Tailor Made Concrete Structures – Walraven & Stoelhorst (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-47535-8

A model for SFRC beams without shear reinforcement

P. Colajanni, A. Recupero & N. SpinellaUniversity of Messina, Messina, Italy

ABSTRACT: In this paper a physical model, for the prediction of ultimate shear strength of Steel FibersReinforced Concrete (SFRC) beams is developed from the plastic Crack Sliding Model (CSM) introducedby Zhang (1997), based on the hypothesis that cracks can be transformed into yield lines. In this work theeffectiveness factors are recalculated for SFRC beams and some further developments are introduced in theCSM, taking into account the fundamental post cracking tensile strength contribute of SFRC. The proposedmodel is validate by a large set of tests collected in literature and some numerical analyses were carried out toshow the influence of fibers on the failure beams mode.

1 INTRODUCTION

Numerous empirical or semi empirical relations havebeen suggested for the prediction of the ultimate shearcapacity of SFRC beams without stirrups. Some ofthem are obtained on the basis known relations pro-posed in literature for plain concrete beams, providingan additional shear strength contribute that dependson the amount and characteristics of the fibers andthe mechanical properties of the concrete matrix. Thiscategory of design equations incorporates fiber prop-erties, which generally is expressed as fiber factorF = βVf (lf /df ), where β is the fiber bond factor; Vfis the fiber volume percentage; and lf /df is the fiberaspect ratio (ratio between length and diameter fiber).Semi empirical models are usually generated by aregression analysis of SFRC beam test data for a fewfiber types and volume percentages, but the numberof beam tests does not cover a wide enough range offiber types and volume percentages.

An appealing alternative approach is provided bythe plastic theory, that has long been applied withgood success to reinforced concrete members (Nielsen1999). Based on this theory and on limit analysisconcepts many rational formulations have been pro-posed in literature to predict the shear capacity of plainconcrete beams.

The usual plastic solution assumes that stress fieldstransfer the load to supports by satisfying the yieldmaterial criteria, but recent works (Zhang 1997;Vecchio 2000a) on plain concrete shear problemsshown that slips along the crack can delay or pre-vent the development of direct strut action spanningbetween the loading and support points of beams.These certainly imply that sliding displacements can

occur along the crack and the failure crack can originfrom a generic section between loading and supportpoint. This failure mechanism is typical of slenderbeams and it is taken into account by plastic theoryin the CSM (Zhang 1997).

In this paper the formulation of CSM proposed byZhang to determine the ultimate shear strength of plainconcrete beams without stirrups is, firstly, improvedto evaluate the shear capacity of short beams, then isextended to fibrous concrete members. The proposedformulation is validated on a large database collectedin literature and a comparison with several known rela-tionships is presented. Finally a set of numericallyanalyses, carried out using the proposed model, arepresented showing the effect of steel fibers in changingthe mode of failure.

2 CRACK SLIDING MODEL

In the application of the theory of plasticity to struc-tural problems, reinforcement is assumed to resistforces in the axial direction only with yield stressfy. Concrete is assumed to behave as a rigid, per-fectly plastic material, obeying the modified Coulombfailure criterion with the associated flow rule.

At failure the cracked concrete in compression issimultaneously subjected to tensile strains in the direc-tion normal to the compression.Therefore, it exhibits areduced strength compared to the uncracked concreteuniaxially compressed. To this, other theories, suchas the modified compression field theory (MCFT),provide a similar interpretation (Vecchio and Collins1986). This is called compression softening and can berecognized in the plastic theory by the effectiveness

619

Figure 1. Typical crack pattern in a beam without stirrupsunder shear load.

factor of concrete. In the usual plastic solution theeffective compressive strength is fc,ef = νcfc, where theeffectiveness factor is given by:

with fc = compressive cylinder concrete strength;h = height of beam’s cross section; r = 100As/bh; anda = shear span. Equation (1) shows that the νc isfunction of shear span-depth ratio a = h.

The question of why it is necessary to include an a/hdependency when the whole range of a/h values has tobe covered was explored and solved by Zhang (1997)in the CSM. The low values of νc for a/h around 2.5are due to sliding in initial cracks.

Due to the dramatically reduced sliding resistancein a crack, sliding along a crack originated in a genericsection of the shear span may be more dangerous thansliding along the theoretical yield line between supportand load point as in the usual plastic solution.

The crack pattern at the state of failure is schemat-ically shown in the Figure 1. The first cracks arenormally formed in the region with maximum momentand are vertical. Then, gradually, diagonal cracksappear in the shear span closer to the support, alonga line that approximately intersect the top face at theloading point.

The load needed to develop these cracks is higher,the less the distance x to support (see the curve markedcracking load in Fig. 1).

The load needed to develop a sliding failure througha crack is lower, the less the distance is from thesupport, like in the usual plastic solution. The shearcapacity curve in Figure 1 shows that higher the shearspan, the lower the load capacity.

According to the plastic theory, when the two curvesintersect the crack may develop, in terms of the plas-tic theory, into a yield line and a shear failure takes

Figure 2. Ideal crack pattern in a beam without stirrupsunder shear load.

Figure 3. Typical crack pattern in a beam without stirrupsunder shear load.

place. The last diagonal crack is referred to as the crit-ical diagonal crack. The cracking load and the shearcapacity curves in Figure 1 do not always intersect,because the cracking load curve can be lower than theshear capacity curve within the x range. In these case,the shear capacity coincides with the value of the usualplastic solution.

In the CSM is assumed that diagonal cracks aredeveloped following straight lines from the bottomface to the loading point, thus the starting cracksections may be individuated by their horizontal pro-jection x. Further is assumed that the beam is overreinforced in the longitudinal direction, then the rel-ative displacement u along the critical diagonal crackto be vertically directed (Fig. 2).

Using the upper bound approach of plastic theoryand on the basis of the beam’s failure mechanism inFigure 2, the work equation Wi =We and the upperbound solution are:

with b = width of cross section, α = (90◦ − β), cot α =(a − x)/h and τu the average shear stress at failure.

The cracking load curve is evaluated in a simpleway. For the beam with a semicircular crack (Fig. 3),

620

the moment equation about point A, with a staticallyequivalent straight tensile stress ft;ef , gives the averagecracking stress τcr:

ft;ef = 0:156f2/3c (h/0.01)−0.3 being the effective tensile

strength.Introducing this new concept Zhang eliminated the

dependence of νc by shear span-depth ratio and pro-posed to evaluated the effectiveness factor for concretein compression as a product of two terms:

where νs = 0:50 is the sliding reduction factor due tothe reduced cohesion of cracked concrete when theyield line follows the diagonal crack path or crossesmany cracks; ν0 is partly adhere to the empirical for-mula obtained in the original plastic solution (Nielsen1999). It’s interesting to observe that recently the dis-turbed stress field model (DSFM), the updating of theMCFT, adopted an analogous coefficient equal to 0.55to take into account the influence of crack sliding onthe compression softening (Vecchio 2000b).

2.1 The arch action contribute

The CSM is a mechanical model to determine theultimate shear load of plain concrete beams withoutstirrups. It has been validate by Zhang (1997) on alarge database of data collected in literature. The testsconsidered by the author for the model corroborationare characterized by values of a/h higher than 2, thusthe most of specimens collapse for diagonal tensionand the beam action is the principal shear resistancemechanism.

The Figure 4 shows the dependence of the relativeflexural capacity (Mu/Mfl) by the shear span-effectivedepth (a/d) for a plain concrete beam, where the nomi-nal flexural capacity is evaluated as suggested by ACI(1983):

fy = yield steel strength and ρ = geometrical per-centage of longitudinal reinforcement. The ultimatemoment Mu is calculated with CSM and by the formu-lation known in literature (Russo et al. 1991) for plainconcrete beams. The Russo et al.’s model provides thecontribution of both beam and arch resistance mech-anisms in the whole range of a/d values. The CSM isin good agreement with numerical results only for a/h

Figure 4. Relative flexural capacity evaluated with CSMand Russo et al.’s model.

values higher than 2 and fails for a/h values lower than2 because it’s not able to furnish a good estimation ofthe arch action. This is due at the choice of Zhang tocompletely eliminate the dependence of the effective-ness factor of concrete in compression by a/h. Thisassumption provides numerical results far from theexperimental values observed for beams with a/h < 2.

In order to eliminate this drawback, the CSM ismodified retaining the correlation of the efficiencyfactor by the a/h ratio for a/h lower than 2, i.e. assum-ing an additional term [1.0 + 0.17(a/h − 2.6)2] in (5)for a/h ≤ 2.6 The accuracy increment obtained by themodified version of the CSM is shown by the solidline in Figure 4, where the assessment of the notice-able increment in the relative flexural capacity for thedeep beams is shown.

3 THE CSM FOR FIBROUS CONCRETEBEAMS

Flatten stress-strain relationship in the post peak rangeof fibrous concrete in compression and tension makethe SFRC more suitable than plain concrete for theapplication of the plastic theory. Moreover, the pres-ence of fibers in the matrix induces the reduction ofthe slips along cracks. To extend the CSM formula-tion to fibrous concrete beams, the most importantissue is the use of reliable constitutive laws for FRCin compression and tension.

For concrete in compression, the main parameter tobe evaluated is the effective compressive strength fc,ef ,related to the cylinder strength fc by the effectivenessfactor νc, which accounts for the limited crack slidingresistance and ductility of material. Few expressionshave been proposed for the effectiveness factors incompression of fiber concrete (Nielsen 1999). A valueof effectiveness factors for fibrous concrete higherthan plain concrete is expected.

621

The residual tensile stress of SFRC also plays animportant role in the shear mechanism failure of beam.The several analytical relationship proposed by Fos-ter et al. (2006), called Variable Engagement Model(VEM06), for fibrous concrete in direct tension isproposed to evaluate the effective tensile strength(ft,ef = νtfct). The VEM06 considers the slip betweenthe fibers and the concrete matrix that occurs beforethat the full bond stress is developed and that the fiberscan fracture themselves before being pulled out acrossa crack.The constitutive tensile law, expressed in termsof tensile tension and crack opening displacements(w), is the simple sum of stresses contribute by matrixand fibers:

According VEM06 the fibers are mechanicallyanchored to the matrix and some slips, betweenfiber and matrix, must occur before the anchorageis engaged. The crack opening w for which thefiber becomes effectively engaged in the tension car-rying mechanism is termed the engagement lengthwe = αtanθ, where α = df /3.5 is a material parameterand θ is the fiber inclination angle evaluated respectto the crack plane.

When w is equal or higher than we the force in a sin-gle fiber is Pf = πdf τf (la − w), with la = initial lengthof embedment of the fiber and τf = mean shear stressbetween the fiber and the matrix measured along theremaining portion of embedded fiber (la − w). Inte-grating the expression of single fiber force, Pf , over aplane of unit area, the tension stress bridging by fibersacross the crack is obtained:

being Fτ a parameter analogous at the fiber fac-tor and Kf (w) the global orientation factor whichdepends by w.

To predict the value of residual tensile strength offibrous concrete at the shear failure of beam, the con-tribute given by the matrix, σc(w), is computed by asimple linear law (Vecchio 2000b), where the energyfracture of plain concrete is evaluated as suggested byMarti et al. (1999). The crack opening at shear col-lapse of the beam (wm) was evaluated by Casanovaand Rossi (1997) on the basis of some experimentalresults on fibrous concrete specimens. They proposedto evaluate wm as the product of the height of the beam(h) and the strain of the longitudinal reinforcement(εs). Assuming a limited value for εsequal at 1%, the

Figure 5. Comparison between experimental and analyticalresults for fibrous concrete beams.

allowable crack opening in shear is wm = 0.01 h. Oncethe value of wm at shear failure is known, the tensionstress bridging by fibers across the crack is calculatedby Equation (8).

It is also interesting observed that a rearrangementof Equation (7) provides the analytical expression ofthe effectiveness tensile factor: νtf = σcf (wm)/fct.

4 CORROBORATION

A large database (109 data) of experimental testsresults on SFRC beams without stirrups was compiledfrom literature to validate the proposed CSMf modelfor prediction of shear strength of rectangular fiberreinforced concrete beams (Narayanan and Darwish1987;Ashour et al. 1992; Imam et al. 1995; Kwak et al.2002). Beam specimens failing in shear, or if with acrack patterns indicating that shear failure mode is pre-dominated, only are added to the database. Moreover,the fiber aspect ratio was limited to a range of 40 to133; the volumetric percentage of fibers between 0.25and 2.00%; the height of cross section to a minimumvalue of 150 mm and a maximum value of 700 mm;and a = d to a range of 1.0 to 3.5 Firstly, for validationof the proposed model (CSMf), data have been spittedin two groups, depending by the concrete compressionstrength. In the Figure 5 the values of the ratio betweenexperimental results given in literature and the analyti-cal values, predicted by using the presented model, arereported together with its mean value and CoefficientOf Variation (COV).

Two different values of crack sliding factor (νsf )have been used: the first (Figs 5a,b) is the original value

622

Figure 6. Statistical comparison of expression for shearcapacity of fibrous concrete beams.

proposed by Zhang for plain concrete beams equal to0.50; the second (Figs 5c,d) is chosen equal to 0.77.The former is too conservative for normal and highstrength concrete, providing also an high value of COVfor both the two concrete compression strengths.

By contrast, the latter provides an accurate predic-tion of experimental results whit a mean value of 1.01and COV value smaller for high strength concrete. Thechoice of a νsf value for fibrous concrete higher thanplain concrete is explained by the capacity of fiber tolimit the crack slips. Finally, for the comparisons ofFigures 5e,f the value of 0.80 for effectiveness fac-tors in compression and tension has been used (Vooet al. 2003). This constant value for both effectivenessfactor is not able to take into account the functionallydependence of effective strength of concrete by dif-ferent parameters and conditions, and the numericalresults overestimated the experimental values.

A comparison of the predicted shear strengthusing some empirical and semi empirical formula-tions known in literature (Sharma 1986; Campioneet al. 2006; Narayanan and Darwish 1987; Al-Ta’anand Al-Feel 1990; Khuntia et al. 1999; Imam et al.1995; Ashour et al. 1992; Kwak et al. 2002) and theexperimental measured failure shear stress has beenperformed. The statistical coefficients are syntheti-cally reported in Figure 6 with the analogues valuesobtained by the proposed model.

The comparison shows that CSMf provides thebest prediction for normal and high strength SFRCbeams. The results in Figure 6 show that only threemodels are able to provide an accurate prediction ofshear strength. Narayanan and Darwish (1987)’s model(ND87) is less conservative than CSMf to predictthe shear capacity of normal strength fibrous con-crete beams. Instead, for high strength fibrous concretebeams, Kwak et al. (2002)’s model (KEKK02) andCampione et al. (2006)’s model (CLP06) give a goodestimation of shear capacity, with the mean value equalto 1.00 for the KEKK02 model.

5 NUMERICAL ANALYSIS

Many experimental tests on fibrous concrete beamswithout stirrups subjected to shear load show that

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 32.5 MPa

ρ (%)

Mu/M

fl

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 65.0 MPa

ρ (%)

Mu/M

fl

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 32.5 MPa

ρ (%)

Mu/M

fl

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 65.0 MPa

ρ (%)

Mu/M

fl

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 32.5 MPa

ρ (%)

Mu/M

fl

1 2 3 4 5 6

0.81

1.21.4

1.6

0.5

0.6

0.7

0.8

0.9

1

a/d

fc = 65.0 MPa

ρ (%)

Mu/M

fl

Figure 7. Valley of diagonal failure for normal and highstrength concrete and for different fiber factor values.

fibers are highly effective in reducing the range of a/dfor which a brittle shear failure is expected.

A numerical analysis carried out with the proposedmodel was performed to reproduce this experimentalevidence and confirm the model’s reliability, and thevalleys of diagonal shear failure are drawn.

The investigation is performed by assuming two dif-ferent typologies of concrete: namely an high strengthconcrete (fcf = 65.0 MPa) and the normal strength(fcf = 32.5 MPa). The longitudinal reinforcement per-centage was limited to a range of 0.75% to 1.50%to reflect practical situations. Finally hooked endedfibers, with a length of 30.0 mm, a diameter of0.50 mm (lf /df = 60) and a yield strength of 1130 MPais considered. The fibre efficiency is quantified bythree different fiber factor (F) values, namely 0.30(low), 0.60 (medium) and 0.90 (high).

To evaluate the relative flexural capacity the bend-ing moment corresponding to flexural failure, Mfl, iscalculated according to the formulation of Imam et al.(1995) for fibrous concrete:

As shown in Figure 7a, the shear failure domains areextend using low fiber factor and high longitudinalreinforcement percentage, for which a minimum value

623

of the relative flexural capacity (Mu/Mfl = 0.6 ÷ 0.7)for a/d ∼= 3 is observed.

Increasing the amount of fibers in the mixture(F = 0.60) the shear failure valley tend to disappear(Fig. 7b). However it is still wide for normal strengthconcrete, while in case of high strength concrete thedomain of shear failure is extended to a/d ratiosbetween 1.0 and 3.5

This trend is confirmed with F = 0.90, where thevalley of diagonal shear failure for normal strengthconcrete members is narrower, while shear collapse isreported just for few a/d ratios and high longitudinalreinforcement ratio. This behavior is emphasized forhigh strength fibrous concrete beams, where the fibershighly help to tighten the shear failure valley.

As seen in Figure 7 the proposed model still pre-dicted a shear failure for very deep beams, witha/d ∼= 1, and normal strength concrete. In these condi-tions experimental tests show a shear capacity higherthan flexural’s one, that depends on the compressivestrength of concrete. The main reason of that is in thenature of the original formulation of CSM, that wasproposed aiming to predict the shear capacity of beamsthat collapse for diagonal tension. In order, to resolvethis drawback, CSM was updated for predicting thebehavior of short beams by introducing an additionalterm, depending of a/h ratio, determined by tests onplain concrete members. This span shear-depth func-tion underestimates the shear capacity of short beams,with a large amount of longitudinal reinforcement andnormal compressive concrete strength.

6 CONCLUSIONS

In the present paper a mechanical model is proposedthat aims at providing the shear capacity of fibrousconcrete beams without stirrups under transversalloads.

The model is based on plastic theory and limitanalysis and takes into account the fiber concrete con-tribute to shear strength including the high residualpost cracking tensile strength of SFRC. At this aimthe constitutive law suggested by Foster et al. (2006)was used.

In the proposed model, the effectiveness factor offiber concrete in compression was modified for deepbeams, by introducing an additional term dependingon the shear span-depth ratio. The reduction slide fac-tor for fiber concrete, νsf , was increased to 0.77, inorder to take into account the ability of fibers in reduc-ing slips along shear cracks. Further study might benecessary to evaluate more accurately the contributeof fiber onto the shear resistance mechanism of shortbeams (arch action).

Numerical analyses indicate that the addition ofsteel fibers enhanced ultimate loads of normal andhigh strength concrete beams. This enhancement is

more prominent when a minimum amount of fiberswith a fiber factor equal to 0.60 is added, or highstrength concrete beams are considered.

REFERENCES

ACI (1983). Building code requirements for reinforced con-crete (ACI318-83). Technical report, American ConcreteInstitute – Detroit – Michigan (USA).

Al-Ta’an, S. A. and J. R. Al-feel (1990). Evaluation of shearstrength of fiber reinforced concrete beams. Cement andConcrete Composites 12, 87–94.

Ashour, S. A., G. S. Hasanain, and F. F. Wafa (1992). Shearbehaviour of high strength fiber reinforced concrete. ACIStruct. J. (2), 176–184.

Campione, G., L. La Mendola, and M. Papia (2006). Shearstrength of fiber reinforced beams with stirrups. StructuralEngs and Mechs (1), 107–136.

Casanova, P. and P. Rossi (1997). Analysis and design of steelfiber reinforced concrete beams.ACI Mat. J. (5), 595–602.

Foster, S. J., Y. L. Voo, and K. T. Chong (2006). FE analysisof steel fiber reinforced concrete beams failing in shear:Variable engagement model., ACI SP-237, 55–70.

Imam, M., L. Vandewalle, and F. Mortelmans (1995). Shearmoment analysis of reinforced high strength concretebeams containing steel fibres. Canadian J. of Civil Engi-neering 462–470.

Khuntia, M., B. Stojadinovic, and C. G. Subhash (1999).Shear strength of normal and high-strength fiber rein-forced concrete beams without stirrups. ACI Struct. J. (2),282–289.

Kwak, Y. K., M. O. Eberhard, W. S. Kim, and J. Kim (2002).Shear strength of steel fiber-reinforced concrete beamswithout stirrups. ACI Struct. J. (4), 530–538.

Marti, P., T. Pfyl, V. Sigrist, and T. Ulaga (1999). Harmonizedtest procedures for steel fiber-reinforced concrete. ACIMat. J. (6), 676–685.

Narayanan, R. and I.Y. S. Darwish (1987). Use of steel fibersas shear reinforcement. ACI Struct. J. (3), 2066–2079.

Nielsen, M. P. (1999). Limit analysis and concrete plasticity(2nd ed.). Boca Raton – Florida: CRC.

Russo, G., G. Zingone, and G. Puleri (1991). Flexure-shearinteraction model for longitudinally reinforced beams.ACI Struct. J. (1), 66–68.

Sharma, A. K. (1986). Shear strength of steel fiber reinforcedconcrete beams. ACI J. (4), 624–628.

Vecchio, F. J. (2000A). Analysis of shear critical reinforcedconcrete beams. ACI Struct. J. (1), 102–110.

Vecchio, F. J. (2000B). Disturbed stress field model for rein-forced concrete: Formulation.ASCE J. Of Struct. Eng. (9),1070–1077.

Vecchio, F. J. and M. P. Collins (1986). The modified com-pression field theory for reinforced concrete elementssubjected to shear. ACI J. (2), 219–231.

Voo, Y. L., S. J. Foster, and R. I. Gilbert (2003). Shearstrength of fibre reinforced reactive powder concrete gird-ers without stirrups. Technical report, School of Civiland Environmental Engineering – The University of NewSouth Wales – Australia.

Zhang, J.-P. (1997). Diagonal cracking and shear strengthof reinforced concrete beams. Magazine of ConcreteResearch (178), 55–65.

624