I N T E R N A T I O N A L S Y M P O S I U M O NE A R T H Q U A K E S T R U C T U R A L E N G I N E E R I N G

Sl. Louis. Missouri, USA, August, 1976

SEISMIC DESIGN OF LIQUID STORAGE TANKSTO EARTHQUAKES

P. C . CHEN and R . B . BARBER

Eng inee r i ng Spec ia l i s t s

Scient i - f ic Development, Bechtel Corporat ion

S a n F r a n c i s c o , C a l i f o r n i a , 9 4 I 1 9 , U . S . A .

SIJMMARY

A r igorous and exper ioental ly ver i f ied technique (Bauer) which f indsEhe solut ion to the Laplace equ,€r t ion and sat is f ies the boundary condi t ionsfor hydrodynanr- ic probleuls was conpared vr i th the currenc tank seismicdesign procedures (Housner) . The current tank design procedures weredeveloped using an approximace approach which did not consider the usualprocedures of deternin ing velociEy potent ia ls for hydrodynanic problens.A- lso, i t only considered the f i rsr node f requency of f lu id osci l la t ion.

Factors inc luded in the compar ison were f lu id osci l la t ion f requencies,f lu id dynanic pressures, equivalent mechanical models, and tank basemoments. The NRC seismic design specEra were used in the compar ison asthe design earchquakes. The resul ts show that for cy l indr ical tanks,the current design approach general ly g ives smal ler cank base moments chanthe r i go rous app roach . Fo r r ec tangu la r t anks , t he d i f f e rences i n cankbase moments between those two approaches are snal l ,

Based upon above resu lEs , imp roved t echn iques we re deve lopedfo r t ank se i sm ic des ign . These s imp l i f i ed t echn iques use t he r i go rousapp roach f o r t ank ana l ys i s and cons ide r h i ghe r mode f r equenc ies o f

f l u i d osc i l l a t l on . Pa rame te r cha r t s we re a l so deve loped wh l ch g rea t l yf ac i l i t a t e t he se i sm ic ana l ys i s and des ign o f l i qu i d s t o rage l - anks .

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INTRODUCTION

The ser ious damage suf fered by sEorage tanks in recent eart .hquakes(4 , 8 , l 0 ) denons t ra tes t he need f o r r e i nves t i ga t i ng cu r ren t l y usedapproaches in designing Eanks to resisE f lu id dynarnic forces induced fyearthquakes. The need is furcher intensi f ied when one real izes Ehac chespen t f ue l poo I , r ad ioac t i ve s to rage t anks , and supp ress ion chambers o fnuclear power p lants are designed usj-ng Ehese same approaches. Theapproaches are based on the assumpt ions Ehat lhe tank is r ig id and onlyt he f i r s t s l osh lng mode i s p resen t i n che dynam ic ana l ys i s ( 7 , 13 ) . Theanalysis i tsel f is carr ied ou! by an approximate nethod which avoids cheusual prdcedures of f inding a solut ion Co che Laplace equat ion thatsat is f ies Ehe boundary condi t ions for hydrodynamic problems.

Since the publ icat ion of current ly used oi l s torage tank designp rocedu res i n 1957 (7 ) , ve r y 1 i t t 1e resea rch wo rk has been doneconcerning s loshing forces in c iv i l -cype structures. In the Deant ime, Ehestate of the ar t has been advanced in the space industry through ef for tst o m in im i ze t he e f f ec t o f p rope l l an t s l osh ing on t he s tab i l i t y o f spaceveh i c l es ( 1 , 3 ,6 ) . The ana l ys i s o f p rope l l an t s l osh ing i s r i go rous , andthe resul ts for cy l indr ical tanks have been exper imencal ly ver i f ied (2) .

The nature of space vehic le propel lant s loshing and f ixed storagetank l iquid s loshlng is the same except for the cause of l iquid d is lur-bance. For the former, the disturbances are gust loads and the mocion ofspace vehic les; for the lat ter , the disturbances are caused by earCh-quakes. Therefore, i t is desirable to compare the Eechniques for spacevehic le fuel tank design wi th the current design approaches for l iquidstorage tanks in order to deternine the adequacy of the lat ter . Becausethe forrnulaLions of and solucions to the problem of l iquid s loshing insidea f lex ib le tank are extremely complex and i ts analysis is st i11 in Chedeve lopmen ta l s cage (5 , 11 , 14 ) , t he compar i son i s r es t r i c t ed co chemoEions of a contained f lu id ins ide r ig id tanks.

This paper compares the resul ts of design procedures for bothrecEangu la r and cy l i nd r i ca l space veh i c l e f ue l t anks and o i l sEo ragecanks. Based on chis compar ison, improved techniques are suggested forthe seismic design of o i l s torage tanks. Parameter charts which faci l i tatethe seismic analysis and design of such tanks are also developed.

BACKGROIJND

The formulat ion of and solut ion to the general problem of f lu idosci l laEions in a lateral ly moving tank are extremely d i f f icul t . Theassumpt ions general ly enployed in such a theoret ical analysis are thacthe tank is r ig id, the f lu id is homogeneous, incompressib le andnonv i scous , t he f l ow i s i r r o t a t i ona l , and t he d i sp l acenen ts , ve l oc i t i e s andand s l opes o f f he f l u i d su r f ace a re s rna l l .

The assump t i on o f i r r o t a t i ona l f l ow i nsu res Ehe ex i sEence o f a s i ng le -value veloci t .y potent ia l in a s i rnply connected region, f rom which theve loc i t y can be de r i ved . Us ing t he equaE ions o f mo t i on f o r a pa rE i c l ein a nonviscous f lu id and consider ing the mass conservat ion and incom-p ress ib i l i t y , Ehe ve loc i t y pocen t i a l i s t he so lu t . i on t o t he Lap lace equa t i on .

W iEh t he bounda ry cond i t i ons t ha t t he no rma l ve l oc i t y a t t he i n t e r f aceof the f lu id surface and tank wal l equals Ehe normal veloci ty of thef l u i d pa r t i c l e a t t haE po in t , t he su r f ace p ressu re j . s ze ro , and t hef l u i d pa rC i c l es mus t scay on t he su r f ace , Ehe ve loc i t y po ten t i a l i s t hende te rm ined . The ve loc i t y componen ts , p ressu re d i s t r i bu t i ons , and t hef lu id surface displacement are then obtained f rom the knor.rn veloci typotent ia l . The forces and momencs act ing on the tank are deterru ined byapp rop r i aEe i n t eg rac i ons o f p ressu re (12 ) .

Using the above basic theory, Graham and Rodr iquez (6) der ivedequaE ions wh i ch dece rn r - i ne t he f r equenc ies , su r f ace d l sp l acemencs ,p ressu res , and f o r ces assoc ia ted w i t h f l u i d s l osh ing i n r ec tangu la rEanks . Baue r ( 3 ) de r i ved t he equ i va lenE equa t i ons f o r cy l i nd r i ca l t anks .The cank geometry and coordinate syst .en for the rectangular and.cyl indr icalEanks are shown in Figures I and 2 respect ively,

The complexi ty of che solucion Eo the s loshing problen in laEeral fyooving tanks resul ted in Housner 's (7) approxinace approaches. Theseapproaches are based upon the s impl i f icat ion thac the pressures act ingon the tank dur ing laceral s loshing can be separat .ed into impuls ive andconvect ive components (Figure 3) . With the separat ion of pressures into tvroparts, each part is developed separately so that the usual procedures ofdeteruin ing veloci ty potent ia ls for hydrodynanic probleurs are not used. Also,only the f i rs t v lbrat ion mode of the f lu id ts considered in che analysis.

The formulas for f requencies. pressures and surface displacementsfor both the rectangular and cyl indr ical tanks using the di f ferent approachesare sunmarized respect ively in Tables I and 2.

The osci l la tory mof ion of f lu id in a tank roay be near resonancewith the orot ion of the support ing structure and Ehus roay lead to dynamici nsEab i l i t i e s o r s t r uccu ra l f a i l u re . I n o rde r t o m in i n i ze poss ib l estructural fa i lure and real is t ical ly determine the coupled structuralresponse Chrough dynamic analysis, i t is somet j .mes desirable to replacethe f lu id r . r i th a dynamical ly equivalent mechanical system and combine i twi th the struccural model . This dynaruic equivalence is taken in the senseo f equa l f r equenc ies , uu l sses , r esu l can t f o r ces , and r cmencs acc i ng on t heEank wa1ls and botcom. The dynanical ly equivalent rnechanical model forf l u i d s l osh ing no rma l l y cons i s t s o f a se r i es o f sp r i ngs and masses a t t achedat speci f ied heights f rorn the tank base, in the plane of syuunetry and int he d i r ec t i on o f base exc i t a t i on .

The equivalent mechanical rnodel-s for laceral s loshing in the rectangularEanks are sumnar ized in Table 3. Their detai led der ivat ions, are givenin References 6 and 13. The diagrams of the equivalenE mechanical modelsare shor^m in Figure 4 for both the Graham and Rodr iquez approach andEhe Housner approach. SiuLi lar ly , the equivalenE mechanical models forI a te ra l s l osh ing i n cy l i nd r i ca l Eanks a re sumnar i zed i n Tab1e 4 , and t he i rder ivaEions are given in References 3 and 13. The diagrams of the equiva-l en t mechan i ca l mode l s f o r bo th t he Baue r and Ehe Housne r app roaches f o rcy l i nd r i ca l Eanks a re s im i l a r t o t hose f o r t he rec tangu la r t ank and a reshoun in Figure 5.

1234

COMPARISON OF TANK DESIGN TECHNIQI'ES

Tank des ign t echn iques f o r bo th t he rec tangu l - a r and t he cy l i nd r i ca lspace veh i c l e p rope l l an t Eanks and o i l s t o rage t anks we re compared us i ngche Grahan and Rod r i quez (G&R) , Baue r , and Housne r app roaches . Thecompar i son i . nc l uded f r equenc ies , p ressu res and t ank base xoomen ts . S incethe resul t j .ng moments about. the tank base are obEained by the integrat iono f p ressu res , t he resu lEs o f p ressu re compar i sons a re seen f r om the com-par ison of resul t ing moments. The equi-valent mechanical rnodels were usedto comPute the base noDencs.

(a) Rectangular Tanks

The f i rs t mode f requency compar ison for uwo fani l ies of rectangulartanks for both the G&R and Housner approaches is g iven in Table 5. Thepredicted f requencies are almosL ident ical for the two methods. Nohigher mode f requenci .es were compared s ince Housnerts approach gives onlythe f i rs t rnode f requency.

Flgure 7 compares lhe resul t ing Eank base rDoments for Ehe Ewofani l ies of tanks studied, One set of l ines in Figure 7 ( ident i f iedby a= l6 r ) i s f o r t he f an i l y o f t anks w i t h a f i xed t ank l r i d ch o f 16 f ee t ,MomenEs f rou the G&R approach and both the IBP (which includes bot tompressures) and EBP (which excludes botEom pressures) cases of theHousner approach are labeled in the f igure. The NRC design spectran (9) ,Figure 6, were used for comput ing the momenEs due to s loshing massesfor both the G&R and Housner approaches. A f lund daroping value of 0.52and a naxlnum ground accelerat ion of l .0g were assurned. I t is interest ingto noce that for the farnl . ly of tanks studied, the resul-Es of the G&Rapproach l ie between Ehose of Ehe IBP and EBP cases of the Housner appThe other set of l ines ln Figure 7 ( ident t f l ,ed by h=20r ) is for thefani ly of tanks wl- th a f ixed tank height of 20 feet . Ttre resul ts show thatmonencs f rom the G&R approach have a near constant value, whi le Che Housnerapproach tends toerard a consEant moment--cl,ose Eo that moment given bythe G&R nethod--as the tank aspect rat io increases. For Eank aspecrrat ios smal ler than 0.5, the Housner approach gives excessively large IBPllpmenEs.

(b) Cyl indr ical Tanks

The f i rs t rDode f requency compar ison of two fani l ies of cy l indr icaltanks for both the Bauer and Housner approaches is shown in Table 6. Thecompar ison shows almosE ident ical resulEs, The higher mode f requenciescould not be compared s ince Housnerts approach only g ives the f i rs t modef requency .

A compar ison of resul t . ing lank base ulomenls for rwo faroi l ies ofcyl lndr lcal tanks is shown in Figure 8. The curves in Figure 8 ( idenci f iedby F l6 ' ) a re f o r t anks w i ch a f i xed cank d i amere r o f 16 f ee r . Theresul ts obtained using the Bauer approach and Ehe IBP and EBP cases of theHousne r app roach a re so l abe ied . The NRC des ign spec t rum w i t h a 0 .52damping value and a maximun ground acceleraf ion of 1.0 g was used forde te rn i n i ng t he momen ts due co t he s l osh ing rDasses . The resu l t s show Ehac 'e x c e p t f o r t a n k a s p e c f r a t i o s s m a l l e r t h a n 1 . 0 , t h e B a r r t ' l - : p D r o ' r ; ' h q i r . r e q

l -- r " ' l i r i : . i , r . a : , l i ' , , r : . - , , i

1235

The cu rves i n F i gu re 8 ( i denE i f i ed by h=20 ' ) a re f o r t he f an i l y

o f t anks w i ch a f i xed t ank he igh t o f 20 f eeE . The momen ts p red i c t ed

i v U t , . Baue r app roach a t f i r s t i n c rease as t he t ank asPecE ra t i o i nc reases

"r ,d th.n approach a conscant value. For t .ank aspecg raElos smal ler than

0.5, the Housner approach gives unreasonably large IBP momenrs. For

t ank aspec t r a t i os l a rge r t han 1 .0 , boCh t he IBP and EBP momen ts t end t o

be cons tan f '

The conpar ison of t .ank design Eechniques indicates EhaE for

ty l indr ical tanks, the current design technique (Housner 's approach)

general ly g ives snel ler base ."T"1! : than the exPer imencal ly ver i f ied

iauer "pproach.

A di f ference of 15l l in base monents has been observed

gure 8) . For reclangular lanks ' the G6R and Housner approaches

generall_y give only small differences in base moments as shoum io Figure

1. This j -s noE surpr is ing, s ince the Housner approach was developed af ter

he G&R approach was publ ished and used the G&R resul ts for ver i f icat ion

(7). For tank asPect rat ios smal ler Ehan 0.5, rhe Housner approach gives

excessively large IBP moments for boch the recEangular and cyl indr ical

tanks. In v iew of these observat ions, the more r iSorous G&R and Bauer

spproaches should be used to achieve more re l iable tank designs.

DESIGN CITARTS

fn order to faci l i taEe tank design using Ehe G&R and Bauer approaches'

aeveral design charts were developed. Figure 9, 10, and l l g ive the

frequencies, u lasses, and mass locaEions in the tank for the equivalent

nical model of lateral s loshing in recEangular tanks- The approxi-

e base Doments for rectangular tanks Lt i th asPect rat ios between 0 '25

6.0 are shown in Figure 12. The f requencies, masses and mass locaEion

the tank for che equivalent oechanical model of lateral s loshing

cyl indr ical tanks are given in Flgures 13, 14 and 15. The approxi-

e base nonents for cy l indr ical tanks wi th aspect rat ios bet ldeen 0.25 and

.0 are shown in Figure 16. The base moments f rom these charts are at

t l0Z snaller than the base moments obtained for a rigorous dynaxtricanalysis using the NRC design spectra wi th a I .0 g naxinum hor izontal

round accelerat ion. I f the ground accelerat ion is smal ler rhan 1.0 g '

the tank base moments for Ehese charts should be scaled down proport i -onal) .y.

CONCLUSION

The compar ison of tank design cechniques indicaEes Ehat for cy l indr i -cal tanks, the current desi .gn technique (Housner approach) general ly

Provides sm. l ler tank base monents than the exper imental ly-ver i f ied Bauerapproach, A di f ference of I5" l in base momencs has been observed. Inrectangular tanks, Ehere is a soal1 di f ference in Eank base momentscalculated using the Graham and Rodr iguez (G&R) aPProach as comPared tothose obtained by t .he Housner approach, The di f ference in base nonentsa l so re f l eccs t he d i f f e rence i n dynam ic p ressu res . Ve ry l i t t l e d i f f e rencein t he f i r s c node osc i l l a c i ng f l u i d f r equency was obse rved .

ir

I 238 1239

Tab le 3 . Equivalent Mechanical Model for LaEeral Sl-oshineRectangular Tank

Tab le 5 . Compar i son o f F requenc j - es ,Conscan t Tank He igh t , h = 20 '

RecEangular Tank

Table 6. Compar ison of Frequencies,Constant Tank Helghc, h = 30r

Cyllndrical Tank

I i r u r i l F r c q u c . , : y

S p r t o a C o i l r r a i l r

R n t € r o f S l o l hd n 3 s r o F l u l d

s a r e s o f F r x c dx r s s t o F l u t d

f , a t r e o f S l o s hl i a s . H c l t h a t oF I u l d D c p r h

of = Xtz* rSE t"^h B z^a ) n a l

K-.**t.s,L

#r-z+*#Yt-

#-,-E*:

b,,_4##'L

r't-,#,E# #

i . ' fa t t ^L r , ss f )

l(,..w,us,z

ftu*!. &Erp.,-tfue; ia.-tuaeur. aitu aar

for LaEeral Sloshing -

R . t l ^ o f i l x c d: l r r . l l e l t \ i t oF t u t d D a p r h

Table 4. Equlvalent MechanLcal ModelCyllndrlcal Tank

lt!_!!lI! I DTI I HOUSXER c&\| Y 6 RODAICU,]Z

1 6 7 t 0 t 8

, 2 0 0 . t 2 0 l l 1 9 5

, 2 5 0 . 1 4 t 6 t149

l l l l 8 t 4 l 8 2 6

. 500 2Ltt 242L

. 6 6 7 2888

I . 0 0 , l 5 8 l t t t 2

r . l l . ( l ( 4

? . 0 0

4 . 0 0 . 7 1 b 0

. 1 0 6 2 |

.r;---l

' i : " { ( eaP)

* ' i [##h, ' , i ( iBP)

u;,{ e^.t ̂Lrc**)

lG-^^u,i.

*.*.- zdJk*4)v1.- a* * (1-t)

H " , 2 a ^/4F tuo "tF

L

f - t - 4t^^L"gr^1,

N a t u r . l F r e q u e n . y

S p r l n t C o n . t a n r

R n t 1 0 o f S l o 3 hY r e i t o F l ! t d

R a t l o o f e l x € dY a s s r o t l u t d

R a : t o o f S l o s hI l s s H e l t h r r of l u t d D € p t h

*.t('tE(H(hb t l o o f F l x c dY ! s s ! L l F h r t oF l ! : d D c p r h

,a,z- ff.7.tn-Ltr.u {-)

Kr =mu, d,L

ft- .ot,a ft tn^ttt a+ {'1

H" t^^JL (l3 x, )^, ' - - : f ;T-

l- r- 44s:A)-:L- ,,u,,L t .sq(*) L&JL(r.s{t) . - .

Lr"=t- f f i#fu)Gse)

L 3T ' B - ( E a P l

4 " = * { . a F i ; - - } . , u o tL d ( . a j ( f 3 + ) ) " -(*-aa

", | ' tzt .o a @ - ( . . ' e d r a b t a - t s r u . < , I g p - D . ( 0 , { 6 E o r r N R €

. 6 J 2 t

. 7 0 t \

[ .

app roaches a re sugges ted t o be used t o ach ieve more re l l ab . l e r ank

design. Design charts are developed to facj - l icate the use of these

approaches for the design of canks subjected to earthquakes.

REFERENCES

7

Abramson, H-.N. ed. , "The Dynamic Behavior of L iquids in MovingQon ta i ne rs , " NASA SP-106 , 1966 .

Ab ranson , H .N . e t a l . , "Wa11 P ressu re D i s t r i bu t i ons Du r i ng S losh ingin R ig i d Tanks , " ARS J . Vo1 . 31 , No . 4 , Ap r i l 1961 .Baue r , H .F . , r rF l u i d Osc i t l a t i ons i n t he Con ta i ne rs o f a SpaceVehic le and lheir Inf luence Upon Stabi l i ty ," NASA TR R-187, 1964.Bug , G . V , and S t ra t t a , J . L . , ' rAncho rage and t he A laska Ea r thquakeo f 1964 , " An re r i can I r on and S tee l I ns t i t u t e , Ju l y , 1974 .Cooperat ive Research Program on the Response of L iquid- f i l1ed StorageTanks to Earthquake Exci t .at ion. Current research act iv i ty ac theUnivers l ty of Cal l fornia, Berkeley, Cal i fornia.Graham, E.W. and Rodr lguez, A. l , l , , ' rThe Character is t ics of FuelMot ion Which Af fect Airp lane Dynamics," J, Appl . Mech., Vo1. 19,N o . 3 , S e p t e n b e r 1 9 5 2 .Housne r , G .W. , t tDynam ic P ressu re on Acce le ra ted F lu i d Con ta i ne rs , t 'BSSA, Vo l . 47 , No . I , Janua ry 1957 .Hus id , R . e t a l . , "Danage E f f ec t s and S t rong Mo t i ons Due t o che L im" ,Peru, Earthquake of Oct .ober 3, L974," che TIst Annual l leet ing of theSeismological Society of Amerlca, l , Iarch 1975.Regulatory Guide 1.60, "Design Response Spectra for Seismic Designo f Nuc lea r Power P lan t s , t t D i r ec to raEe o f Regu la to r y SEanda rds ,U . S . N R C , O c c o b e r , 1 9 7 3 .R inne , J .E . , I 'O i l S t . o rage Tanks , r ' The P r i nce W i l l i au r Sound Ea r thquake1964 and A f t e r shocks , Vo l , I I , pa r t A , U .S . Depa rEmen t o f Couune rce , L967 .

f f ish, W.A., Response of an Empcv Cyl indr ical GroundSuppo rced L i qu id Sco rage Tank t o Base Exc i t a t i on , Eng inee r i ngResea rch I ns t i cu te , Un i ve rs i t y o f MassachuseE ts , Amhers t , Massachuse l t s ,A u g u s t , 1 9 7 5 .S toke r , J . J . , Wa te r Waves , I n t e r sc i ence Pub l i she rs I nc . , New Yo rk ,1957 .T ID 7024 , Nuc lea r Reac to r s and Ea r thquakes , US AEC, 1963 .Wu , C . I . , e t a l . , "Response o f G round Suppo r ted L i qu id S to rageTanks t o Se i s rn i c Exc i t a t i on , " P roceed ings o f t he U .S . Na t i ona lConference on EarEhquake Engineer ing, Ann Arbor, I ' t ich igan, 1975,

' i ; b l e I . i r c q u e r ' r ! y , U y t i a m i c I ' r e s s u r . ' ,

and Free Sur face D isp lacement - Rec tangu lar Tank

Table 2. Frequency, Dynarnlc Pressure,and Free Surface Displacenent - Cyl indrical Tank

?

4 .

6 .

8 .

o

I 0 .

l l .

12.

1 3 .L 4 .

N a t u r r l F r c q u . o c y

Exc l ta i lon

Prcarurc o f i thc! r I l

Pressurc oo tha

Bo i too

D1s?1. !c€ncnt

al,tt*.t^JG-*l

Y ^ ; n tt \o L

pr2'xos-'et a {fr *

? L I@)4^2 aJ'(K^'y^'.. \;A k: _ t ) i G*,) ( r _r?: | < rJ_K; J

pst'ko-;nt o ff, *

+ z . r@^ ' t n_ . _ lto k:-t)I(€^l (r->(1eo:A,(K+l I

sz .z - ; "2 ; " t { f , " .

E-nt'ffiarJ

a;, ff.g t'-Ltle<S)

f4s64tnot 1)"

- t &lJ'/; a; trt I c I a Lt rr E,

P Q b n t A u't n ut @), *','' u,,

po"L tFl *Fs$ ('4 P!i3,8 )

(o q 1 4 "r -*., 6 - t(tU a#-.r. +t *..'-r

a^,. = , "''l:"3,-.=7.\ 6+IE- -rl @..u ( r.vr C-,/

et -'.sz+{ t^Lct.er.$)

C r 3 h r . . n d R o d r t r u . r

\N a t ! r a l F r t q u e n c Y

E x . 1 t . t l o n

P r c a s u r . o d . h .

! a l I

Pr€s l l rc on the

Sot too

f r e . S u r f r c .D i !p l ! . c f tn t

uf . 7 rt* t : I t*. 6r-, a n ( l

X.aiacot

pcu..;urt {t,E'"2#rr. .I qt \ Li^l iz-,!:El ."ot\EL-{Ftr- +r\..d,,/ c-L&La.br

+J

'q".a^"t {x. i'r;#,, ( # -- I

ff i f f i1rf 6^-tlrii'+h,(&.)

n*Lr*.qTq]r

,r.tt/ a^trtT+)Accttana io

p 41^*' t &fd-* LL ({{ f ) t z " *, ", u. 1

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naxtP 41F*:] ,,""u..,u.,o{ ( il A.'u.t7.-}6;"*ft1g.*"

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Xo'rurrpl ^{a'ruoa ot rtc(tatd A. her {ox ? !o rD oGcrL la r€N iaNE

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Figure l. Geonetry andCoordinaEe System, RectangularTank, Graham and RodrlguezApproach

Figure 2. GeometrySysEeo, Cyl indr icalApproach

and CoordinateTank, Bauer

( . f - n f o r c y l l n d r l c a l t 3 r t . s )

C o o r d l n r r € S y . t e h f o r C o n v c c t l v c l ' r e s s u r c

- - - - i - - - ------.6-a

F igu re 3 . GeomeEry and Coo rd ina te Sys tem, Housne r App roach F igu re 5 . Models for Cyl lndr ical Tanks

Figure 4. Equlvalent Hechanlcal

(b ) Hown€r Appro lch

Models for Rectangular Tanks

( : ) Grahan .nd bdr r8 , ' . ! Appro .ch

(a) Daucr ^pproach

Equlvalent Mechanical

it"v

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. , . i

xi< ' . ' . ' ; i ; ' <

; . : ^ ' ^ i ^ ' . ! -( l ' , / \ . . ' . '. ' i , . . Y - ; ^i , ' r '

ii i .';ji..::: .1 l : . ' : - ' '

Frequ( ncy (cps ) " . t

t

FLgure 6. Hor l -zontal Design Spectra For Peak Hor izontalGround Accelerat ion of 1.09

Figure 7. Compar ison of Tank Base Moments, Rectangular TanksRectangular Tanks

l i l ; l i . l

8. Comparison of Tank Base Moment, Cylindrical Tanks

. ' , . i - r f ,

sF- 'xls.';-s t I

--.--i -- rt

, h/a

Flgure 9. Frequencies of Equivalent Mechanical Mode,

4L_____o . o l , o

Ji i

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Flgure 14. Rat io of Sloshlng } lasses, Cyl lndr lcal Tanks

Figure 15. Rat io of Fiass Locat ions, Cyl indr j .cal Tanks

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;-i:'t,1

, - a . s '

. . , -

-': a;l,f

Figure 16. Tank Base Moments, Constant Tank Radius.Gyl indr ical Tank

2p tp

' " ' h / o