Prisco Et Al-2013-Structural Concrete

342 2013 Ernst & Sohn Verlag fr Architektur und technische Wissenschaften GmbH & Co. KG, Berlin Structural Concrete 14 (2013), No. 4

Technical Paper

DOI: 10.1002/suco.201300021

In the fib Model Code for Concrete Structures 2010, fibre-rein-forced concrete (FRC) is recognized as a new material for struc-tures. This introduction will favour forthcoming structural appli-cations because the need of adopting new design concepts andthe lack of international building codes have significantly limitedits use up to now. In the code, considerable effort has been de-voted to introducing a material classification to standardize per-formance-based production and stimulate an open market forevery kind of fibre, favouring the rise of a new technological play-er: the composite producer.Starting from standard classification, the simple constitutivemodels introduced allow the designer to identify effective consti-tutive laws for design, trying to take into account the major con-tribution in terms of performance and providing good orientationfor structural uses. Basic new concepts such as structural char-acteristic length and new factors related to fibre distribution andstructural redistribution benefits are taken into account. A fewexamples of structural design starting from the constitutive lawsidentified are briefly shown.FRC can be regarded as a special concrete characterized by acertain toughness after cracking. For this reason, the most impor-tant constitutive law introduced is the stress-crack opening re-sponse in uniaxial tension. A wide discussion of the constitutivemodels introduced to describe this behaviour, which controls allthe main contributions of fibres for a prevailing mode I crackpropagation, is proposed. The validity of the models is discussedwith reference to ordinary cross-sections as well as thin-walledelements by adopting plane section or finite element models.

Keywords: fibre-reinforced concrete, constitutive equations, identification,modelling, structural characteristic length, structural behaviour, redundancy,structural design

1 Introduction

Fibre-reinforced concrete (FRC) is a composite materialthat is characterized by an enhanced post-cracking resid-ual tensile strength due to the capacity of the fibres tobridge the crack faces.

During the last two decades, a wide range of researchhas been performed on FRC material properties, in boththe fresh and hardened states [19]. The investigations

started in the USA, driven by research into closely spacedwires and random metallic fibres [1018]. This researchwas the basis for a patent on steel fibre-reinforced con-crete (SFRC) based on fibre spacing in 1969 and in 1970[19]. The Portland Cement Association (PCA) started in-vestigating fibre reinforcement in the late 1950s. The prin-ciples of composite materials were applied to analyseFRC. The addition of fibres was shown to significantly in-crease toughness after the onset of the first cracking. An-other patent based on bond and the aspect ratio of the fi-bres was granted in 1972 [19]. Since the time of theseoriginal fibres, many new steel fibres have been produced.The usefulness of SFRC was aided by other new develop-ments in the concrete field. High-range water-reducing ad-mixtures able to improve the workability of some harshSFRC mixtures were formulated and break through thesuppliers and contractors reservations regarding the useof SFRC. Although many experimental campaigns havebeen developed since the 1960s, research on the structur-al response of FRC elements mainly developed over thelast 15 years. As a consequence, there is still a lack of in-ternational building codes for the structural design of FRCelements, even though a number of design guidelines wererecently drawn up. This may explain the limited use ofFRC among practitioners, who hardly accept the adoptionof voluntary guidelines or, even worse, research resultsavailable in scientific papers.

FRC now appears in the fib Model Code 2010 after ahuge amount of research and a historical developmentspanning more than 50 years. Two main reasons justifythe long time needed: a theoretical aspect that forces de-signers to consider fracture mechanics concepts to de-scribe the post-cracking residual strength in tension due tofibre bridging, and the technological aspects mainly relat-ed to workability and fibre alignment, which has askedconcrete chemistry for new products to favour the intro-duction of increasingly large fibre contents in cement-based composites.

Early design considerations were produced byACI 544 [20], and even in ACI 318 [21] some new ruleswere just introduced with reference to minimum shear re-inforcement, while RILEM TC162-TDF produced designguidelines for typical structural elements [22, 23]. After-wards, recommendations were produced by other coun-tries, e.g. France [24], Sweden [25], Germany [26], Austria[27], Italy [28], Japan [29] and Spain [30].

Fibre-reinforced concrete in fib Model Code 2010:principles, models and test validation

Marco di Prisco*Matteo ColomboDaniele Dozio

* Corresponding author: [email protected]

Submitted for review: 8 April 2013Revised: 3 August 2013Accepted for publication: 8 September 2013

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M. di Prisco/M. Colombo/D. Dozio Fibre-reinforced concrete in fib Model Code 2010: principles, models and test validation

Structural Concrete 14 (2013), No. 4

Owing to better knowledge of FRC and the recentworldwide developments in guidelines for structural de-sign, fib Special Activity Group 5 (SAG 5), which preparedthe new fib Model Code, decided to introduce some sections on new materials [31] and in particular on FRCstructure design [32]. The fib Working Groups TG 8.3(Fibre-reinforced concrete) and TG 8.6 (Ultra-high-per-formance fibre-reinforced concrete) prepared these sec-tions of the fib Model Code 2010 concerning FRC designrules to provide guidance to engineers for the proper andsafe design of FRC structural elements at both serviceabil-ity and ultimate limit states, based on state-of-the-artknowledge.

In fib Model Code 2010, FRC is introduced in twosections: 5.6 and 7.7 the former focusing on material be-haviour, the latter on structural behaviour. The basic prin-ciples introduced in these two sections are mainly ob-tained from research on SFRC, but fib Model Code 2010 isopen to every kind of fibre, following a performance-baseddesign approach [33]. Nevertheless, several warnings areintroduced regarding the long-term behaviour of some fi-bres, especially those characterized by a low Youngs mod-ulus value.

This paper aims to present some basic principles gov-erning the structural design of FRC elements made of reg-ular concrete which were mainly introduced by fib TG 8.3.The main concepts were derived from some nationalguidelines for FRC structural design [24, 28] and from theguidelines proposed by RILEM TC162-TDF [22, 23]. Theprinciples discussed here are mainly related to SFRC hav-ing a softening post-cracking behaviour in uniaxial tension(Fig. 1a), even though they can be extended to hardeningmaterials (Fig.1b). Since hardening behaviour in one di-rection is sometimes related to a softening behaviour inthe orthogonal direction [34] for the known materials withaligned fibres, the two fib committees active on softeningFRC materials (TG 8.3) and hardening UHPFRC materials(TG 8.6) are cooperating in the writing of two bulletins inrelation to the design rules in order to favour a unified ap-proach.

2 FRC classification

Classification is an important requirement for structuralmaterials. When referring to ordinary concrete, designerschoose the compressive strength, workability or exposi-tion classes that have to be provided by the concrete pro-ducer.

It is well known that fibres reduce the workability offresh concrete, but workability classes for plain concretecan be adopted for FRC as well [35]. Some studies are stillneeded for exposition classes since fibres may reduce thecrack opening [3638]. Therefore, for the exposition class-es described in EN 206 (2006), different rules may beadopted for FRC structures (i.e. smaller concrete covers,etc.). When using FRC, compressive strength is not partic-ularly influenced by the presence of fibres up to a contentof 1 % by vol., so the classification for plain concrete canbe used. As fibre content grows, the post-peak progressive-ly increases its toughness, becoming ductile for very highfibre contents [3941].

The mechanical property that is mainly influencedby fibres is the residual post-cracking tensile strength, andthat represents an important design parameter for FRCstructures. Owing to the well-known difficulties in per-forming uniaxial tensile tests, standard methods are gener-ally based on bending tests on small notched beams. Sincebending behaviour is markedly different from uniaxial ten-sion behaviour, it may happen that softening materials intension exhibit a hardening behaviour in bending ([42],Fig. 2). In fact, in bending tests, cracks arise before thepeak load is reached and it may happen that softening ma-terials in uniaxial tension exhibit stable crack propagationwith increasing load (hardening behaviour in bending orflexure hardening).

The large number of parameters affecting the fibrepull-out mechanism, and consequently residual strengths,does not allow reliable prediction of FRC response in uni-axial tension based on matrix, fibre mechanical character-istics and fibre content. Experimental evidence suggeststreating this cement-based material as a unique compositewhose characteristics depend on fibre dispersion. The un-known fibre location and post-cracking residual strengthrepresent the most interesting concepts for this material. Ifits constitutive relationships are examined assuming asimple homogeneous material, some links between thematerial and the related structure arise and cannot be ig-nored if we require reliable design predictions.

P

P P

PPcr crP

crack formationcrack

crack formation

localization

Fig. 1. Typical load P vs. deformation curve for FRC: softening (a) andhardening (b) [fib MC2010, Fig. 5.6.2].

a) b)

Fig. 2. Different response of structures made of FRC having a softening orhardening behaviour under uniaxial tension or bending loads [fib MC2010,Fig. 5.6.1]

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Material classification for FRC is based on the nomi-nal properties of the composite material, referring to post-cracking tensile strength, determined from bending testson notched prisms according to EN 14651 (2004 [43],Fig. 3); the diagram of the applied load F vs. the deforma-tion should be produced (Fig. 4). Deformation is ex-pressed in terms of crack mouth opening displacement(CMOD) or mid-span deflection. In order to normalize theload F, the nominal tensile stress N in bending is consid-ered, i.e. the bending moment F * L/4 divided by the elas-

tic modulus in bending, corresponding to that of the criti-cal notched section (Wel = bhsp2/6 and L = 500 mm).

The classification is based on two post-crackingresidual strengths at certain CMODs, which characterizethe material behaviour at the serviceability limit state(SLS; CMOD1 = 0.5mm; fR1k) and at the ultimate limitstate (ULS; CMOD3 = 2.5mm; fR3k). The latter is not in-troduced directly, but the fR3k/fR1k ratio is explicitly indi-cated (Fig. 5). With these assumptions, an FRC materialcan be classified by using a couple of parameters: the firstone is a number denoting the fR1k class, the second is a let-ter denoting the ratio fR3k/fR1k. The fR1k strength valuesindicating the classes are as follows:

1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 [MPa]

whereas the fR3k/fR1k ratio is denoted by the letters a, b, c,d, e corresponding to

a if 0.5 < fR3k/fR1k 0.7b if 0.7 < fR3k/fR1k 0.9c if 0.9 < fR3k/fR1k 1.1d if 1.1 < fR3k/fR1k 1.3e if 1.3 < fR3k/fR1k

The residual flexural tensile strength fRj is defined as

(1) 32,

j

sp2

fF l

bhR jFig. 3. Setup for a three-point bending test [EN 14651, 2004]

Fig. 4. Typical load F vs. CMOD curve for plain concrete and FRC [fib MC2010, Fig. 5.6-6]

Fig. 5. Example of N-CMOD curve with proposed classification rules

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Since brittleness must be prevented in structural mem-bers, fibre reinforcement can only substitute (even partial-ly) rebars or welded mesh at ULS if the following relation-ships are fulfilled:

fR1k/fLk 0.4 (2)

fR3k/fR1k 0.5 (3)

where fLk is the characteristic value of the nominalstrength, corresponding to the peak strength in bending(or the highest nominal stress value in the interval00.05mm of CMOD), determined in a notched beam test(Figs. 3 and 4).

Long-term behaviour of cracked FRC under tensionhas to be properly taken into account for those materialswhose long-term performance is affected by creep and/orcreep failure. The creep effects have not been studiedenough up to now, even though some research is now inprogress at several universities.

3 Constitutive laws in uniaxial tension

The stress-crack opening relationship in uniaxial tensioncan be regarded as the main reference material propertyin the post-cracking range. Two simplified stress-crackopening constitutive laws may be deduced from the bend-ing test results: a rigid-plastic model or a linear post-crack-ing model (hardening or softening), as shown schematical-ly in Fig. 6, where:wu crack opening corresponding to ULSfFts serviceability residual strength, defined as the post-

cracking strength for a crack opening significant forSLS

fFtu residual strength significant for ULS

Both fFts and fFtu are calculated using the residual flexuralstrengths fR1 and fR3 identified in bending.

3.1 Kinematic model, structural characteristic length and ultimate crack opening

When considering softening materials, the definition of astress-strain law in uniaxial tension requires the introduc-tion of a structural characteristic length lcs for the struc-tural element. This basic concept represents a bridge(Fig. 7) to connect continuous mechanics, governed bystress-strain (-) constitutive relationships, and fracturemechanics, governed by a stress-crack opening (-w) law,initially proposed by Hillerborg (1976, [44]). The structuralcharacteristic length is equal to the crack spacing whenmultiple cracking takes place and can be considered asequal to the beam depth when a plane section approach isused in the analysis.

When a finite element (FE) model is used, several ap-proaches related to an internal length defined in relationto physical parameters, such as maximum aggregate sizefor non-local approaches, or element size for local ap-proaches, can be used in order to prevent a mesh depen-dency of the results [4547].

The introduction of the characteristic length allowsthe designer to define the strain as

Fig. 6. Simplified constitutive laws: stress-crack opening (solid and dashedlines refer to softening and hardening materials respectively) [fib MC2010,Fig. 5.6-7]

Fig. 7. Examples of characteristic lengths

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= w/lcs (4)

It is interesting to note that Eq. (4) implies that when aplane section model is taken into account and the crackdistribution is not homogeneous in the cross-section (ow-ing to an inhomogeneous distribution of the reinforce-ment in the cross-section), the same FRC material (de-fined by a unique -w response in uniaxial tension) has tobe described in the different portions of the cross-sectionaccording to different - values because of the differentcrack spacings, which correspond to different lcs values(see examples in Fig. 7).

The ultimate crack width wu can be defined on thebasis of a ductility requirement and therefore as

wu = lcs Fu (5)

by assuming Fu equal to 2 % for a neutral axis crossingthe cross-section and 1 % for a neutral axis external to thecross-section. Moreover, a limited value strictly correlatedto the fibre length should also be considered. In relationto the actual market availability and practical considera-tions regarding RC crack opening observed at ULS, a val-ue of 2.5mm is assumed. Eq. (5) can be rewritten as

wu = min (lcsFu; 2.5 mm) (6)

Multiple cracking occurs in hardening materials. There-fore, the identification of crack openings is not necessarybecause a conventional stress-strain law may be directlydetermined by a uniaxial tension test, typically unnotched(like a dog-bone specimen), by dividing the relative dis-placement by the gauge length. It is interesting to notethat sometimes the same self-compacting concrete can ex-hibit a softening and a hardening behaviour depending onthe stretching direction with reference to fibre alignment[34]. This is one of the main reasons why standards haveto model FRC material behaviour following a similar ap-proach for both hardening and softening materials. More-over, every time fibres are aligned, the material cannot beconsidered isotropic, and suitable anisotropic constitutivelaws should be introduced.

3.2 The -w curve identified from bending tests

Accepting the Hillerborg idea of a cohesive approach todescribe the uniaxial tension behaviour of FRC [44, 48],two possible simplified models can be introduced to de-scribe FRC response after cracking. This is done by em-phasizing that the most significant effect induced by fibresis related to the pull-out mechanism: the rigid-plastic andthe linear elastic-softening models.

The rigid-plastic model requires the identification ofonly one parameter: fFtu. It can be easily identified by rota-tional equilibrium at ULS by assuming that the compressivestress distribution is concentrated at the top fibre, whereas atensile post-cracking residual stress distribution is uniform-ly applied to the overall critical cross-section (Fig. 8).

By equating the internal moment of resistance Mu,intto the external applied moment Mu,ext, it is possible towrite the following equation, which corresponds to the ro-tational equilibrium of the cross-section:

(7)

fFtu = fR3/3 (8)

The linear model identifies two reference values: fFts andfFtu. They can be defined by residual values of flexuralstrengths by using the following equations:

fFts = 0.45fR1 (9)

(10)

where wu is the maximum crack opening accepted instructural design.

The two equations are introduced according to dif-ferent assumptions valid at SLS and ULS respectively andbriefly summarized in Fig. 9. At SLS the constitutive rela-tionship for FRC is assumed to be elastoplastic in uniaxialtension and elastic in uniaxial compression. Two equa-tions can be written to impose longitudinal and rotationalequilibrium. According to the notation used in Fig. 9, andassuming

6 2,

32 2

,intMf bh f bh

Mu extR sp Ftu sp

u

( 0.5 0.2 ) 03

3 1f fw

CMODf f fFtu Fts

uFts R R

; 0.5 ; ;E wyl

x w mm y h x l hscs

s sp cs sp

Fig. 8. Simplified model adopted to compute the ultimate residual tensilestrength in uniaxial tension fFtu by means of the residual nominal bendingstrength fR3 [fib MC2010, Fig. 5.6-8]

(a) (b) (c)

Fig. 9. Stress distributions assumed for determining the residual tensilestrength fFts (b) and fFtu (c) for the linear model [fib MC2010, Fig. 5.6-9]

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then the result is

(11)

(12)

The resolution of the non-linear system of Eqs. (11) and(12) gives a correlation between fFts and fR1 that dependson Youngs modulus E and on the choice of the structuralcharacteristic length lcs:

fFts = ka (E, lcs)fR1 (13)

By assuming a structural characteristic length lcs equal tothe critical cross-section hsp, the factor ka changes withYoungs modulus E, as shown in Fig. 10. The ka valueranges between 0.362 and 0.378 and therefore an averagevalue of 0.37 could be considered.

Let us consider the ULS. In this case the compres-sive stress distribution is concentrated on a very smallportion of the cross-section and therefore, once again, it isassumed that a concentrated compressive force acts at theupper fibre. If at the underside a crack opening of 2.5mmis considered, a linear softening model involves a lineardistribution of stresses that is characterized by a valuekbfR1 for w = 0, kafR1 for wi1 = 0.5 mm and fFt,2.5 forwi2 = 2.5mm. The rotational equilibrium becomes

2 20

23 2 2

23

13 6

2

21

2

bx f bx f b yf

x

f by x yf bx

xf

xf bh

FtsFts

Fts

FtsFts Fts R sp

(14)

By solving Eq. (14), it is possible to compute the unknownfFt,2.5 as

(15)

Now, taking into account that the linear model also has tofit for the point (wi1 = 0.5mm, = 0.37fR1), it is possible toexpress kb as follows:

(16)

If fR3 is considered to be equal to 0.5fR1, then that repre-sents the constraint introduced to define the compositeFRC as a structural composite; kb becomes equal to 0.45and Eq. (15) can be rewritten as

fFt,2.5 = 0.5fR3 0.225fR1 0.5fR3 0.2fR1 (17)

Taking into account the definition of the ultimate crackopening wu previously introduced, and in particular thatintroduced for thin-walled elements, a reduced crackopening value can be considered and Eqs. (9) and (10) de-duced. It is important to note that the shifting of the fFts

0.529 0.143 31

kffbR

R

0.52,2.5 3 1

f fk

fFt Rb

R

2

( )3 6,2.5

2

1 ,2.5

2

3

2

fbh

k f fbh

fbh

Ftsp

b R Ftsp

Rsp

Fig. 10. a) Definition of ka and kb coefficients, b) ka values vs. E values for the linearsoftening model and lcs = h

a) b)

Fig. 11. Spurious results obtained with fFts associated with a crack opening of w = 0.5mm and not w = 0

a) b)

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value at w = 0 prevents some spurious situations where aclass reduction (i.e. FRC composite changes from class bto a, or c to b) could involve a better performance in bend-ing (see Fig. 11) for thin-walled elements where wu is closeto 0.5mm.

Finally, it is worth noting that a CMOD value of2.5mm does not correspond to a CTOD value of 2.5mm(see Eq. (18)), but rather to a CTOD close to 2.1mm ac-cording to a rigid-body assumption. Nevertheless, the ideato arrest the cohesive stress at wu is in this way partiallycompensated by the crack opening translation.

3.3 Three- vs. four-point bending test

In the literature, the three-point bending test is not the on-ly notched test proposed. The debate on the best test foridentifying the uniaxial tension constitutive law -w hasinvolved many researchers and is still in progress [49, 50].It is interesting to observe that the difference between thetwo tests is not so large when a careful non-linear compu-tation is carried out. According to Ferrara and di Prisco[51], for plain concrete, the main differences between thetwo tests are related to the peak strength and the first post-peak slope of the load vs. crack opening displacement

(COD), whose behaviour is dominated by matrix response.A three-point bending test exhibits a weak higher peakstrength and a more brittle first post-peak slope. Ofcourse, there is also a significant difference in the queuevalue for a large COD, but this is due to the confinementin compression obtained in three-point bending tests justbelow the central load knife. However, this significant ef-fect becomes negligible when a fibre pull-out mechanismtakes place. Several SFRC materials denoted according toa unified system (Tables 1 and 2) have been compared byinvestigating their bending behaviour according to UNI11039 [52] and EN 14651 [43]. The designation is ex-pressed by Mn-Fn-Vf, where Mn indicates a certain matrix(n ranges between 1 and 8, Table 1), Fn indicates a certainsteel fibre (n ranges between 1 and 8, Table 2) and Vf indi-cates the fibre volume percentage used in the composite.The experimental comparison highlights what was alreadypredicted in [51], showing very similar pull-out strengths,with the tendency to have something more in three-pointbending test (Fig. 12). It is important to emphasize thatthe measurement of the crack opening is not the same forthe two tests. In fact, whereas for a three-point bendingtest the measured parameter is CMOD, for the four-pointbending test the measured parameter is CTOD. If a linear

Table 1. Mix designs of the different materials considered

M1 M2 M3 M4 M5 M6 M7 M8

Fck [MPa] 50 60 75 65 40 30 55 75

400 400 450 350 300 340 480 380

Cement [kg/m3] CEM I CEM I CEM I CEM I CEM II CEM II CEM I CEM I52.5R 52.5R 52.5R 52.5R 42.5R 42.5R 52.5R 52.5R

Agg. 1 [kg/m3]511 569 620 120 1450 559 620 120

< 20mm < 12mm 03mm 03mm < 12mm 1122mm 03mm 03mm

Agg. 2 [kg/m3]789 403 440 970 420 93 440 970

< 10mm < 8mm 012mm 012mm 36mm 410mm 012mm 012mm

Agg. 3 [kg/m3]375 676 710 815 1201 710 815

< 4mm < 4mm 815mm 815mm < 5mm 815mm 815mm

Filler [kg/m3]98 96 30 60 80 60

(calc.) (calc.) (calc.) (fly-ash) (fly-ash) (fly-ash)

Plasticizer/cement 0.8 % 2.2 % 1.2 % 0.9 % 2.0 % 0.8 % 0.8 % 0.9 %

Water /binder 0.40 0.39 0.41 0.34 0.46 0.50 0.4 0.34

Table 2. Properties of the different steel fibres considered

F1 F2 F3 F4 F5 F6 F7 F8

Df [mm] 0.375 0.62 0.6 0.8 0.6 0.73 1.2 1.0

Lf [mm] 30 30 30 60 30 30 44 60

Aspect ratio 80 48 50 75 50 41 36 60

Tensile strength [MPa] 2300 1250 1192 1192 1100 390 390 390

Type h.e. h.e. h.e. h.e. h.e. c. c. c.

h.e. = hooked end; c. = crimped

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Fig. 12. Three- vs. four-point bending tests: experimental responses for several materials

350



opening of the crack edges is assumed, then that is usuallyaccepted for large crack openings, a rough upper-boundlinear relation between the two COD measures can be in-troduced as

(18)

3.4 Refined constitutive relationships

Once the linear stress-crack opening relationship has beenidentified, the stress-strain relationship can be deduced byintroducing the suitable structural characteristic length lcswith reference to softening materials (Fig. 7, [53]).

At SLS, a more refined curve able to fit the peakstrength of the matrix can be proposed and it is particular-ly suggested for FE analyses. The same constitutive rela-tionship adopted for plain concrete in uniaxial tensioncan be used up to peak strength fct, while in the post-crack-ing stage, a bilinear relation applies (Fig. 13). The residualstrength due to the pull-out mechanism (final branch),which represents the main fibre contribution, is defined bytwo points corresponding to (SLS, fFts = kafR1) and (ULS,fFtU). This simplification, which is easily understood be-cause it takes into account only the most significant fibrecontribution, the pull-out effect, may cause the same prob-lem previously discussed in Fig. 11. For this reason, espe-

150125

1.2CMOD CTOD CTOD

cially when the mechanical behaviour is strongly affectedby small crack openings as for statically redundant struc-tures, the first point, corresponding to SLS, can be shiftedon the first softening branch conserving a value equal tokafR1.

When the identification procedure is carried outstarting from a four-point bending notched test, coefficientka (Eq. (13)) can be set equal to a different value depend-ing on the specific crack opening range adopted in thestandard. If UNI 11039 is considered, a value of 0.39 canbe adopted to take into account that the SLS value coversa range between 0 and 0.6mm of crack tip opening dis-placement (CTOD), which corresponds to an average value of 0.3mm instead of roughly 0.41mm as occurs inEN 14651, where the CMOD value is 0.5mm. In this case,in Eqs. (9) and (10) the residual strengths fRi are substitut-ed by feqi because they are computed as average values incertain CTOD ranges: 00.6mm for SLS and 0.63.0mmfor SLU.

In fib Model Code 2010, the constitutive relation-ships for softening materials (Fig. 13) are presented to-gether with those suggested for hardening materials,where, progressively, the matrix contribution cannot bedistinguished from the fibre one. Only the softening caseis discussed in the following. Further details on hardeningmaterial responses will be discussed in an fib bulletin cur-rently in preparation.

Note that by introducing the peak strength of thematrix it is possible to better induce localization when anFE model is adopted, thus preventing spurious dissipationdue to uncontrolled growths of the cracked band [54].

Finally, the introduction of the fracture energy corre-sponding to the matrix peak strength contribution is notconsidered in the equilibrium equations used to identifyEqs. (9) and (10), but its addition is not generally signifi-cant for typical structures.

3.5 The -w curve identification: theoretical vs. experimental results

A broad experimental campaign was planned in order toascertain whether the constitutive model -w identifiedfrom notched bending tests is reliable. A comparison was

Fig. 13. Simplified relationship suggested for SLS analysis taking matrixpeak strength contribution into account [fib MC2010, Fig. 5.6-10]

Fig. 14. -w curves in uniaxial tension; comparison of constitutive law deduced according to linearsoftening model (Eqs.(9) and (10)) and fixed-end directtensile test carried out on cylindrical specimens (b) directly core-drilled by the notched beam used to characterize the FRC composite (a) [43]

a) b)

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first carried out by casting three specimens made with thesame FRC, belonging to the same batch, and carrying outthree four-point bending tests according to Italian stan-dard UNI 11039 [52]. Three cylindrical specimens werethen core-drilled from one undamaged end (Fig.14a) andtested in uniaxial tension by means of a closed-loop pressable to impose a fixed-end condition [55]. The comparisonis shown in Fig. 14, and is really encouraging. In fact, the

scattering of the results was very much reduced, not onlybetween the LVDTs at 120 in the same test [55], but alsoin the three different tests (see small shadow in Fig.14b).The dashed line shown is computed as the average be-tween the three average responses, computed as the meancurve of the three transducers in each test. Other testswere performed after thermal cycles at different tempera-tures (T = 200, 400 and 600 C). Similar comparisons werealso carried out on prismatic specimens in order to com-pare bending tests on notched and unnotched specimenswith uniaxial tension tests with fixed and rotating endsmade from the same materials [61].

The uniaxial tension test was also simulated bymeans of Diana Finite Element code, introducing as theconstitutive relationship the -w response identified frombending (Fig.15) using Eqs. (9) and (10). The axisymmet-ric mesh (Fig. 16) adopts regular triangular elements [54]and the COD was computed, as in the experimental tests,as the relative displacement between two points at a dis-tance of 50mm astride the notch. The structural charac-teristic length in this case is a localization limiter withlcs = 2A [56]. The results (Fig. 17) are obtained by chang-ing the size of the elements (lcs = 1.163, 2.327 mm). Thehigher curve is the constitutive response identified andused as an input for FE modelling, whereas the two FEstructural responses of the modelled specimens are practi-cally coincident and a little closer to the average experi-mental curve measured with the LVDTs at 120. The maindifference is shown in the post-peak response, where asmaller dissipation should be related to the modelling as-

Fig. 15. Uniaxial tensile constitutive law identified from bending tests: bilinear hardening (bh); bilinear softening (bs) [51]

Fig. 16. FE meshes adopted for simulating the uniaxial tensile test

Fig. 17. FE simulation of uniaxial tension tests in fixed-end condition: a) global response, b) zoom at small COD

a) b)

352



sumption of neglecting defects, which characterize the real specimen.

Using the same constitutive relationship, it is alsopossible to model the original notched beam specimentested in a four-point bending setup. Two meshes were in-troduced: they consider a plane stress approach andquadrilateral elements (mesh 1, Fig. 18, [54]) or triangularelements (mesh 2; Fig. 18, [54]). Even if the same constitu-tive relationship -w considered in the uniaxial tensiontests was assumed and a proper localization limiter wasapplied as suggested by Rots [56] for both meshes, differ-ent post-cracking responses were obtained. In Fig. 19 theFE results are compared in terms of nominal strength Nvs. CTOD with experimental values from the four-pointbending tests. It should be noted that triangular elementsintroduce a non-negligible increase in toughness.

Fig. 18. FE meshes adopted for simulating four bending tests on notched beams

Fig. 19. FE simulation results for notched four-point bending tests

Fig. 20. FE simulation of bending test: a) trilinear constitutive law adopted, b) model reliability with reference to a notched four-point bending test

a) b)

Fig. 21. Notched four-point bending test simulation: multi-linear constitutive law adopted (a) and numerical results (b)

a) b)

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The simplified models introduced can sometimes besubstituted by more complex models identified by meansof back-analysis [57]. With reference to mesh 1 (Fig. 18,[54]), a trilateral softening curve can be introduced asshown in Fig. 20a to reproduce the uniaxial tension testbetter. The improvement involves a more careful fitting ofthe bending response (Fig. 20b). Incidentally, a progres-sive improvement in the identification process is not al-ways associated with the best response if the back-analysisis carried out with reference to a different kinematic mod-

el. In Fig. 21a, a multi-linear -w curve identified by meansof a plane section model can allow the designer to fit per-fectly the response with a plane section approach(Fig. 21b), but the result of the FE analysis with the sameconstitutive law could be far from the ideal fitting(Fig. 21b).

Moving to the use of a plane section model, which ismuch more efficient for the structural design of bent ele-ments, a series of beam specimens made of different ma-terials as defined in Tables 1 and 2 was made to check thereliability of the proposed linear model (le/ls: linear elas-tic pre-peak/linear softening). The same constitutivemodels were adopted in the FE analysis with the properstructural characteristic length by using the two meshesshown in Fig. 18. The results (Fig. 22) confirm good relia-bility for a plane section model and FE mesh 1, whereasFE mesh 2 always gives an over-resistant response. Ofcourse, the fitting is reasonable, it often overestimates thepeak strength and the stiffness before the peak. A com-parison between linear and bilinear relationships at pre-and post-peak is also shown (Fig. 23) with reference tothe plane section and FE models. The main difference forboth the kinematic models appears in the peak region as expected: linear relationships exhibit lower peakstrengths due to the lack of matrix contribution and thesame pull-out strength.

A final investigation in terms of good fitting can beperformed by considering the choice of the structural

Fig. 22. Plane section and FE validation of the linear model proposed in fib MC2010 for several materials defined in Table 1

Fig. 23. Influence of kinematic and constitutive model

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characteristic length. In fact, this variable is not only relat-ed to the ideal condition of the plane section kinematicconstraint, but also plays a role in predicting pre-peak be-haviour and peak strength. By examining different choicesof lcs and two different materials with a linear elastic mod-el in pre-peak and a bilinear softening model (le/bs) inpost-peak, it is possible to highlight the role played by thestructural characteristic length. Fig. 24 clearly highlightsthe value of lcs = h suggested in fib Model Code 2010 as areasonable choice.

4 The structural specimen

The identification of the uniaxial tension constitutive lawin the post-cracking regime is severely affected by fibre dis-tribution and other parameters as observed by many re-searchers [58]. FRC is a cementitious composite and there-fore, to be regarded as a homogeneous material, thespecimens used to characterize its behaviour should havea volume that can be representative of the FRC hetero-geneity grade. Changing casting and handling procedures

Fig. 24. Plane section model: the role of the structural characteristic length for two different materials with bilinear softening model: a), c) structural characteristic length influence for M2-F4-0.62 and M3-F2-0.62 materials respectively; b), d) related zooms

a) b)

c) d)

Fig. 25. Notched standard tests and unnotched structural specimen tests (h = 60 mm; average curves of at least 3 specimens); bending response with thesame materials: tests according to UNI 11039 [49] and UNI 11188 [58]; A and B in (b) refer to specimens tested as cast (A) or turning the specimens upsidedown (B) [56]

a) b)

355



as well as the mixer, as necessitated by larger cast volumes,can drastically change mechanical characteristics in bend-ing [58]. In thin-walled elements the use of fibre reinforce-ment is quite appropriate but, in general, FRC should present a hardening response in bending. The representa-tiveness of standard notched beam specimens, with across-section of 150 150mm, for characterizing the be-haviour of this type of element is questionable. At thesame time, fibre dispersion and orientation are seriouslyaffected by a casting procedure that, in thin elements, isdifferent from that adopted in standard specimens. Fur-thermore, standard specimens are notched and it is diffi-cult to guarantee the hardening behaviour of the material(in statistical terms) by their response. In fact, the notchfavours stable crack propagation and significantly modi-fies the first cracking process, especially if the fibre con-tent is not high enough to change the mechanical behav-iour at peak [5961]. For all these reasons, an unnotchedprismatic specimen cast using the same procedure as forthe structure and with the same thickness is preferred.The specimen can be tested by means of a four-point bend-ing setup that favours a crack propagation starting fromthe weakest cross-section between the load points. A trans-ducer attached between two points on the underside ofthe specimen measures the relative displacement betweenthese points (COD). This type of thin specimen can bevery representative of the behaviour of thin FRC struc-tures; therefore, it can be named structural specimen asfirst suggested in the French guidelines on ultra-high-per-formance fibre-reinforced concrete AFGC-SETRA [24]and in the Italian guidelines on SFRC [62]. In Fig. 25 a setof five different material tests highlights as the structuralspecimen usually gives a weaker response in relation tothe standard notched tests used for classifying the materi-al, even if a higher performance could be expected for themost favourable fibre distribution. For this reason, whenthin-walled elements have to be designed safely, a carefulidentification of the mechanical response by means ofstructural specimens is strongly suggested. This choicecould allow a partial reduction in the safety factors as sug-gested in [28, 58].

5 Reliability of structural behaviour prediction

The constitutive models introduced to describe uniaxialtension can also be used to check the behaviour of beamswith a conventional cross-section. To this end, threebeams 3m long with a 300 300mm square cross-sectionwere tested in a four-point bending test setup to check thereliability of the approach proposed in fib Model Code2010. No conventional reinforcement was introduced [63,64, 54]. The beam geometry and the setup adopted areshown in Fig. 26. The material is M2-F4-0.62, as specifiedin Tables 1 and 2.

According to the measurements indicated in the setup(Fig. 29), both the load vs. vertical displacement and thebending moment vs. measured central curvature are pro-posed in Figs. 27a,b. Careful measurement of the fibre num-bers in the critical cross-sections was also carried out and isindicated in Fig. 28. Fig. 29 also shows the final crack pat-terns. The numerical prediction carried out using the planesection approach fits quite well with the mechanical re-sponse measured experimentally. It is important to empha-size that the comparison is performed by considering the av-

Fig. 26. Geometry and test setup for full-scale beam test

Fig. 27. Full-scale bending test, numerical and experimental results: a) bending moment vs. curvature diagram, b) load vs. displacement curve

a) b)

356



erage curves in the identification process of the uniaxial ten-sion law and not the characteristic ones. The smaller stiff-ness of the global curve (Fig. 27b) is due to the lack of local-ization and to a softening zone assumed to be spreadhomogeneously over the overall zone with the same maxi-mum bending moment. FE analyses were also performed us-ing the same constitutive relationship and quadrilateral ele-ments. In this case average and characteristic curves fit thestructural behaviour quite well, even if the pre-peak responsesimulated by the FE analysis is not able to take into accountthe defect distributions adequately as well as inhomoge-neous shrinkage effects in the cross section, thus exhibiting astiffer behaviour. In order to favour the localization in thecritical section detected experimentally, an initial geometricdefect was introduced by assigning a local width reduction< 10 % in the proximity of the main crack propagation.

To conclude the modelling of unreinforced concretestructures, an example of the role played by the structural

characteristic length is shown in Fig. 30, where the sameuniaxial tension constitutive law is assumed and differentdepths are considered [50]. The structural characteristiclength is therefore able to reproduce a significant size ef-fect without the need for any special coefficient as pro-posed by Rilem TC 162 TDF [23].

The same constitutive relationships were also used tomodel the bending behaviour of prefabricated FRC roof elements, where only prestressed longitudinal reinforce-ment remained all the transverse reinforcement was sub-stituted by different types of steel fibre. Further details forthese cases can be found in [65, 66].

Several research projects are in progress to check theglobal behaviour of R/C structures where an FRC compos-ite is adopted. In section 7.7, the models discussed to re-produce the uniaxial tension behaviour are used coupledwith conventional reinforcement. The bending momentresistance at ULS with longitudinal reinforcement, which

Fig. 28. Fibre distribution in the critical cross-section of the full-scale beams tested

Fig. 29. Crack patterns at failure for the full-scale beams tested

Fig. 30. Size effect introduced with the structural characteristic length: a) bilinear softening model, b) linear softening model

a) b)

357



represents the usual case, can be investigated by means ofthe addition of a fibre contribution as clearly shown inFig. 31. The bending failure stage is supposed to bereached when one of the following conditions applies: attainment of the ultimate compressive strain in the

FRC, cu attainment of the ultimate tensile strain in the steel (if

present), su attainment of the ultimate tensile strain in the FRC, Fu

It is important to emphasize that in this case the structur-al characteristic length usually depends on the crack dis-tance and therefore on the reinforcement ratio and bar diameters used. Furthermore, fibres help to increase theductility of the plastic hinge by increasing the passive con-finement in the compression zone, but this effect is nottaken into account.

Several investigations are in progress to check the ef-fectiveness of fibres in reducing crack distance and open-ings as suggested in fib Model Code 2010, with suitable re-lationships in agreement with conventional reinforcedconcrete [36, 37, 38].

6 Partial safety factors and redundancy coefficients

For ultimate limit states, recommended values of partialsafety factors F are shown in Table 3. For serviceabilitylimit states, the partial factors should be taken as 1.0.However, some special observations have to be made.FRC is scantly homogeneous and isotropic because fibreslocation is random and mainly depends on casting proce-dure, formwork geometry and mix consistency affected byflowability, viscosity and filling ability. Therefore, the scat-tering of its response mainly depends on numbers of fibresin the cracked section, their location and their orientation.On the basis of the previous considerations, the choice ofthe safety factors should take into account the following:

the representativeness of the specimens used to charac-terize the mechanical response of the material, in rela-tion to the structure considered

the number of specimens for mechanical characteriza-tion

the stress redistribution capacity of the structure underconsideration

the fracture volume involved in the failure mechanism

Besides the safety factor indicated in Table 3, suitable co-efficients K, which take into account the representative-ness of the specimen used for the identification in relationto the structure and the casting procedure adopted, are al-so introduced. In general, an isotropic fibre distribution isassumed so that the fibre orientation factor K is equal to 1.For favourable effects, an orientation factor K < 1.0 may beapplied if verified experimentally. For unfavourable ef-fects, an orientation factor K > 1.0 must be verified experi-mentally and applied. The values fFtsd and fFtud shouldthen be modified to

fFtsd,mod = fFtsd/K (19)

fFtud,mod = fFtud/K (20)

A careful analysis of the role played by the safety factorwhen a non-linear mechanical analysis is carried out ac-cording to fib Model Code 2010 is described by Cervenkaet al. [67].

7 Basic aspects for design

Fibre reinforcement is suitable for structures where dif-fused stresses are present. In structures with both local-ized and diffused stresses, which is the usual case, it is bet-ter to base the reinforcement on a combination of rebarsand fibre reinforcement.

In structural elements where fibres aim to replaceconventional reinforcement (even partially), some restric-tions on the minimum residual strength are applied (Eqs.(2) and (3)). This residual strength becomes significant instructures characterized by a high degree of redundancy,where a remarkable stress redistribution occurs. For thisreason, in structures without rebars, where fibres com-pletely replace conventional reinforcement, a minimumredundancy level is required for the structural member.

Fig. 31. ULS for bending moment and axial force: use of simplified stress-strain relationship [fib MC2010, Fig. 7.7-3]

Table 3. Partial safety factors for FRC

Material F

FRC in compression as for plain concrete

FRC in tension (limit of linearity) as for plain concrete

FRC in tension (residual strength) 1.5

358



On the contrary, in structures with rebars, where fibresconstitute additional reinforcement, ductility is generallyprovided by conventional reinforcement that makes a ma-jor contribution to the tensile strength. For hardeningFRCs (in uniaxial tension), fibres can be used as the onlyreinforcement (without rebars), also in statically determi-nate structural elements. The heterogeneity of the me-chanical behaviour in the post-cracking regime is oftensignificantly penalized due to the high scattering mainlyrelated to fibre distribution and orientation. When a sig-nificant redundancy is guaranteed for the structure by itsgeometry and its boundary conditions, and a large volumeof the structure is involved in the failure process, the ex-perimental investigation has highlighted that the averagemechanical behaviour rather than the characteristic one takes place. For this reason, a suitable coefficient KRd,

aimed at increasing the load bearing capacity of the struc-ture, is introduced [54].

Section 7.7, after a rough classification, also intro-duces semi-empirical equations for designing FRC struc-tural elements when subjected to shear according to themulti-level approach [68], punching [69] and torsion.There are even suitable equations for slabs and walls aswell as specific equations to compute crack distance andcrack opening taking into account fibre contribution,which thus allows us to design new FRC structures ac-cording to these principles.

8 Concluding remarks

The implementation of fibre reinforced concrete (FRC) inthe fib Model Code 2010 is a very important milestone. In

Table 4. Experimental results of four-point bending tests

Material Specimen No. fIf,av (std %) [MPa] feq1,av (std %) [MPa] feq2,av (std %) [MPa]

M1-F1-0.62 3 4.9 (12.1 %) 7.43 (19.6 %) 8.11 (23.7 %)

M1-F2-0.62 3 5.15 (4.9 %) 7.02 (19.6 %) 6.16 (5.2 %)

M1-F3-0.32-F4-0.32 3 5.15 (3.6 %) 6.92 (4.68 %) 5.89 (16.5 %)

M2-F4-0.62 9 5.94 (10.1 %) 8.39 (6.3 %) 4.87 (15.6 %)

M3-F2-0.62 3 5.79 (1.1 %) 5.34 (8.5 %) 3.91 (15.8 %)

M4-F2-0.62 7 5.02 (7.9 %) 6.44 (18.0 %) 6.27 (17.0 %)

M5-F5-0.45 8 3.54 (10.7 %) 2.91 (20.6 %) 2.69 (36.6 %)

M6-F6-0.45 6 3.13 (11.4 %) 1.47 (31.2 %) 0.73 (54.5 %)

M6-F6-0.83 6 3.36 (11.6 %) 2.10 (10.1 %) 1.33 (13.2 %)

M6-F7-0.45 6 2.84 (10.8 %) 1.88 (21.4 %) 1.26 (37.6 %)

M6-F7-0.83 6 3.70 (11.0 %) 3.52 (27.1 %) 3.18 (39.0 %)

M6-F8-0.45 6 2.49 (19.8 %) 1.81 (42.0 %) 1.53 (63.9 %)

M6-F8-0.83 6 2.90 (19.8 %) 2.46 (18.0 %) 2.51 (18.8 %)

M7-F2-0.62 8 4.01 (10.3 %) 3.19 (22.2 %) 2.03 (32.5 %)

M8-F2-0.62 6 6.84 (5.8 %) 8.45 (19.5 %) 3.87 (28.4 %)

M8-F1-0.62 6 6.76 (7.1 %) 9.80 (12.2 %) 9.22 (14.8 %)

Table 5. Experimental results of three-point bending tests

Material Specimen No. fctfl,av (std %) [MPa] fR1,av (std %) [MPa] fR3,av (std %) [MPa]

M6-F6-0.45 6 3.54 (9.6 %) 1.30 (23.0 %) 0.68 (42.6 %)

M6-F6-0.83 6 4.12 (4.35 %) 2.07 (16.8 %) 1.41 (20.8 %)

M6-F7-0.45 6 3.41 (14.3 %) 1.42 (15.5 %) 1.21 (11.4 %)

M6-F7-0.83 6 3.95 (20.7 %) 2.54 (38.5 %) 2.31 (40.9 %)

M6-F8-0.45 6 3.20 (7.5 %) 1.92 (27.1 %) 1.97 (38.9 %)

M6-F8-0.83 6 4.15 (10.2 %) 2.90 (28.5 %) 3.21 (30.2 %)

M7-F2-0.62 9 5.06 (13.8 %) 2.53 (30.6 %) 1.86 (35.1 %)

359



the near future it will probably lead to the development ofstructural rules for FRC elements in Eurocodes and na-tional codes.

This paper has carefully discussed the simplifiedmodels suggested for the composite and presented them inthe material section to evaluate the uniaxial tension resid-ual strength, mainly given by fibre pull-out. Their reliabili-ty as well as their limitations are indicated with referenceto several FRC materials, characterized by different ma-trixes, different steel fibres and different fibre contents.The design rules are derived from a unified classificationof FRC composite based on a three-point bending test, al-ready accepted as a European standard.

The identification of the constitutive law is also discussed with reference to the structural characteristiclength concept and two different kinematic models that can be adopted: plane section and finite element approaches. In all the cases discussed, the procedure fits the experimental tests reasonably well, thus showingan appreciable robustness of the whole design ap-proach.

For thin-walled elements, the need for a structuralspecimen to identify the material properties better, takinginto account the real casting procedure, is also highlightedwith reference to various FRC composites. This require-ment becomes essential in the case of self-compacting ma-terials too. Suitable coefficients to take into account inho-mogeneous fibre alignments are also introduced. Due tothe high scatter of FRC responses, a new coefficient ableto consider the beneficial effect of redundancy is also in-troduced.

It is worth noting that although the level of knowl-edge of FRC has increased tremendously over the last 15years, further research is needed to verify and optimizethe proposed design rules, to investigate the long-term be-haviour of different FRCs and other open issues such asthe anisotropic behaviour of FRC, fatigue and multi-axialmechanical behaviour. A new generation of FRCs willsoon enter the market. They are based on a cocktail ofdifferent fibre types (material and/or geometry) to en-hance different structural performance aspects and fibModel Code 2010 should be ready to open up the way fortheir usage.

Acknowledgements

A special vote of thanks goes to professors L. Vandewalleand G. Plizzari for the excellent cooperation to Prof. J.Walraven for the fruitful discussions and to Prof. H. Falkner, who shared with us his considerable design experience. The authors are also indebted to all the mem-bers of fib Task Groups TG 8.3 and TG 8.6 for the constructive discussions during the several meetings,where many ideas presented in this paper took a definitiveshape.

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Matteo Colombo, Ph. D.Assistant ProfessorPolitecnico di MilanoDepartment of Civil and Environmental EngineeringPiazza Leonardo da Vinci, 3220133 Milan, [email protected]: +390223998789, Fax: +390223998377

Daniele Dozio, Ph.D.Senior Structural EngineerArupCorso Italia, 1320122 Milan, [email protected]: +390285979381, Fax: +39028053984

Marco di Prisco, Ph.D.Full ProfessorPolitecnico di MilanoDepartment of Civil and Environmental EngineeringPiazza Leonardo da Vinci, 3220133 Milan, [email protected]: +390223998794, Fax: +390223998377

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