modelling cracking and bending failure of sfrc beams with

VIII International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-8

J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds)

MODELLING CRACKING AND BENDING FAILURE OF SFRC BEAMSWITH CONVENTIONAL REINFORCEMENT

DAVID FALL∗†, RASMUS REMPLING†, ANETTE JANSSON† AND KARIN LUNDGREN†

†Chalmers University of TechnologyCivil and Environmental Engineering, Sven Hultins Gata 8, 412 96 Gteborg, Sweden

e-mail: [email protected], www.chalmers.se

Key words: Steel fibre reinforced concrete, Reinforced concrete beams, Non-linear FE-modelling,Fib Model Code 2010

Abstract. In this study three beams, with varying contents of steel fibre reinforcement, were testedin four point bending and compared with results from FE-analysis. The beams were part of a largerexperimental programme where relevant material properties were investigated.

FE-modelling was performed using a two dimensional model. Concrete was represented by four-node quadrilateral isoperimetric plane stress elements. The smeared crack approach was utilized andthe stress-strain relation describing the tensile behavior of the concrete was calculated from uni-axialtest results, assuming the crack bandwidth to be equal to the element length. In compression, theconcrete was assumed to behave elasto ideal-plastic. The reinforcement was modelled by straight2-node truss elements connected to the concrete by two-dimensional interface elements providing thebond-slip properties. A material model including hardening effects was derived from tension tests ofreinforcement bars and used for modelling the conventional reinforcement. A multi-linear bond-slipmodel was established through pull-out tests. As an alternative, analyses were also performed takinginto account a reduction of the bond stress after yielding of the reinforcement occurred. Loading wasapplied in two phases: the first comprehending only the self-weight, while incremental loading wasapplied by deformation control during the second phase.

General agreement between experiments and FE-analyses was obtained with regard to load-displacement behaviour. By observing the crack patterns, both from FE-analysis and experiments,it can be concluded that the general behaviour agreed; however, in the analyses not all cracks werefully localized. A higher degree of crack localization was obtained when the bond loss at yieldingwas included.

1 INTRODUCTION

The use of steel fibre reinforcement (SFRC)has increased during the last two decades. How-ever, there is a lack of experiments in the scaleof structural elements, in which the materialproperties are well-defined enough to facili-tate model validation. Several researchers havestudied laboratory scale tests with the purposeto determine material properties for SFRC. Ap-proaches of determining the tensile behaviour

of SFRC have been discussed in numerous arti-cles. In RILEM TC 162-TDF [1] and RILEMTC 162-TDF [2], methods based on uniaxialtesting and beam bending, respectively, weresuggested. Such approaches have been studiedby e.g. Kooiman et al. [3] and Giaccio et al.[4]. In addition to these methods, a wedge-splitting procedure was proposed [5]. Largerscale beams have been studied by several re-searchers [6, 7, 8, 9]. The current study was

1

David Fall, Rasmus Rempling, Anette Jansson and Karin Lundgren

performed in order to evaluate the recently pub-lished International Federation for StructuralConcrete (fib) [10] (MC 2010) and to contributeto the field of SFRC-modelling. Establishing abroad knowledge of the structural behavior andmodelling approaches of SFRC is important inorder to enable a wider usage of the technol-ogy. By combining well-defined experimentalwork, non-linear finite element analysis and re-sults obtained in accordance with MC 2010, thispaper ultimately evaluates the structural engi-neering approaches to SFRC.

2 EXPERIMENTAL PROGRAMMEThe experimental programme comprised

uniaxial testing, pull-out tests, reinforcementtension tests and four-point bending test of re-inforced beams. This paper focuses on the con-crete beams; however, the other experiments arealso described here in concentrated form as theyare significant while discussing the validity ofthe model, presented in Section 3. Full descrip-tions of the pull-out tests and unaxial tests areprovided in Jansson et al. [11] and Jansson et al.[12], respectively.

2.1 Four-point bendingThree SFRC beams of fibre content 0.0,

0.25 and 0.5% (percent by volume) were testedin deformation controlled four-point bending.The beams were simply supported and spanned1800 mm, with shear spans of 600 mm, see Fig-ure 1. Each beam was reinforced with threereinforcement bars: φ8 in the beams of fibrecontent 0.0 and 0.5% and φ6 in the beam of fi-bre content 0.25%. Steel quality was B500BT(Swedish quality). Shear reinforcement (stir-rups) was included over the supports. Fi-bre content, reinforcement configurations andtested concrete strengths for all beams are pre-sented in Table 1.

2.2 MaterialThe concrete used for all the experiments

was self-compacting and mixed in a centraldrum-mixer at a ready-mix plant, in batches of 2m3 [11]. The fibres used were end-hooked steel

fibres, Dramix R©RC 65/35-BN from Bekaert,with a tensile strength of 1100 MPa, and theactual fibre content was estimated performingwashout control in accordance with govern-ing standard [13]. The compressive strength(fccm.28d), elastic modulus (Ecm) and splittingtensile strength (fctm.28d) was determined ac-cording to european standards [14, 15, 16]. Tocapture the softening behaviour of the fibre re-inforced concrete (σ − w relation), uni-axialtensile testing was performed on notched cylin-ders, in accordance with RILEM TC 162-TDF[1]. The tests were performed in accordancewith RILEM TC 162-TDF [1] at the TechnicalResearch Institute of Sweden and are further de-scribed in Jansson et al. [12]. In Figure 2 the testsetup is presented.

Figure 2: Geometry and test setup for uni-axialtesting, Jansson et al. [12].

Tension tests were performed to determinethe behaviour of the reinforcement bars. Eachseries (φ6 and φ8) included 5 bars. In addi-tion to this the bond properties between rein-forcement bars and concrete were determinedby pull-out tests, see Figure 3. To avoid walleffects the tested specimens were cut out fromlarger prisms of 110x152x720 mm3; the spec-

2


2000

1800

600 600 600

SYM.LVDTP P

225

150

Reinf. bars

[mm]

Figure 1: Experimental set-up

Table 1: Test beam configurations

Beam No. I II IIIVf , nominal 0.00% 0.25% 0.50%

Vf , actual (mean value from wash-out) 0.00% 0.18% 0.45%Reinforcement 3φ8 3φ6 3φ8

fccm.28d 58.8 MPa 58.1 MPa 57.5 MPafctm.28d 2.9 MPa 2.7 MPa 3.0-3.1 MPaEcm 32.5 GPa 30.5 GPa 31.0 GPa

3


imen dimensions were 112x112x110 mm3. Aribbed φ16 bar of quality B500BT was placed inthe square cross-section centre. Specimen sizewas chosen so that the concrete surface strainswould be measurable while delaying splitting inthe series without steel fibre reinforcement aslong as possible. Five specimens were testedfor each fibre content. Full description of ex-periments and results is given in Jansson et al.[11].

Results from the described uni-axial, tensionand pull-out tests are presented as model inputdata in Section 3.2.

Figure 3: Geometry and test setup for pull-outtesting, Jansson et al. [11].

3 NON-LINEAR FINITE ELEMENTANALYSIS

Non-linear finite element analyses were per-formed using the commercial FE-software TNODIANA [17]. A phased loading procedure wasused. In the first of two loading phases, theselfweight was applied and incremental loadingwas applied by deformation control during thesecond phase. The model geometry and the ma-terial models used are described in the follow-ing two sections, respectively.

3.1 GeometryA dense quadratic mesh (5x5 mm) of four-

node quadrilateral isoperimetric plane stresselements was used for concrete representa-tion. The reinforcement bars were modeledby straight 2-point truss elements connected to

two-dimensional interface elements to whichthe bond-slip properties were assigned. Asa measure for rationalising the computations,only half beam was modelled due to symme-try, see Figure 4. In the symmetry line, allmovement in the horizontal direction was con-strained. Support and loading plates were mod-eled using eccentric tyings, i.e. the verticalmovement of the nodes at the plates was main-tained on a straight line intersecting the platecenter node. Regions acting under these as-sumptions were 150 and 100 mm wide for thesupport node and the loading node, respectively.The centre node in the support was constrainedfrom vertical movement.

3.2 Material models

Nonlinear fracture mechanics with rotatingcracks were used for concrete modeling. Aspreviously mentioned, the tensile properties ofthe concrete were determined by uniaxial ten-sion tests. A smeared crack approach wasutilised. The stress-strain relation used for con-crete in tension was calculated from the crack-widths measured in these tests, wi, in accor-dance with Equation 1. The crack bandwidth, h,was assumed to be equal to the element length.The stress-crackwidth relation can be seen inFigure 5.

εi =ftE

+wi

h. (1)

For compression, an elasto ideal-plasticcompressive behavior was used. In addition tothis relatively simple material model a built-inDIANA function [17], based on the theory ofVecchio and Collins [18], was applied. In shortthis function reduces the compressive stressesin elements with large tensile stresses perpen-dicular to the principal compression direction.Furthermore, the elastic modulus was assumedto be linear during the uncracked state and wasestablished from the uniaxial tension tests, seeTable 1.

4


Figure 4: Model geometry. Eccentric boundary conditions (tyings) and the reinforcement position areindicated with thick black lines.

0 1 2 3 4 5Crack width, w [mm]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Str

ess

[MP

a]

Vf=0.50%

Vf=0.25%

Vf=0%

(a)

0.0 0.2 0.4 0.6 0.8 1.0Crack width, w [mm]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Str

ess

[MP

a]

Vf=0.50%

Vf=0.25%

Vf=0%

(b)

Figure 5: (a) Input used for concrete tensile be-havior. (b) Detail for crack widths up to 1 mm.

A material model that included hardening ef-fects was used for conventional reinforcementbars. In the model, the average nominal diam-eter of the tested reinforcement bars was used.The corresponding input is shown in Figure 6.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Strain, ε [-]

0

100

200

300

400

500

600

700

800

900

Str

ess,σs

[MP

a]

φ6

φ8

Figure 6: Input for the reinforcement bars.

As decribed in Section 2.2, the bond-slip be-havior was measured in the pull-out tests. Note-worthy is that the pull-out tests were performedon specimens with reinforcement bar diameterof 16 mm. However, as the ratio between theconcrete cover thickness and the bar diametersis approximately the same in the pull-out spec-imens and beams, the bond stress versus slip

5


measured in the pull-out tests was assumed tobe directly applicable as input in the modelingof the beams.

0 1 2 3 4 5Slip [mm]

0

5

10

15

20

25

Bon

dst

ress

[MP

a]

Vf=0%

Vf=0.25%

Vf=0.50%

(a)

sy s4 s33 6 9 12Slip [mm]

τf,pl

τf

τy

0

4

8

12

16

20

Bon

dst

ress

[MP

a]

Original model

Engstrom

(b)

Figure 7: (a) Experimentally obtained bondstress versus slip used as input. (b) Bondstress versus slip, original model and modifi-cation when yielding occurs in reinforcement(Vf=0.50%).

Several researchers have observed that thebond stress is reduced after reinforcement yield-ing [19, 20]. To study the effect of of thisphenomenon on the model behaviour analyseswere carried out on two types of models: one inwhich the bond stress depended only on the slip,

and one in which the bond stress was reducedone the reinforcement reached yielding stress.In the latter case, the model of Engstrom [20]was applied, reducing the bond stress linearlytowards the suggested values τf,pl = 0.5τf ands4 = 0.5s3, see Figure 7. It should be pointedout that contrary to what would be expected, theaverage bond capacity of the pull-out tests withVf = 0.50% was lower than the average of theones with less fibre content (Vf = 0.25%). Thisis due to large scatter in the experimental re-sults of the fibre content Vf = 0.50%. In pull-out tests with higher fibre content (Vf = 1.0%),presented in Jansson et al. [11], both the maxi-mum and residual bond capacity increased.

4 RESULTS

The focus of this section is on the compari-son between the performed beam tests and theFE-analyses. First load-deflection curves andcrack patterns are presented and in Section 4.1the effect of bond reduction at yielding are dis-cussed. Noteworthy is that also the results fromcalculations according to MC 2010 are pre-sented. Detailed information on these calcula-tions are presented in Fall et al. [21].

Comparing the load-deflection curves in Fig-ure 8, a general agreement can be seen betweenthe experimental results and the results obtainedtrough FE analysis. For all beams a stiffnessloss can be observed as the first crack devel-ops. Thereafter, the stiffness is relatively con-stant until the reinforcement yields. All threebeams failed in bending. Furthermore, it canbe seen that the calculations in accordance withMC 2010 gives results well on the safe side,both in terms of ultimate load and deflection.For the particular structural member analysed inthis study the magnitude of the underestimationincreases with increasing fibre content. The FEanalysis were generally stable and usually con-verged until some load steps after yielding oc-curred in the reinforcement. For larger deflec-tions, the analyses did not converge in all steps;hence the small disturbances, which can be seenin Figure 8.

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0 5 10 15 20 25Deflection [mm]

0

5

10

15

20

25

30

35

40L

oad

[kN

]

Experiment

FE

MC 2010

(a) Vf=0.00%, 3φ8


0

5

10

15

20

25

30

35

40

Loa

d[k

N]

Experiment

FE

MC 2010

(b) Vf=0.25%, 3φ6


0

5

10

15

20

25

30

35

40

Loa

d[k

N]

Experiment

FE

MC 2010

(c) Vf=0.50%, 3φ8

Figure 8: Load deflection behaviour (mid span):results from experiments, MC 2010 analysisand FE-modeling with improved bond model[20]

Considering the FE analysis of the beamwithout fibres, a sudden loss of stiffness wasobserved just after the yielding of the reinforce-ment (at a load of approximately 27kN), seeFigure 8. It was observed that tensile strains de-veloped locally in the elements surrounding thereinforcement bar, as a continuation of an in-clined crack. The main direction of these crackswere along the bars; thus the crack pattern wassimilar to shear-splitting failure after this point.

In addition to the general agreement previ-ously discussed, also the number of cracks, totalspread of cracks and the distance between themagreed roughly comparing the experiments andresults from the FE-analysis. Some of the dif-fering results can be explained by imperfectionsin the test samples and set-up. A comparison ofmodelled crack pattern and the one obtain in ex-periment can be found in Fall et al. [21]. Someexamples of modelled crack patterns are shownin Figure 9.

4.1 Influence of bond reduction at yielding

As previously described in Section 3.2, twoanalyses of each beam were carried out: onewith original bond-slip model and one wherethe bond stress was linearly decreased with in-creased slip once yielding occurred in the rein-forcement bar [20]. There was no difference incracking before yielding of the reinforcement,as the two bond models were completely equiv-alent up until this point, see Figure 9. As ex-pected the cracks were more clearly localizedwhen the bond stress is reduced at yielding.This is due to the fact that when the bond stressdoes not increase after yielding of the reinforce-ment, no new cracks form.

7


X

Y

Z

Model: FT2Deformation = 17.3LC2: Load case 2Step: 136 LOAD: 2.7Element EL.E1 E1Max = .333E-1Min = -.417E-3

.1E-2

.909E-3

.818E-3

.727E-3

.636E-3

.545E-3

.455E-3

.364E-3

.273E-3

.182E-3

.909E-40

26 AUG 2011 08:53:20 plot/cpat_step136_bwiDIANA 9.4.3-02 : Chalmers University

Crack pattern, step 136

(a) Vf=0.50%, prior yielding with original bondmodel

X

Y

Z

Model: FT2Deformation = 19LC2: Load case 2Step: 134 LOAD: 2.66Element EL.E1 E1Max = .329E-1Min = -.421E-3

.1E-2

.909E-3

.818E-3

.727E-3

.636E-3

.545E-3

.455E-3

.364E-3

.273E-3

.182E-3

.909E-40

30 AUG 2011 09:26:12 plot/cpat_step134_bwiDIANA 9.4.3-02


(b) Vf=0.50%, prior yielding with bond modelaccording to Engstrom [20]

X

Y

Z

Model: FT2Deformation = 3.22LC2: Load case 2Step: 700 LOAD: 14Element EL.E1 E1Max = .194Min = -.928E-3

.1E-2

.909E-3

.818E-3

.727E-3

.636E-3

.545E-3

.455E-3

.364E-3

.273E-3

.182E-3

.909E-40

26 AUG 2011 08:53:55 plot/cpat_step700_bwiDIANA 9.4.3-02 : Chalmers University


(c) Vf=0.50%, after yielding with original bondmodel

X

Y

Z

Model: FT2Deformation = 3.79LC2: Load case 2Step: 650 LOAD: 13Element EL.E1 E1Max = .173Min = -.917E-3

.1E-2

.909E-3

.818E-3

.727E-3

.636E-3

.545E-3

.455E-3

.364E-3

.273E-3

.182E-3

.909E-40

30 AUG 2011 09:26:44 plot/cpat_step650_bwiDIANA 9.4.3-02


(d) Vf=0.50%, after yielding with bond modelaccording to Engstrom [20]

Figure 9: Examples of crack patterns(Vf=0.50%) corresponding to the applieddeformation 2.7 mm (a, b) and 14 mm (c,d). The difference in crack localization afteryielding of the reinforcement occurred isindicated.

4.2 Modelling of beams with equal conven-tional reinforcement

In order to increase the comparability of thevarious beam configurations a modelling seriawas made in which the same dimension of re-inforcement bar was assumed. As expectedthe behaviour were approximately equal for allbeams in the elastic phase. In the phase withcracking and, eventually, yielding of the rein-

forcement the fracture energy was much higherin the fibre reinforced beams. The results fromthis study, in terms of load deflection behaviour,can be seen in Figure 10.

Figure 10: Modelled load deflection behaviour(mid span) for beams reinforced with equalconfigurations of conventional reinforcement(3φ6).

5 CONCLUSIONSTo conclude, it can be seen that the increased

post cracking capacity of SFRC, seen in experi-ments, can be estimated with FE-analysis. Fur-thermore, reasonable agreement was also ob-tained with regards to crack patterns when com-paring experiments and FE-analyses. Utilizinga bond-stress model where the bond stress isreduced post yielding, resulted in more local-ized crack patterns. Modeling fibre reinforcedconcrete with non-linear finite element method,utilizing a 2D-model with plane stress elementswere shown to be successful provided that ma-terial data is chosen with care. In addition tothe study in which experiment and modellingwas compared, models were also establishedwith equal amount of conventional reinforce-ment, but varying content of steel fibre rein-forcement. As expected the post cracking ca-pacity became higher with increasing amount offibres.

However, in the studied beams, Model Code2010 (MC 2010) fails to fully quantify the post

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cracking capacity added by the steel fibre rein-forcement. The method proposed in MC2010underestimates the additional capacity providedby the addition of steel fibres, both with regardsto load and deformation capacities. The under-estimation increases with increased fibre con-tent.

6 ACKNOWLEDGEMENTThe presented experiments were financed by

Thomas Concrete Group AB. All other partsof this research were funded from the Euro-pean Community’s Seventh Framework Pro-gramme under grant agreement NMP2-LA-2009-228663 (TailorCrete). More informationon the research project TailorCrete can be foundat www.tailorcrete.com.

References[1] RILEM TC 162-TDF, 2001. Uni-axial

tension test for steel fibre reinforced con-crete. Mater. and Struct. 34(1):3–6.

[2] RILEM TC 162-TDF, 2002. Bending test.Mater. and Struct. 35(9):579–582.

[3] Kooiman, A. G., Van Der Veen, C. andWalraven, J. C., 2000. Modelling the post-cracking behaviour of steel fibre rein-forced concrete for structural design pur-poses. Heron. 45(4):275–307.

[4] Giaccio, G., Tobes, J. and Zerbino, R.,2008. Use of small beams to obtain designparameters of fibre reinforced concrete.Cem. and Concr. Compos. 30(4):297 –306.

[5] Lofgren, I., Stang, H. and Olesen, J.,2005. Fracture properties of frc deter-mined through inverse analysis of wedgesplitting and three-point bending tests. J.of Adv. Concr. Technol. 3(3):423–434.

[6] Noghabai, K., 2000. Beams of fibrousconcrete in shear and bending: Experi-ment and model. ASCE J. of Struct. Eng.126(2):243–251.

[7] Jansson, A., Lofgren, I. and Gylltoft, K.,2010. Flexural behaviour of members witha combination of steel fibres and conven-tional reinforcement. Nordic Concr. Res.(42):155–171.

[8] Ozcan, D., Bayraktar, A., Sahin, A., Hak-tanir, T. and Turker, T., 2009. Experimen-tal and finite element analysis on the steelfiber-reinforced concrete (sfrc) beams ul-timate behavior. Constr. and BuildingMater. 23(2):1064–1077.

[9] Altun, F., Haktanir, T. and Ari, K., 2007.Effects of steel fiber addition on mechan-ical properties of concrete and rc beams.Constr. and Building Mater. 21(3):654 –661.

[10] International Federation for StructuralConcrete (fib), 2010. Bulletin 55-56:Model Code 2010 First Complete Draft.CEB/FIP.

[11] Jansson, A., Lofgren, I., Lundgren, K. andGylltoft, K., 2012. Bond of reinforcementin self-compacting steel-fibre reinforcedconcrete. Mag. of Concr. Res. 64(7):617–630.

[12] Jansson, A., Flansbjer, M., Lofgren, I.,Lundgren, K. and Gylltoft, K., 2012. Ex-perimental investigation of surface crackinitiation, propagation and tension stiff-ening in self-compacting steel-fibre re-inforced concrete. Mater. and Struct.45(8):1127–1143.

[13] SIS, 2005, SS-EN 14721:2005 Testmethod for metallic fibre concrete - Mea-suring the fibre content in fresh and hard-ened concrete. Swedish Standards Insti-tute (SIS).

[14] SIS, 2009, SS-EN 12390-3:2009 Testinghardened concrete - Part 3: Compressivestrength of test specimens. Swedish Stan-dards Institute (SIS).

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[15] SIS, 2005, SS-EN 137232:2005 Concretetesting - Hardened concrete - Modulus ofelasticity in compression. Swedish Stan-dards Institute (SIS).

[16] SIS, 2009, SS-EN 12390-6:2009 Testinghardened concrete - Part 6: Tensile split-ting strength of test specimens. SwedishStandards Institute (SIS).

[17] TNO DIANA, 2010. Finite element analy-sis, Users Manual Release 9.4.3. TNO Di-ana, Delft.

[18] Vecchio, F. J. and Collins, M. P., 1993.Compression response of cracked re-inforced concrete. J. of Struct. Eng.119(12):3590–3610.

[19] Shima, H., Chou, L. and Okamura, H.,1987. Bond Characteristics in Post-YieldRange of Deformed Bars. Concr. Libr. Int.(10):113–124.

[20] Engstrom, B., 1992. Ductility of Tie Con-nections in Precast Structures. Disserta-tion, Chalmers University of Technology,Gothenburg.

[21] Fall, D., Rempling, R., Jansson, A., Lund-gren, K. and Gylltoft, K., 2012. Steel fi-bres in reinforced beams: Experiments,model code and FE-analyses. Submitted toMater. and Struct .

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modelling cracking and bending failure of sfrc beams with