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