SeismicDesignandAnalysisofTanks
Gian MicheleCalvi
Roberto Nascimbene
IUSS-UniversitySchoolforAdvancedStudies
Pavia
Italy
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Contents
Preface xi
Acknowledgements xv
1 AppealingShellStructures 1
1.1 BeamsandArches 1
1.2 PlatesandVaults 4
1.3 RectangularandCylindricalTanks 8
1.4 SeismicBehaviourofTanks 13
1.5 FieldObservationofDamagetoTanksInducedbySeismicEvents 21
1.6 DesignConsiderations 28
1.7 ASimplifiedDescriptionoftheSeismicResponseofTanks 32
1.8 DiscussionoftheExistingCodes 34
1.9 StructureoftheBook 37
2 Above-GroundAnchoredRigidTanks 39
2.1 Introduction 39
2.2 VerticalCylindricalTanksFullyAnchoredattheBase 39
2.2.1 ImpulsivePressureComponent 41
2.2.2 ConvectivePressureComponent 47
2.2.3 EffectsofVerticalComponentoftheSeismicAction 53
2.2.4 EffectsofTankInertia 54
2.2.5 PeriodsofVibration 55
2.2.6 EffectsofLiquidViscosity 58
2.2.7 EffectsofInhomogeneousLiquids 60
2.2.8 ConvectiveWaveDisplacementandPressure 64
2.2.9 CombinationofPressuresandBehaviourFactor 70
2.2.10 TankForcesandStresses 73
2.2.11 EffectsofRockingMotion 77
2.3 RectangularTanksFullyAnchoredattheBase 80
2.3.1 ImpulsiveandConvectivePressureComponents 80
2.3.2 PeriodsofVibration 84
2.3.3 ConvectiveWaveDisplacement 85
2.3.4 TankForcesandStresses 85
3 Above-GroundUnanchoredRigidTanks 89
3.1 Introduction 89
3.2 VerticalCylindricalTanks 91
3.2.1 AxialMembraneStressinaShellWall 96
3.2.2 ShellUplift 98
3.2.3 RadialMembraneStressatBase 99
3.2.4 PlasticRotationatBase 99
3.3 RectangularTanks 100
4 ElevatedTanks 105
4.1 Introduction 105
4.1.1 FrameElevatedTanks 106
4.1.2 AxisymmetricalTanks 109
4.1.3 CompositeElevatedTanks 111
4.2 SingleLumped-MassModel 111
4.3 TwoUncoupledMassModel 112
4.4 TwoCoupledMassesModel 114
5 FlexibleTanks 121
5.1 Introduction 121
5.2 ImpulsivePressureComponent 123
5.2.1 VerticalCylindricalTanks 124
5.2.2 RectangularTanks 131
5.3 EffectsoftheVerticalComponentoftheSeismicAction 136
5.4 PeriodsofVibration 138
5.5 CombinationofPressures 147
5.6 TankForcesandStresses 153
5.6.1 VerticalCylindricalTanks 154
5.6.2 RectangularTanks 163
5.7 EffectsofRockingMotion 163
6 OtherPeculiarPrinciples 167
6.1 Introduction 167
6.2 EffectsofSoil–StructureInteraction 167
6.3 Flow-DampeningDevices 174
6.4 Base-IsolationDevices 182
6.5 UndergroundRigidTanks 188
6.6 HorizontalTanks 190
6.7 ConicalTanks 195
x Contents
AppendixADimensionlessDesignCharts 277
A.1 Introduction 277
AppendixBCodes,Manuals,Recommendations,Guidelines,Reports 285
B.1 Introduction 285 References 293 Index 331
Preface
Not many books dealing with designing shell structures are available in the international literature. This was the main motivation inducing us to write a book on this subject , published in 2011 , in Italian . That book found its roots in classical texts and in well -established university courses First of all , the fundamen - tal text , Stresses in Shells , published by W. Flügge in 1960 , possibly a compendium of the monumental Statik und Dynamik der Schalen , written when he was still living in Germany . Together with Vlasov , Reissner , Dischinger , and a few others , he had a fundamental role in developing the membrane and flexural solutions for most kinds of shells between the 1930s and the 1950s
In the Preface to his first book in English, mentioned above and written at Stanford, where he moved before the Second World War , he wrote : “ At first sight it may look to many people like a mathematics book, but it is hoped that the serious reader will soon see that it has been written by an engineer and for engineers … The author wishes to assure his readers that nowhere in this book has an advanced mathematical tool been used just for the sake of displaying it . No matter which mathematical tool has been used, it had to be used to solve the problem at hand.”
This book contains all the fundamental equations to solve any static problem of simple and complex shell structures, being clearly and overtly not to be used as class notes, but rather to find specific solutions or as a basis for further research The kind of book that cannot be ignored by designers of complex shells that hide the complicated mathematical nature of their structural responses behind their apparent intuitive simplicity and their aesthetic appeal.
Quite to the contrary , another fortunate book , Thin Shell Concrete Structures , by D . Billington , had been expressly written as a textbook for a graduate course , allowing an easier and faster reading This obviously came at a price , sometimes leaving the reader with an impression of vague or missing information, or with the feeling of some missing link between theory and practice.
Obviously , it was easy for good teachers to bridge this gaps A Scordelis , at the University of California , Berkeley , integrated this text with his notes and papers , but he had participated in the design and analysis of spectacular shell structures , such as the elliptical paraboloid of the Oklahoma State Fair Arena ( 120 m ), the circular paraboloid of the Arizona State Fairgrounds Coliseum at Phoenix (114 m), the reverse dome of the Garden State Art Center in New Jersey (78 m), the roof of the San Juan Coliseum in Puerto Rico (94 m), the roof of St. Mary Cathedral in San Francisco , conceived by Pier Luigi Nervi, and made of eight hyperbolic paraboloids with a height of 42 m. It is not often that a student has a teacher with such experience
Preface
Our book in Italian was something in between, with an extensive presentation of the mathematical apparatus and a number of design examples presented in some detail . However , part of its success (it still sells well) was due to the total absence of any competing reference in Italian.
When we started considering the preparation of an English version , it immediately became clear that there was much less point in revisiting what was available in other books , while the section on seismic design and assessment could have been profitably expanded , since very little information is available on the subject
The relevance of the subject had recently been confirmed by the occurrence of two shocks in north - ern Italy , on 20 and 29 May 2012 (with a magnitude MW = 6.11 and 5.96 ). The affected region , in the Po Valley , is one of the most industrialized zones of Northern Italy . The majority of structures severely damaged were industrial facilities : one -storey pre -cast reinforced concrete structures and nearby storage steel tanks , causing the economic loss of approximately 5 billion Euros , mostly due to the interruption of industrial production . The large number of industrial facilities in the stricken area, in combination with their intrinsic deficiencies , induced damage and losses disproportionately high, compared to the relatively moderate seismic intensity of the events
In the aftermath of the earthquakes, a large reconnaissance effort was undertaken and a clearinghouse (http://www.eqclearinghouse .org/2012-05-20-italy/), hosted by the Eucentre Foundation and the Earthquake Engineering Research Institute (EERI), was prepared The most common types of failures observed in tanks were fracture of anchors and elephant ’s foot buckling near the base of the tanks. In general, ele - phant ’s foot buckling was experienced in squat tanks , while some of the slender tanks surveyed developed diamond -shaped buckling . Total and partial collapse of legged tanks was another common occurrence , induced by shear failure and /or buckling of their legs due to axial forces , resulting from the overturning moment . In some cases , flat -bottomed , steel cylindrical tanks , typically larger than legged tanks , failed in tension at the bottom of the tank wall , where they met the anchor rods or massive concrete pads
It appears that we are still struggling to reach an acceptable quality in design, assessment, and strengthening of tanks and silos , and “ competing against time ”, as G .W . Housner entitled the report on the Loma Prieta earthquake (17 Oct 1989 ) to the Governor of California The damage to infrastructures , freeways, industrial plants had been severe and the scope of the report had been extended from what happened to the measures to be taken to prevent such destruction in future earthquakes . After some thirty years , it is evident that the report title still applies : we can still state “earthquakes will occur , whether they are catastrophes or not depends on our actions ”, but our actions in the past three decades have not been as effective as they should have been.
This book is based on the evidence emerging from a number of structures surveyed following earthquake events, on some significant consulting activity developed in the field of industrial plants, on research developed and published by the authors and other colleagues.
The design and assessment of the expected performance of tanks and silos are presented, considering the following cases:
● above-ground cylindrical and rectangular anchored rigid tanks;
● above-ground cylindrical and rectangular unanchored tanks; ● underground rigid tanks;
● elevated tanks on shaft and frame-type towers;
● flexible tanks.
widelyappliedinancienttimes,itwasRobertHooke1 in1675whofirstclearlyexpressedthebasicconcept thatallowsthedesignofafullycompressedarch.Hisstatementwassimple,thoughverycomprehensive: aperfectlycompressedarchshallhaveashapeinreverseandidenticaltothatwhichasuspendedcable wouldassumeunderthesameloadcombination.Forexample,underauniformlydistributedload,the cablewouldassumeaparabolicshape,withanupwardconcavityandthatshouldbethegeometryofa compressedarch,withtheconcavityorienteddownward.
Asstated,thisconceptualsolutiondoesnotofferarelevantclueastohowtodesignanarchwhen severaldifferentloadconfigurationsareconsidered,nortakesintoaccounttheeffectsofabutmentconstraints,etc.Theproblemisthusfarmorecomplexand,ascommoninthepast(andpresent)building engineeringpractice,crucialsimplificationswereadoptedforsizingandpreliminarydesign,e.g.assumingthatthehorizontalreactionattheabutment(PH )canbeapproximatedbytheequationapplicableto athree-hingedarchcase:
PH = ql2 8h
where q and l aretheloadperunitlengthandthespanlength,and h representstheheightofthearch. Itisinterestingtocompareanarchandabeamusedtospanasimilarlengthsupportingsimilarweights. Considerthusathree-hingedarch,assumethehorizontalforcesareeliminatedattheabutmentsby meansofsomehorizontaltie,andcompareittoabeamofequalspan,simplysupportedatbothends, assumingthatitismadebyanelasticmaterialwithasimilarbehaviourintensionandincompression. Immediatelyonenotesthatthesameexternalmomentinducedatmidspanbytheappliedloads (Mm = ql2 8 ,assumingauniformlydistributedloadperunitlength),mustbeequilibratedbyinternal actioncouplescharacterizedbyquitedifferentarms.Forthecaseofanarch,theinternalcoupleresults fromforceslocatedinthecentreofthemassofthearch(compression)andofthetie(tension),while forthecaseofaelasticbeam,theyareappliedatpointslocatedatadistanceoftwo-thirdsofthebeam height.Whenafullyplasticresponseisassumed,andthusaconstantvalueisassumedforbothtensile andcompressivestresses,thedistancebetweentheresultantforcesisonehalfofthesectiondepth (d,Figure1.1(a)).Inthiscase,thebeam’sinternalactionwouldbe:
Consequently,assuminganidenticalstrengthincompressionandintension, fw ,andagivenwidthof thebeamsection, bb ,therequiredsectiondepth(db )couldbederivedas::
Forthesakeofsimplicity,assumenowthatthearchandthetiearealsomadewithmaterialswiththe samecompressioncapacity(thearch)andtensilecapacity(thetie).Assumingthatbothcompression andtensileforceswillactatthecentreofthecorrespondingelement,eachforcecanbederivedfrom Equation(1.1),andconsequentlytherequireddepthofarch(da )andtie(dt )wouldbe:
1.1 Beamsand Arches 3
Figure1.1 (a)Three-hingedarchwithauniformloadontop;(b)two-hingedparabolicarchandsimplysupported beam.
Assumingthatallconsideredelementshavethesamewidth(ba = bt = bb ),thenthedepthofthearch andthetiecanbecomputedasafunctionofthedepthofthebeam,combiningEquations(1.3)and(1.4):
da,t = d2 b 4h
(1.5)
Itcanimmediatelybeverifiedthatforreasonablevaluesoftheriseofthearchcomparedtoitsspan (la ,e.g. h = la 4 )andoftheheightofthebeam,comparedtoitsspan(lb ,e.g. db = lb 15 ),thedepthrequired forthearchandthetieisatleast10timeslessthantheonerequiredforthebeam.
Applyingthesameuniformloadonatwo-hingedparabolicarch(asshowninFigure1.1(b))andona simplysupportedbeamwiththesamespan(bothwitharectangularsection A = bd),thedeflectionof thearchatthekeystone(PointA)andofthebeamatmidspan(PointB)canbecalculatedasfollows:
The apparent overall stiffness differs by two or three orders of magnitude. This rather trivial example is just a first case study in which the superiority of curved geometry structures is shown in terms of the required material to obtain similar strength or deformation capacities under gravity loads , when compared to similar structures based on straight geometry The more complex case of cylindrical vs. rectangular tanks will offer more, possibly not as trivial, evidence.
1.2PlatesandVaults
Asalreadymentioned,acommontechnologytocoverarectangularareawasbasedonthepropertiesof timber,amaterialreadilyavailable,easytowork,andstructurallyattractive.Thistechnologyismadeby acombinationoflinearelements,overlayinggirders,beamsandjoists,untilreasonablespandimensions areachievedtoapplyboardsofreasonablethickness.
Coveringthesameareausingasingleplatewouldrequiresomehomogeneousmaterialcapableof carryingshearandbendingmomentsintwodirections.Clearlythisisfeasible,thoughimpossiblein practicalterms,usingasteelplate,butbecameaviablealternativeonlywiththeadventofreinforced concrete.Itspotentialforanisotropic(orratherorthotropic)behaviour,thepossibilityofshapingits geometryandtaperingitsthickness,theseparationoftheinternalelementscounteringcompressionand tensilestresses,appeartobeanidealcombinationtobuildanefficienthorizontalslab.
Considerfirstasimplecomparisonbetweenasimplysupportedbeamandasimilarone-wayslabof indefinitewidth.Thebendingmomentwillbeexpressedbythesameequation,whiletheslabstiffness willincreasebecauseofthehinderedtransversaldilatation.Thiseffectwillbeaccountedforbyacorrectionfactortobeappliedtothebeamstiffnessequalto1 �� 2 ,where �� isthePoissoncoefficient,inthe rangeof0.15forconcrete.Thecorrectionwillthusbeintherangeof2%notasrelevant.
Amuchmorerelevanteffectwillbecomeevidentifcomparingaone-wayandatwo-wayresponse, particularlywhenthetwosidesoftheslabwillnotdiffermuch.
Taketheexampleofasimplysupportedsquareplate,withauniformload p,andassumeanisotropic elasticresponse(i.e.inthecaseofconcrete,neglectinganycrackingphenomenon).Inthecaseofa two-wayresponse,themaximumbendingmomentandthedeflectionatthecentreoftheplatewillbe calculatedas:
max = 0.04416 pa2
f = 0.00406 pa4 B = 0.04677 pa4 Es3
isthesideoftheplate,
itsflexuralstiffness.
Consideringthesamegeometryandthesameload,buthingedsupportsontwooppositesidesonly, thebendingmomentandflexuraldeflectionwillbethoseofasimplysupportedbeam(possiblywiththe minorstiffnesscorrectionmentionedabove,notappliedintheequations):
The values of bending moments and deflection calculated for the beam are thus approximately three times those obtained for the bidirectional plate
It is easy to observe that a barrel vault sustained by continuous supports on two sides can be regarded as a tri-dimensional transformation of an arch with the corresponding transversal section. Its basic structural behaviour under gravity loads can thus be derived from that of an arch (Figure 1.2(a)). Things are quite different when a barrel vault is not sustained along the support lines perpendicular to the arch section, but rather along the other two sides (Figure 1.2(b)) or even by punctual vertical support located
axis of simmetry
Figure1.2 Barrelvaultresponse:(a)continuoussupportontwoedges;(b)edgebeamwithoutcontinuouslateral support,withcolumnsand(c)edgebeamwithcolumns.
(a)
(b)
(c)
thetransversalplanesectionwillremainplaneinthedeformedconfigurationandthisisnotalways anacceptableapproximationforbarrelvaults.Actually,refinednumeralanalysishasshownthatthis assumptionisvalidonlyforrelativevaluesofthefundamentalparametersmentionedabove,i.e.essentiallyfor L r > 5.Obviously,thiscanbeobtainedalsobytakingmeasurestorestrainsectiondeformations, suchas,forexample,insertingtransversalwallsorties[499].Itisclearthatinthesecasesalltheadvantagesofadeepbeamwillapply,withalargeleverarmbetweencompressionandtensionresultantsand smalldisplacements.
Inthecaseofunrestrainedrelativelyshortvaults,thesectiondeformationmaydiffersignificantlyfrom astraightlineandtherelatedinternalstressdistributionwillbeaffected(Figure1.4(b)).
Thisproblemisnodifferentfromthatofanydeepbeamwithacomplextransversalsection,which countsonshapemorethanonmassmaterialtoresisttheappliedforces.Aspointedout,amaindesign probleminthesecasesistotakemeasurestocontrolthesectiondeformations.Asecondfundamental problem(inthecaseofreinforcedconcrete)isrelatedtothehighpotentialforsignificantcrackingofthe partssubjectedtotension,whichmaynotbeadequatelycontrolledbymeansofaproperreinforcement distribution.Apossibleviablesolutionmaybefoundinarationalapplicationofpost-tensioning,which willnotonlybeeffectiveincontrollingcracking,butalsoinreducingthedeformation.Possiblequalitative arrangementsofcablesareillustratedinFigure1.5,asafunctionofthepresenceofbeam-likeelements attheedgesofthevault.ItisevidentinFigure1.5(a)thatarationallongitudinaldispositionofthecables inavaultwillimplypotentiallysignificanteffectsonthetransversalresponsesincethesectionalarch sectionswillbesubjectedtovaryingtransverseforcesandbendingmoments,notnecessarilynegligible nornecessarilyfavourable.
Figure1.5 Qualitativedispositionofpost-tensionedcables(a)alongtheshelland(b)alongtheedgebeams.
cables along the shell (a)
cables along the beam (b)
1.3RectangularandCylindricalTanks
Theverticalwallsofarectangulartankcanberegardedasaseriesofverticalplates,usuallyfixedonthree sidesandfree,simplysupportedorfixedonthefourthone(thesidesharedwiththeroof).Eachplateis generallysubjectedtoitsownweight(inplane)andtotheforcesgeneratedbythecontainedmaterial (mainlyoutofplane),whichgenerateavariablepressurealongtheheight.Intheabsenceofinternal friction,themaximumhorizontalpressureatthebaseofeachplateis P0 = �� H ,where �� istheweightper volumeunitofthefluidand H istheheightofthetank.Thispressurevarieslinearlywiththeheight, becomingzeroatthetop.Assuch,thepressurealongtheheightcanbeevaluatedas[184]:
Forsquatandlargetanks,theactingforcesaremainlyequilibratedbyaverticalcantileveractionwhile thecontributionoftheplateeffectinthehorizontaldirectioncanbeignored.Ifthisisthecase,anditis furtherassumedthatnoconstraintisprovidedatthetopside,themaximumbendingmomentperunit lengthatthebaseis:
Withaprogressivereductionofthehorizontalmeasureofeachplatewithrespecttotheirheight,the contributionofthetransversereactionbecomesprogressivelymoreimportant.Forexample,whenthe heightisequaltothehorizontalspan(i.e.thetankassumesacubicshape),thesamemaximummoment aroundthebaselineofofeachplateisreducedtolessthanone-fifth,becoming:
Theestimationofthismaximumbendingmomentisoneofthecrucialissueswhentheflexuralresponse controlsthedesignofthestructureandisoftenusedasabasicpreliminaryparametertodefinethe requiredwallthicknessatthebaseofthetank.
Asanexample,withtheaimofgettingsomefeelingaboutfigures,considerthecaseofacubictank withsidelengthof20m,withaconsequenttotalcapacityof8000m3 ofliquid,assumedtobewater.The resultingbendingmoment M0 atthebaseofeachwallis:
Assumingthatthedesignmomentattheultimatelimitstateisfactorizedto1.3times(M slu 0 = 1.3 ⋅ 2392 = 3110kNm/m)andthattheneutralaxisdepthisapproximatelyequalto �� = x d = 0.35, therequiredconcretethickness s andthecorrespondingamountofverticalreinforcement As canbe estimatedas:
