Construction and Building Materials 23 (2009) 1568–1577

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Construction and Building Materials

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Modeling of debonding failure for RC beams strengthened in shear with NSMFRP reinforcement

Andrea Rizzo, Laura De Lorenzis *

Department of Innovation Engineering, University of Lecce, via per Monteroni, 73100 Lecce, Italy

a r t i c l e i n f o

Article history:Available online 16 May 2008

Keywords:ConcreteFiber-reinforced polymersNear-surface mounted reinforcementShear strengthening

0950-0618/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2008.03.009

* Corresponding author.E-mail addresses: andrea.rizzo@unile.it (A. Rizz

(L. De Lorenzis).

a b s t r a c t

The shear capacity of reinforced concrete members can be successfully increased using near-surfacemounted (NSM) fiber-reinforced polymer (FRP) reinforcement. Tests conducted thus far have shown thatfailure is often controlled by diagonal tension associated to debonding between the NSM reinforcementand the concrete substrate. In absence of steel stirrups and/or when the spacing of the NSM reinforce-ment is large, debonding involves separately each of the bars crossed by the critical shear crack. In orderfor shear strengthening of beams with NSM reinforcement to be safely designed, an analytical model ableto encompass the failure mode mentioned above must be developed. This paper presents two possibleapproaches, a simplified and a more sophisticated one, to predict the FRP contribution to the shear capac-ity. In the first approach, suitable for immediate design use, an ideally plastic bond–slip behavior of theNSM reinforcement is assumed, which implies a complete redistribution of the bond stresses along thereinforcement at ultimate. The second approach, implemented numerically, accounts for detailedbond–slip modeling of the NSM reinforcement, considering different types of local bond–slip laws cali-brated during previous experimental investigations. It also takes advantage of an approach developedby previous researchers to evaluate the interaction between the contributions of steel stirrups and FRPreinforcement to the shear capacity. The paper illustrates the two models and compares their predictions,with the ultimate goal to evaluate whether the first simple model can be used expecting the same safetyin predictions of the second model.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

One of the major applications of fiber-reinforced polymer (FRP)composites for strengthening of reinforced concrete (RC) membersis their use as additional web reinforcement to increase the shearcapacity of the members. Over the last few years, shear strengthen-ing with externally bonded FRP laminates has become a well estab-lished technique upon extensive experimental verification andwith the development of analytical models reflected in the relevantcode provisions. A state-of-the-art review of existing research onthis topic up to 2003 can be found in Ref. [1]. Further papers havebeen published in the past few years (see e.g. [2–8]).

A more recent and less investigated method for shear strength-ening of RC members is the use of near-surface mounted (NSM)FRP reinforcement, in the form of round bars, square bars or rect-angular bars with large width to thickness ratio (also briefly indi-cated as strips). In this paper the term ‘‘bars” is used to refercollectively to all possible types of NSM reinforcement. In theNSM method, the reinforcement is embedded in grooves cut onto

ll rights reserved.

o), laura.delorenzis@unile.it

the surface of the member to be strengthened and filled with anappropriate binding agent such as epoxy paste or cement grout.A review of available research on NSM strengthening of RC struc-tures up to 2005 is reported in Ref. [9]. Further papers have beenpublished more recently [7,10,11]. For shear strengthening withNSM reinforcement, the grooves are cut on the sides of the mem-ber at a desired angle to the beam axis.

Applications of NSM FRP reinforcement for shear strengtheningof RC beams are described in Refs. [7,10–14]. De Lorenzis andNanni [12] carried out tests on large size T-beams, most of whichhad no internal stirrups. Carbon FRP (CFRP) ribbed round bars inepoxy-filled grooves were used as NSM shear reinforcement. Thetest variables included bar spacing and inclination angle, andanchorage of the bars in the flange. The NSM reinforcement pro-duced a shear strength increase which was as high as 106% inthe absence of steel stirrups, and still significant in presence of alimited amount of internal shear reinforcement. Nanni et al. [13]reported the test results of a single full-scale PC girder taken froma bridge and shear-strengthened with NSM CFRP strips. The beamfailed in flexure at a shear force close to the shear resistance pre-dicted by the model given in Ref. [12]. Barros and co-workers[7,10,11] tested beams of different sizes and with differentamounts of longitudinal steel reinforcement. Some of these beams

A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577 1569

were strengthened with NSM CFRP strips of different inclinations,while others were strengthened with equivalent amounts of exter-nally bonded FRP shear reinforcement. The NSM technique provedmore effective than externally bonded FRP in terms of bothstrength and deformation capacity. Rizzo and De Lorenzis [14] car-ried out tests on beams with a limited amount of internal steel stir-rups strengthened with NSM round bars and strips, analyzing theeffect of the spacing and inclination of the FRP strengthening,and of the type of groove-filling epoxy. The increase in shear capac-ity was about 16% for one beam strengthened with externallybonded U-wrapped laminate, and ranged between 22% and 44%for the beams strengthened with NSM reinforcement. The use ofNSM reinforcement was thus more efficient due to early debondingof the externally bonded laminate.

Two different failure modes were identified in Ref. [12] forbeams strengthened in shear with NSM ribbed bars. In absenceof steel stirrups, the failure mode was debonding of the FRP barsby splitting of the epoxy cover and cracking of the surroundingconcrete, associated with the diagonal tension failure of concrete.This failure mode was prevented by providing better anchorageof the NSM bars crossing the critical shear crack, by either anchor-ing the bars in the beam flange or the use of inclined (e.g. 45�) barsat a sufficiently close spacing to achieve a longer total bond length.Once this mechanism was prevented, separation of the concretecover of the steel longitudinal reinforcement became the control-ling failure mode. This second mode, however, may be attributedto the fact that no or very limited steel stirrups were present inthe beams [12], and is unlikely in beams with a more realisticamount of steel stirrups. More recent experiments conducted bythe authors [14] have shown that the presence of steel stirrups,combined with a relatively small spacing of the reinforcement,may originate a third failure mechanism in which the lateral con-crete covers of the steel stirrups detach from the core of the beamwith the NSM reinforcement still embedded. Thus, due to the par-ticular failure mode, decreasing the spacing or increasing the incli-nation of the bars cannot benefit the shear capacity of the beamsince the reduced distance between the bars accelerates the forma-tion of a debonding failure pattern involving all the bars together.Although it has not been observed so far, tensile rupture of theNSM reinforcement is another possible failure mode.

At present, different approaches to compute the capacity ofshear-strengthened beams are available in the literature, most ofwhich are based on the generalization of the truss model usuallyadopted for computation of the shear capacity of RC beams. As deb-onding failure modes often control, and considering that the bondbehavior of NSM reinforcement is markedly different from that ofexternally bonded reinforcement, specific models for NSM-strength-ened beams are needed. The only model specific for NSM reinforce-ment is that by De Lorenzis and Nanni [12]. They proposed a modelbased on the following assumptions: inclination angle of the shearcracks constant and equal to 45�; even distribution of bond stressesalong the FRP bars at ultimate; the local bond strength is reached inall the bars intersected by the critical shear crack at ultimate. Thelast two assumptions in turn derive from the use of an ideally plasticlocal bond–slip curve (with unlimited ductility) for the NSM rein-forcement. In this model, the FRP contribution to the shear strengthof the beam results proportional to the sum of the minimum embed-ment lengths of all the bars intersected by the critical shear crack,computed for the most unfavorable crack position. The same authorsalso proposed that the maximum strain in the bars be limited to4000le in order to avoid the loss of the aggregate interlock due toa large width of the shear crack. The simple equations resulting fromthis model were provided for the cases of vertical bars, for ratios ofbar spacing to beam effective depth ranging from 0.25 to 1.0.

The adoption of an ideally plastic local bond–slip curve in themodel in [12], while yielding simple expressions for the NSM FRP

contribution to the shear capacity, is a crude approximation ofthe real bond–slip behavior of NSM reinforcement and may easilyresult un-conservative. On the other hand, starting from localbond–slip relationships obtained from bond tests, it is possible toimprove the accuracy of the model at the expenses of its simplicity.Moreover, the model in [12], such as virtually all models currentlyavailable on the capacity of RC beams shear-strengthened with FRPsystems, computes the capacity of the strengthened beam as thesum of the contributions of steel stirrups and FRP system. This im-plies the assumption that the peak shear forces that can be resistedby the steel stirrups and by the FRP are reached at the same time,which is generally an un-conservative assumption. Recently,Mohamed Ali et al. [15] proposed a partial-interaction model toquantify the vertical shear interaction between transverse FRPplates and steel stirrups. The analysis of the shear-strengthenedbeam is conducted by gradually increasing the opening of the shearcrack, and computing the corresponding slip and bond stress distri-butions along the NSM reinforcement. The steel stirrups are con-sidered fully anchored at their ends, and bond between the FRPplates and concrete is analyzed by using a linear softening localbond–slip relationship.

In this paper, two models are proposed for the shear capacity ofRC beams strengthened in shear with NSM reinforcement. In bothmodels, the failure mode is assumed to be debonding of the NSMreinforcement. In particular, debonding is assumed to take placein each single NSM bar separately, with no global separation ofthe cover of the internal stirrups. The latter failure mode deservesa specific investigation and is not covered herein.

The first model is a generalization of the model in [12], wherethe assumption of perfectly plastic bond–slip behaviour is main-tained but the angles formed by the critical shear crack. Moreover,by the NSM reinforcement to the beam axis are made variable.Moreover, a wide range of spacings of the NSM reinforcement isconsidered. The second model is based on the approach in [15],where an appropriate local bond–slip model is considered for theshear strengthening system and the possibility of a reducedeffectiveness of the internal stirrups due to interaction with theexternal strengthening is accounted for. The local bond–slipmodels adopted for NSM reinforcement are taken from previouslyavailable bond test results. The purpose of the paper is to compar-atively evaluate the performance of the two models. The final goalis to determine whether, and under which conditions, the first sim-ple model can be used expecting the same safety in predictions ofthe second model.

2. Generalized ideally plastic (GIP) model

2.1. FRP contribution to the shear capacity

As follows, the simplified approach proposed in [12] is general-ized for any value of the FRP spacing, and of the angles formed bythe shear crack and by the FRP strengthening with the horizontaldirection. The formulation is suitable to account for any possibledebonding failure mode, provided that it involves each NSM barseparately and that it displays sufficient pseudo-ductility to makethe assumption of a perfectly plastic bond–slip behavior physicallyreasonable. According to [12], a constant shear stress at failure sf atthe bar-epoxy interface is assumed in all the FRP bars intersectedby the shear crack at ultimate, thus the FRP shear contribution VFRP

can be calculated multiplying sf by the total lateral surface of theminimum embedment lengths of all the bars crossed by the crack.This results in the following equation:

VFRP ¼ 2Xns;f

i¼1

lemb;i

!psf sin a ¼ 2lemb;totpsf sin a ð1Þ

Shear Cracks

1570 A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577

where ns,f is the number of FRP bars intersected by the critical shearcrack, lemb,i is the minimum embedment length of the ith bar (i.e.the minimum between the two lengths ls,i and li,i in which the ithbar is ideally subdivided by the shear crack, see Fig. 1), p is theperimeter along which the bond stress acts (in the case of roundbars and if debonding is critical at the bar – groove filler interface,p = p/ where / is the diameter of the bar), a is the angle of the FRPbars to the horizontal axis z (see Fig. 1), factor 2 accounts for thebars on both sides of the beam, and sina projects the stress resultanton the vertical direction in order to calculate the shear contribution.Eq. (1) is valid provided that lemb,i is smaller than the developmentlength of the bars ldev, i.e. of the value of bond length sufficient tocause failure in tension of the FRP bar prior to its debonding. Forany given geometry of the beam, VFRP depends on a, h (h being theangle between the z-axis and the critical shear crack, see Fig. 1)and the relative position between the shear crack and the strength-ening system. For design purposes, the value of VFRP correspondingto the most unfavourable crack position must be calculated, as willbe described later. The sum in Eq. (1) represents the total embed-ment length and will be indicated with lemb,tot.

For fixed a and h, the shear crack may intersect a variable num-ber of bars depending on its location relative to the strengtheningsystem. The minimum and the maximum number of bars inter-sected by the shear crack (ns,f min and ns,f max, respectively) dependon the ratio between the height of the cross-section and the barspacing. Indicating with jxj the integer part of number x, ns,f min

and ns,f max are given by:

ns;f min ¼ jns;f jns;f max ¼ jns;f j þ 1

�with ns;f ¼

hsf½cot hþ cot a� ð2Þ

where sf is the spacing between the bars measured along the longi-tudinal axis of the beam, and h is the total height of the beam(Fig. 1). The actual number of intersected bars ns,f will be equal toeither ns,f min or ns,f max.

The lengths of the portions of the ith bar situated above and be-low the crack, ls,i and li,i, respectively (Fig. 1), can be calculated asfollows:

li;i ¼ tan hsin aþtan h cos a zi ¼ f ða; hÞzi

ls;i ¼ lf � li;i

(ð3Þ

where zi is the position of the ith bar with respect to the bottom tipof the crack (see Fig. 1), f(a, h) is a known function of angles a and h,and lf is the total length of the bars given by:

lf ¼h

sin að4Þ

It is possible to determine the position zi of the ith bar, and conse-quently lengths li,i and ls,i, when the position z1 of the first bar thatintersects the shear crack is known, as follows:

zi ¼ z1 þ ði� 1Þsf 6 ½cot hþ cot a�h ð5Þ

sc

icz1

α

Shear Crack

d

sf

h

z

i i+1i−1

zi

i,ils,il

θ

FRP Reinforcement

LongitudinalSteel Rebars

Fig. 1. Calculation of the embedment lengths of the FRP bars in the GIP model.

Conservatively, according to the model proposed by [12], theembedment length lemb,i of the ith bar is taken as:

lemb;i ¼ min ls;i �cs

sin a; li;i �

ci

sin a

� �P 0 ð6Þ

where cs and ci are the distances between the centroids of the topand bottom steel reinforcements and the concrete extrados andintrados surfaces, respectively. This reduction of the embedmentlength results from the height of the Mörsch truss extending be-tween the centroid of the steel tension reinforcement and the loca-tion of the resultant compressive force, which is assumed forsimplicity to coincide with the centroid of the steel compressionreinforcement. For the same reason, it is possible to define a re-duced height hnet by subtracting the concrete cover thicknesses cs

and ci from the total height of the cross-section, as follows:

hnet ¼ h� ci � cs ð7Þ

Once z1 and angles a and h are fixed, it is possible to find the index i*

of the last bar for which the minimum embedment length coincideswith the one located below the crack (i.e. with li,i � ci/sina):

i� ¼lf þ ci�cs

sin a

2f ða; hÞsf� z1

sfþ 1

�������� ¼ cot hþ cot a

2hþ ci � cs

sf� z1

sfþ 1

�������� ð8Þ

and the actual number ns,f of bars that intersect the shear crack:

ns;f ¼½cot hþ cot a�h� z1

sfþ 1

�������� ¼ ns;f �

z1

sfþ 1

�������� ð9Þ

The total embedment length lemb,tot needed to calculate the FRPshear contribution for given z1, a and h is then given by:

lemb;tot ¼Xi�

i¼1

f ða; hÞðz1 þ ði� 1Þsf Þ �ci

sin a

h i

þXns;f

i¼i�þ1

lf � f ða; hÞðz1 þ ði� 1Þsf Þ �cs

sin a

h ið10Þ

By varying z1 from 0 to sf, it is possible to find the minimum value oflemb,tot, termed lemb,tot min. In order to find an explicit expression forlemb,tot min, it is useful to introduce the parameters r* and n*:

r� ¼ ½cot hþ cot a�hnet

sfð11Þ

n� ¼ jr�j � i�z1¼0 ¼ jr�j �r�

2þ 1

��������þ 1 ¼ jr�j � r�

2

�������� ð12Þ

where r* has the same meaning as ns;f , but is computed using hnet

instead of h, and n* is the number of the bars for which the mini-mum embedment length coincides with the one located above thecrack for the particular case of z1 = 0. Referring to Fig. 2, it is possibleto note that the trend of the total embedment length versus thelocation of the shear crack is piecewise linear. For the particularcase analyzed in Fig. 2, if the shear crack is in a position between

z

θ

α

Pos. #

FRP Rods

12

3 4

h net

12

3 4 Pos. #

Total Embedment

Length Trend

f

Fig. 2. Calculation of the minimum total embedment length of the FRP bars in theGIP model.

0.00.51.01.52.02.53.03.54.04.55.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

s f / h net

3035404550

,2

FR

Pm

inf

net

Vp

hτ

α = 90o

θ (°)

(a) α = 90o and Variable θ

0.00.51.01.52.02.53.03.54.04.55.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

455060708090

,2

FR

Pm

inf

net

Vp

hτ

θ = 45o

α (°)

[

[

[

[

A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577 1571

#1 and #2, lemb, tot raises from its minimum value reaching its max-imum when the shear crack divides into two equal parts the lengthof a bar; then it decreases until its minimum value corresponding tothe shear crack in position #3. For any other position of the shearcrack, lemb, tot remains constant to its minimum value.

It can be shown that lemb,tot min is given by:

lemb;tot min ¼ n�½cot hþ cot a� � ðn�Þ2 sf

hnet

� �sin h

sinðaþ hÞ hnet ð13Þ

In particular, for a = 90 � and h = 45�, it is:

r� ¼ hnetsf

n� ¼ hnetsf

��� ���� hnet2sf

��� ���lemb;tot min ¼ n�hnet � ðn�Þ2sf

8>>><>>>:

ð14Þ

For a = 45� and h = 45�, it is:

r� ¼ 2hnetsf

n� ¼ 2hnetsf

��� ���� hnetsf

��� ���lemb;tot min ¼ ½2n�hnet � ðn�Þ2sf �

ffiffi2p

2

8>>><>>>:

ð15Þ

To sum up, the computation of the FRP shear contribution VFRP withthe proposed model consists in the following steps:

– computation of hnet from Eq. (7);– computation of r* and n* from Eqs. (11) and (12);– computation of lemb,tot min from Eq. (13);– computation of VFRP min as:

s f / h net

(b) θ = 45o and Variable α

Fig. 3. Normalized shear contribution of the NSM FRP reinforcement according tothe GIP Model.

VFRP min ¼ 2lemb;tot minpsf sin a ð16Þ

VFRP min in Eq. (16) is the value of VFRP computed for the most unfa-vourable position of the crack with respect to the strengtheningsystem.

Fig. 3 reports the normalized FRP shear contribution VFRP min/2psfhnet as a function of the normalized bar spacing for differenta and h values. It can be observed that, for given FRP bar and shearcrack angles, the FRP shear contribution decreases as the sf/hnet ra-tio increases since the number of bars intersected by a shear crack,and hence lemb,totmin, decrease. The slope of the curves decreases forhigher sf/hnet ratios, indicating that the variation of VFRP min is lessaffected by the spacing in the range of large spacings. For the samegeometrical reasons, for given sf/hnet and a, VFRP min is higher forsmaller values of h. Finally, for given values of the remainingparameters, increasing the inclination of the FRP strengtheningraises VFRP min, even though the effect of the longer total embed-ment length is reduced by the opposite influence of the inclinationfor which the stresses in the bars have to be projected on the ver-tical direction.

2.2. Upper limit to the FRP contribution to the shear capacity

The value of VFRP should be bound by an upper limit VFRP inorder to avoid an excessive crack opening with consequent lossof aggregate interlock. The authors in [12], citing previousresearchers, suggest limiting the maximum strain in the bars be-low a value eFRP,max equal to 4000le. While the particular value tobe adopted is a subject of ongoing discussion, the concept of a lim-iting strain is generally agreed upon. This restriction implies thatthe FRP reinforcement cannot develop its maximum tensilestrength, being the ultimate tensile strain of most types of FRP lar-ger than 1%. This limitation can be imposed by using, on the lon-gest among the embedment lengths of the bars, a reducedtangential stress and assuming that tensile stresses in the otherbars are proportional to their embedment lengths [12]. The embed-

ment length li of an FRP bar crossed by the crack corresponding tostrain eFRP,max is given by Eq. (17):

li ¼AFRPEFRPeFRP;max

psfð17Þ

where AFRP and EFRP are the nominal cross-sectional area and mod-ulus of elasticity of the FRP reinforcement.

It can be shown that, when the total embedment length is min-imum, the longest among the minimum embedment lengths lemb,i

of all the bars crossed by the crack, lemb max, is given by:

lemb max ¼hnet

2 sin a½1� ð�1Þjr

�j� þ ð�1Þjr�j n�sf

sin a½cot aþ cot h� ð18Þ

If lemb max 6 li, computation of VFRP is not necessary. Else, a reducedtangential stress sf,red must be computed as:

sf ;red ¼li

lemb maxsf ð19Þ

and the upper limit to the FRP shear contribution is given by:

VFRP ¼ 2plemb;tot minsf ;red sin a ð20Þ

3. Model accounting for local bond–slip behavior

3.1. Load–slip behavior of a single NSM joint

In this section the assumption of an ideally plastic s–s behavioris removed and a numerical approach is presented, applicable toNSM reinforcement featuring any type of bond–slip model.

1572 A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577

The governing differential equation of a bonded joint is thefollowing:

d2

dx2 s� vsðsÞ ¼ 0 ð21Þ

where s and s are local slip and local bond stress, respectively; and xis the coordinate along the bonded joint. This equation can besolved once the local bond–slip model s(s), relating s and s, is de-fined. The constant v is the ratio between the perimeter p alongwhich the bond stress acts, and the product between the cross-sec-tional area AFRP on which the tensile stress acts and the modulus ofelasticity EFRP. For example, for the case of round bars of diameter /it is:

v ¼ pEFRPAFRP

¼ 4EFRP/

: ð22Þ

Of the several local bond–slip models available in the literature, twoare considered in this paper. The first model, s1(s), is:

s1ðsÞ ¼sm

ssm

� �a1for 0 6 s 6 sm

sms

sm

� �a2for sm < s 6 su

8><>: ð23Þ

where sm is the maximum bond stress that occurs at a slip equal tosm; su is the ultimate slip that can assume a finite or infinite value;a1 P 0 and a2 6 0 are coefficients defining the shape of the twobranches of the curve. Fig. 4a illustrates the model for different val-ues of a1 and a2. In particular, a rigid-perfectly plastic behavior isobtained for a1 = 0 and a2 = 0, a linear elastic first branch is obtainedfor a1 = 1, and a drastic drop after the peak can approximately beassociated to a2 < �5.

0.0

0.2

0.4

0.6

0.8

1.0

s /s m

τ / τ

m

( )

1

2

for 0

for

m mm

m mm

ss s

ss

ss s

s

α

α

ττ

τ

≤ ≤

=

<

α2 = -0.5

α2 = -1.0

α2 = -5.0

α1 = 0.10

α2 = 0.0α1 = 0.0

α1 = 1.0

α2

α1

(a) Local bond-slip model #1

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

0 1 2 3 4 5 6

s /s m

τ / τ

m

s ( )( )

2

for 0

for

τ

ττ τ τ

≤ ≤=

−− + < ≤

−

mm

m

u m

u

m m m uu m

s s ss

s

s s s s ss s

τ u = 0.5τ m

τ u = 1.0τ m

τ u = 0.0τ m

(b) Local bond-slip model #2

Fig. 4. Bond–slip models used in the analyses.

The second model, s2(s), consists of two linear branches and isgiven by:

s2ðsÞ ¼smsm

s for 0 6 s 6 sm

su�smsu�sm

ðs� smÞ þ sm for sm < s 6 su

(ð24Þ

with su having a finite value. Fig. 4b illustrates the model for differ-ent values of su. In particular, a perfectly plastic behavior for thesecond branch is obtained for su = sm. The two models above canbe fitted to the experimental results thus calculating the unknownparameters sm, su, a1, a2, sm and su.

In this paper, four different variations of the previous twobond–slip models (sI, sII, sIII and sIV) are used to analyze the influ-ence of the bond–slip model on the shear contribution of the FRPstrengthening system. These variations will also be referred to as‘‘models”. The four curves are illustrated in Fig. 5. sI refers to thefirst bond–slip model (Eq. (23)), where the values of the parame-ters are taken from the experiments in [16], that is sm = 0.25 mm,a1 = 0.55, a2 = � 0.25, sm = 10 MPa. The ultimate slip su was conven-tionally chosen as the value of slip corresponding to a local bondstress equal to 50% of the local bond strength, obtainingsu = 2.5 mm. sII consists of two linear branches according to Eq.(24). The peak corresponds to s = sm and s = sm, while the bondstress is zero corresponding to the maximum slip. Also sIII consistsof two linear branches, but differs from sII in that the maximumslip is maintained constant to the value adopted for sI. The modelsIV neglects the pre-peak linear branch maintaining the maximumbond stress equal to sm = 10 MPa. In order to obtain a meaningfulcomparison between the different models, the unknown parame-ters of the models sII, sIII and sIV were calculated maintaining con-stant the fracture energy Gf found with the first model (sI), equal to17.025 N/mm. Eqs. (25) and (26) give the expressions of the frac-ture energy for the bond–slip models s1(s) and s2(s), respectively:

0123456789

10

0.0 0.5 1 .0 1.5 2.0 2.5 3.0 3.5 4.0

s [mm ]

τ[M

Pa

]

Model GIPModel IModel IIModel IIIModel IV

( )6.81 for 0 2.5

0 for 2.5 GIP

MPa s mms

MPa mm sτ

≤ ≤=

<

( )

0.55

0.25

10 for 0 0.250.25

10 for 0.25 2.50.25

τ−

≤ ≤=

< ≤I

sMPa s mm

ss

MPa mm s mm

( ) ( )40 for 0 0.25

3.170 0.25 10 MPa for 0.25 3.405II

s MPa s mm

ms

s m s mmτ

≤ ≤=

− − + < ≤

( ) ( )40 for 0 0.25

2.657 0.25 10 MPa for 0.25 2.5III

s MPa s mm

ms

s m s mmτ

≤ ≤=

− − + < ≤

( ) 2.937 10 MPa for 0 3.405τ = − + < ≤IV s s s mm

Fig. 5. Bond–slip models used in the analyses (Gf constant).

0

20

40

60

80

0 200 400 600

Embedment Length [mm ]

Max

imum

Loa

d [ k

N] Model I

Model IIModel III

Model IV

GIP Model:τ f = 6.81 MPaG f constant with su = 2.5 mm

Fig. 7. Maximum load at loaded end versus embedment length.

A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577 1573

Bond—slip model s1ðsÞ :

Gf

smsm¼

1a1þ1þ lnðkÞ if a2 ¼ �1

1a1þ1þ ka2þ1�1

a2þ1 if a2–� 1

8<: with su ¼ ksm ð25Þ

Bond—slip model s2ðsÞ :

Gf

smsm¼ kþ ðk� 1Þrs

2with su ¼ ksm and su ¼ rssm ð26Þ

Fig. 5 reports the curves and the explicit equations related to eachbond–slip curve. In the same figure, the GIP model curve is also re-ported. The bond stress sf = 6.81 MPa was obtained by fixing theultimate slip at 2.5 mm and maintaining the fracture energy equalto the value of the remaining models.

A closed-form solution of Eq. (21) can be found just for a fewcases such as, for example, for linear or perfectly plastic bond–slipmodels. In this investigation, a numerical approach based on piece-wise linearization was used for the solution of the differentialequations in the cases where no closed-form solutions wereavailable.

Fig. 6 reports, for each of the four bond–slip curves, the loadversus loaded-end slip (P � sle) curves for two different embed-ment lengths, chosen as 200 and 228 mm (these values will beused again later). These lengths are shorter than all the effectivebond lengths obtained by using the four bond–slip models (equalto 475, 796, 700 and 541 mm, respectively; see Fig. 7). The effectivebond length is the minimum value of embedment length for whichthe maximum debonding load is reached, see [2] for more details.It was assumed to use round CFRP bars with 8-mm diameterand 175 GPa Young’s modulus. Eq. (22) yields v = 2.857 �10�6 MPa�1 mm�1. In the pre-peak region of the P � sle curve,models I, II and III give a very close behavior, whereas model IVyields a stiffer behavior (smaller slips) due to the absence of the

05

101520253035404550

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Slip at Load End [mm ]

Loa

d [

kN]

Model IModel IIModel IIIModel IV

l emb = 200 mm

(a) lemb = 200 mm

05

101520253035404550

Slip at Load End [mm ]

Loa

d [

kN]

Model IModel IIModel IIIModel IV

l emb = 228 mm

(b) lemb = 228 mm

Fig. 6. Load versus. loaded-end slip curves for 200 and 228 mm embedmentlengths.

initial linear branch in the bond-slip curve. The maximum loadaccording to model I is 5.6% and 6.1% smaller than the one pre-dicted by the other models for the 200 and 228 mm embedmentlengths, respectively. This is due to the more rapid decay in bondstress immediately after the peak in model I compared to the othermodels.

Fig. 7 shows the curves giving the maximum load versus theembedment length, obtained using the four different bond–slipmodels in the range of embedment lengths between 0 and 1000mm. For low values of the embedment length, the loads predictedwith models II, III and IV, virtually coincident, are slightly largerthan those obtained from model I. This is again due to the more ra-pid decay in bond stress immediately after the peak in model I. Thedifference between the predictions of models II, III and IV becomeslarger (though always very small) for larger embedment lengths, asthe debonding load for longer embedment lengths is more heavilyinfluenced by the shape of the post-peak branch of the bond–slipcurve. Once the effective bond length of each bar is reached, thedebonding load attained is the same for all models, which resultsfrom the fracture energy of the joint being the same for all models.In the same plot, the curve obtained from a perfectly plastic bond–slip model is also reported. As previously mentioned, the value ofbond strength used in this model was computed by limiting theultimate slip at 2.5 mm and maintaining the fracture energy equalto the value of the other models, which yields sf = 6.81 MPa (Fig. 5).It can be noted that using this model yields conservative predic-tions of the debonding load for embedment lengths smaller thanthe effective bond length, due to the smaller local bond strengthcompared to the other models. The maximum debonding load isobviously the same of the other models as the fracture energy isthe same.

The maximum debonding load, depending on the fracture en-ergy of the joint, may be smaller than the tensile rupture load ofthe FRP bar. For the case analyzed above, the maximum debondingload is 78.7 kN while the load at tensile failure of the FRP bar,assuming a tensile strength of 2214 MPa, is about 98 kN. Hence,in this case, no development length exists, but only an effectivebond length can be found [2].

3.2. Analysis of a strengthened beam

Once the bond–slip behavior of the NSM reinforcement isknown (fracture energy, shape of the bond–slip curve) the analysisof the shear-strengthened beam can be conducted, as proposed in[15], by gradually increasing the opening of the critical shear crackand computing the corresponding slip and bond stress distribu-tions along each NSM bar. These distributions are influenced byseveral parameters related to the geometry of the system (e.g.

1574 A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577

height of the beam, spacing of the NSM reinforcement, location ofthe shear crack).

First of all, once the shear crack position is fixed with respect tothe NSM reinforcement, each FRP bar is divided into two lengths li,iand ls,i for which a separate analysis is done in order to find theP � sle curve. Then, the load supported by each bar (each joint) isrelated to the opening of the shear crack, w, given by the sum ofthe slips at the loaded end (i.e. at the crack location) for the twosides i and s of the bar: w = sle,i + sle,s. Obviously, the shorter lengthcontrols the debonding failure of each bar, which means that thedebonding load of the joint is given by that of the shorter length.It is also assumed that the shear crack width is the same for all barsintersected by the crack, i.e. that the shear crack has constantwidth across the beam height. The effect of different assumedshapes of the critical shear crack on the resulting bond stress dis-tribution in the FRP and on the shear capacity (associated to deb-onding) of the strengthened beam was recently investigated forbeams shear-strengthened with externally bonded FRP laminates[17]. It was found that different crack shapes result in significantlydifferent stress distributions in the FRP, however their effect on thecapacity of the beam is much less significant. Hence it appears thatthe simplest assumption of a constant shear crack width is ade-quate in most situations. In any case, the proposed model can beeasily extended to consider the effect of a variable shear crackwidth.

Fig. 8a shows the application of the described procedure for abar divided by the shear crack into two parts having 200 and228 mm length, respectively. As follows, the shorter length willbe labeled as 1 and the longer one will be labeled as 2, and so willthe corresponding P � sle curves be labeled. Numerically, for eachpoint B1 belonging to curve 1 before the point A1 (corresponding

05

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Slip at Load End, sle /Crack Opening, w [mm ]

Loa

d,P

[ k

N]

C

Snap Back Curve 2l = 228

Curve 1l = 200 mm

Corrections for DisplacementControl Procedure

P -w Curve for the Joint

A2

A1A

BB1

B2 CC2

(a) Procedure to Find the P-w Curve for Model I

05

101520253035404550

Crack Opening, w [mm ]

Loa

d, P

[ k

N]

Model IModel IIModel IIIModel IV

(b) Comparison between P-w Curves Obtained with Different Models

Fig. 8. Load-crack opening curves for a bar with 200 and 228 mm embedmentlengths on the two sides of the shear crack.

to the maximum load capacity), it is possible to find a point B2

on curve 2 (before A2) whose load is the same as in B1: thus thecrack opening in B (on the load versus crack opening curve) isthe sum of the loaded end slips corresponding to B1 and B2. Onthe other hand, for each point C1 belonging to curve 1 after thepoint A1, it is possible to find a point C2 (before A2) on curve 2whose load is the same as in C1: thus the crack opening in C (onthe load versus crack opening curve) is the sum of the loadedend slips corresponding to C1 and C2. It is visible as, after the max-imum load, the ‘‘global” joint tends to behave as its shortestembedment length. In reality, it is very likely that the crack open-ing be continuously growing during loading of the beam. For thisreason, any snap-back of the curve (as happens in the analyzedcase or, in general, when the two portions of the bar separatedby the shear crack have a similar length) can be corrected as shownin Fig. 8a.

A particular case is the one for which the shear crack divides theFRP bar into two equal lengths. In this situation, the points A1, B1

and C1 coincide with the corresponding A2, B2 and C2: thus thecrack opening is twice the loaded-end slip of one length. For thecase analyzed in Fig. 8, this would determine a simultaneous fail-ure of the two FRP lengths at an ultimate crack opening equal to5.0 mm.

Note that the analysis described above applies to cases forwhich at least the shortest of the two embedment lengths on thesides of the crack is shorter than the effective bond length of thejoint (if not, both embedment lengths would fail at the same load)and of the development length (if not, the curve would fallabruptly to zero as soon as the load reaches the rupture load ofthe FRP bar).

Fig. 8b compares the load versus crack opening curves obtainedby using bond–slip models I to IV. The trends are similar to those ofthe load versus loaded-end slip curves, hence the same observa-tions on the differences and similarities between the models canbe applied.

The next step consists of adding the contributions to the shearcapacity of all the FRP bars and steel stirrups intersecting the crack,for each fixed crack opening. The internal steel stirrups can consistof smooth or ribbed bars that are typically fully anchored at theirends being bent around the longitudinal bars. Hence, the steel con-tribution to the shear capacity Vs can be estimated as

V s ¼2nsEs

whs

As if whs6 esy

2nsEsesyAs if esy <whs

(ð27Þ

where ns is the number of steel stirrups intersected by the shearcrack, Es the steel modulus of elasticity, w the crack opening, hs

the distance between the end anchorages, esy the steel yield strainand As the cross-sectional area of each stirrup leg. It is importantto note that the force in each intersected stirrup is independentfrom the position of the critical diagonal crack if its opening is uni-form along its length. In Eq. (27) it was assumed to have verticalsteel stirrups.

4. Numerical application

This section presents the results of the analyses described abovefor a beam with the geometry shown in Fig. 9. The RC beam has arectangular 300-mm � 500-mm cross-section. The steel tensionand compression reinforcements each consist of four steel de-formed bars with 16-mm nominal diameter. The steel shear rein-forcement consists of closed double-legged stirrups with 8-mmnominal diameter spaced at 250 mm o.c. In the next analysis, itis assumed that only one steel stirrup is intersected by the shearcrack, and that it is fully anchored with hs = 452 mm. The yield

Dimensions in mm

Closed StirrupsØ8 @ 250 mm o.c.

z = 50 mm1

Shear Cracks

250

100

100

z = 100 mm1

FRP Rods100 mm o.c.

Closed StirrupsØ8 @ 250 mm o.c.

20

20

4 Ø16

4 Ø16

500428

300

20

20

Fig. 9. Details of the strengthened beam considered in the example.

A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577 1575

strength and modulus of elasticity of steel are fixed at 430 MPa and210 GPa, respectively.

Fig. 10 reports the load versus crack opening curves obtainedfrom the four bond-slip models assuming that the 45� critical diag-onal crack is uniformly opened and is situated at a distance z1 vary-ing between 50 and 100 mm. The spacing of the FRP strengthening(a = 90�) is equal to 100 mm. For z1 = 50 mm the central bar is di-vided into two equal lengths by the shear crack and the sum of

Fig. 10. Load-crack opening curve for the strengthene

the minimum embedment lengths lemb,tot takes its maximum value(470 mm on five intersected bars); for z1 = 100 mm the two centralbars have the same embedment lengths and the sum of the mini-mum embedment lengths lemb,tot takes its minimum value(456 mm on four intersected bars). The trend of the curves is thesame for all models: increasing z1, the maximum load increaseswith a simultaneous reduction in the length of the plateau aroundthe peak point. Note that the maximum difference between the

d side of the beam (FRP and steel contributions).

Table 1Maximum FRP shear contribution for the cases of Fig. 10

z1 (mm) lemb, tot (mm) Model I Model II Model III Model IV GIPa GIPb GIPc GIPd

VFRP (kN)

50 470 163 167 171 166 236 199 161 15860 460 162 167 171 166 231 199 157. 15770 456 164 170 173 168 229 202 156 15680 456 168 175 177 173 229 205 156 15690 456 172 180 182 178 229 205 156 156

100 456 174 183 186 185 229 205 156 156lemb,tot min (mm) VFRPmin (kN)

N/A 456 162 167 171 166 229 199 156 156

a sf = 10 MPa without limitation on the FRP strain.b sf = 10 MPa with limitation on the FRP strain (eFRP,max = 4000le).c sf = 6.81 MPa without limitation on the FRP strain.d sf = 6.81 MPa with limitation on the FRP strain (eFRP,max = 4000le).

050

100150200250300350400450500

0 50 100 150 200 250 300

Bar Spacing, sf [mm ]

VF

RP

min

[ kN

]

Model IModel II

Model III

Model IV

GIP Model:τ f = τ m

GIP Model: τ f = 6.81 MPaG f constant with su = 2.5 mm

GIP Model: τ f = τ m

without strain limitation on FRP

Fig. 11. FRP shear contribution versus FRP bar spacing.

1576 A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577

maximum loads predicted by the different models is less than 5.0%.Finally, it is interesting to note that the maximum shear contribu-tion provided by the FRP strengthening depends on z1 (and henceon lemb,tot) such as for the GIP model but in a different way: in par-ticular, for z1 = 50 mm the shear contribution is minimum while,for the same configuration, it was maximum according to the GIPmodel.

Table 1 reports the predicted FRP shear contribution for the dif-ferent z1 values and bond–slip models. As the crack location rela-tive to the bar is a priori unknown, the FRP shear contribution tobe used for design should be the minimum, VFRP min. Table 1 showsthat the four different bond models give very close values ofVFRP min, the difference between the maximum and minimum pre-dictions being about 5%. Using the GIP model with a shear stressat failure sf equal to the local bond strength sm would clearly giveunconservative results. This is expectable, as the bond stress distri-bution in the bars intersected by the shear crack is highly non-lin-ear, resulting in an average bond stress at failure which issignificantly lower than the local bond strength of the joint. Intro-ducing a limitation to the maximum strain in the FRP bars lowersthe value of VFRP min, which indicates that the shear capacity is con-trolled by the attainment of the limiting strain in the FRP. Finally, ifthe GIP model is used with the reduced value of sf (obtained byimposing that the fracture energy be equal to that of the otherbond models for a given ultimate slip), the predicted VFRP min is veryclose to the value predicted by using the numerical approach.Moreover, this value is not affected by the introduction of a limittensile strain for the FRP.

Fig. 10 indicates that, for the case analyzed herein, the FRP andthe steel contributions to the shear capacity can be added, as thepeak of the FRP contribution is reached when the steel stirrupsare already yielded. This further justifies the use of the GIP modelwhere FRP and steel contributions are considered additive. Theassumption of additive contributions is more likely to be validwhen the fracture energy of the FRP system is larger. For example,the bond fracture energy of FRP laminates externally bonded toconcrete is significantly smaller than that of NSM FRP reinforce-ment, and hence laminates are more likely to debond before thesteel stirrups have yielded, so that the total shear capacity is lessthan the sum of the two maximum contributions of steel andFRP [15].

Fig. 11 plots the FRP shear contribution versus the FRP bar spac-ing comparing the results from the models I–IV with those ob-tained with the GIP model. This figure confirms the trendsalready described above for different spacings of the FRP reinforce-ment. In particular, the GIP model with sf giving equal fracture en-ergy always yields predictions which are close to those of thenumerical approach, and always on the safe side. The validity of

this finding over a wide range of material, geometry and bond-re-lated parameters and its possible limitations should be the subjectof further research.

5. Conclusions

From results of the reported investigation, the following conclu-sions can be drawn:

– two models have been developed and implemented to computethe debonding failure load of RC beams shear-strengthened withNSM reinforcement, assuming that debonding involves eachNSM bar separately. The GIP model uses simplifying assump-tions that allow a closed-form equation to be found for theFRP shear contribution, and hence is simple to use for designpurposes. The model accounting for bond–slip and evolution ofbond stresses during opening of the critical shear crack is moreaccurate but more onerous to use;

– for a given fracture energy, the shape of the local s–s modelslightly influences the expected FRP shear contribution, with amaximum difference in results limited to about 5%;

– using the GIP model with a properly reduced value of designbond strength (namely, the value computed by maintainingthe fracture energy and the ultimate slip of the real s–s model)can offer the same accuracy of the second model while lendingitself to prompt design use.

Further research is needed to extend the conclusions of thepresent study to a wide range of cases, to compare predictions ofboth models to experimental results, and to tackle possible

A. Rizzo, L. De Lorenzis / Construction and Building Materials 23 (2009) 1568–1577 1577

debonding failure modes involving more NSM bars together andthe lateral cover of the steel stirrups.

Acknowledgement

This investigation was supported by the RELUIS project.

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