S C O T TConcrete Society Technical Report No. 43 TR.043 Published 1994 ISBN 0 946691 45 2 Published by The Concrete Society No. 3, Eatongate Slough SL1 2JA Further copies may be obtained from Publication Sales, The Concrete Soaety QThe Concrete Society 1994 All rights reserved, except as permitted under current legislation. No part of this work may be photo- copied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recordedorreproducedinanyform or byany means,withoutthepriorpermission ofcopyright owner. Enquiries should be addressed to The Concrete Society. Although The Concrete Society (limited by guarantee) does its bestto ensure that any advice, reco-m- mendations or information ii may give either in this publication or elsewhere is accurate, no liability or responsibility of any kind(including liability for negligence) howsoever andfrom whatsoever cause arising, is accepted in this respect bythe Society, its servants or agents. CONCRETE SOCXET'II TECHNICAL REPORT POST-TENSIONED CONCRETE FLOORS - DESIGN HANDBOOK This TechnicalReport wasprepared by a Working Party oftheSociety's DesignGroup whichis one ofthespecialist technical groups within The Technical Development Cen:re. Members of the Working Party ME Rai i s (Convenor)ROW&Benaim and Asmiates MA. MSc, DIC. PhD,CEng, MICE, MIStructE G A Bell BSc, CEng, MICE P W Matthew BE, MSc, MIE AusL RT Whitde MA(Cdnmb), CEn& MICE CCL Syslems Ltd Swift Smcrures Ltd Ove Arup and Partners ACKNOWLEDGEMENT Duringthedraftingofthisreporttheworkingpartyreceivedalargenumberofhelpful commenrrcfrommembers of theindustry. Assistanceinpreparationofthereport bythefollowingmembersofArupResearchand Development is gratefully acknowledged: Kate Benton, Ian Feltham, JonathanF~nch, Geoff Lavender, Paula Youngs. CONTENTS 1.Intmduction 1.1Advantages of post-tensioned floors 12Sltuctural types considered 1.4Bonded ar unbonded tendons 1.5Analytical techniques 2.1Effecrs of pascess 2.2One-way and two-wayspanning flmrs 2.3Flexure inone-wayflmrs 2.4-Flexure infIatslab (hvo-way r p n g ). .. . . . .,. 2.4.1Fkt slab critehi2 : ,. 2.5.shear ~. 3.1Column layout 3 3 ' :Floor i h i c b and types .... : , . . . ' 3 3 , ,.. . ? .Effect of reshaint tofloor shortening . . : .,... ... 4.Materials 4.1Concrete 4.2Tenendons 4.2.1S m d4. 22Tendon prowtion 4.2.3Anchorages Un-tensioned reinforcement Cover requirements Thedesign process 1nh.oduction Designflowchart Basic analysis Smh u a llayout Loading Equivalent frame analysis Tendon profile and balanced load Prestress forces and losses Secondary effecks Flexural section design 6.10.1Semceabiity LimitState sfter all losses 6.102Transfer condition 6.103Ultimate Limit State 6.10.4Progressive collapse 6.10.5Designed flexural un-tensioned reinforcement 6.10.6Minimumun-tensioned reinforcement Shear seength 6.11.1Beams and one-wayspanning slabs 6.11.2Flat slabs (punching shear) 6.11.3Openings inslabs Anchorage bursting reinfoicement Reinforcement ktween tendon anchorages Deflection andvibration Lightweight aggrega!e concrete Detailing Tendondistribution Tendonspacing Tendonnotation Tendonsupports Layout ofun-tensioned reinforcement 7.5.1At columns 7.5.2Shear reinforcement 7.5.3Atand bemeen anchorages Penetrations and openings inflwa Conswction derails 7.7.1Extent ofpours 7.7.2Conswction joints 7.7.3Protection ofanchorages 7.74Back-propping 7.75Stressing procedure 7.7.6Soffit d n gDemolition General 8.1.1Structures wilhbandedtendons E.1.2SLmchues wilh unbonded tendons 9.References Appendices Appendix A:Design examples A.1Solid flat slab wilhunbonded tendons A.2One-wayspanning floor wilhbonded and unbonded tendons Appendix E:Calculation of prestiis losses AppendixC:CaIcuMon of tendon geometry Appendix D:Calculation of secondary effects using equivalent loads Appendix E:Calculation anddetailing of anchorage bursting reinforcement AppndixF:Simplified Shear Check: Derivation of Figures17and18 ApziidixG:V1"mlion of post-tensioned concretefloors Appendix B:Advenisemen!s Typicalspanldepth ratios for a varietyofsection21 typesfor multi-spanflwrs. Allowableaverage stresses inflat slabs(two-way40 spanning).analysed usingtheequivalentframe method. Tolerancesontendonpositioning.49 Harbour Exchange Tower1 GrosvenorSquare Car Park-Southampton2 NewOxford Street2 Typicalfla slabs5 Typical one-wayspanning floors6 Wst-tensionedflat slab6 Post-tensionedribbedslab6 Post-tensionedwaffleslab7 Bendingmomentsurfaces and tendondiagrams for9 differenttendonarrangements Appliedloadbending momentsinasolid flat slab10 Distribution ofapplied loadbendingmoments10 acrossthewidthofapanelinasolidflat slab Loadbalancingwithprestresstendonsfor regular11 columnlayouts Tendonsgeometrically bandedineachdirection12 Tendonsfully bandedinone directionand13 uniformly distributed inother directions Loadbalancing with prestressing tendons for14 irregular column layouts Preliminaryselection offlwr thickness for multi-17 spanfloors Preliminaryshear checkfor slab thicknessat18 internalcolumn Ultimateshear check for flat slab at face of19 internalcolumn Restraint lo flwr shortening23 Layoutofunbondedtendons25 Layoutof bondedtendons26 Atypical anchorage for anunbondedtendon26 Atypical anchorage fora bondedtendon27 Designflowchart30 Elastic loaddistribution effects32 Idealised tendonprofile33 Idealisedtendonprofilefortwospanswithsingle35 cantilever Idealisedtendonprotilefortwospanswithpoint35 load Load'dumping' at 'peaks'35 Practicalrepresentation ofidealisedlendonprofile36 Resullantbalancing forces36 Presnessed elemenr as part ofastatically determinate shucture Reactions ona prestressed element dueto secondary effects Zones ofinelasticity requiredforfailure ofa continuous member Section stresses used for thecalculationof un-tensioned reinforcement Burstingstresses inrectangular beam subjected to an axial symeaic fotre Bursting smss dishibution Method ofnotationfor use ontendonlayour dmwings Fiat slab fendon andsupport layout detailing Flat slab reinforcement layout Reinforcement arrangement aL a column Prefabricated shear reinforcement Unseessed areas betweentendons requiring reinforcement Unbonded tendons diverted around an opening Intermediate anchor at a consmction joint Inffilsmp for jackaccess Strand aimming usingadisc cutter Strand aimming using purpose-made hydraulic shears Anchorages for unbondedtendons:fixedto formwork Grease-filled plasticcapto protect strand and wedgegrips Anchorage blcck sealed with mortar Stressing bandedlendons at slab edges Soffit markingusedtoindicate tendon position NOTATION Areaofconcrete Areaofprestressing tendons Areaofun-tensioned reinforcement Area ofs h w reinforcement Drape oftendon Widthofsection Breadthofmember; or for T-,I-andL-beams the breadth ofthen iEffective depthoftensionreinforcement ortendons Depthto the cenaoid ofthe compressionzone Shon-termmodulus ofelasticity ofconcrete Modulusof elasticity ofconcreteat timeofwnsfer Modulus ofelasticity ofprestressingtendons Modulusofelasticity ofun-tensionedreinforcement Eccentricity Compressivestressinconcreteatextremefibreusedtocalculate serviceability un-tensionedreinforcement requisemen~ Concrete suenglh at (initial) transfer (inN/mm? Stressinconcrete at the levelofthetendondue toinitial pr esms anddeadload(in N/mm3 Tensile stress inconcrete at extreme fibre used lo calculate serviceability un- tensioned reinforcement requirements Characteristic concrete cube strength (in N/mmq Tensile stress intendons at (benm) failure (inN/mml) Effective presuess (intendon) after d llosses (inN/mrnz) Characteristic strength ofprestressing steel(inN/mm2) Maximumdesign principaltensile stress (=0.24Jf,.inN/mm2) Characteristic strength ofbondedun-tensionedreinforcement (in N/mm3 Overall depth ofsection Second moment ofarea Effective spanlength Length oftendon Length oftendonaffected bywedgedraw-in Moment due to applied loads Momentnecessarytopmduce zerostressintheconcreteattheextreme tension fibre Secondary moment due toprestress Ultimate resistance moment Resuessforce Slope ofprestress force profile Characteristic strength oftendon(in3N) Prestressing force inthetendonat the jackingend Prestressing force at distance xfrom jack Distance between points ofcontra-flexure intendon Lengthofa critical shear perimeter Shear force due IO ultimateloads Ultimate shear resistance ofconcrete Ultimate shear resistance ofa sectionuncrackedinflexure Ultimate shear resistance of a section cracked inflexure Designeffective shear force Designshear suess at cross-section Design concrete shear strength Uniformlydishiiuted load Neutral axis depth Half theside oftheprestress end block Halftheside of theprestress anchor loaded area Top section modulus 2,Bottomsectionmodulus aAnglechange intendonfmmanchor topointconsidered(radians) a'Average angle change inlendon per unit length(radianslmeue) AWedgedmw-in PCoefficientof fiction 9Creep coefficient Yr Partial safetyfactor for load 7,Partial safetyfactor for materialstrength IiiPrestressedtendon'wobble' factor (radians/metre) INTRODUCTION The use ofpost-tensioned concrets flwrs inbuildings has beenconsistently growinginrecentyears.Thegreatestuseofhistypeofconsmcuonhas b e nintheUSA,andinCaliforniaitistheprimarychoiceforconcrete floors. Post-wsionedfloors have also k n usedinAustralia.Hong Kong, Singapore andEurope. Their useinthe UK is nowincreasing rapidly. Typical applications have been: Offices Car parks Shopping cenms Hospitals ApamenE Indusmal buildings These are illustraed in Figures1, 2 and3. Figure 1:HarbourExchangeTower. Figure t: GrosvenorSquare Car Park. Southamptoo. Figure 3:NewOxford Street. TheConcreteSocietyhaspublishedlhreereponsonthissubject,Technical Report No8"'.The Design of Post-Tensioned Concrete Flat Slabs in Buildings; Technical ReponNo 17'". Flat Slabs inPost-Tensioned Concrete with Particular Regard to t heuse ofunbonded Tendons - Design Recommendationu, Technical ReportNo25"',Post-TensionedFlat-SlabDesignHandbwkTR17wasa revisionofTR8,andTR25ampl i edtherecommendationsofTR17.The purpose ofthecurrent reponistoupdate the information contained inTR17 and TR25 inline withBSgllO,1985"',to combine these two repons into one document andtoexpandsameoftherecommendations inline withcunent practice. Thisreportexplainstheoverallconceptofpost-tensionedconcretefloor consmction as well as giving detailed design recommendations. The intention is to simpliiy the tasks ofthe designer and contractor enabling them to produce effective and economic structures. Post-tensioned flwrs are notcomplex. The lechniques,structural behaviouranddesignaresimpleandverysimilarto reinforcedconcretesmctures.Theprestresstendonsprovideasuspension system within the slab and thesimple arguments ofthe triangle offorces apply with the vertical component df the tendon force carrying part of the dead and liveloadingandthehorizontalcomponentreducingtensilestressesinthe concrete. Two design examples are given inAppendixA. The reportis intended to be readinconjunctionwithBS8110"'.Those areas not covered in BS8110'"are descritedindetail inthe report withreierences givenas appropriate. Tne principles laidout inthe report mayalso be applied todesignsinaccordancewithE mo d eECZ'fl,butsomeofthedetailswill needtobe modified. Two otherConcrete Society publications giveusefulbackgroundinformation to designersofpost-tensionedfloors. Theyare:TechnicalReportNo.21"'. Durability of Tendons inPrestressed Concrete andTechnical Report No. Z3"I. Partial Prestressing. Itshouldbenotedthatsincetheintegrityofthestructmdependsona relativelysmall numberofprestressingtendonsandanchorages theeffect of workmanshipandqualityofmaterialscanbecritical.Thisshouldbe understcod byall parties involved in M design andconstruction. Themainadvantagesofpost-tensionedfloorsoverconventionalreinforced concrete in-situ floors, may be summarised as follows: Increased clear spans Thinner slabs Lighter structures Reduced cracking and deflections Reducedstorey height Rapidconstruction Better watertightness . ' .These advantages canresult insignificant savings inoverallcosts.There are ;. alsosome situations where theheight ofthe building is limited, inwhichthe . : . ? .. . ..reduced. . storey height hasallowed additional storeys tobeconstructed within ', : thebuilding envelope. 12'~lruchY0.lopes considered Thereponisprimarilyconcernedwithsuspendedfloors.However,the recoinmendations apply equally well'tofoundation slabs except that since the -..loadsaregenerallyupwardratherthandownwardthelendonprofiesand locations ofun-tensionedreinforcement aremirrored. The types offloor which can beusedrange fromflat plates toone-way beam and slab structures. An important distinction between structural types is whether hey spanone-way ortwo-ways.Thisisdiscussed ingreater detail inSection 2.2. Amountofpnsbess Theamount ofprestress providedisnotusuallysufficienttopreventtensile stresses occurring in the slab under design load conditions. The structure should therefore beconsidered tobepartially prestressed. Theamountofprestressselectedaffectstheun-tensionedreinforcement requirements. The greater the level ofprestress, the less reinforcement is likely toberequired.Unlikereinforcedconcrete structures,arangeofaccepfable designs are possible fora givengeomeuyandloading.The optimum solution depends on the relative costs of prestressing and un-tensioned reinforcementand onthe ratio oflive load todeadload. Average prestress levels usually varyfrom0.7to 2.5N/mmz for solid slabs and occasionallyupto6N/mm1forribbedorwaffleslabs.However,whenthe prestress exceeds approximately 2N/mmz orthe fl wr isverylong,the effects ofreshainttoslabshoneningbysupportsmaybecomeimponantIfthe supports are stiff asignificant proportionoftheprestressforce goesintothe supponsso thattheeffective prestressingoftheslabisreduced (see Section 3.3). 1.4Bonded or unbonded tendom Post-tensionedflwrs cank constructedusingeitherbondedorunbonded tendons.The relative merits ofthetwotechniques aresubject todebate. The fouowing points may be made infavour of wh:Bonded: - develops higherultimateflexural s~engt h - doesnot dependuponthe anchorage after eouting - localises the effects ofdamage Unbonded: - provides greateravailable lever arm - reduces friction losses - simplifies prefabrication oftendons -grouting not required - can be constructed faster - generally cheaper 15Analytical techniques The design process is described in Section 6. The analyticallechniques are the same as those used for reinforced concretestructures. The structure is normally subdivided into a series ofequivalent frames uponwhichthe analysis is based. Theseframescanbeanalysedusingmomentdistributionorotherhand techniques,howeveritisnowmorecommontomakeuseofa planeframe computer analysis program. In addition tostandard plane frame programs, there areavailableanumberofprograms,specificallywrittenforthedesignof prestressedstructures.Thesepropamsreducethedesigntimebutarenot essential for the design ofpost-tensionedfloors. For more complicated flat slabs orforthosewhichareberepeatedmanytimes,agrillage orfiniteelement analysis oftheflwr maybe more appropriate. 2.STRUCTURAL BEHAVIOUR 2.1Effeectsof prestress Theprimaryeffects ofpresuessareapre-compressionoftheflwr andan upwardloadwithinthespanwhichbalancespartofthe downwarddeadand live loads. Ina reinforced concrete flwr, tensile cracking ofthe concrete is a necessary accompanimenttok generationofeconomic stress levelsinthe reinforcement.Inpost-tensionedflwrsboththepresompressionandthe upwardloadinthespanacttoreducethetensilestressesintheconcrete. However,thelevelofprestressisnotusuallyenoughtopreventalltensile crackingunderfulldesignliveloadingatServiceabilityLimitSlate.Under reduced live load muchofthe cracking willnot bevisible. The act ofpresUessing causes the flwr to bend,shorten, deflect and rotate.If any ofthese effects are restrained, secondary effects ofprestress are set up.As slated above, ifthelevelofprestress does notexceed approximately 2N/mm2 thesecondary effects due tothe restraint toshonening are usuallyneglected. However,unlessthe floor canbeconsideredtobestatically determinate,the displacementsoftheflwrsetsupsecondarymomentswhichcannotbe neglected. Secondary effects are diszussed inmore detail in Section 6.9and the calculation ofthese effects isdescribed inAppendixD. Thereareseveraldifferent typesofpost-tensionedfloor.Someofthemore common layouts are given inFigures 4.5.6.7and 8.Animportant distinction betweentypesoffloorsiswhethertheyareone-wayortwo-wayspanning structures. Solid flatslabSolid flat slab wi th droppanel Cofferedfiatslab Coffered flat slab with solid panelsBanded coffered flat slab Figure 4:Typical flat slabs. Nore:See Section 2.4 forlimiling criteria of two-wayaction. Figure 5:Typicalone-way spanning floors. -. Figure 6:Post-tensioned flat slab. Figure2 Post-tensioned ri bbe slab. Figure 8:Past-tensioned coffered slab. dne-wayflwrs carrythe appliedloadingprimarilyinonedirectionandare mated as beams or plane frames.Onthe other hand.twc-wayspanning flwrs have the ability to sustain the applied loading intwodirections.However, for astructuretobeconsideredtobetwo-wayspanningitmustmeetseveral criteria. These criteria are discussed in Section 2.4. 23Flexure in one-wayfloors One-wayspanningfloorsareusuallydesignedClass3structuresin accordance with BS8110").Although cracking is allowed, it is assumed that the concretesectionisuncradtedandthathypotheticalensilestressescanbe carried atServiceability LimitState.The allowablestressesare discussedin Secdon 6.10.1. The behaviourofone-wayflwrs at loadslessthanthatwhichwouldcause cracking canbeassumedto belinear andelastic. BS811OW1recommendsthat whenthetensilestressesunderdesignpermanentloadsarelessthanthe allowable stresses for Class 2 suuctms, thenthedeflection maybepredicted using grosssection properties, Inother cases calculation ofdeflections should bebasedonthemoment-curvature relationship for cracked sections. 2.4Flerure inflat slobs (two-way spanning) In the context ofthis report, flat slabs are those floors whichcan carryloads in twodifferent directions to discrete column supports.These are definedas flat slabs in BS811OW.It must beemphasised Lhat these skucues are not the same as two-way slabs in accordance withSection3.5ofBS8110'".Two-way slabs in BS8110L"always span onto beams or wal1s.i.e.continuous supporis, and are not considered inthisreport. One misconception held bysome engineers is to consider a reduced loadwhen analysing theslab inone direcuonusing the equivalent h m emethod.Aflat slab supported oncolumns, rather than perimeter beams.canfail as a one-way mechanism justasasingle-wayslab, andit shouldbereinforced toresist the moment Fromthe full load ineachonhogonal direction. Tests and applications have demonsuatedthata post-tensioned flat slabbehaves ns aflat platealmostregardlessoftendonarrangement(see Figure9).The effecrs-ofthe tendons-m;ofcourse, critical t ot he behaviour as they exert loads ontheslabas wellasprovidereinforcemenrThetendonsexertequivalent vertical loads on the slab known as equivalent loads (see Section 6.7).and these loadsmaybeconsideredlike anyotherdeador liveload.Sincethetendon effect is opposite tothe effect of gravity loads,the net load causing bendingis reduced.Anadditional effect of thetendons is the axial precompression which counteracts flexmaltensileskesses.Therefore,atservicedeadload,thenet downward load musing tending inthe slab is normally verylow and the flwr is essentiallyunderuniformaxial compression. Examinationofthedistribution.of moments for aflat plate inEgures10 and 11 revealsthathoggingmomentsacrossapanelare sharply peakedinthe immediatevicinityofthecolumn andthattb moment at thecolumnface is several times the moment midway behveen columns. It should k noted that the permissible snesses given in Table 2 of Section 6.10.1are average sbesses for thefull panel.They are lower than thosefor one-wayfloors Lo allowfor this non-uniformdisnibutionofmomentsacross Lhe panel. a)Fully bandedtendons b)Uniformly distributed tendons e C)50% banded plus 50% evenly distributed tendons overfull width Rigure 9:Bending momentsurfacesandtendondiagramsfor different tendonarrangements Experimental results \A Span Span -i ColumnC Column a)Moments along section onb)Moments halfwaybetween columnlinecolumn lines Figure 10:Appliedloadbending moments in a solid Ratslab. I\ Experimental resuitr Equivalent analysisExperimentalresults + izColumn Panelwidth a)Moments on column lineb)Moments halfwaybetween column lines FigurePI: DMbution ofapp!,"!jdIcedbeadi!zgmoments ecross tbe wjdthof a panel in a solid flat slab. In contrast the sagging moments across the slab inmid-span regions m almost uniformly distributed across thepanel widthas shown inFigurellb. Itishelpful totheunderstandingofpost-tensionedflatslabstoforgetthe arbihary column strip, middle strip and moment percenlage tableswhichhave longbeenfamiliartothedesigner ofreinforced concrete floors. Instead,the mechanics ofthe action ofthe tendonswill be examined first The "load balancing" approach is an even more powerful tool for examining the behaviouroftwo-wayspanningsystemsthanitisforone-wayspanning members.Bythe balancedloadapproach,attentionisfocusedontheloads exerted on the floor by thetendons, perpendicularto the plane of the floor.As for one-wayfloors,thistypically meansauniformload exertedupwardalong themajorportionofthecenuallengthofatendonspan,andstatically equivalentdownwardload exertedover theshort lengthofreversecurvature. In order to apply an essentially uniform upward load over the entire floor panel these tendons should be uniformly dishibuted, and thedownward loads from the tendons should react against another smchual elemenr The additional element couldbeabeamorwallinthecaseofone-wayfloors,orcolumnsina twtiwaysystem. However, a l wk ai aplanviewofaflat slab (see Figure12) revealsthatcolumnsprovideanupwardreactionforonlyaverysmallarea. Thus,tomaintainstaticalrationality,wemustprovide,perpendiculartothe above tendons, a second set of tendonsto provide an upwardload to resist the downward load from the fusi set. Remembering that the downward load of the uniformly distributedtendons occurs over a relativelynarmwwidthunder the reverse curvaturesand thatthe only available exterior reaction,thecolumn,is also relatively narrow.it becomes obvious that the second set of tendons should be innarrowstripsor bands passingoverthecolumns. Tendons uniformly spaced across floor exerting upward loads inthe span and downward loads on thecolumn lines. Tendons concentratedon column lines exerting upward loads In the span t o carry the downward loads of uniformly distributed tendons and downward loads onthe columns. Figure 12:Loadbalancingwithprestresstendonsforregular columnlayouts. There are two ways of accomplishing this two-part tendon system toobrain the nearlyuniform upwardload we desire for ease of analysis. Inthefirst method. tendons are spaced uniformly in each of two directions and react against banded tendons along thecolumn grid lines in eachdirection. This resulrs insome of thetendonsineachdirectionbeingbandedoverthecolumns,andsome uniformly distributed between these bands (see Figure13). This methodworks wellwhere the columns are atiangedona r'ectangular grid. Figure 9 shows thebending moments derived fromgrillage analysis ofsquare panelswithdiffering arrangement oftendons.The balancedload providedby the.tendonsin ea&-directionis equalro the dead load. Figure 9c gives the most uniformdistributionofmomentsandprovidesapracticallayout oftendons. Thisarrangementgives70%ofthetendonsinthebandedzoneandthe remaining 30% betweenthebands. Itshould benotedthat, since thewidthof the bandedzone is 0.4 rimes the width ofthe bay, thisarrangement is identical toproviding 50% ofthetendons evenly distributed overthefullwidthofthe bay in addition to 50% concentrated inthe band. However, as can be seen from Figure 9 the detailed dismbution is not critical providedthat suficienr tendons passthroughthecolumnzonetogiveadaquate protectionagainst punching s ha-and progressive collqse. Figure 13:Tendons geometrically bandedineachdirection. Wherethe -columnmangemernis.irregular.orwhereopeningsorother geometric considerations requireifasecondmethodmaybeused.Inthis methodtheuniformlydistributed tendonsandbandedtendonsmayeachbe placedin one direction only (seeFigure14). The pwerofthe second method becomes very clear by examining afloor which has columns oEimgulx layour. Foranexample.withreferenceto Figure15a.itisassumedthattheodd numbered grid lines are offset one halfbay from the even numberedgrid lines, but columns onagivenletter grid arealigned. I the'columnskip'approachillustratedinFigure15bisretainedas for conventionally reinforced floors, each span which stmed in a column strip ends inamiddlestrip,andtracingofloadpaths,arationalanalysis,and proportioningreinforcemenL becomedifficultifnotimpossibleandforcea renuntothebasicideaofbalancingloadswithtendons.Theuniformly distributed system oftendons (parallel to letter grid lines) can beaccomplished withliule regardforcolumnlocation.It isonly necessarytoplacethehigh points of the tendon prome (where reverse curvature and downward load occur) at the inersection ofthe tendons withthe number gridlines.This systemthen reacts against the bandedtendons placed onthe number gridlines as shown in Figure 1%. Bythis pmwdute, the reaction of the gravity load balanced bythe tendons i scarried direcIly to columns, without any flexural action ofthe flwr. Since this balancedload is typically alargeportionofthepermanent loadon !he flwr, errors in analysis whichare due toincorrect assumptions ofload path areafunctionofrelativelysmallloads,andthusaresmall.Thepossible consequences of such errors can be investigated byexamining the tehaviour of the floor under overloads. Figure14:Tendons fully bandedinonedirectionanduniformly distributed inthe other direction. Flexuralcracking is initiated at column faccs and can occur at load lcvclsinthe sewiccability range. Whilc these andearly radial cracksremainsmall, hcy are unlikelytoaffecth eperformanceoftheslab, Compressionduetoprestress delaystheformationofcracks,butitislessefficient incontrollingcracking thanun-tensionedreinforcementplacedinthetopoffloors,immcdialcly adjacentlo,andabove the column. 2.4.1Flat slab criteria Forapreskessedfloorlobeconsidcrcdasaflatslabthefollowingcriteria apply: Precompressionshould beappliedintwoorlhogonaldirections: Suchaf l wr withno,ormoderate,crackformationperformsasa hdmogeneous elasticplatewilhitsinherent two-waybchaviour.The actual tendonlocation at a given point ina fl wr system is not critical to thc f l wi s two-waybehavioursince prccomprcssion,whichisthedecisivefactor,is commonly appliedlo lhc fl wr at itspcrimcter. Theprecompressionattheedgesoftheslabisconcenuatedbehindthe anchorages,andspreadsintotheflwrwithincreasingdislancefromthe edge.Thisisuueforfloorsofuniformthicknessaswellasflwrswilh beams inh e direction olprecompression. Flwrs wilh bandcd post-tensioning andflwrswithwideshallowbeamsalsoqualifyfortwo-wayactionat regionsawayfromh efreeedgeswhcrc precompression is allained inbolh directions. a)Irregular flat plate column layouL b)Column strips and middlesnips Typical uniformly distributed tendons and loads c)h a d balancing withbonded tendons Figure15:Loadbalancing withprestressing tendons for irregular column layouts Past experience shows that for the precompressiontobeeffective it should beat least 0.7N/mrn2 ineachdirection. Aspect ratio (lengthtowidth)oiany panelshould notbe greater than2.0: Thisapplies losolidflat slabs, supported onorthogonalrows ofcolumns. Foraspectratiosgreater than2.0themiddlesection willLend toacl as a one-way spanning slab. - Stiffness ratios inLWOdirections: The ratio ofthe stiEfness ofthe slab intwo orthogonal directions should no1 bedisproportionate. Thisismorelikelytooccurwithnon-uniformcross- sections suchasribs. Forsquare panelsthisratioshouldnotexceed10.0, otherwise theslab ismorelikely lo behave as one-way spanning. 2.5Shear The method given in BS81101" for calculating shear inbeams and one-way spanning slabs should beused.A methodfor calculating shear for post-tensionedflat slabs is no[ provided in BS8llO.The methodgiveninthis manual (see 6.11.2)combines [he prestresseffccts givcninsection 4 of BS8llOwithLhe methodgivenfor punching shear for reinforced concrete insection 3 ofBSBllO. 3.STRUCTURAL FORM Current experience inmany countries indicalcs a minimumspanof approximately7m tomakepreslressingviableinafloor.However,cxamplcsareknowninwhich prestressedfl wrshavebeencompetitivewhereshortcrspanshavebeenusedfor architectural reasons, but presuessing was then only made viable by chwsing the right slab form. In general the idealsituationis,of course,to 'thinkprestressing'fromthe initial concept ofthebuildingand to chwse suitablylonger spans. In chwsing column layouts ana spans for a prestressedflwr.several possibilitiesmay beconsideredla optimise thedesign,whichinclude: a)Reducethelengthoftheendspansor,ifthearchitectural considerations permit, inset the columns from the building perimeter toprovidesmallcantilevers.Consequently,endspanbending momentswillbereducedandamoreequablebendingmoment configurationoblaincd. b) Reduce,ifnecessary.thestiffness ofthecolumnslominimisethe prestresslost in overcoming the restraint offered ta f l wrshortening (see Section 3.3). C) Where span lengths vary,adjust the tendon profiles andthe number oftendoilsto providetheupliftrequiredfor eachspan.Generally thiswillbeasimilar percentageofthedeadload for eachspan. Oncethecolumnlayouthasbeendetermined.thenextconsiderationisthetype of flwrtobeused.Thisagainisdeterminedbyanumberoffactorssuchasspan lengths,magnitudeofloading.architecturalformanduseofthebuilding,special requirements such as services, location of building, and the cost of malerials available. 3F h r thkktzess md types The sl abthicknessmustmeettwoprimaryfunctionalrequirements-structural strengthand deflection.Vibrationshouldalso k consideredwherethereare onlya fewpanels.The selectionofthicknessor type(e.g.plate withoutdrops.platewith drops,cofferedorwaffle,ribbedorevenbeamandslab)isalsoinfluencedby concrete strengthandloading.There arelikelyto beseveral alternative solutions to thesame problemandapreliminarycostingexercisemaybenecessaryinorderto chwse h emost economical. TheinfomadongiveninFigures16,17 and18 willassia h edesigncrto makea preliminarychoiceofnoorsection.Figure16 (derivedfromTable1)givestypical imposed load capacitiesfor a varietyof'nalslabs and one-wayfloors over a range of spaddcpthratios.Thesefiguresarebasedonpastexperience.Figure16is appropriate for alltypes ofprestressedfloor.Figures17 and18 are onlyappropriate for flatslabs but Figure17 isnotappropriatefor cofferedslabswhichdonothave a solidsectionover he column. At this slage it should bc notedthat the superimposed load usedin Figures16, 17 and 18consislsofallloading(deadandlive)barh cself-weigh~ofh esection.The calculation methods usedfor obtaining thc graphs inFigures17 and18 are described inAppendixF. Total Note: Spanldepthratio - This chart isderived fromthe empericalvaluesgiven inTable1 for multi-spanfloors. For single-spanfloors thedepth should be increasedbyapproximately 15%. Figure16:Preliminaryselectionofflwr thicknessfor multi-spanfloors. Slab thickness adjacent t o columns vc= 0.75N/mm2 Columnsize including head = 300 x 300 mm 203040 50601080WI W110 120 130 140 150 160 Area(rn2) D O D P D D D D D DK : : % 4 9 Z $ 8 $ : Slab thickness adjacent 11 10 Total 9 imposedvc= 0.75N/rnmz Load'Column size including (kN/m2)6head = 500 x 500rnm 5 4 3 2 1 0 20JO4050607080901W 110120 130 I d0 150 160 Area(rn2) O D D D D D D Da m - m - m a m '2s s 1 6 . n - " m T ~ m m ~ ~ ,-Slab thicknessadjacent 14t o columns 13 12 11 10 9 i ota1 imposedvc= 0.75N/mm2 Load'Column size including (kNlm2)head = 700 x700rnm 5 4 3 2 1 0 M24405060708090IM HO120130 140 150 150 Area(mZ) Figure17:Preliminary shearcheck forslabthickness at internal column. 8 E D 8 8 8 ~ ~16NNnnprrulSlab thicknessadjacent 14to C O I U ~ ~ S13 12 'H '10f , , = 40Nlmrn2 Total9Column(inc.head) 300 x300 imposed a D.L.Factor= 1.4 Load7L.L.Factor = 1.6 (kNlm2)6 8 4 3 2 1 , , I I I I I I I I I I - I I zo304060 60m noso iwnorzo1Joraorarsoren Area(m2) Figure18:Ultimate shear checkforflat slab at face of internal column. Notes: 1.For columnsizes other than300 x300 the slab depth should bemultipliedbythe factor (columnperimeter 11200). 2.The maximumshear stress for f.,= 40Nlmm'andmoreis SN1mm2. For f., < 40 Nlmm'themaximumshear stress is0.8I&. For f,.= 35Nlmm'increaseslab depthbyafactor of1.06. For f,,= 30Nlmm'increaseslab depthbyafactor of1.14. 3.The valueofdlhisassumedtobe0.85. 4.The ratioofV,NIVisassumedtobe1.15. 5.These curves do nottakc accountof elastic distributioneffects. See Section6.6. Flatslabse n d 10 exceedpunchingshearlimitsaroundcolumns,andoftenneed additional shear rcinforcemcnl at theselocations.The ,mphsinFigure17 providea prcliminary assessmentas LO whcher shcarreinforcementisneededlor thescction types1, 2,3.5and6(allflar slabs)inTable1. Astheshearcapacity01 aslabis dependent on the dimensions ofthe supponing columns or column heads, each graph hasbeenderivedusingdifferent columndimensions. Inaddition,h eshear capacity at thc face ofthecolumnshould be checked.This can bedone usingh e graphinFigure18. Thc graphhas beendcrivedlor slabs with3W x300mmsupportingcolumns,andloobtaintheimposedloadcapacitiesforslabs wi h olhcrsupporting columnsizes,thevalucsinh e graphshouldbemultipliedby h eratioofrequiredcolumnpcrimcled1200. The followingshouldbe followedwhenusing Tablc1, Figures16, 17 and 18 ta obtainaslab section. a) Knowingthespan andimposedloading rcquiremcnrs.Figure16 or Table1 canbeusedtochoose a suitable spanldepthratioforthe section typcbeing considercd.Table 1 also providesasimple checkfor vibrationeffecs. b) If sectiontype1. 2. 3,5, or 6 hasbcenchoscn,checkLhcshear capacityof the scction.usingonc ofthe graphs in Figurc17 (dependingonwhat size of columnhasbeendccidcdupon).Obtainthcimposedloadcapacitylorthe chosenslabsection.I1thisexcccdstheimposcdload,thenshear rc'inforccmcntis unlikclylobe ncccssary.Ifit docsnot,h e nreinforcement willbercquircd.IfLhcdiffcrcnccisverylargc.thena nincreaseinsection dcpthorcolumnsize shouldbeconsidcred. C) Check the shear capacity at the face ofthe column usingthe graphin Figure 18. Ifh eimposcdloadcapacityis cxceedcd,increascthcslabdcpthand checkagain. It shouldbenotedthatTablcIandFigurc16 areappliwblcformulti-spanfloors only.Forsingle-spanfloorsthedepthshouldbe increasedbyapproximalely15%. Figwes17 and18 arc applicable lor boh floor types and havc b a nderivedusing an avcmgeloadfaclor of1.5(seeAppendixF). Figures17 and18aresetforinternalcolumns.Theymaybeusedforexternal columnsprovidedthat (he loadcdareais doubled for edge andquadrupledfor comer columns. This assumes thar the edge of the slab extendslo a1 leasthe' cenlre line of h ecolumn. Table1:Typicalspanldepthratiosforavarietyofsectiontypesfor multi-spanfloors. *Additional requirements if novibration checktobecarried out fornormaloffice conditions: Aeither the flwr hasat least four panelsandisatleast 250 mmthick or the nwr hasat least eighl panelsandis at least 200 mm thick. Beither theflwr hasat least four panels andisat least 400 mmLhick ortheflwr hasat least eight panels andisat least 300 mmthick Sectiontype 1. Solid flatslab 2. Solid flatslab withdrop panel r------- I I 1 0 ,I II L-----.J 3. Bandedflat slab i I I I I I span16 I 4. Cofferedflat slab ,I .-Ji--JL--JL.. -7r - - 7 r - - 7 r- 111 1 11 5. Cofferedflat slab withsolid panels I I L-- Total imposed loading (kN/ma) 2.5 5.0 10.0 2.5 5.0 10.0 2.5 5 .O 10.0 2.5 5.0 10.0 2.5 5.0 10.0 r A A A B B Spanldepth ratios 6m 5 L 2 13 m 40 36 30 44 40 34 slab 45 40 35 beam 25 22 18 25 23 20 28 26 23 Table1. Continued 1. All panelsasumed tobesquare 2. 'Spanldepthratiosnotaffectedbycolumn head 3.t It may bepossible thatprestressedtendonswillonly berequiredinthe bandedsections andthatuntensionedreinforcementwill suffice intheribs. or viceversa. it The values of spanldepth ratio can vary according to the width of the beam. Sectiontype 6.Coffered slab withbandbeam JL--JL. - 7 - .?---,T' t 7. Ribbed slab I1I / II !II , !I tt 8.One-wayslabwithnarrowbeam II II II II 33E,iectof restraint to f l wr s h o n i n gTotal imposed loading (kNlm3 2.5 5.0 10.0 2.5 5.0 10.0 2.5 5.0 10.0 Post-tensioned floors must be allowedto shorten to enable thepresb-essto be applied to thefloor@s". Shortening occurs becauseof: rl B B A Spanldepth ratios 6 m 5 L 5 13 rn 28 26 23 30 27 24 a) Elasticshortening due to theprestressforce. slab 42 38 34 b) Creep shortening due tothe prestressforce. b-m18 16 13 C) Shrinkage ofconcrete. Theelasticshorteningoccursduringstressingofthetendons,butthecreepand shrinkage are long-term effects. The fl wr will besuppaned on columns or a combination of columns and core walls. Thesesuppons offer a restraint to the shortening ofthe flwr. There are no frmrules which maybeusedto detemine whensuch restraint is significant. As a guide, ifthe presmssis less than2NImmz theflwrisnotverylong andthereisnot more than one stiff resminr (i.e.a IiB shaft)thenthe e f f e c ~ of restraint areusuallyignored. Asimple method ofascertaining the restraintoffered bythe suppons is to calculare theelastic, creep and shrinkage smins expectedinthe slab andthento calculate the forces requiredto deflectthe supports. Figure19 shows two simple framesinwhich thefloorshave shortenedandthecolumns havebeenforcedto deflect. The force in each column maybe calculatedfrom b e amount it hasbeenforcedto deflect andits stiffness. The stiffness may becalculated on the assumptionh a th e column is built-in at bothends. ,..,. . . (a)Symmemcal floor supportedoncolumns @)Fl wrsupponed bycolumns andlifi shaftat one end FigureIF: Restraint to noorshortening. Thecalculation ofelastic.creepandshrinkage strains maybebasedonthevalues given inBS8110''1.The elastic strain should be basedonthemodulus ofelasticity at the time the tendons are stressed. -Ifthis is at seven days after casting the modulus is approximately 80% ofthe modulus at 28days.The creep straindepends onthe age oftheconcretewhenthetendonsarestressed.thehumidityandtheeffective thickness.Thec w pstrainwouldbetypically2.5timestheelasticstrain.The shrinkagestrainwillgenerallybeintherange100to300x1r6,butinsome circumstances it canincrease to 400 x10". Typicalstrains for a 300mminternal floor witha prestressof2 N/mmz wouldbe: Elastic smin100 xlo4 Creep strain250 x10' Shrinkage strain300 x10" Total long-termshah (E,,) 650 xlo4 The following analysis is approximate but conservative and ignores any displacement ofh e foot ofthe columns or rotationofthe ends ofthe columns.A more accurate analysis maybemade usinga plane frame withimposed memberstrains. The force required todeflect eachcolumn.as showninFigure19. maybe assumed to be calculated as follows: For the purposes ofcalculating H,,thevalueofEJ,for thecolumn maybe reduced by creep in the column and in some cases cracking. A reduction of at least 50% from theshort-term elastic propertiesisnormally justifiable. The totaltensioninthe fl wr duetothe restraint toshortening is the sumofallthe column forces to one side ofthe stationary pointInFigure19a. the tensionisH,+ H; inFigure 19b. the tension is H,+ H,+ H,.This lension acu; as a reduction in!he precompressionofthefloor bythe prestress.If thetensionis smallincomparison withtheprestress.itmaybeignored.Ifthetensionforceismoderate.itmaybe necessary to subtract it fromthe prestress toobtain the effective precompression of theflwr.Butiftheresmintissoseverethatflexing ofthevemcallrlembersto accommodate the shortening is not possible. other measures are required.These may include freeing the offending stiff elements during a temporarycondition. However, it must beremembered Lhatcreep andshrinkage will continue to occur forupto30 years. 4.MATERIALS 4.4Concrere Concrete should bemixed,transported and placedinaccordance with BSBIIO, Pan 1. Section614). Choiceofconcrete typeandgradewillbe influencedbydurability. early strength gain requirements, material availability and basic economics. At present concretegradesofC35andC40arethemostcommonlyusedforpost-tensioned floors. Wherelightweightaggregates areused,referencesshouldbemadetoLhe special requirements OF BS8110.Pan 2. Section5l4). 42Tendons 4 . 1 St rand Thetendonmaterialusedforpost-tensioningconcretellwrsisnormally7-wire suand. Thisstrand should comply withType 2 (lowrelaxation) as descr~hdinBS 5896, Table 6"0'. 42.2Tendon protection Unbonded tendons Unbondedtendonsareprotectedbyalayerofgreaseinsideaplasticsheath.An exampleisshowninFigure20.Thesematerialsshouldcomplywiththe recommendations giveninreference 11. Figure 20:Layout oC unbondedtendons. Under normal conditions, the strand is supplied direct from the manufacturer already greased and sheathed. In no circumstances should PVC be used for the plastic sheath, asitissuspected that chloride ions canbe releasedincerlain conditions. Bonded tendons Bondedtendons areplacedinmetalductswhichcanbeeithercircularorovalin om. Anexample is shown inFigure 21.The latter is usedinconjunctionwithan anchoragewhichensureslharupIO fourstrands areretainedinthesameplanein order IO achieve maximumeccentricity. Figure 2i:Layout ofbondedtendons. The ducts are made from either spirally wound or seam folded galvanised mela1 strip. Oncompletionofsmessing,theductsarepumpedfullofcementgroutwhich effectively bondsthe strand tothe structure as well as ensuring corrosion protection. Further information can beobtainedfromreference12. 42.3Anchorages Anchorage companents should comply withBS 4447'L31.Details ofthese areshown i nFigures 22 and 23. In the case of unbonded anchorages corrosion protection should complywithClassAexposureasdefinedinreference14.Inaddition,testsfor unbondedanchoragesshouldincludefatiguetestingconsistingofcyclingthe prestressing force between M) and 65% ofthe characleristic strength ofthe s m d for . two million cycles. Figure 22:Atypicalanchoragefor an unbondedtendon. Figure 23:Atypical anchorage For abondedtendon. 43UE-tensionad reinforcenssn: Un-tensionedreinforcementshall comply with BS1449''5'. 5.COVERREQUIREMENTS Nominalcover is dependent ondurability requirements orfire resistance, whichever conditionis the moreonerous. Bondedtendons:Thecovertothetendonsshouldbeinaccordancewiththe requiremensfor prestressedconcrete inBS8110, Pari 1, Clause 4.12.3'4'thecover beingmeasuredtotheoutside oftheduct.It should be notedLhatthecover tothe centre of the tendon willbe more thanthat to the centre ofthe duct, since the tendon will press againsr thewallof theduct. Unbonded tendons: There is no durability requirement for unbonded tendons protected inaccordancewith4.2.2.Firepmtec6onshallbeprovidedinaccordancewith BS8110, Part 1. Clause 4.12.3.1.3")andthe nominalcover tothe sheath should not beless than25mm.Thetendonis normallyspecifiedasanominaldiameter (e.g. 12.9 or15.7mmfor 7-wiresuperstrand):3mmshould beaddedtothediameter to allowfor thethickness ofsheathing. Un-tensioned reinforcement:The covertotheun-tensioned reinforcement should be in accordance with the requirements for reinforced concrete in BS8110, Part 1, Clause 3.3"'. Anchorages:Thecovertoanchoragesshouldbeas forbondedtendonsgivenin BS8110, Part 1, Clause 4.12.3.1(4). Considerationshouldk givenLO thelayoutofbothtendonsandun-tensioned reinforcementwhen decidingthe critical cover requirements(see Section 7.5). 6.THE DESIGN PROCESS 'Ihissection considers the various stages ofthe design process inmore detail. Asin most reinforcedandprestressed concrete design work, the customarydesign process is ofaniterative nature following the cycle: 1.Preliminary design 2.Check designbyanalysis 3.Revise design as required 4.Repeat steps 2 and3 ifnecessary. The analysis is normally bzsedonsemi-empirical procedures such asthe equivalent frame method. More rigomus analyses based on, for example, finite element methods are md yadopted. They should only be considered for large pmjcts ofunusual form wherethehighdesigncosts andtheinapplicabilityofh eempirical method justify them. ?hedesignisundertakengenerallyinaccordancewithBSgllO'"withadditional guidance giveninthis reportNormallythe flexural capacity at Serviceability Limit State is considered firs4 and thenchecks onflexural andshear capaciry at Ultimare Limit State are carried out. 62Design jlowc i wiAtypicaldesignflowchart isshowninFigure 24. 63Basic a ~ l y s i sThe analysis of post-tensioned flwr systems differs from a reinforced concrete design approachowingtothepositiveeffectlhatthetendonshaveonthestructure.In reinforcedconcretethereinforcementisinidallyunstressed:thesuessinthe reinforcementresults fmm h e deformation and cracking of the structure underapplied load. Inthis waythe reinforcemenr may be considered lo acl passively. Onthe other hand, thetendonsina post-tensioned fl wr are activelystressed byh e jackssothat theyareloadedbeforetheapplicationofotherloads.Theforceinthetendonis chosen by the designer and does not varymuch withthe application of Serviceability Limit State dead andlive loads. -load to be balanced -required presuess forces -number of tendons shears due to externally Check Serviceability deflections and vibrations Consider revisions or refinements to design Figure 24: Design flow chart 30 Theanalysisofequivalentframesmaybeundemkenbyhand,usingmoment distributionorflexibilitymethods.orbycomputerusingplane-frameanalysis programs. There are also available on the market several computer programs specially writtenfor post-tensionedflooringsystems. Theseprogramsnotonlyundenake the analysis oftheframe under appliedloadingandloadingfromthetendons,but also calculate the flexural stresses. For more complex or detailed analysis, grillage or finite element methods may be used. Whichever Whnique is used for the smctural analysis itmust cakeintoaccount not onlythedead andliveloads butalsothe loads which thetendons applyto thestructure (seeSection6.7). Thechoiceoflayout andmemkrsizinghasbeendi s c us 4 inSection3, andis probablythemostimportantdecisioninthedesignprocess.Unlessprevious experienceorovemdingfactorsdictatetheexactformandsection,seveml possibilitiesshouldbestudied,althoughthedesignershouldbeableto limitthe possible solutionsbyconsideringthevariousconsuaintsandbyroughdesignand costing exercises. Withregardtoslabthicknessandconcrete suengths,therelationshipofstructural layout, slab thickness and loading has beenreferredto inSection 3. Adetermination ofatrialmemkrdepthmust be madeat anearlystage inthecalculationprocess. This can often bebest obtained byassuming avalue ofabout 70% oftheequivalent non-prestressed member. TheloadingforServiceabilityLimitStateshouldconsiderthedeadloadand post-tensioning effects actingwiththose combinationsoflive loadswhichresultin themaximumstresses.Unlesstherearespecificabnormalloadspresent,itwill generally besufficient to consider the post-tensioning effects in combinationwilh the live loads as given inBS8110, Part 1, Clause 4.3.3"'. At transfer ofpresuessing only the dead loads present during stressing, togelher wilh the post-tensioning effecls before losses due to creep, shrinkage and relaxation, should beconsideredinobtainingstresses.Wheretheappliedloadschangesignificantly during conshuctionor phasedstressing is employed,thevarious stages should each be checkedfor transfermess limits. At the Ultimate Limit State the loadcombinationsshowninBS8110, Part1, Table 2.1andClause4.3.3'"shallbeconsidered toarriveatthemaximummomentsand shearsatanysection.Secondaryeffectsofprestressingshouldbeincludedinthe applied loadswitha loadfactorof1.0 (see Section 6.9). 6.6Eqdvait-nt frameanaijsis Itisusuallodividethesmchlre inlosubframeelementsineachdirection. Each frame usually comprises one line ofcolumns together with beamlslab elements of one bay width.The frames chosenfor analysis should cover all the element types of the complete smcture. The ends of the columns remote from the sub-frame maygenerally be assumed tobe fixedunlesstheassumptionofapinnedendisclearlymorereasonable(e.g.pad footings). The use of the equivalent frame method does not take account of the two-dimensional elastic loaddisnibution .effects automatically.It will give different support reactions fromthe analysesinthetwoorthogonal directionsunlessthewidthofslab chosen caincides withthe paints ofzero shearintheo h dimtion. Normallyforinternal bays thewidth of slab will bethe full panel width.However for a regular layout, Lhe penultimateframewill pickupmore thanhalfthewidthontheside oftheend bay (seeFigure25).Providedthereactiononeachcolumnistakenasthelarger value fromthetwoanalyseslitrleaccuracywillbelostHoweverwherethesizeand arrangement of edge columns is different from the internal columns the widthof slab shouldbeestimatedmoreaccurately. Thiswillensurethecorrectselectionofthe number of presuess tendons with the profile appropriate for the frame being analysed. It should be noted that these elastic effects are automatically taken into account when thefloorisanalysed usinggrillage or fmite element methods. Irrespective of which analytical technique is used, care should belaken to ensure that theassumptionsmadeareappropriatetothestructureunderconsideration.In particular the prestress appliedto two adjacent frames should not beverydissimilar athenvisetheprestressfmmthemorehighlystressedframewilldissipate intothe adjacent frames. Lines of zeroshear in !ongltudinal direction @--@ End I~enuirfmsre1Internal . . bramibr mi'Freme (a)Equivalent frame widths intransverse direction Lines of zero shear in trsnrvsrie direction h End Frame Internal Frame 1-..v."md k n d Frame (b)Equivalentframe widths inlongitudinal direction Figlare 2 5 Elastic loaddiistributioo ere& BS8110, Past 1, Section 3.72"'gives a clear definition onthe division of a flat slab into sub-frames or'panels'.Other methodsmayalso be used. Itisnowcommontoanalysesmcturesusingplaneframecomputerprograms. However,whenlonghandmomencdisnibution calculations are employed, sliffness. carry-over factors andfued end momentcoefficienrs must te calculated. These can bequitecomplicatedforvaryingsections,columnheadsanddrop-panelsand, althoughoftenignoredinhandcalculations, theeffect onstiffnessofthecomplete beammomentof inertia over thecolumnwidthcan bemost significant, particularly forwidecolumns. It should alsobenotedthatBS8110,Part1, Section 3.7.2.6'"allows reductionof negativemomentsU, thecolumnface,whichequallyappliestopost-tensioned members. 6.7Tendon p mpa d balanced load Ideally thetendonprofileisonewhichwillproduce a bending moment diagramof similar shape. but opposite sign.tothe moments from theapplied loads. This isnor always possible because ofvarying loading conditions and geomemc limitations (see Section 5). Itshouldte notedthatfor bondedsystemsthecentroidofthesnandswillnot coincide with the centroid of the duct.This is particularly true inthe case of circular ducts. Further information maybe available h m the manufacturer'sliterature. Inthesimplestcase,forauniformlyloadedsimply-supportedbeam,thebending moment is parabolic, as isthe idealtendonprofile. The total'sag' inthe parabolais referredto as thetendon'drape' (see Figure26).and is limitedbythesection depth and minimum cover tothetendon. Atthe suppons thelendon hasnoeccentricity and hencethere is nobending momencdue tothetendonforces. Tendon profiles are not always symmewic. However, the point ofmaximum drape is still at thecentre ofthe pointsofinflection, butmaynot correspondtothe pointof maximumsag. .. .. , . . .w/unitlength A t t t t f f t t 4 t ? rI drape = a Bendingmoment Figure 26:Idealised tendonprofile. The upwardforces appliedto the concrete by a parabolic profiledtendon, as shown in Figure 26.areuniformly distributedalongthetendon.At theendsof thetendon downwardforcesare appliedtotheconcretebytheanchorages.I heupwardand downward forces are in equilibrium sn that no external forces occur. The set of forces applied to the member by the tendon are known as the 'equivalent' or 'balanced' loads, inthat theupwardforces counter-balance a proportionofthedownwardforces due to dead and live loads. Fora parabolicprofiletheupwarduniformly distributedload,w,canbecalculated as follows: where:s--distance between pointsofinflection a - - drapeoftendonmeasured at centreof profile between pointsofinflection. Notethat this maynotbe position of maximum sag paw=average presEessing force intendon Usually,incontinuousmembers,themosteffective useofatendoninproducing 'balancedloads'isachievedbyhavingthetendonatitslowestpossiblepointin positivemomentlocations,andatitshighestpossiblepointinnegativemoment locations. Inthis way thedrape, and consequently the'balancedloads',is increased to amaximum. The 'equivalent'or 'balanced' loads may beapplied to the smctural frame inorder to obtainthetotal effectsofprestressing.Thetotal effectsare acombinationofthe PrimaryandSecondary effects as described inSection 6.9. I tis beyond the scope ofthis publication to give anextensivetreatise on prestressing theory or load-balancing design.Funher details maybe oblainedfmm reference16. Inpost-tensioned design it is commontoroughly'balance'equal proponionsofthe dead and applied loads in each span. Some designers set out with a preconceived idea ofwhat loadtheywishtobalance asa proponionofthedeador totalload. Othen balance the minimum amount which will result in the finalstresses due to the out-of- balance loads ki ng as close as pssi bl e to themaximum allowable stresses. This latter approach is usually the most economical overall but may not always k the most suitable for deflection or congestion ofun-tensioned reinforcement. Figure27illusuatesanidealisedtendonprofileforatwo-spanmemkrwitha cantilever. The parabolic profiles result in the balanced loads w,,w,and w,as shown, calculatedfrom thetendonprofile and hencethe'drapes'. Figure2'1:I de a l i d tendonprofile for two spans withsingle cantilever. L , - Figure 28illushates atwo-spanmember withan idealised tendon profile to provide a uniformuplift over span1 anda concenmted uplift inspan2.The concenmed effect is useful inmembers transferring column or similar point loads. Whilethe bendingmoments'peak'overthesupports.itisclear!hatinpracticea tendoncannot dothisandsome approximationmust bemade.Rememberthatthe peak is where the tendon is 'dumping'the load it has picked up byits parabolic shape (Figure 29). In practice, tendonprofiles are oftheform shown inFigure 30. 4 Cantilever Span 1:Span2:Span 3: WT L28,+ 8,W,Lz2e, WJ 4' h?,=- 2~ ( ~ + e ~ ) = - 8~ ( ~ + e ~ ) = ~L ? - Theratio t'/L should generallybekept assmall aspossible andisusually selected as 0.10.Appendix C provides information fromwhich the parabolic tendon geometry canbecalculated. -L 3 Theresultant balancing forces are therefore as shown inFigure 31. Figure28:Idealisedtendonprofilefortwo spans withpoint load I- Rigure 29:Load'dumping'at 'peaks'. 35 L,- -L2 81e2 Span 1: Total drape = el+- 2 &an2: Total drape = e,+-1- Balanced load w,=Pettotal drape x WLI2Balanced load wz= P xtotal drape x 4/Lz P is the prestressing force at ths section under consideration. Note: - thatthecentre ofgravity oftheconcrete and the centre ofgravity ofthe tendoncoincide at the endofthe member so that noequivalent load moments are applied at theendofthe member. Figure 3C:PracticalrepreseabiionOF ideaiised tendonp:o:i!e. Figure 31:Resultantbalancing forces. For the reverse parabala at thesupport the toul loaddownwards: andfor the span parabala the total loadupwards: Ifwe makeL'/L equal to 0.1, a. suggested above, then:= 4s, Since the upwardanddownward loads must beequal, it follows that i h = 37 s1 s, and hencea,=-a, 4 Thebalancingloadsupwardsanddownwardsduetothetendonscanthusbe calculated. 6Prestmss forces and losses Fromthe time that a post-tensioning tendonis stressed, to its find state many years alter stressing, various losses take place which reduce the tension irrthe tendon. nes e losses are groupedintotwocategories, namely: 1.Short-term Losses, whichinclude: a)Friction losses in ihe tendon b)Wedgeset or 'draw-in' c)Elastic shortening ofthestructure. These losses take place during stressingandanchoring ofthetendon. 2.Lang-term Losses, whichinclude: a)Shrinkage oflhe concrete b)C a p ofthe concrete under the effect offhe prestress c)Relaxationofthe steel tendon. Although lhese losses occur over a periodofuplo tenor more years, the bulk occurs in the Grsr two years following stressing. Typically, losses reduce the applied preslress force byapproximately 10% at transfer and20% after aU losses. The calculation of losses is discussedinmore d d l inAppendix B. 69Secondaryeffects The secondary effecrs of presmssing are sometimes called 'parasitic effects' but that impliesthattheeffects areunwantedandharmful. Thisisnotinfactthecase. For most structures the secondary moment will be a sagging moment and will increase the moments duetoapplied loadsat midspan but reduce themomenrs atthe support. In some structures it is possible lo 'tune' the secondary effecrs byadjusting the shape of the tendon profile toobtain the optimum solution. This is more likely to beof usein the design ofbeamsratherthanslabs. Primaryprestressing forces andmoments are fhe direct resultofthe prestress force acting at aneccentricity fromthe section centroidThe primarymoment at a section issimplythesumoftheproducc;ofeachlendonforce withitseccenuicity;h eprimaryshearisthesumoftransversecomponenlsofthetendonforces andthe primaryaxial load is the sum ofrheaxial components of lhetendonforces. Whenanelement ofastructure isprestresseditss hap changes.Itwillalways shorten,and willbendifthe cenuoidof the prestressforce doesnotcoincide at all positions withthe section centroid. (It is possible, however, to select a tendon profile whichresults inno rotationofthe elemenl ends.) If the element is part ofa statically determinate structure thenlhese changes inshape willnotaffect thedistributionofforces andmomenls(Figure 32). G=?lStressed isolated element Unstressed element onsupportsUnstressed isolated element Stressed element still compatible with supports Figure32:Restressedelementas partof a statically determinate structure But when the element forms part ofanindeterminate suucture, the changes inshape resulting from prestressing will modify the support reactions. Additional reactions are requiredtomakeheprestressedmemberpassthroughsupportpointsandhave suitable orientxionwhereappropriate (Figure 33). 37 Unstressed element Reactions applied to -t insrructure through support 7positions Reactions applied Unstressed isolated L-_----_l elemarrt tomakebeamhave compatible rotations Stressed isolated-,4Total secondary -Velementforces and moments - p w - for element Figure 33:Reactions ona prestressedelement due tosecondary effects These secondaryreactions result insecondary forses and moments inthemembers. These are typicallyconsrant axial and shear forces throughout a span anduniformly varying moments.The calculation of thesesecondary effects canbedifficult when staged construction, creep and shrinkage are considered. (Note hat secondary effects cannotdevelopincantileversastheyarestaticallydeterminate.)Methodsof calculating secondary effects are giveninAppendix D. Equivalent loads will automatically generare'the primary and secondary effects when appliedLo thestructure. Seniceability calculations do not require any separation of the primary and secondary effects,andanalysisusingtheequivalentloadsissuaightfonvard.However,at Ultimate Limit State the two effects must beseparated because the secondary effects are treatedas applied loads. 'The primaryprestressing effects aretakeninto account byincluding the tendon force inthe calculation of the ultimate section capacity. The primaryprestressingforcesandmomentsmustthereforebesubtractedfromthe equivalent load analysis togive thesecondary effects. To calculate the ultimate loading oii anelement, rhe secondary forces andmoments arecombinedwiththeultimateforces andmomentshorndeadandliveloads. The HandboaktoBS8llO'"',suggests thatthe partialloadfactor onsecondaryeffects should be 1.0.The totalultimatemoments canbe redishibuted inaccordance with BS8110. Pan1, Section 4.2.3'''. 6.10Flexural section design 6.10.1Serviceability Limit Stateafler al! losses The bending moments calculated from the critical loading conditions given in Section 6.5,including thetendoneffects, providetheserviceability stressesateachsection using: topfibre stress, f, =P + M A,2, boltomfibre smss, f,=P- IVI A,zb where:z, =the topsection modulus zb =the bottomsection modulus M=the tola1 out-of-balance moment e = eccentricity oftendons, takenas positive belowtheneutralaxis MA = applied moment duetodead andliveloads M,= moment fromprestress secondary effects One-way spanning floors Bondedtendons: The maximum allowable concrete compressive andtensile stresses forfloorswithbondedtendonsaregiveninBS8110,Pan1,Section4.3.4.2and 4.3.4.3"'respectively. Most buildings will be satisfactory as Class 3 suuctures andthe- natureoftheloading mustbeconsideredwhendeciding ona0.1or0.2mmcrack width(e.g, frequency and duration). Unbonded tendons: The maximum allowable concrete compressive stresses infloors withunbondedtendonsareasforfloorswithbondedtendonsandaregivenin BS8ll0. Part1, Section 4.3.4.2"'.The maximum concrete tensile stresses should be taken as those given for group (b)in Table 4.2ofthe Standard, with a limiting crack widthof0.lmm.These values must beadjusted for section depthas givenbyTable 4.3oftheStandard.Ifthesuessesareenhancedbyincreasingtheun-tensioned reinforcement asisallowedforbondedrendonsinBS8110"'.crackwidthsand deflections should be rigorously chccked. All concrete tension shall be caniedbyun- tensioned reinforcemenr (see Section 6.10.5). Flatslabs (two-way spanning) Flatslabs maybeanalysedineitheroftwoways.The more commonmethodisto analyse equivalent frames in each direction. In this case some account must betaken of rhe peaking ofthe moments at the columns, described in Section 2.4. The analysis results inmoments and stresses averaged across thewidthofthe panel.The stresses should be limiredtothose giveninTable 2. Grillage orfinite element analysis maybeused,but thisisnormallyonly justified withfloorsofunusualconfiguration orwhereadesignisfobe constructedmany times,suchas inahigh-risebuilding.If suchanalytical techniques areusedwhich takeintoaccount thedistribution ofmomentsandsuesses acrossa panel,thenthe allowable stresses givenforone-wayspanning floors maybe used.Particularcare must betakeninmodelling the columnlfloor intersection and intheinterpolation of theresults obtained. Table2:Allowableaveragestressesinnatslabs,(hvo-wayspanning), analysedusingilleequivalentframe method. Location Support span Note:Bondedreinforcement maybeeitherbondedtendonsorun-tensioned reinforcement. InTable2, thesupportzoneshallbeconsideredas anypartofthespanunder consideration within 02xL of the suppon,where L is the effective span. Outside of this zoneis considered tobethespanzone. - In Compression .- 0.29 kc" Additional designed un-tensioned reinforcement is required in the suppon zone ofall flat slabs, andinthespanzoneoE slabsusing unbonded tendonswherethetensile stressexceeds0.15Jf,.ThedesignofthisreinforcementispresentedinSection 6.10.5. 6.10.2Transfer condition InTension Transfer stresses should be checked for all floors. These are likely to be more onerous for floors withhighimposed loads. withbonded reinforcement 0.454~- 0.45df- Un-tensionedreinforcementshallbecalculatedinasimilarmanneru,the reinforcement for the Serviceability Lirnit'State (seeSection 6.10.5). withoutbonded reinforcement o 0 . 1 5 4 ~One-way spanning floors BS8110,Part1.Clause4.3.5.1"'givessuitablelimits forone-wayfloorconcrete compressivestressesattransferof0.5~"attheextremefibre(or0.4f,fornear uniformstressdistribution) wheref,istheconcretestrengthattransfer.Clause 4.35.2givesthelimitsforallowableconcretetensilestresseswhich,formost buildings, will be 036dfcf,,. Flat slabs (two-wayspanning) The allowable sh'esses given in Table 2for the Serviceability Limit State also apply lothetransferconditionforslabsanalysedusingtheequivalentEramemethod, however,h should be substituted byf,.For slabs analysed bythegrillage or finite element methods, the allowable suesses are those given for one-way spanning floors. 6.10.3Ultimate LimaState AnUltimateLimitStatecheckisnecessaryonallfloorsinadditiontnthe Serviceability Limit State previously covered.In this condition, the factored dead and applied loads are considered together withthesecondary effects ofthe prestressing (seeSection 6.9).The primary prestress effectr are considered as pan ofthe section strength. Additional un-tensioned reinforcement may be required inorder to generate anadequate moment capacity. BS 8110, Part 1, Section 4.3.7'''gives guidance onthe assumptions for calculating the concrete and un-tensioned reinforcement suesses and the allowable design stresses for thetendons.IntheaboveSection,equation52forunbondedtendonshasbeen developed fromthe results of testsinwhichthe stress inthetendonsand the lenglh ofthezoneofinelasticityintheconcretewerebothdetermined.Theflooris considered todevelop bothelastic andinelastic zones andthe lengthof theinelastic zoneistakentobe10 xthe neutralaxis depth. The extension ofthe concrete at the level ofthe tendons isassumedtobenegligible in the elastic zones and the extension i n the inelastic zone is assumed to be mkenup uniformlyover the length.1,ofthe tendon. This is discussed further inreferences 29 and 30. Hence, for asimply supported fl wr there is only one inelastic zone associated with thefailure, butwithacontinuousfloorthenumbee ofinelastic zonesrequiredfor failure is more complex (see Figure 34). The length oftendon. I, inequation 52 can bemodified,bearing inmindthat if the tendondoesnotcontinue the fulllengthof the continuous floor it maynotinclude all theinelastic zones necessary for failure. It is therefore prudenttoassume nomore thanoneinelastic zone perspan, andno morethantwoinelastic zonesforthe fulllength. ---- -plastic hingeWithout Columns WithColumns (a)Ductile failure WithColumns @)BrilUe failure Figure 34:Zones or inelasticityrequiredfor failure ofacontinuousmember. 6.10.4' Progressivecollopse Where p r o ~ v e . c o l l a p e involves the use of unbondedtendonsinkeyelements, the maximum stressin the unbanded tendon shall not exceed 0.85fW This ensures that theanchorages are not over-stressed, and protects against catenary action. Inunbondedmemtersthereisalso theriskthat iftendons are severedaccidentally therewillbea'progression'offailureforthefulllengthofthelendons.Thisis pasticularly relevant for one-wayspanningmemberssuchasbeams.ribsandslabs spanning onto beams or walls. In the case of one-wa), members where horizontal progressive collapse is ofconcern, it is necessary to reiiiforcewiihun-tensionedsteel. This should be provided to satisfy theloadcase ofdeadloadplus onethirdlive load[DL + (I/3)LL]withan overall load factor of1.05,and reduced material factor in accordance with BS8110. Pan 1, Clause 2.4.3.2'"for 'effects ofexceptional loads or localised damage'. Reinforcement shouldbe in accordance withnormalBS8llO limits and arrangements. Experimentalandpracticalevidence inthe USAhasestablishedthatthisproblem does not occur in the iniernal bays of flat slabs due to the overall 'plale' or membrane action.The possibility ofhorizontal progressive collapse ofedge andcomer panels of flat slabs must beconsidered These panels should besupponed forthesituation where the tendons parallel tothe edge have been severed. This support cantypically beprovided bybondedreinforcement inthe panelor anedgebeam. 6.10.5Designedflexuralun-tensioned reinforcement Additional un-tensionedreinforcement shall be designed to cater forthe fulltension forcegeneratedbytheassumedflexuraltensilestressesintheconcreteforthe Iollowingsituations: Alllocations inone-way spanningfloors using unbonded tendons. Aillocationsinme-wayspanning flwrs wheretransferstressesexceed 0.36vfci. Support zonesinallflat slabs. Spanzonesinflatslabsusingunbondedtendonswherethetensile smss exceeds 0.15df,. The reinforcement shall bedesigned, withreference to Figure 35, toact at a stress of (5/8)fy as follows: h-x= -f,,xh f--fa The value off,,willbe negative intension The reinforcementshall be designed for the soesses ar Serviceability Limit State, both aflerallprestress 1osses.andat. ms f erconditions.It shall be placedinthetensile zone,as nearas practicabletotheouterfibre(seeSection7.5).Undermnsfer conditionsanydesignedreinforcementis likelyto beontheopposite facetothar required after all losses. At Ultimare LimitState, additional un-tensioned reinforcement mayalso be required (seeSection 6.10.3).Anyreinforcement providedfortheServiceability LimitState mayalso beusedinthe calculation of themoment capacity at Ultimate Limit Stare. The designed reinforcement shall be checked against the minimum requirements given inSection 6.10.6. Figure 35:Sectionstresses usedfor thecal c~l nt i wof un-tensioned reinforcemenL 6.I0.6Minimumun-tensioned reinforcement Where fire ratings ofgreater than 2 hours are required, it isrecommended that anti- spallingreinforcementbeplacedinthesoffitwhennootherreinforcementis provided. One-way spanningfIoors Bonded tendons: There are no minimum un-tensioned reinforcement requirement5 for one-way spanningfloors with bondedtendons.It is considered thatthese flwrs have sufficient tendon-to-concrete bond to distribute flexural cracking. Care should be taken toensuresufficientreinforcementisprovidedtoguardagainstcrackingbefore stressing, ifearlyphasedsuessing isnot employed. Unbondedtendons:One-wayspanningfloors withunbondedtendonsshouldhave minimum reinforcement in accordance with BSgllO, Part I, Table 3.27, Figures 3.24 and3.25".This reinforcement should be spread evenly across thefull widthof slab in accordance withthespacing rules giveninBS8110, Pan 1.Section 3.12.11'4'. Flatslabs (two-wayspanning) A11 flat slabs shall have minimumun-tensioned reinforcement at column positions to dismbutecracking. The cross-sectional area ofsuch reinforcementshall beatleast 0.075% ofthe gross concrete cross-section (O.WO75xAJ,andshall be concentrated between lines that are 1.5 times the slab depth either side OFthe widlh ofthe column. The reinforcement shall be placed as near as practical to the topofthe nwr,with due regardfor cover andtendonlocation, andshallexrend at least 0.2 xL into the span or as far as necessarybycalculation (see Section6.10.1and6.10.2).The maximum pitchof thereinforcement should be 3Wmm. Inthespanzone,thereare nominimumrequirements. However,whenunbonded tendonsare useditwouldnormally benecessarytoprovidedesignedun-tensioned reinforcement inthe hotlamoftheslab(seeSection6.10.1).Thisreinforcement should extendat leas1 towithina distance of0 2xL,measuredfromthe centre of thesupportItshouldbeplacedataspacingof3xslabthicknessor500mm, whichever isthe lesser. Slabedges Un-tensionedreinforcement shouldbe placedalong edgesofallslabs.Thisshould include U-barslaced withat least two longitudinal bars top and bottom,as shown in Figure 38. See also Section 6.12.Reinforcement should k provided inthe triangular unstressedarea betweenanchorages. See Section6.13. 6.33Shew strength 6.11.1Benmsandone-wny spanning slabs The method inBS8110, Part1. Section 4'"should beused. Where unbonded tendons are used, the value of v,in equation 55 ofBS8110'41should be reduced byafactor of0.9as recommended byRegan"". 6.112Flatslabs(punching shear) BS8110idJdoesnotprovidespecificguidanceforcheckingpunchingshearfor prestressedflat slabs.The working party considered a number of-different methods while preparing this handbook, witha viewIO satisfying the following aims: - Design capacities to be in line withother international standards. Increased punching shenr capacity for bondedtendons. Increasedpunchingshearcapacitywhentendonsareconcenhatedinthe vicinity ofthecolumn. Adesignmethodwhichcomplements BS8110'4' asfar as possible. Adesign method whichallows a smooth transition fromreinforced concrete to prestressed concrete and allows for situations where the slab is prestressed inone direction only. The following method achieves these aims and is recommended. Calculate !he effective shear force, V,,,inaccordance with BSBIIO,Clause 3.7.6. The shear resistance, V. is obtainedbyaddingtogether Lhe contributions fromeach ofthe sides ofthe critical shear perimeter as giveninBS8110,Clause 3.7.7"'.The shearresislanceofeachsideofthecriticalperimetershouldbecalculatedin accordance withBS8110, Clauses 4.3.8and4.4'4J as modifiedbelow. Fiat slabs are generally not heavily prestressed and will therefore be governed bythe design for "sections cracked in flexure", using equation 55 (BS8110, clause 4.3.8.5"3. Equation55doesnot,however,provideasmoothtransitionfromreinforcedto prestressed concrete becauseof theterm: For lighrly presuessed structures the inclusion ofthis termin equation 55 can leadto ashearcapacitylessthanthatwhichwouldbecalculatedforthesameslabbut without prestress.This is obviously incorrectThe British Cement Association @I1 has recently compared various forms ofshear calculation with publishedtest results and concluded that equation 55 would be more consistent with the test results if the above term were omitted.It is thaefore recommended that the shear resistance of each side ofthe critical perimeter becalculatedfrom equation55 moditied'asfollows: where v,,b. andd are fhe valuesfor the relevant side ofthe critical perimeter. The value ofv,should be calculated taking into account both A,and A,,for bonded tendonsinaccordancewithBS8110Clause4.3.8.1.Howeverthepresenceof unbonded tendons should be neglected in thiscalculation ofv,.Nofurther reduction is considered necessary (e.g.as suggested inreference17, page98). Thede-compression moment, M,,,should becalculatedforthewidthoftheside of the critical perimeter under consideration.It should be notedthat the axial effects of prestress,P/A,,areuniformlydistributedoverthewidthoftheslabwhereasthe prestress moment effects (P,+ M,) are concentrated at the lccation ofthetendons at thecritical perimeter.HencethetwocontributionstoM,havetok calculated separately asfollows (for ahogging moment region): where: 0.8=asafety factor on presuess (BS8110, Clause 4.3.8"1 P = thetntal prestress force, over thefull panelwidth, after all losses A, = the concrete section area across thefull panelwidth ' = sectionmodulusforthetopfibreoverthewidthofthesideofthe critical perimeter P'= thetotal prestress force for alltendons passing throughtheside ofthe critical perimeter e' = theeccenhcityoftheprestressforce,P',atthecriticalperimeter, measuredpositive belowthe centroid The value of V/M must be calculated for the load case under consideration, normally that which generates the largest V , . V/h4 should strictly be calculated at the location of the critical perimeter but may be calculated conservatively at the column centreline. For atypical in~ernalcolumnV/M willvaryfrom 5.5&to6.0/L,depending onh eratio of dead load to live load, where L isthe span length. 6.11.3Openings in slabs Tendons should becontinuousanddisplacedhorizontallytoavoid small openings. Iftendonsareterminated at theedgesoflargeopenings, suchasatstairwells, an analysisshould bemadetoensure sufficient strength andproperbehaviour.Edges amund openings may be reinforced similarly to conventionally reinforced slabs: inthe caseoflargeopenings,supplementarypost-tensioningtendonsmaybeusedto strengthenthe edges aroundopenings. 6.12Anchoragebursting reinforcement Reinforcementisusuallyrequiredtoresistthetensilesuessescausedbythe concenlrauon of the forces applied at the anchors. At some distance from the edge of the floor (or the anchorages) it canbe assumed Ihat the distribution of stresses is the classic lineardistribution anddependsonlyonthemagnitudeandpositionofthe resultant ofthe forces applied tothe edge ofthe flwr. Betwen the edge andthe above plane the lines of force are curved and give rise to transversetensilesuessesinbothdirectionsperpendiculartotheappliedforce direction. Figures 36 and 37, adapted from reference 18, illusbate the varying proportions of the presoessing force manifesting itself as a splitting tensile force of magnitude depending ontheanchorage andfloor relative geometries. bursting stress I-AnchoraoeZoned --- Figure 36 Burstingstressesinrectangularbeamsubjectedtoan symmetric force. axial mrnprerslve Fi pre 37:Bursting strtrescdstrib@tioe. 46 Where a group ofanchorages exist,as is often the case for 'banded' slab tendons.the burstingstresszonesforboththeindividualandcollective anchoragesshouldbe considered, and reinforcement placed accordingly. Care should also betakento ensure that the phasing ofthe application ofprestress to anchorage groups does not create a bursting condition whichmaybecritical. Ifthis condition isunavoidable, reinforcement should be added accordingly. BS8110, Part1. Section 4.11"'gives design bursting tensile forces ofa similar nature to Figure 35 andlimits the steel stress to2M)N/mm1 at Serviceability L i i tState. It issuggestedthatbanwithf,=4 6 0 ~ / mm~ areusedforthisreinforcement Alternatively the bursting forces and distribution may be calculated ina more rigorous method,suchassuggestedbyGuyonCm'.Insome casesitmaybeshownthatthe concrete is capable of withstanding bursting without the addition of reinforcement. At Ultimate Limit State forunbondedtendonsonly, reinforcement requirements should becheckedinaccordancewithBS8110,Clause4.11.3.ThisUltimateLimitState check isunlikely tobe governing. Whereanchorages aregrouped,orwherethedistributionofanchorages doesnot reflect the distribution ofconcrete in the cross-section, it maybe necessary Lo include 'equilibrium'reinforcementtopreventsplittingbetweenanchorages.Alsowhen anchorages occur withinthe planarea ofthe floor rather than at the perimeter, it may be necessary u, include 'following' reinforcement This reinforcement runs parallel to thetendonpasttheanchoragetolimitcrackingadjacenttotheanchorage.These effects are discussed inCIRIA Guide No.1"". Post-tensioning system suppliers often test their anchorage systems inconcrete prisms. reinforced ina similar manner to that encountered in practice andusinga orismsize similartothecommonon-sitememberthichess,ek.Suchtestsmaybedeemed under BS8110, Part1. Section 2.6'"to satisfactorily modelthe on-site conditions and thereinforcement maybeconsideredadequate providedsuitable safety factors are observedu4'. Two examples showing the calculation of, and the detailing of, bursting reinforcement are given inAppendix E. 6-13Reinforcementbetween lendon anchomges Figure43showsanareaofslabbetweentendonanchorageswhichrequire reinforcementtospantheunstressedzones.Anypresuessedtendonswhichpass throughthiszone,paralleltotheslabedge,maybeincludedwiththerelevant rcinforccmcnt, providedit isinthelocaltensionzone. Thc area of lension reinforcement (ancl/orpprestressed tendons) provided parallel to the slab edgc should resist bending momenls fmmthe ultimate vertical loads calculaled foracontinuous slabspanningI,. Thisreinforcement should beevenlydisnibuled across awidthequalto 0.71,.andshould be continuous along theedge. The area ofreinforcement placed perpendicular to the slab edge should be the greater of 0.13%bh, or a quarter ofthe reinforcement provided parallel lo the edge. It should beplacedevenly between anchorages, and extend the greater ofI, or 0.7L plus a full anchorage lengthinto Uleslab. Deflection This is a Serviceability Limit State relating to the complete structure. The deflections of a structure, or ofany pans of astructure, should not adversely affect appearance or performance. Thefinalcalculateddeflection(includingtheeffectsoftempmure,creepand shrinkage, andcamber), measured belowthe line betweenthesupports ofthe flwr androof, should not ingeneral exceeds p d 5 0 .Inaddition.where -internalpartitions.claddingandfinishescanbeaffectedby deflection.thedeflectionsshouldbelimitedinaccordancewithBS8110,Clause 3.2.1.2(4'. Asa guidefor a prestressed solid slab, continuous overtwo ormore spans ineach direction, ihe span/depth ratio should notgenerallyexceed 42 for floors and 48for mf s .These limits may be increased to 48 and 52 respectively, if &tailed calculations showacceptable behaviour withregardtoshort- and long-termdeflections, camber andvibration. Lower span/depthratioswill often apply to slabs withhighliveldead load ratio& The span/depth ratiosforwaffleslabs should not generally exceed 35. Vibration Prestressed flwrs are usually thinner or spanfunher thanunpresuessed floors. They hereforelendtohavelower naturalfrequencies and greater consideration must be givento their dynamic performance. TheSteelConstruction Institute haspublishedadesignguideonthevibration of floorsm'.Thisguidecoverssourcesofvibrationexcitationinbuildings.human reaction to vibration, evaluation of natural frequencies, response offloors and design procedures. Althoughitwaswrittenprimarilyforcheckingtheacceptabilityoflightweight concrete composite floors onsteel beams,most ofthe guide is relevant for any floor system.Appendix G gives a procedure, based ontheguide for checking presnessed flwrs witharectangular grid.Vibration should not k aproblemfor general office buildings ifthe totalslab depth is greater or equal tothe values giveninTable 1. Formoresensitivelocations,orforslabsshallowerthantheabovecriteria.an aswsment ofthe dynamic response ofthefloors should bemade. 6.15Lightweight aggregate concrete Additionalconsiderations onthe use oflightweight aggregate concrete are givenin BS8110,Part2,Section5'4',andtheGuidetotheStructuralUseofLightweight Aggregate Concretem1. The allowable tensile stresses giveninSections 6.10.1and 2 should be reducedby a factor of0.8:however,the allowable compressive stresses neednot be reduced. InBS8110, Part 2, Clause 5.4"],the maximum allowable shear smss is givenas Lhelesser of0.63Jf-or 4N/mm2. These values are 0.X times the values givenfor normal weight concrete.However.thislimitisrelatedlocompressive strutfailure, notto tensile failure. Intheviewofh eWorking Party h eLimitations fornormal weight concrete, the lesser of 0.8dfWand M/mm2, may also beused for lightweight concrete. (Referenceshouldalsobemadelo'Standardmethodofdelailingslruclurai concrete'*") 7.4Tendon &tributiQn Variousmethodsfordishibutingthetendonscanbeused.Thesearediscussedin Section2.4. Insituations of ovaload, tendons passingthrough thecolurnn/floor intersectionare more effective thanlendons elsewhere. It is therefore recommended that aminimum oftwotendons should pass throughthissection. For ribbedslabs or beams,thedishibutionoftendons is dictatedbythespacingof members. 7.2Tendon s@g The maximumspacing of uniformly distributedtendons should not exceed sixtimes the slab depth for unbonded tendons oreight times the slab depth for bonded tendons. Unbondedtendonsmaybeplacedingroupsifrequired.Itisrecommendedthat groupedtendons are laidside byside anddo not exceed four lendons pergroup. The minimumhorizontaldistance betweenducts or groups of tendons shouid be the greater of75mmor the gmuplductwidth. Should it be necessary to arrange thetendons invertical layers inbeams or ribs,then it isrecommendedthatthegapbetweenthelayersshouldbeat leastthevenical dimensionofthelendonor ductInthecase ofbondedtendons whereovalmetal ducts are used, it is recommended that their positions are staggered to ease the placing of concrete. Iftolerancesontendonpasitionsarenotstaled,thevaluesinTable3shouldbe adopted. Table 3:Tolerances on tendonpositioning Slab thickness hc 200mm h > Wm mTolerances Vertically f l+ul t 5mm Horizontally * 20mm 5 20mm 73Tendon notation The accepted standard notationor tendons ondrawingsis shown inFigure 38.It is recommendedthatthislegend Figureis included onall tendonlayout drawings. Tendon quantiry,length, colour.code,elongsrlon Ir Placing sequencelwhen requiredl 7Dimenrionrfromreference line /1/to centreline of rendon group One strand Two rtrandr Three rtrandr Four strands Five raandr Y 1x10 67 laddedl2.40+ Oead end RedA = 75 Add rrrandr 3x24 50 ithrul Blue A= 165 LfWillbemarked with one of,heabove rymbolr Note: Whenmore than one symbolappears on atendongroup Edgeof slab the number ofrtrandr equaltherum ofthe symbol derignarionr. Figure 38:Methodofnotationfor useontendonlayoutdrawings. Figure39ashowsanexampleusingthelegendshowinggroupsoftendonsand anchoragestypes,together withthetendon sequence, detailed.This Figureis based uponreference24modified alongLinesrecommended inthisdocument. Tendonprofiiesinthelongitudinalandtransversedirectionsareshownusingan exaggerated scale for the verticaldimensions. These are usually given from the soffit oftheslabtothecentrelineoftheducvsheath andareplottedat intervalsoflm. Closer centres may be necessaryfor sharp verticalcurves. For easeof placement on site,shopdrawingsaredetailedgivingtheverticaltendonpositionfromsoffitm underside oftendon. . me profileofthetendonsiscriticaltothefloorperformance.Itistherefore recommended that the support centres do not exceed lm. For ribbed stabs or beams, support bars can be adequatelyheld byfm wire ties.Spot weldingcanbe usedbut this makesanyadjustment difficult. Figure 39b shows atypical support bar layout. The actual layout may be modifiedby the contractor depending onthe support system adopted, so that thespecified tendon profiles are attained andadequate suppon is provided. Placing sequence not shown Section A.A (a)Flat slab tendon layout Note: - 1. Height given is fromsoffit of slab tounderside 2.Diameter ofsupport bar is 10mm. @)Typical tendon pmfie and support bar layoutfor a flat slab Figure 39:Flat slab tendon and support layout detailing. 'tendon ET12 -.1412 - 81 + Floor plan Section A.Ashowing reinforcement details Figure 40: Flat slab reinforcementlayout 7 5 Layout of un-tcmswned reinforcemni Figure 40 shows an example ofthe reinforcement that is always required at edges and inthetopofflatslabs atcolumns.Italsoshowsthe reinforcement neededinthe boaomoftheslabat midspanforsomedesignapplications.S e Section6.10for details. 7.5.1Atcolumns Reinforcement should be placed intheLop oftheslab over columns. The design of such reinforcement is described inSection 6.10.5withminimum requirements given in Section 6.10.6.Figure 41 shows a typical arrangement oftendons and un-tensioned reinforcement amund acolumn. Figure 41:Reinforcementarrangement a tacolumn. Shear reinforcement Shearreinforcement inflat slabs.ifrequired,isusuallyintheformoflinksor hairpins, although prefabricated shear reinforcement is available. Fabricated steel shear heads mayalsobeused.See Figures 41 and'42andSection 6.11for details. Figure 42:Prefabricated shear reinforcement. Ata nd betweenanchorages Anadequate amount ofreinforcement shouldbeplacedat anchorage endblocksto avoid splitting ofthe concrete. A sample calculation m determine the amount ofthis reinforcement isgiveninAppendix F. Reinforcementshould beprovidedinthe45"wedgeareabetweentheanchorages (Figure 43). Figure 43:Unstressedareas between tendons requiringreinforcement, 7.6Pem~atiomond openings in frwm Unbonded tendons may be diverted around the openings as they are relatively flexible (see Figure 44).The change ofdirectionof thetendonshould occur away from the opening,andtrimmerbarsshouldbeprovidedtoavoidpossiblecrackingatthe comers. Figure 44:Unbondedtendons divertedaround an opening. The oval sheathing used inbonded tendonsisveryrigid in thetransverse direction, and cannot be bent around openings. In this instance, openings should te confined to the areas betweentendons. h e cutting of penemionsinfinished slabs is not a probleminribbedslabs where thetendon positions are, ineffect, defied.Gmuted tendons, providing the gmut is effective, can be cut without significant loss ofprestress. However, whenunbonded tendonshave beenused,caremust betakentolotatethetendons before concrete removal. Tendons canbecut and reinstated but it is recommended that this work be carried out bya specialisr 7.7Construction d@IaiLr 7.7.1Extentof pours With bonded tendons, friction losses usually restrict fhe length of single end stressed tendonsto25m,anddoubleendstressedto5Om.Thelowerfrictionvaluesfor unbondedtendons extend these values to35m and70mrespectively.Longer lengths areachievable butthe frictionlosses should becarefully considered. These limitations usually determine the extent ofpours. Prestressing tendons maybe continuous thmughconstruction jointsallowing larger areaswithout anypermanent joints.Allowances should bemade in accordance with good practice to accommodate temperaturevariations bythe provisionofexpansion jointsonlarger slabs. 7.7.2Consmction joints Generallyconsmctionjointsshouldbemadeinthevicinityofquanerandthird points ofthe spanfromsupports. Shear provision in accordance with good practice should bemadebythe introduction of expanded mesh, by roughening the previously poured surface or by the introduction ofashear key. Inlong slabs, intermediate anchorages maybeintroducedwhichallowthe stressing tobecontinuous throughtheconstruction joint(seeFigure 45).Alternativelyinfill skips canbeused, butit should be notedthatfhesewillnor be prestressed.These suips are cast after the stressing ofthe adjacent sections is complete (see Figure 46). This operation should bedelayed for as long a periodas is reasonable toreduce the effects ofcreep andshrinkage. Figure45:Intermediate anchorat aconstruction joint. 55 Figure 46:Infill@rip for jackaccess. Inassessing the movement of slabs at expansion or confraction jointsfromthe time of polning concrete, a strain of 650 x 10"should be. considered as normal. The drying out effect ofair conditio