Punching Shear Capacity of Fibre Reinforced Concrete …kth.diva-portal.org/smash/get/diva2:610902/FULLTEXT01.pdf · I Punching Shear Capacity of Fibre Reinforced Concrete Slabs with

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Punching Shear Capacity of Fibre Reinforced Concrete Slabs with Conventional Reinforcement

Computational analysis of punching models

ZEINAB TAZALY

Master of Science Thesis Stockholm, Sweden 2011

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Punching Shear Capacity of Fibre Reinforced Concrete Slabs with Conventional Reinforcement

Computational analyses of punching models

Zeinab Tazaly

TRITA-BKN. Master Thesis 334 Structural Design and Bridges, 2011 ISSN 1103-4297 ISRN KTH/BKN/EX-334-SE

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©Zeinab Tazaly 2011 Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Structural Design and Bridges Stockholm, Sweden, 2011 Printed by US-AB, Stockholm, October 2012

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Preface

This Master thesis study was carried out at the Royal Institute of Technology (KTH) in cooperation with the Swedish Cement and Concrete Research Institution (CBI). The investigation was conducted under the supervision of Prof. Håkan Sundquist, Prof. Johan Silfwerbrand and Tekn. Lic. Ghassem Hassanzadeh whom I wish to express my gratitude for their advices, guidance and support.

The performance of this work has given me the opportunity to develop my engineering understanding and to solve a problem that at first seemed very difficult. This has contributed to my personal growth and prepared me to the future profession. Still, this work and also my studies would not have been completed without the support of my family. Therefore, I wish to express my biggest gratitude to my family for all the patience and love they give me.

I would also like to take the opportunity to thank my friends for their help and support and making my years of study to a memorable stage in my life.

Stockholm, October 2011

Zeinab Tazaly

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Abstract

Steel fibre reinforced concrete is not a novel concept, it has been around since the mid-1900s, but despite its great success in shotcrete-reinforced rock walls and industrial floors it has not made any impact on either beams or elevated slab. Apparently, the absence of standards is the main reason. However, the combination of steel fibre reinforced concrete and conventional reinforcement has in many researches shown to emphasize good bearing capacity.

In this thesis, two punching shear capacity models have been analysed and adapted on 136 test slabs performed by previous researchers. The first punching model alternative is proposed in DAfStB – BetonKalender 2011, and the second punching model alternative is established in Swedish Concrete Association – Report No. 4 1994.

Due to missing information of the experimental measured residual tensile strength, a theoretical residual tensile strength was estimated in two different manners to be able to adapt the DAfStB punching model alternative on the refereed test slabs. The first solution is an derivation of a suggestion made by Silfwerbrand (2000) and the second solution is drawn from a proposal made by Choi et al. (2007).

The result indicates that the SCA punching model alternative is easier to adapt and provides the most representative result. Also DAfStb alternative with the second solution of estimating the residual strength contributes to arbitrary result, however due to the uncertainty of the estimation of the residual tensile strength, the SCA punching model is recommended to be applied until further investigation can confirm the accuracy of the DAfStB alternative with experimentally obtained residual tensile strength.

Keywords: Steel fibre reinforced concrete slab, punching capacity, concrete test, residual tensile strength, DAfStB, Swedish Concrete Association, conventional reinforcement, Silfwerbrand.

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Sammanfattning

Stålfiberarmerad betong har funnits sedan mitten av 1900-talet men har trots sin stora framgång inom sprutbetongförstärkt berg och industrigolv på mark inte fått något genomslag i vare sig balkar eller bjälklag. Tydligen är avsaknaden av normer en avgörande orsak. Emellertid har kombination av fiberbetong och slakarmering i plattor påvisat betydligt bättre bärförmåga.

I föreliggande examensarbete har två modeller for genomstansningskapacitet undersökts och analyserats mot 136 försöksplattor, provade av tidigare studier. Den första beräkningsmodellen föreslås i DafStB – BetonKalender 2011, och den andra beräkningsmodellen återfinns i Svenska Betongföreningens rapport nr. 4 om stålfiberbetong, 1994.

Till följd av brist på information om den experimentellt uppmätta residualhållfastheten, har en teoretisk härled residualhållfasthet uppskattats på två olika sätt för att ha möjligheten till att tillämpa DAfStBs beräkningsmodellen på de granskade försöken. Den första lösningen är en härledning av ett förslag publicerat av Silfwerbrand (2000) och den andra lösningen är tagen ur en teknisk rapport av Choi et al. (2007).

Resultatet indikerar att Betongföreningens beräkningsmodell är lättare att anpassa och ger de mest representativa resultaten. Även DAfStB-alternativet med den andra lösningsförslaget på residualhållfastheten uppvisade tämligen goda resultat, men med hänsyn till osäkerheten kring uppskattningen av residualhållfastheten, rekommenderas Betongföreningens genomstansningssmodell till vidare granskningar kan bekräfta noggrannheten av DAfStB alternativet med experimentell uppmätt residualhållfasthet.

Nyckelord: Stålfiberarmerade betongplattor, genomstansningskapacitet, betongprovning, residualhållfasthet, DAfStB, Svenska Betongföreningen, konventionell armering, Silfwerbrand

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Contents

Preface ........................................................................................................................................ i

Abstract .................................................................................................................................... iii

Sammanfattning ....................................................................................................................... v

Nomenclature ............................................................................................................................ x

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

1.1 Background ............................................................................................................. 1

1.2 Purpose .................................................................................................................... 2

1.3 Limitation ................................................................................................................ 2

2 Fibre Reinforced Concrete ............................................................................................ 3

2.1 General .................................................................................................................... 3

2.2 History ..................................................................................................................... 4

2.3 Application .............................................................................................................. 4

2.3.1 Industrial Floors ......................................................................................... 5

2.3.2 Shotcrete Tunnel Lining ............................................................................. 5

2.3.3 Thin Building Component .......................................................................... 5

2.3.4 Repair ......................................................................................................... 5

2.4 Steel Fibre Reinforced Concrete ............................................................................. 6

2.4.1 Mechanical Properties ................................................................................ 7

2.4.2 Mechanism of Crack Formation and Propagation ...................................... 9

2.4.3 Mix Design and Manufacture ................................................................... 10

2.4.4 Concrete Testing ...................................................................................... 10

2.5 Previous Research ................................................................................................. 13

3 Punching Shear Capacity ............................................................................................ 15

3.1 Punching Phenomenon .......................................................................................... 15

3.2 Previous Research ................................................................................................. 16

4 Investigated Test Trials ............................................................................................... 17

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4.1 Description of the experimental test ...................................................................... 17

4.1.1 N. Swamy and S. Ali, 1979 ...................................................................... 17

4.1.2 D. Theodorakopoulos and N. Swamy, 1989 ............................................ 18

4.1.3 S. Alexander and S. Simmonds, 1992 ...................................................... 19

4.1.4 A. Shaaban and H. Gesund, 1994 ............................................................. 19

4.1.5 M. Harajli et al., 1995 .............................................................................. 20

4.1.6 B. Hughes and Y. Xiao, 1995 ................................................................... 21

4.1.7 P. McHarg, 1997 ...................................................................................... 21

4.1.8 G. Hassanzadeh and H. Sundquist, 1998 ................................................. 22

4.1.9 A. Azevedo, 1999 ..................................................................................... 24

4.1.10 S. Ozden et al., 2006 ................................................................................ 24

4.1.11 J. Hanai and K. Holanda, 2008 ................................................................ 25

4.1.12 L. Nguyen-Minh et al., 2011 ................................................................... 25

4.2 Range of Properties ............................................................................................... 26

5 Design Punching Models ............................................................................................. 27

5.1 General .................................................................................................................. 27

5.2 Basic Condition ..................................................................................................... 27

5.2.1 Safety Factors ........................................................................................... 27

5.2.2 Concrete Characteristic Strength .............................................................. 28

5.2.3 Conventional Reinforcement .................................................................... 28

5.3 Punching Shear Capacity with Addition of Steel Fibres ....................................... 29

5.4 Alternative I: DAfStB ........................................................................................... 29

5.5 Alternative II: Swedish Concrete Association ...................................................... 34

5.6 Steel Fibre Reinforced Concrete Slabs without Conventional Reinforcement ..... 35

6 Analysis of the Result .................................................................................................. 37

6.1 Final applied formulas ........................................................................................... 37

6.2 Effect of the Slab Thickness .................................................................................. 38

6.3 Effect of the Fibre Volume Ratio .......................................................................... 39

6.4 Effect of the Fibre Slenderness Ratio .................................................................... 40

6.5 Effect of the Fibre Factor ...................................................................................... 41

6.6 Effect of the Conventional Reinforcement ............................................................ 42

6.7 Effect of the Concrete Compressive Strength ....................................................... 43

6.8 Effect of the Concrete Tensile Strength ................................................................ 44

6.9 Effect of the Computed Residual Strength of DAfStB ......................................... 45

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7 Conclusion .................................................................................................................... 47

8 Recommendation for Future Research ...................................................................... 49

References ............................................................................................................................... 51

Appendix A: Detailed Tables of the Test Slabs .................................................................. 57

A.1 N. Swamy and S. Ali 1979 .................................................................................... 57

A.2 D. Theodorakopoulos and N. Swamy 1989 .......................................................... 60

A.3 S. Alexander and S. Simmonds 1992 .................................................................... 63

A.4 A. Shaaban and H. Gesund 1994 ........................................................................... 64

A.5 M. Harajli, D. Maaloufand H. Khatib 1995 .......................................................... 66

A.6 B. Hughes and Y. Xiao 1995 ................................................................................. 68

A.7 P. McHarg 1997 .................................................................................................... 70

A.8 G. Hassanzadeh and H. Sundquist 1998 ................................................................ 71

A.9 A. Azevedo 1999 ................................................................................................... 73

A.10 S. Ozden, U. Ersoy, and T. Ozturan 2006 ............................................................ 75

A.11 J. Hanai and K. Holanda 2008 ............................................................................... 77

A.12 L. Nguyen-Minh, M. Rovnak, T. Tran-Quoc, and K. Nguyen-Kim 2011 ............ 79

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Nomenclature

Roman letters

Ac cross sectional area of concrete

Ap punching area

As total cross sectional area of reinforcement

Asy,flex flexural reinforcement area in y-direction

Asz,flex flexural reinforcement area in z-direction

C size factor

CRd,c coefficient in Eurocode 2

Ec modulus of elasticity of concrete

Es modulus of elasticity of steel

F vertical point load

Ff fibre factor

I10 toughness index according to ASTM (1992).

Ix ratio between area under the stress-deflection curve between δ = 0 and δ = (X + 1) δcr/2, and the (elastic) area between δ = 0 and δ = δcr

L slab length

Lf steel fibre length

Lf/df fibre slenderness ratio, fibre aspect ratio

Ls standard deviation of the logarithm of individual test results

Pcr first crack load

Py yield load

R ratio (in %) between the average load carrying capacity between certain displacement after cracking and the load at first crack for standard test beam loaded in four point bending

VRd,c punching shear capacity of ordinary reinforced concrete

VRd,cf punching shear capacity of steel fibre reinforced concrete

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VRd,f punching shear capacity achieved from the steel fibres

b slab width

bw effective depth of punching area

c column size

c1 column cube size

c2 column circular size

ch column height

d effective depth of slab

d diameter

df fibre diameter

dfe equivalent fibre diameter for non-circular fibre cross section

f’c concrete compression strength of an equivalent normal concrete without fibres

f fct0,u design value of the flexural strength of the steel fibre reinforced concrete

f fctR,u design value of the post-cracking tensile strength at the ultimate limit state, according to DAfStB

fcc cylindrical compressive strength of concrete

fcc,cube cube compressive strength of concrete

fcfl cylindrical flexural strength of concrete

fcfl,cube cube flexural strength of concrete

fck characteristic compressive strength of concrete

fcsp cylindrical splitting strength of concrete

fcsp,cube cube splitting strength of concrete

fct tensile strength of concrete

ff,ctR the average residual flexural strength of SFRC multiplied with the factor κfG

ffl,cr crack strength at flexural test of concrete

ffl,res,m average residual flexural strength of steel fibre reinforced concrete

ffl,res1 residual flexural strength of steel fibre reinforced concrete at 0.5 mm deflection

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ffl,res6 residual flexural strength of steel fibre reinforced concrete strength at 3.5 mm deflection

fpc post-crack tensile strength of fibre reinforced concrete

ft tensile strength of concrete under pure tension

h slab thickness

k size effect factor

ks constant depending on number of beams tested in bending

u length of control perimeter at which the punching acts

vRd,c punching shear resistance of ordinary reinforced concrete

vRd,cf punching shear resistance of steel fibre reinforced concrete

vRd,f punching shear resistance achieved from the steel fibres

Greek letters

αf coefficient considering long term and unfavourable effects of SFRC

β factor considering the effect of fibre shape and concrete type

γc partial safety factor for concrete

γf partial safety factor of the steel fibre concrete strength

δ load deflection

δcr first crack deflection

δmax maximum load deflection

δy yield deflection

κfF factor considering fibre orientation

κfG factor considering the influence of the component size

λ1 expected pull-out length ratio of steel fibre

λ2 efficiency factor of steel fibre orientation in the concrete crack state

λ3 steel fibre group reduction factor associated with the number of fibres pulling-out from matrix per unit

λf fibre slenderness ratio, fibre aspect ratio (= Lf/df)

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μf bond factor

ξ slab size factor

ρ reinforcement ratio

ρ1 total reinforcement ratio in x- and z-direction

ρf steel fibre volume content (= ρs)

ρs steel fibre volume content (= ρf)

ρy reinforcement ratio in y-direction

ρz reinforcement ratio in z-direction

σcp compressive stress in the concrete from axial load or prestressing

τ average interfacial bond strength of fiber matrix

τf bond strength between fibre and surrounding concrete

Abbreviation

FRC fibre reinforced concrete

HSC high strength concrete

NSC normal strength concrete

SF steel fibre

SFRC steel fibre reinforced concrete

CHAPTER 1. INTRODUCTION

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1 Introduction

1.1 Background

The concept punching refers to a local failure of bi-axial shear characteristics in a flat slab, either below a point load or next to a support. In order to prevent punching shear failure in connection areas, large amount of reinforcement or large energy-absorbing ability are normally required. Punching problem is a very important phenomenon that often represents the design impact on the slabs, not just for the column-supported slab as bridge deck flat slab, but also for the bottom slabs under columns. Such structures need to be heavily reinforced.

Reinforced concrete has been used since the middle of the 19th century, and is now, during the present century, the most used building material. However, reinforced concrete by conventional steel bars is complicated for both engineers and workers. Concrete has also predictable weaknesses, for instance very low tensile strength and brittleness. Moreover, to reinforce the concrete by conventional steel bars is an expensive and time-consuming procedure for both designers and contractors. Elimination of re-bars from the design and construction process does not particularly withstand the stress levels in a construction, although it will increase the tensile strength. Enhanced productivity might be achieved by using non-tensioned re-bars as primary reinforcement and steel fibres in concrete as secondary reinforcement.

The fibrous concrete has been around since the beginning of the 20th century and has had great success in the shotcrete reinforced rock and industrial floors on ground but hardly any impact in either beams or elevated slabs. This is the case despite the fact that a combination of fibrous concrete and non-tensioned reinforcement in slabs has been shown to provide good resistance, where the conventional reinforcement occupies bending moment and the fibres help to increase resistance for punching.

One of the reasons that inhibits the utilization of fibres as reinforcement is the limitation of span length when designing flat slabs without flexural reinforcement. Another reason is the absence of standards directions for designing fibre reinforced concrete structures. In the manuals and literature, there are various proposals for calculation formulas. In our country, the Swedish Concrete Associations published Concrete Report No. 4, entitled "Steel Fibre Concrete - Recommendations for the Design, Implementation and Testing" in 1994. British Concrete Society and American Concrete Institute (ACI) have also published reports on the matter. Recently, even working teams within the international organizations RILEM and fib

Chapter

1.2. PURPOSE

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published or begun work on recommendations. In addition, Eurocode 2 lately published its equations for punching capacity of conventional reinforced concrete slabs. There is therefore a great need to go through this problem in the search of the best to track the best models and equations representing the punching capacity of fibre reinforced concrete.

1.2 Purpose

The thesis aims to carrying out a survey of the formulas found in literature and that are dealing with punching capacity of fibre reinforced concrete slabs with and without conventional reinforcement and to analyse them against experimental data and suggest the equation that best responds to the data. To compare how the different formulas work, a database of slabs tested by different researchers shall be built up. The final aim is to propose the best equation representing the punching capacity of fibre reinforced concrete. The method contains literature studies, selection of the most promising calculation equations, testing the correspondence between these equations and the test data, report writing, and oral presentation.

1.3 Limitation

Since there are a big variety of fibres types used in fibre reinforced concrete, only steel fibres will be attracted in this thesis. The study will analysed slab test with only conventional reinforcement, thus slab with any additional reinforcement such as stirrups, shear reinforcement, prestressed steel bars, etc. will be obliterated from the analyses. Slab tests with missing data of that punching capacity will also be obliterated. There are no specific limits for the size of the slabs.

Due to time constraints, only two alternatives of punching models for fibre reinforced concrete were analysed against the experimental data. These two models are (1) the proposal in the Swedish Concrete Associations Concrete; Report No. 4, 1994 and (2) the proposal in DAfStB; the German Committee for Reinforced Concrete, 2011.

CHAPTER 2. FIBRE REINFORCED CONCRETE

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2 Fibre Reinforced Concrete

2.1 General

Fibre Reinforced Concrete, also known as FRC, is a type of concrete that contains fibrous substances of different variety that increase its structural strength and cohesion. Given that concrete is a quite brittle material with very good compressive strength but comparatively little tensile strength; it makes it likely to crack under many conditions. By adding fibres, not only the strength capacity and the structural integrity will increase, also the post-crack state will improve radically. The main idea of using fibre reinforced concrete is to provide the entire concrete mass with fibres, thereby creating a new building material with its own specific characteristics. In the event of failure, a slab or any other fibre reinforced concrete structures, will only fall a few centimetres before completely break which will prevent endangering anyone’s life (Tepfers, 2010)

Figure 2.1: Steel Fibre Reinforced Concrete (Löfgren 2005).

There are about 19 different types of material used as fibre reinforcement, see Table 1.1. Steel fibres are the most used fibre of all, estimated up to 50 percent of total tonnage used, followed by polypropylene fibres (or synthetic fibres) and glass fibres with 20 percent and 5 percent respectively. The remaining 25 percent belongs to other type of fibres (Banthia, 2008). Current study investigates only steel fibres as reinforced concrete.

Chapter

2.2. HISTORY

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Table 2.1: Physical properties of fibres used as reinforcement in concrete (Löfgren 2005). Fibre type Diameter

[mm] Tensile

strength [MPa] Elastic

modulus [GPa] Ultimate

elongation [%] Steel 5-12 200-3000 197-210 0.5-5 E glass 8-15 2000-4000 72 3.0-4.8 AR glass 8-20 1500-3700 80 2.5-3.6 Acrylic (PAN) 5-17 200-1000 14.6-19.6 7.5-50.0 Aramid 10-12 2000-3500 62-130 2.0-4.6 Carbon 7-18 800-4000 38-800 1.3-2.5 Nylon 20-25 965 5.17 20.0 Polyester 10-8 280-1200 10-18 10.0-50.0 Polyethylene 25-1000 80-3000 5-150 2.9-100 Polypropylene 10-200 310-760 3.5-4.9 6.0-15.0 Polyvinyl acetate 3-8 800-3600 20-80 4.0-12.0 Cellulose (Wood) 15-125 300-2000 10-50 20.0 Coconut 100-400 120-200 19-25 10.0-25.0 Bamboo 50-400 50-350 33-40 - Jute 100-200 250-350 25-32 1.5-1.9 Asbestos 0.02-25 200-1800 164 2.0-3.0 Wollastonite 25-40 2700-4100 303-530 -

2.2 History

Fibres have been used as reinforcement since ancient times, starting from more than 3500 years ago, when mortar and sun-baked mud bricks reinforced with straw were used in brittle matrix material to build the 57 m high hill of Aqar Quf near Baghdad (Hannant, 2000).

For the first time, in 1874, the American A. Berard applied for patent of the idea of combining concrete with spread “fibres” consisting of grains of steel leftovers. About 30 years later, in the early 20th century, asbestos fibres in concrete were encountered. Thereafter, 1918, H. Alfsen patented in France the development of adding fibres of steel, wood or other appropriate materials in concrete to increase its tensile strength. Consequently, fibre reinforced concrete becomes one of the topics of interest in the construction material market.

When asbestos, used in concrete and other building materials, was found to cause health risks as cancer, some new substances had to be found as replacement. Of that cause cellulose fibres, synthetic fibres such as polypropylene and glass fibres have been developed and used for the past 30 years. Research to find new fibre types to reinforce concrete still continues in the present day.

2.3 Application

In Sweden, steel fibre concrete is used for example in the production of floating boat landing, earth basement, septic tanks, cable drums and safety elements between traffic lanes. These are


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three-dimensional products with a traditional look. The more frequent cases of building structures with fibre reinforced concrete are described as follows.

2.3.1 Industrial Floors

Steel fibre concrete is used in ground-floor slabs and paving to control cracks and increase bearing capacity. It is also used to replace conventional steel mesh in industrial slab on grade floor. However, the most important advantage of steel fibres in floor concrete is the simplification of production technique. The exclusion of mesh creates faster production and saves dowel bars, tie wires, reduces labour costs etc., which is economically beneficial.

Fibre dosages of between 15 kg/m3 and 60 kg/m3 are commonly used in floors with slab thicknesses between 120 mm and 200 mm. Although, experience at the Swedish Concrete and Cement Institute indicates that fibre dosages less than 60 kg/m3 often lead to nondurable slab on ground structures that needs reparation.

2.3.2 Shotcrete Tunnel Lining

Shotcrete is mixed concrete-mortar, projected with high pressure on rock surface. Steel fibres can be added to shotcrete and be used on rock wall as rock-slope stabilization. The irregularity of the rock walls and the simplification of manufacturing is the main motivation of using shotcrete. The driving force for shotcrete to become predominant on the market is the idea of being able to substitute steel mesh with shotcrete technique.

2.3.3 Thin Building Component

The simple thin plate structures has been in the market for decades and in various regions. Reinforcement bars are not sufficient in thin sheet material, taking into consideration that conventional reinforcement inhabit a great part of the structural elements area. Thus, fibre constitute the primary reinforcement in such construction, however the fibre concentration is relatively high, typically more than 5 percent by volume (Minelli, 2005). In the beginning, asbestos fibres were used but successively replaced by glass and organic fibres in line with the discovery of health risks caused by asbestos in building material.

By means of fibre in concrete, it is attainable to produce a thin and tough material that can be used as facade elements, balconies, cabinets, safety deposit box, pipes, channel elements, street furniture, decorative details, noise barriers, etc.

2.3.4 Repair

The performance of the homogenous material is mainly controlled by important matrix and steel fibre properties, such as volume percentage of the steel fibres and steel fibre orientation, that have an impact on the mechanical behaviour.

Resistance to high temperature that fibre concrete can persuade makes it suitable as protection material. By polypropylene fibre, prevention of spalling phenomena can be avoided and

2.4. STEEL FIBRE REINFORCED CONCRETE

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structural member can withstand temperature up to 1500 °C (Hannant, 2003). However, at the same time, steel fibre gives the material a certain residual bending resistance even when exposed to high temperature, improving the bearing capacity of the structure itself.

In these cases, initial cost is not the prime consideration if product life can be increased, typically by 100 per cent (Hannant, 2003).

2.4 Steel Fibre Reinforced Concrete

By definition, steel fibre concrete (SFRC) is a composite material made of hydraulic cement, cement water and a dispersion of discontinues steel fibres that are nestled in the cement-matrix. The matrix, that is to say the unreinforced concrete, can consist of fine and coarse aggregate, and sometimes of silica fume and fly ash or any other prescription for the concrete mixture. Initially, the steel fibres and the matrix are bonded and interacting homogenously. When increasing the load, the matrix starts to crack and the fibres will carry the load. Thereafter, the mechanism depends merely on the steel fibres form and shape. Some fibres may fracture and others may pull out, depending on anchorage length, concrete strength and proprietary shape.

The market offers steel fibres in many sizes and shapes including mild steel and high tensile steel. Also, stainless steel fibres are available. The steel itself is produced by a series of hot and cold working methods. In some cases, the steel is chopped from drawn wires and in other cases its slit from sheet or milled from ingots. Steel fibres have as well been produced from hot melt extract (Shah, 1981).

Steel fibres are normally divided into two categories, smooth and deformed. Cross-sectional shapes include circular, rectangular, sickle shaped and mechanically deformed in various ways to improve the bond strength. When it comes to longitudinal shape (Figure 2.2) there are straight, hooked, crimped, curved, paddled, irregular, etc. In the mechanical point of view, the deformed steel fibres are more efficient, thus using greater surface area to increase the cement-matrix bond and creating better pull out resistance, (Hughes & Fattuhi, 1976). The straight fibres on the other hand are only bond to the concrete by friction and chemical adhesion. The steel fibre lengths range from 10 to 65 mm with equivalent diameters between 0.5 and 1.2 mm.

Figure 2.2: Different type of steel fibres (Löfgren 2005 and Minelli 2005).


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2.4.1 Mechanical Properties

The behaviour of the mechanical performance of the homogenous material is mainly controlled by important matrix and steel fibre properties, such as volume percentage of the steel fibres and steel fibre orientation.

Adding steel fibres in the matrix has a minor influence on, for instance the compressive strength, the Poisson’s ratio, modulus of elasticity and porosity, (Vondran, 1991). Hence, before any crack initiation, steel fibres have not a noticeable effect on the concrete behaviour. The main benefit of using steel fibres is the deformation capacity and the crack control as they prevent microcracks from propagating, see Figure 2.3. Equally, by limiting the crack width and crack growth, fibrous concrete help protecting concrete members from exterior as well as harmful environment, such as nitrate and chloride, (Minelli, 2005).

The ultimate elongations at break of the steel fibres are about twofold to threefold greater than the strain when the matrix breaks, and up to twentyfold increase in crack resistance or toughness, (Shah, 1981) thus long before the steel fibre strength is approached the matrix will fail, (Hannant 2003). As long as the loading is small and the structural element is un-cracked, the fibrous concrete acts as any ordinary concrete. It is not until the concrete is in the cracking phase when a significant effect of the steel fibres is obtained.

Figure 2.3: The fracture process in uni-axial tension and the resulting stress-crack opening

relationship in SFRC (Löfgren 2005).

The steel fibres affect the concrete by absorbing the tension and distribute it effectively, and bridging the cracks. The main theoretical benefits and emphasis of the inclusion of steel fibres in hardened concrete relate to the post-cracking state, where the increase in strength, failure strain and toughness of the composite is due to the steel fibres bridging the cracks. While inclusion of steel fibres implies less design work and a reduction of thickness in concrete structures subjected to flexure load.

The tensile strength of steel fibres varies, but is usually around 1200 MPa, which is about threefold higher than that of the average reinforcement bar, but there is also high-strength steel with tensile strength over 3000 MPa in the market, regularly as short steel fibres to counteract small cracks.


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Steel Fibre Pull-out The fibre pull-out mechanism is essential when tensile strength is transmitted from the steel fibres to the surrounding concrete. Therefore, in order to avoid brittle failure, the pull-out length must be taken in cautious consideration. The average fibre pull-out length is l/4 of the steel fibre length, (Hannant, 2003). This indicates that the longer the fibres are the better failure resistance and better mechanical performance is obtained, see Figure 2.4. However, it should be realized that a larger volume of longer steel fibres can not be uniformly distributed. Workability and increased uniform distribution becomes a problem. Deformed steel fibres increase the pull-out strength and subsequently the mechanical properties of the composite, (Shah, 1981).

Figure 2.4: Effect of fibre size on crack bridging (Betterman 1995).

Steel Fibre Bond Strength The bond strength, also in some sentences described as the slenderness ratio, is the strength between steel fibre and matrix, and steel fibre adhesion area in relation to the cross section are also important and are described for the steel fibres as the aspect ratio L / d. L stands for the length of the steel fibre and d stands for the diameter of the steel fibre. Since the failure of steel fibres and steel fibre pull-out depend on the fibre shape and concrete strength, it is not possible to give a generalized formula that can be representative in numerical calculations as a ‘bond strength’. Having a high aspect ratio gives a positive effect on the post-peak behaviour of the steel fibres, so that the slope of the declining stress-strain curve decreases.

Steel Fibre Corrosion Corrosion of steel fibres can occur but is not a major problem. Granju & Balouch (2005) have unexpectedly established that samples of fibre reinforced concrete exposed to one year of salty marine condition were strengthened, according to the hypothesis that the surface area of the fibres becomes unsmooth, and so the pull-out phenomenon becomes more complicated. Moreover, the result indicates that only the fibres crossing the crack within 2 mm to 3 mm zone at the external surfaces demonstrate corrosion and no concrete spalling due to corrosion of the steel fibres was observed.

Stress-Strain Curve It has been proved by Ramakrishnan et al. (1981) that steel fibre reinforced concrete is six-fold better in carrying impact loads than the plain concrete, but since this mechanism is more difficult to analyse, it is easier to study steel fibres stress behaviour under direct tension. For

Large Fibres Microfibres

Microcracks

Short Microfibres

Large Fibres Plain Concrete

Strain

Stress


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that reason, steel fibre reinforced concrete depends on the knowledge of the stress–strain relationship

Typical stress–strain curves for steel fibre reinforced concrete with different volume fraction of fibres and concrete types are illustrated in Figure 2.5. The curve formation depends on several factors, such as the size of the specimen, method of testing, type of concrete stiffness of the testing machine, gage length, and whether single or multiple cracking occurs within the gage length. The first linear ascending part of the curves defines the elastic modulus of the un-cracked composite (Ec) which cracks at its normal cracking stress. The area under these curves defines the toughness of the fibre concrete which is usually well in excess of the area under the first linear part.

Figure 2.5: A typical stress-strain curve of steel fibre reinforced concrete and plain concrete

under uni-axial tension (Choi et al. 2007).

2.4.2 Mechanism of Crack Formation and Propagation

When loading steel fibre reinforced concrete, deflection and resistance can be recorded and thus distinguishing characteristic phases can be observed. Fibrous concrete with low volume percentage, attains similar phases as elastic-phase features, with linearly increasing stress followed by decreasing tension at the cracking stage. At increased steel fibre volume percentage, the first obtained phase is similar to low steel fibre percentage, except followed by a phase of increasing tension with continued crack propagation, to end up with a descending stress phase (Skarendahl, 2004).

Once the steel fibres are mixed in a rotary mixer, the fibre distribution in the concrete develops into randomly three-dimensional direction (Hannant, 2003). Nonetheless, it is confirmed that increased steel fibre volume percentage in the concrete makes the distribution more even. The initial cracks in the concrete occur when the maximum stress is obtained, if the steel fibre content is high, the resistance against the stress will also be high. Where the steel fibre content is low, the resistance will decrease and so the cracks will seek to follow the line of least resistance. For that reason the concretes crack resistance will be controlled by the steel fibre distribution and the steel fibre amount, (Tepfer, 2010). This elucidates the fundamental importance of the post-crack state that steel fibres contribute with.

Considering the small distance between the cracks in steel fibre reinforced concrete, the crack width decreases significantly compared with plain concrete and mesh-reinforced concrete, compared with ordinary reinforcement.


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2.4.3 Mix Design and Manufacture

The mix designs procedure is decisive for the performance of the concrete. Method used to blend the steel fibres with the cement mortar differs and depends on the steel fibre type and on the function of the final product. Additionally, the matrix composition affects the ability to mix steel fibres. For example, polymer impregnation or sulphur impregnation of the matrix improves the contribution of the steel fibres (Shah, 1981). To facilitate the interference of steel fibres in the cementitious matrix and avoiding balling of fibres during mixture, some steel fibres are delivered loosely bonded with water-soluble glue.

For the best result a high proportion of fines should be used, normally a typical mix contains 300-350 kg/m3 cement and 800 kg/m3 river sand, which is usually mixed with pulverized-fuel ash. It is advisable that aggregates that are being used are smaller than 20 mm. The amount of short steel is limited to 1 per cent due to the fact that concrete gets impenetrable, since concrete contains a high percent of aggregate particles, (Hannant, 2003). Longer steel fibres, as mentioned before, gives better reinforcement but reduce the workability and cluster together, if not added properly by sieve through a screen, compromise must be reached usually by using steel fibres that have a low aspect ratio (Shah, 1981).

Water cement ratio should be changed with great care since a ratio increased to more than 0.5 due to water addition may possibly not adjust workability and placing ability under vibration, (Chanh, 2005). It is preferable to utilize water cement ratio with a rate lower than 0.55 by adding plasticizers or super plasticizers (Hannant, 2003).

Normally, the dosage of steel fibres is done manually, but dosage equipment with various degrees of automatics is available. The most common mixers are transit mix truck or revolving drum mixer. The typical mix speed is about 30-40 revolutions per minute which should disperse the steel fibres evenly. Attention must be taken into account for the risk of clustering, hence the ball formation can not be solved by prolonged mixing (Chanh, 2005).

Most commonly, the steel fibres are added last to the fresh concrete. With the support of an appropriate filter, fibres are mixed gradually without clustering together. Alternatively, the fibres can be added to the fine concrete aggregate mix on a conveyor belt during the addition of aggregate to the concrete mix (Hannant, 2003). Shotcrete is more difficult to mix, spray and convey.

Another production process is when pre-placed fibre volumes are mould and then mixed and infiltrated by fine-grained cement-based slurry. This procedure makes it achievable to acquire very high strength and toughness in localized regions such as beam/column intersections. This procedure makes it possible to use tensile strengths up to 16 MPa and flexural strengths up to 60 MPa, (Hannant, 2003). The final point at the mixing stage is principally the same as for ordinary concrete, but with the difference regarding workmanship.

2.4.4 Concrete Testing

There are different ways to test steel fibre reinforced concrete, tests are carried out on both fresh concrete and hardened concrete. Wash out tests on random samples can be performed to check fibre dosage and distribution in fresh concrete. Testing hardened concrete is more


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complex but more decisive to characterise the concrete. Following, the most common tests are described. Concrete Compressive Test Compressive strength is usually determined on individually made test cylinders or cubes, to give an indication of the potential compressive strength of the material rather than the strength achieved in reality. The dimensions of the test cylinder or test cube differ regarding the standards that are to be used. There is no significant impact on the compressive strength with steel fibre addition compared to normal concrete (Døssland, 2008), although adding steel fibres to the concrete alters the behaviour of the concrete from brittle to ductile. Tensile Splitting Test By tensile splitting a sample, the tensile strength can be approximately determined. Usually tensile strength of normal brittle concrete is less imperative, but not in the case of steel fibre reinforced concrete, since steel fibres influence the ductility (Døssland, 2008). The test is conducted by subjecting a cube or cylindrical sample to a longitudinal concentrated load as in Figure 2.6. This gives rise to a large compressive stress at the top and bottom of the specimens, whereas the rest of the object is exposed to high tensile stress. Consequently, by applying a mathematical formula the tensile strength can be achieved.

Figure 2.6: Tensile splitting test and the stress distribution (Holmgren et al. 2008).

Uni-axial Tensile Test (UTT) The stress-crack (σ-ω) opening relation, which is an essential mechanical performance of steel fibre reinforced concrete, can be determined through a uni-axial tension test. The most used UTT method is the one proposed in the RILEM (2001) recommendations. The test procedure is carried out by gluing the flat ends of a notched cylindrical specimen to the fixed loading plates of the testing machine, see Figure 2.7. Thereby, the average crack opening as the control variable is used to establish the concrete tensile strain. However, it is worth mentioning that there are experimental difficulties related to UUT method, such as the demands on highly trained experienced laboratory assistance. It is therefore preferable to assume the uni-axial tensile strength to be 90 percent of the measured tensile splitting strength according to the standard SS-EN 12390-6.


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Figure 2.7: Stress-crack relation for steel fibre reinforced concrete loaded in uni-axial

tension (Löfgren 2005).

Flexural Test - Three Point Bending Test (3PBT) The ultimate flexural strength is a fundamental criterion for various types of structures, and it is usually greater than the uni-axial tensile strength (Hannant, 2003), hence an understanding of the direct tensile mechanism can be of a great importance. By bending a small beam sample and measuring the load–deflection (see Figure 2.8) curve or the crack mouth opening, the tensile strength behaviour can be evaluated. This can be obtained by a so-called three point bending test, illustrate in Figure 2.9. During the test, the cracking load, the ultimate load and the residual load can be determined.

There are different ways to perform 3PBT, depending on what standards or recommendation that are being used. RILEM, the European standards, the Japanese standard (JSCE SF-4) and the Italian standards (UNI 11039), they all applies notched test beams. While ASTM International, the American standards, apply both un-notched beams (ASTM C 1018-97) and notched beams (ASTM 1399-98). The weakness of performing 3PBT with notched beams is the pre-location of crack formation, ending with not knowing where the weakest spot of the beam is. On the other hand, when using un-notched beams the crack propagation starts from the weakest cross section between the load points.

Figure 2.8: Schematic description of plain concrete and steel fibre reinforced concrete

exposed to flexural bending.

Flexural Test - Four Point Bending Test (4PBT) 4PBT is a test method to achieve the ultimate tensile strain, residual strength and toughness of concrete through the maximum bending moment curvature. The small beams are tested with a four point bending set-up, almost similar to 3PBT, that allows constant moment distribution


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between the two loads. The main difference between three point and four point bending test is the stress distribution as illustrated in Figure 2.9.

Figure 2.9: To the left three point bending test and to the right four point bending test.

2.5 Previous Research

In the last four decades, there have been various experiments of steel fibre reinforced concrete reported. Research has been conducted to investigate the various mechanical phenomena that steel fibre contributes with. Swamy & Ali (1979) and Theodorakopoulos & Swamy (1989) observed performance enhancement of shear strength and ductility in slab-column connection by using steel fibre reinforced concrete. Ay (1999) redesigned an ordinary reinforced concrete bridge with steel fibre reinforced high performance concrete. This achievement was found to be very cost and time effective since it made it possible to exclude reinforcement bars in both design and production procedure.

During the last decades, steel fibre reinforcement has become a motivating subject to improve punching shear capacity and crack control of slab-column connections. Researcher such as Alexander & Simmonds (1992), Theodorakopoulos & Swamy (1993), Harajli et al. (1995), McHarg et al. (1997) have verified outstanding test results. Hanai & Holanda (2008) investigated the punching and shear strength of fibre reinforced flat slabs and beams respectively and established that beams and slabs with same height, longitudinal reinforcement ratio and concrete properties have comprehensible similarities. Punching tests in many combinations of ordinary and high strength concrete, steel fibres and shear reinforcement were performed by Azevedo (1999) and Holanda (2002). The outcome confirmed the expectations, also revealing a good competence of steel fibres in high-strength matrix.

Hassanzadeh & Sundquist (1998) preformed an experiment on five slabs on columns to study the influence of fibres on punching shear strength. Thereby indicating the significant increase of the ultimate load capacity and the stiffness of the slabs.

Moreover, steel fibre concrete structures are also an important subject matter for structures sustaining lateral loads due to the ability to absorb energy dissipation of the structures. Cheng & Montesinos (2010) tested slabs under monotonically increased concentrated load and

2.5. PREVIOUS RESEARCH

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established that addition of fibres led to an increase in slab punching shear and deformation capacity, as well as changing the failure mode from punching shear failure to flexural yielding.

Appa Rao & Sreenivasa Rao (2009) carried out a series of test specimens to determine the fracture properties and toughness indices of steel fibre reinforced concrete, to conclude that shear strength and toughness of concrete improve significantly with addition of steel fibres. Additionally, according to Granju & Balouch (2003), cracked steel fibre reinforced concrete samples exposed to marine environment confirmed small sensitivity to corrosion and enhanced the flexural strength after corrosion.

CHAPTER 3. PUNCHING SHEAR CAPACITY

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3 Punching Shear Capacity

3.1 Punching Phenomenon

Punching shear phenomenon emerges in concrete slab structures exposed to high bending moment and concentrated shear stress loads that are either supported on a column or subjected to a point load as shown in Figure 3.1. As an example that can be mentioned is column-supported slabs or bridge deck flat slabs, also foundation slabs under columns are common. The benefit with flat slabs is the exclusion of haunches, capitals, beams and girders, which reduces overall floor depth, thereby creating additional floor space for a given building height.

Figure 3.1: To the left slab-column connection exposed to point load and stress, to right

cross section of punching failure.

Figure 3.2: Failure in the compression zone near the bottom surface of the slab close to the column (Broms 2005).

Chapter

3.2. PREVIOUS RESEARCH

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Punching shear failure is a three-dimensional issue of brittle mode due to the high shear stress. Since punching phenomenon is considered as a combination of shearing and splitting that occurs without crushing, failure is assumed to arise in the compression zone near the bottom surface of the slab close to the column (Broms. 2005), as shown in Figure 3.2, due to the column reaction reaching a critical level.

When only shear forces are applied and no bending moment exists, the cracks will propagate in 30° angle alignment, while pure flexural cracks created by a bending moment with no shear force have 90° angel alignment (Hermansson & Johansson, 2009). Consequently, once punching appears the column and the slab disconnect and the resistance capacity of the structure is radically decreased, hence the failure occurs suddenly and causes hazardous damages. This type of failures must absolutely be avoided since it does not allow any overall development of yielding mechanism.

3.2 Previous Research

Through the years, many researchers have had their main focus on increasing punching shear capacity of slab-column connections and there are several experiments and investigation preformed for further study. The first research found in the literature is an investigation of point load subjected to concrete slab by Forsell & Holmberg (1946). Also Kinnunen & Nylander (1960) were very early with the development of an approach to compute the punching shear strength based on mechanical observations. Other pioneer researcher (Corley and Hawkins, 1968) tested shearheads, while other tested bent-up bars end closed stirrups (Islam and Park, 1976), and shear studs were the main concern for some others, (Dilger and Ghali, 1981).

Hallgren (1996) tested ten slabs with different reinforcement ratio and concrete strength in order to compare the punching capacity. Slabs with shear reinforcement consisting of bent-down bars hade higher ultimate load and better ductility as well as post-punching behaviour. Furthermore, the high strength concrete increased the punching strength and obtained more adequate use of flexure reinforcement compared with normal strength concrete.

Suggestion has been made to use headed-studs to increase the punching strength of flat slab (Feretzakis 2006) which designated the improvement of the punching capacity. The American Concrete Institute (ACI) proposes shear stirrups or studs being placed between the main reinforcement in a cross or L-shape in order to deal with the punching shear forces at internal edge and corner, and they were demonstrated to be very effective, (Goodchild, 2000).

CHAPTER 4. INVESTIGATED TEST TRIALS

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4 Investigated Test Trials

4.1 Description of the experimental test

In this Chapter, test trials of slab specimens conducted by various researchers from different institutions and establishment are described. The plates included are those that reasonably could be considered as possible to treat in accordance with the report limitations. There are more trials published, but those containing either conditions that are not compatible with the limitation of current report or that would not add anything to this investigation. In Appendix A, there is more detailed table information for each test trail.

4.1.1 N. Swamy and S. Ali, 1979

Swamy and Ali performed test on slab-column connections with or without steel fibres. The totals of nineteen specimens were one-to-one full-scale models of flat slab structures, whereas five had no fibre content. The column size was 150×150×250 mm and the size of the slabs was 1800×1800×125 mm with an average effective depth d of 100 mm.

In order to study different variables, the nineteen connections were divided into five series. In series one and three the focus was on volume percentage of the steel fibres and the steel fibre type, respectively. In series two, fibre location and flexural reinforcement distribution were studied. Reinforcement reduction and variation of shear reinforcement were the main studied parameter in series four and five, respectively.

All the connections were compacted on a table vibrator and cured in internal environment condition. After 28 days, the loading tests were carried out. Measuring devices placed at carefully selected points measured deflection, concrete strength, steel strain and rotation. By loading the stub column centrally and beyond the maximum load, the descending post-peak curve was obtained.

The researchers published the result in the ACI Journal 1982, elucidating that fibres increased the ultimate punching shear loads, as well as when doubling the fibre volume ratio content from 0.6 percent (48 kg/m3) to 1.2 percent (96 kg/m3) the maximum load improved from 23 percent to 42 percent. To concentrate the fibres 3h (h = thickness of the slab) from the column faces over the full slab depth confirmed to be as effective as providing fibres in the whole slab depth.

Chapter

4.1. DESCRIPTION OF THE EXPERIMENTAL TEST

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They estimated that hooked fibres were more efficient then plain fibres, yet less effective than crimped fibres. Crimped fibres had better deformation resistance and increased punching shear capacity, ductility, and energy absorption capability.

In Table A.1:2 in Appendix A it can be deduced that the reduction of compression steel in slab S-8 had no effect, although the reduction of tension reinforcement, slab S-9 and slab S-19, reduced the ultimate load and changed the failure mode from punching to flexural. However, when reducing the tension steel and adding 0.9 percent (72 kg/m3) steel fibres to the total concrete amount in slab S-10 and S-16 produced higher maximum load capacity. Thus with fibrous concrete it is possible to attain 30 to 40 percent reduction in tension reinforcement providing less design and labour work.

4.1.2 D. Theodorakopoulos and N. Swamy, 1989

Twenty slab-column connection tests of a flat slab structure were presented by Swamy and Theodorakopoulos to investigate the role and effectiveness of steel fibres in counteracting punching mechanism. The slabs hade a dimension of 1800×1800×125 mm (Figure 4.1), connected centrally to a 250 mm high column stubs with varying size c1, this to study the consequence of different loading areas. Other investigated variables of the experiment were fibre volume, type of fibre, reduction in conventional reinforcement and concrete strength. To merely considering the regions of high critical shear, steel fibres were distributed within a square of 1100×1100×125 mm, expect of on slab, which were casted with fibres in the entire slab.

Figure 4.1: Test setup for Theodorakopoulos and Swamy’s (1989).

As in previous research performed by Swamy, the specimens were divided into five series. Series one and four considered fibre volume and steel fibre type respectively. In series two, the tension and compression reinforcement ratio as well as the fibre location were the vital features. Series three had the loaded area as trial. The last series, series five, examined the variation in cube strength.

All the twenty slabs were simply supported along the edges and loaded monotonically trough the centre of the column, with extensive devices recording deflections, concrete strain, steel strain and rotation. They were also equally cured under uncontrolled laboratory condition with a concreter cover of 15 mm.


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From the data presented in their investigation, also available in Tables A.2:1 and A.2:2 in Appendix A, they draw the conclusion that steel fibres can work effectively as shear reinforcement in punching shear failures. Supplying steel fibre volume ratio of 1.0 percent (80 kg/m3) improved the first crack load, the yield load, and the ultimate punching shear strength by about 30 to 45 percent. Fibres can also delay the formation of inclined shear cracks and changing sudden punching failure into flexural failure by affecting the post-cracking behaviour of the slab-column connection. The punching failures in the steel fibre reinforced slabs were gradual and usually incomplete, with no damage of the concrete cover as well as the structural continuity.

Combining sensible reduction of the conventional reinforcement and adding steel fibres to the concrete mix provided flexural failure without loss in load capacity. By enhancing residual strengths and increasing load requirements to displace the ordinary steel reinforcement, fibres had a distinguishing function in preserving the integrity and continuity of the slab-column connection after failure.

4.1.3 S. Alexander and S. Simmonds, 1992

In a technical paper in the ACI Structural Journal, Alexander and Simmonds described the result of tests to failure of six specimens simulating isolated slab-column connection, that to determine how the punching capacity is effects by adding different volumes of steel fibre in concrete matrix and by varying the concrete cover for the reinforcing bars.

All the flat slabs were 2750×2750×155 mm, connected to 200×200×200 mm columns. The three first slabs (P11F0, P11F31 and P11F66) had a concrete cover equal to 11 mm and the three last slabs (P38F0, P38F34 and P38F69) had a concrete cover equal to 38 mm, which is approximate half and twice the usual concrete cover in slabs. To assure a clear concrete cover, all the specimens were cast upside down. Specimens P11F0 and P38F0 had no fibre content. To facilitate the addition of fibres to the mix, a naphthalene sulfonate-based plasticizer was added.

The tests were performed with a hydraulic jack generating load, acting upwards on the centre of the column stub. Along the four edges of each specimen, positive moments were applied by jacking up on structural steel section arms, which were mounted on roller bearings.

The result confirmed that even diminutive fibre content in concrete would increase the performance of the slab-column connection. Thereby, regardless the thickness of the concrete cover, steel fibres significantly improved the load capacity and the ductility. Increases of 20 to 30 percent of the ultimate load were gained by adding steel fibres to the concrete, and about additional 7 to 8 percent increase when doubling the fibre amount. The deformation resistance were also improved when adding steel fibre. With steel fibre volume ratio of 0.4 percent (30 kg/m3), the resistance improved with approximately 40 percent.

4.1.4 A. Shaaban and H. Gesund, 1994

In order to determine whether addition of steel fibres to the concrete mix could increase the punching shear strength of conventional reinforced concrete slabs, Shaaban and Gesund, tested thirteen slab-column connections.


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The slabs had the size of 1600×1600×83 mm and were connected to a 64 mm square column stub, with a steel fibre volume ratio varying up to 2.0 percent (157 kg/m3). Each slab was subjected to uniform loading, provided by an air bag supporting the entire bottom of the slab. All the slabs were designed to fail in punching shear. The test equipment is illustrated in Figure 4.2.

Figure 4.2: Test equipment of Shaaban and Gesund’s test trial (1994).

The authors concluded from the test outcome that steel fibre reinforced concrete does significantly increase the shear capacity of the slab. The result also declared that only a small region of the slab specimen near the slab-column intersection is highly stressed in tension before rupture, namely punching shear failure occurs in a progressive manner.

4.1.5 M. Harajli et al., 1995

Harajli et al. considered the effect of punching shear capacity by adding fibres on twelve small-scaled slab-column connections, by altering fibre type, content and aspect ratio, as well as the span-depth ratio of the slab. The slabs A6 and B6 are not included in current study, since polypropylene fibres were used in these specimens.

Considering slab thickness variation, the slabs were separated into Series A and Series B, with the thickness of 55 mm and 75 mm in respective series, as can be read in Table A.5:1 in Appendix A. The slabs lengths were 650×650 mm square. To determine the influence of fibres on the ductility of shear failure, loading was continued beyond the maximum peak load. The load was applied centrally via a 100×100 mm column stub, and measurements were recorded automatically without interrupting the load application.

From the analyse of the result, the researchers stated that adding steel fibres up to 2.0 percent (160 kg/m3) of the total volume ratio increased the ultimate punching capacity by about 36 percent compared with plain concrete slabs. The verification of the thickness had no noticeable effect on the ultimate punching capacity. Another important observation made was that steel fibres pushed the failure surface area away from the column face, giving that the steel fibres decreased the angel of shear failure plane. Further, the experimental trial established that the enhanced punching shear load is independent of the length or aspect ratio of steel fibres, but merely the volume percentage of the steel fibres affects the punching load.


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4.1.6 B. Hughes and Y. Xiao, 1995

Another investigation preformed on small-scaled slab-column connections was made by Hughes and Xiao. The variables were the slab depths, the ratio of conventional reinforcement and the inclusion of fibres or stirrups. Regarding the restrictions of current study, the total twenty-two slab specimens were weeded to twelve.

The specimens were 860×860 mm with different thickness of 65 mm and 50 mm. There were three slab specimens without any fibre used as control slabs, more details about these slabs can be found in Tables A.6:1-A.6:5 in Appendix A. The load was applied centrally via a hydraulic jack through the column, while the specimens were simply supported along the perimeter. The test setup is shown in Figure 4.3.

Figure 4.3: Test equipment of Hughes and Xiao’s test trial (1995).

The result indicated that crack resistance was enhanced when steel fibres were added, hence the first crack was delayed, the crack width reduced and the crack propagation was formed and fine. Slabs with duoform steel fibre volume ratio of 1.0 percent increased the punching shear strength with 40 percent. Moreover, addition changed the failure mode to gradual and ductile, and a considerable post-ultimate strength after peak load was reached.

4.1.7 P. McHarg, 1997

P. McHarg reported the result of six two-way slab-column connections in a Ph.D. thesis. The aim of the study was to examine the strategic use of steel fibre reinforced slab-column connection to present the most efficient way to improve the performance. The studied variables were the slab stiffness, the punching shear capacity and the negative moment cracking.

The six slab-column test specimens with the dimension of 2300×2300×150 mm were constructed with 225×225×300 mm reinforced concrete stub columns above and below the slab and were designed such that they would fail in punching shear. The slabs, as can be seen in Figure 4.4, were loaded monotonically with eight equal point loads around the edge. The loads were supplied through jacks connected to hydraulic pumps.

Of the total six slabs, two slabs had no steel fibre content. The specimens were divided into three series with two slabs in each series. The first series, denoted Series N, consisted of Slab NU and Slab NB, had no fibre reinforcement in the concrete and were casted with normal


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strength concrete. Slab NU was reinforced at the top with uniformly distributed reinforcing steel bars according to the U.S. practice. NB was reinforced the same way as NU but with a greater reinforcement in the intermediate column section.

Figure 4.4: Test setup performed by McHarg (1997).

The second and the third series, Series FS and Series FC, hade both steel fibre reinforcement included in the concrete mix. Series FS contained steel fibres only in the column vicinity and through the entire slab depth (h = 150), whereas Series FC had steel fibres placed in the top concrete cover. Specimens FSU and FCU contained uniformly distribution top reinforcement bars, furthermore specimens FSB and FCB had the same top reinforcement but with banded a distribution.

The result of the experimental program clarified the enhancements that can be achieved by adding steel fibres to concrete. By providing steel fibres in Series FS, the cracking load was increased up to 21 percent, comparing with the plain concrete in Series N. Test slabs with steel fibres showed the tendency of the higher punching shear capacity and better crack control. Moreover, slabs with steel fibres and uniform distribution of top reinforcing bars had crack widths that were 20 percent and 25 percent smaller in the vicinity of the column circumference.

4.1.8 G. Hassanzadeh and H. Sundquist, 1998

Hassanzadeh and Sundquist aspired to investigate any notable improvement in the failure behaviour of steel fibre reinforced slab-column connections. Therefore, five test specimens with and without conventional reinforcement were cast and tested to study the concrete load-carrying capacity, the local usage of steel fibres and also the usage of post-tensioned unbounded tendons placed in the slabs without conventional reinforcement.

The slabs had a dimension of 2600×2600×220 mm, and associated centrically to circular column with a diameter of 250 mm. To consider zero-moment distribution around the column, the slabs were loaded at a circle with a diameter of 2380 mm as shown in Figure 4.5. The specimens were separated into series C and series E with two and three specimens in respective series. The steel fibre volume ratio was 1.5 percent (120 kg/m3) equally in all slabs.

Series C was cast with normal strength concrete without ordinary flexural reinforcement in order to observe the influence of steel fibres on punching shear capacity and the deformation capacity of the slabs compared with similar concrete slabs reinforced with ordinary flexural


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reinforcement. As mentioned above, none of the slabs had any flexural reinforcement, although Slab C2 had prestressed tendons in both directions placed along the column band.

Figure 4.5: Test setup and the loaded slab area of Hassanzadeh and Sundquist test trial

(1998).

Series E consisted of three slabs E1, E2 and E3. Steel fibre reinforced high strength concrete was used in the three slabs, but with the difference of E2 which was cast with steel fibre reinforced high strength concrete only within the immediate column vicinity and normal strength concrete for the remaining slab area. Slab E3 had tendons arranged similar to slab C2, whereas the slabs E1 and E3 had ordinary flexural reinforcement.

Three other test slabs, HSC2, N/HSC8 and B1, tested earlier without any steel fibre reinforcement, were also included in the investigation in order to compare the test result. Hallgren (1996) carried out test slabs HSC2 and N/HSC8, and Hassanzadeh (1998) tested Slab B1. Details of these slabs are presented in Appendix A among the details of the other slabs.

The load was applied through the column stub via a hydraulic jack. While increasing the load, slab deflection as well as the maximum crack widths were measured. Also the crack formations were continuously marked.

The outcome of the test indicated that steel fibres can not independently work as structural reinforcement due the inadequacy of the post-cracking tensile strength. This is evident when comparing Slab B1 and Slab C1. Steel fibre reinforced concrete slabs in the investigation, compared with slabs without steel fibres, had an increased first crack load and ultimate load capacity. Additionally, the steel fibre reinforced slabs decreased the crack width and deflection at ultimate loading. Using steel fibre high strength concrete, Slab E1, improved the punching shear capacity with 176 percent, compared with Slab B1. Utilizing steel fibre high strength concrete just around the column area had no notable impact on the stiffness of the slab, however enhanced the maximum deflection and the punching shear capacity with 140 percent comparing to B1. Although, it should not be neglected that the punching shear capacity was reduced with 13 percent for Slab E2 in comparison with E1. This indicates that slabs cast like E2 is an economical solution to increasing the punching shear capacity of slab-column connection locally.


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4.1.9 A. Azevedo, 1999

In a doctoral thesis, Azevedo studied the punching shear behaviour and ductility of twelve slab-column connections. The flat slabs were one-to-one full-scale prototypes with a dimension of 1160×1160×100 mm and a column stub of 80 mm square and 37 mm high. To evaluate the residual strength of the slabs, the researcher also prepared six slab-segments with the size of 1160×330×100 mm to represent a strip of the slab-column connections.

The twelve slabs were divided into two series. With six slabs in each series, Series OSC contained normal strength concrete and Series HSC contained high strength concrete. Every slab had tension reinforcement both on top and on bottom side, and they were designed to fail in punching. All the twelve slabs were simply supported along the edges and loaded centrally through the column via a hydraulic jack. A number of electronic gauges were placed in the vicinity of the column, permitting an evaluation of the post-peak strength behaviour.

The result verified that the application of high strength concrete in addition of steel fibres improved the punching shear capacity and strengthened the slab column connection.

4.1.10 S. Ozden et al., 2006

An experimental program, consisted of circular steel fibre reinforced concrete flat slabs tested under monotonically increasing punching load, was carried out by Ozden et al. to revise the effect of flexural reinforcement and the use of steel fibre reinforcement. The varied parameters were the concrete strength and the eccentricity of loading.

The slabs were circular and had a diameter of 1500 mm and thickness of 120 mm with a concentric 200 mm square column stub on both sides of the slab. Of the total twenty-six slabs, only twelve were steel fibre reinforced, and hence the only ones included in current study. The slabs starting with the letter N refers to normal strength concrete and the slabs starting with the letter H refers to high strength concrete. Furthermore, the letters R and E refer to the reinforcement ratio and the load eccentricity, respectively.

Figure 4.6: Test equipment of Ozden et al (2006).


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To connect the specimens to a concrete reaction block, the slabs were supported along their edges by twelve evenly spaced tie rods. A steel loading arm was used for eccentric loading as shown in Figure 4.6. The displacements were measured at seven different locations along the diameter of the flat slabs. Besides, displacement and load were recorded constantly with no stop during the entire test trial.

The main conclusions drawn from the test result were that concrete strength plays an important role in punching capacity and slab rigidity. The initial and post-cracking stiffness of high strength concrete was higher than those of normal strength concrete. Moreover, addition of steel fibres enhanced the post-peak deformation and residual strength, and the cracks formation were more closely spaced and concentrated around the column stub. Additionally, the increase in flexural reinforcement raised the punching and residual strength.

4.1.11 J. Hanai and K. Holanda, 2008

In an attempt to compare the similarity between steel fibre reinforced concrete slabs and beams, Hanai and Holanda analysed eight slabs and 15 beams in an experimental study. The slabs were 1160 mm square and 100 mm thick. The details of the reinforcement design variation for the slabs are described in Appendix A.

All the slabs were supported along the edge with the load applied upside down through a square steel plate. The slabs were designed to fail by punching failure. Some steel reinforcement bars hade electronic devices to record the strains and forces during the tests and to better evaluate the structural behaviour of the slabs.

The result of the test clarified that the ultimate load performance approached values closer to the estimated load for flexural reinforcement, as more fibres were added the concrete. The result also confirmed that slabs in Series 2, cast with high strength concrete and 2.0 percent (160 kg/m3) steel fibre of the volume ratio, covered the best strength and ductility outcome. Furthermore, it is worth to notify that, the researches main conclusion was that shear tests on prismatic beams provided useful information to choose the type and amount of steel fibres to be used in flat-slabs.

4.1.12 L. Nguyen-Minh et al., 2011

In a technical paper distributed at the conference of the Twelfth East Asia-Pacific Conference on Structural Engineering and Construction, Nguyen et al. describe an experimental trial made on twelve slab-column connections, of which nine were fibre reinforced and three were used as comparison slabs. The investigated variables were the steel fibre amount, the slab size, the punching shear crack behaviour and the resistance of the slab.

The slabs were divided into three groups depending on the size of the slab. The slab size varied and was 900 mm, 1200 mm and 1500 mm. The thickness and effective depth were identical for all specimens and equal to 125 mm and 105 mm, respectively. The load was applied via hydraulic jack centrally through the column and the slabs were simply supported. At each load level, the crack development, concrete and reinforcement strain, and deflection were recorded.

4.2. RANGE OF PROPERTIES

26

Based on the result obtained by the test, the researcher concluded that increasing the steel fibre volume ratio from 0.4 percent (30 kg/m3) to 0.8 percent (60 kg/m3) on slabs with the size 900×900 mm increased the punching shear capacity from 16 percent to approximately 40 percent, and from 9 percent to 24 percent for the other slab sizes. Likewise, steel fibres markedly reduced the average crack width to about 71 percent.

4.2 Range of Properties

The range of the properties varied somehow, depending on the performance of the test as described in section 4.1. In Table 4.1, the range of properties of the composed test specimens is summarized.

Table 4.1: Property ranges of the test samples. Properties Max – Min

L [mm] 650 – 2750

H [mm] 50 – 242

Lf [mm] 25 – 60

df [mm] 0.25 – 1.05

Lf / df [-] 26 – 100

Vf [%] 0.4 – 2.0

fcc,cylinder [MPa] 14 – 100

fcsp,cylinder [MPa] 1.6 – 7.7

fcfl [MPa] 3.0 – 9.7

fct [MPa] 1.7 – 6.5

ρflex [%] 0.18 – 2.0

Ff [-] 0.12 – 1.3

CHAPTER 5. DESIGN PUNCHING MODELS

27

5 Design Punching Models

5.1 General

There are several punching shear strength formulas in various building codes for slab–column connections, nonetheless applicable only for normal concrete with ordinary reinforcement. However, distinguished researchers such as Shaaban and Gesund (1994) and Harajli et al. (1995) suggested a number of pure empirically derived formulas that are uncomplicated to use. The disadvantages of these prescriptions are the experimental uncertainty assessment, as well as some impact may be neglected when performing an experiment analysis. To determine the punching shear mechanism due to only test outcome can give inaccurate result comparing to theoretical analysis.

Contrary, there is one semi-analytical formula derived by Choi et al. (2007). This formula was established by studying the material failure criterion of fibre reinforced concrete. In view of the fact that yielding of tensile reinforcement arises prior to punching shear failure, the formula is convincing only in case of large span to thickness ratio, where the behaviour of the slab is dominated by flexural deformation.

5.2 Basic Condition

Next, some models for punching shear strength of steel fibre reinforced concrete are presented and investigated in order to analyse how adequate they relate to the fibre reinforcement concrete. These equations were also the main implements to compare different test series with each other. First, some basic conditions used for the estimation are defined.

5.2.1 Safety Factors

All partial safety factors such as γc and γf are regarded to be characteristic since it is not necessary to take into account safety margins that consider probabilistic values. Thus, all safety factors are set to be equal to 1.0.

Chapter

5.2. BASIC CONDITION

28

5.2.2 Concrete Characteristic Strength

In all the articles, the concrete compressive strength of the specimens is stated, but not all authors declare the flexural strength and/or the splitting tensile strength. In order to obtain the missing characteristic values some estimation had to be done.

The concrete compressive strength fcc were in some articles tested as cube strength, for these cases the strength given were recalculated as cylindrical:

fcc = 0.8 ∙ fcc,cube [MPa] (5.1)

where,

fcc = cylindrical compressive strength of concrete fcc,cube = cube compressive strength of concrete

The same recalculations were made for the splitting strength fcsp:

fcsp = 0.8 ∙ fcsp,cube [MPa] (5.2)

where,

fcsp = cylindrical splitting strength of concrete fcsp,cube = cube splitting strength of concrete

The priority of the specified tensile strength was in first hand the flexural strength and in second hand the splitting strength. For the specimens with missing values of both flexural strength and splitting strength, the tensile strength fct was used. The tensile strength was attained from the compressive strength through following estimation:

fct = 0.3 ∙ fcc 2/3 [MPa] (5.3)

where,

fct = concrete tensile strength fcc = concrete cylindrical compressive strength

5.2.3 Conventional Reinforcement

Slab specimens with conventional reinforcement are considered and only the reinforcement amounts in the tension zone within the control area equal to 3d in the region of the column, in both y- and z-direction, were taken into account. Specimens with shear reinforcement and stirrups are not included. The reinforcement ratio amount was computed as:

s

w

Ab d

ρ =⋅

(5.4)

where,

As = reinforcement bar area [mm]


29

d = effective depth calculated as mean values of the y- and z-direction [mm], for articles with missing d value it was assumed to be 0.9h (h = slab thickness)

bw = slab width equal to the column width plus 3d on each side [mm]

5.3 Punching Shear Capacity with Addition of Steel Fibres

To calculate the total punching shear capacity VRd,cf in a steel fibre reinforced concrete slab, it is suggested to combine the punching shear capacity of ordinary reinforced concrete slab VRd,c with the shear capacity VRd,f achieved from the steel fibres:

VRd,cf = VRd,c + VRd,f [N] (5.5)

In the following, two different calculation alternatives have been analysed to investigate how adaptable they are to evaluate the capacity of steel fibre reinforced concrete.

5.4 Alternative I: DAfStB

In the German Committee for Reinforced Concrete (Deutschen Ausschuss für Stahlbeton, DAfStB) in section XII Erläuterungen zur DAfStB – Richtlinie Stahlfaserbeton, a punching shear resistance of conventional reinforced slabs without shear reinforcement and a theoretically based approach for steel fibre reinforced concrete slabs are proposed in different sections. This two equation were added together to estimate the punching shear capacity of SFRC slabs.

Initially, the shear resistance for conventional reinforced slabs without shear reinforcement was approximated as in section XI/4.3.4.5.1 in DAfStB (2011), with the multiplication of the punching area to obtain the capacity:

VRd,c = [CRd,c ∙ k (100 ∙ ρ1 ∙ fck)1/3 – 0,12 ∙ σcp] ∙ Ap [N] (5.6)

where,

fck = the characteristic compressive strength. Note, the computed fck is taken as mean value of the plain concrete without fibres.

CRd,c = c0.21 γ

k = 1 200 / 2.0d+ ≤ ρ1= y z 0.02ρ ρ⋅ ≤ ρy and ρz relate to the tension steel in y- and z- directions respectively and are

calculated as equation (5.1.3a). σcp = the mean value of the normal concrete stresses in the critical section in y- and z-directions and is estimated as the quotient of the longitudinal forces divided by the concrete area. The force may be from a load or prestressing action but since there is no such case in the referred test, σcp is set to be equal to zero. and

5.4. ALTERNATIVE I: DAFSTB

30

Ap = u ∙ d [mm2] (5.7)

where,

Ap = the punching area [mm2] assumed to be circular regardless the real shape of the column, since the column for all test was adjusted as circularly shaped.

u = the control perimeter at which the punching acts, and is according to the guidelines set equal to 1,5d (d = the effective depth) measured from the column.

The shear capacity of the steel fibre reinforced concrete, approximated as in section XII/9.1.2 in DAfStB (2011), was determined as the shear resistance and then calculated as shear capacity the same way as in Equation 5.7:

VRd,f = vRd,f ∙ Ap [N] (5.8)

and f

f ctR,uRd,f

f

fv

αγ⋅

= [MPa] (5.9)

where,

γf = partial safety factor of the steel fibre concrete tensile strength αf = coefficient considering long term and unfavourable effects of SFRC, here αf = 1.0 f fctR,u = design value of the post-cracking tensile strength at the ultimate limit state,

derived from the flexural strength test according to section 9.1.2.

To apply the DAfStB equation on the selected slab specimens, an experimental residual tensile strength need to be determined on small test beams:

f fctR,u = κfF ∙ κf

G ∙ f fct0,u [MPa] (5.10)

where,

κfF = factor considering fibre orientation, κf

F = 1.0 for slabs κf

G = factor considering the influence of the component size with the variation of 1.0 + (Ac ∙ 0.9) ∙ 0.5 ≤ 1.7

Figure 5.1: The load-deflection graph of four point bending test (DAfStB 2001).

F [N]

δ [mm] 3.5 0.5


31

However, to determine f fct0,u, a four point bending test according to the guidelines needs to be performed on not less than six prisms with the dimension of 700×150×150 mm, see Figure 5.1. Load deflection at 0.5 mm and 3.5 mm, corresponding to F0.5 and F3.5 respectively, must be recorded for the service limit state (L1) and the ultimate limit state (L2).

Subsequently, by following the given steps of calculation, it’s possible to read f fct0,u from

Table 2.

0.5,ifcflm,L1 2

=1 i i

1 n

i

F lf

n b h⋅

=⋅∑ [MPa] (5.11)

3.5,ifcflm,L2 2

=1 i i

1 n

i

F lf

n b h⋅

=⋅∑ [MPa] (5.12)

( )f fcflm,Li cfl,Li

1 lnLf fn

= ∑ [MPa] (5.13)

f

cflm,Li s s( )f fcflk,Li cflm,Li0.51Lf k Lf e f− ⋅= ≤ ⋅ [MPa] (5.14)

where,

ks = is a constant depending on numbers of test specimens

Ls = ( )( )2f f

cflm,Li cfl,Liln

1

Lf f

n

−

−∑

Table 5.1: Performance category for SFRC in bending (DAfStB 2011). Performance Category of Centric Residual Strength [MPa] Distortion 1 Distortion 2

L1 f fct0,L1 L2 f f

ct0,L2 f fct0,u f f

ct0,s 0 < 0.16 0 - - -

0.4 0.16 0.4 0.10 0.15 0.15 0.6 0.24 0.6 0.15 0.22 0.22 0.9 0.36 0.9 0.23 0.33 0.33 1.2 0.48 1.2 0.30 0.44 0.44 1.5 0.60 1.5 0.38 0.56 0.56 1.8 0.72 1.8 0.45 0.67 0.67 2.1 0.84 2.1 0.53 0.78 0.78 2.4 0.96 2.4 0.60 0.89 0.89 2.7 1.08 2.7 0.68 1.00 1.00 3.0 1.20 3.0 0.75 1.11 1.11

The problem that occurred when applying the DAfStB equation on the referred slab specimens was that only a one of the researchers preformed the residual tensile strength, however not similar to the test corresponding in BetonKalender. To solve this obstacle, a residual strength required to be derived theoretically to replace the experimental f f

ct0,u for each slab specimen. This was done in two different manners.

5.4. ALTERNATIVE I: DAFSTB

32

Solution I The first solution was to obtain a residual strength through a residual strength factor attained by the crack strength. Initially, a residual strength factor R, defined as the ratio between the average load carrying capacity between certain displacement after cracking and the load at first crack for standard test beam loaded in four point bending, needs to be determined:

( )( )

X 1010,X

100100

I IR

X⋅ −

=−

[%] (5.15)

where,

IX = ratio between area under the stress-deflection curve between δ = 0 and δ = (X + 1)δcr/2 and the (elastic) area between δ = 0 and δ = δcr. (see Figure 5.2)

I10 = toughness indices according to ASTM (1992). For a complete elastoplastic material, I10 = 10 and IX = X. The parameter X expresses the ductility demand.

δcr = first crack deflection

Figure 5.2: The stress-deflection graph of flexural bending test beam (Silfwerbrand 2000).

However, it has experimentally been verified by Silfwerbrand (2000) that, by knowing the quantity of fibres, a residual strength factor can be estimated:

R = 20 + ρs (kg/m3) [%] (5.16)

if, 15 kg/m3 < ρs < 80 kg /m3

where,

ρs = steel fibre content

Yet, to have a valid R-estimation, the steel fibres should be hooked or similar, the concrete should be of normal strength range and R = 0 for ρs = 0. Thereafter, crack strength according to Paramasivam et al. (1994) was approximated by the following formula:

fl,cr cc0.33f f= ⋅ [MPa] (5.17)

Max flexural stress

Midspan deflection δ

fflcr

δcr 5.5δcr 10.5δcr 15.5δc

fflu

fflre

Max flexural stress σ [MPa]

Midspan deflection δ [mm]

(X+1)∙δcr/2

fflu

fflre

fflcr

δcr


33

In consequence, by means of the residual strength factor and the crack strength identified, a residual tensile strength ffl,res1 at 0.5 mm deflection, with the abbreviation 1 at the end of the index, was estimated:

fl,res1 fl,cr100Rf f= ⋅ [MPa] (5.18)

where,

f fl,res1 = residual strength at 0.5 mm deflection f fl,cr = crack strength R = residual strength factor

Nevertheless, since the DAfStB direction likewise demands residual tensile strength at 3.5 mm, a correlative residual tensile strength ffl,res6 with the abbreviation 6 was assumed to be one half of the estimated ffl,res1:

f fl,res6 = 0.5 ∙ f fl,res1 [MPa] (5.19)

where,

f fl,res6 = residual strength at 3.5 mm deflection f fl,res1 = residual strength at 0.5 mm deflection

Thus, an average residual strength ffl,res,m was approximated and was used as replacement for the missing experimental post-crack tensile strength f fct0,u in equation (5.3e):

fl,res1 fl,res6fl,res,m 2

f ff

+= [MPa] (5.20)

f fct0,u = ffl,res,m [MPa] (5.21)

Solution II Another method used to substitute the f f

ct0,u was the utilization of the post-crack tensile strength proposed by Choi et al. (2007):

fpc = λ1 ∙ λ2 ∙ λ3 ∙ Vf (Lf/df) ∙ τ ∙ β [MPa] (5.22)

where,

λ1 = expected pull-out length ratio, λ2 = efficiency factor of orientation in the crack state, λ3 = group reduction factor associated with the number of fibres pulling-out per unit β = factor considering the effect of fibre shape and concrete type β = 1.0 for hooked or crimped steel fibres β = 2/3 for plain or round steel fibres with normal concrete β = 3/4 for hooked or crimped steel fibres with lightweight concrete Lf/df = fibre slenderness aspect ratio Vf = fibre volume ratio τ = average interfacial bond strength of fibre matrix estimated as τ = 2 ∙ (0.292 ∙ ft)

5.5. ALTERNATIVE II: SWEDISH CONCRETE ASSOCIATION

34

and

ft = 0,292 f'c0.5

where,

ft = tensile strength of concrete under pure tension f’c = is the concrete compression strength of an equivalent normal concrete without

fibres

Choi et al. derived the equation through a probabilistic investigation of preceding analysis prepared by previous researchers. The coefficient in the formula were reported to be relatively numerically small, and thus estimated as λ1 = 0.25, λ2 = 1.2 and λ3 = 1.0. Thereby, fpc was used as alternative for the residual tensile strength f fct0,u stated in equation (5.3e).

f fct0,u = fpc [MPa] (5.23)

5.5 Alternative II: Swedish Concrete Association

In contrast with DAfStB, The Swedish Concrete Associations (SCA), Concrete Report No. 4, propose to add the expression that takes into account the steel fibres contribution to the shear resistance of conventional reinforced slab:

VRd,cf = [vRd,c + vRd,f] ∙ Ap [MPa] (5.24)

where,

Ap = the punching area [mm2] assumed to be circular and the control perimeter equal to 0.5d from the adjusted circular column.

and

vRd,c = ξ ∙ (1 + 50 ∙ ρ) ∙ 0.30 ∙ fct [MPa] (5.25)

where,

ξ = size factor ξ = 1.4 for d ≤ 0.2 m ξ = 1.6 – d for 0.2 < d ≤ 0.5 m ρ = y z 0.02ρ ρ⋅ ≤ ρy and ρz relate to the tension steel in y- and z- directions respectively and are

calculated as equation (5.1.3a). fct = concrete tensile strength of plain concrete without fibres, computed as equation

(5.1.2d). and

vRd,f = 0.41 ∙ τf ∙ Ff [MPa] (5.26)


35

where,

τf = bond strength factor of fibre matrix, estimated as τ = 4.15 since experimental data is missing

Ff = fibre related factor estimated as Ff = ρf ∙ μf ∙ λf ρf = Vf = fibre volume ratio μf = bond factor μf = 0.5 for plain fibres μf = 0.75 for corrugated fibres μf = 0.9-1.2 for intended or hooked fibres λf = Lf/df, fibre slenderness aspect ratio. For non-circular fibre cross-section df = (4 ∙ Af / π)1/2 and Af = fibre cross-section.

This term was originally developed by Narayanan & Darwich (1987) which in turn refers to Romuladi & Mandel (1964). The factor 0.41 provenances from the orientation of the fibres.

5.6 Steel Fibre Reinforced Concrete Slabs without Conventional Reinforcement

Primarily, the punching shear capacity model of slabs with only steel fibres as reinforcement, purposed by Silfwerbrand (2000), was to be investigated too:

vRd,cf = vRd,f

and

vRd,f = cfl2k C f⋅ ⋅

where, k = 1 200 / 2.0d+ ≤ C = coefficient = 0.45 fcfl = design value of the flexural strength of the steel fibre reinforced concrete

In the original Silfwerbrand paper, another size effect factor than k above was used, but this is a natural step to approach EC2 and its size effect.

However, since only one slab performed by Hassanzadeh and Sundquist had the requested features, the punching model of only steel fibre reinforced slabs was excluded in the analysis.

CHAPTER 6. ANALYSIS OF THE RESULT

37

6 Analysis of the Result

6.1 Final Applied Formulas

After the calculation analysis and that, all expressions combined, the following final formulas were applied on the test data:

• DAfStB VRd,cf = [CRd,c ∙ k (100 ∙ ρ1 ∙ fck)1/3 + αf ∙ (κf

F ∙ κfG ∙ f fct0,u) ] ∙ (u ∙ d) [N] (6.1)

Solution I f fctR,u = ff,R

Solution II f fctR,u = fpc

• SCA Report No. 4 VRd,cf = [ξ ∙ (1 + 50 ∙ ρ) ∙ 0.30 ∙ fct + 0.41 ∙ τf ∙ Ff] ∙ (u ∙ d) [N] (6.2)

The equation models were applied on the test data and analysed in Office Excel. The experimental capacity Vexp was divided by the calculated capacity Vcal to achieve the ratio to relate different variables with. The comparing parameters are:

- slab thickness h - fibre volume ratio Vf - fibre slenderness ratio L/d - fibre factor Ff - conventional reinforcement ratio ρ - concrete compressive strength fcc - concrete tensile strength fct - and the residual strength, compared only for DAfStB alternative, f fct0,u

Chapter

6.2. EFFECT OF THE SLAB THICKNESS

38

6.2 Effect of the Slab Thickness

It is obvious from the graphs of DAfStB alternatives that the ratio of Vexp/Vcal decreases when the slab thickness h increases. There is no great difference between the two solutions, besides that the variation of distribution is slight larger for solution I with about 50 percent of the slab specimens falling under the critical line Vexp/Vcal = 1.0 and the highest value of Vexp/Vcal = 3.25 for a non-fibres reinforced slab, as shown in Figure 6.1.

0,0

1,0

2,0

3,0

4,0

0 50 100 150 200 250

Vex

p/ V

cal

h

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0 50 100 150 200 250

Vex

p/ V

cal

h

DAfStB - solution II

Figure 6.1: Graph of DAfStB model alternative with two different solutions varying with

respect to slab thickness h [mm].

The alternative of SCA has the largest range of values, yet confirming the increase of slab thickness decreases the ratio of the experimental capacity over the calculated capacity. There are four tests that range over Vexp/Vcal = 3.0. Thus, the graphs indicate that DAfStB solution II has the most compact results while the SCA present the most conservative result between the alternatives, according to Figure 6.2.

0,0

1,0

2,0

3,0

4,0

5,0

0 50 100 150 200 250

Vex

p/ V

cal

h

SCA Report No.4

Figure 6.2: Graph of SCA model alternative with respect to slab thickness h [mm].


39

6.3 Effect of the Fibre Volume Ratio

There is no significant differences between the first and the second solution of DAfStB alternative as shown in Figure 6.3, when increasing the fibre volume ratio Vf rather than that there are more tests that fall under Vexp/Vcal = 1.0 for solution I. However, the equation of solution I does not admit fibre volume ratio larger than 1 volume percent (80 kg/m3) and therefore a direct comparison between these two solutions cannot be made. Moreover, solution II has more distributed result and a decreasing trend with increasing fibres amount, as shown in Figure 6.3.

0,0

1,0

2,0

3,0

4,0

0,0% 0,5% 1,0% 1,5% 2,0%

Vex

p/ V

cal

Vf

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0,0% 0,5% 1,0% 1,5% 2,0%

Vex

p / V

cal

Vf



respect to fibre volume ratio Vf [%].

In Figure 6.4, the distribution of the Vexp/Vcal ratio is largeer and the trend for SCA alterative (corresponding to DAfStB alternative in Figure 6.3) is decreasing when increasing the fibre amount. Though, the advantage is that the majority of the tests are on the conservative side of the line.

0,0

1,0

2,0

3,0

4,0

5,0

0,0% 0,5% 1,0% 1,5% 2,0%

Vex

p/ V

cal

Vf

SCA Report No.4

Figure 6.4: Graph of SCA model alternative with respect to fibre volume ratio Vf [%].

6.4. EFFECT OF THE FIBRE SLENDERNESS RATIO

40

6.4 Effect of the Fibre Slenderness Ratio

For the DAfStB alternative, as can be seen from Figure 6.5, the slenderness ratio Lf/df seems not to be preferable the higher it gets.

0,0

1,0

2,0

3,0

4,0

0 20 40 60 80 100

Vex

p/ V

cal

Lf/df

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0 20 40 60 80 100V

exp

/ Vca

lLf/df



respect to fibre slenderness ratio Lf/df.

Also the SCA alternative has a decreasing trend with a rising slenderness ratio but is more conservative than the first solution of DAfStB alternative in Figure 6.5. A rough approximation of the Figure 6.6 is that the SCA alternative is not very dependent of the Lf/df ratio as DAfStB solution I is. Nevertheless, it is worth mentioning that the high aspect ratio for utmost of the refereed test trials are generally attained from the long fibres.

0,0

1,0

2,0

3,0

4,0

5,0

0 20 40 60 80 100

Vex

p/ V

cal

Lf/df

SCA Report No.4

Figure 6.6: Graph of SCA model alternative with respect to fibre slenderness ratio Lf/df.

Observe that both the SCA alternative and DAfStB alternative solution II includes the factor Lf/df, accordingly if the estimation of Lf/df were right the trend line should be horizontal.


41

6.5 Effect of the Fibre Factor

Once observing the influence of the fibre volume and the fibre slenderness ratio it is worth reviewing the fibre factor Ff that includes the two previous variables in addition to the fibre bond factor. The fibre factor, as estimated in Chapter 5.4 equation (5.4c), regards the fibre volume, the fibre bond factor and the slenderness ratio. Thus, the effect of Ff contributes to a direct depiction of the fibre behaviour.

There is a wide distribution of the result in both solutions for DAfStB alternative as illustrated in Figure 6.7. Solution II is estimated with fpc (see Equation 5.22) that contains the same variables as in fibre factor Ff and has a little better evaluation of the fibre factor than solution I that has the majority of the tests falling under the ratio Vexp/Vcal = 1.

0,0

1,0

2,0

3,0

4,0

0,0 0,5 1,0 1,5

Vex

p/ V

cal

Ff

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0,0 0,5 1,0 1,5

Vex

p/ V

cal

Ff



respect to fibre factor Ff.

The descending trend obtained by the effect of the fibre factor for the SCA alternative in Figure 6.8 is almost identical as the trends in Figure 6.7.

0,0

1,0

2,0

3,0

4,0

5,0

0,0 0,5 1,0 1,5

Vex

p/ V

cal

Ff

SCA Report No.4

Figure 6.8: Graph of SCA model alternative with respect to fibre factor Ff.

6.6. EFFECT OF THE CONVENTIONAL REINFORCEMENT

42

6.6 Effect of the Conventional Reinforcement

In Figure 6.9, it is evident that the ratio amount of the conventional reinforcement i.e. the flexural reinforcement does not have any major influence on the punching capacity further than that solution II has a slighter increasing trend than solution I.

0,0

1,0

2,0

3,0

4,0

0,000 0,005 0,010 0,015 0,020

Vex

p/ V

cal

ρ

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0,000 0,005 0,010 0,015 0,020V

exp

/ Vca

lρ



respect to reinforcement ratio amount ρ.

The effect of the reinforcement ratio amount of the conventional reinforcement for SCA calculation model is almost a straight and horizontal line as depicted in Figure 6.10. This indicates that the SCA alterative is not mainly affected by the fibre ratio amount.

0,0

1,0

2,0

3,0

4,0

5,0

0,000 0,005 0,010 0,015 0,020

Vex

p/ V

cal

ρ

SCA Report No.4

Figure 6.10: Graph of SCA model alternative with respect to reinforcement ratio amount ρ.


43

6.7 Effect of the Concrete Compressive Strength

Four test trials (a total of 17 slabs) had a compressive strength higher than 57.0 MPa, which is the upper limit for the compressive strength of normal strength concrete, according to the Swedish Handbook on Concrete Structures (BBK 04). These 17 slabs were included in the scatter of the compressive strength among the other normal strength slabs. However, the most common compressive strength is between 20 and 40 MPa.

Both solutions of the DAfStB alternative in Figure 6.11 confirms a compact distribution of the compressive strength and with a wide distribution. More slabs seem to fall under the critical line Vexp/Vcal = 1 for solution I than for solution II. The trend decreases the higher the compressive strength gets.

0,0

1,0

2,0

3,0

4,0

0 20 40 60 80 100

Vex

p/ V

cal

fcc

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0 20 40 60 80 100

Vex

p/ V

cal

fcc



respect to concrete compressive strength fcc [MPa].

As presented in Figure 6.12 the trend decreases the higher the compressive strength gets just like in the case of DAfStB alternative, thus it is apparent that the analysed equation does not cover the effect of the high strength concrete influence of the punching capacity.

0,0

1,0

2,0

3,0

4,0

5,0

0 20 40 60 80 100

V exp

/ Vca

l

fcc

SCA Report No.4

Figure 6.12: Graph of SCA model alternative with respect to concrete compressive strength

fcc [MPa].

6.8. EFFECT OF THE CONCRETE TENSILE STRENGTH

44

6.8 Effect of the Concrete Tensile Strength

The concrete tensile strength fct was attained as a recalculation from the concrete cylindrical compressive strength as described in Equation (5.3). And the concrete cylindrical compressive strength fct is in turn sometimes recalculated from the concrete cube compressive strength. The means that error is incorporated into the outcome. Hence, it is not strange that the results in graphs in Figure 6.13-6.14 are similar to the result in graphs in Figures 6.11-6.12.

0,0

1,0

2,0

3,0

4,0

0,0 1,0 2,0 3,0 4,0 5,0 6,0

Vex

p/ V

cal

fct

DAfStB - solution I

0,0

1,0

2,0

3,0

4,0

0,0 1,0 2,0 3,0 4,0 5,0 6,0

Vex

p/ V

cal

fct



respect to concrete tensile strength fct [MPa].

The ranges of the result are similar in both Figure 6.13 and Figure 6.14 and there is no direct conclusion to claim about the effect of the concrete tensile strength.

0,0

1,0

2,0

3,0

4,0

5,0

0,0 1,0 2,0 3,0 4,0 5,0 6,0

Vex

p/ V

cal

fct

SCA Report No.4

Figure 6.14: Graph of SCA model alternative with respect to concrete tensile strength fct

[MPa].


45

6.9 Effect of the Computed Residual Strength of DAfStB

The effect of the residual tensile strength was only compared on fibre reinforced slabs hence there is no post-crack strength for non-fibre reinforced slabs. The first solution of obtaining the residual tensile strength (as described in Chapter 5.3) provides a wide range with a high frequent of the result falling under the ratio of the experimental capacity over the calculated capacity equal to 1, as shown in Figure 6.15. The figure also depicts a higher value of the estimated residual tensile strength compared with the second solution.

On the opposite of the first solution, the second solution provides in the graph smaller values of the residual tensile strength with fewer slabs falling under the critical line Vexp/Vcal = 1 than the first solution.

0,0

1,0

2,0

3,0

4,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Vex

p/ V

cal

f fct0,u = ffl,res,m

DAfStB - solution I

Figure 6.15: To the left, graph of DAfStB solution I with respect to the residual strength f fct0,u

[MPa]. To the right, graph of DAfStB solution II model alternative with respect to the residual strength f fct0,u [MPa].

CHAPTER 7. CONCLUSION

47

7 Conclusion

The following conclusions are based on the analysis of the considered punching shear capacity of fibre reinforced concrete slab models and the graphs achieved from applying the models on the test results. Table 7.1 shows an overview of the result for respectively model and solution.

Table 7.1: Summary of the result of the sabs with steel fibres. Max – Min of

Vexp/Vcal ratio Average ratio

of Vexp/Vcal Standard deviation

DAfStB Solution I 3.25 – 0.23 0.90 0.56 Solution II 3.25 – 0.37 1.21 0.52 SCA 4.87 – 0.27 1.16 0.72

From the graphs analysis in Chapter 6 and Table 7.1 the following conclusions can be drawn:

• DAfStB alternative with solution I has the highest percentage of tests falling under the critical line.

• DAfStB alternative with solution II has results almost similar to solution I but is by some means more conservative.

• SCA Report No.4 alternative has the most conservative result with greatest distribution, however it provides the best average ratio of Vexp/Vcal. This alternative has a tendency to underestimate the punching capacity of the slabs, since the ratio of Vcal in some cases is four times larger than Vexp.

• Neither the dimension of the slab nor the conventional reinforcement amount had a major impact on the ratio Vexp/Vcal, other than that models seem not being confident when increasing the slab thickness.

• The presence of steel fibres in the concrete does not influence the concrete compressive strength in relation to the slab punching shear capacity.

The general conclusion is that SCA alternative is based on Naranyan and Darwish’s proposal formula, and is easier to adapt. To elaborate, it only requires data of the fibre properties and a regular concrete compressive test to define the capacity contribution of the SFRC to the total punching shear capacity of the slab.

Chapter

7.1. CONCLUSION

48

SCA alternative relationship is arbitrary as long as the fibre volume ratio is less than one percent (Vf = 1%). Increased steel fibre volume ratio obtained a larger value of the punching capacity achieved from the steel fibres than the punching capacity achieved from the conventional reinforcement. This is evident when the slab dimension increases.

On the other hand, a comprehensive test that accesses detailed data of the residual behaviour needs to be done first, to make it possible to apply the DAfStB model alternative. Unfortunately this kind of a test has not been accomplished, even though the most recently performed test trials has emphasizes the fact that the fracture-mechanism is a vital behaviour of the fibre reinforced concrete. Of the total 12 researches, none of them stated any information about the post-crack strength or the residual strength of the concrete. This demonstrates that the post behaviour of a cracked fibre reinforced element has a greater theoretical importance than a practical one.

It is worth mentioning that the obtained values of the punching shear capacity of a conventional reinforced concrete was very high compared with punching shear capacity of steel fibre concrete. Despite that the compressive strength was estimated without fibres, this exception occurs in both capacity model alternatives. Thus, how this can be remedied is not entirely insignificant since the first term is well established for ordinary concrete and the second term is derived from static and geometric relationships.

Consequently, there are various appropriate correlations between the compressive strength, the tensile strength and the crack strength as described in section 5.2.2. The presented residual strength equations are based on the cylindrical concrete compressive strength that in some cases was gained from the cube compressive strength. However, they are ambiguous, given the fact that many assumptions are made which results in wide error range.

Finally, it is worth mentioning that the DAfStB alternative indicates more probabilistic result to evaluate the punching capacity but does not give satisfying results due to non-experimental obtained residual strength, however it is up to forthcoming studies to evaluate that.

CHAPTER 8. RECOMMENDATION FOR FUTURE RESEARCH

49

8 Recommendation for Future Research

There are various punching shear capacity models for steel fibre reinforced concrete slabs in the literature, not least in standards of countries like US, United Kingdom, Japan and Italy, that need further studies. It is suggested to investigate how useful the models in these standards are. Possibly, they established models that are more current than DAfStB alternative and more precise than SCA alternative.

Nevertheless, there is a necessity to perform tests that cover the residual strength and to publish these result in technical papers. Because the understanding of the post-crack behaviour of fibreous concrete has a great importance not just to estimate the punching capacity, but it can also provide information that improve deciding the quantity and quality of the fibres.

Another recommendation is to develop a semi-empirical estimation of the residual strength that can enhance the assessment of the capacity of fibre reinforced elements. If there are adequate numbers of flexural tests prepared on steel fibre reinforced concrete, a semi-empirical expression of the residual strength can be derived that can be functional on capacity models alike DAfStBs proposal. Otherwise there is a need of developing a European, or possible international, established method to estimate the residual tensile strength, since different testing methods contribute to varying outcomes.

To develop a punching shear capacity model for steel fibre reinforced concrete that takes into account the capacity contribution of the steel fibre and the conventional reinforcement in the same term, instead of having two terms added to gather as the situation is now.

For further analysis of the steel fibre reinforced structural behaviour using software design tools that utilize the finite element method can be accommodating

Chapter

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APPENDIX A

57

Appendix A Detailed Tables of the Test Slabs

A.1 N. Swamy and S. Ali 1979

Table A.1:1 Slab No.

L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcfl (MPa)

fct (MPa)

Asy,flex

Asz,flex

S01 1800 125 150 169 100 750 48,65 38,92 3,85 3,45 12 Ø10 12 Ø10 S02 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S03 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S04 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S05 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S06 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S07 1800 125 150 169 100 750 48,65 38,92 3,85 3,45 12 Ø10 12 Ø10 S08 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S09 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10

S010 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S011 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S012 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 8 Ø10 8 Ø10 S013 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 7 Ø10 7 Ø10 S014 1800 125 150 169 100 750 48,65 38,92 3,85 3,45 6 Ø10 6 Ø10 S015 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 6 Ø10 6 Ø10 S016 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S017 1800 125 150 169 100 750 48,65 38,92 3,85 3,45 12 Ø10 12 Ø10 S018 1800 125 150 169 100 750 48,00 38,40 7,47 3,41 12 Ø10 12 Ø10 S019 1800 125 150 169 100 750 48,65 38,92 3,85 3,45 12 Ø10 12 Ø10

APPENDIX A

58


ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

δcr (mm)

S01 0,0052 0,0052 - 0 0 0 0,0 0 Punching 197700 24,2 35000 1,0 S02 0,0052 0,0052 Crimped 50 0,5 100 0,6 48 Punching 243600 24,5 43100 1,2 S03 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 262900 33,1 56900 1,5 S04 0,0052 0,0052 Crimped 50 0,5 100 1,2 96 Punching 281000 28,2 53100 1,2 S05 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 267200 30,9 53500 1,6 S06 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 239000 22,7 47300 1,1 S07 0,0052 0,0052 - 0 0 0 0,0 0 Punching 221700 27,6 33000 1,0 S08 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 236700 25,6 55000 1,2 S09 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 249000 22,8 56300 1,6

S010 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 262000 35,4 57000 1,6 S011 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 255700 28,2 61300 1,6 S012 0,0042 0,0042 Hooked 50 0,5 100 0,9 72 Flexural 213000 28,2 64000 1,5 S013 0,0031 0,0031 Plain 50 0,6 83 0,9 72 Flexural 203000 30,3 59000 1,4 S014 0,0031 0,0031 - 0 0 0 0,0 0 Flexural 179300 28,6 72700 1,9 S015 0,0031 0,0031 Crimped 50 0,5 100 0,9 72 Flexural 130700 45,7 84700 2,0 S016 0,0052 0,0052 Crimped 50 0,5 100 0,9 72 Punching 334000 29,5 60000 1,5 S017 0,0052 0,0052 - 0 0 0 0,0 0 Flexural 356700 19,5 79000 2,0 S018 0,0052 0,0052 Crimped 50 0,5 100 1,4 110 Punching 265700 33,5 67300 1,7 S019 0,0052 0,0052 - 0 0 0 0,0 0 Flexural 469700 45,5 40500 0,8

Table A.1:3

Slab No.

Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S01 147422 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 0,979 0,979 144355 1,37 S02 147422 1,7 68 2,04 1,39 0,70 1,04 1,15 1,149 0,979 2,128 313671 0,97 S03 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,86 S04 147422 1,7 100 2,04 2,04 1,02 1,53 1,69 1,689 0,979 2,668 393349 0,79 S05 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,88 S06 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,79 S07 147422 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 0,979 0,979 144355 1,54 S08 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,78 S09 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,82

S010 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,86 S011 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 0,84 S012 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,909 2,463 363081 0,73 S013 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,826 2,380 350828 0,80 S014 147422 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 0,826 0,826 121754 1,47 S015 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,826 2,380 350828 0,46 S016 147422 1,7 92 2,04 1,88 0,94 1,41 1,55 1,554 0,979 2,533 373429 1,10 S017 147422 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 0,979 0,979 144355 2,47 S018 147422 1,7 100 2,04 2,04 1,02 1,53 1,68 1,682 0,979 2,661 392353 0,69 S019 147422 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 0,979 0,979 144355 3,25

APPENDIX A

59


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S01 147422 1,82 3,64 0 0,00 0,00 0,00 0,000 0,979 0,979 144355 1,37 S02 147422 1,82 3,64 100 0,01 0,66 0,72 0,722 0,979 1,701 250823 0,78 S03 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,70 S04 147422 1,82 3,64 100 0,01 1,31 1,44 1,444 0,979 2,424 357291 0,71 S05 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,72 S06 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,64 S07 147422 1,82 3,64 0 0,00 0,00 0,00 0,000 0,979 0,979 144355 1,54 S08 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,63 S09 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,67 S010 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,70 S011 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,68 S012 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,909 1,992 293709 0,59 S013 147422 1,82 3,64 83 0,01 0,82 0,90 0,903 0,826 1,729 254839 0,58 S014 147422 1,82 3,64 0 0,00 0,00 0,00 0,000 0,826 0,826 121754 1,47 S015 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,826 1,909 281456 0,37 S016 147422 1,82 3,64 100 0,01 0,98 1,08 1,083 0,979 2,062 304057 0,89 S017 147422 1,82 3,64 0 0,00 0,00 0,00 0,000 0,979 0,979 144355 2,47 S018 147422 1,82 3,64 100 0,01 1,50 1,65 1,649 0,979 2,628 387457 0,68 S019 147422 1,82 3,64 0 0,00 0,00 0,00 0,000 0,979 0,979 144355 3,25

Table A.1:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S01 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,451 1,451 122727 1,61 S02 84590 1,40 100 0,75 0,01 0,45 4,15 0,766 1,451 2,217 187495 1,30 S03 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,20 S04 84590 1,40 100 0,75 0,01 0,90 4,15 1,531 1,451 2,982 252263 1,11 S05 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,22 S06 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,09 S07 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,451 1,451 122727 1,81 S08 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,08 S09 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,13 S010 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,19 S011 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,16 S012 84590 1,40 100 1,05 0,01 0,95 4,15 1,608 1,450 3,058 258676 0,82 S013 84590 1,40 83 0,50 0,01 0,38 4,15 0,638 1,449 2,087 176572 1,15 S014 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,449 1,449 122598 1,46 S015 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,449 2,598 219751 0,59 S016 84590 1,40 100 0,75 0,01 0,68 4,15 1,149 1,451 2,599 219879 1,52 S017 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,451 1,451 122727 2,91 S018 84590 1,40 100 0,75 0,01 1,03 4,15 1,748 1,451 3,199 270614 0,98 S019 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,451 1,451 122727 3,83

APPENDIX A

60

A.2 D. Theodorakopoulos and N. Swamy 1989


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcfl (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

FS-1 1800 125 150 169 100 750 44,20 35,36 3,24 2,85 2,28 3,23 12 Ø10 12 Ø10 FS-2 1800 125 150 169 100 750 42,50 34,00 6,04 4,06 3,25 3,15 12 Ø10 12 Ø10 FS-3 1800 125 150 169 100 750 44,56 35,65 6,15 4,35 3,48 3,25 12 Ø10 12 Ø10 FS-4 1800 125 150 169 100 750 46,67 37,34 6,19 4,52 3,62 3,35 12 Ø10 12 Ø10 FS-5 1800 125 150 169 100 750 47,50 38,00 6,80 4,86 3,89 3,39 8 Ø10 8 Ø10 FS-6 1800 125 150 169 100 750 44,60 35,68 6,52 4,42 3,54 3,25 8 Ø10 8 Ø10 FS-7 1800 125 150 169 100 750 45,80 36,64 6,78 4,56 3,65 3,31 8 Ø10 8 Ø10 FS-8 1800 125 100 113 100 700 45,83 36,66 3,20 2,95 2,36 3,31 12 Ø10 12 Ø10 FS-9 1800 125 100 113 100 700 44,50 35,60 6,56 4,37 3,50 3,25 12 Ø10 12 Ø10

FS-10 1800 125 200 226 100 800 45,50 36,40 3,46 3,12 2,50 3,30 12 Ø10 12 Ø10 FS-11 1800 125 200 226 100 800 42,80 34,24 6,38 4,75 3,80 3,16 12 Ø10 12 Ø10 FS-12 1800 125 150 169 100 750 45,10 36,08 5,20 3,79 3,03 3,28 12 Ø10 12 Ø10 FS-13 1800 125 150 169 100 750 41,85 33,48 6,95 4,36 3,49 3,12 12 Ø10 12 Ø10 FS-14 1800 125 150 169 100 750 43,73 34,98 6,62 4,50 3,60 3,21 12 Ø10 12 Ø10 FS-15 1800 125 150 169 100 750 39,05 31,24 6,47 4,37 3,50 2,98 12 Ø10 12 Ø10 FS-16 1800 125 150 169 100 750 34,90 27,92 6,19 4,14 3,31 2,76 12 Ø10 12 Ø10 FS-17 1800 125 150 169 100 750 58,56 46,85 6,79 5,36 4,29 3,90 12 Ø10 12 Ø10 FS-18 1800 125 150 169 100 750 17,75 14,20 3,93 2,95 2,36 1,76 12 Ø10 12 Ø10 FS-19 1800 125 150 169 100 750 43,12 34,50 3,52 2,82 2,26 3,18 8 Ø10 8 Ø10 FS-20 1800 125 150 169 100 750 46,30 37,04 6,72 4,38 3,50 3,33 8 Ø10 8 Ø10 Table A.2:2

Slab No.

ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

FS-1 0,0052 0,0052 - 0 0,0 0 0,0 0 Punching 173500 9,2 32000 FS-2 0,0052 0,0052 Crimped 50 0,5 100 0,5 40 Punching 225000 9,2 42500 FS-3 0,0052 0,0052 Crimped 50 0,5 100 1,0 80 Punching 247400 9,2 46800 FS-4 0,0052 0,0052 Crimped 50 0,5 100 1,0 80 Punching 224400 0,0 40900 FS-5 0,0042 0,0042 Crimped 50 0,5 100 1,0 80 Punching 198100 7,1 30000 FS-6 0,0042 0,0042 Crimped 50 0,5 100 1,0 80 Flexural 174500 7,1 29000 FS-7 0,0042 0,0042 Crimped 50 0,5 100 1,0 80 Flexural 192400 7,1 30000 FS-8 0,0056 0,0056 - 0 0,0 0 0,0 0 Punching 150300 9,9 31500 FS-9 0,0056 0,0056 Crimped 50 0,5 100 1,0 80 Punching 216600 9,9 41400

FS-10 0,0049 0,0049 - 0 0,0 0 0,0 0 Punching 191400 7,8 36000 FS-11 0,0049 0,0049 Crimped 50 0,5 100 1,0 80 Flexural 259800 7,8 48900 FS-12 0,0052 0,0052 Japanese 25 0,4 60 1,0 80 Punching 217500 9,2 42500 FS-13 0,0052 0,0052 Hooked 50 0,5 100 1,0 80 Punching 235500 9,2 44000 FS-14 0,0052 0,0052 Paddle 53 0,8 70 1,0 80 Punching 239500 9,2 45500 FS-15 0,0052 0,0052 Crimped 38 0,4 89 1,0 80 Punching 238000 9,2 41000 FS-16 0,0052 0,0052 Paddle 53 0,8 70 1,0 80 Punching 227800 0,0 42400 FS-17 0,0052 0,0052 Paddle 53 0,8 70 1,0 80 Flexural 268400 0,0 47500 FS-18 0,0052 0,0052 Paddle 53 0,8 70 1,0 80 Punching 166000 0,0 30500 FS-19 0,0042 0,0042 - 0 0,0 0 0,0 0 Punching 136500 7,1 22500 FS-20 0,0042 0,0042 Crimped 50 0,5 100 1,0* 80 Punching 211000 7,1 31500

* were casted with fibres through the entire slab cross section.

APPENDIX A

61


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

FS-1 147422 1,7 0 1,96 0,00 0,00 0,00 0,00 0,000 0,948 0,948 139812 1,24 FS-2 147422 1,7 60 1,92 1,15 0,58 0,87 0,95 0,954 0,948 1,902 280389 0,80 FS-3 147422 1,7 100 1,97 1,97 0,99 1,48 1,63 1,627 0,948 2,576 379718 0,65 FS-4 147422 1,7 100 2,02 2,02 1,01 1,51 1,67 1,665 0,948 2,614 385332 0,58 FS-5 147422 1,7 100 2,03 2,03 1,02 1,53 1,68 1,680 0,880 2,561 377484 0,52 FS-6 147422 1,7 100 1,97 1,97 0,99 1,48 1,63 1,628 0,880 2,508 369803 0,47 FS-7 147422 1,7 100 2,00 2,00 1,00 1,50 1,65 1,650 0,880 2,530 373011 0,52 FS-8 129697 1,7 0 2,00 0,00 0,00 0,00 0,00 0,000 0,970 0,970 125864 1,19 FS-9 129697 1,7 100 1,97 1,97 0,98 1,48 1,63 1,626 0,970 2,597 336784 0,64

FS-10 165146 1,7 0 1,99 0,00 0,00 0,00 0,00 0,000 0,928 0,928 153289 1,25 FS-11 165146 1,7 100 1,93 1,93 0,97 1,45 1,59 1,595 0,928 2,523 416677 0,62 FS-12 147422 1,7 100 1,98 1,98 0,99 1,49 1,64 1,637 0,948 2,586 381167 0,57 FS-13 147422 1,7 100 1,91 1,91 0,95 1,43 1,58 1,577 0,948 2,525 372308 0,63 FS-14 147422 1,7 100 1,95 1,95 0,98 1,46 1,61 1,612 0,948 2,560 377473 0,63 FS-15 147422 1,7 100 1,84 1,84 0,92 1,38 1,52 1,523 0,948 2,472 364396 0,65 FS-16 147422 1,7 100 1,74 1,74 0,87 1,31 1,44 1,440 0,948 2,389 352127 0,65 FS-17 147422 1,7 100 2,26 2,26 1,13 1,69 1,87 1,866 0,948 2,814 414835 0,65 FS-18 147422 1,7 100 1,24 1,24 0,62 0,93 1,03 1,027 0,948 1,975 291226 0,57 FS-19 147422 1,7 0 1,94 0,00 0,00 0,00 0,00 0,000 0,880 0,880 129790 1,05 FS-20 147422 1,7 100 2,01 2,01 1,00 1,51 1,66 1,659 0,880 2,539 374335 0,56

Table A.2:4

Slab No.

Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

FS-1 147422 1,75 3,49 0 0,00 0,00 0,00 0,000 0,948 0,948 139812 1,24 FS-2 147422 1,75 3,49 100 0,01 0,52 0,58 0,577 0,948 1,525 224822 1,00 FS-3 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,948 2,102 309831 0,80 FS-4 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,948 2,102 309831 0,72 FS-5 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,880 2,034 299809 0,66 FS-6 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,880 2,034 299809 0,58 FS-7 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,880 2,034 299809 0,64 FS-8 129697 1,75 3,49 0 0,00 0,00 0,00 0,000 0,970 0,970 125864 1,19 FS-9 129697 1,75 3,49 100 0,01 1,05 1,15 1,153 0,970 2,124 275442 0,79

FS-10 165146 1,75 3,49 0 0,00 0,00 0,00 0,000 0,928 0,928 153289 1,25 FS-11 165146 1,75 3,49 100 0,01 1,05 1,15 1,153 0,928 2,081 343749 0,76 FS-12 147422 1,75 3,49 60 0,01 0,47 0,52 0,517 0,948 1,466 216077 1,01 FS-13 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,948 2,102 309831 0,76 FS-14 147422 1,75 3,49 70 0,01 0,55 0,60 0,603 0,948 1,552 228737 1,05 FS-15 147422 1,75 3,49 89 0,01 0,94 1,03 1,031 0,948 1,980 291829 0,82 FS-16 147422 1,75 3,49 70 0,01 0,55 0,60 0,603 0,948 1,552 228737 1,00 FS-17 147422 1,75 3,49 70 0,01 0,55 0,60 0,603 0,948 1,552 228737 1,17 FS-18 147422 1,75 3,49 70 0,01 0,55 0,60 0,603 0,948 1,552 228737 0,73 FS-19 147422 1,75 3,49 0 0,00 0,00 0,00 0,000 0,880 0,880 129790 1,05 FS-20 147422 1,75 3,49 100 0,01 1,05 1,15 1,153 0,880 2,034 299809 0,70

APPENDIX A

62


Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

FS-1 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,713 1,713 144884 1,20 FS-2 84590 1,40 100 0,75 0,01 0,38 4,15 0,638 1,713 2,351 198857 1,13 FS-3 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,713 2,989 252831 0,98 FS-4 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,713 2,989 252831 0,89 FS-5 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,642 2,918 246819 0,80 FS-6 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,642 2,918 246819 0,71 FS-7 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,642 2,918 246819 0,78 FS-8 66865 1,40 0 0,00 0,00 0,00 0,00 0,000 1,738 1,738 116223 1,29 FS-9 66865 1,40 100 0,75 0,01 0,75 4,15 1,276 1,738 3,014 201551 1,07

FS-10 102314 1,40 0 0,00 0,00 0,00 0,00 0,000 1,691 1,691 172970 1,11 FS-11 102314 1,40 100 0,75 0,01 0,75 4,15 1,276 1,691 2,967 303535 0,86 FS-12 84590 1,40 60 1,05 0,01 0,63 4,15 1,069 1,713 2,781 235270 0,92 FS-13 84590 1,40 100 1,05 0,01 1,05 4,15 1,787 1,713 3,499 296010 0,80 FS-14 84590 1,40 70 1,05 0,01 0,73 4,15 1,246 1,713 2,959 250274 0,96 FS-15 84590 1,40 89 0,75 0,01 0,67 4,15 1,141 1,713 2,854 241401 0,99 FS-16 84590 1,40 70 1,05 0,01 0,73 4,15 1,246 1,713 2,959 250274 0,91 FS-17 84590 1,40 70 1,05 0,01 0,73 4,15 1,246 1,713 2,959 250274 1,07 FS-18 84590 1,40 70 1,05 0,01 0,73 4,15 1,246 1,713 2,959 250274 0,66 FS-19 84590 1,40 0 0,00 0,00 0,00 0,00 0,000 1,642 1,642 138872 0,98 FS-20 84590 1,40 100 0,75 0,01 0,75 4,15 1,276 1,642 2,918 246819 0,85

APPENDIX A

63

A.3 S. Alexander and S. Simmonds 1992


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcfl (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

P11F0 2750 155 200 226 111 866 41,50 33,20 4,16 3,08 2,46 3,10 12 Ø12 12 Ø12 P11F31 2750 155 200 226 111 866 44,75 35,80 4,25 3,55 2,84 3,26 12 Ø12 12 Ø12 P11F66 2750 155 200 226 111 866 43,75 35,00 4,22 4,71 3,77 3,21 12 Ø12 12 Ø12 P38F0 2750 155 200 226 138 1028 44,50 35,60 4,54 3,35 2,68 3,25 12 Ø12 12 Ø12 P38F34 2750 155 200 226 138 1028 48,00 38,40 5,58 3,55 2,84 3,41 12 Ø12 12 Ø12 P38F69 2750 155 200 226 138 1028 48,13 38,50 5,30 4,40 3,52 3,42 12 Ø12 12 Ø12 Table A.3:2

Slab No.

ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

δcr (mm)

P11F0 0,0042 0,0042 - 0 0 0 0,0 0 Punching 257000 48,0 165000 13,5 P11F31 0,0042 0,0042 Corr.* 50 0,5 100 0,4 31 Punching 324000 69,0 193000 11,2 P11F66 0,0042 0,0042 Corr.* 50 0,5 100 0,8 66 Punching 345000 75,0 210000 13,0 P38F0 0,0028 0,0028 - 0 0 0 0,0 0 Punching 264000 62,0 147000 14,1

P38F34 0,0028 0,0028 Corr.* 50 0,5 100 0,4 34 Punching 308000 85,0 174000 14,5 P38F69 0,0028 0,0028 Corr.* 50 0,5 100 0,9 69 Punching 330000 98,0 184000 14,9

* Corrugated


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

P11F0 194820 1,7 0 1,90 0,00 0,00 0,00 0,00 0,000 0,763 0,763 148579 1,73 P11F31 194820 1,7 51 1,97 1,01 0,50 0,76 0,90 0,900 0,763 1,663 323938 1,00 P11F66 194820 1,7 86 1,95 1,68 0,84 1,26 1,50 1,501 0,763 2,263 440960 0,78 P38F0 277325 1,8 0 1,97 0,00 0,00 0,00 0,00 0,000 0,703 0,703 194888 1,35

P38F34 277325 1,8 54 2,04 1,10 0,55 0,83 0,99 0,987 0,703 1,690 468625 0,66 P38F69 277325 1,8 89 2,05 1,82 0,91 1,37 1,63 1,629 0,703 2,332 646633 0,51

Table A.3:4

Slab No.

Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

P11F0 194820 1,68 3,36 0 0,00 0,00 0,00 0,000 0,763 0,763 148579 1,73 P11F31 194820 1,68 3,36 100 0,00 0,39 0,47 0,466 0,763 1,229 239406 1,35 P11F66 194820 1,68 3,36 100 0,01 0,83 0,99 0,993 0,763 1,755 341953 1,01 P38F0 277325 1,74 3,48 0 0,00 0,00 0,00 0,000 0,703 0,703 194888 1,35

P38F34 277325 1,74 3,48 100 0,00 0,44 0,53 0,529 0,703 1,232 341729 0,90 P38F69 277325 1,74 3,48 100 0,01 0,90 1,07 1,075 0,703 1,777 492888 0,67


Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

P11F0 117405 1,40 0 0,00 0,00 0,00 0,00 0,000 1,610 1,610 189024 1,36 P11F31 117405 1,40 100 0,75 0,00 0,29 4,15 0,494 1,610 2,105 247080 1,31 P11F66 117405 1,40 100 0,75 0,01 0,62 4,15 1,053 1,610 2,663 312628 1,10 P38F0 157668 1,40 0 0,00 0,00 0,00 0,00 0,000 1,521 1,521 239754 1,10

P38F34 157668 1,40 100 0,75 0,00 0,32 4,15 0,542 1,521 2,063 325265 0,95 P38F69 157668 1,40 100 0,75 0,01 0,65 4,15 1,101 1,521 2,621 413292 0,80

APPENDIX A

64

A.4 A. Shaaban and H. Gesund 1994


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

SFO-1 1600 83 64 72 66 457 33,37 26,70 2,76 2,21 2,68 21 Ø9,5* 21 Ø9,5 SFO-2 1600 83 64 72 66 457 39,02 31,22 3,36 2,69 2,97 21 Ø9,5* 21 Ø9,5 SFO-3 1600 83 64 72 66 457 31,03 24,82 3,10 2,48 2,55 21 Ø9,5* 21 Ø9,5 SFO-4 1600 83 64 72 66 457 31,72 25,37 3,10 2,48 2,59 21 Ø9,5* 21 Ø9,5 SF2-1 1600 83 64 72 66 457 34,47 27,58 3,03 2,43 2,74 21 Ø9,5* 21 Ø9,5 SF2-2 1600 83 64 72 66 457 37,23 29,79 3,10 2,48 2,88 21 Ø9,5* 21 Ø9,5 SF2-3 1600 83 64 72 66 457 29,65 23,72 2,76 2,21 2,48 21 Ø9,5* 21 Ø9,5 SF3-1 1600 83 64 72 66 457 37,65 30,12 4,14 3,31 2,20 21 Ø9,5* 21 Ø9,5 SF4-1 1600 83 64 72 66 457 46,75 37,40 5,14 4,11 2,90 21 Ø9,5* 21 Ø9,5 SF4-2 1600 83 64 72 66 457 36,54 29,23 3,90 3,12 3,35 21 Ø9,5* 21 Ø9,5 SF6-1 1600 83 64 72 66 457 22,34 17,87 3,07 2,45 2,85 21 Ø9,5* 21 Ø9,5 SF6-2 1600 83 64 72 66 457 22,06 17,65 3,10 2,48 2,05 21 Ø9,5* 21 Ø9,5

*the varying bar diameter is due to conversion from imperial units

Table A.4:2

Slab No.

ρy,z

ρ1

SF type

Lf (mm)

df (mm)

dfe (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

SFO-1 0,0143 0,0143 - 0 0 0,0 0 0,0 0 - 88964 SFO-2 0,0143 0,0143 - 0 0 0,0 0 0,0 0 - 111206 SFO-3 0,0143 0,0143 - 0 0 0,0 0 0,0 0 - 80068 SFO-4 0,0143 0,0143 - 0 0 0,0 0 0,0 0 - 93413 SF2-1 0,0143 0,0143 Corrugated 25 0,7 1,0 26 0,6 49* - 93413 SF2-2 0,0143 0,0143 Corrugated 25 0,7 1,0 26 0,6 49* - 111206 SF2-3 0,0143 0,0143 Corrugated 25 0,7 1,0 26 0,6 49* - 71172 SF3-1 0,0143 0,0143 Corrugated 25 0,7 1,0 26 0,9 76* - 106757 SF4-1 0,0143 0,0143 Corrugated 25 0,7 1,0 26 1,2 95* - 133447 SF4-2 0,0143 0,0143 Corrugated 25 0,7 1,0 26 1,2 95* - 115654 SF6-1 0,0143 0,0143 Corrugated 25 0,7 1,0 26 1,9 149* - 97861 SF6-2 0,0143 0,0143 Corrugated 25 0,7 1,0 26 2,0 157* - 102309

- there is no information about the failure mode *the dissimilarity of steel fibre content is due to conversion from imperial units.

APPENDIX A

65


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

SFO-1 55234 1,6 0 1,71 0,00 0,00 0,00 0,00 0,000 0,953 0,953 52619 1,69 SFO-2 55234 1,6 0 1,84 0,00 0,00 0,00 0,00 0,000 1,004 1,004 55437 2,01 SFO-3 55234 1,6 0 1,64 0,00 0,00 0,00 0,00 0,000 0,930 0,930 51357 1,56 SFO-4 55234 1,6 0 1,66 0,00 0,00 0,00 0,00 0,000 0,937 0,937 51735 1,81 SF2-1 55234 1,6 69 1,73 1,19 0,60 0,90 0,95 0,949 0,963 1,912 105610 0,88 SF2-2 55234 1,6 69 1,80 1,24 0,62 0,93 0,99 0,986 0,988 1,974 109049 1,02 SF2-3 55234 1,6 69 1,61 1,11 0,55 0,83 0,88 0,880 0,916 1,796 99194 0,72 SF3-1 55234 1,6 96 1,47 1,41 0,70 1,06 1,12 1,120 0,863 1,983 109517 0,97 SF4-1 55234 1,6 100 1,81 1,82 0,91 1,36 1,44 1,445 0,992 2,436 134565 0,99 SF4-2 55234 1,6 100 2,02 2,03 1,01 1,52 1,61 1,610 1,066 2,676 147788 0,78 SF6-1 55234 1,6 100 1,78 1,79 0,89 1,34 1,42 1,421 0,982 2,402 132697 0,74 SF6-2 55234 1,6 100 1,40 1,39 0,69 1,04 1,10 1,103 0,833 1,937 106971 0,96


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

SFO-1 55234 1,51 3,02 0 0,00 0,00 0,00 0,000 0,953 0,953 52619 1,69 SFO-2 55234 1,51 3,02 0 0,00 0,00 0,00 0,000 1,004 1,004 55437 2,01 SFO-3 55234 1,51 3,02 0 0,00 0,00 0,00 0,000 0,930 0,930 51357 1,56 SFO-4 55234 1,51 3,02 0 0,00 0,00 0,00 0,000 0,937 0,937 51735 1,81 SF2-1 55234 1,51 3,02 26 0,01 0,15 0,15 0,154 0,963 1,117 61701 1,51 SF2-2 55234 1,51 3,02 26 0,01 0,15 0,15 0,154 0,988 1,142 63084 1,76 SF2-3 55234 1,51 3,02 26 0,01 0,15 0,15 0,154 0,916 1,070 59093 1,20 SF3-1 55234 1,51 3,02 26 0,01 0,23 0,24 0,239 0,863 1,102 60864 1,75 SF4-1 55234 1,51 3,02 26 0,01 0,28 0,30 0,300 0,992 1,292 71368 1,87 SF4-2 55234 1,51 3,02 26 0,01 0,28 0,30 0,300 1,066 1,366 75468 1,53 SF6-1 55234 1,51 3,02 26 0,02 0,44 0,47 0,470 0,982 1,452 80187 1,22 SF6-2 55234 1,51 3,02 26 0,02 0,47 0,49 0,493 0,833 1,326 73258 1,40


Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

SFO-1 28246 1,40 0 0,00 0,00 0,00 0,00 0,000 1,355 1,355 38279 2,32 SFO-2 28246 1,40 0 0,00 0,00 0,00 0,00 0,000 1,355 1,355 38279 2,91 SFO-3 28246 1,40 0 0,00 0,00 0,00 0,00 0,000 1,355 1,355 38279 2,09 SFO-4 28246 1,40 0 0,00 0,00 0,00 0,00 0,000 1,355 1,355 38279 2,44 SF2-1 28246 1,40 26 0,75 0,01 0,12 4,15 0,205 1,355 1,560 44069 2,12 SF2-2 28246 1,40 26 0,75 0,01 0,12 4,15 0,205 1,355 1,560 44069 2,52 SF2-3 28246 1,40 26 0,75 0,01 0,12 4,15 0,205 1,355 1,560 44069 1,62 SF3-1 28246 1,40 26 0,75 0,01 0,19 4,15 0,318 1,355 1,673 47253 2,26 SF4-1 28246 1,40 26 0,75 0,01 0,23 4,15 0,400 1,355 1,755 49569 2,69 SF4-2 28246 1,40 26 0,75 0,01 0,23 4,15 0,400 1,355 1,755 49569 2,33 SF6-1 28246 1,40 26 0,75 0,02 0,37 4,15 0,625 1,355 1,980 55938 1,75 SF6-2 28246 1,40 26 0,75 0,02 0,39 4,15 0,656 1,355 2,011 56806 1,80

APPENDIX A

66

A.5 M. Harajli, D. Maaloufand H. Khatib 1995


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fct (MPa)

Asy,flex

Asz,flex

A1 650 55 100 113 39 334 37,00 29,60 2,87 5 Ø10 5 Ø10 A2 650 55 100 113 39 334 37,50 30,00 2,90 5 Ø10 5 Ø10 A3 650 55 100 113 39 334 39,25 31,40 2,99 5 Ø10 5 Ø10 A4 650 55 100 113 39 334 30,75 24,60 2,54 5 Ø10 5 Ø10 A5 650 55 100 113 39 334 25,00 20,00 2,21 5 Ø10 5 Ø10 B1 650 75 100 113 55 430 39,25 31,40 2,99 7 Ø10 7 Ø10 B2 650 75 100 113 55 430 39,25 31,40 2,99 7 Ø10 7 Ø10 B3 650 75 100 113 55 430 39,75 31,80 3,01 7 Ø10 7 Ø10 B4 650 75 100 113 55 430 36,38 29,10 2,84 7 Ø10 7 Ø10 B5 650 75 100 113 55 430 36,50 29,20 2,84 7 Ø10 7 Ø10

Table A.5:2

Slab No.

ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

A1 0,0181 0,0181 - 0 0,0 0 0,0 0 Punching 63000 A2 0,0181 0,0181 Hooked 50 0,5 100 0,5 36 Punching 68000 A3 0,0181 0,0181 Hooked 50 0,5 100 0,8 64 Flexural 78000 A4 0,0181 0,0181 Hooked 30 0,5 60 1,0 80 Flex-Punch 69000 A5 0,0181 0,0181 Hooked 30 0,5 60 2,0 160 Flexural 62000 B1 0,0133 0,0133 - 0 0,0 0 0,0 0 Punching 99000 B2 0,0133 0,0133 Hooked 50 0,5 100 0,5 36 Punching 115000 B3 0,0133 0,0133 Hooked 50 0,5 100 0,8 64 Punching 117000 B4 0,0133 0,0133 Hooked 30 0,5 60 1,0 80 Punching 118000 B5 0,0133 0,0133 Hooked 30 0,5 60 2,0 160 Punching 146000

Table A.5:3

Slab No.

Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A1 28160 1,4 0 1,80 0,00 0,00 0,00 0,00 0,000 0,988 0,988 27817 2,26 A2 28160 1,4 56 1,81 1,01 0,51 0,76 0,77 0,771 0,988 1,759 49539 1,37 A3 28160 1,4 84 1,85 1,55 0,78 1,16 1,18 1,184 0,988 2,172 61151 1,28 A4 28160 1,4 100 1,64 1,64 0,82 1,23 1,25 1,247 0,988 2,235 62942 1,10 A5 28160 1,4 100 1,48 1,48 0,74 1,11 1,12 1,125 0,988 2,112 59488 1,04 B1 48007 1,5 0 1,85 0,00 0,00 0,00 0,00 0,000 0,942 0,942 45243 2,19 B2 48007 1,5 56 1,85 1,04 0,52 0,78 0,79 0,794 0,942 1,736 83346 1,38 B3 48007 1,5 84 1,86 1,56 0,78 1,17 1,20 1,198 0,942 2,141 102760 1,14 B4 48007 1,5 100 1,78 1,78 0,89 1,34 1,36 1,364 0,942 2,307 110744 1,07 B5 48007 1,5 100 1,78 1,78 0,89 1,34 1,37 1,367 0,942 2,309 110857 1,32

APPENDIX A

67


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A1 28160 1,59 3,18 0 0,00 0,00 0,00 0,000 0,988 0,988 27817 2,26 A2 28160 1,59 3,18 100 0,00 0,43 0,44 0,436 0,988 1,424 40091 1,70 A3 28160 1,59 3,18 100 0,01 0,76 0,77 0,775 0,988 1,763 49637 1,57 A4 28160 1,59 3,18 60 0,01 0,57 0,58 0,581 0,988 1,569 44182 1,56 A5 28160 1,59 3,18 60 0,02 1,14 1,16 1,162 0,988 2,150 60546 1,02 B1 48007 1,64 3,27 0 0,00 0,00 0,00 0,000 0,942 0,942 45243 2,19 B2 48007 1,64 3,27 100 0,00 0,44 0,45 0,449 0,942 1,391 66793 1,72 B3 48007 1,64 3,27 100 0,01 0,79 0,80 0,798 0,942 1,740 83554 1,40 B4 48007 1,64 3,27 60 0,01 0,59 0,60 0,599 0,942 1,541 73976 1,60 B5 48007 1,64 3,27 60 0,02 1,18 1,20 1,197 0,942 2,139 102710 1,42

Table A.5:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A1 18604 1,40 0 0,00 0,00 0,00 0,00 0,000 2,342 2,342 43577 1,45 A2 18604 1,40 100 1,05 0,00 0,47 4,15 0,804 2,342 3,146 58534 1,16 A3 18604 1,40 100 1,05 0,01 0,84 4,15 1,429 2,342 3,772 70167 1,11 A4 18604 1,40 60 1,05 0,01 0,63 4,15 1,072 2,342 3,414 63519 1,09 A5 18604 1,40 60 1,05 0,02 1,26 4,15 2,144 2,342 4,486 83461 0,74 B1 29000 1,40 0 0,00 0,00 0,00 0,00 0,000 2,047 2,047 59362 1,67 B2 29000 1,40 100 1,05 0,00 0,47 4,15 0,804 2,047 2,851 82677 1,39 B3 29000 1,40 100 1,05 0,01 0,84 4,15 1,429 2,047 3,476 100811 1,16 B4 29000 1,40 60 1,05 0,01 0,63 4,15 1,072 2,047 3,119 90449 1,30 B5 29000 1,40 60 1,05 0,02 1,26 4,15 2,144 2,047 4,191 121536 1,20

APPENDIX A

68

A.6 B. Hughes and Y. Xiao 1995


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

S4 860 65 132 149 59 483 52,00 41,60 5,00 4,00 3,60 10 Ø8 10 Ø8 S5 860 65 132 149 59 483 56,00 44,80 6,25 5,00 3,78 10 Ø8 10 Ø8 S6 860 65 132 149 59 483 45,00 36,00 6,13 4,90 3,27 10 Ø8 10 Ø8 S8 860 65 132 149 59 483 52,00 41,60 8,25 6,60 3,60 10 Ø8 10 Ø8 S9 860 65 132 149 59 483 48,00 38,40 6,75 5,40 3,41 10 Ø8 10 Ø8

S12 860 65 132 149 59 483 39,00 31,20 5,00 4,00 2,97 10 Ø8 10 Ø8 S13 860 65 132 149 59 483 53,00 42,40 8,50 6,80 3,65 10 Ø8 10 Ø8 S16 860 50 132 149 45 402 49,00 39,20 4,50 3,60 3,46 10 Ø8 10 Ø8 S18 860 50 132 149 45 402 37,00 29,60 6,38 5,10 2,87 10 Ø8 10 Ø8 S21 860 65 132 149 59 483 45,00 36,00 4,00 3,20 3,27 10 Ø8 10 Ø8 S22 860 65 132 149 59 483 52,00 41,60 7,13 5,70 3,60 10 Ø8 10 Ø8

Table A.6:2

Slab No.

ρy,z

ρ1

SF type

Lf (mm)

df (mm)

dfe (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

S4 0,0107 0,0107 - 0 0,0 0,0 0 0,0 0 - 89000 7,5 28000 S5 0,0107 0,0107 Round 38 0,4

95 1,0 80 - 108000 9,3 30000

S6 0,0107 0,0107 Round 25 0,3

100 1,0 80 - 106000 12,3 35000 S8 0,0107 0,0107 Duoform 60

0,9 66 1,0 80 - 121000 15,9 30000

S9 0,0107 0,0107 Duoform 40

0,8 50 1,0 80 - 116000 7,7 28000 S12 0,0107 0,0107 Duoform 60

0,9 66 0,5 40 - 105000 11,2 23000

S13 0,0107 0,0107 Duoform 60

0,9 66 1,5 120 - 127000 14,7 30000 S16 0,0167 0,0167 - 0 0,0 0,0 0 0,0 0 - 66000 12,0 17000 S18 0,0167 0,0167 Duoform 60

0,9 66 1,0 80 - 91000 26,8 10000

S21 0,0107 0,0107 - 0 0,0 0,0 0 0,0 0 - 116000 6,8 41000 S22 0,0107 0,0107 Duoform 60 0,9 66 1,0 80 - 108000 16,3 28000

- there is no information about the failure mode Table A.6:3

Slab No.

Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S4 59628 1,5 0 2,13 0,00 0,00 0,00 0,00 0,000 0,963 0,963 57413 1,55 S5 59628 1,5 100 2,21 2,21 1,10 1,66 1,70 1,698 0,963 2,661 158677 0,68 S6 59628 1,5 100 1,98 1,98 0,99 1,49 1,52 1,522 0,963 2,485 148188 0,72 S8 59628 1,5 100 2,13 2,13 1,06 1,60 1,64 1,636 0,963 2,599 154993 0,78 S9 59628 1,5 100 2,04 2,04 1,02 1,53 1,57 1,572 0,963 2,535 151165 0,77

S12 59628 1,5 60 1,84 1,11 0,55 0,83 0,85 0,850 0,963 1,813 108117 0,97 S13 59628 1,5 100 2,15 2,15 1,07 1,61 1,65 1,652 0,963 2,615 155927 0,81 S16 40142 1,5 0 2,07 0,00 0,00 0,00 0,00 0,000 1,069 1,069 63740 1,04 S18 40142 1,5 100 1,80 1,80 0,90 1,35 1,37 1,373 1,069 2,442 98009 0,93 S21 59628 1,5 0 1,98 0,00 0,00 0,00 0,00 0,000 0,963 0,963 38651 3,00 S22 59628 1,5 100 2,13 2,13 1,06 1,60 1,64 1,636 0,963 2,599 154993 0,70

APPENDIX A

69


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S4 59628 1,88 3,77 0 0,00 0,00 0,00 0,000 0,963 0,963 57413 1,55 S5 59628 1,95 3,91 95 0,01 0,74 0,76 0,761 0,963 1,724 102812 1,05 S6 59628 1,75 3,50 100 0,01 0,70 0,72 0,718 0,963 1,681 100252 1,06 S8 59628 1,88 3,77 66 0,01 0,75 0,76 0,764 0,963 1,727 102959 1,18 S9 59628 1,83 3,66 50 0,01 0,55 0,56 0,564 0,963 1,526 91020 1,27

S12 59628 1,75 3,50 66 0,01 0,35 0,36 0,355 0,963 1,318 78598 1,34 S13 59628 1,88 3,77 66 0,02 1,12 1,15 1,146 0,963 2,109 125732 1,01 S16 40142 1,83 3,66 0 0,00 0,00 0,00 0,000 1,069 1,069 42910 1,54 S18 40142 1,75 3,50 66 0,01 0,69 0,71 0,707 1,069 1,775 71272 1,28 S21 59628 1,75 3,50 0 0,00 0,00 0,00 0,000 0,963 0,963 57413 2,02 S22 59628 1,88 3,77 66 0,01 0,75 0,76 0,764 0,963 1,727 102959 1,05

Table A.6:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

S4 38125 1,40 0 0,50 0,00 0,00 4,15 0,000 2,230 2,230 85018 1,05 S5 38125 1,40 95 0,50 0,01 0,48 4,15 0,808 2,230 3,038 115831 0,93 S6 38125 1,40 100 0,50 0,01 0,50 4,15 0,851 2,230 3,081 117453 0,90 S8 38125 1,40 66 1,05 0,01 0,69 4,15 1,178 2,230 3,408 129930 0,93 S9 38125 1,40 50 1,05 0,01 0,53 4,15 0,895 2,230 3,125 119156 0,97

S12 38125 1,40 66 1,05 0,01 0,35 4,15 0,589 2,230 2,819 107474 0,98 S13 38125 1,40 66 1,05 0,02 1,04 4,15 1,767 2,230 3,997 152386 0,83 S16 27419 1,40 0 1,05 0,00 0,00 4,15 0,000 2,666 2,666 73098 0,90 S18 27419 1,40 66 1,05 0,01 0,69 4,15 1,178 2,666 3,844 105398 0,86 S21 38125 1,40 0 1,05 0,00 0,00 4,15 0,000 2,230 2,230 85018 1,36 S22 38125 1,40 66 1,05 0,01 0,69 4,15 1,178 2,230 3,408 129930 0,83

APPENDIX A

70

A.7 P. McHarg 1997


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcfl (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

NU 2300 150 225 254 117 927 37,50 30,00 3,26 3,36 2,90 2,90 14 Ø16 14 Ø16 NB 2300 150 225 254 117 927 38,63 30,90 3,26 3,43 2,95 2,95 14 Ø16 14 Ø16 FSU 2300 150 225 254 117 927 48,75 39,00 3,12 4,00 3,45 3,45 14 Ø16 14 Ø16 FSB 2300 150 225 254 117 927 51,88 41,50 3,22 4,17 3,60 3,60 14 Ø16 14 Ø16 FCU 2300 150 225 254 117 927 46,88 37,50 3,01 3,90 3,36 3,36 14 Ø16 14 Ø16 FCB 2300 150 225 254 117 927 41,75 33,40 4,63 3,61 3,11 3,11 14 Ø16 14 Ø16


ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf/df

Vf (%)

ρf (kg/m3)

Vp,exp (N)

δmax (mm)

Pcr (N)

δcr (mm)

Py (MPa)

δy (mm)

NU 0,0111 0,0111 - 0 0 0 0 0 306000 17,2 80000 1,3 218000 9,9 NB 0,0111 0,0111 - 0 0 0 0 0 349000 15,3 78000 1,0 273000 10,7 FSU 0,0111 0,0111 Hooked 30 0,5 60 0,5 40 422000 36,0 97000 1,1 227000 9,0 FSB 0,0111 0,0111 Hooked 30 0,5 60 0,5 40 438000 32,8 93000 1,0 320000 12,6 FCU 0,0111 0,0111 Hooked 30 0,5 60 0,5 40 329000 18,2 93000 1,3 242000 10,6 FCB 0,0111 0,0111 Hooked 30 0,5 60 0,5 40 361000 16,3 84000 1,0 258000 10,1

- there is no information about the failure mode


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NU 222336 1,8 0 1,81 0,00 0,00 0,00 0,00 0,000 1,026 1,026 228117 1,34 NB 222336 1,8 0 1,83 0,00 0,00 0,00 0,00 0,000 1,026 1,026 228117 1,53 FSU 222336 1,8 60 2,06 1,24 0,62 0,93 1,07 1,071 1,026 2,097 466319 0,90 FSB 222336 1,8 60 2,13 1,28 0,64 0,96 1,11 1,105 1,026 2,131 473835 0,92 FCU 222336 1,8 60 2,02 1,21 0,61 0,91 1,05 1,051 1,026 2,077 461693 0,71 FCB 222336 1,8 60 1,91 1,14 0,57 0,86 0,99 0,991 1,026 2,017 448555 0,80


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NU 222336 1,60 3,20 0 0,00 0,00 0,00 0,000 1,026 1,026 228117 1,34 NB 222336 1,60 3,20 0 0,00 0,00 0,00 0,000 1,026 1,026 228117 1,53 FSU 222336 1,60 3,20 60 0,01 0,29 0,33 0,333 1,026 1,359 302061 1,40 FSB 222336 1,60 3,20 60 0,01 0,29 0,33 0,333 1,026 1,359 302061 1,45 FCU 222336 1,60 3,20 60 0,01 0,29 0,33 0,333 1,026 1,359 302061 1,09 FCB 222336 1,60 3,20 60 0,01 0,29 0,33 0,333 1,026 1,359 302061 1,20


Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NU 136325 1,40 0 1,05 0,00 0,00 4,15 0,000 1,908 1,908 260156 1,18 NB 136325 1,40 0 1,05 0,00 0,00 4,15 0,000 1,908 1,908 260156 1,34 FSU 136325 1,40 60 1,05 0,01 0,32 4,15 0,536 1,908 2,444 333222 1,27 FSB 136325 1,40 60 1,05 0,01 0,32 4,15 0,536 1,908 2,444 333222 1,31 FCU 136325 1,40 60 1,05 0,01 0,32 4,15 0,536 1,908 2,444 333222 0,99 FCB 136325 1,40 60 1,05 0,01 0,32 4,15 0,536 1,908 2,444 333222 1,08

APPENDIX A

71

A.8 G. Hassanzadeh and H. Sundquist 1998


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

ρy,z

ρ1

B1 2600 220 250 282 190 1390 51,20 40,96 4,00 3,20 4,14 0,0029 0,0029 C1 2600 220 250 282 220 1570 52,40 41,92 0,00 0,00 4,20 0,0000 0,0000 C2* 2600 220 250 282 180 1330 52,20 41,76 3,70 2,96 4,19 0,0041 0,0041 E1 2600 220 250 282 184 1354 86,90# 69,52 8,80 7,04 5,89 0,0076 0,0076

E2-NS 2600 220 250 282 184 1354 48,90 39,12 3,70 2,96 4,01 0,0076 0,0076 E2-HS 2600 220 250 282 184 1354 74,80# 59,84 5,50 4,40 5,33 0,0076 0,0076

E3* 2600 220 250 282 190 1390 75,00# 60,00 5,10 4,08 5,34 0,0041 0,0041 HSC2 2540 240 250 282 194 1414 98,80# 79,04 5,70 4,56 6,41 0,0082 0,0082 NSC8 2540 242 250 282 198 1438 34,50# 27,60 3,20 2,56 3,18 0,0080 0,0080 HSC8 2540 242 250 282 198 1438 100,10# 80,08 6,90 5,52 6,47 0,0080 0,0080

- there is no info about the reinforcement bar dimension As,yz * prestressed slabs that are excluded from the analysis # high strength concrete


SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

B1 - 0 0 0 0 0 Punching 437000 33,0 C1 hooked 35/60 0,55/0,75 65/80 1,5 120 Flexural 342000 5,8 C2* hooked 35/60 0,55/0,75 65/80 1,5 120 Flexural 931000 21,8 E1 hooked 35/60 0,55/0,75 65/80 1,5 120 Punching 1210000 24,0

E2-NS hooked 35/60 0,55/0,75 65/80 1,5 120 Punching 1057000 23,0 E2-HS hooked 35/60 0,55/0,75 65/80 1,5 120 Punching 1057000 23,0

E3* hooked 35/60 0,55/0,75 65/80 1,5 120 Flexural 949000 18,0 HSC2 - 0 0 0 0 0 Punching 889000 11,0 NSC8 - 0 0 0 0 0 Punching 944000 14,0 HSC8 - 0 0 0 0 0 Punching 944000 14,0

* prestressed slabs that are excluded from the analysis


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

B1 508619 2,0 0 2,36 0,00 0,00 0,00 0,00 0,000 0,870 0,870 442290 0,99 C1 651130 2,0 100 2,39 2,39 1,19 1,79 2,25 2,253 0,000 2,253 1466839 0,23 C2* 464885 1,9 100 2,38 2,38 1,19 1,79 2,25 2,248 0,968 3,216 1495096 0,62 E1 482152 2,0 100 3,08 3,08 1,54 2,31 2,90 2,901 1,488 4,389 2116086 0,57

E2-NS 482152 2,0 100 2,31 2,31 1,15 1,73 2,18 2,176 1,195 3,371 1625446 0,65 E2-HS 482152 2,0 100 2,85 2,85 1,43 2,14 2,69 2,692 1,488 4,179 2015055 0,52

E3* 508619 2,0 100 2,86 2,86 1,43 2,14 2,70 2,695 1,221 3,916 1991673 0,48 HSC2 526640 2,0 0 3,28 0,00 0,00 0,00 0,00 0,000 1,546 1,546 814164 1,09 NSC8 544963 2,0 0 1,94 0,00 0,00 0,00 0,00 0,000 1,085 1,085 591402 1,60 HSC8 544963 2,0 0 3,30 0,00 0,00 0,00 0,00 0,000 1,548 1,548 843505 1,12


APPENDIX A

72


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

B1 508619 2,09 4,18 0 0,00 0,00 0,00 0,000 0,870 0,870 442290 0,99 C1 651130 1,72 3,43 73 0,02 1,12 1,41 1,407 0,000 1,407 916251 0,37 C2* 464885 1,72 3,43 73 0,02 1,12 1,41 1,407 0,968 2,375 1103996 0,84 E1 482152 2,90 5,80 73 0,02 1,89 2,38 2,381 1,488 3,869 1865480 0,65

E2-NS 482152 1,72 3,43 73 0,02 1,12 1,41 1,407 1,195 2,602 1254646 0,84 E2-HS 482152 2,90 5,80 73 0,02 1,89 2,38 2,381 1,488 3,869 1865480 0,57

E3* 508619 2,90 5,80 73 0,02 1,89 2,38 2,381 1,221 3,602 1832058 0,52 HSC2 526640 2,90 5,80 0 0,00 0,00 0,00 0,000 1,546 1,546 814164 1,09 NSC8 544963 1,72 3,43 0 0,00 0,00 0,00 0,000 1,085 1,085 591402 1,60 HSC8 544963 2,92 5,84 0 0,00 0,00 0,00 0,000 1,548 1,548 843505 1,12



Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

B1 281795 1,40 0 1,05 0,00 0,00 4,15 0,000 1,986 1,986 559582 0,78 C1 347024 1,38 73 1,05 0,02 1,14 4,15 1,943 1,713 3,655 1268516 0,27 C2* 261309 1,40 73 1,05 0,02 1,14 4,15 1,943 2,093 4,036 1054745 0,88 E1 269428 1,40 73 1,05 0,02 1,14 4,15 1,943 3,716 5,659 1524693 0,79

E2-NS 269428 1,40 73 1,05 0,02 1,14 4,15 1,943 2,398 4,340 1169432 0,90 E2-HS 269428 1,40 73 1,05 0,02 1,14 4,15 1,943 3,716 5,659 1524693 0,69

E3* 281795 1,40 73 1,05 0,02 1,14 4,15 1,943 3,245 5,188 1461884 0,65 HSC2 290165 1,40 0 1,05 0,00 0,00 4,15 0,000 3,797 3,797 1101725 0,81 NSC8 298636 1,40 0 1,05 0,00 0,00 4,15 0,000 1,869 1,869 558283 1,69 HSC8 298636 1,40 0 1,05 0,00 0,00 4,15 0,000 3,803 3,803 1135701 0,83


APPENDIX A

73

A.9 A. Azevedo 1999


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcfl (MPa)

fct (MPa)

Asy,flex

Asz,flex

OCS.S1 1160 100 80 90 90 620 54,66 43,73 4,70 3,72 17 Ø10 20 Ø10 OCS.S2 1160 100 80 90 90 620 58,03 46,42 5,50 3,87 17 Ø10 20 Ø10 OCS.S3 1160 100 80 90 90 620 38,50 30,80 6,11 2,95 17 Ø10 20 Ø10 OCS.S4 1160 100 80 90 90 620 48,55 38,84 2,70 3,44 17 Ø10 20 Ø10 OCS.S5 1160 100 80 90 90 620 46,28 37,02 4,39 3,33 17 Ø10 20 Ø10 OCS.S6 1160 100 80 90 90 620 49,65 39,72 5,55 3,49 17 Ø10 20 Ø10 HSC.S1 1160 100 80 90 90 620 108,31# 86,65 4,93 5,87 17 Ø10 20 Ø10 HSC.S2 1160 100 80 90 90 620 102,31# 81,85 7,60 5,66 17 Ø10 20 Ø10 HSC.S3 1160 100 80 90 90 620 99,13# 79,30 8,56 5,54 17 Ø10 20 Ø10 HSC.S4 1160 100 80 90 90 620 103,43# 82,74 6,69 5,70 17 Ø10 20 Ø10 HSC.S5 1160 100 80 90 90 620 91,86# 73,49 7,68 5,26 17 Ø10 20 Ø10 HSC.S6 1160 100 80 90 90 620 89,33# 71,46 9,66 5,17 17 Ø10 20 Ø10

# high strength concrete


ρy

ρz

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

OCS.S1 0,0141 0,0155 0,0148 - 0 0,0 0 0,0 0 - 176480 3,3 OCS.S2 0,0141 0,0155 0,0148 Hooked 30 0,5 67 0,8 60 - 191960 6,3 OCS.S3 0,0141 0,0155 0,0148 Hooked 30 0,5 67 1,5 120 - 197610 4,7 OCS.S4 0,0141 0,0155 0,0148 - 0 0,0 0 0,0 0 - 270440 5,8 OCS.S5 0,0141 0,0155 0,0148 Hooked 30 0,5 67 0,8 60 - 292790 10,8 OCS.S6 0,0141 0,0155 0,0148 Hooked 30 0,5 67 1,5 120 - 329560 9,6 HSC.S1 0,0141 0,0155 0,0148 - 30 0,5 0 0,0 0 - 190720 3,9 HSC.S2 0,0141 0,0155 0,0148 Hooked 30 0,5 67 0,8 60 - 206810 4,1 HSC.S3 0,0141 0,0155 0,0148 Hooked 30 0,5 67 1,5 120 - 293350 6,1 HSC.S4 0,0141 0,0155 0,0148 - 0 0,0 0 0,0 0 - 293350 5,6 HSC.S5 0,0141 0,0155 0,0148 Hooked 30 0,5 67 0,8 60 - 388670 9,4 HSC.S6 0,0141 0,0155 0,0148 Hooked 30 0,5 67 1,5 120 - 439070 17,0

- there is no information about the failure mode Table A.9:3

Slab No.

Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

OCS.S1 101864 1,7 0 2,18 0,00 0,00 0,00 0,00 0,000 1,206 1,206 122895 1,44 OCS.S2 101864 1,7 80 2,25 1,80 0,90 1,35 1,42 1,419 1,184 2,603 265151 0,72 OCS.S3 101864 1,7 100 1,83 1,83 0,92 1,37 1,45 1,445 1,184 2,629 267782 0,74 OCS.S4 101864 1,7 0 2,06 0,00 0,00 0,00 0,00 0,000 1,388 1,388 141406 1,91 OCS.S5 101864 1,7 80 2,01 1,61 0,80 1,20 1,27 1,268 1,184 2,451 249684 1,17 OCS.S6 101864 1,7 100 2,08 2,08 1,04 1,56 1,64 1,641 1,184 2,825 287747 1,15 HSC.S1 101864 1,7 0 3,07 0,00 0,00 0,00 0,00 0,000 1,814 1,814 184770 1,03 HSC.S2 101864 1,7 80 2,99 2,39 1,19 1,79 1,88 1,885 1,504 3,389 345186 0,60 HSC.S3 101864 1,7 100 2,94 2,94 1,47 2,20 2,32 2,319 1,504 3,823 389417 0,75 HSC.S4 101864 1,7 0 3,00 0,00 0,00 0,00 0,00 0,000 1,786 1,786 181948 1,61 HSC.S5 101864 1,7 80 2,83 2,26 1,13 1,70 1,79 1,786 1,504 3,290 335117 1,16 HSC.S6 101864 1,7 100 2,79 2,79 1,39 2,09 2,20 2,201 1,504 3,705 377436 1,16

APPENDIX A

74


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

OCS.S1 101864 1,93 3,86 0 0,00 0,00 0,00 0,000 1,206 1,206 122895 1,44 OCS.S2 101864 1,93 3,86 67 0,01 0,58 0,61 0,610 1,184 1,793 182650 1,05 OCS.S3 101864 1,93 3,86 67 0,02 1,16 1,22 1,219 1,184 2,403 244739 0,81 OCS.S4 101864 1,82 3,64 0 0,00 0,00 0,00 0,000 1,388 1,388 141406 1,91 OCS.S5 101864 1,82 3,64 67 0,01 0,55 0,57 0,574 1,184 1,758 179075 1,64 OCS.S6 101864 1,82 3,64 67 0,02 1,09 1,15 1,149 1,184 2,332 237590 1,39 HSC.S1 101864 2,72 5,44 0 0,00 0,00 0,00 0,000 1,814 1,814 184770 1,03 HSC.S2 101864 2,72 5,44 67 0,01 0,82 0,86 0,858 1,504 2,362 240588 0,86 HSC.S3 101864 2,72 5,44 67 0,02 1,63 1,72 1,716 1,504 3,220 327988 0,89 HSC.S4 101864 2,66 5,31 0 0,00 0,00 0,00 0,000 1,786 1,786 181948 1,61 HSC.S5 101864 2,66 5,31 67 0,01 0,80 0,84 0,838 1,504 2,342 238594 1,63 HSC.S6 101864 2,66 5,31 67 0,02 1,59 1,68 1,677 1,504 3,181 323998 1,36

Table A.9:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

OCS.S1 50970 1,40 0 1,05 0,00 0,00 4,15 0,000 2,718 2,718 138554 1,27 OCS.S2 50970 1,40 67 1,05 0,01 0,53 4,15 0,893 2,616 3,509 178871 1,07 OCS.S3 50970 1,40 67 1,05 0,02 1,05 4,15 1,787 2,616 4,403 224402 0,88 OCS.S4 50970 1,40 0 1,05 0,00 0,00 4,15 0,000 2,512 2,512 128022 2,11 OCS.S5 50970 1,40 67 1,05 0,01 0,53 4,15 0,893 2,616 3,509 178871 1,64 OCS.S6 50970 1,40 67 1,05 0,02 1,05 4,15 1,787 2,616 4,403 224402 1,47 HSC.S1 50970 1,40 0 1,05 0,00 0,00 4,15 0,000 4,288 4,288 218581 0,87 HSC.S2 50970 1,40 67 1,05 0,01 0,53 4,15 0,893 4,224 5,117 260812 0,79 HSC.S3 50970 1,40 67 1,05 0,02 1,05 4,15 1,787 4,224 6,010 306343 0,96 HSC.S4 50970 1,40 0 1,05 0,00 0,00 4,15 0,000 4,158 4,158 211955 1,38 HSC.S5 50970 1,40 67 1,05 0,01 0,53 4,15 0,893 4,224 5,117 260812 1,49 HSC.S6 50970 1,40 67 1,05 0,02 1,05 4,15 1,787 4,224 6,010 306343 1,43

APPENDIX A

75

A.10 S. Ozden, U. Ersoy, and T. Ozturan 2006


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

NR1E0F1 1500 120 200 226 100 800 24,50 19,60 3,05 2,44 2,18 14 Ø10 14 Ø10 NR1E1F1 1500 120 200 226 100 800 24,50 19,60 3,05 2,44 2,18 14 Ø10 14 Ø10 NR1E2F1 1500 120 200 226 100 800 24,50 19,60 3,05 2,44 2,18 14 Ø10 14 Ø10 NR2E0F1 1500 120 200 226 100 800 24,13 19,30 3,11 2,49 2,16 21 Ø10 21 Ø10 NR2E1F1 1500 120 200 226 100 800 24,13 19,30 3,11 2,49 2,16 21 Ø10 21 Ø10 NR2E2F1 1500 120 200 226 100 800 24,13 19,30 3,11 2,49 2,16 21 Ø10 21 Ø10 HR1E0F1 1500 120 200 226 100 800 101,63# 81,30 8,55 6,84 5,63 15 Ø14 15 Ø14 HR1E1F1 1500 120 200 226 100 800 101,63# 81,30 8,55 6,84 5,63 15 Ø14 15 Ø14 HR1E2F1 1500 120 200 226 100 800 101,63# 81,30 8,55 6,84 5,63 15 Ø14 15 Ø14 HR2E0F1 1500 120 200 226 100 800 99,13# 79,30 8,79 7,03 5,54 22 Ø14 22 Ø14 HR2E1F1 1500 120 200 226 100 800 99,13# 79,30 8,79 7,03 5,54 22 Ø14 22 Ø14 HR2E2F1 1500 120 200 226 100 800 99,13# 79,30 8,79 7,03 5,54 22 Ø14 22 Ø14 # high strength concrete


ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf

(%) ρf

(kg/m3) Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

δcr (mm)

NR1E0F1 0,0079 0,0079 Hooked 60 0,8 80 1,0 80 - 266000 7,5 59000 0,3 NR1E1F1 0,0079 0,0079 Hooked 60 0,8 80 1,0 80 - 211000 7,7 55000 0,3 NR1E2F1 0,0079 0,0079 Hooked 60 0,8 80 1,0 80 - 188000 5,8 54000 0,2 NR2E0F1 0,0118 0,0118 Hooked 60 0,8 80 1,0 80 - 245000 5,7 87000 0,5 NR2E1F1 0,0118 0,0118 Hooked 60 0,8 80 1,0 80 - 192000 3,9 79000 0,5 NR2E2F1 0,0118 0,0118 Hooked 60 0,8 80 1,0 80 - 142000 4,2 57000 0,5 HR1E0F1 0,0154 0,0154 Hooked 60 0,8 80 1,0 80 - 576000 10,8 132000 0,5 HR1E1F1 0,0154 0,0154 Hooked 60 0,8 80 1,0 80 - 405000 7,5 126000 0,4 HR1E2F1 0,0154 0,0154 Hooked 60 0,8 80 1,0 80 - 369000 8,7 126000 0,5 HR2E0F1 0,0231 0,0231 Hooked 60 0,8 80 1,0 80 - 691000 7,8 185000 0,6 HR2E1F1 0,0231 0,0231 Hooked 60 0,8 80 1,0 80 - 528000 7,6 160000 0,6 HR2E2F1 0,0231 0,0231 Hooked 60 0,8 80 1,0 80 - 411000 5,8 150000 0,7

- there is no information about the failure mode


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NR1E0F1 165146 1,7 100 1,46 1,46 0,73 1,10 1,18 1,184 0,790 1,974 326005 0,82 NR1E1F1 165146 1,7 100 1,46 1,46 0,73 1,10 1,18 1,184 0,760 1,945 321202 0,66 NR1E2F1 165146 1,7 100 1,46 1,46 0,73 1,10 1,18 1,184 0,750 1,934 319442 0,59 NR2E0F1 165146 1,7 100 1,45 1,45 0,72 1,09 1,18 1,175 0,881 2,056 339591 0,72 NR2E1F1 165146 1,7 100 1,45 1,45 0,72 1,09 1,18 1,175 0,894 2,069 341741 0,56 NR2E2F1 165146 1,7 100 1,45 1,45 0,72 1,09 1,18 1,175 0,882 2,058 339833 0,42 HR1E0F1 165146 1,7 100 2,98 2,98 1,49 2,23 2,41 2,412 1,490 3,902 644392 0,89 HR1E1F1 165146 1,7 100 2,98 2,98 1,49 2,23 2,41 2,412 1,496 3,909 645495 0,63 HR1E2F1 165146 1,7 100 2,98 2,98 1,49 2,23 2,41 2,412 1,496 3,909 645495 0,57 HR2E0F1 165146 1,7 100 2,94 2,94 1,47 2,20 2,38 2,383 1,549 3,931 649194 1,06 HR2E1F1 165146 1,7 100 2,94 2,94 1,47 2,20 2,38 2,383 1,633 4,015 663140 0,80 HR2E2F1 165146 1,7 100 2,94 2,94 1,47 2,20 2,38 2,383 1,633 4,015 663140 0,62

APPENDIX A

76


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NR1E0F1 165146 1,36 2,71 80 0,01 0,65 0,70 0,704 0,790 1,494 246683 1,08 NR1E1F1 165146 1,28 2,57 80 0,01 0,62 0,67 0,666 0,760 1,426 235514 0,90 NR1E2F1 165146 1,26 2,51 80 0,01 0,60 0,65 0,652 0,750 1,401 231452 0,81 NR2E0F1 165146 1,31 2,61 80 0,01 0,63 0,68 0,678 0,881 1,559 257382 0,95 NR2E1F1 165146 1,33 2,67 80 0,01 0,64 0,69 0,693 0,894 1,587 262022 0,73 NR2E2F1 165146 1,31 2,62 80 0,01 0,63 0,68 0,679 0,882 1,562 257903 0,55 HR1E0F1 165146 2,51 5,02 80 0,01 1,21 1,30 1,303 1,490 2,793 461242 1,25 HR1E1F1 165146 2,53 5,06 80 0,01 1,21 1,31 1,312 1,496 2,808 463794 0,87 HR1E2F1 165146 2,53 5,06 80 0,01 1,21 1,31 1,312 1,496 2,808 463794 0,80 HR2E0F1 165146 2,33 4,66 80 0,01 1,12 1,21 1,209 1,549 2,758 455434 1,52 HR2E1F1 165146 2,52 5,05 80 0,01 1,21 1,31 1,310 1,633 2,942 485936 1,09 HR2E2F1 165146 2,52 5,05 80 0,01 1,21 1,31 1,310 1,633 2,942 485936 0,85

Table A.10:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

NR1E0F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,361 2,790 285485 0,93 NR1E1F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,263 2,692 275416 0,77 NR1E2F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,227 2,657 271821 0,69 NR2E0F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,475 2,904 297171 0,82 NR2E1F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,519 2,948 301666 0,64 NR2E2F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 1,480 2,909 297674 0,48 HR1E0F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 3,930 5,360 548357 1,05 HR1E1F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 3,966 5,395 551971 0,73 HR1E2F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 3,966 5,395 551971 0,67 HR2E0F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 4,029 5,458 558408 1,24 HR2E1F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 4,480 5,909 604588 0,87 HR2E2F1 102314 1,40 80 1,05 0,01 0,84 4,15 1,429 4,480 5,909 604588 0,68

APPENDIX A

77

A.11 J. Hanai and K. Holanda 2008


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

L1 1160 100 80 90 90 620 28,88 23,10 2,68 2,14 2,43 9 Ø5 9 Ø5 L2 1160 100 80 90 90 620 30,50 24,40 3,24 2,59 2,52 9 Ø5 9 Ø5 L3 1160 100 80 90 90 620 35,13 28,10 3,73 2,98 2,77 9 Ø5 9 Ø5 L4 1160 100 80 90 90 620 71,25 57,00 4,76 3,81 4,44 9 Ø5 9 Ø5 L5 1160 100 80 90 90 620 74,63 59,70 6,81 5,45 4,58 9 Ø5 9 Ø5 L6 1160 100 80 90 90 620 65,50 52,40 8,24 6,59 4,20 9 Ø5 9 Ø5 L7 1160 100 80 90 90 620 45,75 36,60 4,96 3,97 3,31 9 Ø5 9 Ø5 L8 1160 100 80 90 90 620 57,63 46,10 6,46 5,17 3,86 9 Ø5 9 Ø5

Table A.11:2

Slab No.

ρy

ρz

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

L1 0,0018 0,0018 0,0018 - 0 0,0 0 0,0 0 Punching 137200 L2 0,0018 0,0018 0,0018 Hooked 30 0,6 55 1,0 80 Punching 139550 L3 0,0018 0,0018 0,0018 Hooked 30 0,6 55 2,0 160 Punching 163620 L4 0,0018 0,0018 0,0018 - 0 0,0 0 0,0 0 Punching 192860 L5 0,0018 0,0018 0,0018 Hooked 30 0,6 55 1,0 80 Punching 215140 L6 0,0018 0,0018 0,0018 Hooked 30 0,6 55 2,0 160 Punching 236170 L7 0,0018 0,0018 0,0018 Hooked 50 1,1 48 0,8 60 Punching 182850 L8 0,0018 0,0018 0,0018 Hooked 50 1,1 48 1,5 120 Punching 210900

Table A.11:3

Slab No.

Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

L1 101864 1,7 0 1,59 0,00 0,00 0,00 0,00 0,000 0,480 0,480 48890 2,81 L2 101864 1,7 100 1,63 1,63 0,82 1,22 1,29 1,286 0,480 1,766 179926 0,78 L3 101864 1,7 100 1,75 1,75 0,87 1,31 1,38 1,380 0,480 1,860 189510 0,86 L4 101864 1,7 0 2,49 0,00 0,00 0,00 0,00 0,000 0,649 0,649 66065 2,92 L5 101864 1,7 100 2,55 2,55 1,27 1,91 2,01 2,012 0,649 2,661 271031 0,79 L6 101864 1,7 100 2,39 2,39 1,19 1,79 1,89 1,885 0,649 2,534 258091 0,92 L7 101864 1,7 80 2,00 1,60 0,80 1,20 1,26 1,260 0,649 1,909 194453 0,94 L8 101864 1,7 100 2,24 2,24 1,12 1,68 1,77 1,768 0,649 2,417 246178 0,86

APPENDIX A

78


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

L1 101864 1,40 2,81 0 0,00 0,00 0,00 0,000 0,480 0,480 48890 2,81 L2 101864 1,40 2,81 55 0,01 0,46 0,48 0,483 0,480 0,963 98118 1,42 L3 101864 1,40 2,81 55 0,02 0,92 0,97 0,967 0,480 1,447 147347 1,11 L4 101864 2,20 4,41 0 0,00 0,00 0,00 0,000 0,649 0,649 66065 2,92 L5 101864 2,20 4,41 55 0,01 0,72 0,76 0,759 0,649 1,408 143395 1,50 L6 101864 2,20 4,41 55 0,02 1,44 1,52 1,518 0,649 2,167 220726 1,07 L7 101864 2,20 4,41 48 0,01 0,47 0,50 0,497 0,649 1,146 116698 1,57 L8 101864 2,20 4,41 48 0,02 0,94 0,99 0,994 0,649 1,643 167331 1,26

Table A.11:5

Slab No.

Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

L1 50970 1,40 0 0,00 0,00 0,00 0,00 0,000 0,575 0,575 29283 4,69 L2 50970 1,40 55 1,05 0,01 0,57 4,15 0,974 0,585 1,560 79492 1,76 L3 50970 1,40 55 1,05 0,02 1,15 4,15 1,949 0,613 2,562 130600 1,25 L4 50970 1,40 0 0,00 0,00 0,00 0,00 0,000 0,776 0,776 39570 4,87 L5 50970 1,40 55 1,05 0,01 0,57 4,15 0,974 0,788 1,763 89856 2,39 L6 50970 1,40 55 1,05 0,02 1,15 4,15 1,949 0,755 2,704 137816 1,71 L7 50970 1,40 48 1,05 0,01 0,38 4,15 0,638 0,670 1,308 66660 2,74 L8 50970 1,40 48 1,05 0,02 0,75 4,15 1,276 0,723 1,999 101912 2,07

APPENDIX A

79

A.12 L. Nguyen-Minh, M. Rovnak, T. Tran-Quoc, and K. Nguyen-Kim 2011


L (mm)

h (mm)

c1 (mm)

c2 (mm)

d (mm)

bw (mm)

fcc,cube (MPa)

fcc (MPa)

fcsp,cube (MPa)

fcsp (MPa)

fct (MPa)

Asy,flex

Asz,flex

A0 900 125 150 169 105 780 27,10 21,68 1,95 1,56 2,33 Ø10 Ø10 A1 900 125 150 169 105 780 27,90 22,32 2,23 1,78 2,38 Ø10 Ø10 A2 900 125 150 169 105 780 29,20 23,36 2,42 1,94 2,45 Ø10 Ø10 A3 900 125 150 169 105 780 31,60 25,28 2,57 2,06 2,58 Ø10 Ø10 B0 1200 125 150 169 105 780 27,10 21,68 1,95 1,56 2,33 Ø10 Ø10 B1 1200 125 150 169 105 780 27,90 22,32 2,23 1,78 2,38 Ø10 Ø10 B2 1200 125 150 169 105 780 29,20 23,36 2,42 1,94 2,45 Ø10 Ø10 B3 1200 125 150 169 105 780 31,60 25,28 2,57 2,06 2,58 Ø10 Ø10 C0 1500 125 150 169 105 780 27,10 21,68 1,95 1,56 2,33 Ø10 Ø10 C1 1500 125 150 169 105 780 27,90 22,32 2,23 1,78 2,38 Ø10 Ø10 C2 1500 125 150 169 105 780 29,20 23,36 2,42 1,94 2,45 Ø10 Ø10 C3 1500 125 150 169 105 780 31,60 25,28 2,57 2,06 2,58 Ø10 Ø10


ρy,z

ρ1

SF type

Lf (mm)

df (mm)

Lf /df

Vf (%)

ρf (kg/m3)

Failure mode

Vp,exp (N)

δmax (mm)

Pcr (N)

A0 0,0066 0,0066 - 0 0,0 0 0,0 0 Punching 284000 4,1 20000 A1 0,0066 0,0066 Hooked 60 0,8 80 0,4 30 Punching 330000 5,5 30000 A2 0,0066 0,0066 Hooked 60 0,8 80 0,6 45 Punching 345000 6,8 40000 A3 0,0066 0,0066 Hooked 60 0,8 80 0,8 60 Punching 397000 6,7 45000 B0 0,0066 0,0066 - 0 0,0 0 0,0 0 Punching 301000 11,7 25000 B1 0,0066 0,0066 Hooked 60 0,8 80 0,4 30 Punching 328000 23,2 35000 B2 0,0066 0,0066 Hooked 60 0,8 80 0,6 45 Punching 337000 13,1 40000 B3 0,0066 0,0066 Hooked 60 0,8 80 0,8 60 Punching 347000 14,0 45000 C0 0,0066 0,0066 - 0 0,0 0 0,0 0 Punching 264000 22,1 30000 C1 0,0066 0,0066 Hooked 60 0,8 80 0,4 30 Punching 307000 23,6 46000 C2 0,0066 0,0066 Hooked 60 0,8 80 0,6 45 Punching 310000 23,1 50000 C3 0,0066 0,0066 Hooked 60 0,8 80 0,8 60 Punching 326000 26,5 55000


Ap (mm)

k

R (%)

ffl,cr (MPa)

ffl,res1 (MPa)

ffl,res6 (MPa)

ffl,res,m (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A0 159741 1,7 0 1,54 0,00 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,36 A1 159741 1,7 50 1,56 0,78 0,39 0,58 0,61 0,614 0,754 1,368 218504 1,51 A2 159741 1,7 65 1,59 1,04 0,52 0,78 0,82 0,817 0,754 1,571 250878 1,38 A3 159741 1,7 80 1,66 1,33 0,66 1,00 1,05 1,046 0,754 1,800 287462 1,38 B0 159741 1,7 0 1,54 0,00 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,50 B1 159741 1,7 50 1,56 0,78 0,39 0,58 0,62 0,624 0,754 1,378 220080 1,49 B2 159741 1,7 65 1,59 1,04 0,52 0,78 0,83 0,830 0,754 1,584 252974 1,33 B3 159741 1,7 80 1,66 1,33 0,66 1,00 1,06 1,063 0,754 1,816 290145 1,20 C0 159741 1,7 0 1,54 0,00 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,19 C1 159741 1,7 50 1,56 0,78 0,39 0,58 0,63 0,634 0,754 1,388 221656 1,39 C2 159741 1,7 65 1,59 1,04 0,52 0,78 0,84 0,843 0,754 1,597 255070 1,22 C3 159741 1,7 80 1,66 1,33 0,66 1,00 1,08 1,080 0,754 1,833 292829 1,11

APPENDIX A

80


Ap (mm)

ft (MPa)

τ (MPa)

λf (l/d)

ρf

fpc (MPa)

ff,ctR (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A0 159741 1,36 2,72 0 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,36 A1 159741 1,36 2,72 80 0,00 0,24 0,26 0,257 0,754 1,011 161457 2,04 A2 159741 1,36 2,72 80 0,01 0,37 0,39 0,386 0,754 1,139 181993 1,90 A3 159741 1,36 2,72 80 0,01 0,49 0,51 0,514 0,754 1,268 202529 1,96 B0 159741 1,36 2,72 0 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,50 B1 159741 1,36 2,72 80 0,00 0,24 0,26 0,257 0,754 1,011 161457 2,03 B2 159741 1,36 2,72 80 0,01 0,37 0,39 0,386 0,754 1,139 181993 1,85 B3 159741 1,36 2,72 80 0,01 0,49 0,51 0,514 0,754 1,268 202529 1,71 C0 159741 1,36 2,72 0 0,00 0,00 0,00 0,000 0,754 0,754 120384 2,19 C1 159741 1,36 2,72 80 0,00 0,24 0,26 0,257 0,754 1,011 161457 1,90 C2 159741 1,36 2,72 80 0,01 0,37 0,39 0,386 0,754 1,139 181993 1,70 C3 159741 1,36 2,72 80 0,01 0,49 0,51 0,514 0,754 1,268 202529 1,61


Ap (mm)

ξ

λf (l/d)

μf

ρf (%)

Ff

τf (MPa)

vRd,f (MPa)

vRd,c (MPa)

vRd,cf (MPa)

VRd,cf (N)

Vp,exp/ VRd,cf

A0 90469 1,40 0 1,05 0,00 0,00 4,15 0,000 1,303 1,303 117876 2,41 A1 90469 1,40 80 1,05 0,00 0,32 4,15 0,536 1,303 1,839 166365 1,98 A2 90469 1,40 80 1,05 0,01 0,47 4,15 0,804 1,303 2,107 190609 1,81 A3 90469 1,40 80 1,05 0,01 0,63 4,15 1,072 1,303 2,375 214853 1,85 B0 90469 1,40 0 1,05 0,00 0,00 4,15 0,000 1,303 1,303 117876 2,55 B1 90469 1,40 80 1,05 0,00 0,32 4,15 0,536 1,303 1,839 166365 1,97 B2 90469 1,40 80 1,05 0,01 0,47 4,15 0,804 1,303 2,107 190609 1,77 B3 90469 1,40 80 1,05 0,01 0,63 4,15 1,072 1,303 2,375 214853 1,62 C0 90469 1,40 0 1,05 0,00 0,00 4,15 0,000 1,303 1,303 117876 2,24 C1 90469 1,40 80 1,05 0,00 0,32 4,15 0,536 1,303 1,839 166365 1,85 C2 90469 1,40 80 1,05 0,01 0,47 4,15 0,804 1,303 2,107 190609 1,63 C3 90469 1,40 80 1,05 0,01 0,63 4,15 1,072 1,303 2,375 214853 1,52

APPENDIX A

82

TRITA-BKN. Master Thesis 334,

Structural Design and Bridges 2011

ISSN 1103-4297

ISRN KTH/BKN/EX-334-SE

www.kth.se

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