pw=--p(toh(y/h--(y/h) 2)~/5cos ~
tanh~/5 R(31) I b DYNA~IICPRESSURESONACCELERATEDFLUIDCONTAINERS25
Fr omt hi sexpressi ont her esul t ant force exer t edont hewallis
h2~t a nhx / 3 R P~((pocos~Rd~d~=- p~ 0 ~R2h(33) f r omwhi chi t
isseent ha t t heforce exer t edist hesameasif anequi val ent
massM0 wer emovi ngwi t ht het ank, wher e - R t a nh%/3~- M0=M(34)
Fig.6. Compar i ngwi t hJacobsen(1949),i t isf oundt ha t equat i
on(34)over est i mat esM0 wi t hamaxi mumer r orlesst han4per cent
.Toexer t amoment equal t ot ha t exer t edbyt hefluidpressureont
hewall,t he massM0shoul dbeat ahei ght abovet hebot t om
h0=~h~1.5(35) 26BULLETI NOFTHESEI SMOLOGI CALSOCIETYOFAMERI CA I ft
hemo me n t exer t edb yt hepr essur esont het a n k b o t t o mar
ei ncl uded, t heequi va-l ent mass, M0,mus t beat ahei
ght(34%/3~-1. 5~ ~0=~~ +~, ~ n h ~ / - ~ R - 1 ( ~ ~ ( 3 6 )/ t opr
oducet hepr oper t ot al mo me n t ont het ank. Compa r i ngwi t hJ
acobs en(1949) i t isf oundt h a t equat i on(36)unde r e s t i ma
t e s h0wi t hama x i mu mer r or lesst h a n6per cent .The
freeosci l l at i onsof t hefluid(firstmode) ar edet er mi nedf r
omequat i ons (21), et c. For t hecyl i ndri cal t a n kI~7rR~K==-
427 -~R R y si nh-R --8h s i nh- - R (37) Compa r i ngwi t ht
heexact sol ut i on, La mb (1932),i t isf oundt h a t equat i
on(37) sl i ght l yover es t i mat es ~2 wi t hama x i mu mer r or
lesst h a n 1 per cent .Fr omequat i ons (11)t hepr essur eint
hefluidisgi venb yI x P=- - P3- - g4ROy OyR c o s h - - )- - --0h~
2 si n~ts i nh c s ~ 3 s i - ~ 2 - ) c o s 4( 3 8 )(39) The pr
essur eont hewal l is R 30 ~ (P~ = - PX~ 1 The r es ul t ant hor i
zont al forceexer t edont hewal l is 11 P=- ~r ~,o~2R40hsi n~t_12Ml
gGsin~t11 (40) DYNAMIC PRESSURES ON ACCELERATED FLUID CONTAINERS27
Thisforce isthesameast hat producedbyanequivalent massM1oscillating
ina horizontal planewithmotion x~=Al si n ~t h M1=M~\ 12/A 1 =Oh 11
~t anhR (41) Inorder t hat M1 exert thesame moment as thefluid
pressureonthewall itshould be at an elevation above the bottom of
The pressure exerted on the bottom of the tankis sinh-R ix 4R0~ sin
~t ( 4 2 )This exerts a moment about the z axis equalto
32-55~~rRSp~2 sinhR Including this,thecorrect total moment onthe
tank is produced when cosh~h135 - - ~slnh (43) ELLIPTICAL TANK
Proceeding inthesamewayasforthecylindrical tank,theimpulsive
pressureon the wall is given by equation(14) 1 p w=p ( t o h ( y /
h - - ( y / h )2)% / 3 tanh~/ 5~(44) with asimilar expression for
acceleration inthedirection of the yaxis. 28BULLETI
NOFTHESEISMOLOGICALSOCIETYOFAMERI CA Foroscillationsof t
hefluid,equations(21)appl yandfor thefirst modeabout t he minoraxis
e2=_g542 t anh54h(45) a15~-( b) 5 + ( b) 2 awhere2a is t hemaj
oraxisof theellipse and2b is t heminoraxis.Forh /asmallthis
reducesto 0 3 - -Comparingthiswiththeexactsolution,Jeffreys
(1924),itis foundt hat ~ is slightly overest i mat edwithamaxi
mumerrorless t han1 percent. 2 i \ ~o Fi g. 7. Fi g. 8.' COMP OS I
TET ANKSSymmet ri cal t anksformedofcompositeshapessuchast hat
showninfigure7will
haveimpulsivepressuresgivenbyequation(14)andoscillationsdescribedby
equations(21).Thet ankshowninfigure7has K~=RlS{O.233 ( R ) 5 +
O.627 ( R ) 4 -~l . 3 7 7 1 R ) 3 + O.197( R )~(46) R
~-0.1316~-t-0.016 } I ~ECTANGULARDAM Foradamwi t
hslopingrectangularfaceandconstraintsontheflowasshownin
figure8,theimpulsivepressuresaregivenbyt hefollowingequations:
DYNASTICPRESSURESONACCELERATEDFLUIDCONTAINERS29 du v=( h- y)~xx+uc
s =40exp(--~/3 x / h)Op~_pi; Oy(47) p~=p40h-~/ 3-~cos cos} v'~2
?/Fig.9. Theresul t ant horizontalforce onthedamis s i n0~/ 5 (48)
For90>>55,equat i on(48)overestimatesFhby6.5percent;for