Concrete flat slabs and footings : Design method for punching and detailing for ductility

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Concrete flat slabs and footings

Design method for punching

and detailing for ductility

Carl Erik Broms

Department of Civil and Architectural Engineering

Division of Structural Design and Bridges

Royal Institute of Technology

SE-100 44 Stockholm, Sweden

TRITA-BKN. Bulletin 80, 2005

ISSN 1103-4270

ISRN KTH/BKN/B—80—SE

Doctoral Thesis

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Abstract

Simple but still realistic physical models suitable for structural design of flat concrete plates and column footings with respect to punching are presented.

Punching of a flat plate is assumed to occur when the concrete compression strain at the column edge due to the bending moment in the slab reaches a critical value that is considerably lower than the generally accepted ultimate compression strain 0.0035 for one-way structures loaded in bending. In compact slabs such as column footings the compression strength of the inclined strut from the load to the column is governing instead. Both the strain limit and the inclined stress limit display a size-effect, i.e. the limit values decrease with increasing depth of the compression zone in the slab. Due respect is also paid to increasing concrete brittleness with increasing compression strength.

The influence of the bending moment means that flat plates with rectangular panels display a lower punching capacity than flat plates with square panels – a case that is not recognized by current design codes. As a consequence, punching shall be checked for each of the two reinforcement directions separately if the bending moments differ.

Since the theory can predict the punching load as well as the ultimate deflection of test specimens with good precision, it can also treat the case where a bending moment, so called unbalanced moment, is transferred from the slab to the column. This opens up for a safer design than with the prevailing method. It is proposed that the column rotation in relation to the slab shall be checked instead of the unbalanced moment for both gravity loading and imposed story drift due to lateral loads.

However, the risk for punching failure is a great disadvantage with flat plates. The failure is brittle and occurs without warning in the form of extensive concrete cracking and increased deflection. Punching at one column may even initiate punching at adjacent columns as well, which would cause progressive collapse of the total structure. A novel reinforcement concept is therefore presented that gives flat plates a very ductile behaviour, which eliminates the risk for punching failure. The performance is verified by tests with monotonic as well as cyclic loading.

Keywords: bent-down bars, building codes, cyclic loading, deflection, ductility, earthquake,

flat concrete plates, models, punching shear, shear reinforcement, size effect, stirrups, story drift, structural design, stud rails, tests

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Preface

This thesis is the result of a long process that started in the late 1980´s when the author realized that flat plates are more vulnerable for extreme loads than conventional cast-in-place concrete slabs supported by beams or walls. Specimens with shear reinforcement tested by

Andersson (1963) at the Royal Institute of Technology, KTH, displayed an increased

punching capacity in relation to previously tested slabs by Kinnunen and Nylander (1960), but the failure mode was not ductile enough to constitute a safe structure if overloaded.

The author therefore initiated a test program with different types of shear reinforcement. The tests aimed at achieving flat plates with increased ductility, but they were not successful. The failure modes were brittle despite that the nominal shear capacity of the specimens exceeded the flexural capacity.

In search for an explanation to this disappointing outcome, the punching theory (Paper I) was developed. With improved insight in the punching mechanism the author proposed a second test series with an unconventional reinforcement layout with a combination of bent-down bars and stirrup cages, which turned out to be very successful (Paper II).

Dr. Kent Arvidsson at WSP Sweden AB has supported my endeavours throughout the project. In the late 1990’s he pointed out that the stirrup cages should be improved to facilitate fabrication and erection. This resulted in a new stirrup cage design, the tests of which are described in Papers III and IV.

Many thanks to Professor Håkan Sundquist, who proposed that the above findings should be summarized into a thesis. He also provided valuable advice and proposals during the final preparation.

The thesis as well as the test programs and the papers preceding it have all been developed and written during leisure time – thereof the large time span. My deepest gratitude is therefore directed to my wife Kerstin for her invaluable support and patience during these years.

All the tests were financed by my employer at that time WSP Sweden AB (formerly J&W) and

Fundia Bygg AB provided reinforcement free of charge. The tests described in Paper II were

carried out in the Department of Structural Engineering at the Royal Institute of Technology (KTH), Stockholm. The tests described in Paper III were carried out at the Department of Structural Design at Tallinn Technical University and the cyclic tests in Paper IV at INCERC, National Building Research Institute of Romania. All these contributions are gratefully acknowledged.

Stockholm, February 2005

Carl Erik Broms

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Summary

This thesis is a summary of four papers about prediction of the punching capacity and a method for elimination of the punching failure mode for flat plates. The American notation

flat plate is adopted, which means a slab without drop panels that is supported on columns

without capitals.

The model put forward for concentric punching assumes that failure occurs either when the concrete compression strain in tangential direction near the column reaches a critical value or when the compression strength of a fictitious column capital within the slab is exceeded. The critical value for compression strain is assumed to display a size-effect, i.e. the strain limit decreases with increasing depth of the compression zone at flexure. With slab thickness 200 mm the critical concrete strain becomes round 0.0012, which is considerably less than the value 0.0035 accepted by most concrete design codes as a safe limit in bending – irrespective of the member size.

Likewise, the compression strength of the internal column capital is assumed to decrease with its increasing height. The compression strength is furthermore assumed to decrease with increasing perimeter of the capital in relation to its height.

Comparison with reported test results in the literature demonstrates that these two failure criteria are sufficient to predict the punching capacity as well as the slab deflection and ultimate compression strain – both for slender flat plates and compact column footings. The strain mechanism governs for flat plates and the compression strength of the internal capital is governing for compact slabs like column footings.

Similar approach is applied for flat plates provided with conventional shear reinforcement. The upper bound capacity is governed by an increased critical tangential strain near the column. This strain is assumed to display similar size effect as the limiting strain without shear reinforcement.

The limited flexural compression strain means that the curvature of the slab near the column is limited at the punching failure, which in turn means that the midspan curvature of the slab is limited as well. Too little midspan reinforcement would then adversely affect the punching capacity. Simple expressions are therefore derived for required amount of midspan reinforcement in balance with the reinforcement at the column.

The basic model is valid for concentrically loaded columns in a flat plate with square panels. If the panels are rectangular, then the bending moment in the long direction of a panel increases in relation to the column load. The flexural compression strain in the slab is a function of the bending moment, which means that a flat plate with rectangular panels will have a lower punching capacity than a slab with square panels for a given reinforcement ratio. The punching capacity shall therefore be verified for both reinforcement directions separately. In this context it should be noted that the theory usually calls for more reinforcement for the negative moment within the column strip than would be required according to yield line theory.

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Bending moment – so called unbalanced moment – is often transferred from the slab to the column (or vice versa) in real structures if the panel sizes vary or if the gravity load is not uniformly distributed. Still larger unbalanced moments are transferred due to story drift during earthquakes, i.e. due to lateral displacement difference from one story to the next. The punching capacity of the slab decreases in presence of such unbalanced moment. Most concrete design codes have therefore provisions for this loading type. However, the unbalanced moment is usually a statically indeterminate quantity that cannot be assessed as accurately as for a beam-column frame. A safer method is therefore proposed – rotation capacity of the slab in relation to the column. This rotation can be estimated with better precision than the unbalanced moment, irrespective of the rotation being caused by gravity loading or story drift. The method presupposes that the rotation of the column in relation to the slab that will cause punching can be predicted with sufficient accuracy at both elastic behaviour of the slab and when its reinforcement yields, which is confirmed by comparison with test results.

The story drift capacity of flat plates is in the literature often reported as being a function of the utilization factor, i.e. the column reaction in relation to the nominal punching capacity at concentric loading. Here it is demonstrated that the reinforcement ratio is an equally important – or even more important – factor. The story drift capacity is namely drastically reduced with increasing flexural reinforcement ratio.

The brittle punching failure is a major disadvantage of flat plates. A punching failure at one column will result in increased curvature of the slab at surrounding columns, which implies that punching most probably will occur at these columns as well, which may result in progressive collapse of the entire structure. In order to find a reinforcement layout that would give flat plates the same good ductility (and hence safety against progressive collapse) as cast-in-place slabs supported by beams or walls, different types of shear reinforcement were tested in the late 1980’s. The first test series comprised different types of stirrups that were anchored around the top tension reinforcement in agreement with code provisions. Despite the fact that the stirrups covered a large portion of the test specimens and the resulting nominal shear capacity of the specimens exceeded the load corresponding to yield of all flexural reinforcement, brittle failures occurred. These tests, as well as other tests reported in the literature, demonstrate that stirrups and possibly so-called stud rails can hardly be laid out so that a flat plate displays a ductile behaviour similar to slabs supported by beams or walls. It was found that punching failure could occur due to a steep crack around the column leaving such shear reinforcement elements ineffective.

In a second test series, a combination of bent bars and stirrups was tested. The bent bars were introduced to preclude the failure mode with a steep crack at the column. The stirrups were fabricated from welded deformed wire fabric. They enclosed the compression bottom reinforcement of the slab but did not enclose the tension top reinforcement. This reinforcement system turned out to be very effective in giving the slab the desired property – a ductile failure mode without any tendency for punching failure.

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The stirrup design was later improved to rationalize fabrication and erection. The system is denoted “ductility reinforcement” and is patented in USA and Sweden. All reinforcement is placed in a non-interlocking manner, which means that the stirrups enclose neither the bottom nor the top flexural reinforcement in the slab. Test specimens with this reinforcement system behaved in the same ductile manner as the previous specimens with stirrups enclosing the bottom flexural reinforcement.

Finally, two pilot tests simulating a severe earthquake are presented. As could be expected, the tested specimens with ductility reinforcement could resist the story drift during a severe earthquake with good margin despite the fact that the applied gravity loads were 60 % and 75 % respectively of the load corresponding to yield of all flexural reinforcement. No consideration to unbalanced moment was taken when designing the reinforcement.

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Sammanfattning

Denna avhandling är en sammanfattning och vidareutveckling av fyra uppsatser om pelardäck (Papers I-IV) publicerade under åren 1990 till 2005.

Ett bjälklag utan balkar upplagt på pelare benämns ”pelardäck”. Enkel formsättning, planlösningsflexibilitet och låg våningshöjd eftersom inga balkar utgör hinder för installationer ovan undertaket har bidragit till att bjälklagstypen fått stor användning i kontorshus och sjukhus och på senare tid även i bostadshus.

Försök i USA av Elstner och Hognestad (1956) och av Moe (1961) banade vägen för en förenklad typ av pelardäck utan de kraftiga pelarkapitäl som tidigare ansetts fordras för att förhindra skjuvbrott i plattan. Bjälklagstypen kallas i USA ”flat plate” till skillnad från ”flat slab” som är en platta upplagd på pelare med kapitäl eller som har ökad plattjocklek nära pelaren. Nomenklaturen ”flat plate” har därför använts i denna avhandling.

De amerikanska försöken visade att den nya typen av pelardäck visserligen var känslig för en brottyp runt pelaren som liknade ett vanligt skjuvbrott, men att högre nominella skjuvspänningar kunde tillåtas för sådana pelardäck än för plattor upplagda på väggar eller balkar. I Sverige kallas brottypen ”genomstansning” (engelska punching).

Den nya typen av pelardäck introducerades i Sverige i och med att Kinnunen och Nylander (1960) publicerade försöksresultat och en mekanisk modell med empiriskt bestämda betongegenskaper för dimensionering av pelardäck med hänsyn till genomstansning. Kinnunens och Nylanders dimensioneringsregler antogs av dåvarande Statens Betong-kommitté som utfärdade ”Provisoriska bestämmelser för genomstansning”, K1(1964).

Den teoretiska modell som lanserades i Paper I har stora likheter med Kinnunens och Nylanders mekaniska modell från 1960, men utnyttjar i princip endast de materialegenskaper som av hävd används vid dimensionering av betongkonstruktioner, dvs. betongens och armeringens arbetskurvor som ger sambandet mellan töjning och påkänning. Genom-stansning antas ske antingen om ett gränsvärde för betongens tangentiella stukning på grund av böjmoment överskrids intill pelaren eller om betongens tryckhållfasthet överskrids i ett fiktivt koniskt skal i plattan intill pelaren. Den övre gränsen för betongens tangentiella stukning vid pelaren antas motsvara den stukning då mikrosprickor i betongen utvecklas till makrosprickor. Gränsvärdet antas vara storleksberoende och beroende av betongens sprödhet. Plattans tryckzonshöjd används därvid som jämförelseparameter för storleken och sprödheten antas öka med ökad betonghållfasthet. Enkla jämvikts- och kompatibilitets-ekvationer uppställda med gränsvärdet för betongstukningen som enda brottvillkor visade sig kunna förutsäga publicerade försöksresultat med god precision, alltifrån små försöksplattor till fullskaleprov.

Den förenklade och förbättrade modell för genomstansning av centriskt belastade pelare som beskrivs i denna avhandling är utvecklad från ovannämnda modell. Genomstansning antas även här ske antingen om betongstukningen av plattans böjmoment i tangentiell led uppnår ett kritiskt värde eller om tryckhållfastheten överskrids i ett fiktivt pelarkapitäl inne i plattan.

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Vid normala pelardäck blir enligt modellen gränsvärdet för betongstukningen avgörande för bärförmågan med hänsyn till genomstansning. Gränsvärdet antas vara beroende av plattans storlek och betongens ökande sprödhet med ökad hållfasthet enligt formeln

3 1 1 0 cc cpu 00010 25 0150⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = x . f . . ε (a)

där x är plattans tryckzonshöjd uttryckt i (m) och 0.150 är diametern av en standardcylinder för mätning av betongens tryckhållfasthet. Vid plattjockleken 200 mm blir gränsvärdet ca 0.0012, vilket är betydligt lägre än det vedertagna värdet 0.0035 för betongens maximala stukning vid böjmomentbelastning.

Om böjarmeringshalten är hög nås den kritiska betongstukningen innan böjarmeringen flyter i pelardäcket. Elasticitetsteorins momentfördelning antas då gälla i närheten av pelaren och den kritiska pelarlasten Vu kan beräknas direkt utan iterationer:

2 2 u 2 cpu s s 3 1 1 0 cc cpu c s 1 ln 2 8π 3 1 150 0 25 0010 0 1 2 1 c B B c m V d x d m x x d E x . f . n dn x E E n s . − + ⋅ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ ⋅ ⋅ = − ⋅ ⋅ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + = ⋅ = σ ρ ε σ ε ρ ρ ρ ρ (b)

Vid normala armeringshalter uppnår dock armeringen närmast pelaren flytgränsen innan genomstansning sker. Elasticitetsteorins momentfördelning gäller då inte längre när armeringen intill pelaren börjar flyta. Tilläggsmomentet och tilläggsdeformationen när lasten ökas beräknas i stället under antagandet att en flytled utbildas runt pelaren så att sektorelementen mellan plattans radiella sprickor börjar rotera som styva kroppar kring upplaget på pelarperiferin. Den kritiska betongstukningen εcpu sätter därvid även här en

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Ur ekv. (a) och jämviktssamband kan gränsvärdet för betongstukningen om armeringen flyter härledas till 3 0 cc 3 sy c 6 cpu 10 25 2 150 0 10 . f d . f E ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ = − ρ ε (c) där d är plattans effektiva höjd i (m). Tryckzonshöjden xpu blir c E f d x sy cpu pu = ⋅_ε2ρ ⋅ (d)

Gränsvärdet för betongstukningen definierar därmed också maximal krökning av plattan i tangentiell led intill pelaren:

3 0 cc 2 3 2 2 sy 2 c pu cpu u 00010 25 4 150 0 . " f . d . f E x f _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ = = ρ ε (e)

Kritisk pelarreaktion och tillhörande nedböjning erhålls sedan ur enkla jämviktssamband. Om pelardäcket förses med skjuvarmering tål plattan större tangentiell stukning vid pelaren. Det medför att en större andel av böjarmeringen når sträckgränsen innan genomstansning sker, varvid brottlasten ökar. Även i detta fall begränsas bärförmågan av betongstukningen i tangentiell led intill pelaren. Det visas att genomstansningsbrott vid konventionellt utformad skjuvarmering uppkommer då stukningen når gränsvärdet 0.0015. Gränsvärdet antas vara storleksberoende på samma sätt som gränsvärdet för icke skjuvarmerad platta.

Enligt den lanserade teorin sker alltså genomstansning då plattans krökningskapacitet vid pelarupplaget överskrids. Det medför, såsom tidigare påpekats av Kinnunen och Nylander (1960), att krökningen i fält också är begränsad. Om armeringsmängden i fält då är för liten, så att momentjämvikten inte uppfylls, ökar krökningen vid pelaren och genomstansning inträffar. Därför härleds enkla uttryck för kontroll att fältarmeringen i ett pelardäck harmo-nierar med den fordrade stödarmeringen.

En ny modell för kompakta konstruktioner presenteras, där tryckhållfastheten i ett fiktivt pelarkapitäl inne i plattan avgör bärförmågan. Tryckhållfastheten antas variera med kapitälets slankhet uttryckt som kvoten mellan kapitälets omkrets och plattans tryckzonshöjd. Ju större kapitälets omkrets är i förhållande till tryckzonens höjd, desto lägre antas tryckhållfastheten vara, eftersom spänningstillståndet då alltmer övergår från tvådimen-sionellt till plant. Vid mycket stora pelare i förhållande till plattans tryckzonshöjd antas kapitälets tryckhållfasthet vara ⎟

⎠ ⎞ ⎜ ⎝ ⎛ − 250 1 6

0. f_cc fcc . Vid mycket små pelare i förhållande till tryckzonshöjden antas tryckhållfastheten öka till 1.2fcc. Dessutom antas dessa hållfastheter

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De förenklade och förbättrade modellerna visar sig kunna förutsäga försöksresultat för både slanka pelardäck och kompakta pelarsulor med ännu bättre precision än ursprungsmodellen. Inte bara bärförmågan utan även deformationen och den maximala betongstukningen kan beräknas med god noggrannhet.

Slutligen ges regler för hur modellerna skall användas vid dimensionering med hänsyn till genomstansning, eftersom de strikt gäller för beräkning av den verkliga brottlasten. Vid beräkning av dimensionerande bärförmåga vid given armering beräknas därför först brott-lasten med de karakteristiska värdena på betongens tryckhållfasthet och armeringens sträck-gräns som ingångsparametrar. Dimensionerande bärförmåga i brottsträck-gränstillståndet fås därefter genom att dividera beräknad brottlast med partialkoefficienten för betong i säkerhetsklass 3: γ = 1.2·1.5 = 1.8.

Modellernas ekvationer gäller i sin grundform för centriskt belastade innerpelare i ett pelardäck med kvadratiska plattfält. Om plattfälten är rektangulära ökar böjmomentet per breddenhet i den långa spännviddens riktning som funktion av pelarlasten jämfört med ett pelardäck med kvadratiska plattfält. Eftersom betongstukningen (som är avgörande för bärförmågan i ett pelardäck) beror av böjmomentet, bör ett pelardäck med rektangulära plattfält ha lägre genomstansningskapacitet vid given armeringsmängd än ett däck med kvadratiska plattfält. Därför ges regler ges för hur dimensionerande böjmoment bör beräknas i ett pelardäck med varierande spännvidder och/eller rektangulära plattfält. Som en konsekvens av det sagda skall kapaciteten med hänsyn till genomstansning alltid beräknas i vardera riktningen för sig och inte för ett medelvärde av armeringshalten i de båda riktningarna. I detta sammanhang påpekas att teorin i likhet med de flesta norm-metoder ger mer stödarmering inom c-området än vad som krävs för böjmoment beräknade enligt gängse regler.

Nuvarande regelverk – Boverkets handbok för betongkonstruktioner BBK 04 (2004) – ger bärförmågan med hänsyn till genomstansning som en formell skjuvhållfasthet i ett snitt på avståndet d/2 från pelarkanten i enlighet med ett betraktelsesätt som i princip tillämpas över hela världen. Bärförmågan får dock alternativt beräknas enligt (Nylander & Kinnunens) ”mer nyanserade” metod återgiven i Betonghandboken-Konstruktion (1990). Metoden kallas i fortsättningen Betonghandboks-metoden. Den bygger på den ursprungliga mekaniska modellen från 1960, men har genom vissa approximationer förenklats och anpassats till nuvarande sätt att kontrollera en konstruktions bärförmåga i brottgränstillståndet.

Kontroll mot försöksresultat visar att Betonghandboks-metoden inte kan förutsäga genom-stansningslasten bättre än rent empiriska metoder. Till exempel kan den metod som anges i Model Code 90 (1993) förutsäga bärförmågan med bättre precision. I jämförelse med andra dimensioneringsregler – inklusive teorin som beskrivs i denna avhandling – överskattar Betonghandboks-metoden bärförmågan vid armeringshalter större än cirka 0.7 %.

Modellerna i denna avhandling visas ge nära identisk dimensionerande bärförmåga som funktion av armeringshalt, betonghållfasthet och kvoten B/d som Model Code 90. Detta kan ses som en god verifiering av teorins tillförlitlighet, eftersom Model Code 90 bygger på statistisk bearbetning av en stor mängd försöksresultat. Till skillnad från Model Code 90 beaktas även konstruktionens slankhet, vilket har betydelse framför allt för kompakta konstruktioner såsom pelarsulor. Vidare behandlas storlekseffekten (avtagande nominell skjuvhållfasthet med ökad plattjocklek) på ett mer nyanserat sätt. Vid låga armeringshalter,

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där bärförmågan begränsas av att all armering flyter, fås ingen storlekseffekt. Vid höga armeringshalter erhålls en något större storlekseffekt än vad som anges av BBK 04 och Model Code 90.

I verkliga konstruktioner överförs ofta böjmoment från plattan till pelaren vid ojämnt fördelad last på bjälklaget eller om spännvidderna varierar. Överfört böjmoment uppkommer också av vindlast och framför allt av jordbävning, som ger upphov till skillnad i horisontell förskjutning av de olika våningsplanen. De flesta betongnormer ger därför anvisningar om hur genomstansningskapaciteten minskar vid excentrisk pelarreaktion. Normerna ger emellertid i allmänhet ingen anvisning om hur excentriciteten skall beräknas. Momentet är i de flesta fall en statiskt obestämd kvantitet, som starkt beror av plattans styvhet framför allt i närheten av pelaren. Lösningar enligt elasticitetsteorin ger dålig vägledning eftersom armeringen i normalt utformade pelardäck flyter innan genomstansning sker. Då minskar plattans styvhet och pelarmomentet blir lägre än enligt elasticitetsteorin. Därför lanseras en säkrare metod att ta hänsyn till excentrisk pelarlast – möjlig vinkeländring av plattan i förhållande till pelaren. Vinkeländringen kan nämligen beräknas med bättre precision än det överförda böjmomentet oavsett om vinkeländringen orsakas av last på bjälklaget eller av förskjutningskillnad mellan våningsplanen. Metoden förutsätter att den vinkeländring mellan pelare och platta som ger upphov till genomstansning kan förutsägas med god noggrannhet både vid rent elastiskt beteende och när plattans armering flyter. Jämförelse med försöksresultat visar att så är fallet med den lanserade modellen.

I litteraturen redovisas försök där möjlig förskjutningsskillnad mellan våningsplanen vid jordbävning relateras till utnyttjandegraden, dvs. aktuell pelarreaktion i relation till dimen-sionerande bärförmåga med hänsyn till centrisk genomstansning. Här visas att armerings-halten i plattan är en minst lika viktig parameter eftersom rotationskapaciteten drastiskt minskar med ökande böjarmeringsmängd.

Inte ens de mest nyanserade beräkningsmetoder kan emellertid eliminera nackdelen med pelardäck – risken för ett sprött genomstansningsbrott vid överbelastning. Moderna bygg-nadsbestämmelser kräver att konstruktioner skall vara utformade så att risken för for-skridande ras är ringa som följd av en primär skada. ”Skadan” kan till exempel orsakas av en gasexplosion, byggfel eller dimensioneringsfel. I många länder föreskrivs därför att primärbalkar av betong skall förses med skjuvarmering för att garantera ett segt brott-beteende. Motsvarande krav ställs i allmänhet inte på pelardäck, trots att genomstansning vid en pelare med stor sannolikhet leder till genomstansning vid angränsande pelare med risk för fortskridande ras som följd.

I till exempel USA och Kanada rekommenderas därför en armeringsutformning med koncentrerad underkantsarmering från pelare till pelare, som förmodas kunna förhindra fort-skridande ras (eng. progressive collapse) om genomstansning skulle inträffa vid en pelare. Metoden har nackdelen att den inte kan förhindra att genomstansning överhuvudtaget inträffar eftersom systemet inte träder i funktion förrän en kraftig lokal ”sättning” av plattan inträffar vid pelaren. Risken är därför stor att genomstansning sker även vid angränsande pelare, så att en lokal skada kommer att spridas till en stor del av pelardäcket.

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I syfte att hitta en armeringsutformning så att pelardäck får samma sega brottbeteende och därmed samma goda säkerhet mot fortskridande ras som platsgjutna betongplattor upplagda på väggar eller balkar provades olika typer av skjuvarmering i slutet av 80-talet. Försöks-resultaten redovisas i Paper II. I en första försöksserie provades olika former av byglar, som var förankrade runt överkantsarmeringen i överensstämmelse med gällande normer. Trots att byglarna lades in inom en stor yta runt pelaren och trots att den formella skjuvkapaciteten var större än den last som motsvarade flytning i all böjarmering uppkom spröda skjuvbrott. Byglar och så kallade ”studrails” kan sannolikt inte utformas så att ett pelardäck med säkerhet uppvisar ett lika segt brott som en fyrsidigt upplagd betongplatta eftersom försöken visade att denna typ av skjuvarmering inte förmår förhindra genomstansining på grund av en brant spricka intill pelaren.

I en andra försöksomgång provades nedbockad böjarmering i kombination med byglar. Den nedbockade böjarmeringen avsågs förhindra den ovan beskrivna brottypen intill pelaren. Byglarna var utformade som korgar tillverkade av armeringsnät. De omslöt underkants-armeringen men inte överkantsunderkants-armeringen. Denna armeringsutformning visade sig ge den eftersträvade egenskapen – ett segt (duktilt) brottbeteende utan tendens till genomstansning. En förenklad bygelarmering i form av förtillverkade korgar av armeringsnät har därefter utvecklats för att rationalisera tillverkning och montering. Bygelkorgarna omsluter varken överkants- eller underkantsarmeringen och armeringsutformningen ”ductility reinforcement” är patenterad i Sverige och USA. I Paper III redovisas försök med den armerings-utform-ningen som gav provplattorna samma sega brottbeteende som de tidigare provade plattorna med byglar omslutande underkantsarmeringen. Referensplattor med enbart nedbockad böjarmering utan kompletterande byglar uppvisade ett tämligen sprött brott utan nämnvärd förhöjning av lasten i förhållande till plattor utan skjuvarmering.

Pelardäck i flervåningsbyggnader skall dimensioneras i säkerhetsklass 3, eftersom sprött brott kan befaras vid en eventuell överbelastning. Konstruktioner som uppvisar ett segt brottbeteende får dimensioneras i säkerhetsklass 2, vilket normalt innebär en armerings-besparing om ca 10 %. Detta, i kombination med att stödarmeringen över pelarna inte behöver dimensioneras med hänsyn till genomstansning, innebär att det alltid är ekonomiskt fördelaktigt att förse pelardäck med den nya typ av armering som redovisas i denna avhand-ling. En säkrare konstruktion fås till en lägre kostnad än för ett konventionellt utformat pelardäck.

Slutligen redovisas i Paper IV jordbävningssimulering av pelardäck med den patenterade armeringen. Som väntat kunde de provade plattorna klara normkrav för horisontalförskjut-ningar med god marginal trots att de var belastade med vertikallaster motsvarande mellan 60 % och 75 % av den vertikallast som ger flytning i all böjarmering inom c-området. Försöken bekräftade att pelardäck med ”ductility reinforcement” kan motstå även mycket svåra jord-bävningar utan att kollapsa.

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1 Introduction

The reinforced concrete flat plate is a widely used structural system. It has no beams, column capitals, or drop panels, which renders formwork construction very simple. On the other hand, the flatplate is at disadvantage in comparison to two-way slabs supported by beams or walls, because of the risk of brittle punching failure at the slab-column connection. This subject therefore still attracts attention by code writers and researchers.

The punching failure of flat plates resembles the shear failure of beams in the sense that it is characterized by a “shear crack” from the supporting column up to the top surface of the flat plate. Consequently, the majority of researchers and most building codes define the punching capacity in terms of a nominal shear capacity on a control perimeter at a certain distance from the column perimeter. It is thereby acknowledged and accepted that this method does not reflect the true failure mechanism. The method, for instance, does not give the designer any indication of the limited rotation capacity of the slab at punching. Despite this shortcoming, the design provisions have generally resulted in safe structures in the standard cases that are covered by test results.

The challenge is therefore still there to develop a realistic physical model that can predict the slab behaviour at punching in a way simple enough to be used in the design office – also in non-standard cases. Some researchers have attempted to do it, but none has succeeded so far – with one exception. The mechanical model introduced by Kinnunen and Nylander (1960) has gained worldwide recognition, but their model is complicated and cannot predict the punching capacity with the same accuracy as current purely statistical methods. Anyway, a simplified version of their original model is still used in Sweden for punching design of flat plates. This thesis is an attempt to respond to the challenge to fill the vacuum after Kinnunen and

Nylander and expand the treatment to cover more aspects of flat plate design than just

concentric punching.

1.1 Literature survey

Flat plates seem to have first been constructed in USA in the late 1940’s. Elstner and

Hognestad (1956) realized that the new flat plate concept was rather daring because the

design code provisions for the shear capacity were based on tests with thick column footings,

Talbot (1913) and Richart (1948). They therefore tested 39 flat plate specimens with the

dimensions 6 x 6 ft and thickness 6 in. The major variables in the tests were concrete strength (14 MPa to 50 MPa), percentage of tension reinforcement (0.5 to 3.7 percent), percentage of compression reinforcement, size of column (250 mm and 300 mm), distribution of tension reinforcement and amount and position of shear reinforcement. They concluded, “The shearing strength of slabs is a function of concrete strength as well as several other variables”. Neither compression reinforcement nor concentrated tension reinforcement over the column increased the load capacity. They found that shear reinforcement could increase the ultimate load capacity of slabs as much as 30 % but in no case flexural failure rather than shear failure could be achieved. They concluded: “Even though it would be desirable to fully develop the flexural capacity of relatively thin slabs supported on slender columns, to do so with shear reinforcement may be impractical…. Slab thickness, concrete strength, and column dimension

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should therefore probably be so chosen in design, that only a small amount of shear reinforcement, if any at all, is needed in thin slabs.” This opinion seems to have had a great influence on the development during the years to come.

Throughout the tests, 25-mm or 20-mm reinforcement bars were used, which is an extremely large dimension in slabs with 150-mm thickness and 1.8-m span width. One explanation to their finding that concentration of reinforcement over the columns was not advantageous, but rather the opposite, may be due to bond slip of these too large bars in relation to the slab dimensions. They also tested beam strips with the same thickness and span width as the tested slabs. They found that “tests on beam strips representing a narrow slab section and supported as a beam indicated that the use of such concepts as “beam strip analogy” and “equivalent width” does not necessarily lead to a correct prediction of the mode of failure for the corresponding slabs.”

During the years 1957-1959, Johannes Moe visited USA and the Portland Cement Association where he under the guidance of E. Hognestad carried out a large test series on flat plate specimens, which resulted in the report Moe (1961). The test series comprised 43 slabs of the same size as used by Elstner and Hognestad. Principal variables were effect of holes for utilities near the column, effect of concentration of the tensile reinforcement in narrow bands across the column, effect of special types of shear reinforcement, effect of column size, and effect of eccentric loading. One slab was tested under sustained load.

No test report seems to have had larger impact on punching design than Moe (1961). The proposed design provisions for holes in the slab and for eccentric loading are still considered appropriate by many building codes. He introduced the concept of “eccentricity of shear”, where part of the transferred moment between slab and column at eccentric loading is considered transferred by flexural reinforcement in the slab and the rest by uneven distribution of shear forces around the column. Furthermore, Moe's tests confirmed the test result of Elstner and Hognestad that concentration of flexural reinforcement over the column did not increase the punching capacity – again probably due to bond slip of the large reinforcement bars in relation to the slab dimensions.

One year before Moe published his report Kinnunen and Nylander (1960) published their mechanical model for the punching failure of flat plates. As already mentioned, current building codes such as Model Code 90 can predict the punching capacity with better precision, but this does not belittle their contribution to the understanding of the punching phenomenon. They introduced a completely new approach by studying the sector elements between the radial flexural cracks in the test specimens. Punching occurs, according to their model, when the tangential compression strain and the radial inclined compression stress in the slab near the column simultaneously reach critical values. These critical values were calibrated against their own tests and the tests by Elstner and Hognestad (1956).

These three reports laid the foundation for a successful development of flat plate structures all over the world. Later research has been devoted to expand the validity borders for these tests.

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Narasimhan (1971), Ghali et al (1974, 1976), Islam and Park (1976), Pan and Moehle (1989), Hawkins et al (1989) made tests on specimens with much larger column load

eccentricities than those tested by Moe (1961). The tests by Moe may represent the modest eccentricities that will occur due to gravity loading, whereas the other tests were intended to simulate large eccentricities due to story drift during an earthquake. Only Park and Islam

(1976) presented a different design proposal than the “eccentricity of shear” method.

However, their proposed model has not been commonly accepted, presumably in the light of test evidence.

Sundquist (1978) tested the capacity of flat plates for transient loads produced by for instance

bomb blasts and developed a theoretical model for the impulse resistance of flat plates.

Tolf (1988) demonstrated that a considerable size effect exists, which means that the formal

shear stress at punching decreases with increasing specimen size.

Marzouk and Hussein (1991), Tomaszewicz (1993) and Hallgren (1996) made tests on

concentric punching of high strength concrete specimens and Hallgren (1996) also presented an improved version of the Kinnunen and Nylander (1960) mechanical model based on a non-linear fracture mechanics approach.

All research mentioned above was devoted to slender flat plates. More compact structures such as column footings have been studied by Dieterle (1978), Dieterle and Rostasy (1981),

Hallgren, Kinnunen and Nylander (1983, 1998) and Sundquist and Kinnunen (2004).

Finally, Nölting (1984) contains a summary of numerous published test results that was an invaluable source of information to the author for verification of the presented theory during the first development in 1988.

1.2 Scope of work

One aim of this thesis is to develop a realistic physical model for prediction of the punching capacity that is simple enough to be used in design and which covers both concentric and eccentric punching of slender flat plate structures as well as compact structures such as column footings.

Another aim is to present an improved but still easy-to-install reinforcement detailing that eliminates the brittle punching failure mode of flat plates. In this way the basic integrity requirement for a structure will be fulfilled, i.e. a structure shall be designed so that a local failure due to overloading shall not result in progressive collapse of the building. This seems to be overlooked as regards flat plates by some code writers and many designers.

The issues have been treated in the following papers that form part of this thesis:

Paper I: Broms, C.E. (1990a), “Punching of Flat Plates – A Question of Concrete Properties in Biaxial Compression and Size Effect”, ACI Structural Journal, V. 87, No. 3, pp. 292-304.

Paper II: Broms, C.E. (1990b), “Shear Reinforcement for Deflection Ductility of Flat Plates”, ACI Structural Journal, V. 87, No. 6, pp. 696-705.

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Paper III: Broms, C.E. (2000), “Elimination of Flat Plate Punching Failure Mode”,

ACI Structural Journal, V. 97, No. 1, pp. 94-101.

Paper IV: Broms, C.E. (2005), “Ductility Reinforcement for Flat Slabs in Seismic as well as Non-seismic Areas”, submitted to Magazine of Concrete Research for possible publication.

A theory for concentric punching, inspired by the Kinnunen and Nylander (1960) mechanical model, is presented in Chapter 2. The theory is an improved and simplified version of the theory presented in Paper I and is expanded to cover compact structures such as column footings and is validated by comparison with published test results in the literature. The punching load as well as the accompanying slab deflection and the flexural compression strain can be predicted with good precision.

A completely new theory for eccentric punching is presented in Chapter 3. The relation between unbalanced moment and the corresponding rotation of the column are derived from the relation between load and deflection at concentric punching, which means that the slab rotation in relation to the column is proposed to be the design criterion instead of the current force-based unbalanced moment approach.

Chapter 4 contains a recommended procedure for design with respect to punching in the general case with varying span widths and rectangular slab panels. Comparison of the presented theory is made with the design provisions of existing structural design codes.

The ductility reinforcement concept presented in Papers II and III is summarized in Chapter 5. Finally, in Chapter 6 some comments are added to the earthquake simulation presented in Paper IV.

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2 Theory for concentric punching

The basic principles are described in Paper I, but the theory is here improved and simplified. Punching is assumed to occur either when the concrete strain in the slab due to the bending moment or the inclined compression stress due to the column reaction reaches a critical level.

2.1 General

The reinforcement is assumed to be ideally elastic-plastic with the yield strain

s sy sy E f = ε (2.1)

The modulus of elasticity for reinforcement bars is taken as Es = 200 GPa.

As will be shown in the following, the concrete strain due to the bending moment is so low at punching that the concrete usually behaves elastically:

c c

c ε

σ = E ⋅ (2.2)

The tangent modulus of elasticity Ec0 for concrete at zero strain is taken as the value given in

Model Code 90 (1993): 3 1 cc c0 10 21500 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ = f

E (MPa) with fcc in MPa (2.3)

The concrete secant modulus of elasticity, Ec10, to the strain 0.0010 is defined later in this

chapter; Eq. (2.10).

As long as the reinforcement does not yield, the compression zone depth at flexure is computed by combining the strain compatibility and force equilibrium conditions; see

Figure 2-1. d x c

ε

s

ε

c

σ

F c s F m

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x d x = − s c ε ε (strain compatibility) (2.4) 2 c c10 s s E x dE ⋅ε = ⋅ε ⋅ ρ (force equilibrium) (2.5) n E E = c10 s _(2.6)

Combine Eqs. (2.4), (2.5), and (2.6):

x x d d n x₌ _ρ − 2 0 2 2 2 2 = − + ρ nρ d x n d x ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + = + + − = 2 2 2 1 2 1 ρ ρ ρ ρ ρ n n n n n d x _(2.7)

The bending moment per unit width of a slab, m , is computed by the expression

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = d x d m 3 1 2 s ρσ (2.8)

Extensive flexural cracking will always occur near the column at ultimate loading. The flexural stiffness EI per unit width is therefore computed for a cracked section without any tension stiffening: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ′′ = d x d x d E x d d x d f m EI 3 1 1 3 1 _s 3 s 2 s _ε ρ ρσ (2.9)

where f ′′ is the curvature of the slab due to the bending moment m.

In a flat plate, inclined cracks near the column usually form at a load level of less than 70 % of the ultimate load. Although these cracks can surround the column, the slab is nevertheless stable and can be unloaded and reloaded without any decrease of the ultimate load, Regan and

Braestrup (1985). It is therefore evident that the punching failure mechanism is usually not a

pure “shear failure” governed by the diagonal tensile strength of the concrete.

The punching failure occurs instead when the compression zone with height x adjacent to the column collapses. The model depicted in Figure 2-2 may simulate this zone, where the load from the flat plate is transferred to the column via a column capital within the slab, similar to the conical shell originally proposed by Kinnunen and Nylander (1960).

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V

x internal column capital

Figure 2-2 Transfer of load V to column from the flat plate.

The punching failure is assumed to occur either when the capital collapses when its capacity in compression is reached or when micro cracking at a critical tangential flexural strain softens the concrete at the column edge. The corresponding punching capacities are denoted

Vσ and Vε respectively. These failure modes are analyzed in detail in the following.

2.2 Punching capacity V

ε

Failure occurs when the tangential compression strain in the slab at the column edge reaches a critical value.

2.2.1 Basic assumption

The failure mode is illustrated in Figure 2-3. In contrast to one-way structures, the bending moment capacity in a flat plate can be maintained even if the radial flexural compression stress at the support approaches zero, which is a prerequisite for the following possible scenario.

The support reaction is concentrated to the edge of the column due to the global curvature of the slab. At loads near the ultimate capacity, the compression strain due to the column reaction – in the column as well as the slab – will therefore always exceed the strain corresponding to the peak stress fcc. Then, when the flexural tangential strain in the bottom of

the slab reaches a critical value, the concrete starts to loose its internal bond and an almost vertical “shear crack” opens up at the column/slab interface due to the combined action of the vertical column reaction and the tangential slab strain both of which tend to create a vertical crack in the slab. Once this happens, the column capital will collapse due to a “zip” effect because the inclined compression strut rapidly becomes too weak to resist the support reaction when it is forced to take a flatter load path. The crack propagation is thereby facilitated because the concrete already experiences tension strain in perpendicular direction to the final punching crack due to the shear deformation of the compression zone. This shear deformation is also the reason why the radial flexural strain in the bottom of the slab some distance away from the column ceases to increase with increasing load once inclined circumferential cracks develop around the column.

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Many researchers – as for instance Kinnunen and Nylander (1960) and Hallgren (1996) – report that the radial compression strain near the bottom surface of the slab close to the column suddenly decreases to zero at a load level just below the ultimate punching load. This seems to confirm the scenario described above, that the failure is usually triggered by the formation of a circumferential crack at the slab/column interface and not by propagation of an inclined flexural crack.

x

f_{c c}

Figure 2-3 Failure mode V_ε

.

These general observations lead to the conclusion that the conditions of the concrete at the column edge are decisive for the punching failure capacity Vε, which forms the basis for the

following hypothesis.

Study the stress-strain diagram for concrete with the compression strength 25 MPa according to Figure 2-4. The stress-strain relation is taken from High performance concrete structures

(1998):

(

)

c c1 1 2 1 1 cc c1 c 1 c1 cc c0 0.3 cc c1 for 2 1 0007 0 ε ε η η η σ ε ε η ε ε ≤ − + − ⋅ = = ⋅ = ⋅ = k k f ; f E k ; f . c

At a strain exceeding approximately 0.0010 it is evident that the almost linearly elastic behaviour of the concrete at low strains starts to change – the concrete “softens”. Punching failure of a flat plate is therefore assumed to occur when the tangential concrete strain due to

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the bending moment reaches this critical value adjacent to the column. It is further assumed that this critical strain level decreases with increasing concrete strength because high strength concretes are more brittle.

c

f

MPa c 0 5 10 15 20 25 30

ε

0.0000 0.0 0 1 0 0.0020 3 1 ck c0 10 21500 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = f E E 10

Figure 2-4 Assumed stress-strain curve for concrete strength f_cc =25 MPa.

In the subsequent equations, it is important to estimate the stress-strain relation in the compression zone at flexure correctly. At low concrete grades there is a curved relation between strain and stress already at strains below 0.0010 as indicated in Figure 2-4. The concrete behaves more linearly elastic with increasing concrete grades, which is approximately taken into account by putting the secant modulus Ec10 to the strain 0.0010

equal to c0 4 ck c10 150 1 6 0 1 . f E E _⎟⎟⋅ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = (MPa) (2.10)

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2.2.2 Size effect

The size effect – in this case the decreasing ultimate material strain with increasing structural size – and the varying concrete brittleness are taken into account by the formula

3 1 pu 1 0 cc cpu 00010 25 015_⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = x . f . . ε (2.11)

where εcpu = tangential compression strain at punching

0.15 = diameter of standard test cylinder specimen (m)

xpu = depth of compression zone at flexure when punching occurs (m).

The size effect factor 3

1 pu 15 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ x .

is assumed to affect both strain and stress of the concrete in the same manner. This means that the E-modulus is assumed to be a concrete property that displays no size effect, i.e. it has the same value irrespective of specimen size.

Hillerborg et al (1976) developed the Fictitious Crack Model to explain the size effect for

brittle failures in concrete structures caused by tensile strains. Gustafsson and Hillerborg

(1988) used this model to derive that the shear strength of beams without shear reinforcement

displays a size effect that can be approximated by

25 . 0 ch ct v − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⋅ =k f d_l f (2.12)

with the characteristic length ₂

ct F c0 ch f G E l = ⋅ (2.13)

In the absence of experimental data Model Code 90 recommends the following relations for Ec0, fct and GF : 3 1 cc c0 10 21500 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = f E [MPa] ( = Eq. (2.3)) (2.14) 3 2 ck ct 10 4 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = . f f [MPa] (2.15) 7 0 cc F0 F 10 . f G G ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ = [MPa·mm] (2.16) where ⎪ ⎩ ⎪ ⎨ ⎧ = 038 . 0 030 . 0 025 . 0 F0

G [MPa·mm] for maximum aggregate size ⎪ ⎩ ⎪ ⎨ ⎧ = 32 16 8 a d [mm].

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Insert Eqs. (2.14) to (2.16) into Eq. (2.13) and replace the characteristic value of the compression strength by the recorded value fcc in Eq. (2.15)

3 0 cc F0 3 0 1.33 cc 2 7 0 cc F0 33 0 cc ch 10 10970 10 4 1 21500 . . . . f G f . f G f l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ = ⋅ ⋅ ⋅ ⋅ = [mm] (2.17)

Eq. (2.12) can now be rearranged as

075 0 cc 25 0 F0 ct v 10 . . f G d f k f − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ = (2.18)

that can be used to study the effect of maximum aggregate size. If the beam depth were increased four times without simultaneous scaling of the aggregate size, the formal shear strength fv would be reduced to 4-0.25 = 70.7 % of the strength for the smaller beam.

Simultaneous four-fold scaling of the maximum aggregate size from 8 mm to 32 mm would not eliminate the size effect as maintained by some researchers such as Bažant and Cao

(1987). In this case, the formal shear strength would be reduced to 78.5 % of the strength for

the smaller beam.

According to Eq. (2.12) it is thus evident that the maximum aggregate size has limited effect on the formal shear strength of beams. A doubling of the aggregate size from 16 mm to 32 mm would increase the recorded shear strength by 6.0 % and a reduction from 16 mm to 8 mm would decrease the strength by 4.5 %. It is also evident that an increased concrete compression strength fck has some reduction effect on the formal shear strength versus the

tensile strength fct.

Leonhardt and Walther (1962) made tests on the shear strength size dependence for beams

without shear reinforcement. The shear strength varied approximately proportional to d−0.33 when the reinforcement bars were scaled in proportion to the beam geometry. In a second test series, where a small reinforcement size was kept constant and the number of bars was varied to keep the reinforcement ratio constant when the beam size was increased, the beams displayed no size effect. In this latter case, the better anchoring bond with many small bars instead of few large bars decreased the anchoring slip sufficiently to eliminate the size effect. This demonstrates that tests have to be performed with realistically scaled reinforcement bars whenever reinforcement bond might be of concern for the structural behaviour. Based on the test results, Leonhardt and Walther drew the premature conclusion that the size effect for shear failures would fade out for beams with effective depth larger than round 400 mm because the reinforcement bar size is limited to 25 mm or 32 mm.

The question is; are the findings by Gustafsson and Hillerborg (1988) regarding shear strength of beams applicable also for the punching strength of flat plates despite the fact that the punching failure seems to be more brittle?

Hallgren (1996) used the Fictitious Crack Model to derive an expression for the critical

tangential concrete compression strain at punching. He found it to be proportional to the depth of the compression zone at flexure raised to the power -0.5 for very small depths. The exponent decreases to -1.0 for large compression zone depths. These values seem to be unrealistic – the size effect becomes too large.

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At very brittle failures characterized by a linear stress distribution, the size effect would be described by the Linear Elastic Fracture Mechanics equation for the failure strength

5 . 0 0 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ = d d k f (2.19)

where d is the actual size of the structure and d0 is a reference size.

Most concrete structures display a non-linear stress distribution for brittle fractures, which means that the absolute value of the exponent in the fracture strength equation should be smaller than 0.5 – as in Eq. (2.12) for the shear strength of beams. Theoretically, the more non-linear stress distribution a structure displays, the smaller the absolute value of the exponent becomes – down to zero at a plastic stress distribution (= no size effect). However, Eq. (2.12) with the constant exponent -0.25 is found to be valid for a large range of beam sizes, from small specimens up to beams with effective depth of at least 1000 mm.

The fracture energy GF determined by the RILEM (1985) beam test and the deduced

characteristic length lch according to Eq. (2.13) characterize the relative brittleness of the

concrete at tensile strains. However, they do not give any indication on the exponent to be used in a fracture strength equation. Only experiments with varying specimen size will give a reliable answer.

The punching fracture mode seems to be more brittle than the shear failure mode of beams, because the fracture at punching occurs due to a small shear displacement at high biaxial compression strain, whereas the beam shear failure is usually associated with inclined crack growth due to tensile strains. The absolute value of the exponent for punching should then be larger than the beam-shear exponent 0.25. The chosen exponent 1/3 in Eq. (2.11) therefore seems to be reasonable and can be assumed valid at least for slab sizes covered by the validation of the theory in Section 2.5, i.e. slabs with effective depth varying from 100 mm up to 600 mm. The upper limit 600 mm can most probably be increased because the presented theory presupposes elastic behaviour of the concrete in flexure, which is more realistic the larger the structure becomes. However, thick slabs may display a more pronounced apparent size effect due to possible induced cracks in the compression zone by uneven temperature over the slab depth during the concrete hydration.

The choice of the compression zone depth as reference dimension for the size effect in Eq. (2.11) is a natural consequence of the hypothesis that punching occurs when the compression zone near the column collapses. It is interesting to note that the format of Eq. (2.11) for the punching failure can be derived from the same assumption as Eq. (2.12) for the beam shear failure, i.e. the size effect depends on the relation between a reference size of the structure and lch according to Eq. (2.17):

1 0 cc 3 1 pu 3 1 ch pu cpu 0.0010 0.0010 A 25 . f x l x ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ = − ε (2.11a)

where A is a reference size that should be proportional to the maximum aggregate size factor

GF0 according to Eq. (2.16). However, the reference size in Eq. (2.11) is chosen to be