Cover figure

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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

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

i

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

ii

iii

Table of contents Preface ………………………………………………………………………………………...i

Table of contents......................................................................................................................iii

Notations……………………………………………………………………………………....v

Summary...................................................................................................................................ix

Sammanfattning (Summary in Swedish)…………………………………………….........xiii

1 Introduction ......................................................................................................................1

1.1 Literature survey.........................................................................................................1

1.2 Scope of work.............................................................................................................3

2 Theory for concentric punching......................................................................................5

2.1 General .......................................................................................................................5

2.2 Punching capacity Vε ..................................................................................................7

2.2.1 Basic assumption ................................................................................................7

2.2.2 Size effect .........................................................................................................10

2.2.3 Punching at elastic conditions ..........................................................................13

2.2.4 Yield punching .................................................................................................16

2.2.5 Flat plates with shear reinforcement.................................................................19

2.3 Punching capacity Vσ................................................................................................23

2.3.1 Column footings ...............................................................................................23

2.3.2 Flat plates..........................................................................................................29

2.4 Manual calculation ...................................................................................................29

2.4.1 General .............................................................................................................29

2.4.2 Reinforcement limit ρ 1.....................................................................................30

2.4.3 Reinforcement limit ρ 2.....................................................................................31

2.4.4 Transition zone between ρ 1 and ρ 2..................................................................32

2.4.5 Tabulated values for ρ 1 and ρ 2 ........................................................................33

2.5 Comparison with test results ....................................................................................34

2.5.1 Influence of bending moment...........................................................................34

2.5.2 Influence of concrete mechanical properties ....................................................35

2.5.3 Comparison with test results for flat plates and column footings ....................36

iv

2.5.4 Code predictions .............................................................................................. 43

3 Theory for eccentric punching...................................................................................... 47

3.1 Code approach.......................................................................................................... 47

3.2 Introduction.............................................................................................................. 47

3.3 Approximate theory of elasticity.............................................................................. 48

3.4 Model for eccentric punching of flat plates ............................................................. 51

3.5 Comparison with test results .................................................................................... 58

3.6 Column rotation capacity ......................................................................................... 62

4 Design .............................................................................................................................. 65

4.1 Design of support reinforcement at square panels ................................................... 65

4.2 Bending moments in a continuous flat plate............................................................ 65

4.3 Design of midspan reinforcement ............................................................................ 69

4.4 Comparison with Codes ........................................................................................... 71

4.4.1 Swedish Code for Concrete Structures, BBK 04 ............................................. 71

4.4.2 Swedish Handbook for Concrete Structures .................................................... 72

4.4.3 Model Code 1990, MC 90................................................................................ 72

4.4.4 Building Code Requirements for Structural Concrete, ACI 318-02................ 73

4.4.5 Code comparison.............................................................................................. 73

5 Reinforcement for ductility ........................................................................................... 79

6 Earthquake simulation .................................................................................................. 83

7 Conclusions and summary ............................................................................................ 85

8 References ....................................................................................................................... 89

Appendix A. Punching of flat plate. (No yield punching)….......……………..………..…95

Appendix B. Punching of flat plate. (Yield punching) …..……………… …………….98

Appendix C. Flat plate with shear reinforcement. ………….………………………….. 101

Appendix D. Punching of column footing, surface load. ………………………………..104

Appendix E. Punching of column footing, line load. …………………………………...107

Appendix F. Unbalanced moment loading. …………………………….………………..109

Appended Papers I - IV

Notations

v

Roman upper case letters

B diameter of circular column

Bε diameter of circular column with the same reduction effect on the total bending moment as a square column with width a; Bε = 3πa/8

Bσ diameter of circular column with the same perimeter as a square column with width a; Bσ = 4a/π

D diameter of circular column footing

Ec0 tangent modulus of elasticity of concrete at zero strain

Ec10 secant E-modulus of concrete up to the strain 0.0010

Ec15 secant E-modulus of concrete up to the strain 0.0015 (with shear reinforcement)

Es modulus of elasticity of reinforcing steel

EI flexural stiffness of slab per unit width

EI1 reduced flexural stiffness of slab near the column for unbalanced moment loading

F force

G fracture energy

H horizontal force

L span width, measured centre-to-centre of supports

L1 span width in direction that moments are being determined

L2 span width transverse to L1

Mu unbalanced bending moment

Pσ column load on footing

R radius to centre of gravity for uniformly distributed load outside shear crack

R0 maximum value of sector element reaction due to unbalanced moment

Rb sector element reaction corresponding to tension in bottom reinforcement

Rt sector element reaction corresponding to tension in top reinforcement

V column reaction

V1 column reaction at reinforcement ratio ρ 1

V2 column reaction at reinforcement ratio ρ 2

Vε concentric punching capacity at tangential compression strain failure mode

Vεs upper bound capacity with shear reinforcement

Vσ concentric punching capacity at inclined compression stress failure mode

vi

Vσs upper bound capacity with shear reinforcement

Vy1 column reaction when the reinforcement at the column edge starts to yield

Vy2 column reaction when the reinforcement in tangential direction at the distance c/2 from the column starts to yield

Vu the lesser of Vε and Vσ

Roman lower case letters

a width of square column

b width of square footing

c diameter of circle around the column where the radial bending moment is zero

c0 diameter at reinforcement level of circular punching crack around column

d effective depth

e load eccentricity

f ´´ slab curvature in tangential direction ( = m/EI )

fu´´ slab curvature near column edge at punching

fus´´ slab curvature near column edge at punching with shear reinforcement

fy´´ slab curvature at start of reinforcement yield

fys´´ slab curvature at start of reinforcement yield with shear reinforcement

fcc compressive strength of concrete, measured on standard cylinders with diameter

150 mm and length 300 mm (recorded mean value)

fck characteristic value for compressive strength of concrete

fct tensile strength of concrete (recorded mean value)

fctk characteristic value for tensile strength of concrete

fsy yield strength of reinforcing steel

fv1 one-way shear capacity

fv2 two-way shear capacity

h slab thickness

kI factor for reduced slab stiffness near column due to unbalanced moment, kI =21

1 ⎟⎠⎞

⎜⎝⎛

EIEI

vii

lch characteristic length = 2

ct

Fc

fGE ⋅

m bending moment per unit width

m1 bending moment in tangential direction at column edge

m2 bending moment in tangential direction at the distance c/2 from the column

mr bending moment in radial direction

ms negative strip moment

msc negative bending moment within column strip

msm negative bending moment within middle strip

mt bending moment in tangential direction

my bending moment at reinforcement yield

mys bending moment at reinforcement yield with shear reinforcement

mε bending moment in tangential direction at punching

mεs bending moment in tangential direction at punching with shear reinforcement

n = Es /Ec10

n0 = Es /Ec0

ns = Es /Ec15

r radial distance from column centre

ry radius of circle inside which the reinforcement yields

t depth of inclined compression strut

u effective perimeter of internal column capital

w effective width of strip in a flat plate

x depth of slab compression zone

xs compression zone depth with shear reinforcement

xpu compression zone depth at punching

xpus compression zone depth with shear reinforcement at punching

Greek upper case letters

∆ fictitious deflection of test specimen due to unbalanced moment

viii

∆r radial compression of slab by the horizontal strut due to unbalanced moment

Greek lower case letters

α factor in expression for compression zone force

γ inclination angle for radial compression strut

γm strength reduction factor for material

γn strength reduction factor with respect to safety class (Swedish design method)

δε specimen deflection at punching

δV specimen deflection at column load V

δy1 specimen deflection at start of yield at column edge

δy2 specimen deflection at start of overall yield

εc concrete strain

εcpu concrete strain in tangential direction near the column at punching failure

εcpus concrete strain near the column at punching failure with shear reinforcement

εs strain of reinforcing steel

εsy strain of reinforcing steel at start of yield

θ slab rotation in relation to column (or vice versa) at imposed unbalanced moment

θu rotation capacity of slab in relation to column at imposed unbalanced moment

ξ size-effect factor

ρ reinforcement ratio (= top reinforcement within column strip)

ρ c compression reinforcement ratio (= bottom reinforcement within column strip)

ρ 1 reinforcement ratio above which punching occurs with no reinforcement yielding

ρ 2 reinforcement ratio below which all reinforcement yields at punching

σ c compression strength of internal column capital

σs reinforcement stress

φ average inclination of shear crack at compact slabs or footings

ϕ angle in plane of slab

ψ slab inclination in radial direction at the distance c/2 from the column

ψpu slab inclination at punching

ix

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.

x

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.

xi

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.

xii

xiii

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.

xiv

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

3110

cccpu

15002500100 ⎟⎠⎞

⎜⎝⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

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

2u

2

cpuss

3110

cccpu

c

s

1ln2

8π3

1

15002500100

121

cB

Bc

mV

dxdm

xxdE

x.

f.

ndnx

EE

n

s

.

−+

⋅=

⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅=

−⋅⋅=

⎟⎠⎞

⎜⎝⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

⋅=

σρ

εσ

ε

ρρ

ρρ

(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 gräns för möjlig tillskottsdeformation.

xv

Ur ekv. (a) och jämviktssamband kan gränsvärdet för betongstukningen om armeringen flyter härledas till

30

cc

3

sy

c6cpu

25102150010

.

fd.

fE

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅⋅= −

ρε (c)

där d är plattans effektiva höjd i (m).

Tryckzonshöjden xpu blir

cE

fdx sy

cpupu

2⋅⋅=

ερ (d)

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

30

cc2

3

22sy

2c

pu

cpuu

250010041500

."

f.

d.

fE

xf ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅⋅==

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

160 cccc

ff. . Vid mycket små pelare i förhållande till

tryckzonshöjden antas tryckhållfastheten öka till 1.2fcc. Dessutom antas dessa hållfastheter vara storleksberoende på motsvarande sätt som stukningen enligt ekv. (a).

xvi

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 brottgränstillståndet 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,

xvii

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.

xviii

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 överkantsarmeringen. 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.

1

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 flat plate 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

2

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.

3

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.

4

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.

5

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

sysy 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:

ccc εσ ⋅= 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):

31

ccc0 10

21500 ⎟⎠⎞

⎜⎝⎛⋅=

fE (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

cε

sε

cσ

F

c

s

F

m

Figure 2-1 Depth x of compression zone.

6

xdx −= sc εε

(strain compatibility) (2.4)

2cc10ssxEdE ⋅⋅=⋅ εερ (force equilibrium) (2.5)

nEE

=c10

s (2.6)

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

xxddnx −

= ρ2

0222

2=−+ ρρ n

dxn

dx

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=++−= 121222

ρρρρρ

nnnnn

dx (2.7)

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

⎟⎠⎞

⎜⎝⎛ −=

dxdm

312

sρσ (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:

⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ −=

−⎟⎠⎞

⎜⎝⎛ −=

′′=

dx

dxdExd

dxd

fmEI

311

31 3

ss

2s ρ

ερσ (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).

7

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 Vε

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.

8

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

fc c

Figure 2-3 Failure mode Vε .

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):

( ) c1c1

211

cc

c1

c1

c1cc

c0

0.3ccc1

for21

00070

εεη

ηησ

εε

η

ε

ε

≤−+−

⋅=

=

⋅=

⋅=

kk

f

;

fE

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

9

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.

cfMPa

c

0

5

10

15

20

25

30

ε

0.00

00

0.00

10

0.00

20

31

ckc0 10

21500 ⎟⎠⎞

⎜⎝⎛=

fE

E 10

Figure 2-4 Assumed stress-strain curve for concrete strength fcc =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

4ck

c10 1501601 E

f.E ⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−= (MPa) (2.10)

with Ec0 according to Eq. (2.3); see Figure 2-9.

10

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

31

pu

10

cccpu

1502500100 ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

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 31

pu

150⎟⎟⎠

⎞⎜⎜⎝

⎛

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

chctv

−

⎟⎠⎞⎜

⎝⎛⋅⋅= l

dfkf (2.12)

with the characteristic length 2ct

Fc0ch

fGE

l⋅

= (2.13)

In the absence of experimental data Model Code 90 recommends the following relations for Ec0, fct and GF :

31

ccc0 10

21500 ⎟⎠⎞

⎜⎝⎛=

fE [MPa] ( = Eq. (2.3)) (2.14)

32

ckct 10

41 ⎟⎠⎞

⎜⎝⎛=

f.f [MPa] (2.15)

70

ccF0F 10

.fGG ⎟

⎠⎞

⎜⎝⎛⋅= [MPa·mm] (2.16)

where ⎪⎩

⎪⎨

⎧=

038.0030.0025.0

F0G [MPa·mm] for maximum aggregate size ⎪⎩

⎪⎨

⎧=

32168

ad [mm].

11

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)

30cc

F0301.33cc

2

70ccF0

330cc

ch

10

109701041

21500.

...

f

Gf.

fGfl

⎟⎠⎞

⎜⎝⎛

⋅=⋅⋅

⋅⋅= [mm] (2.17)

Eq. (2.12) can now be rearranged as

0750

cc250

F0ctv 10

..f

Gdfkf

−−

⎟⎠

⎞⎜⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅= (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 33.0−d 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.

12

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

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=

ddkf (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):

10

cc

31

pu

31

ch

pucpu

25A0.00100.0010.

fxlx

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

−

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 constant, independent of the maximum aggregate size, because the aggregate size is seldom

13

reported in the literature. Furthermore, the resulting effect on the critical strain value would anyway be rather marginal.

2.2.3 Punching at elastic conditions

If punching occurs without any yield of the reinforcement (at high reinforcement ratios), then both reinforcement and concrete behave elastically. The depth of the compression zone is then defined by Eq. (2.7) and the critical strain εcpu is defined by Eq. (2.11).

Once the critical strain εcpu is defined, then the bending moment per unit width is defined and the punching load can be estimated if the relation between load and bending moment at the column is known, which will be described hereafter.

A common arrangement for punching testing of flat plates consists of a circular or square slab loaded along its perimeter and supported on a column at its centre; see Figure 2-5. The perimeter of the specimen is intended to reflect the circular line of contra-flexure for bending moment in radial direction in a continuous flat plate. According to the theory of elasticity, this circle has the radius ≈ 0.2 L in a flat plate with square panels, where L is the span width. The following equations assume either a circular or a square specimen arrangement. In the latter case, the diameter of the equivalent circular slab is assumed equal to the width of the square slab if the corners are free to lift in the square specimen.

Up to the load level when the flexural reinforcement starts to yield near the column, the theory of elasticity is assumed valid for the bending moment distribution. Poisson's ratio of the cracked concrete slab is thereby assumed zero.

The column reaction is concentrated to the column perimeter as has been described above. A square column is replaced by a fictitious circular column with the same reduction effect on the total bending moment across the specimen width:

Circular column: π2

∆ BVM = (2.20)

Square column: 163aVM =∆ (2.21)

aB8π3

=∵ (2.22)

where B = diameter of circular column and a = width of square column.

14

c

B

cVπ

BV

π

ψ

δ

1

2

m

m

t

r

r

Figure 2-5 Bending moments and slab deformation according to the theory of elasticity for a circular slab supported on the edge of a circular column.

The following expressions are valid according to the theory of elasticity (with ν = 0) for a circular slab, derived from Eqs. (84) and (85) of Timoshenko and Woinowsky-Krieger (1959); see Figure 2-5:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−+= 2

2

2

2

t4

22

ln2π8 c

Br

Br

cVm tangential moment (2.23)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+= 2

2

2

2

r42

ln2π8 c

Br

Br

cVm radial moment (2.24)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+= 2

2

1 1ln2π8 c

BBcVm tangential and radial moment at column edge (2.25)

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

2

2 1π4 c

BVm tangential moment at the slab edge (2.26)

EIc

cBV

EIcm

21

π42 2

2

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛−==ψ angle of inclination at the slab edge (2.27)

The theoretical deflection δ consists of bending deformation and shear deformation, where the latter is not negligible near the column. When these effects are superimposed the resulting deformation configuration resembles a truncated cone and the deflection at the column edge is consequently assessed as

15

2Bc −

≈ψδ (2.28)

The punching capacity Vε can now be estimated as follows.

The relation between the modulus of elasticity for reinforcement and concrete

c10

sEE

n = (2.6)

Depth of the compression zone in the slab at elastic behaviour

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅= 121

ρρ

nndx (2.7)

Compression strain at the column edge at punching

31

pu

10

cccpu

15002500100 ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

x.

f.

.

ε (2.11)

Reinforcement stress at the column edge at punching

xxdE −

⋅⋅= cpuss εσ (2.29)

If sσ turns out to be larger than the yield stress fsy , then the reinforcement yields before punching occurs and the calculation is performed according to Section 2.2.4.

Bending moment at the column edge at punching

⎟⎠⎞

⎜⎝⎛ −⋅=

dxdm

31s

2ε σρ (2.30)

Finally, the punching load Vε is derived from Eq. (2.25)

2

2εε

1ln2

8π

cB

Bc

mV−+

= (2.31)

No iteration is thus required for determination of the punching load. The calculation is anyhow preferably computerized, see Appendix A, because then the alternative failure mode Vσ is checked automatically as well as the ultimate deflection δ ε.

In flat plates with rectangular panels the above equations have to be modified when checking the punching capacity, see Section 4.2.

16

2.2.4 Yield punching

With medium reinforcement ratios the reinforcement near the column will yield before punching occurs. The bending moments according to the theory of elasticity are then no longer valid for the part of the load that exceeds the load Vy1 when the reinforcement at the column edge just starts to yield. It is instead assumed that a flexural hinge forms at the column edge and the sector elements of the slab between the radial flexural cracks start to rotate as rigid bodies with support on the column edge. Punching is still assumed to occur when the concrete compression strain reaches the critical value εcpu.

The punching strain εcpu at the column edge when the reinforcement yields can then be calculated from

50pucpuc10sy .xEfd ⋅⋅⋅= ερ (force equilibrium) (2.32)

31

pu

10

cccpu

15002500100 ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

x.

f.

.

ε (2.11)

Combine Eqs. (2.32) and (2.11):

30

cc

3

sy

c106cpu

25102150010

.

fd.

fE

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅⋅= −

ρε with effective depth d in (m) (2.33)

c10

sy

cpupu

2Ef

dx ⋅⋅=ε

ρ (2.34)

The ultimate tangential curvature at the column edge, uf ′′ , can then be expressed as

30

cc2

3

22sy

2c10

pu

cpuu

250010041500

.

f.

d.

fE

xf ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅⋅==′′

ρ

ε (2.35)

Bending moment at reinforcement yield

⎟⎠⎞

⎜⎝⎛ −⋅⋅=

dxfdm

31sy

2y ρ (2.36)

The column reactions Vy1 when the reinforcement starts to yield at the column edge and Vy2 when all reinforcement across the slab yields

2

2yy1

1ln2

π8

cB

Bc

mV−+

⋅= (2.37)

cB

mV−

⋅=1

π2yy2 (see Figure 2-6) (2.38)

17

Bc

m r = 0

⎟⎠⎞

⎜⎝⎛ −=

cBV

m 1π2y2

ymy

Figure 2-6 Fan type yield lines.

Curvature in tangential direction at start of yield

xdE

fxdEI

mf

−⋅=

−==′′ 1

s

sysyyy

ε (2.39)

Possible additional curvature at column edge after first yield

yu∆ fff ′′−′′=′′ (2.40)

The circle with radius ry inside which the reinforcement yields is solved from the following equation; see Figure 2-7:

y

2

2

2y

2

y

1y 2

∆4

22

ln2π8 r

BEIfcB

rB

rcV

m y ⋅⋅′′+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−+= (2.41)

The load capacity is equal to the flexural load capacity Vy2 if 2ycr ≥ .

The punching load Vε is calculated by integration of the tangential bending moment curve

over the slab width if 2ycr < ; see Figure 2-7:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅′′+⎟

⎟⎠

⎞⎜⎜⎝

⎛−−++⋅

⋅= ∫

2

2

2

2

2y1

yy

y

y2ε

y

d2

∆4

22

ln2π8

2

c

r

rr

BEIfcB

rB

rcV

rmcm

VV (2.42)

18

The deflection of the slab at punching, δε , is calculated as the sum of the elastic deflection and the additional deflection due to rigid body rotation of the sector elements after first yield at the column:

22∆

221

π4 2

21

εBcBfBc

EIc

cBVy −

⋅⋅′′+−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=δ (2.43)

All equations in this section have to be modified for flat plates with rectangular panels, see Section 4.2.

The calculations have to be computerized, see Appendix B, because Eq. (2.41) can be solved manually by iteration only and Eq. (2.42) is laborious. The computer solution has furthermore the advantage that the alternative failure mode Vσ is checked automatically as well as the ultimate deflection δ ε. However, an approximate manual calculation of Vε is described in Section 2.4.

B

c 2

2

CL

ry

rBEIf2

⋅⋅′′∆

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+= 2

2

2

21

t 42

2ln2

8 cB

rB

rcV

m y

π

r

my

first yield

punching failure

Figure 2-7 Distribution of tangential bending moment at first yield and at punching failure.

19

2.2.5 Flat plates with shear reinforcement

The capacity of the internal column capital will increase when shear reinforcement is provided, because part of the load is transferred by steep compression struts from the shear reinforcement; see Figure 2-8.

CL

xs

Figure 2-8 Model for maximum capacity with shear reinforcement. The favourable inclination of the resulting compression strut means that the critical tangential concrete strain cpusε is assumed to reach the strain 0.0015, which is close to the strain corresponding to the peak stress for concrete grade 25 MPa. The same brittleness and size effect factors as for the strain without shear reinforcement gives

31

s

10

cpus15002500150 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

x.

f.

.

ccε (2.44)

where xs = compression zone depth with shear reinforcement.

The secant modulus c15E to the strain εcpus can with good approximation be represented by

⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛⋅=

19011

250.0015cc

10cccc

c15f

.ff

E.

(2.45)

for concrete grades between 20 MPa and 100 MPa, which is indicated with dots in Figure 2-9 (together with corresponding dots for the secant modulus Ec10 to the strain 0.0010 for slabs without shear reinforcement).

20

The compression zone force due to the tangential bending moment is assessed as

cpusc15c εα ⋅⋅⋅= ExF s (2.46)

with MPa10020100

13050 cc

2cc ≤≤⎟

⎠⎞

⎜⎝⎛ −⋅+= f;

f..α . (2.47)

The compression zone depth xs can then be derived to (compare Eqs. (2.4) to (2.7)):

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅+⋅⋅=

αρααρ

211

41

s2ss n

ndx (2.48)

with ρρ ⋅=c15

ss E

En . (2.49)

fccMPa

100

80

60

40

20

0

0 ε

0.00

1

0.00

2

Ec15

c10E

Figure 2-9 Secant modules Ec10 and Ec15 according to Eqs. (2.10) and (2.45).

21

Punching before reinforcement yield

Reinforcement stress at the column edge at punching

s

scpusss x

xdE

−⋅⋅= εσ (2.50)

If sσ turns out to be larger than the yield stress fsy , then the reinforcement yields before punching occurs and the calculation is performed according to “Punching after reinforcement yield” below.

Bending moment at the column edge at punching

⎟⎠⎞

⎜⎝⎛ −⋅=

dx

dm3

1 ss

2εs σρ (2.51)

Finally, the punching capacity sεV is derived from Eq. (2.25):

2

2εsεs

1ln2

8π

cB

Bc

mV−+

⋅= (2.52)

Punching after reinforcement yield

The forces in the reinforcement and the concrete compression zone are equal:

αερ ⋅⋅⋅= puscpusc15sy xEfd (2.53)

Combine Eqs. (2.53) and (2.44):

30

cc

3

sy

c156cpus

2515150010.

fd.

fE

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

⋅⋅= −

ραε with effective depth d in (m) (2.54)

c15

sy

cpuspus E

fdx ⋅⋅=ε

ρα

(2.55)

The ultimate tangential curvature at the column edge, usf ′′ :

30

cc2

3

2

2

2sy

2c15

pus

cpusus

25001501500.

f.

d.

fE

xf ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅

⋅⋅==′′

ραε

(2.56)

Bending moment at reinforcement yield

⎟⎠⎞

⎜⎝⎛ −⋅⋅=

dx

fdm3

1 ssy

2ys ρ (2.57)

22

The column reactions Vy1 when the reinforcement starts to yield at the column edge and Vy2 when all reinforcement across the slab yields:

2

2ysy1

1ln2

8π

cB

Bc

mV−+

⋅= (2.58)

cB

mV−

⋅=1

π2ysy2 (2.59)

Curvature in tangential direction at start of yield:

ss

syys

1xdE

ff

−⋅=′′ (2.60)

Possible additional curvature at column edge after first yield:

ysus∆ fff ′′−′′=′′ (2.61)

The distance ry and punching capacity Vεs is then determined from Eqs. (2.41) and (2.42).

The calculations are preferably computerized, see Appendix C.

The upper limit for punching capacity derived above presupposes that the punching failure occurs within the zone with shear reinforcement. It is further assumed that the shear reinforcement is designed for at least 60 % of the total column reaction and stirrups or stud rails are well anchored outside the innermost top and bottom reinforcement layers. The shear reinforcement must extend far enough from the column to exclude a shear failure outside the shear reinforced area, which is preferably checked in accordance with Model Code 90.

Larger capacity can be achieved with inclined stirrups. The stirrups in the first row outside the column act as hangers that transfer their load directly to the column without affecting the internal column capital if the upper end of the stirrups is anchored inside the column edge. The total punching capacity can therefore be assessed by adding the vertical component of the hanger force to the above capacity Vεs, but not to more than the load corresponding to overall yield of the flexural reinforcement. The nominal ultimate stress in the hangers should thereby be limited to round 350 MPa.

Still larger capacity in combination with ductile behaviour can be achieved with the “ductility reinforcement” described in Chapter 5.

23

2.3 Punching capacity Vσ

Punching occurs when the compression stress in the fictitious internal column capital of the slab reaches a critical value.

2.3.1 Column footings

Consider a relatively compact circular test specimen according to Figure 2-10. The specimen could simulate a column footing.

The shear force V is transferred to the column via a column capital within the slab and punching occurs when the stress at the upper edge of the capital reaches the compression strength σc.

D

R

x

dB

V

2γ1x∆

γ

t

cσ

0

Figure 2-10 Failure mode Vσ . Definitions of parameters.

Square columns are replaced by equivalent circular columns with the same perimeter and square slabs are replaced by equivalent circular slabs with the same area. The diameter of a circular slab is denoted D.

24

Part of uniformly distributed load will fall within the final shear crack. That part of the load does not affect the punching capacity; see Figure 2-11. The shear crack is assumed steep in compact foundations and the inclination angle φ should not be taken less than round 45° in slender foundations. A reasonable expression for the angle φ is

123

4.1tan ≥−

=BD

dφ (2.62)

RD

kD

φ

Vσ

=c0

Figure 2-11 Definition of angle φ and shear load Vσ .

The diameter of the circle within the fictitious shear crack at the flexural reinforcement level is denoted c0

φtan2

0dBkDc +== (2.63)

The radius to the centre of gravity for uniformly distributed load outside the fictitious shear crack can be shown to be

)1

1(3

2

kkDR+

+= (2.64)

The column capital forms part of a compression strut from the load to the column. Punching occurs when the capital fails in compression so that a diagonal shear crack forms. It is easily shown that the capacity of the capital is at maximum when the angle γ1 is equal to γ, where 2γ is the angle to the horizontal of the punching crack near the column; see Figure 2-10.

25

γγγγ

2sin2sinsin 11

0V)(

xkV−⋅

⋅=

( ) ( ) ( )[ ] 02cossin2sincos2sin 1111

0V

1=−−−

⋅= γγγγγγ

γγxk

ddV

( ) γγγγγ ==− 111 ;tan2tan

The inclination angle γ of the compression strut is determined by

BR

xd5.0

)(tan−

∆−=γ (2.65)

γ2

0

cos41∆

xx ⋅= (2.66)

Eliminate ∆x from Eqs. (2.65) and (2.66):

( )00

20

2 2142tanx

BRxd

xBR −

−−+−

=γ (2.67)

The average upper diameter of the capital that supports the inclined compression strut from the load is

( ) ( )γγ tan22tan00 xx

B ++ (2.68)

The effective perimeter u of the capital is thus

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛++=

γγ tan22tanπ 00 xx

Bu (2.69)

The compression strength “σc” of the capital is assumed to vary with the slenderness u/x of the cantilevering part of the capital; see Figure 2-12.

cc

2

0ccc 21007019060 f.

xu...f ≤

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=σ (2.70)

For small values of u/x, it is evident from Figure 2-12 that the compression zone of the surrounding slab confines the capital. That effect decreases with increasing u/x

until ⎟⎠

⎞⎜⎝

⎛ −=250

160 ccccc

ff.σ , which is the generally accepted uniaxial compression strength in

cracked zones, (= fcd2 according to Model Code 90).

The upper limit 1.2 fcc in Eq. (2.69) represents the concrete compression strength in bi-axial compression, when the perpendicular compression stress is moderate; see Nilsson (1983).

26

B

d

γγ

B/d = 1.0

B/d = 2.5

tσc

σc

Figure 2-12 Confinement of internal column capital by surrounding slab.

r

rB2c0 ⋅ε

2B

20c

rc2

0s0 ⋅ε

s0ε

0cε

2D

x

CL

φ 0

Figure 2-13 Strain distribution in compact footing.

27

In compact footings, the sector elements between the radial flexural cracks are assumed to rotate as rigid bodies even before yielding of the flexural reinforcement, which affects the depth of the compression zone; see Figure 2-13.

Study a sector element under the shear crack with sector angle ϕ∆ :

r

cxdr

Bx 22

0

0

s0c0 ⋅−

=⋅εε

(2.71)

⎟⎠⎞

⎜⎝⎛ +⋅⋅⋅⋅=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⋅⋅⋅⋅= ∫ BDBx

Err

BBxEF

D

Bln1

22∆d

222∆ 0

c0c0

2

2

0c0c0c εϕεϕ (2.72)

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⋅⋅⋅⋅=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⋅⋅⋅⋅⋅= ∫0

0s0s

2

2

00s0ss ln1

2∆d

22∆

0cDc

dErr

ccdEF

D

cρεϕρεϕ

(2.73)

Combine Eqs. (2.71), (2.72), (2.73), and put ρρ ⋅+

+⋅=

BDcD

EE

nln1

ln10

c0

s0 : (2.74)

(D shall be replaced by the slab width b in square footings)

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 121

00

0ρ

ρn

ndx

(2.75)

The punching capacity can then be determined as

31

cσ150.0)sin( ⎟

⎠⎞

⎜⎝⎛⋅⋅⋅⋅=

tutV γσ (2.76)

where ( )γcos20x

t = = depth of compression strut

31

150.0⎟⎠⎞

⎜⎝⎛

t = size effect factor with dimension t in (m)

0.150 = diameter of standard test cylinder specimen (m).

Finally, the total load capacity with respect to punching is determined as

2σ

σ1 k

VP

−= (2.77)

28

For geometrical reasons the angle γ is limited to 45° corresponding to a vertical shear crack through the compression zone.

The angle γ need not be taken less than 25°, which agrees with the average shear crack inclination 30° observed at slender test specimens, see for instance Kinnunen and Nylander (1960); see Figure 2-14.

γ2

γd tan 30°

γ

°=→+=°

25tan

7.02tan

3.030tan

γγγddd

d3.0≈

Figure 2-14 Angle γ for flat plates.

The flexural capacity and the concrete strain of a square column footing are checked as follows.

⎟⎠⎞

⎜⎝⎛ −=

π28εBbPM (2.78)

⎟⎠⎞

⎜⎝⎛ −⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅

=

dx

cbcd

M

31ln1

00

2s

ρσ (2.79)

where M = total bending moment over footing width

b = width of square footing

3110

cccpu

s

s

ε

0c

15025001002

⎟⎠⎞

⎜⎝⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=≤⋅

−⋅

+=

x.

f.

Exdx

xBc

.

εσ

ε (2.80)

with x according to Eq. (2.75) and with Ec10 instead of Ec0 in Eq. (2.74).

Equation (2.80) may govern the capacity at high reinforcement ratios in combination with high strength concrete.

The expressions for the capacity Vσ presuppose that the flexural reinforcement in the footing does not yield, which has to be considered when designing such reinforcement.

29

2.3.2 Flat plates

In flat plates, the flexural reinforcement near the column will often yield before punching occurs. Eq. (2.7) is then no longer valid for the depth of the compression zone near the column for tangential bending moments – the depth decreases. When punching occurs, the compression zone near the column has decreased to the value given by Eq. (2.34).

However, when the flexural reinforcement starts to yield near the column then the sector elements between the radial cracks in the slab start to rotate as rigid bodies. Additional deflection will then cause only limited increase of the radial curvature of the sector elements. The compression zone depth in radial direction will therefore not decrease below the value given by Eq. (2.75), which value shall be used when calculating Vσ for flat plates. This is confirmed by Figure 4-11 of Hallgren (1996) that shows the recorded radial strain distribution over the compression zone for specimen HSC1. The depth remained constant – conforming to Eq. (2.75) – up to the punching load.

The lesser of Vσ and Vε governs the punching capacity of flat plates. If Vσ turns out to be governing, then the displacement of the flat plate is computed according to Section 2.2 with a decreased critical value εcpu so that Vε becomes equal to Vσ However, experience from published test results simulating slender flat plates demonstrates that Vσ is governing only when columns are small in relation to the slab thickness and the concrete compression strength is low, which is also evident from Figure 4-7. For flat plates with shear reinforcement, Vσs shall be determined with the angle γ = 45°.

2.4 Manual calculation

2.4.1 General

The relation between punching capacity of flat plates and flexural reinforcement ratio is typically as depicted in Figure 2-14. Two limit values for the reinforcement ratio can be identified. When the flexural reinforcement ratio exceeds the value ρ 1, punching occurs without any reinforcement yielding and the punching capacity can be easily determined by the equations given in Section 2.2.3. When the flexural reinforcement ratio is less than the value ρ 2, punching occurs after all reinforcement has reached the yield limit. The punching capacity is then equal to the flexural capacity of the slab.

Between these two limits part of the reinforcement yields. The “exact” estimation of the punching capacity in this region leads to rather complicated equations, the solution of which requires computerized calculations as described in Section 2.2.4. However, if the curved relation is replaced by a linear relation as indicated in Figure 2-15, the punching capacity can be easily determined even in this region.

30

VkN

ε

1500

1000

500

00 0.5 1.0 1.5 %ρ

V1

2

ρ

V

ρ12

Figure 2-15 Punching capacity Vε versus reinforcement ratio. (fcc = 30 MPa, fsy =500 MPa, d = 0.25 m, B = 0.5 m, c = 3.2 m)

2.4.2 Reinforcement limit ρ 1

The reinforcement limit ρ 1 is estimated by trial and error calculations until σs is equal to fsy:

Without shear reinforcement

31

cc4

ccc10 10

21500150

1601 ⎟⎠

⎞⎜⎝

⎛⋅⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −−=ff

.E (2.10)

1c10

s1 ρρ ⋅=

EE

n (2.6)

xxdE

x.

f.

nndx

.

−⋅⋅=

→⎟⎠⎞

⎜⎝⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=→⎟

⎟⎠

⎞⎜⎜⎝

⎛−+⋅=

cpuss

3110

cccpu

11

1502500100121

εσ

ερ

ρ (2.81)

The punching capacity for reinforcement ratio exceeding ρ 1 is then determined according to Section 2.2.3.

31

With shear reinforcement

2

cc100

-10.30.5 ⎟⎠⎞

⎜⎝⎛+=

fα (2.47)

⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛⋅=

19011

250.0015cc

10cccc

c15f

.ff

E.

(2.45)

1c15

s1s ρρ ⋅=

EE

n

→⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=→⎟

⎟⎠

⎞⎜⎜⎝

⎛−

⋅+⋅=

31

s

10

cccpus

1s21ss

1502500150211

41

x.

f.

nndx

.

εαραα

ρ

sxxd

E scpusss

−⋅⋅= εσ (2.82)

The punching capacity for reinforcement ratio exceeding ρ 1 is then determined according to Section 2.2.5.

2.4.3 Reinforcement limit ρ 2

Without shear reinforcement

The additional tangential curvature 2∆f ′′ at the slab edge at punching due to the rigid body rotation after first yield at the column can be derived from Eqs. (2.33) and (2.34):

30

cc2

3

sy

c106cpu

25102150010

.

fd.

fE

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅⋅= −

ρε (2.33)

c10

sy

cpu

2pu

2Ef

dx ⋅⋅=ε

ρ (2.34)

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=′′→

xdxB

f sy

pu

cpuε22 c

∆εε

ρ (2.83)

with 2c10

s2 ρρ ⋅=

EE

n and ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅= 121

22 ρ

ρn

ndx

The additional curvature at the slab edge can also be expressed as

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ +−−

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−⋅

−−

−=′′

c1501

c1

2ππ4

11∆ εsy2

2

sy2

B.

xdBcB

xdf

εε

ε

ε

(2.84)

32

Only a few iterations are normally required to determine the reinforcement ratio ρ 2 that makes the two curvature expressions equal. The curvature according to Eq. (2.83) decreases rapidly with increasing ρ, and the curvature according to Eq. (2.84) increases slowly with increasing ρ.

The punching capacity up to the reinforcement ratio ρ 2 is equal to the flexural capacity at overall yield, Vy2:

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−⋅⋅⋅

−==

cc

y2sy

εy2ε 501

1

π2f

f.df

cB

VV sρρ 2ρρ ≤ (2.85)

With shear reinforcement

The calculation is performed in the same way as without shear reinforcement:

30

cc2

3

sy

c156cpus

2515150010.

fd.

fE

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

⋅⋅= −

ραε (2.54)

c15

sy

cpus

2pus E

fdx ⋅

⋅⋅=

εαρ

(2.55)

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=′′→

s

sy

pus

cpusε22 c

∆xdx

Bf

εερ (2.86)

with 2c15

s2s ρρ ⋅=

EE

n and ⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅+⋅=

αρααρ

211

41

2s22s n

ndx s (2.48)

The additional curvature at the slab edge can also be expressed as

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +−

−=′′

c1501∆ ε

s

sy2

B.

xdf

ε (2.87)

2.4.4 Transition zone between ρ 1 and ρ 2

The punching capacity Vε is determined by linear interpolation:

( )2121

22ε VVVV −⋅

−−

+=ρρρρ (2.88)

33

2.4.5 Tabulated values for ρ 1 and ρ 2

In order to facilitate calculations the limit values ρ 1 and ρ 2 can be tabulated for common standard designs. Examples are given in Tables 2-1 and 2-2.

The reinforcement ratios ρ 2 in Table 2-2 are especially interesting because they represent the limits below which the flexural reinforcement within the column strip can be fully utilized without correction with respect to punching.

Table 2-1 Reinforcement ratio ρ 1 % for fsy = 500 MPa

Table 2-2 Reinforcement ratio ρ 2 % for fsy = 500 MPa.

d

(m) cBε Without shear reinforcement

Concrete cylinder strength, fck (MPa)

With shear reinforcement

Concrete cylinder strength, fck (MPa)

20 30 40 50 60 20 30 40 50 60

0.20

0.1

0.2

0.3

0.375

0.496

0.577

0.462

0.613

0.712

0.535

0.710

0.825

0.596

0.791

0.919

0.646

0.857

0.996

0.697

0.926

1.078

0.893

1.184

1.376

1.032

1.366

1.586

1.129

1.493

1.732

1.195

1.579

1.831

0.25

0.1

0.2

0.3

0.338

0.447

0.520

0.416

0.552

0.642

0.482

0.640

0.744

0.537

0.713

0.829

0.581

0.772

0.899

0.627

0.834

0.971

0.805

1.068

1.241

0.930

1.232

1.432

1.018

1.347

1.564

1.077

1.425

1.654

0.30

0.1

0.2

0.3

0.310

0.411

0.478

0.382

0.507

0.590

0.442

0.588

0.684

0.493

0.655

0.762

0.534

0.709

0.826

0.575

0.766

0.892

0.739

0.981

1.141

0.854

1.133

1.317

0.935

1.239

1.439

0.990

1.311

1.522

d

(m)

Without shear reinforcement

Concrete cylinder strength, fck (MPa)

With shear reinforcement

Concrete cylinder strength, fck (MPa)

20 30 40 50 60 20 30 40 50 60

0.20

0.25

0.30

0.821

0.742

0.683

1.014

0.916

0.843

1.176

1.062

0.977

1.310

1.183

1.089

1.421

1.283

1.180

1.416

1.283

1.183

1.856

1.681

1.550

2.175

1.970

1.816

2.401

2.174

2.004

2.558

2.316

2.135

34

2.5 Comparison with test results

2.5.1 Influence of bending moment

50865865

4579153000

457 915

160

v

m 1m1

Figure 2-16 Test set-up, Regan (1986)

Regan (1986) made a very illustrative test with specimens subjected to a bending moment at the slab boundary as shown in Figure 2-16. The specimens were 100 mm thick and their central panels were reinforced with φ 10 c/c 75 (fsy = 525 MPa) both ways in the top and φ 8 c/c 75 (fsy = 510 MPa) both ways in the bottom. The average effective depths were 80 mm and 82 mm respectively.

An upward load was applied at the centre through a 160 mm square plate and downward line loads were applied at the four sides of the 1.83 m square. The assembly was supported on rollers positioned 0.457 m beyond the downward loads. By varying the ratios of the upward and downward loads differing reactions could be produced at the roller supports, thus changing the ratio between the central load (V) and the restraining moments (m1) at the edges of the 1.83 m square.

35

All slabs failed in punching and the test data are summarized in Table 2-3 together with predictions according to the Model Code 90 and the theory of this thesis. The Model Code 90 may represent the common code approach where the punching failure load is related to formal shear strength irrespective of the bending moment in the slab near the column. The presented theory on the other hand assumes that punching occurs when the tangential strain in the concrete near the column reaches a critical value. That strain is a function of the bending moment in the slab near the column.

Table 2-3 Comparison of test results by Regan (1986) with predictions of Model Code 90 and the presented theory.

Slab fcc

(MPa)

m1/V

Vtest

(kN)

MC 90

Vcalc (kN)

Theory

Vcalc (kN)

MC 90

Vtest/Vcalc

Theory

Vtest/Vcalc

IV/1 26.2 0 190 193 180 0.984 1.056

IV/2 34.0 0.017 236 210 217 1.124 1.088

IV/3 28.3 0.036 248 198 224 1.252 1.107

IV/4 31.3 0.049 262 205 254 1.278 1.031

It is evident from Table 2-1 that the punching failure cannot be treated as a pure shear force failure; the bending moment in the slab at the column plays a decisive role for the punching failure mechanism and the resulting punching capacity.

The punching capacity in the tests increased when the bending moment in the slab at the column versus the column reaction V decreased. Most probably, the opposite is also valid, i.e. the punching capacity will decrease if the bending moment in the slab versus the column reaction V increases. That occurs for instance for the bending moment in the long direction in flat plates with rectangular panels, which is confirmed by specimen S1 in Kinnunen et al (1980) (see Table 2-4). Nylander and Sundquist (1972) concluded that if flexural reinforcement has to be added due to punching, then the required flexural reinforcement ratios ρ x and ρ y in the two orthogonal directions shall be increased with the same factor to k·ρ x and k·ρ y, because an increase of only the lesser of the two reinforcement ratios did not increase the punching capacity in their tests. These findings have unfortunately never been incorporated in Swedish concrete codes and handbooks.

2.5.2 Influence of concrete mechanical properties

The punching failure modes for slender flat plates and compact footings are fundamentally different. Slender flat plates usually display a sudden brittle failure – often characterized as explosive. Compact footings display a gradual failure similar to the failure of cylinders for testing of concrete compression strength.

36

Flat plates seem to fail when the tangential strain in the concrete reaches a critical value. Compact footings seem to fail when the inclined compression stress reaches the failure stress in bi-axial compression.

These observations indicate that the concrete E-modulus has influence on the punching strength of flat plates. This property is traditionally recorded neither for test specimens nor for actual structures. The relation between compression strength and E-modulus as given by Model Code 90 has therefore been used for the verification in Table 2-4. This relation is at best a good approximation, which is confirmed by those tests where the E-modulus was actually recorded. The recorded values by for instance Hallgren (1996) were consistently lower than the values derived from the compression strengths according to Model Code 90. The difference was still larger for the specimen described by Ožbolt et al (2000). The recorded E-modulus was there only 79 % of the value according to Model Code 90, probably due to a concrete mix design with aggregates from sedimentary rock. This had a large impact on the calculated punching capacity where the theoretical capacity with the recorded E-modulus was only 83 % of the capacity with E-modulus according to Model Code 90.

The E-modulus of concrete thus seems to be an important concrete property for prediction of the punching strength. Not only the compression strength but also the E-modulus should therefore be recorded for test specimens and should be specified on structural drawings for flat plates.

2.5.3 Comparison with test results for flat plates and column footings

The theory is validated by comparison with published test results in Tables 2-4 to 2-7. Only specimens with normal density aggregates are included.

The column size in relation to slab depth is represented by the parameter Bσ/d, where a square column with width a is replaced by a circular column with the same perimeter, π/4σ aB = .

The slab slenderness is represented by the expression (c – Bε)/2d, where a square column is replaced by a circular column with the same bending moment reduction effect, Bε = 3πa/8.

The test specimens simulating flat plates listed in Table 2-4 cover a very wide range of conditions. The reinforcement ratio varies from 0.35 % up to 3.7 % and the yield strength varies from 300 MPa up to more than 700 MPa. Concrete grades vary from 14 MPa up to more than 100 MPa. The effective depth of the specimens varies from 70 mm up to 619 mm and the column width versus the effective depth of the slab varies between 1.2 and 4.0.

Some of the duplicated tests by Kinnunen and Nylander (1960), Tolf (1988) and Tomaszewicz (1993) display a capacity scatter, which is larger than the usual scatter observed for cylinder compression tests. The variation coefficient 0.073 for the punching capacity predictions in Table 2-4 must therefore be regarded as a good verification of the theory for flat plates, with a prediction scatter approaching the inevitable material strength scatter. Furthermore, the simple and comprehensible failure model is based on recordable data for the stress-strain relation of concrete in uniaxial compression combined with prevailing knowledge of concrete properties in biaxial compression. These base properties can simply not be “manipulated” or “tuned”, they are directly related to the recorded compression strength of the test specimens.

37

The theory for column footings is more intricate, because it utilizes a rough estimate of the decreasing compression strength of the column capital with increasing perimeter versus the compression zone depth. However, the variation coefficient 0.092 in Table 2-5 indicates that the presented strut-and-tie model seems to describe the structural behaviour well enough to give a reasonably good estimate of the punching capacity, which at least is a better prediction result than any existing concrete code method. The small scatter in Table 2-6 for specimens with shear reinforcement is partly due to the fact that several specimens failed at loads close to the load corresponding to overall yield of the flexural reinforcement – a case that is trivial for the presented theory. However, the failure capacity is predicted even for those specimens where the reinforcement did not reach overall yield. The two specimens of Sundquist (1977) displayed a very ductile behaviour with overall yield, but they did not reach the theoretical yield capacity, which indicates that the bent down reinforcement bars were not fully active in resisting the bending moment as assumed in Nylander and Kinnunen (1990). Some of the slabs in Table 2-6 were provided with an extremely large amount of shear reinforcement, but punching failures still occurred within the shear reinforced zone, which demonstrates that the code approach with the capacity taken as scu 750 VV.V += must be utilized with caution. The upper bound 1.6Vc according to Model Code 90 appears well advised, where Vc is the nominal capacity without shear reinforcement. However, larger capacity than 1.6Vc can be achieved if the slab is provided with ductility reinforcement described in Chapter 5.

Predicted deflection and concrete strain in the tangential direction of flat plate specimens are in Table 2-7 compared to recorded values, whenever reported. The good agreement between theory and reality confirms that the presented model can predict the punching capacity as well as the slab deflection and the concrete strain near the column, which forms the prerequisite for the approach in Chapter 3 about eccentric punching.

It is noteworthy that the theory can predict the large deflection and the sudden punching failure in slabs where all the reinforcement yields across the slab width. This is a strong support for the hypothesis that punching of flat plates occurs when the flexural compression strain in the slab reaches a critical value, which is further supported by the tests with varying bending moment described in Section 2.5.1.

Figure 2-15 displays a typical curve for the punching capacity versus the flexural reinforcement ratio in a flat plate. It is evident that if the theory can predict the capacity for ρ >ρ 1, then any reasonable transition curve between ρ 1 and ρ 2 will give a good estimate of the punching capacity in this range as well, because the capacity when all reinforcement yields (for ρ < ρ 2) is well-defined by the fan-type yield line configuration. That is why it is most important that a theory for punching capacity should primarily have the ability to predict the punching failure at such high reinforcement ratios that no reinforcement yields before punching. It is then logical that the moment distribution according to the theory of elasticity should be applied in that case. This moment distribution differs radically from the moment distribution corresponding to rigid body sector elements rotating around a support perimeter near the column edge, which is used by Kinnunen and Nylander (1960) and Hallgren (1996) even for elastic conditions with no reinforcement yielding.

38

Table 2-4 Test results, flat plate specimens. For explanations, see next page. Authors

Test slab No.

fcc MPa

fsy MPa

ρ %

d mm

c mm

Column size mm

Bσ d

c−Bε 2d

σs MPa

Vcalc kN

Vtest kN

calc

testVV

Elstner, Hognestad (1956)

A-1b A-1c A1-d A-1e A-2b A-2c A-7b A-3b A-3c A-3d A-4 A-5 A-6 A-13 B-1 B-2 B-4 B-9 B -14

25.2 29.0 36.8 20.3 19.5 37.4 27.9 22.6 26.5 34.5 26.1 27.8 25.0 26.2 14.2 47.6 47.7 43.9 50.5

332 " " " 321 " " " " " 332 321 " 294 324 321 303 341 325

1.16 " " " 2.50 " " 3.74 " " 1.18 2.50 3.74 0.554 0.476 " 1.01 2.00 3.02

118 " " " 114 " " " " " 118 114 " 121 114 " " " "

1780 " " " " " " " " " " " " " " " " " "

254 " " " " " " " " " 356 " " " 254 " " " "

2.74 " " " 2.84 " " " " " 3.84 3.98 " 3.75 2.84 " " " "

6.27 " " " 6.50 " " " " " 5.76 5.97 " 5.62 6.50 " " " "

fsy " " " 256 fsy 297 198 212 237 fsy 297 207 fsy " " " " 324

350 356 354 335 381 512 449 432 465 526 393 530 535 178 136 139 270 511 599

365 356 351 356 400 467 512 445 534 547 400 534 498 236 178 200 334 505 578

1.043 1.000 0.992 1.063 1.050 0.912 1.140 1.030 1.148 1.040 1.018 1.008 0.931 1.3264 1.3094 1.4394 1.2374 0.988 0.965

Kinnunen, Nylander (1960)

5 6 24 25 32 33

26.8 26.2 26.4 25.1 26.3 26.6

441 454 455 451 448 462

0.80 0.79 1.01 1.04 0.49 0.48

117 118 128 124 123 125

φ1710 " " " " "

φ150 " φ300 " " "

1.28 1.27 2.34 2.42 2.44 2.40

6.67 6.61 5.51 5.69 5.73 5.64

fsy

" " " " "

263 266 446 419 231 241

255 275 430 408 258 258

0970 1.034 0.964 0.974 1.1174 1.0714

Moe (1961)

R2 M1A

26.5 20.8

328 481

1.38 1.50

114 "

1780 "

152 305

1.70 3.41

7.02 6.23

fsy 390

303 394

311 433

1.026 1.099

Schaeidt et al (1970)

28.5

555

1.31

240

φ2650

φ500

2.08

4.48

381

1470

1694

1.152

Marti et al (1977)

P-2

34.6

558

1.44

145

φ2600

φ300

2.07

7.93

453

569

600

1.054

Pralong et al (1979)

P-5

26.2

515

1.34

154

φ2600

φ300

1.95

7.47

421

551

569

1.033

Kinnunen, Nylander, Tolf (1980)

S1

30.6

621

0.82 0.40

619

4680 2340

φ800

1.29

3.13 1.24

404

4780

4915

1.028

Tolf (1988)

S1.1 S1.2 S2.1 S2.2 S1.3 S1.4 S2.3 S2.4

28.6 22.9 24.2 22.9 26.6 25.1 25.4 24.2

706 701 657 670 720 712 668 664

0.80 0.81 0.80 0.80 0.35 0.34 0.34 0.35

100 99 200 199 98 99 200 197

φ1190 " φ2380 " φ1190 " φ2380 "

φ125 " φ250 " φ125 " φ250 "

1.25 1.26 1.25 1.26 1.28 1.26 1.25 1.27

5.38 5.43 5.38 5.40 5.49 5.43 5.38 5.46

699 584 483 465 fsy " " "

2291

1891

6301

5981

1471

1401

4531

4351

216 194 603 600 145 148 489 444

0.943 1.026 0.957 1.003 0.986 1.057 1.079 1.021

39

Table 2-4 Continued from previous page. Authors

Test slab No.

fcc MPa

fsy MPa

ρ %

d mm

c mm

Column size mm

Bσ d

c−Bε 2d

σs MPa

Vcalc kN

Vtest kN

calc

testVV

Marzouk, Hussein (1991)

HS 1 HS 2 HS 3 HS 4 HS 5 HS 6 HS 7 HS 8 HS 9 HS 10 HS 11 HS 12 HS 13 HS 14 HS 15 NS 1 NS 2

67 70 69 66 68 70 74 69 74 80 70 75 68 72 71 42 30

490 " " " " " " " " " " " " " " " "

0.49 0.84 1.47 2.37 0.64 0.94 1.19 1.11 1.61 2.33 0.95 1.52 2.00 1.47 " " 0.94

95 " " 90 125 120 95 120 " " 70 " " 95 " " 120

1500 " " " " " " " " " " " " " " " "

150 " " " " " " " " " " " " 220 300 150 "

2.01 " " 2.12 1.53 1.59 2.01 1.59 " " 2.73 " " 2.95 4.02 2.01 1.59

6.96 " " 7.35 5.29 5.51 6.96 5.51 " " 9.45 " " 6.53 6.04 6.96 5.51

fsy " " 460 fsy

" " " " 446 fsy " " " " " "

144 242 353 367 322 425 323 469 557 626 148 212 232 414 469 305 349

178 249 356 418 365 489 356 436 543 645 196 258 267 498 560 320 396

1.2364 1.029 1.008 1.139 1.1344

1.150 1.102 0.927 0.975 1.030 1.3243,4 1.2173 1.1513 1.203 1.145 1.049 1.103

Tomaszewicz (1993)

65-1-1 95-1-1 115-1-1 95-1-3 65-2-1 95-2-1D 95-2-1 115-2-1 95-2-3 95-2-3D 95-2-3D+ 115-2-3 95-3-1

64 84 112 90 70 88 87 119 90 80 98 108 85

500 " " " 500 " " " " " " " "

1.49 " " 2.55 1.75 " " " 2.62 " " " 1.84

275 " " " 200 " " " 200 " " " 88

2500 " " " 2200 " " " 2200 " " " 1100

200 " " " 150 " " " 150 " " " 100

0.93 " " " 0.96 " " " " " " " 1.27

4.12 " " " 5.06 " " " " " " " 4.91

443 476 fy

325 449 475 474 fsy 354 345 361 401 fsy

1955 2113 2245 2409 1159 1233 1230 1310 1349 1308 1375 1403 340

2050 2250 2450 2400 1200 1100 1300 1400 1450 1250 1450 1550 330

1.049 1.065 1.091 0.996 1.035 0.892 1.057 1.069 1.075 0.956 1.055 1.105 0.971

Hallgren (1996) 2)

HSC0 HSC1 HSC2 HSC4 HSC6 HSC9 N/HSC8

90 91 86 92 109 84 95

643 627 620 596 633 634 631

0.80 0.80 0.82 1.19 0.60 0.33 0.80

200 200 194 200 201 202 198

φ2400 " " " " " "

φ250 " " " " " "

1.25 1.25 1.29 1.25 1.24 1.24 1.26

5.38 " 5.54 5.38 5.35 5.32 5.43

fy

" " " " " "

1057 1051 921 1169 954 565 1042

965 1021 889 1041 960 565 944

0.922 0.971 0.965 0.891 1.006 1.000 0.906

Ožbolt et al (2000)

21

569

0.80

190

φ2400

400

2.68

5.08

442

6685

806

615

0.921

Sundquist, Kinnunen (2004b)

C1 C2 D1

24.0 24.4 27.2

718 " "

0.80 0.80 0.64

100 100 125

φ 1190 " "

φ 250 φ 250 φ 125

2.50 2.50 1.00

4.70 4.70 4.26

688 692 583

302 304 2411

270 250 265

0.894 0.822 1.100

Mean value Vtest / Vcalc = 1.021(1± 0.073)

1) Failure mode Vσ governing. 2) Recorded Ec0-values used instead of Eq. (2.3). 3) Not included in the statistical evaluation due to the small effective depth 70 mm. 4) Overall yield with membrane action and strain hardening, therefore not included in the statistical evaluation. 5) Recorded Ec0 value 21.7 GPa used instead of Eq. (2.3) which would give Ec0 = 27.5 GPa and Vcalc = 806 kN.

40

Table 2-5 Test results, column footings. Authors

Test slab No.

fcc

MPa

fsy

MPa

ρ %

d

mm

Line load

c mm

Slab width

b mm

Column size mm

Bσ d

c−Bε 2d

Vcalc kN

Vtest kN

calc

testVV

Dieterle (1978) Dieterle, Rostasy (1981) surface load

B-1 B-2 B-3 B-4 V-2 B-4/2 B-4/3 B-4/4 C-1 C-3 H-2 H-3 S1-H

23.5 23.6 28.1 24.1 25.9 25.3 24.3 24.8 28.1 28.7 29.4 26.2 30.6

444 433 407 387 477 449 455 387 564 572 572 510 512

0.208 0.434 0.642 0.866 0.501 0.784 0.805 0.830 0.275 0.430 0.333 0.390 0.862

296 294 293 292 294 290 294 292 290 290 375 450 290

1500

" " " " " " " " " " " "

300

" " " " " " "

150 450 300

" "

1.3 " " " " " " "

0.66 1.98 1.02 0.85 1.32

1.3 " " " " " " "

1.64 1.03 1.03 0.86 1.34

907

1405 1905 1845 1621 1839 1829 1863 633

2646 2255 2773 2249

1034 1493 2025 1865 1765 2050 2028 1853 859 2367 2234 3116 2368

1.1402

1.063 1.063 1.011 1.089 1.115 1.109 0.995 1.357 0.895 0.991 1.124 1.053

Hallgren, Kinnunen, Nylander (1983, 1998) line load surface load

S1 S2 S3 S4 S7 S8 S9 S12 S13 S11 S14

40

28.4 29.8 25.7 14.4 31.4 25.5 27.3 19.8

28.2 21.4

621

" " " " " " " " " "

0.401 0.399 0.388 0.659 0.395 0.239 0.398 0.413 0.416

0.40 0.39

242 243 250 232 246 245 244 242 244

235 240

600

" " " " " "

φ674 "

850

" " " " " "

φ960 "

850 "

φ250

" " " " " " " " " "

1.03 1.03 1.00 1.08 1.02 1.02 1.02 1.03 1.02

1.06 1.04

0.89 0.89 0.87 0.93 0.88 0.88 0.89 0.79 0.78

0.83 0.81

1309 967

1021 994 532 880 881 959 727

1296 1035

1363 1015 1008 992 622 915 904 1049 803

1190 1103

1.041 1.050 0.987 0.998 1.169 1.040 1.026 1.094 1.105

0.918 1.066

Sundquist, Kinnunen (2004a)

LBU1 LBU2 LBU3 KBU1 KBU2 KSU1 KSU2 KSU3

26.6 32.6 30.0 35.4 24.6 26.7 29.0 27.7

679 700 699 679 687 689 689 695

0.400 0.372 0.394 0.400 0.392 0.573 0.584 0.565

205 220 208 205 209 210 206 208

φ1600

" " " " " " "

φ2000

" " " "

φ2300 "

φ1730

φ1000

" " " "

φ500 " "

4.88 4.55 4.81 4.88 4.78 2.38 2.43 2.40

1.46

" " " "

2.64 " "

1412 1859 1593 1784 1364 992

1053 1004

1406 1725 1763 1607 1448 1039 1017 875

0.996 0.928 1.107 0.901 1.062 1.047 0.966 0.872

Timm (2004)

Ti-1A Ti-2A Ti-3A

40.7 36.0 32.8

500

" "

1.25 1.25 1.18

172 172 246

φ560 φ800 φ800

760 1000 1080

φ175 φ175 φ250

1.02 1.02 1.02

1.18 1.82 1.18

958 624

1407

789 668 1356

0.824 1.071 0.964

Mean value Vtest / Vcalc = 1.032(1± 0.095)

1) d

Bd

D23

ε− for footing with surface load.

2) Overall yield with strain hardening, therefore not included in statistical evaluation.

41

Table 2-6 Test results, flat plates with shear reinforcement. Authors

Test slab No.

fcc MPa

fsy MPa

ρ %

d mm

c mm

column size mm

Bσ d

c−Bε 2d

Vy2 kN

Vcalc kN

Vtest kN

calc

testVV

Andersson (1963)

62 1) 63 1) 64 1)

65 1)

66 2)

67 2)

76 1)

77 1)

78 1)

79 1)

80 1)

81 1)

82 2)

83 2)

26.3 26.4 26.4 26.3 26.8 27.4 27.0 28.0 28.2 28.5 28.1 25.9 27.1 23.5

439 435 435 437 438 434 457 453 461 469 436 440 442 442

0.94 0.94 1.05 1.05 0.78 0.77 1.22 1.20 1.41 1.42 1.06 1.07 1.08 1.09

120 120 120 121 119 121 122 125 120 119 121 120 120 119

φ 1710 " " " " " " " " " " " " "

φ 150 " " " " " φ 300 " " " " " " "

1.25 " " " " " 2.50 " " " " " " "

6.50 " " " " " 5.88 " " " " " " "

364 360 400 409 299 303 555 569 622 627 457 456 463 456

364 360 399 406 299 303 555 569 618 621 457 456 463 456

346 353 371 373 292 294 534 549 606 612 453 471 459 459

0.950 4) 0.981 4) 0.930 4) 0.919 4) 0.977 4) 0.970 4) 0.962 4) 0.965 4) 0.981 4) 0.986 4) 0.991 4) 1.033 4) 0.991 4) 1.007 4)

Hallgren (1966)

HSC3s 1) HSC5s 1) HSC7s 1)

92 91 85

632 604 630

0.82 1.18 0.63

200 201 200

φ 2400 " "

φ 250 " "

1.25 " "

5.38 " "

1338 1831 1033

1338 1687 1033

1329 1631 1106

0.993 0.967 4) 1.071

Sundquist (1977)

D 1) E 1)

26.8 20.4

454 457

0.71 0.71

169 168

φ 1710 "

200 φ 250

1.51 1.49

4.36 4.35

605 601

605 601

580 560

0.959 5) 0.932 5)

Andrä et al (1979)

1 3) 2 3) 3 3) 4 3)

40.0 36.0 21.6 36.0

450 " " "

1.12 1.14 1.14 1.27

230 225 220 267

2000 " " "

300 " " φ 500

1.66 1.70 1.74 1.87

3.58 3.66 3.74 2.81

1819 1765 1653 3023

1819 1765 1540 3022

2119 1904 1537 2956

1.165 5) 1.079 0.998 0.978

Broms (1990b)

7 1+ 2)

20

691

0.99

150

2165

250

2.12

6.23

981

710

1006

1.417 6)

Yamada et al (1992)

K5 2) K7 2)

26.0 27.8

568 "

1.53 "

164 "

1500 "

300 "

2.33 "

3.50 "

1662 1667

1349 1397

1440 1498

1.067 7) 1.072 7)

Beutel, Hegger (2000)

PI-I 2) PI-II 2) P2-I 2) P2-II 2) P2-III 2) P3-I 2) P6-I 2) P7-I 2)

27.3 26.2 37.9 29.8 37.5 23.2 46.3 40.0

550 550 550 550 550 550 550 550

0.806 0.806 0.806 0.806 0.806 1.15 1.753 1.301

190 190 190 190 190 220 220 230

φ 2400 " 8) " " " " " "

400 " " " " 320 " "

2.68 " " " " 1.85 " "

5.08 " " " " 4.60 " "

1287 1283 1299 1289 1298 2286 3497 2868

1283 1277 1299 1289 1298 1635 2522 2375

1151 1055 1326 1109 1276 1624 2349 2117

0.897 4) 0.826 4) 1.021 0.860 0.983 0.993 0.931 7) 0.891 7)

Krüger et al (2000)

PP0B 2)

37.7

500

1.30

121

2750

300

3.16

9.90

609

603

579

0.960

Hegger et al (2001)

Z3 3) Z4 3) Z5 3) Z6 3)

24.0 29.2 25.3 37.0

890 890 562 562

0.80 0.80 1.25 1.25

250 " " "

φ 2400 " 9) " "

φ 200 φ 200 φ 263 φ 200

0.80 0.80 1.05 0.80

4.40 4.40 4.27 4.40

3407 3428 3385 3337

1558 1720 1954 2071

1616 1646 2024 1954

1.037 0.957 1.036 0.944

Mean value Vtest/Vcalc = 0.993(1± 0.063)

1) Bent bars as shear reinforcement. 2) Vertical stirrups as shear reinforcement. 3) Studs as shear reinforcement. 4) Failure outside shear reinforcement, therefore not included in statistical evaluation. 5) Overall yield with strain hardening therefore not included in statistical evaluation. 6) High capacity due to “ductility reinforcement”, not included in statistical evaluation. 7) Extremely high shear reinforcement ratio. 8) Slab width 2750 mm with reinforcement over the whole width. 9) Slab width 3000 mm with reinforcement over the whole width.

42

Table 2-7 Tangential concrete strain and deflection at punching. Comparison with test results.

Authors Specimen fcc

MPa ρ

% ε cpu ·103 ε test·103 δ calc 3)

mm δ test mm

Kinnunen Nylander (1960)

5 6 24 25 32 33

26.8 26.2 26.4 25.1 26.3 26.6

0.80 0.79 1.01 1.04 0.49 0.48

1.97 1.93 1.64 1.64 2.42 2.40

1.7

4.0!!

10.6 10.2 9.5 9.5 28.5 27.5

13.0 12.9 10.4 10.4 27.2 26.4

Andersson (1963)

63 1) 65 1) 67 2) 76 1) 78 1) 80 1) 82 2) 83 2)

26.4 26.3 27.4

27.0 28.2 28.1 27.1 23.5

0.94 1.05 0.77

1.22 1.41 1.06 1.08 1.09

3.38 3.18 3.76

2.89 2.72 3.21 3.15 3.04

4.0 3.8 5.2

2.3

4.3

19.9 16.6 28.0

19.5 16.7 27.4 25.5 22.1

13.2 11.2 14.0

17.1 18.0 13.0 13.0 18.0

Sundquist (1977)

D 1) E 1)

26.8 20.6

0.71 0.71

3.22 3.00

6.0!! 3.4

22.0 17.0

24 18

Tolf (1988)

S1.1 S1.2 S2.1 S2.2 S1.3 S1.4 S2.3 S2.4

28.6 22.9 24.2 22.9 26.6 25.1 25.4 24.2

0.80 0.81 0.80 0.80 0.35 0.34 0.34 0.35

1.58 1.45 1.17 1.15 2.09 1.93 1.28 1.25

1.8 2.8 1.2 1.8 2.4 2.4 1.2 1.2

5.8 5.0 8.2 8.0 10.2 9.1 12.0 11.7

5.5

7.5 15.5

16.0

Hallgren (1996)

HSC0 HSC1 HSC2 HSC3s 1) HSC4 HSC5s 1) HSC7s 1) HSC9 N/HSC8

90 91 86 92 92 91 85 84 95

0.80 0.80 0.82 0.82 1.19 1.18 0.63 0.33 0.80

1.45 1.46 1.38 2.67 1.21 2.27 3.00 2.17 1.47

1.55 1.66 1.35 4.0

1.26 2.25 3.1

1.98 1.64

12.7 12.9 11.8 26.8 9.7 17.3 40.1 34.7 13.1

13.5 12.5 10.5 30 10 16 33 29

13.5

Hassanzadeh (1996)

NS

29.5

0.80

1.31

1.50

9.3

9.3

Hassanzadeh (1998)

B1

45.6

0.31

2.2

3.0

34.8

35

Beutel, Hegger (2000)

P2-II 2) P6-I 2)

29.8 46.3

0.806 1.753

2.65 1.80

29.7 11.1

25 14

Krüger et al (2000)

PP0B 2)

37.7

1.30

2.88

45.8

36

1) Bent bars as shear reinforcement. 2) Vertical stirrups as shear reinforcement. 3) Calculated deflection at calculated punching load.

43

2.5.4 Code predictions

The recorded ultimate loads for flat plates shown in Table 2-4 are in Table 2-8 compared to ultimate load predictions according to the design codes ACI 318-02, Model Code 90 and BBK 04.

These codes treat punching as a form of shear failure. Punching is assumed to occur when the shear stress at a control section on a certain distance from the column reaches a critical value.

Ultimate punching capacity according to ACI 318-02

The control section is placed at the distance 0.5d from the column edge. A control section with four straight sides is permitted for square and rectangular columns.

( ) columns squarefor4v2u dadfV +⋅⋅= ; ( ) columnscircular for πv2u dBdfV +⋅⋅=

daf

f 4for 3

ckv2 ≤= and fck ≤ 69 MPa

Ultimate punching capacity according to Model Code 90

The control section is placed 2 d outside the column edge.

( ) columns squareforπ44vu dadfV +⋅= ; ( ) columnscircular for 4πvu dBdfV +⋅⋅=

( )31

ckv 10012.05.1 ff ⋅⋅⋅⋅= ρξ ; (upper limit for fck = 80 MPa disregarded in Table 2-8)

(m)in with 200.01 dd

+=ξ

Ultimate punching capacity according to BBK 04

The control section is placed 0.5 d outside the column edge.

( ) columns squareforπ4v2u dadfV +⋅= ; ( ) columnscircular for πv2u dBdfV +⋅⋅=

( ) 01.0with 45.0501 ctkv2 ≤⋅⋅+= ρρξ ff

m5.00.2for 6.1 ≤≤−= ddξ

( ) cube c,ck32

ctk 8.0 with 19.0 ffff ck =≈ and fck ≤ 48 MPa

( ) ( )[ ] cube c,ckck32

ckctk 8.0 with 48008.0119.0 fffff =−−⋅≈ and 48< fck ≤ 64 MPa

44

The size effect factor ξ is in design taken as 1.4 for d ≤ 0.2 m according to BBK 04. In order to get realistic evaluation of test specimens with d less than 0.2 m, the expression for ξ is assumed valid also for d < 0.2 m.

All values for Vu are intended to reflect the ultimate capacity according to the different codes. That is why fv is multiplied by 1.5 for Model Code 90, since the design strength instead of the ultimate strength is given in this code.

The load factors are 1.2 for dead load and 1.6 for live load according to ACI 318-02. The corresponding values are 1.35 and 1.5 for Model Code 90. The strength reduction factors in design differ also. They are 0.75 for the American Code and 1/1.5 = 0.67 for the European Code. The Swedish load factors are 1.0 and 1.3, which is compensated by the strength reduction factor 1/(1.2·1.5) for brittle punching failure mode. If the total load comprises 50 % dead load and 50 % live load the total safety factors γ become:

ACI 318-02: ( ) 87.175.0/6.12.15.0 =+=γ

Model Code 90: ( ) 142515135150 ..... =+=γ

BBK 04: ( ) 0725121310150 ...... =⋅+=γ

Model Code 90 predicts the ultimate capacity with a small scatter, which is no wonder because the code expressions are based on regression analysis of a large amount of test results.

BBK 04 displays a larger scatter and a very conservative estimate of the ultimate capacity. The American Code displays the largest scatter, because the code considers neither the strength increase with increasing flexural reinforcement ratio nor the strength reduction with increasing specimen size.

When comparing Table 2-4 and Table 2-8 it is evident that the presented theory can predict the punching capacity of flat plates better than the studied design codes.

No comparison is made for column footings because the code provisions seem to be unrealistic for compact slabs.

45

Table 2-8 Observed ultimate loads of flat plate specimens compared to predictions according to the codes ACI 318-02, Model Code 90, BBK 04. For explanations see next page.

Vtest /Vcalc Authors

Test slab No.

fcc

MPa

fsy

MPa

ρ %

d

mm

c

mm

Column size mm

Vtest kN

ACI 318-02 MC 90 BBK 04

Elstner, Hognestad (1956)

A-1b A-1c A1-d A-1e A2-b A-2c A-7b A-3b A-3c A-3d A-4 A-5 A-6 A-13 B-1 B-2 B-4 B-9 B -14

25.2 29.0 36.8 20.3 19.5 37.4 27.9 22.6 26.5 34.5 26.1 27.8 25.0 26.2 14.2 47.6 47.7 43.9 50.5

332

" " "

321 " " " " "

332 321

" 294 324 321 303 341 325

1.16

" " "

2.50 " "

3.74 " "

1.18 2.50 3.74 0.554 0.476

" 1.01 2.00 3.02

118

" " "

114 " " " " "

118 114

" 121 114

" " " "

1780

" " " " " " " " " " " " " " " " " "

254

" " " " " " " " "

356 " " "

254 " " " "

365 356 351 356 400 467 512 445 534 547 400 534 498 236 178 200 334 505 578

1.241 1.130 0.989 1.348 1.619 1.365 1.736 1.673 1.854 1.663 1.050 1.416 1.395

2) 2) 2) 2)

1.361 1.456

0.971 0.904 0.822 1.017 0.937 0.881 1.064 0.867 0.987 0.927 0.899 0.954 0.806

2) 2) 2) 2)

0.973 0.926

1.367 1.211 1.020 1.541 1.852 1.398 1.862 1.862 2.015 1.731 1.130 1.504 1.505

2) 2) 2) 2)

1.358 1.417

Kinnunen, Nylander (1960)

5 6 24 25 32 33

26.8 26.2 26.4 25.1 26.3 26.6

441 454 455 451 448 462

0.80 0.79 1.01 1.04 0.49 0.48

117 118 128 124 123 125

φ1710

" " " " "

φ150

" φ300

" " "

255 275 430 408 258 258

1.509 1.618 1.458 1.478

2) 2)

0.973 1.050 1.089 1.085

2) 2)

1.635 1.774 1.493 1.522

2) 2)

Moe (1961)

R2 M1A

26.5 20.8

328 481

1.38 1.50

114

"

1780

"

152 305

311 433

1.495 1.493

0.963 1.088

1.663 1.672

Schaeidt et al (1970)

28.5

555

1.31

240

φ2650

φ500

1694

1.706

1.337

1.866

Marti et al (1977)

P-2

34.6

558

1.44

145

φ2600

φ300

600

1.511

1.040

1.493

Pralong et al (1979)

P-5

26.2

515

1.34

154

φ2600

φ300

569

1.517

1.018

1.585

Kinnunen, Nylander, Tolf (1980)

S1

30.6

621

0.574

619

4680

φ800

4915

0.966

1.051

1.687

Tolf (1988)

S1.1 S1.2 S2.1 S2.2 S1.3 S1.4 S2.3 S2.4

28.6 22.9 24.2 22.9 26.6 25.1 25.4 24.2

706 701 657 670 720 712 668 664

0.80 0.81 0.80 0.80 0.35 0.34 0.34 0.35

100 99

200 199 98 99

200 197

φ1190

" φ2380

" φ1190

" φ2380

"

φ125

" φ250

" φ125

" φ250

"

216 194 603 600 145 148 489 444

1.714 1.748 1.300 1.339 1.229 1.276 1.029 0.978

1.064 1.037 0.945 0.966 0.993 1.028 1.004 0.938

1.815 1.921 1.523 1.579 1.576 1.644 1.430 1.362

46

Table 2-8 Continued from previous page. Vtest /Vcalc

Authors Test slab No.

fcc

MPa

fsy

MPa

ρ %

d

mm

c

mm

Column size mm

Vtest kN

ACI 318-02 MC 90 BBK 04

Marzouk, Hussein (1991)

HS 1 HS 2 HS 3 HS 4 HS 5 HS 6 HS 7 HS 8 HS 9

HS 10 HS 11 HS 12 HS 13 HS 14 HS 15 NS 1 NS 2

67 70 69 66 68 70 74 69 74 80 70 75 68 72 71 42 30

490

" " " " " " " " " " " " " " " "

0.49 0.84 1.47 2.37 0.64 0.94 1.19 1.11 1.61 2.33 0.95 1.52 2.00 1.47

" "

0.94

95 " "

90 125 120 95

120 " "

70 " "

95 " "

120

1500

" " " " " " " " " " " " " " " "

150

" " " " " " " " " " " "

220 300 150

"

178 249 356 418 365 489 356 436 543 645 196 258 267 498 560 320 396

2)

0.961 1.380 1.786

2) 1.355

4) 1.214

4) 4) 1) 1) 1)

1.469 1.327 1.592 1.451

2)

0.852 1.015 1.111

2) 1.161 1.064 0.984 1.058 1.082

1) 1) 1)

1.211 1.186 1.077 1.245

3) 3) 3)

1.944 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3)

1.608 1.877

Tomaszewicz (1993)

65-1-1 95-1-1

115-1-1 95-1-3

65-2-1

95-2-1D 95-2-1

115-2-1

95-2-3 95-2-3D

95-2-3D+ 115-2-3

95-3-1

64 84 112 90

70 88 87 119

90 80 98 108

85

500 " " "

500 " " " " " " " "

1.49

" "

2.55

1.75 " " "

2.62 " " "

1.84

275

" " "

200 " " "

200 " " "

88

2500

" " "

2200 " " "

2200 " " "

1100

200

" " "

150 " " "

150 " " "

100

2050 2250 2450 2400

1200 1100 1300 1400

1450 1250 1450 1550

330

1.472

4) 4) 4)

1.536 4 4) 4)

4) 4) 4) 4)

4)

1.150 1.152 1.047 1.004

1.078 0.916 1.086 1.054

1.047 0.939 1.018 1.053

1.024

1.889

3) 3) 3)

3) 3) 3) 3)

3) 3) 3) 3)

3) Hallgren (1996)

HSC0 HSC1 HSC2 HSC4 HSC6 HSC9

N/HSC8

90 91 86 92 104 84 95

643 627 620 596 633 634 631

0.80 0.80 0.82 1.19 0.60 0.33 0.80

200 200 194 200 201 202 198

φ2400

" " " " " "

φ250

" " " " " "

965

1021 889

1041 960 565 944

4) 4) 4) 4) 4) 2) 4)

0.977 1.030 0.949 0.916 1.011

2) 0.924

3) 3) 3) 3) 3) 2) 3)

Ožbolt et al (2000)

21

569

0.80

190

φ2400

400

615

0.898

0.874

1.147

Sundquist, Kinnunen (2004b)

C1 C2 D1

24.0 24.4 27.2

718

" "

0.80 0.80 0.64

100 100 125

φ 1190

" "

φ 250 φ 250 φ 125

270 250 265

1.500 1.381 1.550

1.134 1.046 1.023

1.646 1.506 1.791

Mean value Vtest / Vcalc = 1.41(1± 0.17) 1.02(1± 0.10) 1.60(1± 0.15) Compare thesis: Mean value Vtest / Vcalc = 1.021(1± 0.073) 1) Not included in the statistical evaluation due to the small effective depth 70 mm. 2) Overall yield with membrane action and strain hardening, therefore not included in the statistical evaluation. 3) fcc (= 0.8 fc,cube) is larger than 64 MPa (= upper limit according to BBK 04). 4) fcc is larger than 69 MPa (= upper limit according to ACI 318-02).

47

3 Theory for eccentric punching

When determining the punching capacity of a flat plate existing design codes presuppose that the transferred moment between slab and column is defined. That bending moment is, however, normally a statically indeterminate quantity, which cannot be estimated as accurately as for a beam-column frame. Therefore, another concept is proposed here – imposed slab rotation in relation to the column. Both the imposed rotation and the rotation capacity of a flat plate can be assessed with good accuracy. Conservative results are achieved if the column is considered stiff in relation to the slab.

3.1 Code approach

Transfer of moment between slab and columns – so called unbalanced moment – can occur due to gravity loading or due to story drift, i.e. the lateral displacement between stories caused by wind or earthquake.

As described previously, most codes assume that punching occurs when the shear stress at a control section on a certain distance from the column reaches a critical value. An unbalanced moment is thereby considered partly transferred by “eccentricity of shear”. The shear stress at the control section due to this part of the unbalanced moment plus the shear stress caused by concentric loading shall fall below the shear stress capacity defined by the code.

However, no generally accepted method for assessment of the unbalanced moment seems to exist. When caused by gravity loading or story drift, the unbalanced moment is a statically indeterminate quantity that has to be determined by some form of approximate frame analysis where due respect should be paid to the fact that the flexural reinforcement at the column usually yields before punching occurs. This means that a flexural hinge forms at the column, which in turn implies that an analysis based on elastic conditions cannot correctly describe the true behaviour of the system in the strength limit state.

3.2 Introduction

Due to shortcomings of the code approach, a safer concept is proposed here – imposed rotation of the column in relation to the slab (or vice versa). A simple example may describe the principle:

Study the first interior column of a flat plate structure with equal span widths in both directions. The column is assumed stiff in relation to the slab:

21070 qL.m = (strip moment per unit width assuming strip acting as beam simply supported on the columns)

21250 qL.m = (strip moment at support assuming zero support rotation)

48

( ) 22 018010701250∆ qL.qL..m =−=

q = 15 kN/m2 ; 321=L

h

3

33

321212 ⋅==

LhI

E = 10⋅106 kN/m2

3337

310533212

103150180

3∆ −⋅=⋅⋅

⋅⋅

⋅⋅=⋅= .

LL.

EILmθ

In this simplified example, the slab shall be able to resist an imposed slab rotation in relation to the column equal to 3.5⋅10−3 radians. At least six times larger rotations of the column in relation to the flat plate may be imposed due to story drift during a severe earthquake.

3.3 Approximate theory of elasticity

Figure 3-1 depicts a common test set-up for eccentric punching. It resembles the one used for concentric punching described in Chapter 2. The influence of the unbalanced moment is supposed to be mainly concentrated to the close vicinity of the column and therefore the same specimen size as for concentric loading seems to be a reasonable choice. This assumption will be evaluated later in this chapter.

H

V

H

c

Figure 3-1 Test set-up for eccentric punching.

The fan-type crack pattern at concentric loading is assumed to remain when the column is forced to rotate. The sector elements between the radial flexural cracks will then deflect with varying fictitious deflection ∆ in relation to the column as shown in Figure 3-2. The torsional resistance of the sector elements is considered negligible.

49

θ

sinϕ

c

M

ϕ

M 0 ϕcπ

R sin

∆

Figure 3-2 Definition of parameters.

The support reaction of the sector elements is proportional to their deflection. The total reaction R for each half of the specimen is

∫ =⋅⋅

⋅=π

0

00π

d2π

sin Rcc

RR ϕ

ϕ (3.1)

The quantity R is consequently the total shear force that is transferred to the each half of the column due to the column rotation.

The relation between unbalanced moment Mu and the maximum value R0 of the support reaction along the slab edge can be expressed as

∫ ⋅=⋅=⋅⋅⋅

=π

000

0u82

ππ4

d2

sin2π

sin2

;cRcRccc

RMϕϕ

ϕ40ucRM ⋅= (3.2)

50

The deflection ∆ due to a concentric load R0 can be derived from Eqs. (2.27) and (2.28):

22π422

1π4

02

20 Bc

EIcRBc

EIc

cBR −

⋅⋅≈−

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=∆ (3.3)

The rotation due to an unbalanced moment is

EI

MEI

cRBc π2π8

2 u01 =

⋅=

−=

∆θ (3.4)

L

2

R R

L L2

L 2

2L

Figure 3-3 Unbalanced moment due to story drift.

Story drift during earthquakes or wind load as illustrated in Figure 3-3 is a common cause for unbalanced moment. The broken lines represent the size of test specimens that are assumed to simulate the behaviour of the slab near the column. The effect of the column rotation within the broken lines is evaluated above. The reaction R according to Eq. (3.1) is conservatively assumed to act at the distance L/4 from the column. If the slab width resisting this force R is assumed equal to the column strip width (L/2), the additional rotation of the column can be assessed as

51

( )EI

MEI

RLL.EI

LR

π43250162 u

2

2 ==⋅

=θ becauseπ0R

R = and c

MR u

04

= (3.5)

504π2π

1

2 .==∴θθ ;

EIM

.π2

51 u21 =+= θθθ (3.6)

EIM ⋅⋅= θ2.4u (3.7)

Compare Aalami (1972) who used the theory of elasticity for an isotropic thin plate to derive

EIM ⋅⋅= θ10.4el u, (for a/L = 0.05) (3.8)

The simple model shown in Figure 3-2 thus seems to be accurate enough to form basis for a developed model that can describe the non-linear behaviour of a reinforced concrete flat plate subjected to gravity load plus unbalanced moment.

3.4 Model for eccentric punching of flat plates

The model described in the previous section only reflects the global elastic behaviour of the system; it does not consider the local effects of force transfer from the column to the slab or vice versa. Figure 3-4 shows a possible load path for these effects.

The total unbalanced moment is assumed transferred to the slab by a strut-and-tie system similar to the model often used for beam-column connections. There is a difference, however. The horizontal compression strut forces in the slab are larger than the tension tie forces from the reinforcement that passes through the column. Circumferential cracking around the column isolates other reinforcement bars from participating in the moment transfer. In-plane forces in the slab therefore balance the compression struts. Any transfer of unbalanced moment by “eccentricity of shear” in the slab is neglected. Such force effects should be regarded as fictitious quantities that in reality are replaced by the two horizontal compression struts.

52

0.5 Mu

0.5 Mu

Figure 3-4 Unbalanced moment transfer by strut-and- tie system.

The half of the slab where the unbalanced moment causes additional tension in the top reinforcement for negative moment in the slab is denoted “the negative slab half”. The opposite half where the unbalanced moment can cause tension in the bottom reinforcement of the slab is consequently called “the positive slab half”.

A sector element in the negative slab half is depicted in Figure 3-5. The large radial compression strut at the column connection is balanced by the tangential reinforcement and to a lesser degree by the few radial reinforcement bars passing through the column or within its close vicinity. The radial compression stress near the column is consequently much larger than at concentric gravity loading.

Corresponding forces act on the positive slab half. The tangential tension strains due to the unbalanced moment initially reduce the tangential flexural compression strain due to gravity loading before any tension stress develops in the tangential bottom reinforcement.

Figure 3-5 The slab resists unbalanced moment by radial concrete compression and tangential reinforcement.

53

The relation between concentric column load and slab deflection within the circle with diameter c is depicted in Figure 3-6. The broken line illustrates the behaviour of the slab due to a column rotation. The concentric gravity load V causes the slab deflection δV. A column rotation will cause non-uniform reaction intensity along the circle with diameter c as described in the previous section and illustrated in Figure 3-7. The sector element reactions are denoted Rti in the negative slab half and Rbi in the positive slab half. The reactions Rti and Rbi denote the column reactions for a uniform slab deflection ∆sinφi (all around). Punching failure is assumed to occur when the sum of the fictitious deflection ∆ of the slab in the negative slab half due to a column rotation θ and the deflection δV due to concentric gravity loading V reaches the ultimate deflection δε that is associated with concentric punching failure.

ε

V

Rt i

V

Vε

∆

Rb

i

y2δt i

negative slab halfpositive slab half

1

1k

δδδ

δ

isinϕ

isinϕ∆

b i

I

∆∆

Figure 3-6 Fictitious column reactions Rti and Rbi due to overall slab deflection ∆sinφi.

54

The tangential flexural stiffness of the slab near the column for column rotation is reduced because the lever arm for the reinforcement is reduced due to the position of the radial compression strut. The reduced stiffness EI1 can be assessed in accordance with Eq. (2.9) with x replaced by h/2:

⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛ −⋅⋅⋅=

dh

dhdEEI s 6

12

131 ρ (3.9)

The mean value 1EIEI ⋅ , which is equal to kI·EI, is assumed to be representative for the overall behaviour of the slab due to a column rotation where

21

1Ik ⎟

⎠⎞

⎜⎝⎛=

EIEI (3.10)

M

ϕϕsin∆

ϕ ϕ ϕ

ϕ

1 2i

n

c

Figure 3-7 Variation of fictitious slab deflection ∆ due to column rotation.

Furthermore, the radial compression struts from the column cause a larger radial curvature of the sector elements near the column than at concentric loading. The additional curvature results in an additional column rotation; see Figure 3-8.

55

Relation between unbalanced moment Mu and maximum value R0 of support reaction along slab edge:

40

ucR

M⋅

= (3.2)

Radial bending moment per unit width along axis x:

⎟⎠⎞

⎜⎝⎛ −=

⋅⎟⎠⎞

⎜⎝⎛ −⋅=

crM

rrcR

m 21π∆

12

∆π2

u0r ϕ

ϕ (3.11)

Radial compression strain in the slab along axis x due to the horizontal compression struts in the slab:

2c10

rc

6hE

m⋅

≥ε (3.12)

The major part of the surface shortening r∆ due to the compression strain εc along axis x will occur at the column and only a minor part at the slab edge:

∫ ∫ ⎟⎠⎞

⎜⎝⎛ +−

⋅⋅=⎟

⎠⎞

⎜⎝⎛ −

⋅⋅=⋅≈→

⋅≥

2

2

2

22

10

u2

10

ucr2

c0

rc 1ln

π6

d21π

6d6 /c

/B

/c

/B cc cB

Bc

hEM

rcrhE

Mr∆

hEm

εε

⎟⎠⎞

⎜⎝⎛ +−

⋅⋅=≥

cB

Bc

hE

Mh c

1lnπ

1223

ur2

∆θ (3.13)

r

ϕπ

∆2

0R m r

c 2

B

ϕ∆

cε

cε

θ

h

M2

r∆

u

x

Mu

Figure 3-8 Column rotation due to radial curvature of sector elements.

56

With these assumptions it is possible to determine the flat plate capacity for unbalanced moment for a given concentric column reaction V. The calculation procedure may be best illustrated by a numeric example taken from a well-documented test, Ghali et al (1976), Specimen SM 1.0; see Appendix F and Table 3-1.

The calculation steps are:

1. Perform the normal punching evaluation for concentric loading in accordance with Appendix A or B.

2. Determine the deflection δV due to the actual column load V.

3. Guess the additional overall deflection 2∆Μ due to the imposed ultimate column rotation. Half of this deflection is assumed to affect the slab before column rotation and the other half is assumed to affect the slab after full column rotation in order to simulate the continuously increasing deflection when the column rotates.

4. Determine the additional fictitious varying deflection ∆·sinφ along the circle with diameter c due to a column rotation, where ∆ = (δε − δV - 2∆M). (3.14) Divide each half-circle in “n” equal parts corresponding to the angels iϕ ; see Figure 3-7.

⎟⎠⎞

⎜⎝⎛ −=

nnii 2

ππϕ (3.15)

The corresponding total deflections of the sector elements are thus, with regard to the overall deflection ∆M at this stage : δ ti = δV + ∆sinφi + ∆M. (3.16) (index “t” stands for deflection causing tension in top reinforcement)

5. Determine the fictitious reactions Rti for unbalanced moment on the negative slab half due to overall deflections δti in step 4. Correct result is achieved by calculating the reactions from the curve for concentric loading for the deflections δti = δV + kI·∆sinφi + ∆M (3.17)

with the factor kI according to Eq. (3.10) and Rti = V{δti} - V

6. Determine the total real reaction Rt for the negative slab half due to column rotation and additional deflection ∆Μ:

∑=n

in

RR

1

tt 2

(3.18)

7. Determine the part of the total unbalanced moment caused by the reactions Rti:

∑ ⋅⋅=n

ii c

nR

M1

tt sin

22ϕ (3.19)

57

8. Determine the deflections on the positive slab half: δbi = δV - ∆sinφi + ∆Μ (3.20)

9. Determine the concentric column reactions Rbi corresponding to deflections in step 8. Observe the reduced stiffness once tension in the bottom reinforcement occurs; see Figure 3-6.

10. Determine the total reaction Rb for the positive slab half due to column rotation and additional deflection ∆Μ:

∑=n

in

RR

1

bb 2

(3.21)

11. Check force equilibrium by determining A:

A = Rb – Rt -∆Μ·Vy1/δy1 (3.22)

12. Repeat the calculation from step 3 with a larger value of ∆Μ if A > 0 until A = 0. If A < 0 decrease ∆Μ.

13. Determine the part of the unbalanced moment caused by the reactions Rbi:

∑ ⋅⋅=n

ii c

nRM

1

bb sin

22ϕ (3.23)

14. Determine the unbalanced moment capacity btu MMM += (3.24)

15. Determine the column rotation neglecting additional radial curvature of sector elements:

Bc −

⋅=∆θ 2

k1

I1 (3.25)

The factor 1/kI takes the effect of the reduced tangential flexural stiffness near the column into account.

16. Determine the column rotation due to radial curvature of the slab sector elements due to the radial compression strut according to Eq. (3.13):

⎟⎠⎞

⎜⎝⎛ +−

⋅⋅=

cB

Bc

hEM

c

u 1lnπ

123

102θ (3.13)

17. Determine the rotation capacity of the system due to deformations of the slab within the circle with diameter c: 21 θθθ +=u (3.26)

58

3.5 Comparison with test results

Table 3-1 Unbalanced moment. Test results. Authors

Test slab No.

fcc

MPa

fsy

MPa

ρ / ρc

%

d h

mm

c

m

Column

size

mm

Vtest

kN

e test

mm

testtest

δθ

% / mm

Vcalc kN calc

calcδ

θ

% / mm calc

testVV

Moe (1961)

M2

M3

M6

M7

M8

M9

M10

25.7

22.8

26.5

25.0

24.6

23.2

21.1

481

"

327

"

"

"

"

1.50 / 0.0

"

1.34 / 0.0

"

1.34 / 0.57

1.34 / 0.0

1.34 / 0.57

152114

1.78

"

"

"

"

"

"

305

"

254

"

"

"

"

292

207

239

311

150

267

178

196

338

168

61

437

127

308

269

202

272

322

180

277

198

0.9 / 0

1.2 / 0

1.0/ 0

0.6 / 0

1.6 / 0

0.8 / 0

1.3 / 0

1.086

1.025

0.879

0.966

0.8331

0.964

0.899

Narasimhan (1971)

L1 26.6 398 1.05 / 1.05170143 2.0

305 399 306 -- 338

1.2 / 0

1.180

Ghali et al (1974)

B5NP 28.3 345 1.39 / 1.39152115 1.8 305 100 1960 -- 74.5 2.6 / 0 1.342

Ghali et al (1976)

SM0.5

SM1.0

SM 1.5

36.8

33.4

39.9

476

"

"

0.53 /0.18

1.05 / 0.35

1.58 / 0.53

152121 1.8

"

"

305

"

"

129

"

"

775

984

1031

6.5/6

2.7

2.0

126

122

127

6.9/1.8

2.6 / 0

1.9/ 0

1.024

1.057

1.016

Islam, Park (1976)

2

31.9

374

1.0 6 / 0.53 89

70 1.143 229 28 1346

5.0/0

24.2

6.2 / 0

1.157

Elgabry,Ghali (1987)

1 35 452 1.07 / 0.46152116 1.8 254 150 867 -- 126 3.0 / 0 1.190

Pan , Moehle 1989

AP1

AP3

29.3

31.7

484

"

0.86 / 0.29

" 121103 1.83

"

274

"

104

53

548

1536

1.7

3.4

137

61

2.8 / 0

4.2 / 0

0.7592

0.8692

Hawkins et al (1989) h=152 h = 114 h = 152

6AH

9.6AH

14AH

6AL

9.6AL

14AL

7.3BH

9.5BH

14.2BH

7.3BL

9.5BL

14.2BL

6CH

9.6CH

14CH

6CL

14CL

31.3

30.7

30.3

22.7

28.9

27.0

22.2

19.8

29.5

18.1

20.0

20.5

52.4

57.2

54.7

49.5

47.7

472

415

420

472

415

420

472

472

415

472

472

415

472

415

420

472

420

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.73 / 0.40

0.95 / 0.48

1.42 / 0.75

0.73 / 0.40

0.95 / 0.48

1.42 / 0.75

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.60 / 0.28

1.40 / 0.63

121

118

114

121

118

114

83

83

79

83

83

79

121

118

114

121

114

1.83

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

305

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

169

187

205

244

257

319

80

94

102

130

142

162

186

218

252

273

362

535

522

489

134

135

136

488

483

500

98

117

129

511

519

529

135

136

176

181

189

259

315

315

91

89

101

142

157

151

201

242

240

297

400

4.2 / 2.4

2.7 / 0.5

1.6 / 0

2.0 / 1.4

1.4 / 0.9

0.8 / 0

4.7 / 1.4

3.1 / 0

3.1 / 0

2.1 / 1.0

1.5 / 0.6

1.2 / 0

6.5 / 5.4

4.0 / 2.2

2.4 / 0

4.6 / 3.2

1.1 / 0.6

0.960

1.033

1.085

0.942

0.816

1.013

0.879

1.056

1.010

0.915

0.904

1.073

0.925

0.901

1.050

0.919

0.905

1) Presupposes restraint for uplift. Mean value Vtest / Vcalc = 1.006(1± 0.112)

2) Cyclic loading, not included in statistical evaluation.

59

The tests by Ghali et al (1976), Islam and Park (1976), and Pan and Moehle (1989) are especially interesting because they also report the column rotations. It is evident from the table that the presented theory can predict the unbalanced moment capacity and the corresponding rotation with acceptable accuracy. The tests by Pan and Moehle (1989) were cyclic load tests simulating story drift during an earthquake. That explains why the recorded ultimate unbalanced moments were lower than the calculated values for monotonic loading.

The recorded unbalanced moments are in Table 3-2 compared to predictions according to the design codes ACI 318-02, Model Code 90 and BBK 04 in the same way as for concentric loading in Section 2.5.4.

According to ACI 318-02, the shear stress due to concentric column load and unbalanced moment is calculated as

vv fW

eVAV

≤⋅

⋅+= γτ (3.27)

where 4.0;3

;6

)(34;)(4 v

ckv

32 ==++⋅=+= γ

ffddadWdadA

WAe

AfV

⋅+

⋅=∴

4.01

vu (3.28)

The corresponding values for Model Code 90 are:

( ) ( )( ) ( )

(mm)in with 200160

1001205116π245144

v

31

ckv22

dd

;.

;f..f;dada.dW;ddaA

+==

⋅⋅=+++=⋅+=

ξγ

ρξπ

WAe

AfV

⋅+

⋅=∴

6.01

vu (3.29)

The approach in BBK 04 is similar to the approach by Moe (1961):

( )

( )

MPa 6448for )48(008.01 MPa; 48for 1

19.0;5.11

1;m0.50.2for 6.1

;45.0501;)4(;

ckckck

32

ckctk

ctkvu

≤≤−−=≤=

⋅≈

++

=≤≤−=

⋅⋅+=+=⋅⋅=

ffkfk

fkf

dae

dd

ffdadAAfV

ηξ

ρξπη v

dae

AfV

++

⋅=∴

5.11

vu (3.30)

60

The size effect factor ξ is equal to 1.4 for d ≤ 0.2 m according to BBK 04. In order to get realistic evaluation of test specimens with d less than 0.2 m the expression for ξ is assumed valid also for d < 0.2 m.

All values for fv are intended to reflect the ultimate strength according to the different codes. That is why fv is multiplied by 1.5 for Model Code 90, because this code gives the design shear strength instead of the ultimate strength.

When comparing the results due respect should be paid to the total safety factors, which were derived in Section 2.5.4.

ACI 318-02: γ = 1.87

Model Code 90: γ = 2.14

BBK 04: γ = 2.07

Model Code 90 displays a very good prediction result with small scatter. The mean value of Vtest / Vcalc is less than 1.0, however. Both BBK 04 and ACI 318-02 show a larger scatter, which is partly compensated by the mean values being larger than 1.0.

61

Table 3-2 Unbalanced moment test results. Comparison with code predictions.

Authors Test slab

No. fcc

MPa

fsy

MPa

ρ / ρc

%

d h

mm

c

m

Column

size

mm

ACI 318

Vtest / Vcalc

MC 90

Vtest / Vcalc BBK 04

Vtest / Vcalc

Moe (1961)

M2

M3

M6

M7

M8

M9

M10

25.7

22.8

26.5

25.0

24.6

23.2

21.1

481

"

327

"

"

"

"

1.50 / 0.0

"

1.34 / 0.0

"

1.34 / 0.57

1.34 / 0.0

1.34 / 0.57

152114

"

"

"

"

"

1.78

"

"

"

"

"

"

305

"

254

"

"

"

"

1.407

1.334

1.279

1.331

1.302

1.397

1.380

0.985

0.889

0.878

0.949

0.827

0.952

0.876

1.664

1.659

1.517

1.521

1.652

1.669

1.761

Narasimhan (1971)

L1 26.6 398 1.05 / 1.05 170143

2.0

305 1.639 1.245

2.032

Ghali et al (1974)

B5NP 28.3 345 1.39 / 1.39 152115

1.8 305 1.911 1.242 2.487

Ghali et al (1976)

SM0.5

SM1.0

SM 1.5

36.8

33.4

39.9

476

"

"

0.53 / 0.18

1.05 / 0.35

1.58 / 0.53

152121

"

1.8

"

"

305

"

"

0.978

1.216

1.152

0.946

0.909

0.772

1.404

1.512

1.393

Islam, Park (1976)

2

31.9

374

1.0 6 / 0.53

8970

1.143 229 1.131 0.783

1.383

Elgabry, Ghali (1987)

1 35 452 1.07 / 0.46 152116

1.8 254 1.674 1.160 2.077

Pan , Moehle (1989)

AP1

AP3

29.3

31.7

484

"

0.86 / 0.29

" 121103

1.83

"

274

"

1.0121

1.0631

0.7941

0.8051

1.3001

1.4061

Hawkins et al (1989) h=152 h = 114 h = 152

6AH

9.6AH

14AH

6AL

9.6AL

14AL

7.3BH

9.5BH

14.2BH

7.3BL

9.5BL

14.2BL

6CH

9.6CH

14CH

6CL

14CL

31.3

30.7

30.3

22.7

28.9

27.0

22.2

19.8

29.5

18.1

20.0

20.5

52.4

57.2

54.7

49.5

47.7

472

415

420

472

415

420

472

472

415

472

472

415

472

415

420

472

420

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.73 / 0.40

0.95 / 0.48

1.42 / 0.75

0.73 / 0.40

0.95 / 0.48

1.42 / 0.75

0.60 / 0.28

0.96 / 0.50

1.40 / 0.63

0.60 / 0.28

1.40 / 0.63

121

118

114

121

118

114

83

83

79

83

83

79

121

118

114

121

114

1.83

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

305

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

"

1.095

1.251

1.396

1.024

0.991

1.336

0.989

1.223

1.186

0.926

1.006

1.239

0.907

1.065

1.338

0.777

1.141

1.010

0.988

0.977

0.968

0.834

0.982

0.842

0.936

0.851

0.803

0.809

0.873

0.914

0.933

1.029

0.837

0.922

1.542

1.545

1.692

1.407

1.146

1.537

1.344

1.566

1.394

1.195

1.190

1.438

1.211

1.279

1.566

0.950

1.194

Mean value 1.23 (1± 0.20) 0.93 (1± 0.13) 1.51 (1± 0.20)

Compare Thesis: Mean value Vtest / Vcalc = 1.006(1± 0.112)

1) Cyclic loading, not included in statistical evaluation.

62

3.6 Column rotation capacity

Flat plates display a much more pronounced non-linear behaviour a both gravity loading and story drift than beam-column frames. Many methods have been proposed to solve the problem of estimating a design value for the unbalanced moment – with limited success. The code ACI 318-02 for instance allows flat plates to be designed according to the “Equivalent Frame Method”, which introduces a torsional member between the slab and the column to simulate the flexible force transfer of unbalanced moment between column and slab. This approach may seem elegant, but it cannot handle the decreasing slab stiffness at increasing gravity load or increasing unbalanced moment because the stiffness of the torsional member is assumed constant irrespective of the load level.

These shortcomings are overcome with the approach described in Section 3.4. The calculation procedure is laborious and is only included here for verification of the model, it is not intended for use in the design office. However, as proposed in Section 3.2, flat plates should be checked for rotation capacity rather than unbalanced moment capacity. The reason for this is two-fold; the actual rotation can be determined with good precision by means of standard methods as indicated in Section 3.2 and it is simple to determine a conservative value for the rotation capacity of the column in relation to the slab in a flat plate structure.

The rotation capacity was derived in Section 3.4, conservatively expressed as

⎟⎠⎞

⎜⎝⎛ +−

⋅⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−⋅=

cB

Bc

hEM

Bc c1ln

π12

1k2

310

u

ε

Vε

Iu δ

δδθ (3.31)

or more conveniently as

⎟⎠⎞

⎜⎝⎛ +−

⋅⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−⋅=

cB

Bc

hEM

VV

Bc c1ln

π12

1k2

310

u

ε

ε

Iu

δθ (3.32)

with kI according to Eq. (3.10) and Mu taken as the lowest value according to Eqs. (3.33) to (3.37).

Vε and δ ε are output values from the concentric punching check described in Chapter 2. The punching deflection εδ can always be determined without any iteration.

When δ ε approaches or exceeds δy2 (at low reinforcement ratios), then Eq. (3.32) becomes very conservative. In that case the more exact Eq. (3.31) is recommended.

The computed values Vε and δ ε shall be divided by the strength reduction factor γn·γm in order to receive the design rotation capacity, see also Chapter 4.

An upper bound for the unbalanced moment can be assessed by combining Eqs. (3.4), (3.10), and (3.13):

3c10I

u

π

1ln12

kπ21

hEcB

Bc

EI

M

⋅⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛

+⋅⋅

=θ (3.33)

63

However, the unbalanced moment is limited by the lesser of the punching capacity, the flexural yield capacity of the slab, or the local compression strength of the horizontal compression struts:

1. The punching failure load limits the unbalanced moment according to the lesser of Mu1a and Mu1b that are derived from Eq. (3.2):

( )4

4k

y2u1b

y1

y1εIu1a

cVVM

cVV

M

⋅−=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅=

δδ

(3.34)

2. The capacity corresponding to overall yield of the positive (bottom) reinforcement may be governing:

42y

c2u

cVVM ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅≤

ρρ

(3.35)

where

cB

mV−

⋅=1

π2y2y and ρc = bottom reinforcement ratio.

3. If the flat plate is provided with shear reinforcement or if the flexural reinforcement ratio is so low that punching occurs with yield of all flexural reinforcement, then the sum of negative and positive flexural capacities defines an upper bound for the unbalanced moment:

( ) =⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=⋅⋅⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅+−≤

81

21

4c

2yy2c

2y3ucVcVVVVM

ρρ

ρρ

cB

cm−

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=

14π1 c

y ρρ

(3.36)

4. The local compression strength at the column connection for the horizontal compression struts in the slab may limit the unbalanced moment:

31

2ck4u 50

15004

3 ⎟⎠⎞

⎜⎝⎛⋅

⋅⋅⋅≤

h..hafM (3.37)

where h = slab thickness [m] and a = column width.

64

The ultimate rotation capacity for a slender flat plate structure versus reinforcement ratio and column size is displayed in Figure 3-9, which is derived from the “exact” expressions in Section 3.4. The span width is 7.0 m and the effective depth is 0.20 m corresponding to a slenderness L/d equal to 35. The factored uniformly distributed load in an office building would typically be 11 kN/m² (with Swedish load factors). The column reaction would then be 11·7.02 = 539 kN, which corresponds to a required ultimate punching capacity 1.2·1.5·539 = 970 kN with a required reinforcement ratio ρ = 0.8 % according to Figure 3-9 (interior column with θ = 0).

The figure demonstrates that it may be unfavourable to add support reinforcement in a flat plate in order to resist imposed column rotation due to for instance pattern loading. It is also evident that flat plates with moderate reinforcement ratio can resist large imposed column rotations, which was found experimentally already by Ghali et al (1976), and such flat plates may display no reduction in punching capacity when subjected to imposed column rotation.

0.04

0.03

0.02

0.01

01000 15000 500

VkN

θ

ρ

0.0080.010

0.012

u

= 0.006

Figure 3-9 Column rotation capacity versus reinforcement ratio. (c = 2.8 m, L = 7.0 m, d = 0.20 m, h = 0.23 m, column 0.5x0.5 m², fck =30 MPa, fsy=420 MPa, ρ’ = 0.5ρ)

65

4 Design

This Chapter demonstrates how the presented theory shall be applied for design of flat plates. Comparison is made with current structural design codes.

4.1 Design of support reinforcement at square panels

In design, the equations in this thesis should be used to first calculate the ultimate punching capacity or the ultimate rotation capacity of the slab using the characteristic strength values fck for concrete and fsy for reinforcing steel and nominal dimensions. The design punching capacity is then taken as the calculated ultimate punching capacity divided by the applicable strength reduction factor for concrete. In Sweden this factor should be γn·γm = 1.2·1.5 = 1.8, where the factor γn = 1.2 corresponds to Swedish safety class 3, which is applicable if the probable failure mode is brittle. The design rotation capacity is calculated in a similar manner. The quantities Vε and δ ε in Eqs. (3.31) and (3.32) shall be divided by the strength reduction factor γn·γm to derive the design value of the rotation capacity at factored loading.

The punching failure in flat plates usually occurs when the tangential compression strain at the column edge due to the bending moment reaches a critical value. It is therefore essential that this bending moment be estimated in a correct way.

The basic case – a flat plate structure with square panels – is treated in Section 2.2. The bending moment distribution near the column is assumed polar-symmetric within a circle with the diameter c, where c/2 is the distance from the column to the line where the radial bending moment is zero. The equations in Section 2.2 will then give correct results for interior columns in flat plates with square panels if c is taken as 0.4L, where L is the span width.

4.2 Bending moments in a continuous flat plate

A rational method for calculating the bending moments in the general case with varying span widths and rectangular panels is described in the following. The flat plate structure is divided into strips in accordance with Figure 4-1. Each strip is assumed pin-supported on the columns and the lines of zero shear for the perpendicular strips bound the strip laterally. The bending moments per unit width in the strips are calculated according to the theory of elasticity with due respect paid to the effect of pattern loading. The negative strip moments can normally be determined for full load on all bays.

66

L1

2L2L0.5w

Figure 4-1 Definition of strip parameters.

The negative bending moment per unit width in a strip is denoted ms. Since the strip is supported on columns, ms is not uniformly distributed over the width of the strip, it is concentrated toward the columns.

If the strip is unsymmetrical in relation to the columns the following procedure presupposes that the calculation is performed for a symmetrical strip with width two times the width of the larger of the two half-strips. Only reinforcement within the effective width w of the strip is considered active, where w is the lesser of the width of the strip and the span width L1.

The average negative bending moment per unit width within the width c - the column strip - is denoted msc, and the average negative bending moment per unit width on the remaining effective width of the strip - the middle strip – is denoted msm.

The following provisions are adapted to the approach in Chapter 2, where the polar symmetric conditions within the circle with diameter c were studied. The width c is taken as 0.4w.

67

Let us first study the basic case, a continuous flat plate with square panels. The average bending moment msc within the column strip with width 0.4w corresponds to the fan-type yield line depicted in Figure 4-2. The average bending moment msm within the remaining width – the middle strip – is determined by the conditions of moment equilibrium. Please note that the moment reduction due to the column extension is concentrated to the column strip only:

V.mw.mw.VwVwqLwm 0330604021212 smsm

2

s −=→⋅+⋅−=⋅−=⋅−=⋅π

(4.1)

⎟⎠⎞

⎜⎝⎛ −−=

cBVm ε

sc 12π

(4.2)

The bending moment within the column strip, msc, is identical to the average bending moment within the circle with diameter c according to Chapter 2. Since punching normally occurs before all reinforcement reaches the yield limit, more reinforcement will be required within the column strip than corresponding to the average bending moment according to Eq. (4.2).

Bc

m r =0

⎟⎠⎞

⎜⎝⎛ −−=

cBVm 1

2π

Figure 4-2 Fan-type yield lines.

In the general case with rectangular slab panels, the part of the strip moment sm that exceeds

the bending moment 12Vm = shall be evenly distributed over the width w:

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ −−=

121

2 sε

scVm

cBVm

π (4.3)

⎟⎠⎞

⎜⎝⎛ ++−=

12033.0 ssm

VmVm (4.4)

68

Observe that the bending moment ms is a quantity with negative sign and that the term

⎟⎠⎞

⎜⎝⎛ +

12sVm shall be omitted if it turns out to be positive, which for instance occurs when

L1 < L2 .

Example 1. Interior panel of a flat plate with span width 7.2 m in both directions.

Column size = 400x400 mm

Total factored load = 12 kN/m2.

kNm/m88228804711

π2622

kNm/m5206220330

m8822740mm4714008π3

kN6222712

kNm/m851122712

sc

sm

ε

2

2

s

.m

..m

...c;B

.V

..m

−=⎟⎠⎞

⎜⎝⎛ −−=

−=⋅−=

=⋅===

=⋅=

−=−=

Example 2 Interior panel of flat plate with span widths 7.2 and 4.8 m in the two directions.

Column size 400x400 mm

Total factored load 18 kN/m2

mm4714008π3

m9218440kN622842718

direction)short (in the kNm/m634128418

direction) long (in the kNm/m877122718

ε

2

s

2

s

==

===⋅⋅=

−=−=

−=−=

B

....c..V

..m

..m

kNm/m77419204711

π2622

kNm/m5206220330:directionshort in the

kNm/m710012622877

19204711

π2622

kNm/m546126228776220330

:direction long in the

sc

sm

sc

sm

.m

..m

..m

...m

−=⎟⎠⎞

⎜⎝⎛ −−=

−=⋅−=

−=⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −−=

−=⎟⎠⎞

⎜⎝⎛ +−+⋅−=

69

The column reactions in the two examples are identical (622 kN). The bending moments at the column differ, however. The average bending moment within the column strip in the long direction in Example 2 is approximately 22 % larger than for the flat plate with square panels in Example 1. In Section 2.5.1, it was demonstrated that the bending moment at the column plays a decisive role for the punching capacity. It is then evident that it cannot be correct to check the punching capacity of Example 2 presupposing a relation between bending moment and column reaction valid for square panels as in Nylander and Kinnunen (1990). Therefore, in cases where the required flexural reinforcement ratio differs in the two directions, the punching capacity shall be checked for each direction separately.

It can be shown that Eq.(4.3) yields a very good estimate of the average bending moment within the width 0.4 w according to the theory of elasticity for 1 < L1/L2 < 2. The quantity

121

Vs −=

mA shall therefore be added to all expressions for bending moments – as for

instance Eqs. (2.23) and (2.25) – when checking the punching capacity according to Chapter 2. Please note that the negative sign for the bending moment is omitted in that chapter:

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−−+= A

cB

rB

rcVm π8

42

2ln2

π8 2

2

2

2

t tangential moment (2.23a)

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−+= A

cB

BcVm π81ln2

π8 2

2

1 tangential moment at column edge (2.25a)

4.3 Design of midspan reinforcement

Kinnunen and Nylander (1960) realized that from the poor rotation capacity at the columns in a continuous flat plate follows that the midspan reinforcement has to be designed in balance with the support reinforcement. Eq. (4.3) for the negative bending moment at the column presupposes that the midspan flexural moment per unit width is at least qL2/24 for interior panels and 0.07qL2 for exterior panels, when full loading is applied on all panels of the flat plate structure. These static equilibrium conditions can be checked in accordance with Figure 4-3.

70

L 0

ψψ

curvature withparabolic variation

L

Exterior panel

0.2 L0.2 L

0.25 L L0

L

ψ

= 0.75 L

ψ3

= 0.6 L

Interior panel

Figure 4-3 Static equilibrium and compatibility conditions.

Interior panels

Lxd

xdL.Lf

fpusf

f

sf0pu

5

360

31

−=

−=⋅′′=

ψε

εψ

(4.6)

Exterior panels

Lxd

xdL

fpusf

f

sf0pupu

316

331

−=

−+−=

ψε

εψψ

(4.7)

where εsf = midspan reinforcement strain

=puψ slab inclination at the distance c/2 from the column when punching occurs

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 121

fff ρ

ρn

dnx

fρ = midspan reinforcement ratio.

The inclination puψ is determined from Eq. (2.27) if punching occurs without any reinforcement yielding. If some or all reinforcement yields before punching then the inclination puψ is determined from δε according to Eq. (2.43):

ε

εδψBc −

=2

pu (4.8)

71

If the strain in the midspan reinforcement εsf corresponds to a flexural moment larger than qL2/24 and 0.07qL2 respectively, then the design is safe because the negative flexural moment would be less than given by Eq. (4.3) and the punching capacity would be larger than calculated. The opposite is valid if the strain in the midspan reinforcement corresponds to a flexural moment less than qL2/24 or 0.07qL2 respectively. Additional midspan reinforcement has then to be provided until the described equilibrium and compatibility conditions are fulfilled.

The midspan reinforcement is usually designed for the effect of pattern loading. Additional midspan reinforcement as described here is therefore normally required only at such high flexural reinforcement ratio at the column that punching would occur without yielding of any reinforcement near the column.

4.4 Comparison with Codes

The theory is in Figure 4-4 compared to some common codes for design of flat plates, the design provisions of which is briefly summarized hereunder.

The Swedish load factors are 1.0 and 1.3 for dead load and live load respectively. The average load factors for the two codes Model Code 90 and ACI 318-02 are approximately 20 % larger. The design strength for these two codes is therefore divided by 1.2 in the Figures 4-4 to 4-6 in order to make them comparable with the Swedish approach.

The chosen notations are identical for all the codes:

dufVR ⋅⋅= v2 (4.9) where

VR = design punching capacity fv2 = two-way shear strength u = length of control perimeter at the distance 0.5d from the column d = average effective depth

4.4.1 Swedish Code for Concrete Structures, BBK 04

nm

ctkv2 γ145.0)501(⋅

⋅+=γ

ρξ ff (4.10)

where

fctk is the characteristic tensile strength of concrete ( given as tabulated values for the cube strengths K8 to K80)

ξ is a size-effect factor ξ = 1.4 for d ≤ 0.2 m ξ = 1.6 - d for 0.2 m ≤ d ≤ 0.5 m ξ = 1.3 – 0.4d for 0.5 m ≤ d ≤ 1.0 m ξ = 0.9 for 1.0 m ≤ d

ρ is the reinforcement ratio within the circle with diameter c and ρ is limited to maximum 0.01 in Eq. (4.10).

72

γm = strength reduction factor for concrete = 1.5 γn = safety class related strength reduction factor = 1.2 for safety class 3 (= brittle failure mode)

The control perimeter is placed 0.5d outside the column edge:

u = π(B+d) for circular columns and u = 4a + πd for square columns

4.4.2 Swedish Handbook for Concrete Structures

The Handbook gives a simplified design method – Nylander and Kinnunen (1990) – based on the original mechanical model by Kinnunen and Nylander (1960). It is described in detail in Hallgren (1996).

4.4.3 Model Code 1990, MC 90

The Model Code 90 defines the punching shear capacity along a control perimeter at the distance 2d from the column edge. The formal punching shear strength is then assumed equal to the shear strength for one-way structures such as beams. This is unfortunate for two reasons. Firstly, it gives false information about the punching failure mode and secondly it cannot be applied to compact structures such as footings, where the control perimeter would fall outside the structure. The deficiency can be overcome, however, if the approach proposed in Paper III is applied. The control perimeter is – as in most other codes – proposed to be placed 0.5d from the column edge (instead of 2d) and the punching shear strength – also called the two-way shear strength – is taken as the one-way shear strength multiplied by a correction factor. This approach has furthermore the advantage that it is possible to establish a more realistic upper limit for the two-way shear strength than the present value in Model Code 90.

( ) MPaff 31

ckv1 10012.0 ⋅⋅= ρξ (one-way design shear strength) (4.11)

d

2001+=ξ with d in mm (size effect) (4.12)

v1v1v2 5.2 ffu

duf ≤⋅⋅+

=α (two way design shear strength) (4.13)

dau π4 += square columns (4.14)

( )dBu += π circular columns (4.15)

π3=α interior columns (4.16)

73

4.4.4 Building Code Requirements for Structural Concrete, ACI 318-02

ckv1 61 ff = (4.17)

v1v1v2 220 ffu

duf ≤⋅+

= (4.18)

The control perimeter is placed 0.5d outside the column edge. A control section with four straight sides is permitted for rectangular columns:

)(4 dau += square columns (4.19)

( )dBu += π circular columns (4.20)

Furthermore, a reduction factor is given when the aspect ratio of a rectangular column is larger than 2. The strength reduction factor is 0.75 for punching and shear failure and 0.80 for flexural failure.

4.4.5 Code comparison

Hallgren (1996) found that Model Code 1990 predicts punching test results with very good accuracy. The code expressions are purely empirical, based on regression analysis of many test results. It is therefore encouraging that the theory in this thesis displays a similar design capacity curve at the concrete characteristic cylinder strength 24 MPa and B/d ≈ 2.9; Figure 4-4. If these two curves are assumed to represent the true design punching strength, the following conclusions can be drawn.

The very simple expression in BBK 04 for punching capacity seems to reflect the influence of the reinforcement ratio in a correct way. However, the resulting safety factor is unnecessarily high and the limit 1 % for the reinforcement ratio seems to be too cautious.

The Handbook method overestimates the punching capacity for reinforcement ratios exceeding 0.7 %. It is therefore amazing that this method still is classified in the Swedish Concrete Code BBK 04 as being more profound than the simple BBK-method despite convincing evidence on the contrary, here and in other evaluations.

The code ACI 318-02 gives a single value for the punching shear strength, only depending on the square root of the compression strength and independent of reinforcement ratio and size effect. The code overestimates the punching capacity at low reinforcement values, but the reinforcement ratio in flat plates is usually high in North America because the best economy is achieved if the slab is made as thin as possible and the code ACI 318-02 allows very slender two-way slabs. This is probably the reason why the code provisions for punching still are considered appropriate in USA. However, the ACI code is also used in many other parts of the world where practice often calls for thicker slabs with less reinforcement. In such cases the code provisions will result in structures with a low safety against punching, which even has been a partial cause of a serious progressive collapse with many casualties, Gardner et al (2000).

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V

0ρ

1000

500

0.005 0.010 0.015

kN

0

12

3

4

5

1. BBK 042. MC 90/1.23. Thesis/1.84. Handbook5. ACI 318-02/1.2

R

Figure 4-4 Code comparison. Design capacity at concentric punching versus flexural reinforcement ratio. ( c = 3.6 m, d = 0.26 m, a = 0.6 m, fck = 24 MPa, fsy =500 MPa)

The straight lines from the origin of coordinates in Figure 4-4 represent the flexural capacity. They are derived from the bending moment within the column strip according to the fan-type yield lines in all cases except for ACI 318-02:

⎟⎠⎞

⎜⎝⎛ −⋅−=

cBVm 1

π2sc (fan type yield lines) (4.21)

( ) 5.112

2

sc ⋅−

−=aLqm (ACI 318-02) (4.22)

Figure 4-4 reveals an inconsistency with the curves 3 and 4 for the Thesis and the Handbook. The strength reduction factor for brittle concrete failure is there used throughout, even for the part of the curves where the flexural yield capacity governs. This is discussed in the following.

The deflection of the slab within the circle with diameter c is δε at punching and δy2 when the reinforcement in tangential direction at the distance c/2 from the column has just reached the yield limit. If δε < δy2 (at normal to high reinforcement ratios) then it is obvious that the capacity is punching-controlled. The design capacity is then derived from the theoretical ultimate capacity by division with the strength reduction factors for concrete and safety class 3, γm· γn = 1.5·1.2 = 1.8.

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Tests by for instance Kinnunen and Nylander (1960) have demonstrated that if δε > δy2, which means that punching occurs after all reinforcement has reached the yield limit, a punching failure still occurs suddenly with little warning of impending failure. The structure can therefore not be defined as flexure-controlled until δε >> δy2, say δε > 3δy2, so that extensive cracking and large deflection will give ample warning of impending failure. The strength reduction factors for reinforcement (= 1.15) and safety class 2 (= 1.1) would then be appropriate.

The described approach is applied on the Thesis curve 3 of Figure 4-4 and the result is shown in Figure 4-5. The reinforcement ratio and the capacity Vy2 corresponding to point B is easily calculated by trial and error in Appendix B until δ ε = 3δy2 and point A corresponds to the reinforcement ratio when Vε = Vy2. A linear transition between points A and B corresponds to a gradual change of the punching-controlled strength reduction factor to the flexure-controlled. It is evident that this refined approach is of limited value because the flexure-controlled behaviour occurs at very low reinforcement ratios seldom encountered in practice. The refinement will therefore not be used in the following.

A

B

0 0.005 0.010

1000

500

0

VRkN

ρ

Figure 4-5 Design capacity with varying strength reduction factors. Above point A = punching controlled capacity (γm·γn =1.5·1.2). Below point B = flexure controlled capacity (γm·γn =1.15·1.1). Linear transition between A and B.

In order to verify that the close agreement between the Thesis and Model Code 90 in Figure 4-4 is not just a coincidence, comparison is made with varying concrete grades and column sizes in Figure 4-6.

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VkN

1500

1000

500

00 0.005 0.010 0.015

ρ

80

fck

50

24

Thesis/1.8

MC 90/1.2

MPaa = 600 mm

B/d = 2.9c - B

=2d 5.6

R

VkN

1500

1000

500

00 0.005 0.010 0.015

ρ

80

fck

50

24

Thesis/1.8

MC 90/1.2

MPa

a = 300 mm

B/d = 1.5c - B

=2d 6.2

R

Figure 4-6 Design capacity at concentric punching. Comparison between Thesis and Model Code 90. (c = 3.6 m, d = 0.26 m, fsy = 500 MPa)

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All major concrete codes use the same approach by expressing the punching capacity as formal shear strength along a control perimeter at a certain distance from the column edge. The codes can therefore neither differentiate between slender and compact structures nor identify the influence of the bending moment on the punching capacity. This is partly illustrated in Figure 4-7. A column-supported structure, a continuous flat plate or a single foundation, is studied. The punching capacity is expressed as the design shear strength along a control perimeter at the distance 0.5d from the column.

V

Vε

σ

1.0

1.5

2

57

10

Bd

1050

0

0.5

1.0

MPa

1. BBK 042. Model Code 90/1.23. ACI 318-02/1.24. Thesis/1.8

1.

2.

3.

4

fv2

(B+d)d

2.5 f

≥

c-B2d

VR = fv2 * πB

B+dc

d

VR

1.2

4.

4.

1.5

2.0

2. fv1

v1

Figure 4-7 Effect of (c-B)/2d and B/d on design strength for punching. (d = 0.3 m, fck =30 MPa, fsy = 500 MPa, ρ = 0.7 %)

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The code methods give a shear strength that is independent of the slenderness of the flat plate structure, which means that the strength of compact slabs such as footings is underestimated. The high allowable shear stress at large columns by the code ACI 318-02 is remarkable, however. (The curve represents the capacity for a square column).

The Thesis on the other hand gives a significant punching strength dependence on the

slendernessdBc

2− . Just as for the shear strength of beams, the punching strength of slabs

increases with decreasing slenderness. The decreasing strength of the Vσ -curves for B/d < 3 is a consequence of the chosen position of the formal control section. If it were placed close to the column, corresponding to the diameter of the internal column capital, the Vσ -curves would be continuously increasing with decreasing column size.

Finally, the size-effect is illustrated in Figure 4-8. The two codes BBK 04 and Model Code 90 give a size effect that depends only on the effective depth of the slab. The Thesis theory on the other hand displays a dependence also on the reinforcement ratio, with no size effect if all reinforcement yields before punching.

ξ

1.0

0.5d (m)0 0.5 1.0

ρ

0.002

0.005

0.004

0.003

0.01≥

MC 90

BBK 04

Figure 4-8 Normalized size-effect (ξ =1.0 for d = 0.2 m) for punching ultimate capacity versus effective depth of the slab and reinforcement ratio. Comparison with Model Code 90 and BBK 04. (fck =30 MPa, fsy = 500 MPa, B/d = 1.9, (c-B)/2d = 5.4)

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5 Reinforcement for ductility

In the preceding chapters, a theory for prediction of punching capacity is presented and validated. It is demonstrated that the theory can predict the capacity and deflection of test specimens with good accuracy. It is commonly accepted that such test specimens do simulate the behaviour of continuous flat plates near the columns. The theory can therefore be applied for verification of existing structures and for design purpose.

However, it must be emphasized that even the most accurate theory cannot eliminate the disadvantage of flat plates, namely the risk of a brittle punching failure in the event of overloading. In this context, it should be remembered that modern building codes agree with what is stated in Eurocode 2 (1991). That code requires a structure to be designed is such a way “that it will not be damaged by events like explosions, impact, or consequences of human error, to an extent disproportionate to the original cause”. In other words, a local failure shall not spread over a large portion of the structure and shall not trigger a progressive collapse.

It is therefore surprising that the same code – in the detailing chapters – requires a least amount of shear reinforcement in primary beams in order to prevent a brittle failure, but no similar requirement is put on flat plates despite the fact that a punching failure of a flat plate may lead to worse consequences than a shear failure of a beam. Punching usually occurs when the concrete strain near the column due to the bending moment in the slab exceeds a critical value, see Chapter 2.2. If a punching failure occurs at one column due to a local overloading, then the slab inclination and hence the concrete strain will increase at the adjacent columns, which in turn most probably will result in punching at these columns as well. A progressive collapse of the entire building is then impending.

Consequently, it should be a code requirement that a flat plate structure in a multi-story building in case of overloading displays a ductile failure mode. A flat plate should behave in the same manner as a cast-in-place concrete slab supported by beams or walls. Such a slab displays a very ductile flexural failure mode without risk for brittle shear failure.

One solution would be to provide the flat plate structure with some form of shear reinforcement in order to prevent the brittle punching failure mode. Please observe that this approach differs from the current perception by codes, researchers and designers, who all seem to utilize shear reinforcement merely for increasing the punching capacity – not for creating a ductile structure.

In order to find a reinforcement system that could result in the desired ductile behaviour, slabs with various forms of stirrups were tested (Paper II). The stirrups were anchored around the top reinforcement of the slab in accordance with code provisions. Despite the fact that the formal shear capacity exceeded the yield capacity of the specimens, the failure was brittle.

80

Three test specimens failed due to a steep shear crack near the column leaving the stirrups ineffective. One slab with stirrups arranged in the form of a cross failed due to a shear crack outside the shear-reinforced zone. The stirrups were in the latter case obviously too far apart to cause a uniform shear stress in the slab along a critical perimeter outside the shear reinforced zone. This configuration is currently standard practice in USA and in Canada for so-called stud rails. The shear capacity is according to US and Canadian Codes calculated assuming a uniform stress along the critical perimeter outside the stud rails, irrespective of the distance between the outermost studs. Most European codes apply a more restrictive approach.

The outcome of the stirrup test described in Paper II was thus very disappointing. It seemed impossible to achieve ductile flat plates with intermediate or high flexural reinforcement ratios. Three important conclusions could be made, however:

1. Stirrups and stud rails may increase the punching capacity of the slab, but they cannot prevent a steep shear crack from forming near the column when the stability of the compression zone of the slab decreases due to high flexural compression strain.

2. The shear reinforcement should be well distributed along the outer perimeter in order to achieve a uniform shear stress along that perimeter.

3. The shear reinforcement should extend far enough from the column to preclude a shear failure outside the shear reinforced area.

Bent down flexural reinforcement constitutes another shear reinforcement possibility. From the tests with stirrups, it was learnt that the bent bars should not be detailed according to current practice in order to achieve a ductile behaviour. That practice prescribes that the bars should be bent down at a certain distance outside the column perimeter and some of the bars should be placed outside the column. Obviously, this configuration aims at making the shear-reinforced zone around the column as large as possible in order to maximize the possible shear capacity. However, there is risk that such a layout would result in the same type of ultimately brittle failure as experienced with stirrups. A steep shear crack could develop inside the bent bars. Furthermore, it is evident that bent bars anyhow do not reach far enough away from the column to exclude the possibility of a shear failure outside the bent bars. This scenario is confirmed by the Hallgren (1996) tests. Some of the specimens were provided with bent bars as shear reinforcement. These slabs had higher capacity than the corresponding slabs without shear reinforcement, but the failure mode was still a sudden punching failure. Steep shear cracks developed inside the bent bars in the slabs HSC3s and HSC7s. The shear crack developed outside the shear reinforcement in specimen HSC5s with a high reinforcement ratio. From Figure 4-21 of Hallgren (1996) it is evident that only specimen HSC7s with reinforcement ratio 0.63 % displayed some ductility before the sudden punching failure. The ultimate deflection was in the order of two times the deflection at overall yield. A structure with such a low ductility (δu/δy = 2) is normally not considered ductile, however.

Therefore, after evaluation of the stirrup tests, a second test series was performed with a combination of bent bars and stirrups; see Paper II. The bent bars were all placed within the column width and were bent down at the column edge at a shallow slope in order to bridge over the zone with large circumferential cracks around the column at flexural yielding. The bent bars were designed as hangers with the vertical component of their yield capacity in balance with the column reaction at overall yield of the flexural reinforcement. Stirrup cages were added in order to exclude a shear failure outside the bent bars.

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That concept turned out to be very effective in creating an extremely ductile structural system without any punching tendency even at high flexural reinforcement ratios. The concept was later on further developed (Paper III), where the stirrup cages were simplified as regards both fabrication and installation. Furthermore, the zone with stirrups was reduced in relation to Paper II.

The design calculations of the mature concept “ductility reinforcement” are very simple and described in detail in Paper IV. The bent bars and the stirrups shall be designed for the column reaction corresponding to the formation of yield lines over the supports and the midspans at uniform loading. Alternatively, the column reaction can be taken as five times the contribution from the worst adjacent panel. In this way, respect is paid to the fact that the flexural reinforcement might be “over designed” and that pattern loading has been considered when designing the midspan reinforcement.

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83

6 Earthquake simulation

Since ordinary flat plates have a very limited ductility, they should be used with caution in seismic areas. On the other hand, flat plates that are provided with ductility reinforcement described in Chapter 5 display such good ductility that they should be well suited also in seismic areas, if the building stability does not rely on frame action with the flat plate as horizontal member. Stability should be provided by shear walls or equivalent systems.

Most seismic codes seem to agree upon that the stabilizing system shall be designed so that the story drift ratio is limited to 2.5 % or less. The slab rotation in relation to the column at story drift resembles the deflection inclination of a concentrically loaded slab. Paper III demonstrated that a flat plate with ductility reinforcement displays an inclination capacity of about 9% at monotonic loading. The drift capacity at cyclic loading could therefore be expected to be in the order of half that value.

In order to examine if the ductility reinforcement used for test slabs in Paper III also could be effective at seismic cyclic loading, two pilot tests were performed and reported in Paper IV.

The specimens were loaded to a concentric load corresponding to 60 % and 75 % respectively of their flexural yield capacity. Then a cyclic imposed story drift was applied up to a story drift ratio of 7 %.

The resulting hysteresis curves are displayed in Paper IV. It is evident that the specimens could withstand a story drift ratio of more than 4 %, which demonstrates that flat plates with ductility reinforcement are safe even in regions of high seismic risk. It should be noted that demands on the ductility reinforcement for seismic loading are identical to the demands for normal gravity loading. No flexural reinforcement has to be added to cater for unbalanced moment due to story drift.

84

85

7 Conclusions and summary

The punching failure of flat plates resembles the shear failure of beams in the sense that an inclined “shear crack” constitutes the failure. The failure mechanism is different, however. Inclined circumferential cracks down to the neutral axis form around the column already at a load level of less than 70 % of the ultimate load. The slab is nevertheless stable and can be loaded and reloaded without any decrease of the ultimate load. Punching occurs instead when the compression zone of the slab near the column collapses.

The presented models are based on information that can be gained from the stress-strain relation of concrete in uniaxial compression. Low strength concretes start to “soften” at a compression strain of about 1.0 per mille. This level for the flexural compression strain is therefore regarded to be critical for the stability of the compression zone near the column of a flat plate, but it is assumed to slowly decrease with increasing concrete strength to account for the increasing brittleness of concrete with increasing strength. If the slab is provided with adequate amount of conventional shear reinforcement the critical concrete strain is assumed to increase to 1.5 per mille, which is close to the strain at the peak stress for low strength concretes. It should be observed that these critical strain levels are considerably lower than the generally accepted ultimate strain 3.5 per mille for uniaxially spanned members in bending.

In compact slabs such as column footings the compression strength of the inclined compression strut from the load to the column is found to be governing. The thickness of the compression strut near the column is limited by the compression zone depth in radial direction. If the column is small in relation to the compression zone depth, the compression strength of the strut is assumed to reach the value 1.2 fcc corresponding to the strength of concrete in biaxial compression with moderate perpendicular compression stress. If the

column is very large, the compression strength is assumed reduced to ⎟⎠

⎞⎜⎝

⎛ −250

160 cccc

ff. , which

is the generally accepted value for the uniaxial compression strength in cracked zones. It is interesting to note that the latter strength corresponds to a compression strain of about 0.75 per mille – the same for all concrete grades.

The above critical strain and stress levels are assumed to display a size effect that is inversely proportional to the cube root of the compression zone depth and the thickness of the inclined compression strut respectively – an approach that was utilized already in Paper I of 1990. As a consequence, the size effect decreases with decreasing amount of flexural reinforcement. It should be observed that the apparent size effect factor may increase for thick slabs where cracks in the compression zone may be induced due to uneven temperature effects during the concrete hydration.

A strong support for the hypothesis that the concrete strain in tangential direction plays a decisive role is given by the fact that the theory can predict the deflection at the sudden punching failure of flat plate specimens with all flexural reinforcement yielding. In this case it is obvious that the failure is not caused by the shear force, it is caused by the limited curvature capacity of the slab. That capacity depends in turn on a limited concrete strain capacity. To crown everything, this strain limit is found to be a sufficient criterion for prediction of the punching capacity and deflection of a large variety of flat plate specimen types reported in the literature.

86

As stated in Paper I: “The basic assumptions behind the theory are, in reality, very simple and straightforward. Nevertheless, the theory is able to predict reported test results … with amazing accuracy, which demonstrates, above all, that the punching failure mechanism … is perhaps not as complex as many researchers claim.” The presented models do not explain the failure mechanisms in detail, which is similar to the case that the compression strength tested on a cylinder specimen does not explain the failure mechanism, which is mainly a tensile failure in lateral direction. Similarly, the presented size dependent strain limit in a flat plate happens to capture the conditions when the concrete near the column edge becomes unstable, which initiates the punching failure due to a “zip” effect.

Equation (2.35) is informative as regards the parameters that affect the curvature capacity of the slab near the column in the normal case with the flexural reinforcement yielding at the column before punching occurs:

30

cc2

3

22sy

2c10

pu

cpuu

250010041500

."

f.

d.

fE

xf ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅==

ρ

ε (2.35)

A high concrete E-modulus is favourable, which means that the curvature capacity increases with increasing concrete grade. A high strength concrete slab has therefore a better rotation capacity than a normal strength slab despite that the high strength concrete matrix is more

brittle. (Observe that ( )31

ccc10 fkE ⋅≈ ). It is also evident that the curvature capacity of a flat plate rapidly decreases with increasing reinforcement ratio and increasing effective depth.

The dependence on the concrete E-modulus indicates that capacity predictions will be uncertain if only the compression strength of the concrete is recorded. It is therefore recommended that the E-modulus shall be specified for flat plate structures, which is especially important if the coarse aggregates in the concrete mix do not emanate from primitive rock.

Flat plates where the support moments differ in the two directions (as for slabs with rectangular panels) shall not be checked for a mean value of the reinforcement ratios in the two directions, but for each direction separately. This follows from the hypothesis that the concrete compression strain in flexure is decisive for the punching capacity.

If so called unbalanced moment is transferred from the slab to the column or vice versa, then it is safer to check the rotation capacity of the slab in relation to the column instead of the unbalanced moment capacity of the slab, because the imposed slab rotation can be estimated with much better certainty than the imposed unbalanced moment. A conservative value for the rotation capacity is derived from the slab behaviour at concentric punching.

The capacity increase for flat plates with conventional shear reinforcement can be attributed to the fact that the compression zone can endure an increased tangential strain. The curvature of the slab at failure will then increase in relation to a slab without shear reinforcement. The increased curvature means that more flexural reinforcement will reach the yield limit before punching occurs, which in turn means that the capacity increases. However, the failure mode cannot be classified as ductile because the ultimate deflection usually does not even reach two times the deflection at overall yield of the reinforcement.

87

Flat plates provided with a novel reinforcement concept denoted “ductility reinforcement” display an extremely ductile behaviour. They can therefore be classified as having no risk for brittle punching failure and can be designed in Swedish safety class 2, which means a reinforcement saving of about 10 % in relation to flat plates with conventional shear reinforcement (and still larger saving in comparison to flat plates without shear reinforcement).

The complete calculation steps for prediction of the punching failure are demonstrated in Appendices A to E. The program Mathcad is used for this purpose, but all calculations are possible to perform manually except for the punching load at partial yield of the flexural reinforcement in Appendix B. However, an approximate manual method for this case is described in Section 2.4.

Two examples of flat plates are treated in Appendices A and B. Flat plates with shear reinforcement are treated in Appendix C. The displayed calculations are valid for interior square panels. The required modifications for other cases are described in Chapter 4.

Column footings are treated in Appendix D and footing specimens with line load in Appendix E.

The laborious calculations for unbalanced moment are shown in Appendix F just for documentation purpose. In practice, the rotation of the column in relation to the slab shall be checked instead, which is described in Section 3.6.

The simple design procedure for flat plates with ductility reinforcement is described in detail in Paper IV.

88

89

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92

Narasimhan, N. (1971) “Shear Reinforcement in Reinforced Concrete Column Heads”, PhD-Thesis, Imperial College, London.

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94

95

Appendix A. Punching of flat plate.

Tolf (1988), S2.1(no yield punching)

d

B

c

fc

fy

ρ

0.20

0.25

2.38

24.2

657

0.008

m

m

m

MPa

MPa

Bε3 π

8a.a Bσ

4 aπ

a

Bε B Bσ B

Ec0 21500 fc10

13

(Model Code 90) Ec0 2.88654 104=

Ec10 1 0.6 1 fc150

4Ec0.

Ec10 2.02972 104= MPa

nρ200000Ec10

ρ. nρ 0.07883=

x d nρ. 1 2nρ

1. x 0.0652= m

EI ρ d3. 200. 106. 1 xd

. 1 x3 d

. EI 7.68998 103= kNm

1. PUNCHING CAPACITY Vε

Guess factor k to make V ε equal to or less than Vσ: k 0.883

1.1 No yield punching

ε cpu k 0.001. 0.150x

13

. 25fc

0.1. ε cpu 1.16949 10 3=

Recorded: ε = 0.0012

σs 200000ε cpu d x( )x

. σs 483.62138= MPa

fy 657= MPa

If σs > fy go to 1.2

96

Vε ny 8 π. 103.ρ σs. d2. 1 x

3 d..

2 ln cBε

. 1 Bε2

c2

.Result Vε ny 630.82819=

Deflection

δε ny Vε ny4 π

1 Bε2

c2. c

2 EI. c Bε

2. δε ny 8.1819 10 3= m

Recorded δ = 7.5 mm

1.2 Yield punching (not governing in this case)

ε cpu 10 6 Ec10fy

0.075d

. k 10.( )3

ρ. 25

fc

0.3.. ε cpu 1.00338 10 3=

xpu d 2 ρ

ε cpu. fy

Ec10. xpu 0.10323= m

my ρ d2. fy. 1 x3 d

. 103. my 187.39524= kN

Vy1 my 8 π

2 ln cBε

. 1 Bε2

c2

. Vy1 856.98057= kN

Vy2 my 2 π

1 Bε

c

.Vy2 1.31564 103= kN

f´´u ε cpuxpu

f´´u 9.71972 10 3=

f´´y myEI

f´´y 0.02437=

∆ f´´ f´´u f´´y ∆ f´´ 0.01465=

Guess

ry Bε

2Given

my Vy18 π

2 ln c2 ry

. 2 Bε2

4 ry2

Bε2

c2. ∆ f´´ Bε

2 ry. EI.

ry find ry( )find ry( ) ry 0.125=c2

1.19=

If ry > c/2 then V ε = Vy2

97

Vε y Vy2

my c2

.my ry.

ry

0.5 c

rVy18 π

2 ln c2 r

. 2 Bε2

4 r2

Bε2

c2. ∆ f´´ Bε

2 r. EI. d.

Result Vε y 669.77791= kN

Deflection

δy1 Vy14 π

1 Bε2

c2. c

2 EI. c Bε

2. δy1 0.01112= m

δε y δy1 f´´u f´´y( ) Bε

2c Bε

2. δε y 9.16496 10 3= m

2. PUNCHING CAPACITY Vσ

c0 Bε 2 d c0 0.65=

n0ρ200000

Ec0ρ.

1 ln cc0

1 ln cBε

.n0ρ 0.03915=

x0 d n0ρ. 1 2n0ρ

1. x0 0.04868=

u π Bσx0

tan 50 deg( )x0

2 tan 25 deg( ).. u

x022.13876=

t x02 cos 25 deg( )

t 0.02686=

Vσ 0.6 0.9 1 0.007 ux0

.2

fc. t. sin 25 deg( ). u. 0.150t

13

. 103.Vσ 652.64153=

Vσmax 1.2 fc t. sin 25 deg( ). u. 0.150t

13

. 103.Vσmax 630.23351=

3. PUNCHING CAPACITY WITHOUT SHEAR REINFORCEMENT

k 0.883= Vε ny 630.82819= kN

Vσmax 630.23351= kN

Vtest = 603 kN

98

Appendix B. Punching of flat plate.

Hallgren (1966), HSC1(yield punching)d

B

c

fc

fy

ρ

0.20

0.25

2.4

91

627

0.008

m

m

m

MPa

MPa

Bε3 π

8a.a Bσ

4 aπ

a

Bε B Bσ B

Ec0 21500 fc10

13

(Model Code 90) Ec0 4.48868 104=

Ec0 42900 (recorded value) Ec0 4.29 104=

Ec10 1 0.6 1 fc150

4Ec0.

Ec10 4.22839 104= MPa

nρ200000Ec10

ρ. nρ 0.03784=

x d nρ. 1 2nρ

1. x 0.04797= m

EI ρ d3. 200. 106. 1 xd

. 1 x3 d

. EI 8.95203 103= kNm

1. PUNCHING CAPACITY Vε

Guess factor k to make V ε equal to or less than Vσ: k 1

1.1 No yield punching (not governing in this case)

ε cpu k 0.001. 0.150x

13

. 25fc

0.1. ε cpu 1.28508 10 3=

σs 200000ε cpu d x( )x

. σs 814.56107= MPa

fy 627= MPa

If σs > fy go to 1.2

99

Vε ny 8 π. 103.ρ σs. d2. 1 x

3 d..

2 ln cBε

. 1 Bε2

c2

.Result Vε ny 1.09336 103=

Deflection

δε ny Vε ny4 π

1 Bε2

c2. c

2 EI. c Bε

2. δε ny 0.0124= m

1.2 Yield punching

ε cpu 10 6 Ec10fy

0.075d

. k 10.( )3

ρ. 25

fc

0.3.. ε cpu 1.46474 10 3=

xpu d 2 ρ

ε cpu. fy

Ec10. xpu 0.0324= m

my ρ d2. fy. 1 x3 d

. 103. my 184.59891= kN

Vy1 my 8 π

2 ln cBε

. 1 Bε2

c2

. Vy1 841.60164= kN

Vy2 my 2 π

1 Bε

c

.Vy2 1.29474 103= kN

f´´u ε cpuxpu

f´´u 0.04521=

f´´y myEI

f´´y 0.02062=

∆ f´´ f´´u f´´y ∆ f´´ 0.02459=

Guess

ry Bε

2Given

my Vy18 π

2 ln c2 ry

. 2 Bε2

4 ry2

Bε2

c2. ∆ f´´ Bε

2 ry. EI.

ry find ry( ) ry 0.4738=c2

1.2=

If ry > c/2 then V ε = Vy2

100

Vε y Vy2

my c2

.my ry.

ry

0.5 c

rVy18 π

2 ln c2 r

. 2 Bε2

4 r2

Bε2

c2. ∆ f´´ Bε

2 r. EI. d.

Result Vε y 1.05141 103= kN

Deflection

δy1 Vy14 π

1 Bε2

c2. c

2 EI. c Bε

2. δy1 9.54612 10 3= m

δε y δy1 f´´u f´´y( ) Bε

2c Bε

2. δε y 0.01285= m

Recorded δ = 12.5 mm

2. PUNCHING CAPACITY Vσ

c0 Bε 2 d c0 0.65=

n0ρ200000

Ec0ρ.

1 ln cc0

1 ln cBε

.n0ρ 0.02637=

x0 d n0ρ. 1 2n0ρ

1. x0 0.04096=

u π Bσx0

tan 50 deg( )x0

2 tan 25 deg( ).. u

x025.18017=

t x02 cos 25 deg( )

t 0.0226=

Vσ 0.6 0.9 1 0.007 ux0

.2

fc. t. sin 25 deg( ). u. 0.150t

13

. 103.Vσ 2.0393 103=

Vσmax 1.2 fc t. sin 25 deg( ). u. 0.150t

13

. 103.Vσmax 2.02129 103=

3. PUNCHING CAPACITY WITHOUT SHEAR REINFORCEMENT

k 1= Vε y 1.05141 103= kN

Vσmax 2.02129 103= kN

Vtest = 1021 kN

101

Appendix C. Flat plate with shear reinforcement.

Hallgren (1996)HSC5s

d

B

c

fc

fy

ρ

0.201

0.25

2.4

91

604

0.0118

m

m

m

MPa

MPa

Guess k 1 (to make V ε equal to or larger than Vσ)

Bε3 π

8a.a Bσ

4 aπ

a

Bε B Bσ B

Ec0 21500 fc10

13

. (Model Code 90) Ec0 4.48868 104=

Ec0 43000 MPa (Recorded value) Ec0 4.3 104=

Ec10 1 0.6 1 fc150

4. Ec0. Ec10 4.23825 104=

1. No yield punching Vεsny (not governing in this case)

α 0.5 0.3 1 fc100

2. α 0.50243=

Ec15 fc0.0015

fc25

0.1. 1.1 fc

190. Ec15 4.28735 104=

ns ρ200000Ec15

ρ. ns ρ 0.05505=

xs d ns ρ. 1

4 α2

1α ns ρ.

12 α

. xs 0.05642= m

EI ρ d3. 200. 106. 1 xsd

. 1 xs3 d

. EI 1.24948 104= kNm

ε cpus k 0.0015. 25fc

0.1 0.150xs

13

. ε cpus 1.82609 10 3=

σs 200000ε cpus d xs( )xs

. σs 935.7871= MPa

fy 604= MPa

If σs > fy go to Section 2

102

Vε sny 8 π. 103.ρ σs. d2. 1 xs

3 d..

2 ln cBε

. 1 Bε2

c2

.Result Vε sny 1.84358 103= kN

Deflection

δε sny Vε sny4 π

1 Bε2

c2. c

2 EI. c Bε

2. δε sny 0.01498= m

2. Yield punching Vεsy

ε cpus 10 6 Ec15fy

α 0.15.

d. 153

ρ. 25

fc

0.3.. ε cpus 2.27296 10 3=

Recorded ε = 0.00225

xpus dα

ρ

ε cpus. fy

Ec15. xpus 0.02926= m

mys ρ d2. fy. 1 xs3 d

. 103. mys 261.00196= kN

Vy1 mys 8 π

2 ln cBε

. 1 Bε2

c2

. Vy1 1.18993 103= kN

Vy2 mys 2 π

1 Bε

c

.Vy2 1.83061 103= kN

f´´u ε cpusxpus

f´´u 0.07768=

f´´y fy200000

1d xs

. f´´y 0.02089=

∆ f´´ f´´u f´´y ∆ f´´ 0.0568=

Guess

ry Bε

2

Given

mys Vy18 π

2 ln c2 ry

. 2 Bε2

4 ry2

Bε2

c2. ∆ f´´ Bε

2 ry. EI.

ry find ry( ) ry 0.73126=c2

1.2=

If ry > c/2 then V εs = Vy2

103

Vε sy Vy2

mys c2

.mys ry.

ry

0.5 c

rVy18 π

2 ln c2 r

. 2 Bε2

4 r2

Bε2

c2. ∆ f´´ Bε

2 r. EI. d.

Result Vε sy 1.68702 103= kN

Deflection

δy1 Vy14 π

1 Bε2

c2. c

2 EI. c Bε

2. δy1 9.67012 10 3= m

δε sy δy1 f´´u f´´y( ) Bε

2c Bε

2. δε sy 0.0173= m

Recorded δ =16 mm

3. Punching capacity Vσs

Ec0 4.3 104= MPa

n0ρ200000

Ec0ρ. n0ρ 0.05488=

x0 d n0ρ. 1 2n0ρ

1. x0 0.05647= m

u π Bσx0

tan 90 deg( )x0

2 tan 45 deg( ).. u 0.8741= m

t x02 cos 45 deg( ) t 0.03993= m

Vσs 0.6 0.9 1 0.007 ux0

.2

fc. t. sin 45 deg( ). u. 0.150t

13

. 103.Vσs 4.59288 103=

Vσsmax 1.2 fc t. sin 45 deg( ). u. 0.150t

13

. 103.Vσsmax 4.18953 103=

4. Maximum punching capacity with shear reinforcent

k 1= Vε sy 1.68702 103=

Vσsmax 4.18953 103=

Vtest = 1631 kN

104

Appendix D. Punching of column footing, surface load.

Dieterle (1978) S1-Hd

a

b

fc

fy

ρ

0.29

0.3

1.5

30.6

512

0.00862

m

m

m

MPa

MPa

1. Punching capacity Vσ

Bσ BB Bε BB

Bσ4 aπ

Bσ 0.382= m

D 2 b

π

D 1.693= m

Bε3 π

8a. Bε 0.353= m

Inclination φ of fictitious shear crack

tanφ1.4 d

D3

Bε

2

tanφ 1.048=

If tanφ < 1 put tanφ = 1

Diameter kD = c0 of circle within fictitious shear crack

kBε

2 dtanφ

Dk 0.536=

c0 k D. c0 0.907= m

Radius R to center of gravity for load outside shear crack

R D3

1 k2

1 k. R 0.67= m

105

Inclination γ of compression strut

Ec0 21500 fc10

13

. Ec0 3.121 104=

n0ρ200000

Ec0ρ.

1 ln bc0

1 ln bBε

. n0ρ 0.034=

x0 d n0ρ. 1 2n0ρ

1. x0 0.066= m

Guess

γ 30 deg

Given

tan γ( ) 2 R Bε( )2

x024 d

x01 2 R Bε

x0

γ Find γ( ) γ 0.499=

tan γ( ) 0.545=

If tan(γ) < tan(25 deg) = 0.466 put γ = 0.436 = (25 deg)

If tan(γ) > 1 put γ = 0.785 = (45 deg)

Shear capacity V σ

u π Bσx0

tan 2 γ( )x0

2 tan γ( ).. u 1.526= m

t x02 cos γ( )

t 0.038= m

ux0

22.996=

Vσ 0.6 0.9 1 0.007 ux0

.2

fc. t. sin γ( ). u. 0.150t

13

. 103. Vσ 1.648 103=

Vσmax 1.2 fc t. sin γ( ). u. 0.150t

13

. 103. Vσmax 1.603 103=

106

Load capacity P σ

PσVσmax

1 k2 Pσ 2.249 103= kN

Ptest = 2368 kN

Flexural capacity

Ec10 1 0.6 1 fc150

421500 fc

10

13

.Ec10 2.369 104= MPa

nρ200000Ec10

ρ.1 ln b

c0

1 ln bBε

.nρ 0.045=

x d nρ. 1 2nρ

1. x 0.075= m

M Pσb8

Bε

2 π. M 295.214= kNm

σs M

ρ d2. c0. 1 ln bc0

. 1 x3 d

.10 3. σs 326.778= MPa

ε c c0Bε 2 x

σs200000

. xd x

. ε c 1.023 10 3=

ε cpu 0.0010 25fc

0.1 0.15x

13

. ε cpu 1.236 10 3=

107

Appendix E. Punching of column footing, line load.

Hallgren, Kinnunen, Nylander (1983, 1998) S12

d

B

c

D

fc

fy

ρ

0.242

0.25

0.674

0.96

27.3

621

0.00413

m

m

m

m

MPa

MPa

Bσ4 aπ

a Bσ B Bσ 0.25=

Bε3 π

8a.a Bε B Bε 0.25=

D 3 π

8b.b D 0.96=

R c2

R 0.337=

c0 B 2 d c0 0.734=

Ec0 21500 fc10

13

. Ec0 3.00487 104= MPa

n0ρ200000

Ec0ρ.

1 ln Dc

1 ln DB

.n0ρ 0.01587=

x0 d n0ρ. 1 2n0ρ

1. x0 0.03944=

Ec10 1 0.6 1 fc150

421500 fc

10

13

.Ec10 2.19765 104= MPa

nρ200000Ec10

ρ.1 ln D

c

1 ln DB

.nρ 0.02169=

x d nρ. 1 2nρ

1. x 0.04543= m

108

Calculate γ

Guess

γ 30 deg

Given

tan γ( ) 2 R Bε( )2

x024 d

x01 2 R Bε

x0

γ Find γ( ) γ 0.80706=

tan γ( ) 1.04428=

If tan(γ) < tan(25 deg) = 0.466 put γ = 0.436 = (25 deg)

If tan(γ) > 1 put γ = 0.785 = (45 deg)

γ 0.785

Calculate V σ

u π Bσx0

tan 2 γ( )x0

2 tan γ( ). u 0.8475=

t x02 cos γ( )

t 0.02788=

ux0

21.48887=

Vσ 0.6 0.9 1 0.007 ux0

.2

fc. t. sin γ( ). u. 0.150t

13

. 103.Vσ 998.2513=

Vσmax 1.2 fc t. sin γ( ). u. 0.150t

13

. 103.Vσmax 958.62469=

Vtest = 1049 kN

Flexural capacity

M Vσmax2 π

1 Bε

c. c. M 64.68962= kNm

σs M

ρ d2. c. 1 ln Dc

. 1 x3 d

.10 3. σs 312.70442=

ε c cB 2 x

σs200000

. xd x

. ε c 7.14512 10 4=

ε cpu 0.0010 25fc

0.1 0.15x

13

. ε cpu 1.47603 10 3=

109

Appendix F. Unbalanced moment loading.

Ghali et al 1976, SM1.0

h

d

a

c

fc

fy

ρ

ρc

0.152

0.121

0.305

1.8

33.4

476

0.0105

0.0035

m

m

m

m

MPa

MPa

B 3 π

8a. B 0.35932=

Ec 1 0.6 1 fc150

421500. fc

10

13

Ec 2.50977 104=

nρ200000

Ecρ. nρ 0.08367=

x d nρ. 1 2nρ

1. x 0.0404=

EI ρ d3. 200. 106. 1 xd

. 1 x3 d

. EI 2.20237 103= kNm

EI1 ρ d3. 200. 106. 1 h2 d

. 1 h6 d

. EI1 1.0939 103=

kI EI1EI

12

kI 0.70476=

1. LIMIT FOR REINFORCEMENT YIELD, ρ1

Guess

ρ1 0.01 εclim 0.001 xlim x

Given

εclim 0.0010 25fc

0.1 0.150xlim

13

.

xlim εclim

εclim fy200000

d.

ρ1 0.5 xlim. Ec. εclim.

d fy.

110

Limit for reinforcement yieldρ1

xlim

εclim

Find ρ1 xlim, εclim,( )ρ1 0.01437=

ρ 0.0105=

2. PUNCHING CAPACITY Vε

εcpu 10 6 Ecfy

0.075d

. 103

ρ. 25

fc

0.3.. εcpu 1.68922 10 3=

xpu d 2 ρ

εcpu. fy

Ec. xpu 0.02853= m

my ρ d2. fy. 1 x3 d

. 103. my 65.03184= kN

Vy1 my 8 π

2 ln cB

. 1 B2

c2

. Vy1 390.7489= kN

Vy2 my 2 π

1 Bc

.Vy2 510.51769= kN

f´´u εcpuxpu

f´´u 0.05921=

f´´y myEI

f´´y 0.02953=

Guess

ry c2

Given

my Vy18 π

2 ln c2 ry

. 2 B2

4 ry2

B2

c2. f´´u f´´y( ) B

2 ry. EI.

ry find ry( ) ry 0.55508= 0.5 c 0.9=

If ry > c/2 then V ε = Vy2

VεVy2

my c2

.my ry.

ry

0.5 c

rVy18 π

2 ln c2 r

. 2 B2

4 r2

B2

c2. f´´u f´´y( ) B

2 r. EI. d.

Vε 473.82889=

111

3. DEFLECTIONS

δy1 Vy14 π

1 B2

c2. c

2 EI. c B

2. δy1 8.78857 10 3=

δε δy1 f´´u f´´y( ) B2

. c B2

. δε 0.01263=

δy2 δy1 f´´y c B( )2

4. δy2 0.02411=

4. UNBALANCED MOMENT CAPACITY Mu

4.1 Insert column reaction V and guess value for ∆M

V 122 Vy1 390.7489= Vy2 510.51769= Vε 473.82889=

δy1 8.78857 10 3=

∆M 0.00048

4.2 Deflection due to load V

V < Vy1

δV V4 π

1 B2

c2. c

2 EI. c B

2.

δV 2.74398 10 3=

4.3 Tension in top reinforcement of slab

Fictitious slab deflections δt along circle c due to column rotation

∆c δε δV 2 ∆M( ) ∆c 0.01085=

δt1 δV ∆M 0.195 ∆c kI.δt1 3.75451 10 3=

δt2 δV ∆M 0.555 ∆c kI. δt2 6.50628 10 3=

δt3 δV ∆M 0.831 ∆c kI. δt3 8.61596 10 3=

δt4 δV ∆M ∆c kI. δt4 9.90776 10 3=

δy1 8.78857 10 3=

δy2 0.02411=

Elastic behaviour for reactions Rt1 to Rt3 because the deflections δt are less than δy1

112

Reactions Rt at deflections δt

Rt1 δt1δy1

Vy1. V Rt1 44.92956=

Rt2 δt2δy1

Vy1. V Rt2 167.27575=

Rt3 δt3δy1

Vy1. V Rt3 261.0745=

Guess

f´´v 2 f´´y ry c2

Given

my Vy18 π

2 ln c2 ry

. 2 B2

4 ry2

B2

c2. f´´v f´´y( ) B

2 ry. EI.

δt4 δy1 f´´v f´´y( ) B2

. c B2

.

f´´v

ryFind f´´v ry,( ) f´´v 0.03818= ry 0.35529= 0.5 c 0.9=

Rt4 Vy2

my c2

.my ry.

ry

0.5 c

rVy18 π

2 ln c2 r

. 2 B2

4 r2

B2

c2. f´´v f´´y( ) B

2 r. EI. d. V

Rt4 302.7886=

Shear force Rt and unbalanced moment Mt

Rt Rt1 Rt2 Rt3 Rt4( ) 18

. Rt 97.00855=

Mt 0.195 Rt1 0.555 Rt2 0.831 Rt3 Rt4( ) c16

. Mt 69.90084=

4.4 Tension in bottom reinforcement of slab

δb1 δV ∆M 0.195 ∆c δb1 1.49032 10 4=

δb2 δV ∆M 0.555 ∆c δb2 3.75548 10 3=

δb3 δV ∆M 0.831 ∆c δb3 6.74894 10 3=

δb4 δV ∆M ∆c δb4 8.58189 10 3=

δV 1 kIkI

. 1.14949 10 3=

113

Rb1 Vδb1 1 kI

kIδV.

δy1Vy1. ρ

ρ. kI. Rb1 81.31145=

Rb2 Vδb2 1 kI

kIδV.

δy1Vy1. ρc

ρ. kI. Rb2 149.21922=

Rb3 Vδb3 1 kI

kIδV.

δy1Vy1. ρc

ρ. kI. Rb3 180.48547=

Rb4 Vδb4 1 kI

kIδV.

δy1Vy1. ρc

ρ. kI.

Rb4 199.63038=

Rbmax V ρcρ

Vy2. Rbmax 292.17256=

Rb Rb1 Rb2 Rb3 Rb48

Rb 76.33081=

Equilibrium check

Rb Rt 20.67774=

Rb Rt ∆M Vy1δy1

. 0.66356=

Shear force Rb and unbalanced moment Mb

Rb 76.33081=

Mb 0.195 Rb1 0.555 Rb2 0.831 Rb3. Rb4( ) c16

. Mb 50.4322=

4.5 Unbalanced moment capacity Mu

Mu Mt Mb Mu 120.33304=

e MuV

e 0.98634=V 122=

etest =0.984

θ1 2kI

∆cc B

.θ1 0.02136=

θ2 12 Mu

π Ec. h3.ln c

B1 B

c. 10 3. θ2 4.22909 10 3=

θu θ1 θ2 θu 0.02559=

δM 2 ∆M δM 9.6 10 4=

114

4.6 Simplified approach

θ1a 2kI

δε

c B. 1 V

Vε. θ1a 0.01847=

θ1b 2kI

δε

c B. 1 δV

δε. θ1b 0.01947=

Mu θu 103.

12 π kI. EI.

12 ln cB

1 Bc

.

π Ec. h3.

Mu 726.09793=

Mumax1 kI δεVy1δy1

. V. c4

. Mumax1 139.39621=

Mumax2 Vy2 V( ) c4

. Mumax2 174.83296=

Mumax3 ρcρ

Vy2. V c4

.Mumax3 131.47765=

Mumax4 π

4c

1 Bc

. my. 1 ρcρ

. Mumax4 153.15531=

Mumax5 3 fc a. h2

4. 0.15

0.5 h.

13

103. Mumax5 221.42193=

θ2a 12 Mumax3

π Ec. h3.ln c

B1 B

c. 10 3. θ2a 4.62077 10 3=

θu θ1b θ2a θu 0.02409=