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Shear Capacity of Steel Fibre Reinforced Concrete Beams without Conventional Shear
Reinforcement
Tvärkraftskapacitet hos fiberbetongbalkar utan konventionell armoring
ELEONORA MONDO
Master of Science Thesis Stockholm, Sweden 2011
Shear Capacity of Steel Fibre Reinforced Concrete Beams without Conventional Shear Reinforcement Tvärkraftskapacitet hos fiberbetongbalkar utan konventionell armoring
Eleonora Mondo
©Eleonora Mondo, 2011 Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Structural Design and Bridges Stockholm, Sweden, 2011 Politecnico di Torino Dipartimento di Ingegneria Strutturale
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TRITA‐BKN. Master Thesis 331 Structural Design and Bridges, 2011
ISSN 1103‐4297 ISRN KTH/BKN/EX‐331‐SE
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Preface
This master thesis is the result of my studies at Politecnico di Torino and at the Royal Institute of Technology of Stockholm where I did my last year of M.Sc. within the Erasmus Exchange program.
After the first five months in this University, the knowledge and the availability of the Professors, the friends I met and the great experiences I had, led me to extend my studies here for doing my master thesis.
Professor Johan Silfwerbrand, Professor Bernardino Chiaia and Professor Alessandro Pasquale Fantilli gave me the possibility to do this thesis on the field of the shear of SFRC beams. I would like to express my sincere gratitude to Professor J. Silfwerbrand three times, for this opportunity, for bringing me to know this innovative material and for the time spent on my work. Further I would like to thank Professor B. Chiaia and Professor A.P. Fantilli for their great co‐operation, supervision and help.
I would like to thank the employees at the Swedish Cement and Concrete Research Institute for their assistance and positive attitude and the Politecnico di Torino for the technical background provided to me and the scholarship I received.
During this unique experience I met special people that became friends, I saw amazing places and I discovered interesting cultures. They will always stay in my heart.
Finally, I wish to express my greatest thanks to my family and friends who gave me the support I needed without any exception.
Stockholm, September 2011
Eleonora Mondo
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Abstract
While the increase in shear strength of Steel Fibre Reinforced Concrete (SFRC) is well recognized, it has yet to be found common application of this material in building structures and there is no existing national standard that treats SFRC in a systematic manner.
The aim of the diploma work is to investigate the shear strength of fibre reinforced concrete beams and the available test data and analyse the latter against the most promising equations available in the literature. The equations investigated are: Narayanan and Darwish’s formula, the German, the RILEM and the Italian guidelines. Thirty articles, selected among over one hundred articles taken from literature, have been used to create the database that contains almost 600 beams tested in shear. This large number of beams has been decreased to 371 excluding all those beams and test that do not fall within the limitation stated for this thesis. Narayanan and Darwish’s formula can be utilized every time that the fibre percentage, the type of fibres, the beam dimensions, the flexural reinforcement and the concrete strength class have been defined. On the opposite, the parameters introduced in the German, the RILEM and the Italian guidelines always require a further characterization of the concrete (with bending test) in order to describe the post‐cracking behaviour. The parameters involved in the guidelines are the residual flexural tensile strengths according to the different test set‐ups. A method for predicting the residual flexural tensile strength from the knowledge of the fibre properties, the cylindrical compressive strength of the concrete and the amount of fibres percentage is suggested. The predictions of the shear strength, obtained using the proposed method for the residual flexural tensile strength, showed to be satisfactory when compared with the experimental results.
A comparison among the aforementioned equations corroborate the validity of the empirical formulations proposed by Narayanan and Darwish nevertheless only the other equations provide a realistic assessments of the strength, toughness and ductility of structural elements subjected to shear loading. Over the three investigated equations, which work with the post‐cracking characterization of the material, the Italian guideline proposal is the one that, due to its wide domain of validity and the results obtained for the gathered database of beams, has been selected as the most reliable equation.
Keywords: Steel Fibre Reinforced Concrete, shear, Narayanan and Darwish’s Equation,
RILEM, CNR, DafStB, post‐cracking, flexural tensile strength.
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Sammanfattning
Trots att fiberbetongens utökade tvärkraftskapacitet är väl känd har fiberbetongen ännu inte uppnått en ställning som ett vanligt alternativ inom husbyggnad. Det finns heller inga nationella standarder som behandlar fiberbetong på ett systematiskt sätt.
Målet med föreliggande examensarbete är att undersöka tvärkraftkapaciteten hos fiberbetongbalkar genom tillgängliga försöksresultat samt analysera försöksresultaten mot de mest lovande ekvationerna som är tillgängliga i litteraturen. De undersökta ekvationerna är följande: Narayanans och Darwishs ekvation samt de tyska, RILEMs och de italienska rekommendationernas ekvationer. Trettio artiklar, som valts ut bland över 100 artiklar från litteraturen, har använts för att skapa en databas som innehåller över 600 balkar som provats i skjuvning. Detta stora antal har reducerats till 371 genom att alla balkar som faller utanför avhandlingens begränsningar uteslutits. Narayanans och Darwishs ekvation kan användas så fort man känner fiberinnehållet, fibertypen, balkdimensionerna, böjarmeringen och hållfasthetsklassen. I motsats till det kräver de tyska, RILEMs och de italienska rekommendationerna en mer utförlig beskrivning över fiberbetongen (genom böjprovning) för att beskriva hur fiberbetongen fungerar efter uppsprickning. Den nödvändiga parametern i rekommendationerna är fiberbetongens residualhållfasthet bestämd genom olika varianter av böjning av fiberbetongbalkar. I rapporten har författaren utvecklat en ett förslag till metod med vars hjälp man kan uppskatta residualhållfastheten ur uppgifter om fiberns egenskaper, betongens cylinderhållfasthet samt fiberinnehåll. Bestämningen av skjuvhållfastheten, genom användning av denna metod, visade sig stämma väl överens med experimentella resultat.
En jämförelse mellan de ovan nämnda ekvationerna visar giltigheten hos Narayanans och Darwishs empiriska ekvation, även om enbart de andra tre ger en realistisk bedömning av bärförmåga och seghet hos konstruktionselement som belastas i skjuvning. Bland de tre undersökta ekvationerna, som beaktar materialets verkningssätt efter uppsprickning, är de italienska rekommendationernas metodik de som förespråkas här, eftersom den har en bred giltighet och stöd i databasen.
Nyckelord: Fiberbetong, skjuvning, Narayanans och Darwishs ekvation, RILEM, CNR,
DafStB, verkningssätt efter uppsprickning, böjdraghållfasthet.
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Sommario
Il calcestruzzo fibrorinforzato è un materiale composito ottenuto introducendo fibre corte d’acciaio in una matrice di calcestruzzo. L’aggiunta di queste fibre ad una matrice cementizia,che è caratterizzata da un comportamento fragile a trazione, rallenta il processo di fessurazione aumentando la duttilità e la capacità di assorbimento di energia del materiale composito. Inoltre, come è ampiamente riconosciuto dalla letteratura scientifica, le fibre aumentano la capacità a taglio degli elementi strutturali. Tuttavia, resta la necessità di stabilire meccanismi di resistenza e di redigere normative nazionali riguardanti il taglio che permettano di trattare il problema in modo affidabile e sistematico.
La presente tesi intende esaminare le molteplici formulazioni analitiche per il calcolo della resistenza ultima a taglio, testimoniate dalle molte pubblicazioni scientifiche e non, presenti in letteratura. Nello specifico, sono state confrontate la formula di Narayanan e Darwish e le equazioni proposte dagli enti normativi Tedesco (DafStB), Italiano (CNR) e dal RILEM. Lo studio è fondato su trenta campagne sperimentali, selezionate tra un centinaio di pubblicazioni, per un totale di 600 elementi trave di dimensioni reali testati a taglio. Escludendo i test su elementi le cui proprietà non rientrano nel dominio investigato da questa tesi, si è ottenuto un database di 371 elementi trave.
La formula di Narayanan and Darwish ha la peculiarità di poter essere utilizzata tutte le volte che sono note le dimensioni della trave, la percentuale in volume di fibre, le caratteristiche fisiche‐meccaniche delle fibre, il quantitativo di armatura longitudinale e la classe del calcestruzzo. Le linee guida del DafStB, del RILEM e del CNR, invece, contengono all’interno delle loro formulazioni un parametro legato al comportamento del materiale nella fase posteriore alla fessurazione, ovvero la resistenza residua desunta da prove a flessione. Questi tre metodi, pertanto, richiedono una caratterizzazione del materiale che consiste in un ulteriore test a flessione (differente per ogni ente normativo) su un provino di piccole dimensioni, dal quale si possono desumere le resistenze residue del materiale. Al fine di proseguire nel confronto, che è precipuamente l’oggetto di questa tesi, la scarsità di questa tipologia di dati é stata superata performando una back analysis; la formulazione analitica ottenuta, una per ciascun ente, si basa sulla percentuale di fibre, sulle caratteristiche delle fibre e sulla resistenza cilindrica a compressione del calcestruzzo, restituendo il valore della tensione residua mancante. La capacità a taglio, basata sulle tensioni residue calcolate con la formulazione analitica presentata in questo lavoro, mostra risultati soddisfacenti quando viene comparata con la capacità a taglio misurata durante le prove sperimentali riportate nel database.
Ampie interpretazioni e discussioni generate dai risultati ottenuti con le sovramenzionate equazioni, confermano la validità della formulazione empirica di Narayanan e Darwish commisurando difficoltà e risultati; nonostante tutto, le altre tre
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equazioni sono in grado di dare maggiori indicazioni relative alla resistenza, alla tenacità e alla duttilità dell’elemento trave soggetto ad azioni taglianti. Tra queste ultime la proposta italiana, grazie al suo esteso dominio di validità ed ai risultati ottenuti, viene indicata come la piú promettente formula per la computazione della capacità a taglio di una trave fibrorinforzata in assenza di armatura specifica.
Parole chiave: Calcestruzzo fibrorinforzato, sforzi di taglio, Narayanan e Darwish,
RILEM, DafStB, CNR, post fessurazione, tensione residua.
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Contents
Preface .......................................................................................................................................... iii
Abstract .......................................................................................................................................... v
Sammanfattning ....................................................................................................................... vii
Sommario ..................................................................................................................................... ix
Contents ........................................................................................................................................ xi
Notations ................................................................................................................................... xiii
Abbreviations ............................................................................................................................ xv
Introduction ......................................................................................................... 1 Chapter 1
1.1 Background ............................................................................................................................... 1
1.2 Scope of the Thesis ................................................................................................................. 3
1.3 Limitation .................................................................................................................................. 3
1.4 Outline of the Thesis .............................................................................................................. 4
Theoretical Background .................................................................................. 7 Chapter 2
2.1 Fibre Reinforced Concrete: Material, Geometries and Physical Properties. ..... 7
2.2 Mechanisms of Crack Formation and Propagation ................................................. 13
2.3 Test Methods ......................................................................................................................... 20
Shear Capacity .................................................................................................. 27 Chapter 3
3.1 Alternative I: Narayanan & Darwish ............................................................................. 31
3.2 Alternative II: Equation Developed from the German Committee for
Reinforced Concrete (DAfStB)......................................................................................... 36
3.3 Alternative III: RILEM TC 162‐TDF (2003) ................................................................. 38
3.4 Alternative IV: Italian Guideline CNR DT 204‐2006 ................................................ 42
Analysis and Comparison of the Specimens .......................................... 45 Chapter 4
4.1 Introduction .......................................................................................................................... 45
4.2 Presentation of all the Data ............................................................................................. 48
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4.3 Classification of the Specimens Based on the Main Properties........................... 57
4.3.1 Test Methods & Post‐Cracking Parameters ..................................................................... 57
4.3.2 Modes of Failure ......................................................................................................................... 66
4.3.3 Strength of Concrete ................................................................................................................. 67
4.3.4 Fibres & Volume Percentage ................................................................................................. 68
4.3.5 Longitudinal Reinforcement .................................................................................................. 68
4.3.6 Specimen Dimensions .............................................................................................................. 68
4.3.7 Other Properties ......................................................................................................................... 69
4.4 Data Processing .................................................................................................................... 70
4.5 Comparison between the Test Data from Literature and the Theoretical
Formulas of Chapter 3 ........................................................................................................ 94
4.5.1 Effect of a/d Ratio ...................................................................................................................... 95
4.5.2 Effect of the Maximum Aggregate Size .............................................................................. 96
4.5.3 Effect of Fibres Percentage ..................................................................................................... 97
4.5.4 Effect of Longitudinal Reinforcement ................................................................................ 98
4.5.5 Effect of the Cylindrical Compressive Strength ............................................................. 99
4.5.6 Effect of the RI and the Fibre Factor F ............................................................................ 100
4.5.7 Effect of the Effective Depth d ............................................................................................ 102
Conclusion and Future Perspectives ...................................................... 105 Chapter 5
5.1 Discussion of the Results ................................................................................................ 105
5.2 Proposal of the Best Alternative. ................................................................................. 106
5.3 Future Perspectives .......................................................................................................... 108
References ............................................................................................................................... 109
Appendix A ............................................................................................................................... A‐1
Appendix B SFRC Papers and References ............................................................... B‐1
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Notations
General notations
(.)c property (.) referred to concrete
(.)d design value of property (.)
(.)exp experimental value of property (.)
(.)F property (.) referred to fiber‐reinforced concrete
(.)k characteristic value of property (.)
(.)m medium value of property (.)
(.)R property (.) as resistance
(.)s property (.) referred to steel
(.)S property (.) as demand
(.)u ultimate value of property (.)
[AXX] article number XX
Uppercase Roman letters
A' nondimensional constant in Narayanan&Darwish's Eq.
Ab total bond area of the fibres across the inclined cracked section [mm2]
Ac area of the concrete cross‐section [m2]
Af area of a single fibre [mm2]
As area of tensile reinforcement [mm2]
A's area of compression reinforcement [mm2]
B' dimensional constant in Narayanan&Darwish's Eq.
Vf Fibre content by volume
Ff Fibre Factor
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EC elasiticiy modulus of concrete
Pu observed ultimate load
Lowercase Roman letters
a shear span [mm]
a/d shear span/effective depth ratio
b width of the beam [mm]
d effective depth of the cross‐section [mm]
da maximum aggregates size [mm]
df fiber diameter (equivalent) [mm]
df bond factor in Narayanan&Darwish's Eq.
fcc,cube 28 compressive cube strength of concrete [MPa]
fcsp 28 splitting tensile strength of concrete [MPa]
fsy Yield stress of reinforceing steel [MPa]
lf/df Aspect ratios of fibres
fcfl flexural tensile strength of concrete [MPa]
fcsp splitting tensile strength of concrete [MPa]
Lowercase Greek letters
φ diameter of rebar [mm]
ρf fiber mass density [kg/m3]
τ fibre‐matrix interfacial bond stress [MPa]
ρ'flex percentage of area of tensile reinforcement = (A/bd) x 100
ρ flex percentage of area of compression reinforcement = (A's/bd)/100
ρ tot total ratio of reinforcement
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Abbreviations
3PBT Three Point Bending Test
4PBT Four Point Bending Test
ASTM American Society for Testing and Materials
CEN Comitato Normativo Europeo ‐ European Committee for Standardization
CNR Consiglio Nazionale delle Ricerche ‐ National Research Council
CRC Chemical Rubber Company
CSTR The Concrete Society Technical Report (U.K.)
CUR Centre for Civil Engineering Research and Codes (Netherlands)
DAfStB the German Committee for Structural Concrete
DBV DEUTSCHER BETON‐ UND BAUTECHNIK‐VEREIN German Society for concrete and Technology
DT Diagonal tension ‐ Mode of Failure
e.g. exempli gratia, for example
EC 2 Eurocode 2
EN Euro Norm
F Failure in Flexure
FRC Fibre Reinforced Concrete
FRP Fiber Reinforced Polymer
FT Flexural Tension ‐ Mode of Failure
i.e. id est, that is
IB Information Bulletin
JCI Japan Concrete Institute
JSCE Japan Society of Civil Engineers
NA Failure Mode not Available
PC Plain Concrete
PCA Portland Concrete Association (U.S.A.)
PPR Partial Prestressing Ratio
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RC Reinforced Concrete
RILEM
Réunion Internationale des Laboratoires et Experts des Matériaux, Systèmes de Construction et Ouvrages; International Meeting of Experts and Laboratories for Materials, Building Systems and Structures
S Shearing ‐ Mode of Failure
SC Shear Compression ‐ Mode of Failure
SCA Swedish Concrete Association
SLS Serviceability Limit State
ST Failure in Shear Tension
TC Technical Committee
ULS Ultimate Limit State
UNI Ente Nazionale Italiano di Unificazione, Italian Organization for Standardization
UTT Uni‐axial Tensile Test
WC Web Crushing ‐ Mode of Failure
WST Wedge Splitting Test
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Chapter 1
Introduction
1.1 Background
In the second half of the XIX° century there were the first patents of the reinforced concrete, when the Parisian gardener Joseph Monier incorporated the metal cage used to shape his flower pots and he understood that this strengthens the concrete in tension. He got is patents in the 1867.
After this first experience concrete‐steel other products started to be realized in this manner: pipes, tanks, flat and curved slabs, stairs etc... In 1855, at the Universal Exhibition in Paris a small boat, built by the French lawyer J. L. Lambot, with a metal structure covered with concrete was exposed (Brencich A., 1992).
In the year of 1874, 19 years later the Lambot’s boat, A. Berard first patented fibre Reinforced Concrete (Balaguru et al., 1992).
Since then a new concept of discrete reinforcement done with “fibres” born and a lot of other fibres, than the steel one, were tried like silicon, carbon, ceramics, glass, nylon, polypropylene, asbestos, silicon carbide etc..
For long time the FRC was useless due to its high material costs, missing theoretical knowledge and simultaneous development of the RC. But when, at the end of 1950’ and beginning of 1960’, Romualdi, Batson and Mandel published papers about the fracture mechanics design approach for FRC, it became to draw the modern countries (Romualdi and Batson, 1963; Romualdi and Mandel, 1964). The modern era of research of FRC began interesting more and more users all over the word. Since the beginning its most common utilization is in the field of shotcrete applications.
The expression “fiber‐reinforced concrete” is by ACI 116 (2000) and the definition of the Italian National Agency for Standardization states: “composite material made from concrete base in which a fibrous widespread and evenly distributed reinforcement is embedded” (UNI 11039‐1, 2003).
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There are mainly four kinds of FRC: (1) Steel Fibre Reinforced Concrete (SFRC), (2) Glass Fibre Reinforced Concrete (GFRC), (3) Natural Fibre Reinforced Concrete (NFRC) and (4) Synthetic Fibre Reinforced Concrete (SNFRC).
Today, after realizing that asbestos is harmful to one’s health and the establishment of glass fibre is mostly limited to the production of cladding material, the better known is the SFRC. As more experience is gained with SFRC, thanks to its applicability in the field of the civil engineering construction, more applications and data are accepted by the engineering community. SFRC is widely used in structure where fibre reinforcement is not essential for integrity and safety (i.e. slabs on grade, rock slope stabilization and repair) but then the industry and the researchers are testing fibres as substitutes of the shear reinforcement and they are trying to cover the lack of codes.
It is now recognized that the paramount effect of fibres is that it enhances the post‐cracking behaviour and the toughness – i.e. the capacity of transferring stresses after matrix cracking and the tensile strain at rapture‐ rather than the tensile strength. The addition of fibres also slightly improves compressive strength, elastic modulus, crack resistance, crack control, durability, fatigue life, resistance to impact and abrasion, shrinkage, expansion, thermal characteristics, and fire resistance (ACI 544, 1996).
The first works of Romualdi, Batson and Mandel (Romualdi and Batson, 1963; Romualdi and Mandel, 1964) were focused on the thought that the fibres improve the tensile strength and delay the widening of microcracks; they found confirmation during the interpretation of the indirect test methods (i.e. splitting test and flexural test that highlight the increase of toughness) used to determine the tensile strength.
On the other hand, also new modern research postulated the influence of the fibres on the delay of the widening of microcracks like Nelson et al. (2002) and Lawler et al (2003).
From a practical viewpoint, however, the use of steel fibres became attractive in case where they can completely replace bar‐type shear reinforcement. The research points out that this is possible: FRC is able to replace the minimum quantity of stirrups request and to well perform connection between slabs and columns (De Hanai et al, 2008).
Considerable research, development, and applications of FRC as shear reinforcement
are taking place throughout the world. The numerous research papers, articles, international symposia and state‐of‐the‐art demonstrate the increasing interests on the industry and the potential business of FRC development. The ACI Committee 544 published its first state‐of‐the‐art report in 1973. RILEM’s technical committee 19‐FRC on fibre reinforced cement composites published a report, too. The latest Fib’s Model Code 2010, intended to serve as a basis for future codes and an operational document for normal design situations, gives an extensive state‐of‐the‐art regarding SFRC and their shear capacity.
Despite the lack of codes, the literature is full of authors that, in evaluating their experimental data, have proposed different analytical equations to estimate the shear capacity of FRC; each theoretical model agrees well with the experimental data from which the model equations are derived but they don’t show the same degree of agreement when applied to other published data.
The evaluation of the shear capacity could follow two different thoughts: one, most common in the past but still utilized today and recommended by RILEM (2000a), that estimates the contribution of the fibres to the shear capacity as an independent addend (see Narayanan & Darwish (1987), RILEM TC 162‐TDF (2003), DAfStB guidelines (2011), Etc.) and the other thought that states the contribution of concrete and fibre are
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coupled and, thus, must be solved simultaneously considering the toughness of the material (Fib Bulletin 57, 2010).
1.2 Scope of the Thesis
Today fibre reinforced concrete is in its fourth decade of development, after Romualdi et al. (1963 and 1964) research, and it has established itself as one of the major building material but nevertheless compared with its high performances it is still not widely used, not common in building structures with an inexplicable lack of national and international standards able to treat it in a systematic manner.
The interests on fibre reinforced concrete are increasing day by day considering the economics advantages that can arise from its use like the substitution of the transversal reinforcement and the earn in terms of saving labour time and increasing durability of the structures.
The scope of this thesis is to compare the most used formulas for the calculation of
the shear strength; they well agree with the experimental data from which the model equations are derived but they don’t work as well as when applied to other data; a further scope is to emphasise the difference between them.
Last but not least the aim of the work is to point out the formula that better performed shear in a fibre reinforced beam considering the different key‐parameters.
1.3 Limitation
To deal with such a huge topic, several limitations have to be set. There are a lot of different fibres (glass, steel, natural, synthetic, etc.) and even the
hybrid combination of metallic and non‐metallic fibres can offer potential advantages in improving concrete properties as well as reducing the overall cost of concrete production in this work, we will go exclusively through the Steel Fibre Reinforced Concrete abbreviated in SFRC. Section 2.1 shows all the varieties of fibres but normally the samples contain steel end‐hook, crimped or straight fibres.
Herein, only beams with flexural reinforcement and fibres are analysed while beams with fibres, shear reinforcement and flexural reinforcement are not considered.
The samples investigated exhibits a strain‐softening behaviour and all the formulas are designed for this kind of material.
With regard of the compressive strength of the samples it is worth to stress that, from the beginning, it is not possible to make a categorical exclusion of some compressive classes of concrete. The decision of which compressive strength can be
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accepted depend on the limitations of the formulas (if present) but at first we must define how the concrete can be classified. Normally it is divided into two big categories: “normal strength” and “high strength”.
The perception of what level of compressive strength constitutes “high strength” has been continually revised upwards over the past 20 years or so and may well continue to rise in the near future. A simple definition would be “concrete with a compressive strength greater than that covered by current codes and standards”. In the UK this would include concrete with a characteristic compressive cube strength of 60 MPa or more, but Eurocode 2 already includes concrete with characteristic cube strengths up to 105 MPa and Sweden and German had risen this limit up to 115 MPa, even this simple definition is not really adequate. Therefore for the purposes of this thesis, concrete with compressive (cube) strength smaller than 105 MPa will be considered as “normal strength”.
This thesis considers mainly normal strength but even a minority of high strength concrete is gathered in order to investigate their shear behavior.
The formulas utilized come from the European setting (Germany, Italian and Swedish guidelines) even if they are well‐known all over the world.
1.4 Outline of the Thesis
When specimens are tested there are a lot of parameters that influence the results, especially when the formulas used have an empirical background and they had been performed on a limited bunch of samples.
This thesis consists on graphs that show the ratio of the shear strength obtained from the test and the shear strength coming from the formula related to the key‐parameters that influence the shear response of concrete members with special emphasis of FRC toughness and size effect.
The thesis consists of five chapters. In Chapter 2 a theoretical background is given to better understand the subject
treated in the next chapters; FRCs are described with their geometrical and physical properties and the mechanisms of crack formation and propagation are explained. In addition different test methods are presented.
In Chapter 3 the importance of the shear strength is pointed out and four different alternatives to quantify it are presented, described and theoretical notions are given for each of the four alternatives.
Chapter 4 is the soul of the thesis. After a brief introduction about the importance of the research in the shear field, in Section 4.2 all the articles gathered are presented and their main features are explained. The following subchapter (Section 4.3) shows the domain investigated in this work according to the main characteristics (percentage of fibre, maximum aggregate size, type of failure, test methods, etc.). In Section 4.4 a method to obtain the post‐cracking parameters in case of absence of standard bending tests is proposed and also the data processing is therein explained. In the next subchapter (Section 4.5) the theoretical results coming from the formulas of Chapter 3 are compared with the test results and the trend of each formula in different field tests is
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evaluated. All the work is done paying particular attention to the different performance of the four alternatives.
Chapter 5 leads a discussion among the results obtained on the previous chapters and suggests the formula that best suits the specimens analysed; in addition future perspectives of the work are presented.
The thesis concludes with an alphabetical list of bibliographical sources cited in the text.
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Chapter 2
Theoretical Background
The continuous research in the construction field improves the properties and the qualities of the concrete available on the market.
First of all researchers tried to increase the strength but then they focused on other properties like the weight, the workability, the permeability, the ductility and the toughness.
The fibre reinforced concrete responds really well to the latter two needs. The main benefits of the inclusion of fibres in hardened concrete can be appreciated
in the post‐cracking state, where the fibres, bridging the cracks, contribute to increase (1) the strength, (2) the failure strain and (3) the toughness of the composite. In tension, SFRC fails only after the steel fibres break or are pulled out of the cement matrix (ACI Committee 544, 2002).
2.1 Fibre Reinforced Concrete: Material,
Geometries and Physical Properties.
Fibre reinforced concrete is a concrete containing dispersed fibres. The concept of discrete reinforcement finds its root in 1874 when A. Berard patented
it for the first time (Balaguru et al., 1992). Compared to the conventional reinforcement, the fibre reinforcement is:
‐ distributed throughout a cross section (whereas bars are only place where needed);
‐ relatively short and closely spaced (while bars are continuous and not as closely spaced);
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‐ not comparable, in term of area, to the one of the bars.
As stated before, the addition of fibres to plain concrete totally changes the post‐cracking behaviour leading to a softening branch after the peak load. Moreover the fibres ‐bridging the cracks‐ contribute to increase the strength, the failure strain and the toughness of the composite.
The toughness is significantly increased obtaining, thus, a really versatile construction material; but, fibre reinforced concrete becomes more and more attractive when it is able to totally replace transversal reinforcements that are one of the more labour‐cost activities necessary for concrete structures. This technology also improves the durability of concrete structures.
The fibre reinforced concrete is not a recent concept, but, due to the lack of national and international standards, it is not used in really significant structural applications. Nowadays it is mainly used in non‐structural elements like:
‐ slabs and pavements in which fibres are added as secondary reinforcement and with the aim of withstanding the crack induced by the humidity and the temperature variation (crack for which the conventional reinforcement is not effective);
‐ tunnel linings, precast piles (that have to be hammered in the ground) and blast resistance structures that have to carry high load or deformation;
‐ thin sheets or elements with complicated shape where the conventional reinforcement cannot be used and, in any case, due to the thin concrete cover, it will be difficult to preserve from corrosion.
In most of the applications, the function of the fibres does not consist into increasing the strength (although an increase of tensile strength is a consequence) but just to control and delay both widening cracks and the behaviour of the concrete after the crack of the matrix.
In a simple view, the elements involved in the system FRC are three: the concrete, the
bond and the fibres.
The Concrete
The moderate addition of fibres has no effect on the mechanical material properties of plain concrete before cracking unless the fibre dosage exceeds around 80 kg/m3 (Technical Report No. 63). The design of such high performance composites is not covered in this thesis. In consequence the addition of fibres does not change the compression strength after 28 days, that is the main characteristic used to classify concrete (see Table 2.1).
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Table 2.1 ‐ Strength Concrete Classification (PCA, 1994).
Conventional
concrete
High‐strength
concrete
Very high
strength concrete
Ultra high
strength concrete
Strength, MPa (psi) < 50 ‐7250
50 ÷100 (7250÷14500)
100÷150 (14500÷21750)
> 150 ‐21750
Water‐cement ratio > 0.45 0.45÷30 0.30÷0.25 < 0.25
Chemical Admixtures Not necessary WRA/HRWR HRWR HRWR
Mineral Admixtures Not necessary Fly ash Silica Fume Silica Fume
Permeability coefficient
(cm/s) > 10‐10 10‐nov 10‐dic < 10‐13
Freeze‐thaw protection Needs air entrainment
Needs air entrainment
Needs air entrainment
No freezable water
However, fibre addition causes a less brittle failure; this is due to the fact that
compression failure of concrete is related to its tension failure, since tensile stresses cause growth of the pre‐existing microcracks in the concrete (and tanks to the fibre bridging the stresses continue to increase).
Nevertheless, the addition of fibres changes the consistence. The mutation depends upon the aspect ratio of the fibres that is defined as the ratio of its length lf to its diameter df (when cross section is not circular, diameter is substituted by equivalent diameter). It is physically difficult to include fibres with an aspect ratio of more than 50 because concrete contains about 70 % by volume of aggregate particles which, obviously, cannot be penetrated by fibres.
Longer fibres of smaller diameter will be more efficient in the hardened FRC, but will make the fresh FRC more difficult to cast. This explains why the mix design of FRC often requires additives for obtaining the consistence needed.
Another important factor that could not be ignored is the maximum aggregate size; every time that a concrete matrix is designed, particular attention should be paid to the determination of this parameter that influences the phenomenon of interlock.
The Bond
The bond between the matrix and the fibres influences the performances of the FRC. The value of the bond strength for a straight round steel fibre rarely exceeds 4 MPa, but, with mechanical deformation of the fibre or devices (anchors), the bond slip can be avoided during the fibre failure.
It is really difficult to predict the behaviour of the fibre; this is because it depends on both fibre shape and concrete strength. Therefore it is not possible to give a generalized “bond strength” which could be used in numerical calculations. The only certainty in the fibre behaviour is that the fibre bond lies between fibre slip and fibre failure.
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The Fibres
Based on industrial sources, the amount of fibres used worldwide is estimated at 300,000 tons per year, and is projected to increase. In North America, the growth rate has been placed at 20% per year. However, it should be pointed out that FRC remains a small fraction of the amount of concrete used each year in the construction industry (Li, V. C., 2002). As shown in Table 2.2, there are many different types of fibres in commerce with different properties. On the whole, steel fibres remain the most used fibres (50 % of total tonnage used), followed by polypropylene (20 %), glass (5 %) and other fibres (25 %) (Banthia, 2008).
Table 2.2 ‐ Properties of fibres used as reinforcement in concrete (Banthia, 2008).
Fibre type
Tensile
strength
(MPa)
Tensile
modulus
(GPa)
Tensile strain
(%) Fibre
diameter
(μm)
Alkali
stability
(relative) min max
Asbestos 600÷3600 69÷150 0.1 0.3 0.02÷3 excellent Carbon 590÷4800 28÷520 1 2 7÷18 excellent Aramid 2700 62÷130 3 4 11÷12 good Polypropylene 200÷700 0.5÷9.8 10 15 10÷150 excellent Polyamide 700÷1000 3.9÷6 10 15 10÷50 ‐ Polyester 800÷1300 up to 15 8 20 10÷50 ‐ Rayon 450÷1100 up to 11 7 15 10÷50 fair
Polyvinyl Alcohol 800÷1500 29÷40 6 10 14÷600 good
Polyacrylonitrile 850‐1000 17÷18 9 19 good
Polyethylene 400 2÷4 100 400 40 excellent Polyethylene pulp
(oriented) 400 2÷4 100 400 1‐20 excellent
High Density Polyethylene 2585 117 2.2 38 excellent
Carbon steel 3000 200 1 2 50÷85 excellent Stainless steel 3000 200 1 2 50÷85 excellent AR‐Glass 1700 72 2 12÷20 good
The type of fibres to be used depends mainly upon the application of the FRC.
Asbestos fibres have been used for a long time in pipes and corrugated or flat roofing sheets. Glass fibres find their application as reinforcing materials in automotive and naval industries or like cladding materials. Vegetable fibres have been used in low cost buildings. Synthetic fibres like polyethylene (PE), polypropylene (PP), acrylics (PAN), polyvinylacetate (PVA), polyester (PES) and carbon are incorporated in the cement matrix mainly for reducing plastic shrinkage cracking and for increasing the resistance to fire spalling.
At any rate, the most interesting fibres in the building materials sector are those made out of metal. They improve the toughness and reduce the crack widths. Surely, along the years, thanks to the new technology, their shape is changed and today modern steel fibres have higher slenderness and more complex geometry.
This master thesis deals only with steel fibres that are more developed into structural applications.
In particular, metallic fibres are made of either carbon steel or stainless steel and
their tensile strength varies from 200 to 2600 MPa.
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The European Standard EN 14889‐1:2006 (CEN, 2006) says that “[...] steel fibres are straight or deformed pieced of cold‐drawn steel wire, straight or deformed cut sheet fibres, melt extracted fibres, shaved cold drawn wire fibres and fibres milled from steel block which are suitable to be homogeneously mixed into concrete or mortar [...]”. Moreover, in that norm, steel fibres are divided into five general groups and are defined in accordance with the basic material used for the production of the fibres according to:
‐ Group I, cold‐drawn wire; ‐ Group II, cut sheet; ‐ Group III, melt extracted; ‐ Group IV, shaved cold drawn wire; ‐ Group V, milled from blocks.
There are also many other classifications made by other standard bodies that consider different fibres features. The Japanese Society of Civil Engineers (JSCE) has classified steel fibres according to the shape of their cross‐section:
‐ Type 1: Square section; ‐ Type 2: Circular section; ‐ Type 3: Crescent section.
ASTM A 820 provides a classification for four general types of steel fibres based upon the product used in their manufacture:
‐ Type I—Cold‐drawn wire; ‐ Type II—Cut sheet; ‐ Type III—Melt‐extracted; ‐ Type IV—Other fibers.
For steel fibres, three different variables are used for controlling the fibres performance: (1) the aspect ratio; (2) the fibre shape and surface deformation (including anchorages that increase their performance) and (3) the surface treatments (Löfgren, 2005).
For fibres, in order to be effective in cementitious matrices, it has been found (by
both experiments and analytical studies) that they should have the following properties (Naaman, 2003): (1) a tensile strength significantly higher than the matrix (from two to three orders of magnitude); (2) a bond strength with the matrix preferably of the same order as, or higher, than the tensile strength of the matrix; (3) an elastic modulus in tension significantly higher than that of the matrix (at least three times) and (4) enough ductility so that the fibre does not fracture due to its abrasion or bending. In addition, the Poisson ratio (ν) and the coefficient of thermal expansion (α) should preferably be of the same order of magnitude for both fibre and matrix (Löfgren, 2005).
A great variety of fibre shapes and lengths are available depending on the
manufacturing process as it is represented in Figure 2.1.
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Figure 2.1‐ Variety of fibres (Di Prisco, 2007 & IB39, 2009).
The cross‐section of an individual fibre could be circular, rectangular, irregular, flat or any substantially polygonal shape. Mechanical deformation along their length can improve the bond strength producing smooth, indented, deformed, crimped, coiled and twisted fibres. Also different shaped ends could improve the bond strength (end‐paddles, end‐buttons, end‐hooks or other anchorages). Steel fibres can also have coatings like zinc (for improving corrosion resistance) or brass (for improving bond characteristics).
Fibre length ranges from 10 to 60 mm with equivalent diameters between 0.5 and 1.2 mm (0.15‐0.40 mm thickness and 0.25‐0.90 mm in width) and an aspect ratio less than 100 (typically ranging from 40 to 80).
While the straight fibre is only anchored in the matrix by friction and chemical adhesion, all other fibers, which have a deformation along their axis, develop greater bond properties. In order to utilize the usually high tensile strength of fibres it is important that fibres are well anchored in the concrete matrix.
Hughes and Fattuhi (1976) indicated that crimped fibres show a better workability compared to straight or other forms of fibers for a similar fibre aspect ratio (Minelli, 2005).
The orientation and the distribution of fibres are worth mentioning. They play an
important role for the mechanical performance of the FRC. Body random orientation is characterised by equi‐probable and unlimited (free) distribution of short fibres throughout the body of the concrete (in three dimensional space). Plane random orientation occurs in thin walled elements (flat sheets, plate, thin walls, etc.). The smaller the cross‐section is, the more restricted the possibilities of free orientation of the fibres and a three dimensional bulk are. The mechanical behaviour must include the orientation of the fibre in order to quantify the fibres bridging the crack. For this purpose, it is common to define the fibre efficiency factor (b) as the efficiency of bridging, in terms of the amount of fibres bridging crack, with respect to orientation effects.
The fibre content in a mixture, when steel fibres are used, usually varies between
0.25 and 2 % by volume, i.e. from 20 to 160 kg/m3. Normally lowest percentage is used
13
in slabs on grade while the upper value is used for structurally more complicated applications.
Nowadays, it is believed that a proper characterization of fibers should be undertaken by considering the post‐cracking behaviour itself rather than the geometry and the amount of fibres provided in the matrix. In fact, the same amount of fibres in different types of concrete give quite different post‐cracking behaviours of the composites (Minelli, 2005).
2.2 Mechanisms of Crack Formation and
Propagation
The first crack that appears in a beam normally is in correspondence of the region where the bending moment is maximum and the shear force is small; the cracks are aligned whit each other and, more or less, perpendicular to the flexural stress; they are, therefore, in mode I condition. As visible in a normal load‐deflection plot, at a certain point, the behaviour from linear becomes non‐linear (Figure 2.2, Karyhaloo, 1993).
Figure 2.2 – A longitudinal reinforced concrete beam in three‐point bending. First flexural cracks appear in the region of maximum bending moment (a) accompanied by nonlinearity in load‐deflection response (denoted by an asterisk in (b)). More flexural cracks appear away from the region of maximum moment under increasing load (c), and a dominant crack propagates towards load point until ultimate failure by crushing of compressive concrete (d) (Karihaloo, 1993).
Increasing the load more cracks are formed away from the region of maximum bending moment and also the non‐linearity increases; these further cracks are along region where the shear forces are no longer small, for this reason they are in mixed mode condition (mode I + II), but always normal to the major tensile principal stress.
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These cracks, growing, follow a bent path (due to mode II) and they are no more parallel to the direction of the applied load (sometimes these are incorrectly called shear cracks). Mode II is also responsible of the sliding of the crack faces. Longitudinal bars, transversal bars and fibres counteract the opening of the crack, but it is difficult to separate their effects for quantifying the contribution of each element.
Further increasing of load does grow a dominant crack towards the reduced compression zone until failure take place. The response is generally ductile.
Changing the beam dimensions or the reinforcement, another kind of collapse could be appreciated that consists into the formation of a secondary crack which crosses the first flexural cracks (Figure 2.3). This mode of failure (due to the combination of shear and normal stresses) is often sudden and unstable and it is called the diagonal tension mode (or, incorrectly, shear mode).
Figure 2.3‐ A secondary crack crossing the flexural cracks leads to sudden brittle failure (Karihaloo, 1993).
The sliding displacement of the inclined crack faces bring into play the aggregate interlock which gives a contribution to the total shear strength. A further contribution comes from the longitudinal reinforcement that acts as a dowel. Bond stresses also act between reinforcement and surrounding concrete caused by slip due to the opening of the crack (Karyhaloo, 1993). The contribution of dowel action, aggregate interlock and bond stress due to slip are very hard to quantify and even fracture mechanics is not able to describe the crack propagation in a correct way.
The most significant effect of the presence of steel fibres is the cracking behaviour; the beams made out of FRC display an increased number of both flexural and shear cracks at closer spacing than the corresponding beams without fibres. Normally, also a reduction of spalling in the vicinity of the support and bond cracking can be found. The addition of fibres could (not always) lead to achieve the flexural failure; but, although fibres beneficially and substantially improve the crack and deformational behaviour as well as the ultimate strength, this does not always happen.
The tensile fracture mechanism of concrete is a complex phenomenon and still it has not been fully elucidated. The post‐cracking behaviour, as shown in Löfgren (2005), is affected by two different mechanisms:
‐ Aggregate bridging that is always present in the plain concrete ‐ Fibre bridging that contributes to energy dissipation in FRC concrete.
The fibre bridging is always predominant, but the final bearing in uni‐axial tension is the combination of both the two mechanisms; aggregate bridging decays to zero for a crack opening of around 0.3 mm. The addition of fibres increases the work of fracture (represented by the area under the stress‐crack opening curve) and the critical crack
15
opening (from approximately 0.3 mm to half the fibre length – for steel fibres usually 10 to 30 mm).
Aggregate Bridging
Aggregate bridging is the major toughening mechanism for plain concrete; an aggregate that bridges the crack until 0.3 mm works almost in the same way as fibres do. The concrete crack bridging is the coalescence of microcracks in the matrix due to the development of bond cracks between aggregate and matrix and the frictional pull‐out of aggregates (Löfgren, 2005). In plain concrete, in addition to aggregate bridging, many different mechanisms are involved:
‐ crack shielding: the nucleation of many microcracks, around the tip of a propagating crack, has a significant influence on the propagation of the main crack. It reduces the stress intensity factors of the main crack (Loehnert et al., 2007) ;
‐ crack deflection: at the interface of dissimilar materials the crack can arrest or advance by either penetrating the interface or deflecting into the interface (He et al., 1989);
‐ crack surface roughness‐induced closure: the mechanisms of crack closure arising from microscopic roughness of the fatigue fracture surfaces are not fully understood. It’s known to strongly influence fatigue crack growth rates (Várkoly, 2001);
‐ crack tip blunted by void; ‐ crack branching: the tip of the crack shares in two different branches, the
main and the secondary crack tip.
Figure 2.4 ‐ Some toughening mechanisms in plain concrete (Shah et al., 1995).
The toughening mechanisms can be divided into crack frontal, crack tip and crack wake mechanisms (where the aggregate bridging , but more fibre bridging, mechanism is developed); they can also be classified in long‐range effect over a large crack extension distance (e.g. microcracking and aggregate and fibre bridging) and short‐
16
range effect over a small crack extension distance (e.g. crack deflection, bowing and pinning).
The major toughening mechanism of plain concrete is the aggregate bridging and a lot of experimental and numerical observations support this hypothesis. The stress‐crack opening relationship has been investigated changing all the parameters that influence it. In particular, several researchers have investigated the effects that aggregates play changing the type, the size, the shape and the volume fraction (see Tasderi and Karihaloo, 2001; van Mier, 1991, 1997; Giaccio and Zerbino, 1997). After these numerous studies the uniaxial behaviour can be depicted as in the Figure 2.5. It has been observed that, even before any stresses have been applied, pre‐existing microcracks exist within the concrete, and this is due to the internal restrain caused by the aggregate and both shrinkage and thermal deformations. With the development of externally caused stresses, the microcracks start to grow, at first between the cement paste and the aggregates (A) and later also into the mortar (B). After the peak stress (C), microcracks propagate in an unstable manner and crack localisation occurs; at this time macro‐cracks propagate through the specimen with the stress‐drop consequence (D). The toughening action of the aggregates and crack branching are responsible for the long softening tail (D‐E) observed during experiments.
Figure 2.5 ‐ Schematic description of the fracture process in uni‐axial tension and the resulting stress‐crack opening relationship (Löfgren, 2008).
The graph shown in Figure 2.5 can change shape considering lightweight concrete or high‐strength concrete because the aggregates may became the weak link and aggregate rupture may occur, which reduce the bridging effect and results in a more brittle fracture process.
Fibre Bridging
The fibre bridging, like the aggregate one, depends on many parameters and, for simplicity, an isolated fibre is investigated along a crack. The fibre contributes to dissipate energy thanks to: (1) matrix fracture and matrix spalling, (2) fibre‐matrix interface debonding, (3) post‐debonding friction between fibre and matrix (fibre pull‐out), (4) fibre fracture and (5) fibre abrasion and plastic deformation (or yielding) of the fibre.
17
Figure 2.6 ‐ (a) A schematic illustration of some of the toughening effects and crack front debonding, the Cook‐Gordon effect, and debonding and sliding in the crack wake. (b) Matrix spalling and matrix cracking. (c) Plastic bending (deformation) of inclined fibre during pull‐out – both at the crack and at the end‐anchor (Löfgren, 2008).
The mechanical behaviour of FRC depends surely on the amount of fibre (which shows benefits from 1 % until 15 %, for engineered cementitious composites ECC), on the orientation of the fibres and largely on the pull‐out versus load (or load‐slip) behaviour of the individual fibres. In particular, the pull‐out depends on the type and the mechanical/geometrical properties of the fibres, on the mechanical properties of the interface between fibre and matrix, on the angle of inclination of the fibre with respect to the direction of loading and on the mechanical properties of the matrix. A large amount of literature covers this subject.
The fibre pull‐out behaviour is the gradual debonding of an interface surrounding the fibre, followed by frictional slip and pull‐out of fibre.
The bond (responsible of the forces transmission between fibre and matrix) has different components:
‐ the physical and/or chemical adhesion between fibre and matrix; ‐ the frictional resistance; ‐ the mechanical component (arising from the fibre geometry, e.g. deformed,
crimped or hooked‐end); ‐ the fibre‐to‐fibre interlock;
Several pull‐out models exist, the simplest ignore the elastic stress transfer and the matrix deformation (e.g. Hillerborg (1980) and Wang (1989)) while other models assume elastic interfacial shear bond stresses that gradually change into a frictional forces because of the debonding of the interface (e.g. Gopalaratnam and Shah (1987)).
The debonding criterion can be described with two different approaches:
‐ strength‐based criterion (or stress‐based) where it is assumed that the debonding initiates when the interfacial shear stress exceeds the shear strength;
‐ fracture‐based criterion that considers the debonding zone as an interfacial crack together with the evaluation of fracture parameters and energy consideration;
18
Figure 2.7 ‐ Different debonding models for fibre pull‐out (Löfgren, 2008).
Once debonding has taken place, stress transfer develops owing to frictional resistance that, in its turn, can be described, as depicted in Figure 2.7, with the following different relationships:
‐ constant friction ‐ decaying friction (or slip softening) ‐ slip hardening friction.
In the literature, there can be found huge differences on the interfacial shear bond strength, ranging from 1 up to 10 MPa (Minelli, 2005); moreover interfacial shear friction capacity (ranging from 0,5 to 20 MPa) makes the correct interpretation of the pull‐out test difficult (Löfgren, 2005).
The dissipated energy is equal to the area beneath the load‐displacement (slip) curve. The pull‐out energy (both debonding and friction) increases with the embedment length, unless the embedment becomes too long and the fibre breaks, and depends on the end of the fibres (crimped, straight, hooked, etc.) as shown in Figure 2.8.
Figure 2.8 ‐ Typical fibre pull‐out relationship between end‐slip and load for straight and end‐hooked fibre (Löfgren, 2008).
The behaviour of different fibres during the pull‐out test depends on both their mechanical and geometrical properties, as well as on their chemical affinity to the
19
matrix. The pull‐out behaviour of a hooked‐end fibre differs from straight, crimped/corrugated, indented fibres or the one named Torex (with polygonal cross section twisted along its axis).
When the fibre is not perpendicular to the concrete block, the pull‐out energy is also influenced from the angle of inclination, and, in this case, it is related to the matrix strength. In particular, it increases for flexible fibres (e.g. synthetic) and stiff but ductile fibres (e.g. steel, but only up to about 45°, Löfgren, 2008) while it decreases for brittle fibres (e.g. carbon).
It is well known that a fibre‐reinforced concrete consists of several fibres which, in most cases, have random orientation; throughout this observation, Bentur and Mindess (1990) explained that the process of debonding and pull‐out is quite different in an actual fibre‐reinforced specimen compared to a simple pull‐out test on a reinforcing bar. It is also important to note is that the pull‐out behaviour and maximum load depend upon the spacing of the fibres.
Aggregate and Fibre Bridging, a Cooperative Mechanisms.
The two phenomena explained above during the cracks formation act simultaneously; fibre act as an additional bridging mechanism; the final result is that the critical crack opening increases by a factor larger than 10 and, consequently, the fracture energy increases too.
To summarise the fibre‐reinforced concrete behaviour is a combination of the effects caused by aggregate and fibre bridging where the former has a relatively short working range in comparison with the latter.
Figure 2.9 depicts a schematic description of the effect of the fibres on the fracture process in uni‐axial tension; three distinct zones are pointed out (Löfgren, 2005):
‐ a traction‐free zone, which occurs for relatively large crack opening; ‐ a bridging zone , where stress is transferred by fibre pull‐out, and
aggregate bridging; ‐ a zone of microcracking and microcrack growth.
Figure 2.9 ‐ Schematic description of the effect of fibres on the fracture process in uni‐axial tension (Löfgren, 2008).
The contribution from fibre bridging comes gradually and it is not until crack opening reaches, at least, 0.05 mm that it has any major influence.
Obviously, the kind, the mechanical characteristic, the percentage and the aspect ratio of the fibres can change the shape of this graph.
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2.3 Test Methods
It should now be clear that the properties of an FRC could not be represented by a single characteristic (compression strength) as it happens for normal concretes. In particular, seeing that the addition of fibres increases significantly the toughness leaving the compression strength almost unchanged, for fibre‐reinforced concrete some sort of toughness property is required, and thus other tests have to be used to characterise it.
The moderate addition of fibre (<1 %) does not change significantly the compressive strength and the pre‐peak properties, so, as suggested by the RILEM Technical Committee TDF‐162, “Test and design methods for SFRC”, the compressive strength of SFRC should be determined by means of standard tests that, in the case of Eurocode 2 (CEN, 2006), could be done on either concrete cylinders or cubes. Furthermore, the concrete is classified according to the same strength classes as in Eurocode 2 (CEN, 2006). As it is shown in Figure 2.10, in case of addition of microfibres and for high fibre volumes (>1 %), it is possible to appreciate an increase of the compressive strength.
Figure 2.10 ‐ Schematic description of the behaviour of concrete and FRC in compression (Löfgren, 2008).
In the last forty years the researchers experienced different methods to characterise the tensile behaviour of a fibre‐reinforced concrete like determining dimensionless toughness indices (as prescript in ASTM C 1018) as well as, determining the residual flexural strength for specified deflection or measuring the flexural strength.
The presence of fibres mainly affects ductility and this influence is strongly dependent on the fibre content and fibre type (Fib Bulletin 57, 2010).
The main test‐set‐ups used are:
‐ uni‐axial tension test or direct tensile test (UTT); ‐ flexural test:
Three Point Bending Test (3PBT) (notched/unnotched): it is the most widespread method on beam/prism specimens; it is suggested also by RILEM TC 162‐TDF (2002b) for SFRC;
Four Point Bending Test (4PBT) (notched/unnotched); ‐ panel test or plate test (used for shotcrete and for specific load condition
that simulates a design situation in a real structure); ‐ wedge‐splitting test (WST) that sometimes is an alternative to the UTT and
the 3PBT.
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The RILEM Technical Committee TDF‐162, “Test and design methods for SFRC”, proposed the UTT and the 3PBT on notched beams.
Uni‐axial Tensile Test
As proposed in the RILEM Recommendations (2000b) the influence of the fibres on a strain softening FRC, after the crack, can be determined directly in terms of stress‐crack opening (‐) relation with the uni‐axial tensile test (UTT). The method, however, is not intended for the determination of the tensile strength that is recommended to be determined independently (Døssland, 2008).
Cast specimens have to follow geometrical condition as depicted in Figure 2.11.
Figure 2.11 ‐ UTT specimen as proposed by RILEM TC‐162 TDF.
This test consists of a controlled tensile displacement imposed at the end of a notched cylindrical specimen; it is characterized by a high rotational stiffness (provided by four turnbuckles).
Typical results, from the UTT experiments, show, in the (‐) graph, that in the pre‐peak region the curve is linear up to the stress level of about 70 % of the peak‐stress, and after this point the curve deviates and a non‐linear behaviour can be observed. The magnitude of the deformation in the pre‐peak non‐linear zone is quite small but after the peak‐stress a softening response is observed and large displacements arise.
Figure 2.12 ‐ Typical results from the UTT experiments: (a) stress‐deformation response in the pre‐ and immediate post‐peak behaviour; (b) Stress‐crack opening relationship (Löfgren, 2005).
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This test is rarely utilized due to the complex test set up. The test is also time‐consuming and difficult to carry out; it demands highly trained and experienced personnel. Other tests (e.g. Wedge Splitting Test, WST, and Bending Test, BT) are more economical without compromising the reliability of the method. They are, normally, used for the determination of the (‐) relationship in advanced design procedures.
Three Point Bending Test (3PBT)
In the three points bending test proposed by RILEM, the tensile behaviour is evaluated in terms of the load bearing capacity at a certain deflection or crack mouth opening on a notched specimen (RILEM TC 162‐TDF, 2002a).
This test method can be used for determination of: (1) the limit of proportionality, (2) the equivalent flexural tensile strength and (3) the residual flexural tensile strength. Furthermore, it evaluates the flexural performance of toughness parameters derived from fibre‐reinforced concrete in terms of areas under the load‐deflection curve. When toughness is determined in terms of areas under the load‐deflection curve, it is an indication of the energy absorption capability of the particular test specimen during deformation, and, consequently, its magnitude depends directly on the geometrical characteristics of the test specimen and the loading system (ASTM C1018‐97).
Figure 2.13 shows the specimen geometry and loading conditions in the three point bending test according to RILEM Recommendation (RILEM TC 162‐TDF, 2002a). The suggested standard test specimen is not intended for concrete with steel fibres longer than 60 mm and/or aggregates larger than 32 mm; the beam are cast in moulds, cured and notched using wet sawing.
Figure 2.13 ‐ Test set‐up for the three‐point bending test in notched beams according to RILEM TC 162‐TDF.
The results of this standard test method are dependent on the size of the specimen, it follows that those obtained using a certain size moulded specimen may not correspond to those obtained from larger/smaller moulded specimens, concrete in large structural units or specimens sawn from such units.
This difference may occur because the degree of preferential fibre alignment becomes more pronounced in moulded specimens containing fibres that are relatively long compared with the cross‐sectional dimensions of the mould (ASTM C1609‐ C1609M‐10).
The advantage of using notched specimen is that the crack will form in a predefined position and not in the weakest section. Consequently, notched beam tests tend to give higher values of flexural strength than un‐notched beam tests but with a lower coefficient of variation.
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The tests are normally performed under crack mouth opening displacement (CMOD).
Four Point Bending Test (4PBT)
Appendix of the JCI guidelines (JCI 2007) presents a method of calculating the tensile strength and ultimate tensile strain of fibre‐reinforced cementitious composites using the maximum bending moment and curvature.
Corresponding methods of the JCI bending test are used in the Norwegian design rule draft, in the Italian standards (UNI, 2003), in the German guideline (DAfStB, 2011b), in the Swedish Concrete Association design rule and in the design guidelines for Dramix steel fibres.
Normally, as displayed in Figure 2.14, the dimension of the specimen are 150 x 150 x 600 mm (these are the exactly dimensions according to DAfStB (2011b) and CNR DT 204‐2006) and, similarly to the 3PBT, can be used un‐notched or notched beam, depending on the standard to which they refer; the beams are loaded up to failure under four point bending across a span of 450 mm. The bending moment along the span, in the middle of the two point loads, is constant; this is, on one hand, an advantage because the crack will appear at the weakest section (incorporating the effect of variation in the material’s strength) but, on the other hand, a disadvantage because the position of the crack cannot be predicted making harder the measurement of the crack opening deflections.
Figure 2.14 ‐ (a, b) Specimen geometry and (c, d) testing setup for the larger and the smaller beams (Sorelli et al., 2005).
Toughness and the equivalent flexural strength can be calculated from the mid‐span deflection that is measured during the whole test.
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According to the Italian guideline, specimens can have different dimensions than those above‐mentioned; in order to obtain a more clear comparison between different specimen geometries, experimental results from bending tests are reported in terms of nominal stress N defined according to a linear stress distribution asEquation Chapter (Next ) Section 1Equation Chapter (Next) Section 1
sp
N 2
b b 0‐
FL
B H a (Eq. 2.1)
where F = force; Bb and Hb = beam thickness and depth, respectively; Lsp = span length; and a0= notch depth.
Experimental results from fibre reinforced concrete with low fibre contents are sensitive to the number of fibres in the cracked sections which have a higher degree of variation in the smaller surface areas, especially when notched specimens are adopted.
The fibre orientation has to be considered by cutting a block from the beam
specimen and making an average of the number of fibres crossing the two cross‐sections (only the longitudinal fibres). The block is sawn from the middle part of the beam between the twin loads at a minimum distance from the crack of 2/3 lf, where lf is the length of the fibre. It is important to highlight that this valuation does not consider that some fibres may be ineffective due to reduced anchorage. These data are totally missed in the specimens treated in Chapter 4.
Wedge Splitting Test (WST)
The wedge splitting test is interesting since it does not require sophisticated test equipment; it is also time‐and‐cost efficient, with good reproducibility. Despite all these good characteristics it is still not widely applied (Löfgren, 2008).
Figure 2.15 ‐ (a) Schematic view of the equipment and test set‐up and (b) photo of test set‐up (Löfgren, 2008).
Since the beginning it has had a wide range of application; in fact, it can be used to evaluate fracture properties, fatigue crack growth in high‐strength concrete, stress‐crack opening relationships for plain and fibre‐reinforced concrete.
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Figure 2.16 – Specimen with notches on the sides of the specimen to prevent horizontal cracks (Löfgren, 2008).
Figure 2.15 and Figure 2.16 clarify the geometry and the loading procedure; the specimen has a groove (for the application of the splitting load), a starter notch (for the crack propagation) and a guide notch (for preventing horizontal crack in case of high fibre volume fractions). Two steel plates with roller bearing are placed partly on top of the specimen and partly into the groove; the splitting force, Fsp, is applied through a wedging device. During the test, are monitored both the load in the vertical direction (Fv) and the crack mouth opening displacement on top of the specimen (CMOD); moreover, the load is applied in a deformation controlled way and Fv is related with Fsp.
For steel fibre‐reinforced concrete, a small number of references about specimen size and experimental interpretations can be found. In 2004, Löfgren provided some recommendations for using WST for FRC; in that research he pointed out that the specimen size, in order to avoid the wall effects and provide a larger fracture surface as well as reducing the scatter, should follow these recommendations: (1) the outer dimension of the specimen should be at least 3 times the fibre length and/or 5 times the maximum aggregate size; (2) the length of the ligament should be at least 1,5 times the fibre length and/or 5 times the maximum aggregate size.
The same research found that the WST, compared to UTT and 3PBT, has lower scatter, although the scatter in general is large for all methods.
The test methods used for the determination of the parameters included in the
formula presented in Chapter 3 will be deeply discussed in section 4.3.1.
Equation Chapter 3 Section 1
27
Chapter 3
Shear Capacity
A really short historical review is here necessary to better understand the importance of the shear design. Hennebique and Ritter at the end of the 19th century were the pioneers in the study of stirrups and shear. In 1908 Mörsch stated the simple or multiple truss system model where the concrete is the compressed strut and stirrups or bent‐up bars are the tensile member. Later Kupfer proposed a variable strut inclination (even smaller than 45 degrees within following limits: with α = inclination of the compressive strut to the axis of the member). Worthy to note are models developed by Kani (with is comb‐like mechanism), Leonhardt (that with the web compression failure giving the upper limit of shear resistance), Thürliman and Warlaven (who corroborated the positive influence of prestressing on the shear capacity). Moreover, Collins and Vecchio in 1986 developed the modified compression field theory for reinforced concrete elements. Nowadays, fib is working to find the best procedure for shear design based on earlier experiences with MC78, MC90 as well as MC2010.
The majority of the analytical models, introduced here before, have to be re‐arranged when steel fibres are added in concrete matrix, because they considerably influence shear behaviour as well as the shear capacity (fib Bulletin 57, 2010).
At any rate, along these decades, year by year, different experiments done by researchers all over the word, have tried to predict the shear capacity of members with and without transverse reinforcement.
As it can be seen in Chapter 4, each researcher after his own studies usually
proposed empirical relationship that is based on a limited batch of specimens with similar characteristics. This relationship fits really well with the set of shear test results from which they come from. In reality, when those formulas are applied to other specimens the results are not as satisfactory as expected.
The growing interest of the researcher community can be observed looking at the
number of papers on shear design published in ACI Journal since the beginning of the last century.
0 25 tgα 1.00.
28
Figure 3.1 ‐ Number of papers published in ACI Journal since the beginning of the last Century (Minelli, 2005).
Since catastrophic shear failure happened (like the failure of roof beams in Air Force warehouse on August 1955 in Ohio) the common aim was to fully understand and avoid the brittleness of these events (Minelli F., 2005).
In spite of the huge amount of material available in literature, when a shear scholar try to collect a systematic presentation of the influence of the basic parameters, like concrete strength, percentage of longitudinal reinforcement, shear length, volume of fibres, this turn out to be almost impossible to witness that the shear mechanism is difficult to be totally understood.
Kani, from the University of Toronto, in 1966 stated that “The primary reason for this
limited understanding of the problem of diagonal failure is the great number of parameters influencing the beam strength: grade of steel, percentage of steel, grade of concrete, shape of the cross section, shear arm ratio, type of web reinforcement, the type of loading […], the type of beam […], and prestress in the longitudinal, transverse and vertical direction which, of course, create additional parameters”.
Kani postulated the shear domain and investigated the influence of the reinforcement ratio. Moreover he theorized the “beam behaviour” for beams with a shear‐to‐dept ratio greater than about 2.5 (Kani, 1964).
Further researchers gave their contribute to the shear design and the most significant are presented in Chapter 4 (e.g. Narayanan, Darwish, Voo, Foster, Gilbert, Swamy, Bahia, etc.).
In the last decades particular attention was devoted to FRC that nowadays are largely
available in the market. They are more and more utilized thanks to the opportunity to totally substitute the transverse reinforcement, that require a time and cost consuming work and to the improved performance of FRC after cracking.
Several equations based on test data and theoretical analyses have been proposed for calculating the shear capacity of SFRC beam. These equations can be divided into two categories. The first category assumes that steel fibres give shear strength in addition of
29
the shear strength provided by the plain concrete and the stirrups that the fibre can totally substitute. This kinds of formulas have a basic format as follow:
c s f V V V V (Eq. 3.1)
Where Vf, Vc and Vs are the shear strength carried by, respectively, the fibres, the concrete and (when present) the stirrups.
The second category considers that the steel fibers directly influence the shear capacity of concrete (this influence is determined by appropriate tests e.g. split tensile cylinder and modulus of rupture tests) and it does not explicitly consider the characteristics of the fibres.
This kind of formula includes in the concrete member the characteristics obtained from the fibre addition and it is based on observation that fibres of the same type can achieve different results if added in different matrix, formwork, etc. as well as different fibres can get the same results; the inventors of this second category of formulas strongly believe that it is impossible to draw out the shear force carried out by the fibres only knowing the kind of fibre and its percentage.
The aim of this thesis is to find the formula, among all those present in literature, which best mirrors the results of perform all the specimens collected in the database presented in Chapter 4 [Appendix A].
The characteristics that a good formula has to supply are listed below:
‐ it has to give lower scatter compared with other formula applied on the same batch of specimens;
‐ it should be developed from a consistent batch of specimens; ‐ it has to be easy to handle in order to be widely used from the community
of engineer and designers; ‐ it has to involve parameters plainly available for the users; ‐ it should be valid for a large variety of parameter values. ‐ mechanically sound
Rules of design of SFRC have been drafted in almost all European countries; Switzerland produced its Recommendation SIA 162/6 in 2008; Italy followed with the CNR DT 204‐2006, Italian guidelines; in Sweden the Swedish Concrete Association (SCA) developed its first recommendations for SFRC in 1995 (SCA, 1997) and at that moment they were considered to be one of the most cutting‐edge recommendations; Austria has its own “Fibre‐reinforced concrete” guideline (Österreichische Vereinigung für Beton‐ und Bautechnik “Richtlinie Faserbeton”); Netherlands also set out recommendations for the testing and dimensioning of steel‐fibre‐reinforced concrete based on the CUR (the Centre for Civil Engineering Research and Codes) rules and RILEM Recommendation; United Kingdom historically adopted the Japanese beam test JCI‐SF4 but recently the RILEM beam test has been incorporated almost totally into BS EN 14651; moreover the Concrete Society with its Technical Report No. 63 (2007) summaries the current applications for SFRC considering practical aspects such as production and quality control; it does not give a definitive design guidelines but the information for the designers to exercise judgement in this area of evolving technology; Germany has its DAfStB guideline for steel‐fibre‐reinforced concrete (DAfStB, 2011b); The Norwegian Concrete Association set out its first recommendation in the Technical Specification and Guidelines in 1993; Steel‐fibre‐reinforced concrete has also been included in the fib Model Code 2010 (fib Bulletin 55, 2010).
30
After these considerations four different formulas have been chosen for being compared in this thesis. Worthy to note is the distinction between formulations that are empirical or semi‐empirical model based on test results and those that are analytical model based on theoretical studies. Alternative I is an empirical model, and it is already included in the Swedish Concrete Report No. 4 (SCA, 1997), while Alternative II, III and IV are analytical formulations included in the DAfStB guideline, in the RILEM TC 162‐TDF and in the CNR DT 204‐2006, respectively.
31
3.1 Alternative I: Narayanan & Darwish
In 1987, two researchers, Narayanan and Darwish, postulated their formula that, in the following years, has been one of the most used and has later been shown to be one of the best alternatives through comparison with published data (Hällgren, 1997).
R. Narayanan obtained his first civil engineering degree in India and a master’s degree from the University of London. Since his PhD, he worked in the University of Manchester and University of Wales (Cardiff), with more than 25 years of experience working in the construction industry.
I. Y. S. Darwish obtained his first degree from the University of Damascus and his master’s degree from the University College (Cardiff). At the time of the publication, he was taking his PhD at the University College.
Their formula was presented, for the first time, in the ACI Structural Journal, May‐June 1987 volume, into the article “Use of Steel Fibers as Shear Reinforcement” (Narayanan et al., 1987).
Their purpose was to investigate the behaviour of steel fibre reinforced concrete beams subjected to predominant shear. After their investigation they presented the semi‐empirical equations that are tools to be used for design purposes.
These predictive equations are suggested for evaluating (1) the cracking shear strength and (2) the ultimate shear strength of fibre reinforced concrete beams.
In the paper mentioned above they established that the inclusion of steel fibres in RC beams results in a substantial increase in their shear strength (e.g. when 1 % volume fraction of fibres was used, an increase of up to 170 % in the ultimate shear strength was observed).
The test program consisted on fabricating 49 beams having identical rectangular
cross section of 85 x 150 mm, and testing them under four symmetrically placed concentrated loads.
Four clear spans and four shear spans were employed. Three different types of beam were tested: (1) beams without web reinforcement, (2) beams with conventional stirrups and (3) beams containing crimped steel fibres as web reinforcement.
The fabrication data, the material used and all the characteristics that could influence the shear behaviour of the specimens, available on the article, are quoted in Appendix A. Test result showed that the first‐crack shear strength fcr increased significantly due to the crack‐arresting mechanism of the fibres. Even for a fibre volume fraction of 1 %, which was the optimum percentage, the ultimate shear strength improvements were of the same order as those obtained from conventional stirrups.
They recognized that the shear force V withstood by a beam could have the following form:
a b c dV V V V V (Eq. 3.2)
where Va is the vertical component of the interlocking force, which results from interlocking of aggregate particles across a crack; Vb is the vertical component of the fibre pull‐out forces along the inclined crack; Vc is the shearing force across the compression zone and Vd is the transverse force induced in the main flexural reinforcement by dowel action. However, it should be noted that the above four shear forces are not necessarily additive when failure is imminent. In the formula hereinafter
32
shown, the contribution of the aggregate interlocking has been ignored (this assumption gives a safe prediction).
Figure 3.2 ‐ Free body diagram of part of the shear span of a simple supported beam fibre reinforced concrete beam (Narayanan et al., 1987).
To use properly this formula some terms have now to be defined:
Fibre factor F given by
f f
LF d
D
(Eq. 3.3)
Where L/D is the fibre aspect‐ratio, f is the fibre volume fraction and df is the bond factor that accounts for differing fibre bond characteristics; based on a large series of pull‐out tests, df was assigned a relative value of 0.5 for round fibres, 0.75 for crimped fibres and 1.0 for indented fibres.
Split cylinder strength of fibre concrete ( fspf)
Direct tension tests, modulus of rupture tests and cylinder split tests have all been employed to measure the tensile strength of FRC.
The determination of true tensile strength by direct tensile test of fibre concrete is not easy because of the stress concentration at the grips of the testing machines. Moreover, it is affected by machine stiffness, specimen alignment, the size and the shape of the specimen, the fibre orientation and so on, giving, thus, a wide scatter of results.
A quick and safe estimation of split cylinder strength fspf of FRC that relates it with its compressive strength and fibre factor has been found:
cufspfc
ff B C F
A (Eq. 3.4)
Where fcuf is the cube strength of fibre concrete, A is a nondimensional constant having a value of ! = (20 − √(), B is a dimensional constant having a value of 0.7 N/mm2 and C is a dimensional constant having a value of 1 N/mm2; the formula has been obtained by a regression analysis, thus:
33
cuf
pfc 0.7 120
s
ff F
F
(Eq. 3.5)
The ultimate shear strength (N/mm2), as mentioned before, consists of three terms, I, II and III, according to Eq.3.2, they will be described in the following paragraphs.
Part I
b 0.41v F (Eq. 3.6)
To evaluate the vertical component Vb [N], one may start with the number of fibres at a cross section nw, which according to Romualdi et al. (1964) is given by
fw
1.64n
D
(Eq. 3.7)
where f is the volume fraction of fibres and D is the diameter of the fibres. Assuming that the shear crack will have an inclination of α to the horizontal (see FIGURA[1]) and therefore a length equal approximately to jd/sinα, the total number of fibres at the inclined cracked section of the fibre reinforced concrete beam will be:
w sin
jdn n b
(Eq. 3.8)
where b is the width of the beam. The total bond area of fibres across the inclined cracked section is thus
b 4
DLA n
(Eq. 3.9)
where L/4 is assumed to be the average pull‐out length since the latter may range between 0 and L/2. Assuming that the forces of the fibres are normal to the crack, the total force Fb developed is given by
b bF A (Eq. 3.10)
where is the average fibre matrix interfacial bond stress. The fibre pull‐out force Vb is given by
b b cos [kN]V F (Eq. 3.11)
or in term of strength
bb 2
cos N
mm
Fv
b j d
(Eq. 3.12)
assuming α = 45 degrees, the stress vb can be written from Eq. (3.8), (3.9), (3.10) and (3.12) as
b f0.41L
vD
(Eq. 3.13)
To allow its application for the different pull‐out resistance offered by different types of fibres, it is essential that this is modified by introducing the bond factor df into Eq. (3.13).
34
b f f0.41 Lv dD (Eq. 3.14)
or, introducing the fibre factor F, Eq. (3.14) becomes as the Eq. (3.6). With steel fibres in cementitious composites, the fibre matrix interfacial bond is
mainly a combination of adhesion and friction and mechanical interlocking (see Section 2.2). The available investigations on the fibre bond resistance have shown a large scatter of test results (Narayanan et al., 1987). However, the indirect methods adopted by Swamy, Mangat and Rao (1974) seem to be more realistic and the value of 4.15 N/mm2 suggested by them for the ultimate bond stress was adopted in the Narayanan & Darwish study. For this reasons it will also be adopted in Chapter 4.
Part II
c spfc'v e A f (Eq. 3.15)
This term considers the shear span ratio a/d and the split cylinder strength fspfc calculated by Eq. (3.5). All the units are in Newton (N) and millimetres (mm). The term e is a non‐dimensional factor that takes into account the effect of arch action and is given by
1.0 when / 2.8
2.8 when / 2.8
e a d
de a d
a
(Eq. 3.16)
It is emphasized that e is a non‐dimensional factor, and it is unaffected by the system of units. In reinforced concrete beam, Zsutty (1971) has obtained the value e to be 2.5 d/a when a/d 2.5. But as the inclusion of fibres improves the arch action, through enhancing the split compressive strength of concrete, a higher value of e is not unreasonable.
A’ is a non‐dimensional constant having a value of 0.24; e and A’ (like the constant B’ presented in equation (3.17)) were evaluated by a regression analysis of the test data through computer. The purpose of such statistical analysis was to determine that combination of the constants which, when applied to published data, would yield an average value of the ratio of observed to predicted ultimate shear loads equal to one with a minimum standard deviation.
Part III
d 'd
v e Ba
(Eq. 3.17)
The last and third term, considers the dowel action provided by the amount of longitudinal tensile reinforcement = As/bd with the shear span ratio a/d; B’ is a dimensional constant having the value of 80 N/mm2.
The data reported in the article present a wide range of variables, such as concrete strength, a/d ratio, fibre factors, amount of tensile reinforcement, shapes and cross section (rectangular, I or T section) and size of specimen.
35
Worthy to note is the consistency of the formula that when f = 0 or F = 0 predicts appropriate values for reinforced concrete. The authors also suggested one equation for predicting the cracking shear strength of fibre reinforced concrete beams.
The predicted values obtained from the equation for the ultimate shear strength gave to Narayanan and Darwish (1987) acceptable results compared with available test data of beams collapsing by shear failure. The mean value of the ratio between the observed ultimate shear and the predicted ultimate shear of 91 tests was 1.09 with a standard deviation of 0.157. This means that the formula is applicable for a wide range of parametric variations and is validated by tests carried out by the authors and by more than 30 years of its utilization.
36
3.2 Alternative II: Equation Developed from
the German Committee for Reinforced
Concrete (DAfStB)
The German Committee for Reinforced Concrete (DAfStB) was founded in 1907 and is a nationally and internationally recognized and respected professional body for the promotion of concrete structures. The scope of the work focuses on research activities, the preparation of guidelines for concrete structures and the documentation of the information in its own publications. Essential characteristic of the bodies of DAfStB is its composition in the form of a "round table" on which the balance is needed between representatives from the different fields as:
‐ Contractors and construction supervision, ‐ Building materials and construction, ‐ Science and engineering consultants.
The results of the research activities are often implemented in the DAfStB guidelines that are usually introduced by the concrete construction bodies and in this case ‐ as well as relevant standards ‐ recognized rules of the art (DAfStB, 2011a).
Before the publication of the DAfStB Guidelines, the DBV “Steel‐fibre‐reinforced concrete“ recommendation (Deutscher Beton‐ und Bautechnik‐Verein e V., DBV‐ Merkblatt Stahlfaserbeton. October 2001) has been available in Germany for the design of steel‐fibre‐reinforced concrete elements. This was based on the DBV “Tunnel Engineering” recommendation and includes European developments in the field of standardisation. The DBV “Steel‐fibre‐reinforced concrete” recommendation provides a well‐founded aid to the design of steel‐fibre‐reinforced concrete. Anyway, this code does not have the character of a standard. In view of this fact, the DAfStB decided to draft a Guidelines orientated around DIN 1045‐1. The final edition of the DAfStB “Steel‐fibre‐reinforced concrete” Guidelines was published in 2011. These Guidelines replace and complement parts of DIN 1045‐1 and after being taken up in the List of Building Materials they have a status of a code (Tunnel, 2011).
Fibres for concrete (steel fibres and polymer fibres) have now been standardized throughout Europe and their use in concrete, according to EN 206‐1/DIN 1045‐2, is allowed. Other fibres, according to DAfStB standard, may be added to the concrete; however, their load‐bearing effect may not be considered. Polymer fibres and steel fibres formed into bundles in a metering package require National Technical Approval for proving that they can be mixed evenly throughout the concrete (VDZ, 2009).
The DAfStB doesn’t change the validity of the:
‐ DIN 51220, “Werkstoffprüfmaschinen – Allgemeines zu Anforderungen an Werkstoffprüfmaschinen und zu deren Prufung and Kalibrierung“;
‐ DIN RN 12390 – 5, “Prüfung von Festbeton – Teil 5: Biegezugfestigkeit von Probekörpern“ and
‐ DVB – “Merkblatt Stahlfaserbeton“ Ausgabe 2001‐10.
The guideline consists of three parts:
‐ explanation about the size/shape of the beam ‐ production, design and conformity of materials
37
‐ practical description of the work.
The Guidelines has a limitation that consists to be valid for normal concrete until the class C50/60.
The formula for the calculation of the shear capacity in a SFRC beams contains two
terms:
f fRd,ct Rd,ct Rd,cfV V V (Eq. 3.18)
where VfRd,ct, VRd,ct, VfRd,cf are the total shear capacity of the element, the shear capacity carried out by the concrete and the shear capacity due to the fibres, respectively. Moreover:
f fc ctR,u
Rd,cf fct
wf b hV
(Eq. 3.19)
where αfc is a coefficient taking into account long term and unfavourable effects of SFRC (preferable value = 1.00), ffctR,u is a characteristic value of residual tensile strength of SFRC at the larger displacement value and fct is a partial safety factor for SFRC that could be chosen according to Section 2.4.2.4 of the EC 2.
38
3.3 Alternative III: RILEM TC 162‐TDF (2003)
RILEM (which acronym comes from the name in French Réunion Internationale des Laboratoires et Experts des Matériaux, systèmes de construction et ouvrages) was founded in June 1947, with the aim to promote scientific cooperation in the area of construction materials and structures.
Moreover, the mission of the association is to advance scientific knowledge related to construction materials, systems and structures and to encourage transfer and application of this knowledge worldwide.
This mission is achieved through collaboration of leading experts in construction practice and science including academics, researchers, testing laboratories, authorities and a constant production of materials easily available on the web (RILEM, 2011).
In a context where empirical and semi‐empirical design methods bind the designer to certain types of fibres and do not allow them to develop a rational optimized process, in April 1995 the RILEM Technical Committee 162‐TDF (Test and Design Methods for Steel Fibre Reinforced Concrete) has been setup. Most of the members were already active in standardization with regard to SFRC in their own country.
The objectives of RILEM TC 162‐TDF are: ‐ to develop design methods to accurately evaluate the behaviour of SFRC in
structural applications (both in SLS and ULS) and to supply the lack of national and European building code requirements for this material;
‐ to make recommendations for appropriate test methods to characterize the parameters that are essential in the design methods (toughness) and not only take pre‐peak behaviour into account (typically Young’s modulus and compressive strength) as happened up to now.
The work of the Technical Committee found its result in RILEM Recommendations published in Materials & Structures in 2000 and 2001 and later in RILEM Final Recommendations in 2002 and 2003. The RILEM Workshop in Bochum (Germany, 20‐21 March 2003) gave background information to the previous Recommendations.
The RILEM Technical Committee based its first approach on the experimental results
of Vandewalle and Dupont (2000), which performed experiments on 43 full‐scale beams. The former, due to her valid contribution for the research in SFRC materials, was
elected Chairlady of the here before mentioned committee. During the RILEM Workshop in Bochum the results, of tests carried out for
investigating the design proposal, were analysed. Due to the results found within the Brite/Euram project (about 38 beams tested) it can be concluded, that the shear design proposed by RILEM TC 162‐TDF (RILEM, 2000b) is a simple way to calculate the shear resistance with a sufficient margin of safety. However, in the final draft of the RILEM Recommendations (RILEM TC 162‐TDF, 2002) the equivalent flexural tensile strength is replaced by the residual flexural tensile strength; the equivalent flexural tensile strength is derived from the contribution of the steel fibres to the energy absorption capacity (area under the load‐deflection curve) while the residual flexural tensile strength is derived from the load at a definitely crack mouth opening displacement (CMOD) or midspan deflection (δR). The value which is used for the ULS is fRk,4 (CMOD4 = 3.5 mm or δR,4 = 3.0 mm) is related to the strain of 2.5 %. Also the factor that takes into account the height of the member is replaced by the factor used in the final draft of the EC 2.
39
The RILEM Design Method is based on the European pre‐standard ENV 1992‐1‐1 (Eurocode 2). This method calculates the shear capacity V as consisting of 3 separate contributions:
c w f V V V V (Eq. 3.20)
This is the equation given in the first draft of EC 2 (1993) with the addition of the term for the contribution of the fibres Vf. However, the shear resistance of the plain concrete Vc is taken from the second draft of the EC 2 (2001) with the partial safety factor c = 1.5.
Table 3.1 ‐ Shear design for steel fibre reinforced concrete members according to RILEM Recommendations (RILEM TC 162‐TDF, 2003a).
1/3cd l cpc
0.18100 0.15
2001 2
fck wV k f b d
kd
(Eq. 3.21)
fd f l fd wV k k b d (Eq. 3.22)
fd R,4
f ff f
w
l
0.12
1 and 1.5
2001 2
f
h hk n k
b d
kd
(Eq. 3.23)
With:
2ck charactersitics cylinder compressive strength [N/mm ];f
width of the beam [mm];b
effective depth of the beam [mm];d
sll
w
0.02;A
b d
(Eq. 3.24)
f factor for taking into account the contribution of the flanges in T‐section; it is equal
to 1 for rectangular sections;
k
f the height of the flange;h
f the width of the flange;b
w the width of the web;b
40
shear span;a
f w w
f f
33 and
b b bn n
h h
(Eq. 3.25)
In the RILEM Recommendations, dated 2000, the influence of the height for the shear resistance due to the steel fibers Vfd is taken into account by the factor defined in Eq. (3.27).
l
1600 >1 factor for taking into account the size effect of the member;1000
dk (Eq. 3.26)
This is the factor kd used in the first draft of the EN 1992‐1 (EC 2, 1993). It is proposed to use the factor
l
2001 2k
d
used in the formula for the shear resistance of the plain concrete because this is closer to the final draft of the Eurocode 2 (EC2, 2001).
2df Rk,40.12 [N/mm ];f
R,4R,4 2
sp
3
2
F Lf
b h
(Eq. 3.27)
where FR,4 is the load corresponding to a CMOD of 3.5 mm in the 3PBTs performed, L is the span of the specimen (500 mm), b the width of the specimen and hsp is the distance between the tip of the notch and the top of the cross section (125 mm).
The test programme involved both plain concrete and steel fibre reinforced concrete. The variables were the content of steel fibres, the longitudinal reinforcement ratio, the conventional shear reinforcement ratio (stirrups) and the cross section shape (T or rectangular section). The used fibre type for all specimens was Dramix RC‐65/60‐BN. It was planned to check if there are influences of the addition of steel fibres on the shear resistance of the plain concrete Vcd or the shear resistance due to the stirrups Vwd and if there are influences of the varied parameters (a/d, l, w, h, cross section shape) on the shear resistance due to the fibers Vfd. The amount of the tested beams was 38. All specimen were single‐span beams. The reinforcements were chosen in that way, that nearly all beams were expected to fail in shear. A few of the beams were foreseen to fail in flexure to check the sensible range of the use of steel fibres as shear reinforcement. The variation of the parameters is shown in Appendix A.
It is really interesting to observe that the crack propagation occurred in different ways. The course of the crack propagation was smoother for the beams with fibres. Furthermore, the time dependent crack propagation in the region of the uncontrolled crack propagation (see Figure 3.3) could be followed with the eyes for the beams with fibres while this was not possible for the beams without fibres. For beams without fibres the last step before the compression zone was chopped through, the crack propagation grows in a sudden way.
41
Figure 3.3 ‐ Failure mechanism observed at beams with and without steel fibres, respectively, and without stirrups (RILEM TC 162‐TDF, 2003a).
A minimum shear reinforcement is not necessary for steel fibre reinforced concrete members. Anyway it must be guaranteed that the fibre dosage has a significant influence on the shear resistance. This can be assumed if the residual flexural tensile strength is at least fR,4 = 1.0 N/mm2. Similar proposals were made in the German DBV‐guideline (DBV, 2001), in the DAfStB guideline (Section 3.2) and in the Italian guideline (Section 3.4).
Rosenbusch and Teutsch, after their studies (2003), established that the RILEM proposal with the two additional terms is a simple way to calculate the shear resistance with a sufficient margin of safety and due to the fact that it is a conservative design method (standard method). Moreover, they said that the proposal also leads to a sufficient margin of safety for the cases of higher fibre contents, shear reinforcement ratios and longitudinal reinforcement ratios.
This formula is suitable for both rectangular and T cross sections and in presence or absence of stirrups. The presence of a flange, in a T‐section, increases the ultimate shear load‐carrying capacity significantly in comparison with a rectangular beam. The test results suggest that there is a limit in the flange depth beyond which there is a significant increase in the load‐carrying capacity and ductility. For beams with lower flange depth and rectangular beams there can be found no significant influence of the flange depth on the first‐crack and maximum load while there is a big increase of the loads for the beam with a large flange depth. The exact depth limit is not defined within this master thesis.
42
3.4 Alternative IV: Italian Guideline CNR DT
204‐2006
The Italian National Research Council (CNR) is the largest public research institution in Italy, it is the only one under the Research Ministry performing multidisciplinary activities. It was founded on November 18 of 1923. Since 1945, the National Research Council (CNR) is a public organization.
Its mission is to perform research in its own Institutes, to promote innovation and competitiveness of the national industrial system, to promote the internationalization of the national research system, to provide technologies and solutions to emerging public and private needs, to advice Government and other public bodies, and to contribute to the qualification of human resources. Since 14 July 2004 Prof. Fabio Pistella has been CNR Chairman.
CNR is framed in departments that are organizational units, structured by macro‐areas of technological and scientific research, with the task of planning, coordinating and monitoring research activities in the affiliated institutes, by assuring them the necessary financial resources. Each department furthermore has its national and international relations, dealing with its macro–area of interest. Every department sets up its own research strategies and programmes, also in cooperation with other departments, and follows up their implementation through specific research projects. The department decides, together with its institutes, single project’s scientific lines, identifying the research groups to be entrusted with the relevant research tasks, at the same time providing them with the necessary resources. Each group of researchers, in charge of carrying out a single scientific line, thus gives its contribution to the achievement of the project goals.
The 11 departments are: (1) Agrifood, (2) Cultural Heritage, (3) Cultural Identity, (4) Earth and Environment, (5) Energy and Transport, (6) Information and Communication Technologies, (7) Life Sciences, (8) Materials and Devices, (9) Medicine, (10) Molecular Design, (11) Production Systems.
CNR is distributed all over Italy through a network of institutes aiming at promoting a wide diffusion of its competences throughout the national territory and at facilitating contacts and cooperation with local firms and organizations (CNR, 2011).
The Council was motivated by the belief that the development of design codes for construction plays a crucial role in the outgrowth of a modern industrial community. Furthermore, the council thinks that guidelines help meeting the safety requirements, promoting the transfer of technological innovation, and opening the global market to fair and equitable competition.
Within this context, the National Research Council (CNR) has played an active role in the technical culture of Italy since its foundation. For more than fifty years, the CNR activity, which resulted in the formulation of Design Codes, Instructions and specific Recommendations, has been supported by general agreement.
Since the publication of the CNR‐DT 200/2004, concerning coating of reinforced and pre‐stressed concrete as well as masonry structures through the use of long fibres reinforced composite materials (FRP), CNR started its activity in the composite materials like SFRC, arriving at the publication of CNR‐DT 204/2006: Guide for the Design and Construction of Fiber‐Reinforced Concrete Structures.
43
After its publication, CNR‐DT 204/2006 was subject to a public hearing and after that some modifications and integrations have been made to the document including corrections of typos, additions of subjects that had not been dealt with in the original version and elimination of others deemed not to be relevant.
This Technical Document has been approved as a final version on Nov. 28, 2007, including the modifications derived from the public hearing, by the “Advisory Committee on Technical Recommendation for Construction” (CNR, 2008).
The design value for the shear resistance in members with conventional longitudinal
reinforcement and without shear reinforcement is given by: 13
FtukRd,F 1 ck cp w
c ctk
0.18100 1 7.5 0.15
fV k f b d
f
(Eq. 3.28)
Where:
c is the partial safety factor for the concrete matrix without fibres;
k is a factor that takes into account the size effect and equal to 200
1 2;kd
d is the effective depth of the cross section;
l= sl
w
0.02A
b d
is the reinforcement ratio for longitudinal reinforcement;
Asl is the cross sectional area of the reinforcement which is bonded beyond the considered section;
fFtuk is the characteristic value of the ultimate residual tensile strength for the FRC, by considering wu=1.5mm;
fctk is the characteristic value of the tensile strength for the concrete matrix in accordance to the current Codes;
fck is the characteristic value of cylindrical compressive strength in accordance to the current Codes;
cp=NEd/Ac is the average stress acting on the concrete cross section,
Ac, for an axial force NEd due to loading or prestressing actions (shall be considered positive compression stresses);
bw is the smallest width of the cross‐section in the tensile area.
The shear resistance VRd,F, is assumed to be not less than the minimum value, VRd,Fmin, defined as:
Rd,Fmin min cp0.15 wV v b d (Eq. 3.29)
with:
3 12 2
min 0.035 ckv k f (Eq. 3.30)
For members with loads applied on the upper side within a distance 0.5d ≤ a ≤ 2d from the edge of a support (or centre of bearing where flexible bearings are used), the
44
acting shear force may be reduced by β = a/(2d). This is only valid provided that the longitudinal reinforcement is fully anchored at the support. For a ≤ 0.5d the value a = 0.5d should be used.
When point loads close to the support or in diffusive regions are present, the verification can be carried out with strut-and-tie models.
Equation Chapter (Next) Section 1
45
Chapter 4
Analysis and Comparison of the
Specimens
4.1 Introduction
The determination of the shear behaviour of the FRC is challenging due to the large number of parameters involved, e.g. shear span to depth ratio, scale effect, type of fibre, fibre content and orientation, bonding between fibre and concrete and also contribution from any longitudinal reinforcement bars placed to sustain the flexural moments.
There are several methods for determining the shear capacity as show in Table 4.1. Many of these formulas are just empirical equations based on fitting of a limited series of experiments that do not properly account for all parameters that influence the shear capacity. Moreover these formulas differ due to the different data required in them.
One hundred and sixty‐five shear failure tests are recorded in 28 references from previously conducted shear failure tests in SFRC beams without stirrups, and fifty‐eight shear failure tests on reinforced concrete beams without any shear reinforcement are included in the database of this thesis. A complete list of the tests used and their references are tabulated in Annex B.
46
Table 4.1 ‐ Articles and models collected in the thesis database. Paper
Design equations for the shear strength [MPa] References
1
tot w c
w cu w
c R
fcu f f c
f
fu fcu fu f f c
f
0.9
3.75
0.41 for
0.41 1 for 4
w
V V V
V b d
V b d
lV l l
d
dV l l
l
R. Narayan Swamy, R. Jones and Andy T.P. Chiam
(Eq. 4.1)
3
u 0.7 ´ 7 17.2c
d dv f F
a a
Modification of ACI Building equation by Ashour et al. (1992)
(Eq. 4.2)
1
33
u c
1
33
u c b
for / 2.5
2.11 ´ 7
for / 2.5
2,52.11 ´ 7 2.5
a d
dv f F
a
a d
d d av f F v
a a d
Modification of Zsutty´s equation by Ashour et al. (1992)
(Eq. 4.3)
6
f w f f0.24
where
ultimate interfacial bond stress fibre‐matrix
V U b d l
U
Kaushik et al. (1987)
(Eq. 4.4)
9
u c w tu
c
w
tu
0.504 176 / /
where
cylinder compressive strength of PC
ratio of area if tension steel to area of web
/ ratio of moment to shear
ultimate tensile strength offered by fi
V f p d M V bd
f
P
M V
bre
Saluja et al. (1992)
(Eq. 4.5)
10
0.25
uf t
dv k f
a
Sharma (1986) (Eq. 4.6)
47
11
For beams with a/d<3
n sp b0.22 217 0.834d
v f va
For beams with a/d>3
n sp b0.19 93 0.834d
v f va
Shin et al. (1994)
(Eq. 4.7)
12 u cu
cu
0.517 0.283
where concrete flexural strength
v
Swamy et al. (1985)
(Eq. 4.8)
15
2.330.46 '1/2 0.91 '0.38 0.96
uf s c s c yl
fb st yst f
f
0.97 0.2
1.75 0.5 ctg
av f f f
d
lI f V
d
Ding et al. (2011)
(Eq. 4.9)
16 u f c0.325 0.15 0.51 /10V V f bd
Hanai et al. (2008)
(Eq. 4.10)
22
13
uc c s s
1 43 3
uc c s s
fuf cu f
f
u uc uf
10 ' / 2.5
160 ' / 2.5
0.5
v f d a a d
v f d a a d
h c h clv v
d d d
v v v
Al‐Ta’an et al. (1990)
(Eq. 4.11)
29
1 32/3
u bspfc
b
3.7 0.8
0.41
1 for / 3.4
3.4 for / 3.4
dv ef v
a
v F
a d
e da d
a
Kwak et al. (2002)
(Eq. 4.12)
All test beams were loaded with one or two point loads and they were provided with
high longitudinal reinforcement ratios ρ sufficient to secure shear failure that involves web crushing rather than flexural or combined flexural and shear failure of tested beams. Nonetheless, in 343 tests, other than aforementioned, failures are caused by flexure or the combined effects of shear and flexure, which are not considered in this study. That is because this study only addresses shear failures in order to facilitate SFRC ultimate shear strength prediction.
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4.2 Presentation of all the Data
The intention, here, is to do a brief summary of all articles, papers and books included in Appendix B highlighting the most salient and interesting concepts for the purposes of this thesis.
Paper 1, Swamy et al. (1993).
This article concerns about SFRC beams done in lightweight concrete I‐beams (fly ash, PFA, was used as a replacement for both the cement and the sand); it underlines that most of the test reported so far are on rectangular beams, which are not the best cross sections for flexural members. Fibres are also less effective in such members. In account of the reduced modulus of elasticity and the lower tensile strength of lightweight concrete, the benefit of adding a relatively high modulus fibre, such steel, on the strength and deformation characteristics may be more pronounced than for normal weight concrete. The purpose of the paper is thus to assess the effectiveness of steel fibres as shear reinforcement in lightweight concrete beams. A simple theoretical model is presented to compute the ultimate strength in shear. This model leads to simple equations to predict ultimate shear strength of lightweight and normal weight concrete beams (see Table 4.1). The fibre concrete beams displayed an increased number of both flexural and shear cracks at closer spacing than the corresponding concrete beams without fibres. The cracking behaviour clearly showed the ability of the steel fibres in mobilizing the tension zone of the beam in resisting the shear forces. The method presented by the authors was also valid to predict the shear strength of normal weight fibre concrete beams containing steel fibres as shear reinforcement.
In this article previous tests done by Muhidin and Regan (1977) and La Fraugh and Moustafa (1975) are used to compare the validity of the formula proposed by the authors.
Paper 2, Dinh et al. (2010). In this paper 28 relatively large SFRC beams subjected to shear, simply supported,
are investigated with the aim to investigate the effectiveness of the fibre as shear reinforcement in a beam without stirrup reinforcement. The result showed that the use of hooked steel fibres in a volume fraction equal or greater than to 0,75 % led to multiple diagonal cracking and substantial increase in shear strength compared to reinforced concrete (RC) beams without stirrup reinforcement. Moreover, all SFRC beams sustained a peak shear stress of at least 0.33,-′/. The test result also indicated that the hooked steel fibres evaluated in this investigation can safely be used as minimum shear reinforcement in RC beams constructed with normal‐strength concrete and within the range of member depths considered. Data presented herein provide information on the effect of parameters such as fibre geometry, strength, volume fraction and longitudinal reinforcement ratios on the shear behaviour of relatively large SFRC beams. The flexural behaviour of the SFRCs was evaluated through ASTM C1609 (2005) four‐point bending test 150 x 150 x 510 mm beams (455 mm clear span). Each test was continued up to a midspan deflection of 1/150 of the span length (3 mm).
49
Figure 4.1 ‐ Deflection under the loading point and load along the horizontal axis and vertical axis, respectively (Dinh et al., 2010).
These responses represent the average of three or more individual tests, except for Beam 27‐5, for which only two beams were tested. In this graph the effect of different addition of fibres can be seen.
Paper 3, Ashour et al. (1992).
The authors present test results on 18 rectangular high‐strength fibre reinforced concrete beams subjected to combined flexure and shear. All beams, were singly reinforced and without shear reinforcement. The main variables were the steel fibre content, the longitudinal steel ratio and the shear‐span/depth ratio. The concrete matrix compressive strength was about 93 MPa containing only one type of fibre. Two empirical equations are proposed to predict the shear strength of high‐strength fibre reinforced concrete beams without shear reinforcement. It is worth noting that HSFRC is more difficult to mix efficiently than conventional concrete because of the relatively low water content, high cement content, absence of large coarse aggregate and presence of fibres. For these reasons a superplasticizer was used and the mixing time was increased to produce a uniform mix without segregation. The authors performed the calculations of the shear capacity of the specimens with Narayanan and Darwish’s equation and Sharma’s equation, too. Moreover, as shown in Table 4.1, the authors purposed two modifications of the ACI building Code equation and of Zutty’s equation; these were defined after that a regression analysis was carried out on the 18 test results. Even the
50
good prediction of the shear strength for the tested beam the two equations are no longer used.
Paper 6, Kaushik et al. (1987).
This paper is an attempt to study the ultimate strength of fibre reinforced concrete beams vis‐a‐vis shear failure. The scope of the study was limited to observation on the gain in strength compared to ordinary R.C. beams without fibres, looking at the deflections, the curvatures, the rotations and the crack pattern, too. The resistance offered by the fibres crossing a major diagonal crack was evaluated using an effective fibre‐spacing equation and the total force developed through the fibre lying in a vertical plane. The test program consisted of 10 series, with 2 beams in each series, classified as A, B, C, D, E, F, G, H, I and J. All the details are reported in Appendix A. The fibres utilized in the beams were obtained by cutting black annealed mild steel wires (26SWG) of suitable length.
Paper 7, Murty et al. (1987).
This investigation was designed to provide a comprehensive experimental and analytical evaluation of steel fibres as shear reinforcement. To obtain this information, a batch of rectangular beams was tested to failure. Variables that were studied included shear span/depth ratio, aspect ratio of the fibre and volume percentage of fibres. The experimental programme involved tests on eleven reinforced concrete rectangular beams of same cross‐section under two point loading; out of eleven beams, two beams were without any web reinforcement, two had conventional stirrups and the remaining seven were provided with fibres in the test zone. The non‐test zone was reinforced against shear failure by providing web reinforcement in the form of vertical stirrups.
Paper 8, Narayanan et al. (1987a).
The research reported in this paper established the formula discussed in Section 3.1. The authors investigated 49 shear tests carried out on simply supported rectangular beam under symmetrically placed concentrated loads; out of 49 beams, 10 beams contained conventional stirrups and 33 were reinforced with crimped steel fibers instead of web reinforcement. The parameters varied were the volume fraction f of the fibres, fibres aspect ratio L/D, the concrete strength fcu, the amount of longitudinal reinforcement and the shear‐span/effective depth ratio a/d. Beams B1 to B6 were similar except for the volume fraction of fibres which was increased from 0.5 percent, Beam B1 to 3.0 percent in Beam B6. Beam B1 was observed to fail in shear while Beam B2, which has f = 1 %, exhibited a flexural‐shear failure. Increasing the volume fraction of fibres above 1 percent (Beam B3 to B6) the failure resulted in a predominantly flexural mode. How the mode of failure changed from the shear to the flexural type when the volume fraction of fibres was significantly increased can be seen in Figure 2.1. FRC beams having a low volume fraction of fibres (i.e., less than 1 percent by volume) exhibited a sudden failure at the ultimate stage, although this was less catastrophic compared with conventionally reinforced beam without any shear reinforcement.
51
Figure 4.2 ‐ Crack patterns for beams B1, B2, B4 and B7 (Narayanan et al, 1987).
Narayanan and Darwish observed that the crack pattern that develops in SFRC beams subjected to shear is similar to that observed in the corresponding reinforced concrete beams with conventional stirrups. This remark comes from comparison between the performances of beams reinforced only with stirrups and of beams without stirrups but prepared with SFRC using fibres in a percentage equivalent to that of the stirrups in the shear span of the corresponding conventionally reinforced beam: the improvement in the ultimate mean shear strength is not significant, but the first crack shear strength increased noticeably. The same authors, in a subsequent paper (Narayanan et al., 1988) came to the conclusion that the fibres cannot entirely replace the conventional shear reinforcement when the structural elements are subjected to very high shear stress.
Paper 9, Saluja et al. (1992).
In this paper, attempt has been made to suggest a method to have a reasonable estimation of shear strength of fibre reinforced concrete beams (as reported in Table 4.1). On account of the lack of fibre characteristic used in the tests; in order to work in the safe side the following assumptions have been taken: aspect ratio lf/df =100 and shape of the fibre as “round”.
Paper 10, Sharma (1986).
The aim of the paper was to show that steel fibres added at the normal shear reinforcement can be effectively used for increasing the shear strength of concrete; tests have shown that a combination of stirrups and fibre reinforcement forms an effective system of shear reinforcement in a structural member. At any rate the only beam included in this thesis are the one in PC and SFRC without any stirrups; out of seven beams, the ones these characteristics were three.
Paper 11, Shin et al. (1994).
This paper reports the results of an investigation on the strength and ductility of fibre reinforced high strength concrete beams (with concrete compression strength equal to 80 MPa) with and without steel fibre reinforcement, the diagonal cracking strength as well as the nominal shear strength of the beams were determined. 22 beam specimens were tested under monotonically increasing loads applied at mid‐span. The major test parameters included the volumetric ratio of steel fibres, the shear‐span‐to‐
52
depth ratio, the amount of longitudinal reinforcement and the amount of shear reinforcement.. Empirical equations are suggested for evaluating the nominal shear strength of SFR high‐strength concrete beams (Table 4.1).
Paper 12, Swamy et al. (1985).
The aim of this study was to quantify the contribution of steel fibres to the shear resistance of concrete members, in order to develop the design rules for such members. The tests reported in this paper were designed to clarify further the role of steel fibre in shear failures, and in particular, to evaluate the effectiveness of steel fibres in shear strength and shear deformation of reinforced concrete beams. The tests were conducted on T‐beams and rectangular beams, 3.4 m long. Nine T‐beams and two rectangular beams were tested in this study. All the beams were simply supported with a clear span of 2.8 m and a moment/shear ration equal to 4.5.
Paper 13, Tan et al. (1995).
This paper presents a systematic study on the behaviour of partially pre‐stressed SFRC beams subjected to shear. A simple approach is developed to determine the contribution of steel fibres, when used as partial or complete replacement of stirrups, to the shear‐carrying capacity of partially pre‐stressed beams. A test program was carried out with the partial pre‐stressing ratio, the shear span‐to‐effective depth ratio and the steel fibre content of the beam as major parameters. The partial pre‐stressing ratio (PPR) is a quantity used to represent the extent of pre‐stressing in a beam and it is defined as the ratio of the ultimate moment of resistance due to the pre‐stressed reinforcement (Mu)p to the moment of resistance due to all tensile steel reinforcement (Mu)p+s.
Paper 14, Rosenbusch et al. (2003).
This paper mainly deals with the work carried out within the framework of subtask 4.2 “Trial Beams in shear” of the Brite/Euram project BRPR‐CT98‐0813 and with the change between the RILEM Recommendation TC 162‐TDF (2000) and the final Recommendations (2003). In the latter the equivalent flexure tensile strength was replaced by the residual flexural tensile strength and the factor which takes into account the height of the member was replaced by the factor used in the EC 2 (Eq.3.26).
Rosenbusch and Teutsch looking for a coefficient to convert -01,3 to -4,5 established that the conversion factor could be taken equal to 1. The tests were conducted on 38 T‐beams and rectangular beams. Out of 38 beams, eight beams were with steel fibre and stirrups, five beams were without any shear reinforcement and the left beams were with fibres.
Paper 15, Ding et al. (2011).
This paper presents the results of an experimental research program on the shear behaviour of steel fibre reinforced SCC beams. The major aims of this program are to evaluate the possibility of replacing stirrups by steel fibres, to study the hybrid effect of steel fibres and stirrups on the mechanical behaviour of beams, and to analyze the influence of steel fibres in the failure mode and shear strength. The beams studied in the test program had a cross section of 200 mm x 300 mm and 2400 mm length. They were tested on a span of 2100 mm having two stirrups ratio and two fibre contents. They were 9 beams, but only 3 over these were suitable for the investigation carried in this work. Moreover the authors investigated the validity of the existing semi‐empirical
53
equation for predicting the shear strength and they suggested a new formula (Table 4.1). This formula is used for predicting the values from data from different sources included, in their turn, in the database (appendix A). The article reports the data of the works of Ashour (already in paper 3), Kwak (partially included in paper 29), Noghabai and Zhang.
Paper 16, Hanai et al. (2008).
This paper discusses the influence of steel fibre on both punching strength of flat slabs and shear strength of concrete beams. Similarities in the structural behaviour of analogous slab and beam were observed in many experimental analyses present in literature, even in the DAfStB formulas for beams and slabs there is a strong analogy. The authors designed concrete mixtures to attain different strength levels, from ordinary to high‐strength range. Ultimate load capacity and ductility of analogous slabs and beams showed the same performance tendencies as the fibre content varied from 0 to 2 %. The main conclusion of the study is that shear tests on prismatic beams provide useful information for SFRC mixture design for slab application. Five series of analogous slabs and beams (S1 to S5) were tested for a total of 15 beams designed with four different concrete mixes and with the inclusion of hooked steel fibre in a 0 %, 0.75 %, 1.50 % volume fraction. The ACI 318M‐02 equation to evaluate the ultimate shear force for beams without stirrups, in order to consider the steel fibre effect, has been modified as reported in Table 4.1.
Paper 17, Lim et al. (1999).
The purpose of this study was to explore the shear characteristics of reinforced concrete beams containing steel fibres. The tests reported in this article consist of nine beams reinforced with stirrups and steel fibres. The main aims of this study were to investigate (1) the mechanical behaviour of reinforced concrete beams containing steel fibres under shear, (2) the potential use of fibres to replace the stirrups and (3) the combinations of stirrups and steel fibres for improvements in ultimate and shear cracking strengths as well as ductility. A method of predicting ultimate shear strength of beams, when reinforced with stirrups and steel fibres, is proposed.
Paper 18, Sachan et al. (1990).
This paper describes an experimental investigation to study the strength and behaviour of SFRC deep beam. In total 14 beams were tested. The variation of the fibre content, the percentage of longitudinal reinforcement and the type of loading were investigated. The ultimate load carrying capacity, the mode of failure and the load deflection behaviour are reported. A simple model is proposed to predict the load‐carrying capacity of the beams (Table 4.1).
It is worth noting that the non‐linear nature of concrete together with the cracking owing to the low tensile strength makes the behaviour of reinforced deep beams more complex than can be predicted by theoretical studies of linear homogeneous elastic material. Some really important information like the kind of steel fibres and the presence of stirrups are missing. Even making some assumptions on the safe side the results obtained performing Alternative I are not consistent and not included in the database.
Paper 19, Cucchiara et al (2004).
The aim of the paper consists of the evaluation of the improvement in the post‐peak behaviour due to the presence of fibres and in particular to the coupled effects of fibres
54
and stirrups. The load‐deflection graphs recording the post‐peak branch, allowing the conclusion that the inclusion of fibres can modify the brittle shear mechanism into a ductile flexural mechanism, thus allowing a larger dissipation of energy necessary especially in seismic resistant reinforced concrete framed structures. The tests were carried out by considering two different values of shear span, different volume of fibre and stirrups, for two series of eight beams.
Paper 20, Roberts et al. (1982).
The article presents the investigation of nine deep SFRC beams. The beams contained conventional tensile steel reinforcement but different percentages of steel fibres in place of conventional shear reinforcement. All nine beams were simply supported and loaded to failure by a central load distributed through two bearing plates. The results confirm that steel fibres can prevent shear failure in deep beams and also indicate the various modes of failure of deep beams.
Paper 21, Adhikary et al. (2006).
The paper presents the development of artificial neural network models for predicting the ultimate shear strength of SFRC beams. Neural networks are being applied to an increasing large number of real word problems. Neural networks constitute an information processing techniques based on the way biological nervous systems, such as the brain, process information. The fundamental concept of neural networks is the structure of the information processing system. Composed of a large number of highly interconnected processing elements or neurons, a neural network system uses the human‐like technique of learning by example to solve problems. This thesis does not focus on the neural network but the data used in the article, the ones which come from the literature, can be included in the database of Appendix A. Even some parameters are totally missing this bunch of tests can be used to perform the Narayanan and Darwish’s formula.
Paper 22, Al‐Ta’an et al. (1990).
In this paper predictive equations are suggested for evaluating the cracking and ultimate shear strength of rectangular fibre‐reinforced concrete beams without stirrups (see Table 4.1). The method shows good agreement with the published test results of 89 beams which failed in shear. The data relative to these beams were reported in Appendix A but utilized for performing only the Alternative I owing to the lack of parameters needed to run the other alternatives.
Paper 23, Tan et al. (1993).
This study presents an investigation of the behaviour of SFRC beams subjected to predominant shear. Although several semi‐empirical relations have been suggested to determine the ultimate shear capacity, the author wanted an analysis on the complete shear response with detailed strain measurements in the beams.
Tests were conducted on six simply supported beams in order to obtain strain measurements in the steel reinforcement and the web of the SFRC beams under predominant shear and giving more attentions on the post‐cracking tensile strength.
Paper 24, Batson et al. (1972).
The purpose of this study was to investigate the various shapes of steel fibres, the fibre size, the fibre volume concentration and the effectiveness when the fibres
55
substitute the vertical stirrups in conventional reinforced beams. The shear span ratio, a/d, was decreased with increasing steel fibre content. The 72 beam have all 101 x 152 mm cross section with a clear span of 915 mm.
The authors, after their investigation, concluded that the replacement of vertical stirrups by round, flat or crimped steel fibre provides effective reinforcement against shear failure.
Paper 25, Londhe (2010).
The main aim of this paper was to study the performance of the RC beams in shear experimentally reinforced with longitudinal tension steel only, and reinforced with steel fibres and to know the relative contribution of the different mechanism through which shear are transmitted between two adjacent plane in a reinforced concrete beam.
In this work an attempt is made to investigate shear strength and ductility of fiber reinforced concrete beams by using hooked steel fibres. All the test beam specimen were 100 mm in width, 150 mm in depth and 1200 mm in length and the primary variable investigated were percentage of fibres (0.5 to 5 %), percentage longitudinal tension steel (0.8 to 3.22 %) and cube compressive strength of concrete (in the range of 34 of 41 MPa) for a total of twenty beams. The shear span‐depth ratio was kept constant at 3.20. All the beam specimens were tested under four‐point loading test set‐up and the failure load, crack pattern and deflections were recorded. Thirty six beams were cast and tested for various fibre contents and longitudinal reinforcements.
Paper 26, Narayanan et al. (1987b).
In their previous study Narayanan and Darwish (1987) were not able to locate the influence of the fibres on the behaviour of pre‐stressed concrete beams under predominant shear loading. The studies reported in this paper are aimed at investigating the combined (beneficial) influence of fibre reinforcement and pre‐stressing. The results of 36 shear tests on pre‐stressed concrete beams, containing steel fibres as web reinforcement are presented and discussed. One analytical method of predicting the shear capacity of pre‐stressed concrete beams is developed to include the effect of fibre incorporation (Table 4.1).
Paper 27, Voo et al. (2003).
This study reports the results of testing seven pre‐stressed beams failing in shear. The beams were cast using 150‐170 MPa steel fibre reinforced reactive powder concrete and were designed to assess the ability of steel fibre reinforced reactive powder concrete to carry shear stresses in thin webbed pre‐stressed beams without shear reinforcement. The authors, comparing the crack patterns of specimens, observed that the quantity and the types of fibres in the concrete mix did not significantly affect the initial shear cracking load but, increasing the volume of fibres, the failure load is increased. Moreover the tests were also analyzed using finite element modelling and the variable engagement constitutive model (VEM).
Paper 28, Pauw et al (2008).
In this experimental program the behaviour of precast pre‐tensioned concrete beams made with steel fibre concrete and without ordinary shear reinforcement is compared with the behaviour of a standard beam made with concrete without fibres but with stirrups as shear reinforcement. Furthermore a beam made of plain concrete without shear reinforcement is tested to investigate the effect of the shear reinforcement. The
56
beams were loaded to failure in a three point bending test. The results showed that the beams made with the steel fibre concrete can resist shear forces as well as the standard beam with the stirrups. Four I‐shaped beams with a total length of 10.9 m were manufactured. This paper is really interesting because it reports both test results on the small scale tests and on the full scale beams; the former provide the value on the compression strength and the equivalent and flexural strength coming from both the 4PBT (according to the Belgian standard NBN B15‐238) and the 3PBT (according to the RILEM method); the latter, on the other hand, give the most important load values for each beams like the load at first bending crack, the load at first shear crack, the load at final manual measurement and the failure load. The value of 678 = 9:;/!7 is not directly explicit in the text but it is deduced by a value that comes from other calculation; 9:; is the axial force in the cross–section due to loading or pre‐stressing [in N] (9:; > 0 for compression) and !7 is the area of the concrete cross‐section [mm2].
Paper 29, Kwak et al (2004).
Twelve reinforced concrete beams were tested to failure to evaluate the influence of fibre‐volume fraction, a/d and concrete compressive strength on beam strength and ductility. The beams denoted by the letters FHB (fibre‐reinforced, higher‐strength concrete beams) were constructed with concrete having a compressive strength near 65 MPa while the one denoted by FNB2 had an average compressive strength of 31 MPa. No stirrups were included in the shear span, only behind the supports in order to preclude the possibility of anchorage failure of the longitudinal bars.
The authors’ results demonstrated that the nominal stress at shear cracking and the ultimate shear strength increased with increasing concrete compressive strength, fibre volume, and decreasing shear span‐depth ratio. Moreover the results of 139 tests of FRC beams without stirrups were used to evaluate existing and proposed empirical equation for estimating shear strength (Table 4.1). The evaluation indicated that the equation developed by Narayanan and Darwish and the equation proposed herein provided the most accurate estimates of shear strength.
Paper 30, Kearsley et al (2004).
The authors of this paper investigated the effect of stainless steel fibres by casting nine different series of beams.
Three beams in each series were cast, resulting in a total of 27 beams. Out of nine series, one series was done in plain concrete without any shear reinforcement, three series were with fibre reinforcement and five series were with both fibre and conventional shear reinforcement. To evaluate the effect of the fibres, the equivalent flexural tensile strength of the SFR‐concrete was determined applying load in control of displacement on 150 x 150 x 750 beams in four point bending test (clear span equal to 600 mm).
The test values confirm that the stainless steel cast fibres are significantly less effective in providing post‐cracked concrete strength than draw wire fibres.
From Table 4.1 it can be observed that the contribution of fibres to the ultimate shear strength depends essentially on the volume fraction of the fibres, on their geometric characteristics and on the fibre‐matrix interfacial bond that determines the resistance to fibre pull‐out.
57
4.3 Classification of the Specimens Based on
the Main Properties
4.3.1 Test Methods & Post‐Cracking Parameters
Fibres have no effect on the pre‐cracking mechanical material properties of plain concrete unless the fibre dosage exceeds around 80 kg/m3 (~1 %). The design of high performance composites (Vf >1 %) is not covered in this thesis although some tests incorporate fibres up to 2 % (volume percentage). In this way the material properties of un‐cracked SFRC can be estimated by treating it as plain concrete (e.g. applying the formulas given in Eurocode 2). The material properties of SFRC in tension are discussed below.
The axial tensile strength and the flexural strength of SFRC with a softening behaviour do not change from the ones of plain concrete. The differences between PC (plain concrete) and SFRC (steel fibre reinforced concrete) arise beyond these two strengths. It follows that the axial strength and the flexural strength of SFRC are determined in the same way of the PC using, for example, the formula proposed by the EC 2:
2 3ctk(0.05) ck0.21f f (Eq. 4.13)
2 3ctm ck
cmctm
0.3 C50/60
2.12 ln(1 C50/6010
f f
ff
(Eq. 4.14)
where -7FG(H.HI) is the lower characteristic tensile strength of concrete, -7FJ is the mean tensile strength of concrete and -7G is the characteristic cylinder strength in compression. The “true” tensile strength of concrete can be determined in direct or indirect (splitting) tests (see Section 2.3).
The flexural strength of SFRC is calculated from the failure load in a standard beam tests, making the usual assumption that the stress distribution is linear over the depth of the section. The assumption of a linear stress distribution is reasonable up to first cracking but not at the peak load, which corresponds to the flexural strength. Eurocode 2 defines the flexural strength of concrete in terms of the tensile strength as follow:
ctk,fl ctk(0.05) ctk(0.05)max 1.6 ;1000
hf f f
(Eq. 4.15)
ctm,fl ctm ctmmax 1.6 ;1000
hf f f
(Eq. 4.16)
where -7FG,KL is the lower characteristic flexural strength, -7FJ,KL is the mean flexural strength, h is the section depth in mm and -7FJ is the mean value of axial tensile strength of concrete calculated like -7FJ = 0,30-7G
(M 3⁄ ) ≤ C50/60. Specific tests for the qualification of the SFRC material ductility are:
‐ beam tests for determining the residual flexural strength and, in some tests, a measure of toughness, as already introduced in Section 2.3;
‐ plate tests for determining the toughness in terms of energy absorption
58
capacity and for more realistic models the biaxial bending that can occur in some applications (they are not deal in this thesis).
The results from these two kinds of test are not directly comparable in terms of specifying ductility. Designers should be careful not to specify demanding requirements that may not be directly relevant to a particular design. Normally, the residual strength is required where the concrete characteristics are used in a structural design model, whereas the energy absorption is required in more relevant situations such as rock bolting in conjunction with shotcrete, i.e. where energy has to be absorbed during deformation under service conditions.
The residual flexural strength of SFRC after cracking depends on the fibre type, fibre dosage and concrete strength. It is determined experimentally since it cannot be calculated reliably in terms of the properties of the plain concrete matrix and the steel fibres. Standard test methods are available to determine the residual strength in bending and tension as its toughness. Standard flexural test procedures have been proposed by several organisations including RILEM, the Japanese Concrete Institute (JCI), ASTM, the German Committee for Structural Concrete (DAfStB), the EFNARC (the European Federation of Producers and Applicators of Specialist Products for Structures) and the Italian National Research Council (CNR)
Theoretically, uniaxial tension tests are preferable to beam tests since they can be used to characterise the stress –crack opening (σ‐w) response of SFRC, which is needed in advanced design methods. Commonly, in practice, beam tests are preferred because they are simpler to execute than tension tests and simulate the conditions in many practical applications. For this reason the majority of the formulas concerning the post‐cracking behaviour of SFRC are based on the parameters that can be determined throughout the beams tests. Herein only the test methods proposed by DAfStB, RILEM and CNR are presented, because they are necessary to perform calculations using Alternatives II, III and IV (see Sections 3.2, 3.3 and 3.4) which correspond respectively to their proposed formula for the evaluation of the shear capacity in SFRC beam without shear reinforcement.
1) German Committee for Structural Concrete (DAfStB)
The identification of the parameters necessary to characterize the post‐cracking softening behaviour of SFR mix is described in the DAfStB Guidelines proposed in 2011 (Section 9. Baustoffe, DAfStB, 2011b). The characterization is determined through four‐point bending test. The test specimen is a concrete beam of 150 x 150 mm cross‐section with a fixed clear span of 600 mm and a total length of 700 mm (see Figure 4.3). The beam is unnotched and the distance between the two loads is equal to one third of the clear span.
Figure 4.3 – Position of the load and supports of the beam specimen (DAfStB, 2011).
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During the test, the load at predefined displacements, different from the SLS (identified by the abbreviation L1) and the ULS (L2), are recorded for each specimen. On the load‐midspan deflection graph (Figure 4.4) the load at the displacement of 0.5 mm (F0.5) and the load at the displacement of 3.5 mm (F3.5) are read‐out for the SLS and the ULS, respectively.
Figure 4.4 ‐ Load‐deflection diagram (DAfStb).
To obtain the tensile strength of the concrete (-7FS,TK ) requested in the Eq. 3.19 for the evaluation of the shear capacity the following procedures has to be done:
‐ plot the load‐deflection graph for the n tested specimens (n ≥ 6); ‐ read‐out the value of F3.5 from each graph; ‐ calculate -7KLJ,UMK using the following formula
3.5,fcflm,L2 2
1
1 ni
i i i
F lf
n b h
(Eq. 4.17)
where l is the clear span of the beam, b is the width of the beam, h is the depth of the beam and n is the number of specimens;
‐ calculate -7KLG,UMK
fcflm,L2 sf f
cflk,L2 cflm,L2e 0.51Lf k Ls
f f
where:
f fcflm,L2 cfl,L2,i
3.5,fcfl,L2,i 2
2f fcflm,L2 cfl,L2,i
1ln
ln
1
i
i i
Lf fnF l
fb h
Lf fLs
n
VW is the standard deviation from Table 4.2;
60
Table 4.2 ‐ Values of ks based on the number of tests (DAfStb).
‐ with the value of -7KLG,XY3.I,UMK determine L2 as the closest value of the ones standing on Table 4 (Figure) and take the correspondent value of -7FH,TK ;
Figure 4.5 ‐ Table for the characterization of the SFRC mix (DAfStb).
‐ determine -7FS,TK with the following formula
f f f fctR,u F G ct0,uf f
fF
f fG ct
fct
0.5 for walls
1 for slabs 5
1.0 0.5 1.70
0.9 c
b h
A
A A
where Z[K is the factor to take into consideration the dimension of the test, Z\K is the factor to take into consideration the fibre orientation and !7FK is the area between two cracks.
61
2) The International Union of Laboratories and Experts in Construction Materials, Systems and Structures (RILEM, from the name in French).
This test method evaluates the tensile behaviour of steel fibre‐reinforced concrete either in terms of areas under the load‐deflection curve or by the load bearing capacity at a certain deflection or crack mouth opening displacement (CMOD) obtained by testing a simply supported notched beam under three‐point loading (Figure 4.6).
Figure 4.6 – Position of the load and supports of the beam specimen (RILEM 2002b).
This standard is not intended to be applied in the case of shotcrete. This test method can be used for the determination of: (i) the limit of proportionality (LOP), i.e. the stress which corresponds to the point on the load‐deflection or load‐crack mouth opening displacement (CMOD) curve (U (that is equal to the highest value of the load in the interval (] or CMOD) of 0.05 mm)(see Figure 4.7); (ii) two equivalent flexural tensile strengths (-̂ _,` and -̂ _,5) which identify the material behaviour up to the selected deflection; (iii) four residual flexural tensile strengths which identify the material behaviour at selected deflections or CMODs. In the final RILEM Recommendations TC 162‐TDF (2003) the equivalent flexural tensile strength -̂ _,3 is replaced by the residual flexural tensile strength -S,5; due this change it might be necessary to adjust the design formulas: the relation between -̂ _,3 and -S,5 for the SFRCs used within the Brite/Euram project was found to be -S,5 = 0.87-̂ _,3 but in a case that the branch in the post cracked region is nearly on the same level, the value for -̂ _,3 and -S,5 will be approximately the same, while in case of a great decrease in the post‐cracked branch the ratio -S,5/-̂ _,3 may be lower than 0.87. So it is proposed to replace -̂ _,3 by -S,5 without adjusting the design formula (i.e. -̂ _,3=-S,5).
Figure 4.7 ‐ Load‐CMOD diagram (RILEM, 2003b).
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The test specimen is a concrete beam of 150 x 150 mm cross section with a minimum length of 550 mm; moreover the beam is notched using wet sawing and the notch is not larger than 5 mm and the beam has an unnotched depth ℎc8 of 125 mm ± 1 mm. The clear span length of the three‐point loading test is 500 mm (Figure 4.8).
The residual flexural tensile strengths -S,` and -S,5, respectively, are defined at the following crack mouth opening displacement (CMODi) or mid span deflections (]S,e):
CMOD1=0.5 mm ]S,` = 0.46 mm CMOD2=3.5 mm ]S,M = 3.00 mm
and, assuming a linear stress distribution among the cross section, can be determined by means of the following expression:
R,iR,i 2
sp
3
2
F Lf
b h
(Eq. 4.18)
where:
g = width of the specimen [mm]
ℎc8 = distance between tip of the notch and top of cross section [mm] (often called ligament)
y = span of the specimen [mm]
The relation between “characteristic” and “mean” residual flexural tensile strength, according to EC 2 in case of lack of tests for developing statistic values, is:
fctk,fl fctm,fl0.7f f
Hardened SFRC is classified using two parameters that are determined from the residual flexural strengths -S,` and -S,5. The first parameter FL0.5 is given by the value of fR,1 reduced to the nearest multiple of 0.5 MPa, and can vary between 1 and 6 MPa. The second parameter FL3.5 is given by the value of fR,4 reduced to the nearest multiple of 0.5 MPa, and can vary between 0 and 4 MPa. These two parameters denote the minimum guaranteed characteristic residual strengths at CMOD values of 0.5 and 3.5 mm, respectively. The residual strength class is represented as FL (yH.I (y3.I⁄ , with the corresponding values of the two parameters. For example, a SFRC with a characteristic cylinder compressive strength of 30 MPa, -S,` = 2.2 MPa and -S,5 = 1.5 MPa would have (yH.I = 2.0 MPa, (y3.I = 1.5 MPa and will be classified as C30/37 FL 2.0/1.5.
Figure 4.8 ‐ Arrangement of displacement monitoring gauges and specimen dimensions (RILEM, 2003b).
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Figure 4.9 ‐ Detail of the notch (RILEM, 2003).
The RILEM beam test has almost been incorporated totally into EN 14651.
3) Italian National Research Council (CNR)
For the Italian National Research Council (CNR‐DT‐204/2006) the identification of the constitutive parameters of softening behaviour material through bending tests can be done with a linear elastic model or with a rigid‐plastic model.
The linear elastic model identifies two reference values, -[Fc and -[FT, concerning SLS and ULS behaviour, respectively. They have to be defined through equivalent values of flexural strength using the following equations:
Fts eq1 = 0.45f f (Eq. 4.19)
uFtu Fts 2 1
i2
= 0.5 0.2 0Fts eq eq
wf k f f f f
w
(Eq. 4.20)
Where: -̂ _` is the post cracking equivalent strength useful for SLS; -̂ _M is the post cracking equivalent strength useful for ULS; V is a coefficient equal to 0.7 for cross section fully subjected to tensile stresses and equal to 1 in other cases; ~eM is the mean value of the crack opening at the endpoints of the interval where -̂ _M is evaluated (see Figure 4.10(b)).
Figure 4.10 ‐ Tensile Strength determined through bending test in softening materials (CNR‐DT 204, 2006).
Equations 4.17 and 4.18 are still valid when the local values -̀ and -M, are considered instead of the average values, under the condition that ~eM is assumed equal to the largest value of the considered interval (CTOD2, Figure 4.10(a)). These equations may be
64
deduced through simple equilibrium considerations concerning rectangular section under bending, corresponding to the critical section of the tested specimen assuming (for crack opening values typical of serviceability conditions ~ ≤ 0.6 mm) the following assumptions:
‐ plane sections remain plane after bending; ‐ elasto‐plastic tensile behaviour (with maximum value equal to -[Fc; Figure
4.10(b)); ‐ linear‐elastic compressive behaviour (Figure 4.11(a)).
Figure 4.11 ‐ Stress diagrams for the determination of the tensile strength (CNR‐DT 204, 2006).
The Equation 4.17 is obtained considering a linear constitutive law between points with abscissa wi1 and wi2, up to the point with abscissa ~T (Figure 4.10(b)).
The rigid‐plastic model identifies a unique reference value, -[FT, based on the ultimate behaviour that is determined as:
eq2Ftu 3
ff (Eq. 4.21)
Equation 4.19 is obtained from the equilibrium as in the previous case (with reference to ULS).
In accordance to the standard UNI 11188 and using the symbol of the standard UNI 11039 it is assumed:
65
Table 4.3 ‐ Four point bending test on notched and unnotched specimens with the parameters involved (CNR‐DT 204/2006).
‐For notched specimen ‐For unnotched specimen
-̂ _`G = -̂ _(H�H.Ä)Å evaluated in the interval 0 ≤ ~ ≤ 0.6 mm -̂ _MG = -̂ _(H.Ä�3.H)Å evaluated in the interval 0.6 ≤ ~ ≤ 3.5 mm In order to consider the notch, the value of the tensile strength may be assumed:
-[F = 0.9-7F
-̂ _`G = -̂ _(H�H.Ä)Å = -̀ F[G evaluated in the interval 3~É ≤ ~ ≤ 5~É -̂ _MG = -̂ _(H.Ä�3.H)Å = -[F[G evaluated in the interval 0.8~T ≤ ~ ≤ 1.2~T where ~É is the crack opening calculated at cracking at the maximum recorded load in the interval 0 ≤ ~ ≤ 0.1 mm and ~T = 3 mm
For structures subjected to bending with section depth less than 150 mm (or for
hardening bending behaviour) it is better to carry out the identification process of material properties by taking into account the casting direction and the small thickness of the structure without notching the specimens.
As shown in Table 4.3, specimens can have different dimensions but always the same proportions; in order to obtain a more clear comparison between different specimens geometries, experimental results from bending tests are reported in terms of nominal stress Ñ defined according to a linear stress distribution as reported in Eq. 2.1.
Correlations between different test methods could significantly improve the confidence level between the results and performance classification. It also would help if the results obtained with one specific test method could be used to deduce parameters needed for another design method (for another application). This would drastically reduce the number of tests needed to explore specific applications for the material. The results presented in the literature show that these correlations exist and depend on the deflection/cracking load.
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Table 4.4 ‐ Comparison between flexural test methods according to DAfStb, RILEM and CNR recommendations.
DAfStb RILEM CNR notched CNR unnotched SCA
Type of bending 4‐point bending (4PB)
3‐point bending (3PB)
4‐point bending (4PB)
4‐point bending (4PB)
4‐point bending (4PB)
Notch depth [mm] 0 25 a 0 0 Span length [mm] 600 500 3l 3l 450 Beam length [mm] 700 550 4l not specified 500 Beam width [mm] 150 150 not specified not specified 125 Beam height [mm] 150 150 h+a=l h 75 Net height [mm] 150 125 h h 75 Slenderness l/h 3 3.33 3 3l/h 6
4.3.2 Modes of Failure
When beams are tested under a load up to the failure it is essential to record the mode in which the beam fails observing the crack path and position at failure. The types and formation of cracks depend on both the span‐to‐depth ratio of the beam and the load. These variables influence the moment and shear along the length of the beam. For a simply supported beam under uniformly distributed load or concentrated load at the midspan, without pre‐stressing, three types of cracks are identified (Amlan et al, 2011):
‐ Flexural cracks: these cracks are located near the midspan; they start from the bottom of the section and propagate vertically upwards.
‐ Web shear cracks: these cracks are located near the neutral axis and close to the support, they propagate inclined to the beam axis.
‐ Flexure shear cracks: these cracks start at the bottom of the beam due to flexure and propagate inclined due to both flexure and shear.
Beams with low span‐to‐depth ratio or inadequate shear reinforcement often present a shear failure. A failure due to shear is sudden if compared to a failure due to flexure, cracks are more localized and most of them are located above and along the inclined line joining the support with the point at which the load is applied. The following five modes of failure due to shear are identified:
a) Diagonal tension failure: in this mode, an inclined crack propagates rapidly due to inadequate shear reinforcement;
b) Shear compression failure: there is crushing of the concrete near the compression flange above the tip of the inclined crack;
c) Shear tension failure: due to inadequate anchorage of the longitudinal bars, the diagonal cracks propagate horizontally along the bars;
d) Arch rib failure: for deep beams, the web may buckle and subsequently crush. There can be anchorage failure or failure of the bearing;
e) Web crushing failure: the concrete in the web crushes owing to inadequate web thickness;
The modes are shown through sketches in Figure 4.12.
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Figure 4.12 ‐ Modes of failure: (a) diagonal tension failure, (b) shear compression failure, (c) shear tension failure, (d) arch rib failure and (e) web crushing failure (Amlan et al, 2011).
The occurrence of a mode of failure depends on the span‐to‐depth ratio, loading, cross‐section of the beam, amount and anchorage of reinforcement as well as the concrete strength. The flexural failure, opposite to the shear one, is characterized by crack that appear progressively, allowing the beam to reach significant ductility (Figure 4.13)
Figure 4.13 ‐ Crack patterns at flexural ultimate condition (Chucchiara et al., 2003)
4.3.3 Strength of Concrete
For the specimens included in the database the SFRC specified compressive strength for standard cylinders, -7Ö, ranges from 12.3 to 171 MPa. 327 specimens have normal specified compressive strength (less than 60 MPa), and the remaining 47 specimens have high compressive strength -7Ö. The SFRC compressive strength for standard cube, -7,7TÜ^Ö , ranges from 14.8 to 187 MPa. The tensile strength obtained from splitting tests range from 1.87 to 10.9 MPa
Alternative II, III and IV suggest that the ultimate shear capacity predictions for steel fibre reinforced concrete (SFRC) beams can be made based on the post‐cracking tensile
68
strength while the Alternative I recommends the use of tensile strength obtained using split‐cylinder tests.
Table 4.5 ‐ Ranges of the concrete properties from the database.
Min Max
áàâ,àäãåçéèêÖ [N/mm2] 12.3 171
áàâ,àâëèÖ [N/mm2] 14,8 187
áíìîàÖ [N/mm2] 1.87 10.9
ïà [N/mm2] 18700 26300
4.3.4 Fibres & Volume Percentage
The six different types of steel fibres that were used in the test are hooked, crimped, round, cut wire, stainless and duoform. Steel fibre tensile strength in most of the tests was greater than 1100 MPa. The fibre volume fractions ranged from 0.20 to 3.00 % (per unit volume of cured SFRC). The ratio of fibre length to fibre diameter, Lf/Df, varies from 37 to 133 and the fibre lengths ranges between 13 and 60 mm.
Table 4.6 ‐ Ranges of the fibre properties from the database.
Min Max
ñî [mm] 0.20 1.05
óî [mm] 13 60
óî ñî⁄ 37 133
òî [%] 0.20 3
ïí [GPa] 210 231
4.3.5 Longitudinal Reinforcement
The longitudinal reinforcement ratio ôflex ranges from 0.80 to 9.4 %, depending on the cross section of the beam (Table 4.7).
Table 4.7 ‐ Ranges of the longitudinal reinforcement from the database.
Min Max
úîãèù [%] 0.80 9.4
ú′îãèù [%] 0.13 3.7
ïí [GPa] 200 235
4.3.6 Specimen Dimensions
The beam widths ranged from 50 to 300 mm and the effective depths ranged from 80 to 810 mm. The beam clear spans ranged from 274 to 10300 mm. The span‐depth ratio (a/d) ranged from 0.81 to 6 (Table 4.8).
69
Table 4.8 ‐ Ranges of the geometrical properties of the specimens collected in the database.
Min Max
û ñ⁄ 0.81 6
ü [mm] 100 900
üî [mm] 50 230
ñ [mm] 80 810
† [mm] 50 300
†î [mm] 140 1000
ó [mm] 274 10300
° [mm] 455 10900
4.3.7 Other Properties
Other properties stand in Table 4.9.
Table 4.9 ‐ Further properties of the specimens included in the database.
Min Max
¢ £⁄ §û•¶ß 0.19 0.70
Age test [é®ä] 7 60
Max aggr. size [©©] 6.30 20
™â [´¨] 30.9 1081
≠â [¨/©©Æ] 0.14 19,9
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4.4 Data Processing
Before the beginning of the calculations, the database has to be cleaned up from the specimens that for different reasons do not fulfil the following requests:
‐ The materials have to be classified with the compressive strength (either cubes or cylinder), the fibre volume and the fibre aspect ratio ØK ∞K⁄ .
‐ The cross‐sections of the beams have always to be reported in the articles. ‐ The beams must not contain stirrups or other shear reinforcement
different than the fibres along the span. ‐ The mode of failure must be in shear.
Particularly attention should be paid to the last statement: the aim of the thesis is to investigate the shear behaviour of the SFRC beams but the failure of a beam can depend upon other phenomena, like the flexural failure, not directly related to the shear. Normally the failure mode changes from the shear to the flexure at the increase of the fibre percentage. Omitting those beams that do not have the characteristics stated above, the number of specimens becomes 573 instead of 370.
Not all the beams contained in the reduced database are suitable to develop the back analysis; this because only some articles (see Figure 4.14 ‐ Group 1) contain both RC beams and the respective SFRC beams.
At the moment of the evaluation of the shear strength according to the four formulas reported in Chapter 3, it can easily be noted that, while the Narayanan and Darwish’s formula can be immediately applied to all those beams contained in the database, many of the papers are not reporting the main parameters that are needed for using the DAfStB formula as well as the RILEM one and the CNR one. The parameter omitted from all the articles (with the exception of article 14 for the RILEM formula) is the residual flexural tensile strengths) estimated according to the different tests set‐up as shown in Section 4.3.1.
The direct compression test on concrete is well established and standardized and is found by casting either cubes or cylinders and testing them in direct compression. Unfortunately it is not the same regarding the post‐cracking characteristics. This lack is evident if we think that among a selection of about one hundred articles only five characterize the specimens according, at least, to one of the standards explained in Section 4.3.1 and among these only one is suitable for the shear investigation carried out in this thesis. This scarceness of data changes what was the first aim of the thesis, i.e. a comparison between shear strength coming from the application of different formulas, adding one more piece of work: try to get the missing value using a new relation based on the available data.
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Figure 4.14 ‐ Division of the database in three different groups to satisfy the analysis developed in this Section.
In this section, in order to obtain the missing parameters, a back analysis has been developed. From the back analysis it is expected to obtain a relation able to describe the post‐cracking behaviour, or, at least, an estimation of the residual flexural tensile strength, necessary to use the Alternatives II, III and IV and to estimate the shear capacity of SFRC beam.
The Back Analysis
The paramount concept on which the back analysis is based is that the SFRC has the same safety factor as the corresponding RC beam; the safety factor ±̂ ≤8 ±F≥F⁄ is the ratio between ±̂ ≤8, the shear force experienced during the test, and ±F≥F, the shear force calculated by the formula. The prior statement about the safety factor means to assume that beams designed, performed and tested in the same laboratory and by the same authors are affected mainly by the same variables. Moreover all the three formulas have
72
the purpose to be feasible even in case of RC without any addiction of steel fibres. This fact reinforces the previous assumption as it is normal to think that a formula has been constructed in order to have a safety margin ±̂ ≤8 ±F≥F⁄ almost constant all‐interior of its domain of validity. Furthermore this seems to be the only possible starting point of the analysis related to the database possessed while, with a database with more articles that report the residual flexural tensile strengths, a direct linear regression on these could also have been a good alternative.
To better understand the process that has been used look at Figure 4.15. Once that the shear force, under which the beam failed, is known (from the run of the
test) and the safety factor is fixed for all the three formulas:
DafStb
f f1/3 c ctR,u
Rd,F l fck cpf fct ct
0.15100 0.15 w
w
f b hV k f b d
RILEM
1/3
Rd,F l fck cp f l Rk,4fct
0.18100 0.15 0.12 wV k f k k f b d
(Eq. 4.22)
13
CNR
FtukRd,F 1 ck cp w
c ctk
0.18100 1 7.5 0.15
fV k f b d
f
the only unknowns remain respectively -7FS,TK , -SG,5 and -[FTG. Applying the following inverse formulas the unknowns can be obtained for each
specimen:
DafStb
f1/3Rd,Ff ct
ctR,u l fck cpf fct c
0.15100 0.15
w w
Vf k f
b d b h
RILEM 1/3Rd,FRk,4 l fck cpf
f lct
0.18 1100 0.15
0.12w
Vf k f
b d k k
CNR
3Rd,F cctk
Ftukl ck w
11
7.5 100 0.18
Vff
f k b d
They are for the DafStB, RILEM and CNR alternative, respectively. The process is depicted in Figure 4.15; once that all the residual flexural tensile
strengths have been calculated there is the need to find the parameters that influence the post‐cracking behaviour and that will be included like variables of the formulation. The variables that seem to affect mostly the shear capacity and the post‐cracking behaviour are:
‐ The ratio index defined as ¥µ = ±K∂∑∏∑ that includes the effects of the fibre
percentage and their aspect ratio; it is expressed in percentage. ‐ The cylindrical compressive strength of the concrete -′7 used to make the
beam tests.
The ratio index RI is used as variable along the x axis while the compressive strength -′7 is used to obtain a nondimensional factor; the nondimensional factor, that varies along the y axis, is defined as -π^c -′7⁄ where -π^c is the residual flexural tensile strength
73
at the defined displacement. It is good to remind that the process herein explained is done three times and that each time, according to the Guidelines considered, the
nondimensional factor ∫ªºΩ∫Öæ is specifically
∫æø¿,¡∑
∫Öæ, ∫¿¬,√∫Öæ and
∫ƒø¡¬∫Öæ
for DafStB, RILEM and CNR
Guidelines, respectively. On the defined graph all the SFRC beams are included that come from the same
article; doing a regression analysis on the bunch of tests it is possible to get two different lines:
‐ ≈ = ∆« + g that defines a line with an intercept equal to b; for convenience this kind of lines are identified with the uppercase letter “I” (Intercept).
‐ ≈ = ∆« that defines a line with an intercept equal to 0, crossing the origin of the Cartesian plane; for convenience this kind of lines are identified with the uppercase letter “O” (Origin).
Requiring the passage of the line through the origin of the Cartesian plane leads to lines with a higher slope m than the line of I type and, moreover, farther from the realistic value that the specimens show (especially for high values of RI). Anyway, this line is necessary for connecting the I line to the origin: it is easy to understand that the formula, in case of absence of fibre added in the mix (RI = 0 because Vf = 0), have to satisfy the boundary condition with the RC beams. Looking at the Equations 4.22, this is possible only by establishing a relation that for RI = 0 returns to a residual flexural tensile strength value equal to 0. The process is explained in Figure 4.15.
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Figure 4.15 – Description of the process followed for the creation of a clean bundle of lines.
Every article will have six equations, three of the I‐type and 3 of the O‐type, two for every code. As first check the formulas obtained are applied back on the specimens which they come from like in a design process (γDafStB = 1.25 and γRILEM = γCNR = 1.50). Obviously the results are on the safe side with values of Vexp/Vtot almost all the time higher than 1 (thanks to the application of the safety coefficient γ).
Plotting all the equations obtained on the same graph, one for each code, dividing the I lines from the O lines, we obtain the graphs depicted in Figure 4.17, Figure 4.18 and Figure 4.19.
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Figure 4.16 ‐ Development of the bundle of lines.
76
Figure 4.17 ‐ Graphs of the DafStB Code. The first three graphs are obtained by the process described in Figure 4.15 while the bottom right graph comes from the process
depicted in Figure 4.20.
Analyzing Figure 4.17 it seems that there is a light dependence on the cylindrical compressive strength f’c (as expected) being most of all the lines placed in descending order of f’c from the bottom to the top. From the graphs we can immediately notice that articles A07 and A26 are practically horizontal, articles A19 and A28 have a negative slope and that articles A02, A14, A13 and A15 are positioned far from the bundle of lines that is delimited from articles A23 and A11. Figure 4.16 shows the process followed to obtain the bundle of lines.
Looking at the I‐lines, Figure 4.17, these are the conclusions:
‐ A02 cannot be included, the cause could be the presence of stirrups along half of the span of the beams;
‐ A14 presents a wide range of beam characteristics (like the longitudinal reinforcement ôKL^≤ that ranges from 1,15 to 3,56) for this reason it is divided in sub‐populations with similar ôKL^≤; however the new equations are not contained in the bundle of lines and they are deleted. This article is handled more in detail in Section 4.4 “The Check” because it is the only one, among all the articles, that contains the values of the residual flexural tensile strength according to RILEM Code.
‐ A13 and A15 contain only two beams, this could be the reason why the linear regression does not give satisfactory results, with equations that go out of the bundles of lines; nothing can be done more than exclude these to articles.
77
‐ A19 has a negative slope m that is not a realistic value; in fact the shear strength is supposed to improve at the increasing of the RI or at least up to the limit value.
‐ A08 and A16 are divided in sub‐populations according to the compressive strength of their beams; the equations obtained are satisfactory.
‐ A26 needs to be divided in sub‐populations because its beams differ to much in cross section and flexural reinforcement ôKL^≤; only one of three equations has been included in the bundle of lines.
‐ A28 as A19 have negative slope and only two beams that do not permit the creation of any sub‐populations.
No analysis and reasoning can be done on the second graph “DafStB‐O” (Figure 4.17 top right corner) more than deleting the respectively O‐lines of the ones deleted in the graph “DafStB‐I”.
78
Figure 4.18 ‐ Graphs of the RILEM Code. The first three graphs are obtained by the process described in Figure 4.15 while the bottom right graph comes from the process
depicted in Figure 4.20.
The analysis for the RILEM bundle of lines (see Figure 4.18) reaches the same results of the DafStB one: with the same deleted articles and the same sub‐divisions. The only difference is that A16 is not divided in sub‐populations because equations stand out of the bundle (by the way also this sub‐division shows to be under the influence of the compressive strength f’c).
This similarity on the behaviour might be attributed at the similarities in the structure of the two formulas (DafStB and RILEM) that add the contribution of the fibres at the shear strength of members not requiring the design of the shear reinforcement (according to EC 2, Equation 6.2.a).
79
Figure 4.19 ‐ Graphs of the CNR Code. The first three graphs are obtained by the process described in Figure 4.15 while the bottom right graph comes from the process depicted
in Figure 4.20.
The graph “CNR‐I” looks different than the previous graphs (see Figure 4.19); this might be because the formula is slightly different and the contribution of the fibres is taken into account modifying the equation for the shear strength of members not requiring the design of the shear reinforcement (EC 2, Eq. 6.2.a).
The graphs “CNR‐I” and “CNR‐O” show fewer differences between themselves compared to the other graphs (“DafStB‐I” vs “DafStB ‐O” and “RILEM‐I” vs “RILEM ‐O”) probably due to the structure of the formula (Eq. 4.22).
Anyway the articles that give problems standing out of the delineated bundle of lines are always the same; this is comfortable because it means that those articles for different reason (like the test set‐up or because outside the domain investigated), are not suitable for this investigation through the different Codes.
For CNR the articles A13, A14 and A15 make an exception being included for the calculation of the average equation line (Aaverage).
Consideration: The dislocation of the lines within the bundle has been investigated according to the compressive strength f’c, the a/d ratio and the longitudinal reinforcement ôKL^≤ but they do not look affected by these quantities except that for the f’c; most of all the lines are placed in descending order of f’c from the bottom to the top especially in the DafStB and RILEM graphs.
Now that the bundles are defined from these it is useful to get a unique equation named Aaverage; it is obtained by the weighted average on the number of beams from which every equation come from.
80
Another investigation that can be developed is the one on the cloud of points. In the
cloud of points it is less interesting to consider from which article the beams come from. During this step the criteria for deleting specimens from the cloud (and get new equation from the linear regression on the whole batch of tests) were:
‐ fres > fres,lim; ‐ fres < 0; ‐ both the aforementioned criteria at the same time.
The process is schematized in Figure 4.20; from this four formulas are obtained: (1) Equation B from the whole cloud of points; (2) Equation C from the cloud of points without the specimens that have a computed residual flexural strength higher than the limit value (according to the different codes); (3) Equation D from the cloud of points without the specimens that have a computed residual flexural strength less than zero and (4) Equation E from the cloud of points without the specimens previously deleted for the Equation C and D.
Figure 4.20 ‐ Process followed for the analysis of the cloud of points.
Now we have five equations, four (B, C, D and E) plus one (A) from the weighted average among the bundle of lines, for each of the three alternatives; their values are shown in Table 4.10.
81
Equations A‐E have been drawn in Figure 4.17, Figure 4.18 and Figure 4.19 (bottom right corner) the differences among them are few but not negligible
Table 4.10 ‐ Equations A, B, C, D and E for the DafStB, RILEM and CNR's Guidelines.
After the previous statements and considerations it is clear that the shape of the equation that will describe the post‐cracking behaviour of a SFRC material will be a straight line with three branches as the one drawn in Figure 4.21:
(a) the first branch is the O‐line that link its point of intersection with the I‐line to the origin;
(b) the second branch is the I‐line: (c) the third branch is an horizontal line that represents the upper limit
that cannot be overcame.
Figure 4.21 ‐ Ideal shape of the researched equation for the characterization of the post‐cracking behaviour of a SFRC beam. It consist in three branches:(a), (b) and (c).
With regard to the first branch (a) some further explanations are necessary. It should be built only with the specimens contained in its range of validity (about 0 < RI ≲ 50); one attempt in this way shows that not all the articles contain test in this range and that even the ones that have the tests, give a slope of the lines almost close to the one that comes from the inverse analysis on the whole batch of specimens (that is the one performed, finally).
m b m b m b
A I 0,010 0,852 0,045 2,610 0,038 0,795(average lines) O 0,021 0,000 0,078 0,000 0,044 0,000
B I 0,006 1,279 0,029 4,537 0,027 2,194(cloud points) O 0,019 0,000 0,073 0,000 0,048 0,000
C I 0,003 1,026 0,016 3,434 0,023 1,636(<f ct0,u) O 0,014 0,000 0,054 0,000 0,039 0,000D I 0,005 1,393 0,026 4,933 0,025 2,400(>0) O 0,019 0,000 0,074 0,000 0,048 0,000E I 0,004 1,288 0,013 3,795 0,022 1,800
O 0,016 0,000 0,055 0,000 0,039 0,000
DafStB RILEM CNR
'
0 c
'
res c res,limmin ; ;/ /f f m RI m RI b f f
82
Multiplying the nondimensional factor for the cylindrical compressive strength -′7 the equation varies as shown in Figure 4.22.
Figure 4.22 ‐ Development of the researched equation for the different cylindrical compressive strength -′/ .
' '
0 c cres res,limmin ; ;f m RI f m RI b f f
83
The Check
Before applying the formulas back on Group 3 and Group 1 it is suitable to check the results and their goodness for the only article that presents the post‐cracking characterization: A14 (Table 4.11).
Table 4.11 ‐ Values reported in the article A14 for the characterization of the concrete according to the RILEM Recommendations and evaluation of the value of Vexp/Vtot due to the measured value of -4Å,5.
As it is possible to see in Table 4.12 the values are quite different and the equivalent flexural strengths reported into the article (recognizable by the star “*”) are rather
beamf eq,3
MPa
f Rk,4
MPa
V*
kNV exp/V*
f Rk,4< 4
MPa
V*
kNV exp/V*
f Rk,4/f 'c
(%)
1.2/1 0,00 0,00 95,91 1,89 0,00 95,91 1,89 0,001.2/1 0,00 0,00 95,91 1,89 0,00 95,91 1,89 0,001.2/2 1,49 1,30 120,78 1,82 1,30 120,78 1,82 2,661.2/3 3,05 2,65 142,36 1,69 2,65 142,36 1,69 5,851.2/4 4,85 4,22 173,07 1,79 4,00 169,21 1,83 8,432.2/1 0,00 0,00 173,32 2,42 0,00 173,32 2,42 0,002.2/1 0,00 0,00 173,32 2,42 0,00 173,32 2,42 0,002.2/2 1,91 1,66 241,58 2,32 1,66 241,58 2,32 3,892.2/3 5,60 4,87 371,31 1,62 4,00 335,74 1,79 11,652.3/1 0,00 0,00 99,78 1,57 0,00 99,78 1,57 0,002.3/1 0,00 0,00 99,78 1,57 0,00 99,78 1,57 0,002.3/2 1,35 1,17 131,98 1,25 1,17 131,98 1,25 2,832.3/3 4,13 3,59 197,30 1,09 3,59 197,30 1,09 8,952.4/1 0,00 0,00 116,06 2,07 0,00 116,06 2,07 0,002.4/1 0,00 0,00 116,06 2,07 0,00 116,06 2,07 0,002.4/2 1,35 1,17 148,26 1,46 1,17 148,26 1,46 2,832.4/3 4,13 3,59 213,39 1,35 3,59 213,39 1,35 8,952.6/1 0,00 0,00 80,03 1,87 0,00 80,03 1,87 0,002.6/1 0,00 0,00 80,03 1,87 0,00 80,03 1,87 0,002.6/2 1,91 1,66 111,55 1,48 1,66 111,55 1,48 3,892.6/3 5,60 4,87 171,45 1,36 4,00 155,02 1,51 11,653.1/1 5,45 4,74 194,10 0,97 4,00 179,02 1,06 12,12
3.1/1 F2 5,58 4,85 197,34 1,14 4,00 179,96 1,25 12,063.1/2 5,45 4,74 281,85 0,88 4,00 260,27 0,96 12,1220*50 5,58 4,85 285,43 0,95 4,00 260,14 1,05 12,063.1/3 5,45 4,74 343,58 0,77 4,00 316,72 0,84 12,12
3.1/3 F2 5,58 4,85 349,34 1,10 4,00 318,38 1,20 12,068*50 5,58 4,85 340,43 0,99 4,00 308,04 1,10 12,063.2/1 5,45 4,74 349,35 0,82 4,00 318,95 0,90 12,12
10*50 F2 5,58 4,85 355,48 0,75 4,00 320,44 0,83 12,063.2/2 5,45 4,74 369,78 1,21 4,00 336,18 1,33 12,12
15*50 F2 5,58 4,85 376,40 0,73 4,00 337,68 0,82 12,0623*50 F2 5,58 4,85 392,27 1,09 4,00 350,75 1,22 12,063.2/3 5,45 4,74 385,27 1,13 4,00 349,25 1,25 12,123.2/4 5,45 4,74 385,27 1,07 4,00 349,25 1,18 12,12
1,36 1,41
f Rk,4<4 MPa
Average Average
84
bigger than the one calculated with the five different formulas (A – E) developed for the RILEM Code.
Actually the reported values were the equivalent flexural tensile strength -̂ _,3 (according to the first draft of the RILEM Code) that the authors of the paper tune into the residual flexural tensile strength -SG,5 (from the last draft) with the relation -SG,5 = 0.87-̂ _,3.
Even through this reduction the computed values are too far from the test values and this fact can be attributed to the fact that at the beginning of the back analysis a safety factor for the SFRC beams was imposed equal to the respectively RC beams and almost
of all of the latter had a safety factor ºÀà øÕø
= 2.
It is worth nothing that ºÀà ∗ has an average value equal to 1.41 using a limit for the
value of -SG,5 (as imposed by the RILEM Recommendation (2003) -SG,5 < 4) or equal to 1.36 using the value found by the tests (Table 4.11).
85
Table 4.12 ‐ Comparison between the values reported in the article A14 and the values estimated by using the formulas obtained from the back analysis.
At this point it is interesting to try to do almost the same analysis done previously for the other articles but in this case the safety factor Vexp/Vtot it substituted by Vexp/V* that is not affected by the one of the formula for members not requiring design shear reinforcement in absence of fibres. Plotting the point on the graph (RI,-SG,5/-′7) and getting the linear regression the results obtained are not satisfactory; this because RI ranges between 17 and 50 % and being a limited interval the line has a big slope that for values of RI = 100 % gives improbable high values (Figure 4.23). A logarithmic regression seems to fit better the behaviour of the fRk,4 at the increasing ratio index RI. The equation has to be translated on order to satisfy the request of fRk,4 = 0 for RI = 0(at the argument of the natural logarithmic is added the value of RI for which ln(RI) = 0 RI = 9.2). Moreover, looking at the shape of the equation (Figure 4.25,(I)), once that they are multiplied by the cylindrical compressive strength, for f’c = 30 MPa and RI > 50 the limit value is reached incurring into an overestimation of the fRk,4; for that reason the
f* V f * V e/V* f A f*‐f A V e/V f B f*‐f B V e/V f C f*‐f C V e/V f D f*‐f D V e/V f E f*‐f E V e/V
MPa kN ‐ MPa kN ‐ MPa kN ‐ MPa kN ‐ MPa kN ‐ MPa kN ‐
0,00 0 1,89 0,00 0,00 1,89 0,00 0,00 1,89 0,00 0,00 1,89 0,00 0,00 1,89 0,00 0,00 1,890,00 23 1,82 1,41 ‐0,11 1,79 0,61 0,69 2,02 0,45 0,85 2,08 0,61 0,69 2,02 0,45 0,84 2,081,30 47 1,69 2,44 0,21 1,73 1,13 1,52 2,08 0,84 1,82 2,17 1,14 1,51 2,07 0,84 1,81 2,172,65 70 1,83 3,29 0,71 1,98 1,87 2,13 2,35 1,38 2,62 2,52 1,88 2,12 2,35 1,40 2,60 2,514,22 0 2,42 0,00 0,00 2,42 0,00 0,00 2,42 0,00 0,00 2,42 0,00 0,00 2,42 0,00 0,00 2,420,00 68 2,32 1,21 0,45 2,51 0,52 1,14 2,87 0,39 1,28 2,95 0,53 1,14 2,87 0,39 1,27 2,950,00 163 1,79 2,74 1,26 2,11 1,56 2,44 2,54 1,15 2,85 2,73 1,56 2,44 2,54 1,16 2,84 2,731,66 0 1,57 0,00 0,00 1,57 0,00 0,00 1,57 0,00 0,00 1,57 0,00 0,00 1,57 0,00 0,00 1,574,87 32 1,25 1,18 0,00 1,25 0,51 0,67 1,45 0,38 0,80 1,50 0,51 0,66 1,45 0,38 0,80 1,500,00 99 1,09 2,63 0,97 1,26 1,49 2,10 1,55 1,10 2,49 1,68 1,50 2,09 1,54 1,11 2,48 1,670,00 0 2,07 0,00 0,00 2,07 0,00 0,00 2,07 0,00 0,00 2,07 0,00 0,00 2,07 0,00 0,00 2,071,17 32 1,46 1,18 0,00 1,46 0,51 0,67 1,66 0,38 0,80 1,71 0,51 0,66 1,66 0,38 0,80 1,713,59 99 1,35 2,63 0,97 1,54 1,49 2,10 1,85 1,10 2,49 1,99 1,50 2,09 1,85 1,11 2,48 1,980,00 0 1,87 0,00 0,00 1,87 0,00 0,00 1,87 0,00 0,00 1,87 0,00 0,00 1,87 0,00 0,00 1,870,00 31 1,48 1,21 0,45 1,60 0,52 1,14 1,83 0,39 1,28 1,89 0,53 1,14 1,83 0,39 1,27 1,881,17 75 1,51 2,74 1,26 1,78 1,56 2,44 2,15 1,15 2,85 2,31 1,56 2,44 2,14 1,16 2,84 2,303,59 81 1,06 2,10 1,90 1,35 0,98 3,02 1,61 0,72 3,28 1,68 0,98 3,02 1,61 0,73 3,27 1,680,00 81 1,25 2,17 1,83 1,58 1,00 3,00 1,89 0,74 3,26 1,98 1,01 2,99 1,89 0,75 3,25 1,980,00 116 0,96 2,10 1,90 1,21 0,98 3,02 1,45 0,72 3,28 1,51 0,98 3,02 1,44 0,73 3,27 1,511,66 118 1,05 2,17 1,83 1,32 1,00 3,00 1,59 0,74 3,26 1,66 1,01 2,99 1,58 0,75 3,25 1,664,87 145 0,84 2,10 1,90 1,07 0,98 3,02 1,28 0,72 3,28 1,34 0,98 3,02 1,28 0,73 3,27 1,344,74 145 1,20 2,17 1,83 1,52 1,00 3,00 1,82 0,74 3,26 1,91 1,01 2,99 1,82 0,75 3,25 1,914,85 152 1,10 2,17 1,83 1,42 1,00 3,00 1,74 0,74 3,26 1,83 1,01 2,99 1,74 0,75 3,25 1,834,74 164 0,90 2,10 1,90 1,19 0,98 3,02 1,47 0,72 3,28 1,55 0,98 3,02 1,47 0,73 3,27 1,554,85 164 0,83 2,17 1,83 1,08 1,00 3,00 1,34 0,74 3,26 1,42 1,01 2,99 1,34 0,75 3,25 1,424,74 181 1,33 2,10 1,90 1,78 0,98 3,02 2,24 0,72 3,28 2,38 0,98 3,02 2,24 0,73 3,27 2,374,85 181 0,82 2,17 1,83 1,08 1,00 3,00 1,37 0,74 3,26 1,45 1,01 2,99 1,36 0,75 3,25 1,454,85 194 1,22 2,17 1,83 1,63 1,00 3,00 2,08 0,74 3,26 2,22 1,01 2,99 2,08 0,75 3,25 2,214,74 194 1,25 2,10 1,90 1,70 0,98 3,02 2,16 0,72 3,28 2,30 0,98 3,02 2,16 0,73 3,27 2,304,85 194 1,18 2,10 1,90 1,60 0,98 3,02 2,04 0,72 3,28 2,17 0,98 3,02 2,03 0,73 3,27 2,16
1,41 1,08 1,61 1,97 1,87 2,20 1,96 1,97 1,87 2,19 1,96Average
86
formula has been tuned imposing a coefficient of reduction µ that has an inverse proportion with the cylindrical compressive strength f’c. The coefficient of reduction is chosen fixing the coefficient of reduction for the f’c = 20 MPa and f’c = 90 MPa. Two different possibilities are investigated:
‐ µ20 = 0.80 and µ90 = 0.35 (Figure 4.25, (II)). ‐ µ20 = 0.70 and µ90 = 0.35 (Figure 4.25, (III)).
The formula of the equation is:
Rk,4 c6.346 ln( 9,2) 14.126
'100
RIf f (Eq. 4.23)
Where μ is the coefficient of reduction as imposed (Table 4.13). Its trend is depicted in Figure 4.24 while in Figure 4.25 (II) and (III) can be seen the results of the application of µ.
Figure 4.23 ‐ Analysis and regression on the article A14. Table 4.13 ‐ Values of the coefficient of reduction μ for the logarithmic formula at the variation of the cylindrical compressive strength -′/ .
f' c μ II μ III
0,8 20 0,80 0,70
0,35 25 0,77 0,6830 0,74 0,6535 0,70 0,63
‐0,006428571 40 0,67 0,600,928571429 45 0,64 0,58
50 0,61 0,5555 0,58 0,5360 0,54 0,5070 0,48 0,4580 0,41 0,4090 0,35 0,35
87
Figure 4.24 –Variation of the coefficient of reduction µ according to the two different alternatives.
88
(I)
(II)
(III)
Figure 4.25 – Different possibilities at the variation of the imposed coefficient of reduction µ. In (I), where there is no reductions, it can be seen that for f’c > 30 MPa the limit of fRk,4 = 4 MPa is reached already for RI > 50 and this is not realistic. (II) and (III) have been modelled with two different coefficients of reduction µ in order to obtain a more realistic shape of the graph.
c
Rk,4 20 90
'min 6.346 ln 9.2 14.126 ; 4 1.0 & 1.0
100
ff RI
c
Rk,4 20 90
'min 6.346 ln 9.2 14.126 ; 4 0.7 & 0.35
100
ff RI
c
Rk,4 20 90
'min 6.346 ln 9.2 14.126 ; 4 0.8 & 0.35
100
ff RI
89
As reported in Table 4.14 the two equations are able to decrease the difference between the measured values and the estimated ones. The equation with an interval of the coefficient of reduction, that ranges from 0.80 to 0.35, has a better performance with an average value of Vexp/Vtot equal to 1.52 against the value of 1.41 that comes from the data of the article.
Table 4.14 ‐ Comparison between the values reported in the article A14 and the values estimated by using the formulas obtained from the back analysis performed only on the
specimens contained in A14.
For the RILEM alternative there are now seven equations, five coming from the back analysis overall the articles and the data processing of the cloud of points and two coming from the back analysis of the article A14, only. Comparing the formula, applying them on the specimens of article A14, the most promising one is the logarithmic equations with µ20 = 0.80 MPa and µ90 = 0.35 MPa but a further step is the comparison over the whole database. This is carried out in the Section “The Choice of the Best Formula”.
f* V f * V e/V* f II f*‐f II V e/V f III f*‐f III V e/V
MPa kN ‐ MPa kN ‐ MPa kN ‐
0,00 0 1,89 0,00 0,00 1,89 0,00 0,00 1,890,00 23 1,82 1,96 ‐0,67 1,66 1,77 ‐0,48 1,701,30 47 1,69 2,81 ‐0,16 1,65 2,53 0,12 1,712,65 70 1,83 3,60 0,40 1,91 3,26 0,74 1,984,22 0 2,42 0,00 0,00 2,42 0,00 0,00 2,420,00 68 2,32 1,81 ‐0,15 2,26 1,62 0,04 2,330,00 163 1,79 3,26 0,74 1,96 2,92 1,08 2,061,66 0 1,57 0,00 0,00 1,57 0,00 0,00 1,574,87 32 1,25 1,78 ‐0,60 1,11 1,59 ‐0,42 1,150,00 99 1,09 3,18 0,41 1,16 2,84 0,75 1,220,00 0 2,07 0,00 0,00 2,07 0,00 0,00 2,071,17 32 1,46 1,78 ‐0,60 1,31 1,59 ‐0,42 1,353,59 99 1,35 3,18 0,41 1,43 2,84 0,75 1,490,00 0 1,87 0,00 0,00 1,87 0,00 0,00 1,870,00 31 1,48 1,81 ‐0,15 1,44 1,62 0,04 1,491,17 75 1,51 3,26 0,74 1,66 2,92 1,08 1,743,59 81 1,06 2,58 1,42 1,26 2,30 1,70 1,310,00 81 1,25 2,63 1,37 1,48 2,35 1,65 1,540,00 116 0,96 2,58 1,42 1,14 2,30 1,70 1,181,66 118 1,05 2,63 1,37 1,24 2,35 1,65 1,294,87 145 0,84 2,58 1,42 1,00 2,30 1,70 1,044,74 145 1,20 2,63 1,37 1,43 2,35 1,65 1,484,85 152 1,10 2,63 1,37 1,32 2,35 1,65 1,384,74 164 0,90 2,58 1,42 1,10 2,30 1,70 1,154,85 164 0,83 2,63 1,37 1,00 2,35 1,65 1,054,74 181 1,33 2,58 1,42 1,64 2,30 1,70 1,724,85 181 0,82 2,63 1,37 1,00 2,35 1,65 1,054,85 194 1,22 2,63 1,37 1,50 2,35 1,65 1,584,74 194 1,25 2,58 1,42 1,56 2,30 1,70 1,644,85 194 1,18 2,58 1,42 1,47 2,30 1,70 1,54
1,41 0,66 1,52 0,89 1,57Average
90
The Choice of the Best Formula
For the choice of the best formula among all those obtained it is essential test it on:
‐ Group 3 (which contains all the articles that do not satisfy the requests and cannot be added in Group 1 and Group 2);
‐ Group 1 (because also some tests contained in this articles have not been included in the back analysis);
‐ the whole data base.
With a first glance it is possible to individuate the articles for which the formulas found are not suitable (for the DafStB those specimens that has Vexp/Vtot higher than 3.0 and lower than 0.8 are considered not suitable while, for RILEM and CNR; the ones with Vexp/Vtot higher than 2.5 and lower than 0.66). This diversification on the Vexp/Vtot is done taking into account the security coefficient that should be applied in the design stage, equal to 1.25 for DafStB and to 1.5 for RILEM and CNR. Picking out these article it is feasible to decrease the average value of Vexp/Vtot and its standard deviation as well as putting the finishing touches to the domain of application of the formulas.
Before to get any statistical analysis these articles have to be deleted from the database; the articles are:
‐ A02 due to the presence of stirrups on half of the span; ‐ A13 gives too high safety factor owing to its a/d ratio equal to 2.5 and
perhaps because the pre‐stressed reinforcement has much more importance than the flexural one.
‐ A14 has only two tests that do not perform well with the formula; this because they have a a/d ratio equal to 1.54.
‐ A20 confirms that the addition of fibres can improve shear strength even in deep beam but when the a/d ratio is reduced to value minor than 1.60 the formulas do not work anymore.
‐ A24 does not fit the formulas mainly for three specimens that has a a/d ratio that ranges from 1.20 to 1.80 moreover it is worth noting that those beams have a percentage of fibre Vf equal to 1.76.
‐ A27 shows a diffuse trend to minimize the shear strength that reactive powder concrete can carry obtaining value of Vexp/Vtot always higher than 2.3; the compressive strength around 150‐170 MPa, more than the presence of pre‐compression, could be the reason of the under‐estimation of the residual flexural tensile strength.
The CNR alternative, might due to its structure, seems to fit more articles without giving value of Vexp/Vtot too high and so the only articles that give problems are A02, A20 and A24.
Looking at DafStB, Table 4.15, it is possible to see that the equation that better perform the value of Vexp/Vtot is Equation A with an average value of Vexp/Vtot equal to 1.95 and a standard deviation equal to 0.59. Equation A is the equation obtained, in the previous section “The Back Analysis”, as the weight average of all the parameters of the lines found on the suitable articles (Group 1). The equation is drawn in Figure 4.26.
91
Table 4.15 ‐ Comparison of the average values of the safety factor Vexp/Vtot and its standard deviations among the five possible equations for the DafStB alternative.
Figure 4.26 – Equation of the DafStB alternative for the different cylindrical compressive strength. f’c.
In Table 4.16, for the RILEM alternative, it can be seen that the equation that better perform the value of Vexp/Vtot is Equation (II) with an average value of Vexp/Vtot equal to 1.451 and a standard deviation equal to 0.45. Equation (II) is the equation obtained in the previous section “The Check” (at page 83) that has a logarithmic shape translated in the origin and tuned with a coefficient of reduction µ that change linearly from 0.80 to 0.35 for the cylindrical compressive strength equal to 20 and 90, respectively. Figure 4.27 shows the plot of the selected equation.
Table 4.16 ‐ Comparison of the average value of the safety factor Vexp/Vtot and its standard deviation among the seven possible equations for the RILEM alternative.
Eq. A Eq. B Eq. C Eq. D Eq. E
average value 1,95 1,98 2,10 1,98 2,04standard deviation 0,59 0,60 0,63 0,60 0,62average value 1,92 1,95 2,08 1,95 2,01
standard deviation 0,58 0,59 0,63 0,59 0,61average value 2,06 2,09 2,20 2,09 2,15
standard deviation 0,56 0,57 0,58 0,57 0,58
Whole database
Group 1
Group 3
DafStB
Eq. A Eq. B Eq. C Eq. D Eq. E Eq. (II) Eq. (III)
average value 1,38 1,65 1,76 1,65 1,76 1,40 1,43standard deviation 0,44 0,49 0,52 0,49 0,52 0,42 0,43average value 1,34 1,63 1,75 1,63 1,75 1,39 1,43
standard deviation 0,41 0,48 0,52 0,48 0,52 0,39 0,40average value 1,48 1,73 1,83 1,73 1,83 1,45 1,49
standard deviation 0,45 0,48 0,49 0,48 0,49 0,45 0,45
Whole database
RILEM
Group 1
Group 3
' '
DafStB 0 c c res,limmin ; ;f m RI f m RI b f f
92
Figure 4.27 ‐ Equation of the RILEM alternative for the different cylindrical compressive strength. f’c.
For CNR alternative, as shown in Table 4.17 , the equation that better perform the value of Vexp/Vtot is Equation D with an average value of Vexp/Vtot equal to 1.49 and a standard deviation equal to 0.45. Equation D is the equation obtained in the previous section “The Back Analysis”; this, as the DafStB one, is a three branches line defined from the cloud of points deleting those tests that have fFtuk <0 (see Figure 4.20). The equation is depicted in Figure 4.28.
Table 4.17 ‐ Comparison of the average values of the safety factor Vexp/Vtot and its standard deviations among the five possible equations for the CNR alternative.
Eq. A Eq. B Eq. C Eq. D Eq. E
average value 1,52 1,50 1,57 1,49 1,56standard deviation 0,46 0,45 0,47 0,45 0,46average value 1,50 1,48 1,56 1,48 1,55
standard deviation 0,51 0,49 0,52 0,49 0,52average value 1,60 1,58 1,64 1,57 1,64
standard deviation 0,43 0,43 0,44 0,43 0,44Group 3
CNR
Whole database
Group 1
c
Rk,4 20 90
'min 6.346 ln 9.2 14.126 ; 4 0.7 & 0.35
100
ff RI
93
Figure 4.28 ‐ Equation of the CNR alternative for the different cylindrical compressive strength. f’c.
Now that the formulas for the determination of the DafStB, the RILEM and the CNR parameters have been chosen the comparison between the test data from literature and the theoretical formulas of Chapter 3 can be carried out.
For completing this analysis it is here finally reported the value of Vexp/Vtot for the Narayanan and Darwish’s alternative; the performance of the formula on the database, excluding the pre‐stressed beams, gives and average value of Vexp/Vtot equal to 1.58 with a standard deviation equal to 0.60.
res,lim
' '
CNR 0 c cmin ; ; ff m RI f m RI b f
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4.5 Comparison between the Test Data from
Literature and the Theoretical Formulas of
Chapter 3
Is it good to make here a precise resume of the equations that will be used to develop the comparison between the test data and the calculated ones.
‐ Alternative I: Narayanan & Darwish (1987)(Section 3.1)
±F≥F = –— ∙ ”!Ö-c8K7 + ‘Öô∏’÷ + 0.41◊(ÿ gŸ∞
‐ Alternative II: DafStB (2011) (Section 3.2)
±F≥F =0.15⁄7FK
∙ V ∙ ¤(100ô-7G)`/3 + 0.15678‹gŸ∞ +›7K ∙ -7FS,TK ∙ gŸℎ
⁄7FK
where
-7FS,TK = fimin fl0.0186 ∙ ¥µ; 0.0097 ∙ ¥µ + 0.8516;1.15 ∙ 100
-7Ö·‚ ∙
-7Ö
100
‐ Alternative III: RILEM (2003) (Section 3.3)
±F≥F = „H.`‰Âæ
∙ V ∙ ¤(100ô-7G)`/3 + 0.15678‹ + 0.12VKV`-SG,5Ê gŸ∞
where
-SG,5 = –min ”[6.346 ∙ ln(¥µ + 9.2) − 14.126] ∫æÁ
`HH∙ Ë; 4÷ÿ
‐ Alternative IV: CNR (2010) (Section 3.4)
±F≥F = ÈH.`‰Âæ
∙ V ∙ fi100ô ”1 + 7.5 ∫ƒø¡¬∫æø¬
÷ -7G ‚`/3
+ 0.15678Í gŸ∞
where
-[FTG = –min ”0.0484 ∙ ¥µ; 0.0252 ∙ ¥µ + 2.3997; Î∙`HH∫æÁ÷ÿ ∙ ∫æ
Á
`HH
Narayanan and Darwish’s formula can be applied only on the articles that contain RC or SFRC beams but it is not designed for elements that have pre‐compression. The same authors proposed another formula for pre‐stressed SFRC beams.
Since they contain pre‐tension forces, the articles A01, A13, A26, A27 and A28 are not investigated for this option.
Narayanan and Darwish’s equation on the database herein studied present an average value of the Vexp/Vtot equal to 1.53 and a standard deviation of 0.60.
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4.5.1 Effect of a/d Ratio
Figure 4.29 ‐ Graphs of the four alternatives at the variation of the parameter: a/d.
As depicted in Figure 4.29 the Narayanan and Darwish’s alternative shows a sensible increasing of the Vexp/Vtot for higher value of the a/d ratio. DafStB, RILEM and CNR do not show any influence of this parameter.
From these graphs it is possible to figure out how the four different formulas spread the Vexp/Vtot factor along the vertical axis; Narayanan and Darwish’s formula is the one that reaches the lowest values; DafStB is the alternative that has the larger range of Vexp/Vtot and that reaches the highest value of Vexp/Vtot; RILEM one has a quite compact distribution of the Vexp/Vtot values but a lot of tests fall below the line Vexp/Vtot =1; finally CNR alternative has the most of the tests with Vexp/Vtot between 1 and 2, with the highest value close to 3 and few specimens with Vexp/Vtot smaller than 1.
It is worth noting that all the values herein reported have been developed with all the safety factor γ = 1 and that, differently than the other ones, DafStB in the design should have 1.25 instead of 1.50. Thinking to be on the design stage, the Vexp/Vtot ranges should be almost equalized among the DafStB, the RILEM and the CNR alternatives.
96
4.5.2 Effect of the Maximum Aggregate Size
Figure 4.30 ‐ Graphs of the four alternatives at the variation of the parameter: maximum aggregate size [mm].
Many articles of the database miss the information about the maximum aggregate size used into the concrete mix. These cannot be included in this analysis.
Narayanan and Darwish’s graph shows a scatter diminution of the Vexp/Vtot values for higher values of the maximum aggregate size.
DafStB, RILEM and CNR graphs do not show any trend.
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4.5.3 Effect of Fibres Percentage
Figure 4.31 ‐ Graphs of the four alternatives at the variation of the parameter: fibres percentage [%].
The Vexp/Vtot ratio seems to decrease at the increasing of the fibres percentage. This trend is more visible in the Narayanan and Darwish’s graph than in the other
three. Moreover it can be noted that the DafStB, the RILEM as well as the CNR alternative for the tests that have the values of Vf equal to 1.00‐1.50‐2.00‐2.50 show a decrease of the correspondent value of Vexp/Vtot, too. The points standing between Vf =0 and Vf =1.00 look more untidy but always with a feeling of descending trend; this range (Vf = 0‐1.00) is the one mostly investigated.
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4.5.4 Effect of Longitudinal Reinforcement
Figure 4.32 ‐ Graphs of the four alternatives at the variation of the parameter: flexural reinforcement ρflex[%].
From the graphs depicted in Figure 4.32 no trend can be drawn except that for Narayanan and Darwish’s. It has a light growing trend at the increasing of the flexural reinforcement.
99
4.5.5 Effect of the Cylindrical Compressive
Strength
Figure 4.33 ‐ Graphs of the four alternatives at the variation of the parameter: cylindrical compressive strength [MPa].
Also from the Figure 4.33 no trend can be drawn. It can be seen that among 30 articles the most investigated concrete is the one with a
cylindrical compressive strength that ranges between 30 MPa and 60 MPa. Among these tests it is possible to note how much the value of Vexp/Vtot can vary and how the shear behaviour is influenced for sure by other parameters.
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4.5.6 Effect of the RI and the Fibre Factor F
Figure 4.34 ‐ Graphs of the four alternatives at the variation of the parameter: ratio index RI [%].
All the four graphs reported in Figure 4.34 show a dependence on the ratio index RI. The trend is a decrease of the Vexp/Vtot ratio at the increase of RI. Narayanan and
Darwish’s graph shows a more sharp influence of this effect. For the other three graphs it may be just an optical effect due to the concentration of the tests in the left part of the graphs.
The graphs look different than the ones drawn in Figure 4.31 because the ratio index includes also the aspect ratio of the fibres giving a more disperse cloud of points. The most common RI ranges from 30 % to 80 %.
In Figure 4.34 and Figure 4.31 check the behaviour of the formula in absence of fibres. In fact for ±∫ = 0 or ¥µ = 0 the values of the RC beams are reported along the y axis.
It is worth noting that the Narayanan and Darwish’s formula contains in itself the
fibre factor f f/F L D d that it is the ratio index RI but multiplied for the term
fd
that is the bond factor that accounts for different bond characteristics of the fibre (with values from 0.50 to 1.00). Fantilli et al. (2008) demonstrated how the different bond characteristics of the fibre act only when the fibres are perfectly orthogonal to crack surfaces; otherwise, in the case of fibres randomly inclined with respect to crack surfaces, the effect of the kind of the end of the fibre is replaced by the so‐called frictional snubbing effect.
101
In addition, since Vexp/Vtot decreases with increasing RI, Narayanan and Darwish’s formula seems to exaggerate the RI influence.
This is why in this thesis the ratio index RI has been considered more reliable than the fibre factor F.
Figure 4.35 ‐ Graphs of the four alternatives at the variation of the parameter: ratio index F.
However, since the fibre factor F is directly included into the Narayanan and Darwish’s formula, it is also interesting see the trend at the variation of F. Even in this case, as depicted in Figure 4.35, Narayanan and Darwish’s formula has a marked decreasing trend when the value of F increases while the other three alternatives keep always the same steady trend.
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4.5.7 Effect of the Effective Depth d
Figure 4.36 ‐ Graphs of the four alternatives at the variation of the parameter: effective depth d [mm].
No trend can be drawn from the graphs in Figure 4.36. It is worth noting that the majority of the tests have an effective depth that ranges
from 80 mm up to 250 mm.
103
Get dependences and relations from the plotting of these graphs is hard and complicated.
Anyway, some differences among the four formulas can be understood:
‐ Narayanan and Darwish’s formula looks to be more affected by the variation of the parameters investigated; in fact it shows a slightly dependence on the flexural reinforcement ρflex, on the a/d ratio, on the maximum aggregate size, on the Vf and the ratio index RI.
‐ On the contrary the other three alternatives are less affected by the variation of the investigated parameters and this is a good quality because it means that the formula used is already able to catch the parameters variation within itself.
‐ DafStB, RILEM and CNR alternatives in many graphs seem to have almost the same dislocation of the points but only tuned by a scale factor; moreover they show a more uniform values of Vexp/Vtot at the variation of the parameters investigated.
‐ Since the literature is full of articles that show a good agreement of their proposed formula (performed on the same batch of specimens they come from) it is worth highlighting that the herein proposed formulas have the same values of Vexp/Vtot both for the data they come from (Group 1) as well as the Group 3, that was not utilized during the back analysis.
Make a clear decision in the next chapter will not be feasible because all the calculations have been performed with approximation related to the purposed formula for the missing values.
On top of that the Narayanan and Darwish’s formula is completely different than the other three because it does not take care of the post‐cracking behaviour of the concrete while the other three alternatives are related to it.
105
Chapter 5
Conclusion and Future
Perspectives
5.1 Discussion of the Results
Nowadays, fibre reinforced concrete is in its fourth decade of development and it has established itself as one of the major building material; nevertheless, compared with its high performances, it is still not widely used. One of its greatest qualities is the improvement in shear capacity and the possibility of the substitution of the transversal reinforcement, earning, in this way, in terms of saving labour time and increasing durability of the structures.
In 1987 Narayanan and Darwish, after their studies, proposed a formula for the evaluation of the shear strength of SFRC beam (Section 3.1) that it is still competitive when compared with the most modern formulas ‐ like DafStB (2011), RILEM (2003) and CNR (2010).
This work is focused on how the Narayanan and Darwish’s formula still gives good results, even if it is a semi‐empirical formula, when compared with other formulas that are based on the post‐cracking characterization of the SFR concrete.
Notwithstanding the effort done by the standardization bodies like DafStB, RILEM and CNR, two problems still influence the research in this field. Firstly, the majority of the researchers and designers are used to Narayanan and Darwish’s formula for which all the data are easily available. Secondly the scientific community has not yet produced such a number of tests that it permits a direct comparison between empirical formulation and codes based on the post‐cracking characterization of the concrete.
The two problems are closely related since the Narayanan and Darwish’s formula gives, with less efforts, an average value of Vexp/Vtot not too far from the one that has
106
been found for the other three alternatives; this fact did not encourage the scientific community to deeply investigate the others formulas.
The second problem has been bypassed, in this thesis, using a back analysis for estimating the residual flexural tensile strength that was missed in all the articles except one. The results of this back analysis show a good agreement with the tests reported in the database; anyway the formulas that have been found are only a palliative to estimate the residual flexural tensile strength in case of absence of the characterization of the material according to one of the standards available in literature, it is worth to stress that these formulas are not a design instruments but just a tool here developed to have the possibility to compare the four alternatives.
It is supposed that the major factor contributing to the differences and variation between the estimated and the measured residual flexural tensile strength is related to the assumption done at the beginning of Section 4.4 about the safety factor Vexp/Vtot.
In particular it has been assumed that the SFRC beams should have the same Vexp/Vtot as the RC beams without fibres almost always tested as a comparison by the researchers. It is well known that the formula for the shear strength of member not requiring shear reinforcement has a different background than the part added in almost all of the “new generation” formulas for keeping into account the fibre effect. This aspect could probably lead to higher value of Vexp/Vtot.
Article A14 , that is the only one that reports both the post‐cracking characterization and the shear tests, shows the real potential of those formulas that use the residual flexural tensile strength to evaluate the shear capacity.
5.2 Proposal of the Best Alternative.
With the inverse analysis it has been possible to determine a three linear relationship between the fibre ratio index RI and the relative residual strength fres/f’c for the DafStB alternative and the CNR alternative while for the RILEM alternative there has been found a logarithmic relationship with an upper limit that is depicted as an horizontal branch. The average value of Vexp/Vtot for the four formulas are reported in Table 5.1.
Table 5.1 – Average values and standard deviations of Vexp/Vtot for the four alternatives
N&D DafStB RILEM CNR
Average value of Vexp/Vtot 1.58 1.95 1.40 1.49
Standard deviation of Vexp/Vtot 0.60 0.59 0.42 0.45
Since the comparison is affected by the inverse analysis it is difficult to state which
formula that is the best. However, further observations can be done analysing the graphs performed in Section 4.5:
‐ The Narayanan and Darwish’s formula, in most of these graphs, shows the aptitude to change the range of Vexp/Vtot at the variation of the investigated parameter. The other three alternatives do not show the same aptitude.
107
This is a drawback of the Narayanan and Darwish’s formula that is not able to keep almost a constant range of the safety factor Vexp/Vtot when the variables change; moreover, Narayanan and Darwish’s formula has not been designed for beam with pre‐stressing forces.
‐ RILEM and DafStB alternatives have a similar behaviour probably due to the structure of their formulas that add the fibre contribute as a separate addendum. RILEM has the lowest value of the average of Vexp/Vtot and standard deviation among the four investigated equations; the two values are 1.40 and 0.42 respectively. DafStB, on the opposite, has the highest values of these parameters (see Table 5.1).
‐ CNR alternative has the most compact range of Vexp/Vtot (between 1 and 2) but an average value of Vexp/Vtot (among the database) slightly higher than the RILEM one; the average value of Vexp/Vtot is equal to 1.49 with a standard deviation equal to 0.45.
In conclusion the advantages of the Narayanan and Darwish’s formula are that they do not require any sophisticated testing equipment because it is enough to consider the measurement of the concrete compressive strength. Moreover, even if it does not take care of the post‐cracking behaviour of the material neither its toughness, it has shown a good average value of Vexp/Vtot (1.58 with a standard deviation equal to 0.60).
However, it seems that the RILEM and CNR alternatives have a high potential to describe and estimate the shear capacity of SFRC beam; especially because they are based on the same fundamentals as the design of normal reinforced concrete. This belief is also supported by the analysis carried out in Section 4.4, where the average value of Vexp/Vtot (that comes from the measured values of the residual flexural tensile strength) is equal to 1.41. In addition, since different mixes can have the same residual flexural tensile strength, the advantage of determining it is that this quantity can be viewed as an intrinsic material property although it also (besides the concrete mix) depends on the specimen size, fibre percentage and kind of fibres; moreover, once that the material has been chosen and the mechanism behind the shear behaviour has been identified, the design optimization to achieve the desired shear strength performance of the beam can be realized by optimizing only the geometry. This is what already happens when we work with concrete with guaranteed performance. The designer should not think about the materials that have to be included in the mix but he must indicate the material (and its compressive strength) that he used in the design phase.
Another advantage is that they can include the pre‐stressing force when it is present. Since the RILEM Recommendation set up the limit of 50/60 MPa for the compressive strength of the concrete for which its formulation is still valid (as well as the DafStB alternative does), the CNR covers a larger domain of concrete because it does not establish any limit; among the three “new generation” formulas CNR alternative seems to be the most reliable.
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5.3 Future Perspectives
Although the research literature on Steel Fibre Reinforced Concrete (SFRC) is extensive, there is still need of further studies especially for those codes that require the post‐cracking characterization of the concrete. In fact, as aforementioned, there is a strong need of increasing the number of tests that, as article A14 does, report the post‐cracking properties. These tests could be used to carry out a more precise comparison of the formulas for the evaluation of the shear capacity.
In any case, there is still room for further improvements regarding design rules and the main challenges is to try to get a unique test set up that can be included in EC 2.
If all the needs will be satisfied and a unique formula will be recognized by all the research community and the normative bodies, it will be possible to cover the lack of Eurocode 2 about the determination of the shear capacity of steel fibre reinforced concrete elements.
Regarding this thesis, the future works that can be developed are:
‐ Check the validity of the suggested approach for the evaluation of the residual flexural tensile strength on a wider population of beams that have their own characterization (according to Section 4.3.1);
‐ Develop a design software and tools that practising engineer can use. ‐ Provide further developments towards standardization of the methods
DafStB, RILEM and CNR in order that they can be used indifferently, as a natural step to determine material properties.
‐ Investigate high‐strength fibre reinforced concrete beams subjected to shear. Since high‐strength concrete is a brittle material the addition of fibre is a good compromise between strength and ductility that is worth to be further analyzed.
‐ Extend the field of investigation/validity for the phenomenon of the punching in slab that, as pointed out by De Hanai et al. in the papers here contained in Appendix B [A16] and by the DafStB Guideline too, can be represented by the same strength mechanisms as the shear in beams.
Future studies should approach the recent Model Code 2010, too; it seems to be successful on the harmonization of different ideas about the design procedure for shear and punching in RC as well as SFRC.
109
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Romualdi, J.P & Batson, G.B. (1963): Mechanics of Crack Arrest in Concrete. Proceeding, American Society of Civil Engineering, Journal, Engineering Mechanics Division, Volume 89, EM3, pp. 147‐168.
112
Romualdi, J.P. & Mandel, J.A. (1964): Tensile Strength of Concrete Affected by Uniformly Distributed and Closely Spaced Short Lengths of Wire Reinforcement. ACI Journal Proc, Volume 61, No. 6, Detroit, USA, 1964, pp. 657‐671.
SCA (1997): Steel Fibre Reinforced Concrete ‐ Recommendations for Design, Construction and Testing. (Stålfiberbetong ‐ rekommendationer för konstruktion, utförande och provning). Concrete Report No. 4, 2nd Edn, Stockholm, Sweden, 1997. (in Swedish)
Shah, S. P., Ouyang, C. & Swartz S. E. (1995): Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and other Brittle Materials. John Wiley and Sons, New York, USA.
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Sivakumar, A. & Manu, S. (2007): Mechanical Properties of High Strength Concrete Reinforced with Metallic and Non‐Metallic Fibre. Cement & Concrete Composites, Volume 29, pp. 603‐608.
Sorelli, L. G., Meda, A. & Plizzari, G.A. (2005): Bending and Uniaxial Tensile Tests on Concrete Reinforced with Hybrid Steel Fibers. Journal of Materials in Civil Engineering ASCE, Volume 17, No. 5, pp. 519‐527.
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113
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A‐1
Appendix A
The articles are here reported with a code (e.g. A01). All the references are reported in Appendix B.
The following legend should be used to read the database.
A‐2
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
A01 1TL‐1 0,40 28 14 4,31 530 2,00 I 300 52 265 55 295 3000 3500 0,00 3,98 36,8* 44,3 49,3 3,38 4PBT S1TLF‐1 0,40 28 14 4,31 530 2,00 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 36,9* 44,5 80,3 5,51 4PBT S1TL‐2 0,40 28 14 4,31 530 3,43 I 300 52 265 55 295 3000 3500 0,00 3,98 34,9* 42,0 19,0 1,30 4PBT S1TLF‐2 0,40 28 14 4,31 530 3,43 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 42,4* 51,1 59,0 4,05 4PBT S1TL‐3 0,40 28 14 4,31 530 4,91 I 300 52 265 55 295 3000 3500 0,00 3,98 35,4* 42,6 17,5 1,20 4PBT S1TLF‐3 0,40 28 14 4,31 530 4,91 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 37,4* 45,0 42,5 2,92 4PBT S2TL‐1 0,40 28 14 2,76 530 2,00 I 300 52 265 55 295 3000 3500 0,00 3,98 37,8* 45,6 36,5 2,50 4PBT S2TLF‐1 0,40 28 14 2,76 530 2,00 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 39,2* 47,2 71,9 4,93 4PBT S2TL‐2 0,40 28 14 2,76 530 3,43 I 300 52 265 55 295 3000 3500 0,00 3,98 34,6* 41,7 17,8 1,22 4PBT S2TLF‐2 0,40 28 14 2,76 530 3,43 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 34,4* 41,4 45,6 3,13 4PBT S2TL‐3 0,40 28 14 2,76 530 4,91 I 300 52 265 55 295 3000 3500 0,00 3,98 37,4* 45,1 15,5 1,06 4PBT S2TLF‐3 0,40 28 14 2,76 530 4,91 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 37,3* 44,9 42,8 2,94 4PBT S‐F3TL‐1 0,40 28 14 1,55 530 2,00 I 300 52 265 55 295 3000 3500 0,00 3,98 38,8* 46,7 31,5 2,16 4PBT S3TLF‐1 0,40 28 14 1,55 530 2,00 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 37,0* 44,6 67,8 4,65 4PBT S‐F3TL‐2 0,40 28 14 1,55 530 3,43 I 300 52 265 55 295 3000 3500 0,00 3,98 34,0* 41,0 15,0 1,03 4PBT S3TLF‐2 0,40 28 14 1,55 530 3,43 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 35,8* 43,1 41,5 2,85 4PBT F3TL‐3 0,40 28 14 1,55 530 4,91 I 300 52 265 55 295 3000 3500 0,00 3,98 35,1* 42,3 15,1 1,04 4PBT S3TLF‐3 0,40 28 14 1,55 530 4,91 I 300 52 265 55 295 3000 3500 crimped 0,50 50 100 1,00 3,98 33,7* 40,6 29,5 2,02 4PBT FTF1 2,00 2,80 61 duoform 93,8 2,13 4,98 37,9* 45,7 71,3TF2 2,00 2,80 43 duoform 63 2,25 4,98 41,3* 49,8 81,0TF3 2,00 2,80 14 duoform 63 0,75 4,98 54,9* 66,1 64,5TF4 2,00 2,80 28 duoform 63 1,50 4,98 44,9* 54,1 61,0TF5 2,00 2,80 57 duoform 63 3,00 4,98 43,6* 52,5 78,9TF6 2,00 2,80 55 duoform 80 2,25 4,98 35,3* 42,5 59,6TF7 2,00 2,80 64 duoform 93,8 2,25 4,98 35,1* 42,3 76,4TF8 2,00 2,80 35 duoform 80 1,50 4,98 43,3* 52,2 65,0TF9 2,00 2,80 36 duoform 80 1,50 4,98 20,2* 24,3 55,0TF10 2,00 2,80 36 duoform 80 1,50 4,98 63,3* 76,3 69,5TF11 2,00 2,80 51 duoform 80 1,50 4,98 45,7* 55,0 93,8TF12 2,00 2,80 22 duoform 80 1,50 4,98 45,7* 55,0 55,4TFL1 2,00 2,80 35 duoform 80 1,50 4,98 50,7* 61,1 62,4TFL2 2,00 2,80 41 duoform 80 1,50 4,98 40,7* 49,0 81,4TFL3 2,00 2,80 7 duoform 80 0,30 4,98 40,8* 49,1 36,9TFL4 2,00 2,80 18 duoform 80 0,75 4,98 34,9* 42,1 45,0TFL5 2,00 2,80 27 duoform 80 1,13 4,98 43,8* 52,8 57,4TFL6 2,00 2,80 53 duoform 80 2,25 4,98 44,9* 54,1 60,6
1 2,00 2,80 73 plain 62,5 1,00 4,15 40,3* 48,6 215,82 2,00 2,80 117 plain 100 1,00 4,15 47,8* 57,6 191,4
A02 B18‐0a 41 MPa 10 2,70 3,43 R 455 381 152 2136 2440 ‐ ‐ 0 0,00 42,8 51,6* #### 1,10 3PBT DTB18‐0b 41 MPa 10 2,70 3,43 R 455 381 152 2136 2440 ‐ ‐ 0 0,00 42,8 51,6* 99,11 1,10 3PBT DTB18‐1a 41 MPa 10 2,00 3,43 R 455 381 152 2136 2440 hooked 0,55 30 55 0,75 44,8 54,0* #### 2,90 3PBT SC+STB18‐1b 41 MPa 10 2,00 3,43 R 455 381 152 2136 2440 hooked 0,55 30 55 0,75 44,8 54,0* #### 2,80 3PBT ST+DTB18‐2a 41 MPa 10 2,00 3,50 R 381 152 2236 2440 hooked 0,55 30 55 1,00 38,1 45,9* #### 3,00 3PBT ST+DT
Cross‐sectionSpecimens
Gravel
By
Lytag‐Sand cube strngth 40/45 N/mm2
Grav
el
16503,2716492,4516550,9316521,2316507,1116500,1516485,7916054,4216507,94
23083,909886,6216009,0718503,4016099,7716029,1416002,4215985,4776205,4576209,15
FibresMix
Beam size
Flexural Reinforcement
16507,94
Concrete Shear Force at [kN]
A‐3
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
B18‐2b 41 MPa 10 2,00 3,50 R 381 152 2236 2440 hooked 0,55 30 55 1,00 38,1 45,9* #### 3,10 3PBT ST+DTB18‐2c 41 MPa 10 2,70 3,50 R 381 152 2236 2440 hooked 0,55 30 55 1,00 38,1 45,9* #### 3,50 3PBT NAB18‐2d 41 MPa 10 2,70 3,50 R 381 152 2236 2440 hooked 0,55 30 55 1,00 38,1 45,9* #### 2,60 3PBT NAB18‐3a 41 MPa 10 2,70 3,43 R 381 152 2136 2440 hooked 0,55 30 55 1,50 31,0 37,3* #### 2,60 3PBT ST+DTB18‐3b 41 MPa 10 2,70 3,43 R 381 152 2136 2440 hooked 0,55 30 55 1,50 31,0 37,3* #### 3,40 3PBT SC+STB18‐3c 41 MPa 10 2,70 3,43 R 381 152 2136 2440 hooked 0,55 30 55 1,50 44,9 54,1* #### 3,30 3PBT ST+DTB18‐3d 41 MPa 10 2,70 3,43 R 381 152 2136 2440 hooked 0,55 30 55 1,50 44,9 54,1* #### 3,30 3PBT ST+DTB18‐5a 41 MPa 10 2,70 3,43 R 610 152 2136 2440 hooked 0,75 60 80 1,00 49,2 59,3* #### 3,00 3PBT DTB18‐5b 41 MPa 10 2,70 3,43 R 610 152 2136 2440 hooked 0,75 60 80 1,00 49,2 59,3* #### 3,80 3PBT ST+DTB18‐7a 41 MPa 10 2,70 3,43 R 610 152 2136 2440 hooked 0,38 30 80 0,75 43,3 52,2* #### 3,30 3PBT ST+DTB18‐7b 41 MPa 10 2,00 3,43 R 610 152 2136 2440 hooked 0,38 30 80 0,75 43,3 52,2* #### 3,30 3PBT ST+DTB27‐1a 41 MPa 10 2,00 3,50 R 610 203 3558 4320 hooked 0,55 30 55 0,75 50,8 61,2* #### 2,90 3PBT ST+DTB27‐1b 41 MPa 10 2,00 3,50 R 610 203 3558 4320 hooked 0,55 30 55 0,75 50,8 61,2* #### 2,70 3PBT DTB27‐2a 41 MPa 10 2,00 3,50 R 610 203 3558 4320 hooked 0,75 60 80 0,75 28,7 34,6* #### 2,80 3PBT SC+STB27‐2b 41 MPa 10 2,00 3,50 R 610 203 3558 4320 hooked 0,75 60 80 0,75 28,7 34,6* #### 2,80 3PBT DTB27‐3a 41 MPa 10 1,60 3,50 R 610 203 3558 4320 hooked 0,55 30 55 0,75 42,3 51,0* #### 2,70 3PBT FB27‐3b 41 MPa 10 1,60 3,50 R 610 203 3558 4320 hooked 0,55 30 55 0,75 42,3 51,0* #### 2,80 3PBT SC+STB27‐4a 41 MPa 10 1,60 3,50 R 610 203 3558 4320 hooked 0,75 60 80 0,75 29,6 35,7* #### 2,10 3PBT ST+DTB27‐4b 41 MPa 10 1,60 3,50 R 610 203 3558 4320 hooked 0,75 60 80 0,75 29,6 35,7* #### 1,80 3PBT ST+DTB27‐5 41 MPa 10 2,10 3,50 R 610 203 3558 4320 hooked 0,55 30 55 1,50 44,4 53,5* #### 3,50 3PBT SC+STB27‐6 41 MPa 10 2,10 3,50 R 610 203 3558 4320 hooked 0,75 60 80 1,50 42,8 51,6* #### 3,40 3PBT ST+DTB27‐7 41 MPa 10 1,60 3,50 R 610 203 3558 4320 hooked ‐ ‐ 0 0,00 37,0 44,6* #### 1,30 3PBT DTB27‐8 41 MPa 10 1,60 3,50 R 610 203 3558 4320 stirrup ‐ ‐ 0 0,00 37,0 44,6* #### 1,80 3PBT DT
A03 B‐2‐1.0‐L 0,25 10 0,37 470 2 R 250 215 125 1360 1680 30hooked 0,80 60 75 1,00 76,4* 92,0 90,2 1,68 4PBTB‐4‐1,0‐L 0,25 10 0,37 470 4 R 250 215 125 2220 2540 30hooked 0,80 60 75 1,00 76,9* 92,6 48,0 0,89 4PBT SB‐6‐1,0‐L 0,25 10 0,37 470 6 R 250 215 125 3080 3400 30hooked 0,80 60 75 1,00 77,8* 93,7 30,1 0,56 4PBTB‐1‐0,5‐A 0,25 10 2,84 470 1 R 250 215 125 930 1250 30hooked 0,80 60 75 0,50 82,2* 99,0 480,0 9,09 4PBTB‐2‐0,5‐A 0,25 10 2,84 470 2 R 250 215 125 1360 1680 30hooked 0,80 60 75 0,50 82,3* 99,1 265,0 4,82 4PBTB‐4‐0,5‐A 0,25 10 2,84 470 4 R 250 215 125 2220 2540 30hooked 0,80 60 75 0,50 79,2* 95,4 122,0 2,27 4PBT SB‐6‐0,5‐A 0,25 10 2,84 470 6 R 250 215 125 3080 3400 30hooked 0,80 60 75 0,50 79,5* 95,8 112,0 1,95 4PBTB‐1‐1,0‐A 0,25 10 2,84 470 1 R 250 215 125 930 1250 30hooked 0,80 60 75 1,00 79,1* 95,3 680,0 12,74 4PBT FB‐2‐1,0‐A 0,25 10 2,84 470 2 R 250 215 125 1360 1680 30hooked 0,80 60 75 1,00 79,1* 95,3 325,0 6,06 4PBTB‐4‐1,0‐A 0,25 10 2,84 470 4 R 250 215 125 2220 2540 30hooked 0,80 60 75 1,00 80,9* 97,5 172,0 3,17 4PBT SB‐6‐1,0‐A 0,25 10 2,84 470 6 R 250 215 125 3080 3400 30hooked 0,80 60 75 1,00 83,4* 100,5 103,0 1,96 4PBT FB‐1‐1,5‐A 0,25 10 2,84 470 1 R 250 215 125 930 1250 30hooked 0,80 60 75 1,50 80,0* 96,4 745,0 13,95 4PBTB‐2‐1,5‐A 0,25 10 2,84 470 2 R 250 215 125 1360 1680 30hooked 0,80 60 75 1,50 80,2* 96,6 385,0 7,21 4PBTB‐4‐1,5‐A 0,25 10 2,84 470 4 R 250 215 125 2220 2540 30hooked 0,80 60 75 1,50 80,6* 97,1 187,0 3,51 4PBT SB‐6‐1,5‐A 0,25 10 2,84 470 6 R 250 215 125 3080 3400 30hooked 0,80 60 75 1,50 84,1* 101,3 105,0 1,98 4PBTB‐2‐1,0‐M 0,25 10 4,58 470 2 R 250 215 125 1360 1680 30hooked 0,80 60 75 1,00 78,4* 94,5 360,0 6,73 4PBT SB‐4‐1,0‐M 0,25 10 4,58 470 4 R 250 215 125 2220 2540 30hooked 0,80 60 75 1,00 77,9* 93,8 208,0 3,88 4PBT SB‐6‐1,0‐M 0,25 10 4,58 470 6 R 250 215 125 3080 3400 30hooked 0,80 60 75 1,00 78,9* 95,0 156,0 2,93 4PBT F
A04 B3 4,40 R 0,22 33,6 40,5*C2 4,20 R 0,22 33,6 40,5*
Cross‐sectionSpecimens
By
FibresMix
Beam size
Flexural Reinforcement
Portland cement High‐Strength Fiber Reinforced Concrete.
Concrete Shear Force at [kN]
A‐4
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[Mpa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[Mpa]
f' c.
cyl,m
[Mpa]
f cu.28,m
[Mpa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
D2 4,30 R 0,22 33,6 40,5*F1 4,00 R 0,44 40,2 48,4*G3 4,40 R 0,22 33,2 40,0*L1 4,00 R 0,22 33,2 40,0*M1 4,60 R 0,22 33,2 40,0*M3 4,40 R 0,22 33,2 40,0*N1 5,00 R 0,22 33,2 40,0*N2 4,80 R 0,22 33,2 40,0*P2 4,20 R 0,44 40,2 48,4*R1 3,20 R 0,88 39,7 47,8*R2 3,40 R 0,88 39,7 47,8*S3 3,40 R 0,88 39,7 47,8*W1 1,20 R 1,76 39,8 47,9*W2 1,20 R 1,76 39,8 47,9*U1 2,80 R 1,76 39,8 47,9*V2 1,80 R 1,76 39,8 47,9*A1 2,37 R 152 203 2000 0,40 36,5 44,0* 115,6 SA2 3,56 R 152 203 2000 0,40 36,5 44,0* 93,4 SA3 4,74 R 152 203 2000 0,40 36,5 44,0* 81,8 SB1 2,37 R 152 203 2000 0,80 37,2 44,9* 161,0 SB2 3,56 R 152 203 2000 0,80 37,2 44,9* 106,8 SB3 4,74 R 152 203 2000 0,80 37,2 44,9* 0,9 FC1 2,37 R 152 203 2000 0,40 42,7 51,5* 1,7 SC2 3,56 R 152 203 2000 0,40 42,7 51,5* 1,4 SC3 4,74 R 152 203 2000 0,40 42,7 51,5* 1,2 SD1 2,37 R 152 203 2000 0,80 40,0 48,2* 1,7 SD2 3,56 R 152 203 2000 0,80 40,0 48,2* 1,4 SD3 4,74 R 152 203 2000 0,80 40,0 48,2* 1,3 S
A05 1 0,43 R 0,00 0,2 0,3* 4PBT2 0,46 R 2,50 0,4 0,5* 4PBT3 0,47 R 2,00 0,4 0,5* 4PBT4 0,46 R 1,40 0,4 0,5* 4PBT
A06 A 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 21,0 25,3* 22,09 3PBT SB 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 46 100 1,50 1,52 27,8 33,5* 36,9 3PBT SC 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 46 100 1,00 1,52 24,1 29,0* 33,0 3PBT SD 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 46 100 0,50 1,52 21,9 26,4* 29,8 3PBT SE 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 37 80 1,50 1,52 22,5 27,1* 34,8 3PBT SF 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 37 80 1,00 1,52 23,0 27,7* 32,7 3PBT SG 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 37 80 0,50 1,52 22,1 26,7* 31,9 3PBT SH 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 28 60 1,50 1,52 22,6 27,2* 33,3 3PBT SI 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 28 60 1,00 1,52 21,7 26,2* 31,2 3PBT SJ 0,62 10 1,72 670 2,34 R 150 131 100 1080 1220 0,46 28 60 0,50 1,52 21,2 25,6* 28,4 3PBT S
Cross‐sectionSpecimens
By
FibresMix
Beam size
Flexural Reinforcement
Portland cement
Concrete Shear Force at [kN]
Black annealed steel wire
(26 SWG)
A‐5
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
A07 A21 18 1,26 510 2,00 R 200 180 100 1296 1600 ‐ 0,00 1,87 18,7* 22,5 19,2 4PBT SA22 25 1,26 510 2,00 R 200 180 100 1296 1600 0,54 27 50 0,50 2,65 23,8* 28,7 30,5 4PBT SA23 25 1,26 510 2,00 R 200 180 100 1296 1600 0,54 54 100 0,50 2,88 26,7* 32,2 36,1 4PBT SA24 23 1,26 510 2,00 R 200 180 100 1296 1600 0,54 27 50 1,00 2,67 24,1* 29,0 40,5 4PBT SA25 23 1,26 510 2,00 R 200 180 100 1296 1600 0,54 54 100 1,00 2,90 27,1* 32,6 50,8 4PBT FA26 25 1,26 510 2,00 R 200 180 100 1296 1600 stirrups 0,00 1,93 22,8* 27,5 51,6 4PBT FB31 23 1,26 510 3,00 R 200 180 100 1646 1950 ‐ 0,00 1,90 22,2* 26,8 22,3 4PBT SB32 23 1,26 510 3,00 R 200 180 100 1646 1950 0,54 27 50 1,00 2,87 26,6* 32,1 29,5 4PBT SB33 25 1,26 510 3,00 R 200 180 100 1646 1950 0,54 54 100 1,00 2,88 26,8* 32,3 36,8 4PBT SB34 23 1,26 510 3,00 R 200 180 100 1646 1950 0,54 27 50 1,50 2,92 27,2* 32,8 45,1 4PBT FB35 23 1,26 510 3,00 R 200 180 100 1646 1950 stirrups 0,00 1,91 21,2* 25,6 35,7 4PBT F
A08 SP1 A 0,40 2,00 530 2,00 R 150 130 85 900 50,0* 60,2 29,3 2,30 4PBTSP2 A 0,40 2,00 530 2,50 R 150 130 85 1030 50,0* 60,2 20,8 1,63 4PBTSP3 A 0,40 2,00 530 3,00 R 150 130 85 1160 50,0* 60,2 20,1 1,58 4PBTSP4 B 0,55 2,00 530 2,00 R 150 130 85 900 35,9* 43,3 23,7 1,86 4PBTSP5 B 0,55 2,00 530 2,50 R 150 130 85 1030 30,5* 36,7 18,7 1,47 4PBTSP6 B 0,55 2,00 530 3,00 R 150 130 85 1160 35,9* 43,3 21,8 1,71 4PBTSS1 A 0,40 2,00 530 2,00 R 150 130 85 900 62,3* 75,0 37,0 2,90 4PBTSS2 A 0,40 2,00 530 2,50 R 150 130 85 1030 50,0* 60,2 28,8 2,26 4PBTSS3 A 0,40 2,00 530 3,00 R 150 130 85 1160 50,0* 60,2 28,3 2,22 4PBTSS4 B 0,55 2,00 530 2,00 R 150 130 85 900 35,9* 43,3 32,0 2,51 4PBTSS5 B 0,55 2,00 530 2,50 R 150 130 85 1030 30,5* 36,7 29,8 2,34 4PBTSS6 B 0,55 2,00 530 3,00 R 150 130 85 1160 35,9* 43,3 30,0 2,35 4PBTS18 A 0,40 5,72 530 3,10 R 150 126 85 1160 47,5* 57,2 41,1 3,22 4PBTS24 A 0,40 5,72 530 3,10 R 150 126 85 1160 47,5* 57,2 59,5 4,67 4PBTS25 A 0,40 5,72 530 3,10 R 150 126 85 1160 47,5* 57,2 75,6 5,93 4PBTS26 A 0,40 5,72 530 3,10 R 150 126 85 1160 47,5* 57,2 86,3 6,77 4PBTSF1 A 0,40 2,00 530 2,00 R 150 130 85 900 0,30 30 100 0,25 4,15 51,2* 61,7 37,7 2,96 4PBT SSF2 A 0,40 2,00 530 2,50 R 150 130 85 1030 0,30 30 100 0,25 4,15 51,2* 61,7 34,0 2,67 4PBT SSF3 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 0,25 4,15 51,2* 61,7 35,3 2,77 4PBT SSF4 B 0,55 2,00 530 2,00 R 150 130 85 900 0,30 30 100 0,25 4,15 33,1* 39,9 34,6 2,71 4PBT SSF5 B 0,55 2,00 530 2,50 R 150 130 85 1030 0,30 30 100 0,25 4,15 33,1* 39,9 26,4 2,07 4PBT SSF6 B 0,55 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 0,25 4,15 33,1* 39,9 24,7 1,94 4PBT SB1 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 40 133 0,50 4,15 51,2* 61,7 41,2 3,23 4PBT SB2 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 40 133 1,00 4,15 59,3* 71,5 46,7 3,66 4PBT F‐SB3 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 1,50 4,15 63,7* 76,7 46,4 3,64 4PBT FB4 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 2,00 4,15 66,0* 79,5 47,7 3,74 4PBT FB5 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 2,50 4,15 64,1* 77,2 47,9 3,76 4PBT FB6 A 0,40 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 3,00 4,15 62,9* 75,8 49,1 3,85 4PBT FB7 B 0,55 2,00 530 3,00 R 150 130 85 1160 0,30 40 133 0,50 4,15 35,1* 42,3 25,1 1,97 4PBT SB9 B 0,55 2,00 530 3,00 R 150 130 85 1160 0,30 30 100 1,00 4,15 34,4* 41,4 37,9 2,97 4PBT SB11 A 0,40 2,00 530 2,00 R 150 130 85 900 0,30 40 133 0,50 4,15 51,2* 61,7 58,9 4,62 4PBT SB12 A 0,40 2,00 530 2,50 R 150 130 85 1030 0,30 40 133 0,50 4,15 51,2* 61,7 47,0 3,69 4PBT S
crimped
Cross‐sectionSpecimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
STIRRUP spacing 80/130
mm with rv=0,25/2 and
f=3/6
chopping
binding
wire
(circular)
choppi
ng
binding
wire
A‐6
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
B13 A 0,40 2,00 530 3,50 R 150 130 85 1290 0,30 40 133 0,50 4,15 46,2* 55,7 33,3 2,61 4PBT SB14 A 0,40 2,00 530 2,00 R 150 130 85 900 0,30 40 133 1,00 4,15 55,8* 67,2 71,0 5,57 4PBT SB15 A 0,40 2,00 530 2,50 R 150 130 85 1030 0,30 40 133 1,00 4,15 55,8* 67,2 56,4 4,42 4PBT SB16 A 0,40 2,00 530 3,50 R 150 130 85 1290 0,30 40 133 1,00 4,15 59,7* 71,9 37,9 2,97 4PBT SB17 A 0,40 3,69 530 3,00 R 150 128 85 1160 0,30 40 133 0,50 4,15 46,2* 55,7 37,7 2,96 4PBT SB18 A 0,40 5,72 530 3,10 R 150 126 85 1160 0,30 40 133 0,50 4,15 46,2* 55,7 45,3 3,55 4PBT SB19 B 0,55 3,69 530 3,00 R 150 128 85 1160 0,30 40 133 0,50 4,15 35,1* 42,3 28,6 2,24 4PBT SB20 B 0,55 5,72 530 3,10 R 150 126 85 1160 0,30 40 133 0,50 4,15 35,1* 42,3 29,7 2,33 4PBT SB23 A 0,40 3,69 530 3,00 R 150 128 85 1160 0,30 40 133 1,00 4,15 59,7* 71,9 55,7 4,37 4PBT SB24 A 0,40 5,72 530 3,10 R 150 126 85 1160 0,30 40 133 1,00 4,15 59,7* 71,9 63,8 5,00 4PBT SB25 A 0,40 5,72 530 3,10 R 150 126 85 1160 0,30 30 100 1,50 4,15 55,6* 67,0 61,8 4,85 4PBT SB26 A 0,40 5,72 530 3,10 R 150 126 85 1160 0,30 30 100 2,00 4,15 46,4* 55,9 62,9 4,93 4PBT SB27 A 0,40 3,69 530 3,00 R 150 128 85 1160 0,30 30 100 1,50 4,15 55,6* 67,0 56,9 4,46 4PBT SB28 A 0,40 5,72 530 2,00 R 150 126 85 900 0,30 30 100 0,50 4,15 49,2* 59,3 69,6 5,46 4PBT SB29 A 0,40 5,72 530 2,00 R 150 126 85 900 0,30 30 100 1,00 4,15 49,8* 60,0 86,3 6,77 4PBT SB30 A 0,40 5,72 530 2,00 R 150 126 85 900 0,30 30 100 1,50 4,15 55,6* 67,0 91,2 7,15 4PBT SB31 A 0,40 5,72 530 2,00 R 150 126 85 900 0,30 30 100 2,00 4,15 46,4* 55,9 80,3 6,30 4PBT S
A09 1 0,58 28 1,18 2,00 R 150 133 100 800 850 0,00 2,36 13,1* 15,8 19,9 1,492 0,58 28 1,18 2,00 R 150 133 100 800 850 100 0,25 2,36 13,5* 16,3 22,5 1,693 0,58 28 1,18 2,00 R 150 133 100 800 850 100 0,50 2,36 13,8* 16,6 25,0 1,874 0,58 28 1,18 2,00 R 150 133 100 800 850 100 0,75 2,36 14,0* 16,9 27,3 2,055 0,58 28 1,18 2,00 R 150 133 100 800 850 100 1,00 2,36 14,3* 17,3 29,8 2,236 0,58 28 1,18 2,00 R 150 133 100 800 850 100 1,25 2,36 14,5* 17,4 32,0 2,407 0,58 28 1,18 2,00 R 150 133 100 800 850 100 1,50 2,36 14,7* 17,7 34,0 2,55
A10 S1 28 1,34 1,90 R 300 263 150 1600 42,3 51,0* 64,5 1,57 4PBT SS2 32 1,34 1,90 R 300 263 150 1600 43,2 52,0* 126,9 2,82 4PBT SS3F 34 1,34 1,90 R 300 263 150 1600 collated 0,60 50 80 0,96 48,6 58,6* 136,4 3,03 4PBTD1 30 2,18 1,90 R 300 263 150 1600 47,7 57,5* 128,3 2,85 4PBTD2 32 2,18 1,90 R 300 263 150 1600 46,8 56,4* 139,1 3,1 4PBTD3F 31 2,18 1,90 R 300 263 150 1600 0,60 50 80 0,96 47,7 57,5* 171,9 3,82 4PBTD4F 31 2,18 1,90 R 300 263 150 1600 0,60 50 80 0,96 43,2 52,0* 182,3 4,05 4PBT
A11 1 0,30 13,00 4,71 2,00 R 200 175 100 700 0,00 80,0 96,4* 125,0 6,25 3PBT2 0,30 13,00 4,71 2,00 R 200 175 100 700 round straigth 0,40 40 100 0,50 80,0 96,4* 136,8 6,84 3PBT3 0,30 13,00 4,71 2,00 R 200 175 100 700 round straigth 0,40 40 100 1,00 80,0 96,4* 148,0 7,40 3PBT4 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,00 80,0 96,4* 57,4 2,87 3PBT5 0,30 13,00 4,71 3,00 R 200 175 100 1050 round straigth 0,40 40 100 0,50 80,0 96,4* 63,8 3,19 3PBT6 0,30 13,00 4,71 3,00 R 200 175 100 1050 round straigth 0,40 40 100 1,00 80,0 96,4* 82,0 4,10 3PBT7 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,00 80,0 96,4* 132,4 6,62 3PBT8 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,50 80,0 96,4* 134,0 6,70 3PBT9 0,30 13,00 4,71 3,00 R 200 175 100 1050 1,00 80,0 96,4* 131,8 6,59 3PBT10 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,00 80,0 96,4* 143,4 7,17 3PBT11 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,50 80,0 96,4* 140,0 7,00 3PBT12 0,30 13,00 4,71 3,00 R 200 175 100 1050 1,00 80,0 96,4* 146,4 7,32 3PBT
Portland cement
collated steel with
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
crimped
Specimens
By
Cross‐section
Type 2 Portland cement HPSC 80
MPa
A‐7
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
13 0,30 13,00 4,71 3,00 R 200 175 100 1050 0,00 80,0 96,4* 147,2 7,36 3PBT14 0,30 13,00 9,42 3,00 R 200 175 100 1050 0,00 80,0 96,4* 63,0 3,15 3PBT15 0,30 13,00 9,42 3,00 R 200 175 100 1050 0,00 80,0 96,4* 149,8 7,49 3PBT16 0,30 13,00 9,42 3,00 R 200 175 100 1050 0,00 80,0 96,4* 152,8 7,64 3PBT17 0,30 13,00 4,71 4,50 R 200 175 100 1575 0,00 80,0 96,4* 51,8 2,59 3PBT18 0,30 13,00 4,71 4,50 R 200 175 100 788 round straigth 0,40 40 100 0,50 80,0 96,4* 55,6 2,78 3PBT19 0,30 13,00 4,71 4,50 R 200 175 100 788 round straigth 0,40 40 100 1,00 80,0 96,4* 68,8 3,44 3PBT20 0,30 13,00 9,42 4,50 R 200 175 100 1575 0,00 80,0 96,4* 52,4 2,62 3PBT21 0,30 13,00 4,71 6,00 R 200 175 100 2100 0,00 80,0 96,4* 46,6 2,33 3PBT22 0,30 13,00 9,42 6,00 R 200 175 100 2100 0,00 80,0 96,4* 54,6 2,73 3PBT
A12 B51 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 0,00 39,4* 47,5 65,0 1,79 4PBT DTB52 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 crimped 0,50 50 100 0,40 36,9* 44,4 79,5 2,19 4PBT DTB53 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 crimped 0,50 50 100 0,80 38,8* 46,8 114,0 3,14 4PBT S, SCB54 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 crimped 0,50 50 100 1,20 41,3* 49,8 115,0 3,17 4PBT S, SCB55 0,45 28 10,00 3,05 445‐490 4,50 T 250 50 207 175 500 2800 3400 crimped 0,50 50 100 0,80 39,6* 47,7 118,2 3,26 4PBT SC, FTB56 0,45 28 10,00 1,95 445‐490 4,50 T 250 50 207 175 500 2800 3400 crimped 0,50 50 100 0,80 43,4* 52,3 96,4 2,66 4PBT FTB57S 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 stirrup 0,80 39,8* 47,9 142,0 3,91 4PBT SC, FTB58S 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 stirrup 0,80 37,7* 45,4 145,0 3,99 4PBT SC, FTB59 0,45 28 10,00 4,00 445‐490 4,50 T 250 50 207 175 500 2800 3400 0,00 39,0* 47,0 69,0 1,90 4PBT DT
28 445‐490 4,50 207 175 2800 3400 crimped 0,50 50 100 0,80 38,6* 46,5 0,00 4PBTB61R 0,45 28 10,00 1,95 445‐490 4,50 R 250 207 175 2800 3400 0,00 44,6* 53,7 57,9 1,60 4PBT DTB63R 0,45 28 10,00 1,95 445‐490 4,50 R 250 207 175 2800 3400 crimped 0,50 50 100 0,80 36,4* 43,9 75,5 2,08 4PBT FT
A13 TB21 0,45 7 9,50 3,25 530 2,50 T 250 60 165 75 250 1125 1950 0,00 33,7 40,6* 80,1 4PBT WCTB22 0,45 7 9,50 3,25 530 2,50 T 250 60 165 75 250 1125 1950 0,50 30 60 0,50 34,7 41,8* 87,5 4PBT DTTB23 0,45 7 9,50 3,25 530 2,50 T 250 60 165 75 250 1125 1950 0,50 30 60 1,00 40,5 48,8* 113,0 4PBT DTTB24 0,45 7 9,50 3,25 530 2,00 T 250 60 185 75 250 1040 1865 0,50 30 60 1,00 12,3 14,8* 156,4 4PBT WCTB25 0,45 7 9,50 2,90 530 3,00 T 250 60 185 75 250 1310 2135 0,50 30 60 1,00 38,2 46,0* 80,0 4PBT DTTB26 0,45 7 9,50 4,87 530 2,50 T 250 60 165 75 250 1125 1950 0,50 30 60 1,00 39,1 47,1* 110,1 4PBT DTTB27 0,45 7 9,50 6,50 530 2,50 T 250 60 165 75 250 1125 1950 0,50 30 60 1,00 38,5 46,4* 91,2 4PBT DTTB28 0,45 7 9,50 0,00 530 2,50 T 250 60 165 75 250 1125 1950 0,50 30 60 1,00 34,0 41,0* 93,5 4PBT DTTB23‐A 0,45 7 9,50 3,25 530 2,50 T 250 60 165 75 250 1125 1950 stirrup 0,50 31,5 38,0* 93,5 4PBT DTTB23‐B 0,45 7 9,50 3,25 530 2,50 T 250 60 165 75 250 1125 1950 stirrup 0,00 34,5 41,6* 99,0 4PBT DT
A14 1.2/1 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,00 45,7* 55,0 181,0 3,48 3PBT S1.2/2 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,25 48,7* 58,7 220,0 4,23 3PBT S1.2/3 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,50 45,4* 54,7 240,0 4,61 3PBT S1.2/4 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,75 50,1* 60,3 310,0 5,96 3PBT S1.3/1 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,00 51,7* 62,3 240,0 4,61 3PBT S1.3/2 3,56 3,46 R 300 260 200 1800 0,90 60 67 20,00 52,0* 62,7 295,0 5,67 3PBT S1.3/3 3,56 3,46 R 300 260 200 1800 0,90 60 67 40,00 45,1* 54,3 356,0 6,84 3PBT S1.3/4 3,56 3,46 R 300 260 200 1800 0,90 60 67 60,00 52,3* 63,0 395,0 7,59 3PBT S1.4/1 3,56 3,46 R 300 260 200 1800 0,90 60 67 0,00 50,4* 60,7 340,0 6,54 3PBT S1.4/2 3,56 3,46 R 300 260 200 1800 0,90 60 67 20,00 50,6* 61,0 435,0 8,36 3PBT S1.4/3 3,56 3,46 R 300 260 200 1800 0,90 60 67 40,00 42,9* 51,7 440,0 8,46 3PBT S
Type 2 Portland cement
HPSC 80 Mpa
FibresMix
Beam size
Flexural Reinforcement
Type 1 Portland cement 40‐45 MPa
Concrete Shear Force at [kN]Specimens
By
steel hook‐ended
Cross‐section
Dramix RC‐65/60‐
BN
A‐8
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
1.4/4 3,56 3,46 R 300 260 200 1800 0,90 60 67 60,00 52,6* 63,3 500,0 9,61 3PBT S2.2/1 1,81 1,54 R 300 747 200 2300 0,90 60 67 0,00 42,4* 51,0 420,0 2,81 3PBT S2.2/2 1,81 1,54 R 300 747 200 2300 0,90 60 67 0,20 42,7* 51,5 560,0 3,75 3PBT S2.2/3 1,81 1,54 R 300 747 200 2300 0,90 60 67 0,75 41,8* 50,4 600,0 4,02 3PBT S2.3/1 1,15 2,50 R 300 460 200 2300 0,90 60 67 0,00 41,6* 50,2 157,0 1,71 3PBT S2.3/2 1,15 2,50 R 300 460 200 2300 0,90 60 67 0,25 41,5* 50,1 165,0 1,79 3PBT S2.3/3 1,15 2,50 R 300 460 200 2300 0,90 60 67 0,70 40,1* 48,4 216,0 2,35 3PBT S2.4/1 1,81 2,50 R 300 460 200 2300 0,90 60 67 0,00 41,6* 50,2 240,0 2,61 3PBT S2.4/2 1,81 2,50 R 300 460 200 2300 0,90 60 67 0,25 41,5* 50,1 216,0 2,35 3PBT S2.4/3 1,81 2,50 R 300 460 200 2300 0,90 60 67 0,75 40,1* 48,4 288,0 3,13 3PBT S2.6/1 1,81 4,04 R 300 285 200 2300 0,90 60 67 0,00 42,4* 51,0 150,0 2,63 3PBT S2.6/2 1,81 4,04 R 300 285 200 2300 0,90 60 67 0,25 42,7* 51,5 165,0 2,90 3PBT S2.6/3 2,83 4,04 R 300 285 200 2300 0,90 60 67 0,75 41,8* 50,4 234,0 4,11 3PBT S3.1/1 2,83 3,50 T 300 0 314 200 200 2200 0,90 60 67 0,50 39,1* 47,1 189,0 3,01 3PBT S
3.1/1 F2 3,09 3,50 T 300 0 314 200 200 2200 0,90 60 67 0,50 40,3* 48,5 225,0 3,58 3PBT S3.1/2 2,73 3,34 T 450 0 494 200 200 3300 0,90 60 67 0,50 39,1* 47,1 249,0 2,52 3PBT S20*50 2,73 3,37 T 500 0 504 200 200 3400 0,90 60 67 0,50 40,3* 48,5 272,0 2,70 3PBT S3.1/3 2,73 3,48 T 600 0 647 200 200 4500 0,90 60 67 0,50 39,1* 47,1 265,0 2,05 3PBT S
3.1/3 F2 2,80 3,48 T 600 0 647 200 200 4500 0,90 60 67 0,50 40,3* 48,5 383,0 2,96 3PBT S8*50 2,80 3,37 T 500 80 564 200 500 3800 0,90 60 67 0,50 40,3* 48,5 338,0 3,00 3PBT S3.2/1 2,80 3,37 T 500 100 564 200 500 3800 0,90 60 67 0,50 39,1* 47,1 286,0 2,54 3PBT S
10*50 F2 2,80 3,37 T 500 100 564 200 500 3800 0,90 60 67 0,50 40,3* 48,5 265,0 2,35 3PBT S3.2/2 2,80 3,37 T 500 150 564 200 500 3800 0,90 60 67 0,50 39,1* 47,1 446,0 3,96 3PBT S
15*50 F2 2,80 3,37 T 500 150 564 200 500 3800 0,90 60 67 0,50 40,3* 48,5 276,0 2,45 3PBT S23*50 F2 2,80 3,37 T 500 230 564 200 500 3800 0,90 60 67 0,50 40,3* 48,5 427,0 3,79 3PBT S3.2/3 2,80 3,37 T 500 150 564 200 750 3800 0,90 60 67 0,50 39,1* 47,1 437,0 3,88 3PBT S3.2/4 2,80 3,37 T 500 150 564 200 1000 3800 0,90 60 67 0,50 39,1* 47,1 412,0 3,65 3PBT S
A15 SFSCCB0 0,50 10,00 2,80 460 3,00 R 300 263 200 2100 2400 0,00 32,6* 39,3 79,3 1,51 4PBT SSFSCCB25 0,50 10,00 2,80 460 3,00 R 300 263 200 2100 2400 hooked 0,55 35 65 0,32 34,7* 41,8 105,0 2,00 4PBT SSFSCCB50 0,50 10,00 2,80 460 3,00 R 300 263 200 2100 2400 0,55 35 65 0,64 40,3* 48,5 142,3 2,71 4PBT S
A16 V1A 0,65 6,30 1,54 3,53 R 100 85 120 600 0,00 23,1 27,8* 24,9 3PBT SV1B 0,65 6,30 1,54 3,53 R 100 85 120 600 0,00 23,1 27,8* 29,7 3PBT SV2A 0,65 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 1,00 24,4 29,4* 43,7 3PBT SV2B 0,65 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 1,00 24,4 29,4* 47,2 3PBT SV3A 0,65 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 2,00 28,1 33,9* 55,1 3PBT SV3B 0,65 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 2,00 28,1 33,9* 51,1 3PBT SV4A 0,34 6,30 1,54 3,53 R 100 85 120 600 0,00 57,0 68,7* 36,3 3PBT SV4B 0,34 6,30 1,54 3,53 R 100 85 120 600 0,00 57,0 68,7* 36,4 3PBT SV5A 0,34 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 1,00 59,7 71,9* 72,8 3PBT SV5B 0,34 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 1,00 59,7 71,9* 66,6 3PBT SV6A 0,34 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 2,00 52,4 63,1* 57,2 3PBT SV6B 0,34 6,30 1,54 3,53 R 100 85 120 600 0,55 30 55 2,00 52,4 63,1* 53,9 3PBT SV7A 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,00 57,0 68,7* 54,8 3PBT S
Portland
C42,5 R
Cross‐section
Dramix RC‐65/60‐
Dramix RC‐65/60‐
Dramix RC‐65/60‐
Dramix RC‐65/60‐BN
Type IS Portland
25 MPa
Type IS Portland
56 MPa
hooked circular section
hooked circular section
Dramix RC‐65/60‐
Specimens
By
FibresMix
Beam size
Flexural Reinforcement
28‐120
Concrete Shear Force at [kN]
A‐9
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
V7B 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,00 57,0 68,7* 46,4 3PBT SV8A 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,55 30 55 1,00 59,7 71,9* 68,3 3PBT SV8B 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,55 30 55 1,00 59,7 71,9* 80,1 3PBT SV9A 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,55 30 55 2,00 52,4 63,1* 81,2 3PBT SV9B 0,34 6,30 1,59 3,55 R 170 155 130 1100 0,55 30 55 2,00 52,4 63,1* 104,9 3PBT SVP1A 0,50 6,30 1,54 3,53 R 100 85 120 600 0,00 36,1 43,5* 28,4 3PBT SVP1B 0,50 6,30 1,54 3,53 R 100 85 120 600 0,00 36,1 43,5* 27,0 3PBT SV10A 0,50 6,30 1,54 3,53 R 100 85 120 600 1,05 50 48 0,75 36,6 44,1* 42,7 3PBT SV10B 0,50 6,30 1,54 3,53 R 100 85 120 600 1,05 50 48 0,75 36,6 44,1* 39,0 3PBT SV11A 0,50 6,30 1,54 3,53 R 100 85 120 600 1,05 50 48 1,50 46,1 55,5* 50,0 3PBT SV11B 0,50 6,30 1,54 3,53 R 100 85 120 600 1,05 50 48 1,50 46,1 55,5* 61,8 3PBT SV12A 0,34 6,30 1,84 1,94 R 100 155 110 600 0,00 75,3 90,7* 64,4 3PBT SV12B 0,34 6,30 1,84 1,94 R 100 155 110 600 0,00 75,3 90,7* 50,8 3PBT SV13A 0,34 6,30 1,84 1,94 R 100 155 110 600 0,67 25 37 0,75 73,5 88,6* 62,6 3PBT SV13B 0,34 6,30 1,84 1,94 R 100 155 110 600 0,67 25 37 0,75 73,5 88,6* 51,4 3PBT SV14A 0,34 6,30 1,84 1,94 R 100 155 110 600 0,67 25 37 1,50 73,1 88,1* 67,5 3PBT SV14B 0,34 6,30 1,84 1,94 R 100 155 110 600 0,67 25 37 1,50 73,1 88,1* 55,1 3PBT S
A17 S0.00V0 10,00 3,30 420 3,33 R 180 122 100 1300 1700round straight0,70 42 60 0,00 34,0 41,0* 43,4 3,56 4PBT SS0.50V0 10,00 3,30 420 3,33 R 180 122 100 1300 1700 50% stirrup 60 0,00 34,0 41,0* 65,3 5,35 4PBT SS0.75V0 10,00 3,30 420 3,33 R 180 122 100 1300 1700 75% stirrup 60 0,00 34,0 41,0* 77,6 6,36 4PBT SS1.00V0 10,00 3,30 420 3,33 R 180 122 100 1300 1700 stirrup 60 0,00 34,0 41,0* 85,4 7,00 4PBT SS0.00V1 10,00 3,30 420 3,33 R 180 122 100 1300 1700round straight0,70 42 60 1,00 38,7 46,6* 54,8 4,49 4PBT FS0.50V1 10,00 3,30 420 3,33 R 180 122 100 1300 1700 50% stirrup 60 1,00 38,7 46,6* 69,9 5,73 4PBT FS0.75V1 10,00 3,30 420 3,33 R 180 122 100 1300 1700 75% stirrup 60 1,00 38,7 46,6* 85,4 7,00 4PBT FS0.00V2 10,00 3,30 420 3,33 R 180 122 100 1300 1700round straight0,70 42 60 2,00 42,4 51,1* 69,9 5,73 4PBT FS0.00V2 10,00 3,30 420 3,33 R 180 122 100 1300 1700 50% stirrup 60 2,00 42,4 51,1* 83,0 6,80 4PBT F
A18 DB1000 0,60 20,00 0,00 433 2,00 300 280 100 600 800 100 0,00 20,90 50,0 3PBT FDB1005 0,60 20,00 0,00 433 2,00 300 280 100 600 800 100 0,50 24,50 50,0 3PBT FDB1010 0,60 20,00 0,00 433 2,00 R 300 280 100 600 800 100 1,00 32,60 75,0 3PBT FDB1020 0,60 20,00 0,00 433 2,00 R 300 280 100 600 800 100 2,00 25,50 80,0 3PBT FDB1200 0,60 20,00 0,36 433 2,00 R 300 280 100 600 800 100 0,00 20,90 125,0 3PBT FDB1205 0,60 20,00 0,36 433 2,00 R 300 280 100 600 800 100 0,50 24,50 155,0 3PBT FDB1210 0,60 20,00 0,36 433 2,00 R 300 280 100 600 800 100 1,00 32,60 165,0 3PBT FDB1220 0,60 20,00 0,36 433 2,00 R 300 280 100 600 800 100 2,00 25,50 230,0 3PBT FDB1300 0,60 20,00 0,54 433 2,00 R 300 280 100 600 800 100 0,00 20,90 180,0 3PBT SDB1305 0,60 20,00 0,54 433 2,00 R 300 280 100 600 800 100 0,50 24,50 200,0 3PBT SDB1310 0,60 20,00 0,54 433 2,00 R 300 280 100 600 800 100 1,00 32,60 240,0 3PBT FDB1320 0,60 20,00 0,54 433 2,00 R 300 280 100 600 800 100 2,00 25,50 270,0 3PBT FDB2310 0,60 20,00 0,54 433 0,67 R 300 280 100 600 800 100 1,00 32,60 270,0 4PBT SDB1510 0,60 20,00 0,90 433 2,00 R 300 280 100 600 800 100 1,00 32,60 270,0 3PBT S
A19 A00 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 0,00 41,2 49,6* 40,5 1,23 4PBT SA01 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 0,00 41,2 49,6* 89,5 2,72 4PBT SA02 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 0,00 41,2 49,6* 114,0 3,47 4PBT F
Cross‐section
Portland Pozzolana Cement
Type I Prdinary Portland
cement 35MPa
Type IS Portland
40 MPa
Type III Portland
73 MPa
hooked circular section
hooked circular section
dato mancante: hp =round steel fibre
Type IS
Portland 56
MPa
Portland
Cement
Type
hooked circular section
Specimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐10
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[Mpa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[Mpa]
f' c.
cyl,m
[Mpa]
f cu.28,m
[Mpa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
B00 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 0,00 41,2 49,6* 49,5 1,51 4PBT SB01 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 0,00 41,2 49,6* 60,2 1,83 4PBT SB02 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 0,00 41,2 49,6* 142,6 4,34 4PBT SA10 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 96,4 2,93 4PBT SA20 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 2,00 43,2 52,1* 103,3 3,15 4PBT FA11 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 99,7 3,03 4PBT SA12 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 115,8 3,53 4PBT FA21 0,36 10,00 1,91 610 2,80 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 2,00 43,2 52,1* 123,0 3,74 4PBT FB10 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 115,1 3,50 4PBT SB20 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 2,00 43,2 52,1* 115,5 3,52 4PBT SB11 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 120,8 3,68 4PBT SB12 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 1,00 40,9 49,2* 156,6 4,77 4PBT FB21 0,36 10,00 1,91 610 2,00 R 250 219 150 2300 2500 hooked‐end 0,50 30 60 2,00 43,2 52,1* 173,3 5,28 4PBT F
A20 PCB1 0,50 60 10,00 2,37 2,41 R 200 170 50 820 1000 0,38 38 100 0,00 32,5* 39,2 23,5 2,76 3PBT SF30B1 0,50 60 10,00 2,37 2,41 R 200 170 50 820 1000 0,38 38 100 0,85 32,5* 39,1 32,8 3,86 3PBT SF45B1 0,50 60 10,00 2,37 2,41 R 200 170 50 820 1000 0,38 38 100 1,30 39,8* 47,9 36,2 4,26 3PBT F‐SPCB2 0,50 60 10,00 2,37 1,62 R 200 170 50 552 725 0,38 38 100 0,00 32,5* 39,2 45,3 5,33 3PBT SF30B2 0,50 60 10,00 2,37 1,62 R 200 170 50 552 725 0,38 38 100 0,85 32,5* 39,1 50,9 5,99 3PBT FF45B2 0,50 60 10,00 2,37 1,62 R 200 170 50 552 725 0,38 38 100 1,30 39,8* 47,9 54,1 6,37 3PBT FPCB3 0,50 60 10,00 2,37 0,81 R 200 170 50 274 455 0,38 38 100 0,00 32,5* 39,2 74,5 8,76 3PBT SF3OB3 0,50 60 10,00 2,37 0,81 R 200 170 50 274 455 0,38 38 100 0,85 32,5* 39,1 81,2 9,55 3PBT SF45B3 0,50 60 10,00 2,37 0,81 R 200 170 50 274 455 0,38 38 100 1,30 39,8* 47,9 107,6 12,66 3PBT F
A21 2,20 3,00 R 102 60 1,00 22,7 27,3* 3,16 S1,10 3,00 R 102 60 1,00 22,7 27,3* 2,43 S1,10 1,50 R 102 60 1,00 22,7 27,3* 5,64 S2,20 3,00 R 102 100 1,00 26,0 31,3* 3,55 S2,20 3,00 R 204 60 1,00 22,7 27,3* 3,05 S1,34 2,00 R 197 60 0,50 29,1 35,1* 2,54 S1,34 2,80 R 197 60 0,50 29,1 35,1* 1,78 S1,34 3,60 R 197 60 0,50 29,1 35,1* 1,52 S2,00 2,80 R 197 60 0,75 29,1 35,1* 2,20 S2,00 2,80 R 197 60 0,75 20,6 24,8* 2,03 S2,00 2,80 R 197 60 0,75 33,4 40,2* 2,91 S1,10 2,50 R 221 60 0,50 34,0 41,0* 1,73 S2,20 1,50 R 221 60 0,50 34,0 41,0* 4,02 S2,20 2,50 R 221 60 0,50 34,0 41,0* 1,90 S2,20 3,50 R 221 60 0,50 34,0 41,0* 1,47 S2,20 1,50 R 221 60 1,00 34,0 41,0* 4,39 S2,20 2,50 R 221 60 1,00 34,0 41,0* 2,46 S2,00 2,00 R 130 100 0,25 61,0 73,5* 2,96 S2,00 3,00 R 130 100 0,25 61,0 73,5* 2,77 S2,00 3,00 R 130 133 0,50 36,0 43,4* 1,97 S2,00 3,00 R 130 100 1,00 36,0 43,4* 2,97 S
Cross‐section
crimped and
hooked steel
hooked
hooked
Li, W
ard and
Ham
zaMansur, Ong and
Param
asivam
Lim, Param
avisam
and Lee
Narayanana
and Darwish
crimped
Duoform brass coated
Portland Cement Type 42,5
Specimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐11
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
f ys
average
[Mpa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[Mpa]
f' c.
cyl,m
[Mpa]
f cu.28,m
[Mpa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
2,00 3,50 R 130 133 0,50 49,0 59,0* 2,61 S2,00 2,00 R 130 133 1,00 57,4 69,2* 5,57 S3,69 3,00 R 130 133 0,50 36,0 43,4* 2,24 S5,72 3,10 R 130 133 0,50 36,0 43,4* 2,33 S5,72 3,10 R 130 133 1,00 57,4 69,2* 5,00 S2,84 1,00 R 215 75 0,50 99,0 119,3* 9,09 S2,84 2,00 R 215 75 0,50 99,0 119,3* 4,82 S2,84 1,00 R 215 75 1,00 95,0 114,5* 12,74 S2,84 1,00 R 215 75 1,50 96,0 115,7* 13,95 S2,84 2,00 R 215 75 1,50 96,0 115,7* 7,21 S4,58 2,00 R 215 75 1,00 94,0 113,3* 4,89 S4,58 4,00 R 215 75 1,00 94,0 113,3* 3,88 S4,00 4,50 R 210 100 0,40 44,4 53,5* 2,16 S4,00 4,50 R 210 100 0,80 46,8 56,4* 3,10 S4,00 4,50 R 210 100 1,20 49,8 60,0* 3,13 S3,55 0,70 R 350 100 0,50 60,0 72,3* 9,42 S3,55 0,47 R 350 100 1,00 60,0 72,3* 13,16 S3,55 0,92 R 350 100 1,00 60,0 72,3* 9,97 S3,55 0,70 R 350 100 1,00 67,0 80,7* 11,48 S3,55 0,70 R 350 100 1,00 38,0 45,8* 8,52 S3,55 0,70 R 350 100 1,00 42,0 50,6* 9,65 S3,55 0,70 R 350 100 1,25 68,0 81,9* 11,39 S3,59 2,00 R 175 100 0,50 80,0 96,4* 6,84 S3,59 3,00 R 175 100 0,50 80,0 96,4* 3,19 S3,59 4,50 R 175 100 0,50 80,0 96,4* 2,78 S3,59 2,00 R 175 100 1,00 80,0 96,4* 7,40 S3,59 3,00 R 175 100 1,00 80,0 96,4* 4,10 S3,59 4,50 R 175 100 1,00 80,0 96,4* 3,44 S2,15 1,35 R 557 60 0,75 54,0 65,1* 3,30 S2,15 1,35 R 557 60 1,50 50,0 60,2* 3,87 S2,15 1,35 R 557 60 0,40 55,0 66,3* 2,44 S2,15 1,35 R 557 60 0,60 56,0 67,5* 2,77 S2,15 1,35 R 557 100 0,40 47,0 56,6* 2,95 S1,22 2,00 R 186 50 0,50 28,7 34,6* 1,64 S1,22 2,00 R 186 100 0,50 32,2 38,8* 1,94 S1,22 2,00 R 186 50 1,00 29,0 34,9* 2,18 S1,22 3,00 R 186 50 1,00 32,1 38,7* 1,58 S1,22 3,00 R 186 100 1,00 32,3 38,9* 1,98 S1,22 3,00 R 186 50 1,50 32,8 39,5* 2,42 S3,89 2,00 R 340 60 0,50 35,0 42,2* 10,68 S3,89 2,00 R 340 60 0,75 33,0 39,8* 8,87 S3,89 2,00 R 340 60 1,00 36,0 43,4* 10,31 S3,89 2,50 R 340 60 1,00 36,0 43,4* 7,56 S3,89 1,50 R 340 60 1,00 36,0 43,4* 15,05 S
plain
hooked
Cross‐sectionAdebar,
Mindess, St.
Pierre and
Olund
Murthy and
Venkatachartulu
Tan,
Murugappan
and
Param
asivam
hooked
Ashour, Hasanain and
Wafa
Swam
y and
Bahia
Narayanan and
Darwish
Shin, Oh and Goosh
hooked
crimpe
dcrimped
plain
Narayanana
and Darwish
crimped
Specimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐12
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
4,31 2,00 R 265 100 1,00 44,5 53,6* 5,51 S4,31 3,43 R 265 100 1,00 51,1 61,6* 4,05 S4,31 4,91 R 265 100 1,00 45,0 54,2* 2,92 S2,76 3,43 R 265 100 1,00 41,4 49,9* 3,13 S1,55 2,00 R 265 100 1,00 44,6 53,7* 4,65 S
A22 2,20 1,50 R 200 182 100 60 0,75 3,78 53,0 63,9* 99,0 5,44 S2,20 2,00 R 200 182 100 60 0,75 3,78 53,0 63,9* 65,0 3,57 S2,20 2,50 R 200 182 100 60 0,75 3,78 53,0 63,9* 62,1 3,41 S2,00 2,50 R 300 280 100 60 0,75 3,78 53,0 63,9* 90,2 3,22 S2,00 3,00 R 300 280 100 60 0,75 3,78 53,0 63,9* 73,6 2,63 S3,54 1,50 R 100 80 100 60 1,50 3,78 50,0 60,2* 52,2 6,52 S3,54 2,00 R 100 80 100 60 1,50 3,78 50,0 60,2* 44,6 5,58 S1,84 1,50 R 100 85 100 60 1,50 3,78 50,0 60,2* 46,8 5,51 S2,20 1,00 R 200 182 100 60 1,50 3,78 50,0 60,2* 131,8 7,24 S2,20 1,50 R 200 182 100 60 1,50 3,78 50,0 60,2* 109,6 6,02 S2,20 2,00 R 200 182 100 60 1,50 3,78 50,0 60,2* 77,0 4,23 S2,00 2,00 R 300 280 100 60 1,50 3,78 50,0 60,2* 106,7 3,81 S1,16 2,00 R 150 130 100 53 0,75 3,78 48,0 57,8* 31,1 2,39 S2,20 1,50 R 200 182 100 53 0,75 3,78 48,0 57,8* 82,8 4,55 S2,20 2,00 R 200 182 100 53 0,75 3,78 48,0 57,8* 57,3 3,15 S2,20 2,50 R 200 182 100 53 0,75 3,78 48,0 57,8* 46,0 2,53 S2,20 3,00 R 200 182 100 53 0,75 3,78 48,0 57,8* 41,9 2,30 S2,20 3,50 R 200 182 100 53 0,75 3,78 48,0 57,8* 36,8 2,02 S2,00 2,00 R 300 280 100 53 0,75 3,78 48,0 57,8* 68,6 2,45 S1,16 1,50 R 150 130 100 53 1,50 3,78 54,0 65,1* 52,0 4,00 S2,20 1,50 R 200 182 100 53 1,50 3,78 54,0 65,1* 97,4 5,35 S2,20 2,00 R 200 182 100 53 2,00 3,78 54,0 65,1* 60,1 3,30 S2,20 2,50 R 200 182 100 53 2,50 3,78 54,0 65,1* 58,2 3,20 S2,00 2,00 R 300 280 100 53 2,00 3,78 54,0 65,1* 101,9 3,64 S1,55 2,50 R 150 135 75 50 0,75 3,61 31,4 37,8* 21,8 2,15 S1,55 2,50 R 150 135 75 63 0,75 3,61 30,6 36,9* 24,0 2,37 S1,55 2,50 R 150 135 75 83 0,75 3,61 29,2 35,2* 27,5 2,72 S1,55 2,50 R 150 135 75 100 0,75 3,61 31,2 37,6* 27,3 2,70 S1,34 2,00 R 225 197 150 0,50 30 60 0,50 3,30 29,1 35,1* 75,1 2,54 S1,34 2,80 R 225 197 150 0,50 30 60 0,50 3,30 29,1 35,1* 52,6 1,78 S1,34 3,60 R 225 197 150 0,50 30 60 0,50 3,30 29,1 35,1* 44,9 1,52 S1,34 2,00 R 225 197 150 0,50 30 60 0,75 3,30 29,9 36,0* 85,1 2,88 S1,34 2,80 R 225 197 150 0,50 30 60 0,75 3,30 29,9 36,0* 60,0 2,03 S2,00 2,80 R 225 197 150 0,50 30 60 0,75 3,30 29,9 36,0* 65,0 2,20 S1,34 2,80 R 225 197 150 0,50 30 60 0,75 3,30 20,6 24,8* 44,9 1,52 S2,00 2,80 R 225 197 150 0,50 30 60 0,75 3,30 20,6 24,8* 60,0 2,03 S2,00 2,80 R 225 197 150 0,50 30 60 0,75 3,30 33,4 40,2* 86,0 2,91 S
A23 1 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,00 34,0 41,0* 63,15 0,31 4PBT S
Uomoto et al. (1986) Indented cut wire
crimped
Cross‐section
Sheared
Kadir
&Saeed
(1986)
Mansur et al. (1986)
Duoform
Hooked end
Swam
y, Jones
and Chiam
Specimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐13
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
2 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 0,50 34,0 41,0* 109,0 0,53 4PBT S3 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 0,75 34,0 41,0* 90,5 0,44 4PBT S4 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 105,2 0,52 4PBT S5 0,50 7 19,00 3,90 460 2,50 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 77,1 0,38 4PBT S6 0,50 7 19,00 3,90 460 1,50 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 153,5 0,75 4PBT S
A24 A1 0,60 28 3,12 4,80 R 152 127 101 1966 1981 0,00 35,1 42,3* 42,4 3,30 4PBT SA2 0,60 28 3,12 4,80 R 152 127 101 1967 1982 0,00 35,1 42,3* 41,4 3,23 4PBT SA3 0,60 28 3,12 4,80 R 152 127 101 1968 1983 0,00 35,1 42,3* 45,6 3,56 4PBT SB3 0,60 28 3,12 4,40 R 152 127 101 1969 1984 round 0,25 25 102 0,22 33,2 40,0* 55,6 4,33 4PBT SC1 0,60 28 3,12 4,20 R 152 127 101 1970 1985 round 0,25 25 102 0,22 33,2 40,0* 55,4 4,32 4PBT SC2 0,60 28 3,12 4,20 R 152 127 101 1971 1986 round 0,25 25 102 0,22 33,2 40,0* 49,1 3,82 4PBT SC3 0,60 28 3,12 4,20 R 152 127 101 1972 1987 round 0,25 25 102 0,22 33,2 40,0* 44,2 3,45 4PBT SD2 0,60 28 3,12 4,30 R 152 127 101 1973 1988 round 0,25 25 102 0,22 33,2 40,0* 52,0 4,05 4PBT SD3 0,60 28 3,12 4,30 R 152 127 101 1974 1989 round 0,25 25 102 0,22 33,2 40,0* 49,2 3,84 4PBT SF1 0,60 28 3,12 4,00 R 152 127 101 1975 1990 round 0,25 25 102 0,44 40,7 49,1* 58,2 4,54 4PBT SF2 0,60 28 3,12 4,00 R 152 127 101 1976 1991 round 0,25 25 102 0,44 40,7 49,1* 54,8 4,27 4PBT SF3 0,60 28 3,12 4,00 R 152 127 101 1977 1992 round 0,25 25 102 0,44 40,7 49,1* 58,2 4,54 4PBT SG1 0,60 28 3,12 4,40 R 152 127 101 1978 1993 round 0,25 25 102 0,22 33,2 40,0* 49,8 3,88 4PBT SG3 0,60 28 3,12 4,40 R 152 127 101 1979 1994 round 0,25 25 102 0,22 33,2 40,0* 47,3 3,69 4PBT SL1 0,60 28 3,12 4,00 R 152 127 101 1980 1995 crimped 0,51 25 50 0,22 33,2 40,0* 52,9 4,13 4PBT SL2 0,60 28 3,12 4,00 R 152 127 101 1981 1996 crimped 0,51 25 50 0,22 33,2 40,0* 53,1 4,14 4PBT SL3 0,60 28 3,12 4,00 R 152 127 101 1982 1997 crimped 0,51 25 50 0,22 33,2 40,0* 58,2 4,54 4PBT SM1 0,60 28 3,12 4,60 R 152 127 101 1983 1998 crimped 0,51 25 50 0,22 33,2 40,0* 45,5 3,54 4PBT SM2 0,60 28 3,12 4,40 R 152 127 101 1984 1999 crimped 0,51 25 50 0,22 33,2 40,0* 47,6 3,71 4PBT SM3 0,60 28 3,12 4,40 R 152 127 101 1985 2000 crimped 0,51 25 50 0,22 33,2 40,0* 45,2 3,52 4PBT SN1 0,60 28 3,12 5,00 R 152 127 101 1986 2001 crimped 0,51 25 50 0,22 33,2 40,0* 42,8 3,34 4PBT SN2 0,60 28 3,12 4,80 R 152 127 101 1987 2002 crimped 0,51 25 50 0,22 33,2 40,0* 47,3 3,69 4PBT SO1 0,60 28 3,12 4,00 R 152 127 101 1988 2003 crimped 0,51 25 50 0,44 40,7 49,1* 55,3 4,31 4PBT SP1 0,60 28 3,12 4,20 R 152 127 101 1989 2004 crimped 0,51 25 50 0,44 40,7 49,1* 59,3 4,63 4PBT SP2 0,60 28 3,12 4,20 R 152 127 101 1990 2005 crimped 0,51 25 50 0,44 40,7 49,1* 52,9 4,13 4PBT SP3 0,60 28 3,12 4,20 R 152 127 101 1991 2006 crimped 0,51 25 50 0,44 40,7 49,1* 57,1 4,46 4PBT SR1 0,60 28 3,12 3,20 R 152 127 101 1992 2007 crimped 0,51 25 50 0,88 40,2 48,5* 64,5 5,03 4PBT SR2 0,60 28 3,12 3,40 R 152 127 101 1993 2008 crimped 0,51 25 50 0,88 40,2 48,5* 60,3 4,70 4PBT SS1 0,60 28 3,12 3,40 R 152 127 101 1994 2009 crimped 0,51 25 50 0,88 40,2 48,5* 58,2 4,54 4PBT SS2 0,60 28 3,12 3,40 R 152 127 101 1995 2010 crimped 0,51 25 50 0,88 40,2 48,5* 73,7 5,74 4PBT SS3 0,60 28 3,12 3,40 R 152 127 101 1996 2011 crimped 0,51 25 50 0,88 40,2 48,5* 69,4 5,41 4PBT SV2 0,60 28 3,12 1,80 R 152 127 101 1997 2012 crimped 0,51 25 50 1,76 40,3 48,6* 135,5 10,56 4PBT SW1 0,60 28 3,12 1,20 R 152 127 101 1998 2013 crimped 0,51 25 50 1,76 40,3 48,6* 255,4 19,91 4PBT SW2 0,60 28 3,12 1,20 R 152 127 101 1999 2014 crimped 0,51 25 50 1,76 40,3 48,6* 245,3 19,12 4PBT SX1 0,60 28 3,12 4,80 R 152 127 101 2001 2016 crimped 0,51 25 50 0,22 33,2 40,0* 42,7 3,33 4PBT SX2 0,60 28 3,12 4,80 R 152 127 101 2002 2017 crimped 0,51 25 50 0,22 33,2 40,0* 41,1 3,20 4PBT SX3 0,60 28 3,12 4,80 R 152 127 101 2003 2018 crimped 0,51 25 50 0,22 33,2 40,0* 45,8 3,57 4PBT S
Ordinary
Portland
Cement
hooked‐
ended steel
fibres
Cross‐sectionSpecimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐14
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
A25 A C35 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 0,00 32,5* 39,2 14,4 1,15 SB 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 0,50 34,1* 41,1 15,3 1,22 SC 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 1,00 35,3* 42,5 18,5 1,48 SD 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 1,50 35,9* 43,2 20,5 1,64 SE 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 2,00 36,6* 44,0 21,1 1,69 SF 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 2,50 37,9* 45,6 21,4 1,71 SG 0,47 28 0,00 0,80 3,20 R 150 125 100 1000 1200 hooked 1,00 51 50 3,00 38,3* 46,1 21,5 1,72 S
A26 P1 0,40 6,27 3,13 530 3,00 R 150 114 85 900 0,30 30 100 0,00 44,7 53,9* 27,0 2,79 4PBTP2 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,30 55,0 66,3* 36,0 3,72 4PBTP3 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,60 56,0 67,5* 40,0 4,13 4PBTP4 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,90 54,4 65,5* 50,0 5,16 4PBTP5 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 1,20 56,8 68,4* 58,6 6,05 4PBTP6 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 1,50 55,3 66,6* 55,4 5,72 4PBTP7 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 2,00 66,2 79,8* 51,1 5,27 4PBTP8 0,40 6,27 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 3,00 0,0 0,0* 0,0 0,00 4PBTP9 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,30 65,0 78,3* 35,1 3,62 4PBTP10 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,60 67,8 81,7* 40,0 4,13 4PBTP11 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,90 59,1 71,2* 55,4 5,72 4PBTP12 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 1,20 61,3 73,9* 50,1 5,17 4PBTP13 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 2,00 66,2 79,8* 49,5 5,11 4PBTP14 0,40 4,71 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 3,00 0,0 0,0* 0,0 0,00 4PBTP15 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,30 63,3 76,3* 30,0 3,10 4PBTP16 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,60 66,3 79,9* 42,3 4,37 4PBTP17 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 0,90 53,8 64,8* 46,3 4,78 4PBTP18 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 1,20 54,8 66,0* 49,8 5,14 4PBTP19 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 2,00 61,5 74,1* 52,0 5,37 4PBTP20 0,40 3,14 3,13 530 3,00 R 150 114 85 900 crimped 0,30 30 100 2,50 63,5 76,5* 54,5 5,62 4PBTP21 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 0,30 63,3 76,3* 46,4 5,46 4PBTP22 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 0,60 66,3 79,9* 53,5 6,29 4PBTP23 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 0,90 53,8 64,8* 50,3 5,92 4PBTP24 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 1,20 54,8 66,0* 51,3 6,04 4PBTP25 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 2,00 60,0 72,3* 60,9 7,16 4PBTP26 0,40 7,84 4,47 530 3,00 R 150 100 85 900 crimped 0,30 30 100 2,50 66,0 79,5* 57,4 6,75 4PBTP27 0,40 3,14 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 0,00 55,4 66,7* 34,3 3,54 4PBTP28 0,40 3,14 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 1,00 56,3 67,8* 59,3 6,12 4PBTP29 0,40 3,14 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 2,00 61,5 74,1* 59,7 6,16 4PBTP30 0,40 3,14 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 2,50 63,5 76,5* 68,1 7,03 4PBTP31 0,40 7,84 4,47 530 2,00 R 150 100 85 900 crimped 0,30 30 100 0,00 55,4 66,7* 53,4 6,28 4PBTP32 0,40 7,84 4,47 530 2,00 R 150 100 85 900 crimped 0,30 30 100 1,00 56,3 67,8* 71,1 8,36 4PBTP33 0,40 7,84 4,47 530 2,00 R 150 100 85 900 crimped 0,30 30 100 2,00 60,0 72,3* 75,8 8,92 4PBTP34 0,40 7,84 4,47 530 2,00 R 150 100 85 900 crimped 0,30 30 100 2,50 66,0 79,5* 69,3 8,15 4PBTP35 0,40 0,00 3,13 530 2,00 R 150 114 85 900 52,0 62,7* 34,8 3,59 4PBTP36 0,40 0,00 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 1,00 53,0 63,9* 36,4 3,76 4PBT
no coarse aggregate
Cross‐sectionSpecimens
By
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
A‐15
Article
No. type
w/c ratio
Age at test
(days)
Maximum
aggregate
s size
σ cp
[N/mm2]
ρ flex
(%)
fys
averag
e
[MPa]
a/d
Kind of cross‐
section
h
[mm]
h f
[mm]
d
[mm]
b w
[mm]
b f
[mm]
l
clear
span
[mm]
L
[mm]SF type
d f
[mm]
l f
[mm]l f /d f
V f
(%)
τ
[MPa]
f' c.
cyl,m
[MPa]
f cu.28,m
[MPa]
Shea
r
force
[kN]
Ultimate
shear
stress
[MPa]
kind
of test
Mode of
failure
S= Shear
F=Flexure
P37 0,40 0,00 3,13 530 2,00 R 150 114 85 900 crimped 0,30 30 100 0,00 52,0 62,7* 54,3 5,60 4PBTP38 0,40 0,00 3,13 530 2,00 R 150 114 85 900 53,0 63,9* 59,9 6,18 4PBT
A27 1 0,20 90 0,00 7,26 3,33 I 650 100 600 50 400 4000 4500 straight 0,20 13 65 2,50 161,0 176,0 430,0 3PBT S2 0,21 85 14,83 7,26 3,33 I 650 100 600 50 400 4000 4500 straight 0,20 13 65 2,50 160,0 178,0 497,0 3PBT S3 0,22 65 7,15 7,26 3,33 I 650 100 600 50 400 4000 4500 straight 0,20 13 65 2,50 149,0 166,0 428,0 3PBT S4 0,20 58 7,15 7,26 3,33 I 650 100 600 50 400 4000 4500 straight 0,20 13 65 1,25 164,0 180,0 336,5 3PBT S5 0,19 49 7,15 7,26 3,33 I 650 100 600 50 400 4000 4500 mix 0,32 19,80 62 2,50 171,0 187,0 440,0 3PBT S6 0,22 34 7,15 7,26 3,33 I 650 100 600 50 400 4000 4500 hooked 0,50 30 60 2,50 157,0 168,0 330,0 3PBT S7 0,21 34 7,15 7,26 3,33 I 650 100 600 50 400 4000 4500 mix 0,32 20 62 2,50 169,0 185,0 400,0 3PBT S
A28 1 27 14,00 13,08 1,58 2,35 I 900 100 810 80 300 10300 10900 52,3* 63,0 437,6 0,68 3PBT S2 26 14,00 13,08 1,58 2,35 I 900 100 810 80 300 10300 10900 54,3* 65,4 528,5 0,82 3PBT S3 38 14,00 13,08 1,58 2,35 I 900 100 810 80 300 10300 10900 hooked 0,75 60 80 0,51 51,8* 62,4 542,4 0,84 3PBT S4 37 14,00 13,08 1,58 2,35 I 900 100 810 80 300 10300 10900 hooked 0,75 60 80 0,76 46,2* 55,7 509,4 0,79 3PBT S
A29 1 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,00 34,0 41,0* 63,15 0,31 4PBT S2 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 0,50 34,0 41,0* 109,0 0,53 4PBT S3 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 0,75 34,0 41,0* 90,5 0,44 4PBT S4 0,50 7 19,00 3,90 460 2,00 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 105,2 0,52 4PBT S5 0,50 7 19,00 3,90 460 2,50 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 77,1 0,38 4PBT S6 0,50 7 19,00 3,90 460 1,50 I 375 50 340 60 140 1910 0,50 30 60 1,00 34,0 41,0* 153,5 0,75 4PBT S
A30 1 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 0,00 44,8 54,0* 42,5 0,14 4PBT S2 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 stirrup 0,69 42 60 0,00 44,8 54,0* 50,0 0,16 4PBT F3 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 stirrup 0,69 42 60 0,00 44,8 54,0* 52,0 0,17 4PBT F4 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 0,69 42 60 0,57 45,2 54,5* 48,5 0,15 4PBT S5 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 0,69 42 60 1,15 40,7 49,0* 66,5 0,21 4PBT F6 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 0,69 42 60 1,72 34,3 41,3* 74,0 0,24 4PBT F7 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 stirrup 0,69 42 60 0,57 45,2 54,5* 65,5 0,21 4PBT F8 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 stirrup 0,69 42 60 1,15 40,7 49,0* 71,5 0,23 4PBT F9 0,70 9,50 1,08 450 2,87 R 230 209 150 1800 stirrup 0,69 42 60 1,72 34,3 41,3* 74,0 0,24 4PBT F
hooked‐
ended steel
fibres
Ordinary
Portland Cement
Cross‐sectionSpecimens
ByPortland
Cement R
52,5
FibresMix
Beam size
Flexural ReinforcementConcrete Shear Force at [kN]
B‐1
Appendix B
SFRC Papers and References
[A01] Swamy, N.R., Roy, J. & Chiam A.T.P (1993): Influence of Steel Fibers in the Shear Resistance of Lightweight Concrete I‐ Beam. ACI Structural Journal, Volume 90, No. 1, pp. 103‐114.
[A02] Dinh,H. H., Parra‐Montesinos, G.J. & Wight, J.K. (2010): Shear Behavior of Steel Fiber‐Reinforced Concrete Beams without Stirrup Reinforcement. ACI Structural Journal Volume 107, No. 5, pp. 597‐606.
[A03] Ashour, S.A., Hasanain, G.S. & Wafa, F.F. (1992): Shear Behavior of High‐Strength Fiber Reinforced Concrete Beams. ACI Structural Journal Volume 89, No. 2, pp. 176‐183.
[A04] Batson, G.B. & Youssef, A.G. (1994): Shear Capacity of Fiber Reinforced Concrete Based on Plasticity of Concrete: A Review. Fiber Reinforced Concrete: Developments and Innovations, SP‐142 (Eds.: Daniel, J.I & Shah, S.P.) American Concrete Institute, Detroit, USA, pp. 141‐165.
[A05] Henager, C.H., Asce, M. & Doherty, T.J. (1976): Analysis of Reinforced Fibrous Concrete Beams. Journal of the Structural Division, ASCE Volume 102, No. 1, pp. 177‐188.
[A06] Kaushik, S.K., Gupta, V.K. & Tarafdar, N.K. (1987): Behaviour of Fiber Reinforced Concrete Beams in Shear. Proceedings of the International Symposium of Fibre Reinforced Concrete, Madras, India, December 16‐19, pp. 133‐149.
[A07] Murty, D.S.R. & Venkatacharyulu, T. (1987): Fibre Reinforced Concrete Beams Subjected to Shear Force. Proceedings of the International Symposium of Fibre Reinforced Concrete, Madras, India, December 16‐19, pp. 125‐131.
[A08] Narayanan, R. & Darwish, I.Y.S. (1987): Use of Steel Fibers as Shear Reinforcement. ACI Structural Journal, Volume 84, No. 3, pp. 216‐227.
B‐2
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TRITA‐BKN. Master Thesis 331,
Structural Design and Bridges 2011
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