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
7 GeneralDesignPrinciples 201
7.1 Introduction 201
7.2 RequirementsforSteelTanks 201
7.2.1 BasePlate 202
7.2.2 Sidewall 205
7.2.3 Openings 209
7.2.4 Roof 212
7.2.5 Foundation 218
7.2.6 Stiffeners 223
7.2.7 Rings 223
7.2.7.1 In-PlaneBucklingofIsolatedRing 224
7.2.7.2 Out-of-PlaneBucklingofIsolatedRing 227
7.2.7.3 BucklingofaLightRing-StiffenedCylindricalWall:LimitingStiffness 228
7.2.7.4 BucklingofaLightRing-StiffenedCylindricalWall:FlexuralBucklingoftheRing Stiffeners 232
7.2.7.5 BucklingofLightRing-StiffenedCylindricalWall:ShellWallLocalBuckling 233
7.2.7.6 BucklingofaLightRing-StiffenedCylindricalWall:RingLocalBuckling 233
7.2.7.7 BucklingofaLightRing-StiffenedCylindricalWall:Out-of-PlaneBucklingoftheRing 235
7.2.7.8 HeavyRingStiffeners 236
7.2.8 Stringers 237
7.2.8.1 LocalBucklingoftheStringers 238
7.2.8.2 LocalBucklingoftheShellPanels 240
7.2.8.3 GlobalBucklingoftheStiffenedShell 240
7.2.9 BucklingLimitState 243
7.2.9.1 MeridionalBuckling 247
7.2.9.2 CircumferentialBuckling 252
7.2.9.3 ShearBuckling 253
7.3 RequirementsforConcreteTanks 253
7.3.1 ServiceabilityLimitState 254
7.3.1.1 Leakage 254
7.3.1.2 Durability 257
7.3.1.3 Deformability 259
7.3.2 UltimateLimitState 259
7.4 DetailingandParticularRules 260
7.4.1 Walls 260
7.4.2 Slabs 262
7.4.2.1 FlexuralReinforcementofSlabs 263
7.4.2.2 ShearReinforcementofSlabs 270
7.4.3 Joints 271
7.4.3.1 ConnectionsSubjectedtoNegativeMoment 272
7.4.3.2 ConnectionsSubjectedtoPositiveMoment 272
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.
Preface xiii
Thispossiblyartificialcategorizationhasbeenfoundtobeconvenientwithreferencetothemainresponse parameterstobeconsidered.
Theeffectsofliquidviscosity,non-homogeneousliquids,soil–structureinteraction,theintroduction ofdampingdevicesandisolationsystemsarepresentedanddiscussed.
Thisbookisintendedprimarilyforteachingcoursesonseismicdesignandanalysisoftankstograduate studentsandforprofessionaltrainingcourses.However,itisexpectedthatthistextcanbeeffectiveand practicalasadesignandanalysisreferenceforresearchersandpractisingengineers.
Gian MicheleCalvi RobertoNascimbene IUSS-UniversitySchoolforAdvancedStudies
Pavia
Italy
Acknowledgements
ThisbookemergesfromalongstorythatdatesbacktooneofthefirstItalianuniversitycoursesondesign ofshells,taughtattheUniversityofPaviaintheearly2000s.Atthattime,oneoftheauthorsofthisbook wastheprofessorincharge,theotheronehisassistant.Afriendlyrelationshipbetweenthetwoofthem survivedthechallengesoflifeandmadethisbookpossible,favouringacontinuous,enjoyableatmosphere overthelongyearsofitsgestationandcompletion.
Manystudentshavebeenfundamentalinprovidingcriticismandsuggestionstoimprovetheclass notesthatwereusedastheoriginofthisendeavourandforseveralyears.Thelistoftheirnamescannot bemadeexplicithere,buteachoneofthemisfelttobepartofthiseffort.
Wementionheretwonamesonly,notofstudentsorcolleagues,butratherofthetwopeoplewhohave beenessentialtocompleteadecentproduct:GiuliaFagàandGabrieleFerro,fortheircontinuoussupport indrawing,refining,andcommentingonthefigures,whichmakethisbookmoreunderstandableand readingitmorepleasurable.
AppealingShellStructures
Afterreadingthischapteryoushouldbeableto:
● Listthemainsubsectorsandcomponentsofatank’sdesignandanalysis
● Explainthefunctionofeachelement
● Identifythebehaviourrelatedtoseismicandstaticperformance
1.1BeamsandArches
Thestructuraldesignprocesshastraditionallybeen,andstillis,essentiallycarriedoutforelements subjectedtobendingactions(beamsandslabs),generallycontrolledbyaflexuralbehaviour,uponwhich thedesignisbased.Onceflexuralresistancehasbeenensured,thesemembersareverifiedtoprevent excessivedeformationorshearfailure.Inthecaseofmembersloadedbyacombinationofbending momentsandaxialcompressiveloads(columnsorwalls),thepreliminarydesignisoftenbasedonthe axialcomponentonlyandthecombinationwithflexuralactionisthenverified(inthecaseoftanks, bendingissometimesconsideredjusttocheckthepossibilityofbuckling).
Thesesimplifiedapproachesassumethepresenceofstructuralcomponentsabletoresisteithertensile orcompressivestresses,suchasconcreteandsteelbarsinreinforcedconcreteelements.Thedesignis thusbasedonanestimateoftheloadstobecarriedbyeachmember,andsubsequentlyonthedesignof sectionswherethemaximumresultingbendingmomentisexpected.Asanexample,considerabeam wherethemomentactingonasectionisestimatedas M = ql2 �� ,where q istheappliedloadperunitlength, l is the length of the element and �� is a coefficient that depends on the end constraints . This acting moment has to be balanced by a couple, estimated as the result of internal tensile and compressive actions, equal to each other and multiplied by the distance between their approximate points of application to compute a resisting moment.
It is thus quite understandable that in ancient times the material mainly used for roofing systems was timber, combined in boards, joists, beams, and girders with progressively increasing capacity The only viable, though more complex, alternative was to resort to an arch, which was able to cover long spans using materials able to resist only compressive actions, such as brick or stone masonry. Though
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)
Flatroofplateonsidebeams.
atthecorners(Figure1.2(c)).Thestructuralsystembecomesmuchmorecomplex,withthebarrelvaults forcedtoactsomehowasalongitudinalbeam,coupledwithtransversalarches.Thisresponsecanbe comparedwiththatofastructuremadeoflongitudinalbeamsandatransverseslab(Figure1.3).
Thestructuralresponseofthebarrelvaultcanbedecomposedintotwointeractingmechanisms:an archactioninthetransversaldirectionandabeamactioninthelongitudinaldirection.
Whileinthecaseofslabandbeamsthecomponents’responsesareessentiallydecoupled,thebehaviour ofthebarrelvaultismorecomplex,becausethetwosystemsinteractbetweenthem,witharesulting combinedresponsethatdependsmainlyontheratiobetweenthelongitudinallengthofthevault(L)and itsradiusofcurvature(r ).
Thetraditionalbeamtheorycanbeappliedasafirstapproximationtocalculateinternalforcesand deformationofthevaultstructureinthelongitudinaldirection.Theappliedbendingmomentcanthusbe readilycomputedandsectionequilibriumwillallowthedeterminationofstressesalongthearch-shaped sectionofthebeam.Ingeneral,iffreerotationisassumedattheends,thestressdistributionwilllook likethatshowninFigure1.4(a).Unfortunately,onefundamentalassumptionofbeamtheoryisthat
Figure1.4 Qualitativedistributionofinternalstressesinatransversalarchsectionofabarrelvault,for(a)large or(b)small L r ratios.
Figure1.3
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:
1 3 Rectangular and Cylindrical Tanks 9 where fcd is the concrete design strength , fyd is the yield strength of steel , b is the unitary width of the wall, and d is the depth of the section excluding the concrete cover, assumed to be 35 mm The resulting vertical reinforcement percentage is 1.14%. The acting bending moment reduces to about 1000 kNm / m at midheight , hence the section can be tapered , the reinforcement can be reduced , or both . A possible solution is to reduce the wall thickness to 550 mm and the vertical reinforcement ratio to about 1 % Considering a linear tapering , the wall depth at the top will be 200 mm The bending moment acting at midheight around a vertical axis is about 1400 kNm / m and will require a horizontal reinforcement ratio approximately equal to 1.4%. Cracking width and distance at the base can be checked by applying one of the several formulations proposed in different codes of practice Considering as an example the equations recommended by the Italian code , NTC 2018 , the serviceability limit state bending moment ( ∼ 2400 kN / m ) will be used , estimating the maximum distance between cracks as Δmax = 162 and their width as approximately 0.2 mm.
Simple calculations can estimate the total amount of concrete required to cast the four walls of the tank, of about 880 m3 A proper evaluation of the required reinforcement will imply some assumptions about appropriate minimum percentages for the compression sides of the sections, for corner detailing, for low stress regions . A reasonable total amount will be around 2200 kN , which corresponds to an average steel weight of 2 5 kN/m3 or to an average reinforcement percentage in both directions of about 1 6%
A tank with a similar volume capacity can be designed as a cylindrical structure, with a diameter of 20 m and a height of 25 m. In order to complete some simplified design as done above in the case of a square tank , and thus be able to compare the required material quantities , it is necessary to anticipate the basic features of the response of cylindrical tanks This can be regarded as one of the simplest cases of a shell structure , which can be used as a case study for the development of a more general theory, to be applied to more complex geometrical shapes.
For this purpose, a number of key assumptions have to be accepted (similar to those applied to the simplified theory of plates):
1) the thickness is small compared to the smallest radius of curvature;
2) the resulting displacements will be small with respect to the shell thickness;
3) the cross-sections normal to the midsurface of the shell will remain straight and normal to the midsurface after deformation;
4) the component of internal stress normal to the midsurface is negligible if compared to the tangential components;
5) the stress distribution along the shell thickness is linear, and its value is zero at mid-surface Under
these assumptions, the response of a cylindrical tank can be simplified in the interaction of two different reaction systems: a flexural response of vertical cantilevers, and a retaining action of a system of horizontal rings. The solution of this redundant problem must obviously respect both equilibrium and compatibility and was ingeniously developed between the end of the nineteenth and the beginning of the twentieth century
To solve the problem it was first assumed one should ignore any flexural reaction at the base of the tank, i.e. to allow the base of the cylinder to rotate and to move freely along the horizontal plane. In this case, the horizontal acting forces will have to be equilibrated by the ring action only , while the vertical action will be simply equilibrated by a vertical reaction component This situation is defined as a membrane solution , since bending is excluded , and shear in the plane tangent to the shell must also be zero because
1 Appealing ShellStructures
oftheaxi-symmetricconditionofbothloadsandstructure.Consideringatankcontainingaliquidwith unitweight �� withnointernalfrictionorcohesion,withheightwithnointernalfrictionorcohesion,with height H ,radius r ,andatotalselfweight(ortotalappliedverticalload) R,thesolutiondescribedcanbe expressedbythefollowingequations,asafunctionofinternalvertical(N ′ y )andhorizontal(N ′ �� )actions perunitlength:
where y istheverticalcoordinate,beingequaltozeroatthebaseandto H atthetop.
Thehorizontaldisplacement(w)androtationaroundthetangenthorizontalline(��′ y )requiredby compatibilityinthisspecificcasewerealsocalculated,as:
where h isthethicknessoftheshelland E istheYoungmodulusofitsconstituentmaterial.Notethatthe rotationisconstantalongtheheightandthedisplacement,consistently,varieslinearly,fromamaximum valueatthebasetozeroatthetop.Assumingaconcreteshell,withaYoungmodulus E = 30000MPa, athickness h of200mmanduncrackedsections,thedisplacementatthebasewouldbe4.2mmandthe constantrotation16.7 × 10 5 rad,whichwillleadtozerodisplacementatthetop.Inthecaseofsome flexuralandshearconstraintatthebasethatwouldnotalloweitherrelativedisplacementorrotationor both,someflexuralactionwilloccur,andthedisplacementandrotationresultingfromthisactionwill havetoproperlycombinewiththemembraneresultstoassurecompatibility.
Theflexuralsolutionoftheproblemisfullyderivedandexplainedin[66],herewewillsimplymention anduseit,asexpressedbythefollowingfourth-orderdifferentialequationanditssolution:
where pz isthehorizontalpressure, B = Eh3 12(1 �� 2 ) istheflexuralstiffnessofaplatewiththicknessequal tothatoftheshelland4�� 4 istheratiobetweenthehorizontalstiffnessoftherings(kr )andtheflexural stiffnessoftheplate,expressedbythefollowingequations:
In a mathematical context, the function f ( y) is defined as “particular integral” and can here be represented by the membrane solution found earlier , which is “particular ” in the sense that corresponds to the specific situation described above with reference to the base constraints . Again , in mathematical terms , the first term of the solution is the “ general integral ” of the associated homogeneous equation
1.3 Rectangularand CylindricalTanks 11 (i.e.withoutanyappliedpressure).Thispartofthesolutiondependsontheboundaryconditions(inthis case,fromthebaserestraints,whichwilldeterminethevaluesoftheconstant C3 and C4 ).
Itisintuitivethattheparameter4�� 4 ,i.e.theratiobetweenthecantileverandtheringstiffness,determineswhichoneofthetwosystemswilldominatethestructuralresponse.Equation(1.17b)isthus indicatingthatlargeradius,thickertanksaremoreaffectedbytheflexuralresponse,whilesmall,thin shellsaredominatedbythering’sreaction.Itisinterestingtoobservethatthisisnottheonlyrelevant roleplayedbytheparameter �� .
Actually,thewavelengthofthedampedsinusoidexpressedbythegeneralintegralis �� = 2�� �� andthe dampingfactorexpressedbythenegativeexponentof e isalinearfunctionof �� .Botheffectshaveaclear physicalmeaning:ifthestiffnessoftheringssystemprevailsontheflexuralresponse, �� ishigherand consequentlytheperiodofthesinusoidisshorteranditseffectisdampedmorerapidly.Thedamping factorcanbeexpressedasafunctionofthewavelength,replacing �� with 2�� �� ,andobtaining e 2�� �� y .This expressionindicatesthereductionfactortobeappliedtothesinusoidisaround5,adistanceequaltoone quarterofthewavelength(i.e.at y = �� 4 ),around23atadistanceofhalfwavelength(y = �� 2 ),andaround 535at y = ��.
Inthecasestudywewereconsidering,itisimmediatelyobviousthattocalculatethat4�� 4 = 2.87m 4 and �� = 0.92m 1 (aPoissonratio �� = 0.2hasbeenadopted),thewavelengthofthesinusoidisthusaround 6metres,andtheflexuraleffectcancertainlybeignoredatadistanceofabout3metresfromthebase. Whilethevaluesofbendingmoment(M )andshearforce(Q)alongtheheightcanbecalculatedby applyingtheusualrelationsbetweendisplacement,curvature,bendingmomentandshear,i.e.:
Itappearsthatonlythemaximumvaluesofbendingmoment(Figure1.6)andshearforcearerelevantto designtherequiredamountofverticalreinforcementatthebase,areinforcementthatwillbeneededfor afewmetresonly.
Asalreadypointedout,thecombinationofthemembraneandtheflexuralsolutionsaimstoobtaintotal displacementandrotationatthebasecompatiblewiththerestraintofeachspecificcase.Forafixed-base
Figure1.6 Qualitativevariationofmembraneaction N ′ �� (hoopforce)andbendingmoment M y or M0 alongthe heightofthecontainer.
tank,thedisplacementsandrotationsgeneratedbytheflexuraleffect(M0 and Q0 )mustbeequaland oppositetothosegeneratedbythemembranesolution,i.e.:
Inthecasestudy,thebasebendingmomentandshear,calculatedbyapplyingagaina1.3protectionfactor,are M0 = 183kNm/mand Q0 = 345kN/m.Thesevaluesarecompatiblewiththeassumedsection depth(0.20m),requiringatotalamountofreinforcementof3254mm2 ,orageometricalsteelpercentage �� = 1.63%.Atheightsabove5mfromthebaseandonthecompressionside,theminimumallowedreinforcementpercentagecanbeadopted(possibly0.3%).Thebendingmomentcomputedatthebaseofthe cylindricaltankisaboutseventeentimeslowerthanthatobtainedforthecubictank,butunfortunately thisisnotthegoverningactiontodimensiontheshell.Actually,whenconsideringthecircumferential membraneforces,itisimmediatelyrealizedthatarelevantcrackpatternwillbeinduced.Forexample, ataheightofabout3metresthemembraneactionis N ′ �� = 1.3 ⋅ 2300 = 3000kN/m(Figure1.6),which wouldrequireabout7670mm2 ofsteel,correspondingtoaveryhighgeometricalpercentageof3.8%.The resultingtheoreticalcrackopeningwidthwouldbe0.4mmwithanaveragecrackmaximumdistanceof Δmax = 315mm.Thiscrackpatternwouldnotbeacceptableinmostcases.
Themostobviouscorrectionofthepreliminarydesigntoredirecttheresulttoperformancessimilar tothoseobtainedinthecaseofarectangulartankwith900mmthickwallswillsimplybetoincrease theshellthickness.Infact,increasingtheshelldepthfrom200to400mmwillinduceamodificationof thecoefficient �� = 0.65m 1 ,thusincreasingtheflexuralstiffnesswithrespecttothemembraneone. Thepreviouscalculationswillberepeated,obtaining M0 = 358kNm/m, Q0 = 480kN/mand N ′ �� = 1.3 ⋅ 2000 = 2600kN/m.Whilebendingmomentandshearactionwillbeeasilyabsorbedbytheincreased resistingsection,thetheoreticalmeancrackwidthwillnowbe0.17mm.Thetotalrequiredamountsof concreteandsteelwillstillbearound30%lowerthanthoseestimatedinthecaseofarectangulartank, thusresultinginacompetitivedesign.
Asecond,moresophisticated,designsolutioncouldbebasedontheinsertionofpost-tensioningcables toreducethetensilemembraneforcesanddisplacementalongthecircumferentialdirection.Apossible wayofestimatinganeffectivepost-tensioningforcecouldbetoequilibratethetensilecircumferential resultantduetotheinternalfluidpressure p = 1.3 10 25 1 = 325kN/mactingoneachringofunitary height.Inthiscase,apreliminaryestimateoftherequiredpost-tensioningsteelarea(Asp )canbeobtained asfollows:
where f ′ syd isthesteelstressafterprestresslossesoccurred,estimatedaround65%ofthesteelcableyield strength.Areasonablecorrespondingconcretesectionisthenobtainedassumingthatatyieldconditions ofthesteelcablestheconcreteaveragestressisstillacceptable,i.e.,lowerthanthecompressiondesign strength:
