On the competition between delamination and shear failure in retrofittedconcrete beams and related scale effects

A. Carpinteri, G. Lacidogna & M. PaggiPolitecnico di Torino, Department of Structural and Geotechnical Engineering, Torino, Italy

ABSTRACT: In this paper we deal with the most commonly reported failure modes related to interfacial stressconcentrations at the FRP cut-off points, i.e. diagonal (shear) crack growth and FRP delamination. Dependingon the mechanical properties of the tested beams, their geometry and size, a prevalence of a given failure modeto the other is very often experimentally observed. To analyze this failure mode competition, a combinedanalytical/numerical model is proposed for the determination of the critical loads required for the onset ofdelamination or shear failure. In this way, the experimentally detected failure modes observed in RC and FRCbeams are reexamined and interpreted in this new framework.

1 INTRODUCTIONStructure rehabilitation is required whenever designmistakes, executive defects or unexpected loadingconditions are assessed. In these cases, the use of astrengthening technique may be required in order toeither increase the loading carrying capacity of thestructure, or to reduce its deformations. The choiceof the proper rehabilitation technique and the assess-ment of its performance and durability clearly repre-sent outstanding research points. Among the differ-ent rehabilitation strategies, bonding of steel plates orFRP sheets on the concrete members is becoming in-creasingly popular (Hollaway & Leeming 1999).

In these situations, the main observed failure modescan be summarized as follows: (a) flexural failure byFRP yielding (Arduini et al. 1997), (b) flexural failureby concrete crushing in compression (Arduini et al.1997), (c) shear failure (Ahmed et al. 2001), (d) con-crete cover separation (David et al. 1993), (e) FRPdelamination (Leung 2004a; Leung 2004b), and (f)intermediate crack induced debonding (Alaee & Kar-ihaloo 2003; Wang 2006). Among them, shear fail-ure, concrete cover separation and FRP delaminationhave been far more commonly revealed in experimen-tal tests. In these cases, damage initiates near the FRPcut-off points due to the presence of a stress concen-tration or even a stress intensification. As a conse-quence, either a pure FRP delamination or a diago-nal crack growth can occur. Moreover, in the lattercase, depending on the amount of steel reinforcementand thickness of concrete cover, the diagonal crackmay give rise to either shear failure, or concrete cover

separation. Therefore, since concrete cover separationoccurs away from the FRP-concrete interface, thisfailure mode should be carefully distinguished fromthe pure FRP delamination. Therefore, it seems to bemore appropriate to interpret failure modes (c) and (d)in the same framework.

As regards the mathematical models available inthe Literature, most of them focus on the problemof delamination in steel plated and FRP strengthenedbeams (Smith & Teng 2002; Smith & Teng 2002b).Shearing and peeling stresses in the adhesive layer ofa beam with a strengthening plate bonded to its soffitwere determined in (Taljsten 1997; Malek et al. 1998;Ascione & Feo 2000; Smith & Teng 2001). The anal-ysis of interface tangential and normal stresses in FRPretrofitted RC beams was also recently reexaminedin (Rabinovitch & Frostig 2001; Rabinovitch 2004),along with a fracture mechanics model for the predic-tion of FRP delamination.

Comparatively, a little attention has been di-rected toward the analysis of shear crack growthand to the competition between FRP delaminationand shear failure. The problem of size-scale effectis also an open issue (Maalej & Leong 2005). Todeal with these problems, we propose a combinedanalytical/numerical model to describe the failuremode competition between FRP delamination andshear failure in reinforced concrete (RC) and fiber-reinforced concrete (FRC) beams. In this way, the ex-perimentally detected failure modes are reexaminedand interpreted in this new framework. As regards RCbeams, we refer to the data on four-point bending tests

reported in (Ahmed et al. 2001), whereas the experi-mental results on FRC beams have been determinedaccording to a new testing programme carried out inour laboratory.

2 ANALYTICAL MODELIn this section, we consider the typical three-pointbending and four-point bending tests carried out in thelaboratories to assess the mechanical performance ofretrofitted beams (see Fig. 1).

h

t

FRP ( h f )s

P

l FRPs

l

h

t

FRP ( h f )s

P /2

l FRPs

P /2

l

Figure 1: Schemes of three- and four-point bendingtests.

2.1 Stress-singularities and generalized stress-intensity factors

According to Linear Elastic Fracture Mechanics,the FRP cut-off point can be a source of stress-singularities due to the mismatch in the elastic prop-erties of concrete and FRP. The geometry of aplane elastostatic problem consisting of two dissim-ilar isotropic, homogeneous wedges of angles equalto γ1 = π and γ2 = π/2 perfectly bonded along theirinterface is schematically shown in Figure 2.

O

r

θγ 1

γ 2

Concrete

FRP

Figure 2: Scheme of the bi-material wedge composedof FRP and concrete.

In this general case, the singular components of the

stress field can be written as follows:

σij = K∗r(Reλ−1)Sij(θ), (1)

where K∗ is referred to as generalized stress-intensityfactor (Carpinteri 1987). The parameter λ defines theorder of the stress-singularity and can be obtained ac-cording to an asymptotic analysis of the stress field,see e.g. (Williams 1952; Bogy 1971; Carpinteri &Paggi 2005; Carpinteri & Paggi 2007) for similar ap-plications.

According to this approach, the parameter λ is de-termined by solving a non-linear eigenvalue problemresulting from the imposition of the boundary con-ditions. In the present problem, they consist in thestress-free boundary conditions along the free edges,and in the continuity conditions of stresses and dis-placements along the bi-material interface. Since weare interested in the analysis of the the singular termsof the stress field, we are concerned only with thosevalues of λ which may lead to singularities. This fact,together with the condition of continuity of the dis-placement field at the vertex where regions meet, im-ply that we are seeking for eigenvalues in the range0 < Reλ < 1.

Function Sij(θ) in Eq. (1) is the eigenfunction ofthe problem and it locally describes the angular vari-ation of the stress field near the singular point, O. Ithas to be remarked that, since both λ and Sij(θ) aredetermined according to the asymptotic analysis, theysolely depend on the boundary conditions imposed inproximity of the singular point.

Moreover, from dimensional analysis arguments(Carpinteri 1987), it is possible to consider the fol-lowing expression for the generalized stress-intensityfactor:

K∗ =Pl

th1+Reλf

(hf

h,l

h,lFRP

h,Ef

Ec

), (2)

where function f depends on the boundary conditionsfar from the singular point and can be determined ac-cording to a FE analysis on the actual geometry ofthe tested specimen. For the sake of generality, thisfunction depends on the relative thickness of the re-inforcement compared to the beam depth, hf/h, onthe ratio between the span and the depth of the beam,l/h, on the length of the FRP sheet, lFRP/h, and onthe modular ratio between FRP and concrete, Ef/Ec.Parameters P and t denote, respectively, the appliedload and the beam thickness.

The critical load corresponding to the onset of de-lamination can be determined by setting the general-ized stress-intensity factor equal to the critical stress-intensity factor for the interface. This approach, well-established for the analysis of bonded joints (Reedy& Guess 1993; Qian & Akisanya 1999; Carpinteri &

Paggi 2006), yields to the following equation:

P delC = K∗

C,intth1+Reλ

l

1

f. (3)

An analogous reasoning can be proposed for theanalysis of the onset of shear failure, i.e. before thedevelopment of the crack-bridging effect due to steelreinforcement. In this case, we can postulate the exis-tence of a small vertical crack into concrete at the FRPcut-off point simulating an initial defect. This crackmay result in a sudden diagonal propagation leadingto premature failure of the beam.

The stress field at the crack tip is again singular, butwith the order of the singularity typical of a crack in-side a homogeneous material (Carpinteri 1987; Boccaet al. 1990):

σij = Kr−1/2Fij(θ), (4)

where function F locally describes the angular vari-ation of the stress field near the crack tip. From di-mensional analysis considerations, it is possible towrite the following expression for the Mode I stress-intensity factor:

KI =Pl

th3/2g

(a0

h,hf

h,l

h,lFRP

h,Ef

Ec

), (5)

where function g depends again on the boundary con-ditions far from the singular point and can be deter-mined according to a FE analysis on the actual geom-etry of the tested specimen. The additional parametera0 with respect to FRP delamination denotes the ini-tial crack length.

Crack propagation in this case takes place underMixed Mode, although the Mode I stress-intensityfactor is numerically prevailing. Under such assump-tions, the critical load corresponding to the onset ofshear crack propagation is reached when the Mode Istress-intensity factor equals the critical value of con-crete. This condition yields to the following equation:

P shearC = KIC

th3/2

l

1

g. (6)

2.2 Size-scale effects and failure modes competitionFor a given tested beam, i.e., for a given beam geom-etry, the ratio between the critical loads for delamina-tion and shear failure can be written as:

P delC

P shearC

=

(K∗

C,int

KIC

g

f

)h(Reλ−1/2), (7)

which is a non-linear function of the beam depth. Inaddition, it is possible to recast Eqs. (3) and (6) in a

logarithmic form:

logP delC = log

(tK∗

C,int

lf

)+ (1 + Reλ) logh, (8)

logP shearC = log

(tKIC

lg

)+

3

2logh. (9)

These equations are qualitatively plotted as func-tions of the beam depth in Figure 3. As expected, thehigher the beam depth, for a given ratio l/h, the higherthe critical load of failure. The intersection point be-tween the two curves defines the critical beam sizecorresponding to the transition from pure delamina-tion to shear failure. Moreover, since the real part ofthe eigenvalue λ is usually higher than 0.5, we ex-pect a prevalence of shear failure in larger beams. Infact, if we consider Ef = 200 GPa and Ec = 30 MPaas the values representative of the Young’s moduli ofconcrete and FRP, then the asymptotic analysis givesλ = 0.58.

log h*

log Pc

log h

h *

Delamination

Shear failure

1

Reλ+1

1

3/2

O

Figure 3: Competition between delamination andshear failure: dashed line corresponds to delamina-tion, solid line corresponds to shear failure.

3 REINFORCED CONCRETE BEAMSIn this section we propose numerical simulations ofshear failure and FRP delamination in FRP-retrofittedRC beams. Concerning the tested geometry, we referto the extensive test programme carried out by Ahmedet al. (2001). More specifically, they tested a set ofrectangular beams (h = 0.225 m, l = 1.5 m, t = 0.125m) under four point bending and considered differ-ent FRP lengths (lFRP = 0.70m; 0.65m; 0.60m and0.55m). Independently of lFRP, all the beams faileddue to shear crack propagation originating from theFRP cut-off point. Numerical predictions and experi-mental results are shown and compared in the sequel.

3.1 Shear failureThe simulation of shear failure in RC beams can beperformed according to different numerical strategies.For example, Gustraffson (1985) considered a FE for-mulation taking into account a softening cohesive lawfor concrete and truss elements connected by springsto the concrete blocks to model the presence of thereinforcement.

To avoid finite element computations, Jenq & Shah(1989) and So & Karihaloo (1993) proposed a semi-analytical model where the concrete contribution toshear strength was evaluated according to LEFM, andthe effect of steel-concrete interaction was includedusing an empirical relationship.

In the present approach, we consider an initialcrack length equal to a0 in correspondence of theFRP cut-off point. Simulation of crack growth is thenperformed using the FRANC2D finite element code(Wawrzynek & Ingraffea 1987). At each step, thestress-intensity factors are computed using the dis-placement correlation technique and the direction forcrack propagation is determined according to maxi-mum circumferential stress criterion. This approachpermits to take into account the contribution of con-crete to shear strength, as also done in the approx-imate model by Jenq & Shah (1989). Since the ef-fect of reinforcement is not considered, the proposedmethod predicts an unstable crack growth, see e.g.Figure 4 for the case with lFRP = 0.70 m.

0

30

60

90

120

150

180

0.0 0.2 0.4 0.6 0.8 1.0

a / h

Pcshear [kN]

a0/h

h

Figure 4: Critical load for shear failure vs. non-dimensional crack length.

The computed critical force corresponding to a0,i.e. to the onset of diagonal failure, is shown in Fig-ure 5 for different FRP lengths. Experimental resultsby Ahmed et al. (2001) are also reported in the samediagram. As expected, the good agreement betweenthe numerical predictions and the experimental resultsdemonstrate that this approach is suitable for the com-putation of the value of the critical force correspond-ing to the onset of crack propagation. Moreover, Fig-

ure 5 shows the shorter the FRP sheet, the lower thecritical force.

0

30

60

90

120

150

180

0.7 0.75 0.8 0.85 0.9 0.95

l FRP / l

Pcshear [kN]

LEFM results

Experimental data

Figure 5: Critical load for shear failure vs. non-dimensional FRP length.

3.2 DelaminationThe computation of the critical load required for de-lamination can be performed according to the ana-lytical model illustrated in Section 2. In this case,the critical stress-intensity factor of the interface canbe estimated as K∗

C,int∼= √

GC Ea, where GC and Ea

are, respectively, the interface fracture energy and theYoung’s modulus of the adhesive. These parameterscan be determined either from experiments (Ferretti &Savoia 2003), or estimated according to the prescrip-tions reported in design codes and standards (ACI440R-96 1996; fib Bulletin 2001; JCI 2003).

It is important to observe that the values of P delC

computed for different FRP lengths, say l′FRP < lFRP,can also be obtained from the numerical simulationof the delamination process in a reference retrofittedbeam with l′FRP = lFRP. In fact, considering an inter-face crack length equal to a, the debonded portionof the FRP sheet is stress-free. Under these condi-tions, the critical load for interface crack propagation,P del

C (a), corresponds to that for the onset of delamina-tion in a retrofitted beam with a shorter reinforcementlength, i.e. P del

C (lFRP − a).The progress of FRP delamination can be numeri-

cally simulated using the finite element method with acohesive model for the description of the mechanicalbehavior of the interface. Cohesive models were intro-duced by Hillerborg et al. (1976) to the analysis of thenonlinear fracture process zones in quasi-brittle ma-terials. Carpinteri firstly applied a cohesive formula-tion to the study of ductile-brittle transition and snap-back instability in concrete (Carpinteri 1985; Carpin-teri et al. 1986; Carpinteri 1989). More recently, wehave proposed the use of this approach for the analy-sis of snap-back instability during FRP delamination

(Carpinteri et al. 2006).Numerical results for the beam with lFRP = 0.70 m

are reported in Figure 6 in terms of the applied load,P , vs. the mid-span deflection, δ. When the peak loadis achieved, point (A), delamination takes place and,using the crack length as a driving parameter, we ob-serve that both the external load and the mid-span de-flection of the beam are progressively reduced up topoint (B). After that, the progress of delamination re-quires an increase in the external load, tending asymp-totically to the mechanical response of the undamagedRC beam without FRP. From the engineering point ofview, this brittle mechanical response is particularlydangerous, since it corresponds to a severe snap-backinstability. A deformed mesh showing the progress ofdelamination is also shown in Figure 7.

0

30

60

90

120

150

180

0 3 6 9 12 15

δ δ δ δ [mm]

P [kN]

NLFM results

Unreinforced beam (exp. data)A

B

C

Figure 6: Load vs. non-dimensional deflection duringdelamination.

P/2

Figure 7: Deformed mesh showing delamination.

The structural response can be represented in termsof critical load vs. crack length or, equivalently, crit-ical load vs. FRP length (see Fig. 8). During crackpropagation, i.e. from (A) to (B), we have an unstablecrack growth, since the external load is progressivelyreduced. An increase in the critical load is observedafter point (B), i.e. for a/lFRP

∼= 0.8. In any case, forshorter FRP lengths, the actual failure load is certainlybounded by flexural or diagonal failures of the con-crete beam (see e.g. Figures 6 and 8 where the criti-

cal load corresponding to flexural failure of the beamwithout FRP is reported for comparison).

0

30

60

90

120

150

180

0 0.2 0.4 0.6 0.8 1

a/l FRP or 1 l FRP/l

Pcdel [kN]

A

B

C

Pcflex

Figure 8: Critical load for delamination vs. non-dimensional crack advancement. P flex

C is the ultimateload of the reference beam without retrofitting.

A comparison between Figures 4 and 8 shows thatboth the critical loads for shear failure and those forFRP delamination are decreasing functions of lFRP inthe usual range of FRP lengths. Moreover, for thecase-studies herein analyzed, the critical loads forshear failure are less than those for FRP delamination.This numerical prediction is in agreement with the ex-perimental findings by Ahmed et al. (2001), where thereinforced beams failed due to shear failure. Accord-ing to Figure 3, a pure FRP delamination is expectedfor smaller beams.

These results permit to interpret the common ex-perimental observation that shear failure and con-crete ripping are the most frequently observed failuremodes in RC beams tested in laboratory. FRP delami-nation is less frequent and should be ascribed to weakbonding properties.

4 FIBER-REINFORCED CONCRETE BEAMSIn this section, we show some of the main results ofan experimental programme on fiber-reinforced con-crete beams (FRC) retrofitted with FRP. The testshave been conducted in the Laboratory of Materialsand Fracture Mechanics of the Department of Struc-tural and Geotechnical Engineering of the Politecnicodi Torino.

In this respect, we notice that most of the currentresearch studies on retrofitting techniques deal withstandard RC members, whereas FRC beams are ana-lyzed only in a few studies (Yin & Wu 2003). Fromthe engineering point of view, FRC beams can beused for applications requiring high durability and theproblem of retrofitting may arise when we are lookingat another dimension of such members, namely, their

upgrading capability (Shah & Ouyang 1991; Wegian& Abdalla 2005).

In this programme, seven FRC beams have beentested under three-point bending up to failure un-der displacement control (see Fig. 9). The beams aremade of high-strength concrete reinforced with stan-dard steel fibers produced by Bekaert. These fibershave a length equal to 50 mm, a diameter equal to0.50 mm, and a content of 40 kg/m3. The concreteYoung’s modulus is equal to 35 GPa, with a Pois-son’s ratio of 0.18. FRP sheets have a Young’s modu-lus equal to 165 GPa, a tensile strength of 2300 MPaand a maximum tensile strain at failure of 1.8%. Con-cerning the geometrical parameters, we have l = 100cm, lFRP = 70 cm, h = t = 15 cm, s = 7.5 cm andhf = 1.4 mm.

Figure 9: Photo of the testing apparatus.

Four retrofitted beams failed due to delamination(B2, B3, B5 and B6), two experienced shear failure(B1 and B4) and the remaining one was tested withoutFRP (B7) to establish the behavior of the unreinforcedbeam (see the test results shown in Figure 10). Photosof shear failure and FRP delamination are also shown,respectively, in Figures 11 and 12.

Experimental results indicate that mixing of steel-fibers affects the cracking behavior of concrete, giv-ing rise to distributed crack patterns. The higher fre-quency of FRP delamination suggests that the failuremode changes from shear failure to pure FRP delam-ination, as compared to standard RC beams. This re-sult is fully consistent with the analytical model inSection 2. In fact, the use of steel-fibers results intoan increased concrete toughness as compared to regu-lar concrete. As a consequence, higher values of KICcorrespond to the higher critical loads required forshear crack propagation (see Eq. (6)). Moreover, thisconfirms that the transition from FRP delamination toshear failure is expected for larger beams.

0

10

20

30

40

50

60

70

80

90

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

δδδδ [mm]

P [kN]

B 4

B 1B 2

B 3B 5

B 6

B 7

B1-4: shear failure

B2-3-5-6: FRP delamination

B7: FRC beam w/o FRP

Figure 10: Results of the experimental tests: appliedload vs. mid-span deflection.

Figure 11: Photo of a failed beam due to shear crackgrowth.

5 CONCLUSIONS

In this paper we have proposed a combined analyt-ical/numerical approach for the analysis of failuremodes in concrete beams. Numerical predictions andexperimental results show that shear failure and con-crete ripping are more likely to occur in RC beams. Infact, for the analyzed case-studies, the critical load forthe onset of shear crack growth is found to be lowerthan that corresponding to FRP delamination, inde-pendently of the reinforcement length. On the otherhand, the results of the experimental programme onFRC beams show that in these cases FRP delamina-tion is more frequent than shear failure. This differentbehavior as compared to RC beams has to be ascribedto the increased value of concrete fracture toughnessdue to the steel-fiber bridging effect.

Concerning the issue of stability of crack propa-gation, it has been shown that the process of FRPdelamination leads to severe snap-back instabilities,thus resulting into a brittle mechanical response ofthe retrofitted concrete member. From the numericalpoint of view, it has been shown that the snap-backinstability can be followed either under crack lengthcontrol, or by computing the critical loads for de-lamination in correspondence of concrete beams withshorter and shorter FRP lengths. This numerical result

Figure 12: Photo of a failed beam due to FRP delam-ination.

gives also the possibility to experimentally follow thesnap-back instability by testing concrete beams withdifferent reinforcement lengths.

ACKNOWLEDGEMENTS

The financial support provided by the Italian Min-istry of University and Research is gratefully ac-knowledged. The Authors would like to thankMr. V. Di Vasto for the technical support providedduring the experimental programme and Ing. R. Got-tardo, Technical Manager of the D.L. Building Sys-tems DEGUSSA Construction Chemicals Italia, forproviding the FRP sheets used in the testing pro-gramme.

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