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Review article

Bond and size effects on the shear capacity of RC beams without stirrups

Jacinto R. Carmona, Gonzalo Ruiz

E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

a r t i c l e i n f o

Article history:

Received 26 July 2013Revised 27 December 2013

Accepted 31 January 2014

Keywords:

Shear strengthSize effectBond effect

a b s t r a c t

The paper presents a model which allows studying the influence of bond between the reinforcing barsand the concrete matrix and the size effect on the evaluation of shear strength in reinforced concrete

beams without stirrups. The formulation assumes that shear failure is caused by the propagation of flex-ural cracks. When the crack length reaches a certain depth, the so-called critical depth, the section col-lapses. This depth depends on the position of the section being studied, the external load, the beamboundary conditions and geometry. Non-linear concepts of Fracture Mechanics are used to model con-crete behavior in tension during the crack propagation. Size effect is reproduced through Bazants law(Bazant and Pfeiffer, 1987). Bond slip is considered by a rigid-plastic bondslip curve. The results ofthe model can provide an understanding of the influence of steel-to-concrete bond on the shear strengthand the size effect exhibited in test results and its asymptotic behavior. All these topics are of the utmostimportance to concrete technology, yet they are not satisfactorily dealt with by construction codes.

2014 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452. Modeling shear crack propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1. Plain concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2. Reinforced concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3. Failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504. Crack shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515. Results and discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1. Model response and experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2. Size effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3. Bond effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1. Introduction

The evaluation of the load that causes shear failure due to diag-onal tension in RC elements is a problem that has yet to be satisfac-torily resolved. A consensus over a mechanical model, which maysimply and reasonably explain the behavior of the RC elementsexperiencing this type of failure and the influence of the bond be-tween concrete and steel, has not been reached as of the presenttime. Nevertheless, the interest in this subject is apparent from

the hundreds of publications written about it over the last fiftyyears. Moreover, the ability of the proposed models to predicthas gradually increased. The models to determine the failure loaddue to diagonal tension in elements without stirrups have evolvedfrom hypotheses based on empirical statistics[13]to truss mod-els based on plasticity[48].

A new perspective for analyzing the problem was introduced byReinhardt in the 1980s[9]. He stated that models and formulas fordetermining diagonal traction failure should be based on FractureMechanics (FM). Indeed, the brittle nature of diagonal tension fail-

http://dx.doi.org/10.1016/j.engstruct.2014.01.054

0141-0296/2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +34 616021731.

E-mail address:[email protected](J.R. Carmona).

Engineering Structures 66 (2014) 4556

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ure, together with the size effect observed in tests [10,11]and thefact that failure is associated with crack propagation through theconcrete element [12], suggest that shear failure can be studiedthrough theories generated within the framework of FractureMechanics. In fact, several models have been proposed to increase

our understanding of the causes and variables that intervene in theproblem. Special emphasis must be placed on the works byGustafsson[13] and Hillerborg and Gustafsson[14], who appliedthe cohesive crack model, formulated by Hillerborg et al. [15]tothe study of diagonal tension failure and the works about size ef-fect on shear failure of RC beams performed by Bazant and Yu[16,17]. Different formulations have also been proposed withinthe field of Linear Elastic Fracture Mechanics for the purpose ofproviding simple mechanical models that do not need to use finiteelements. Among them, we highlight the works by Jenq and Shah[18]and by Carpinteri et al.[19]. Both models introduce the ideathat failure is caused by a flexural crack that propagates in thebeam, which demonstrates the existing relationship between thefailure due to bending and diagonal tension (shear). This relation-

ship is disregarded by most standards since both phenomena areaddressed independently.

Another important landmark in the study of shear failure wasthe work done by Kani and Wittkopp[20]. Based on experimentalobservations, they noted that the type of failure developed in rect-angular RC beams without web reinforcement was strongly relatedto the shear span to depth ratio. They detected a minimum in thecurve shear strength against slenderness, the well known Kanisshear valley[10]. Related to this effect, Niwa [21,22], in a finite-ele-ment numerical study, reported the existence of a minimum in thefailure shearing force when the crack initiation position is changedalong the shear span. The same conclusion was drawn by Carpinte-ri et al.[23]. They used an analytical LEFM model to show failuremode transitions by varying the controlling non-dimensional

parameters. They demonstrated that shear cracks initiating aroundthe middle of the shear span need less load to grow in an unstable

manner than cracks near the beam midspan (flexural cracks) orthan cracks near the support. The presence of stirrups avoids theunstable crack growth and changes the collapse mode from shearto bending. Thus, the connection between crack propagation andelement collapse is key to model the shear strength.

Crack opening and propagation depend on bond conditions. Forthat reason, an understanding of its effect on crack propagation isnecessary to evaluate the failure load. Within the framework ofFracture Mechanics, the influence of bond between concrete andsteel has been studied for lightly-reinforced beams failing by bend-ing [2427], but there are no references of previous analyticalmodels implementing the effect of the influence of steel-to-con-crete bond on the shear behavior. Thus, there was a need for astudy that would cover such a topic.

Therefore, Fracture Mechanics provides a sound theoreticalframework in order to study the failure by diagonal tension (shearfailure), its size effect and the influence of bond properties. A cor-rect analysis of size effect is important, since most of the tests usedto validate shear strength of beams were performed on small spec-

imens reinforced with adherent bars. Therefore, they cannot be di-rectly used to derive empirical formulas because the shearbehavior is bond and size dependent. Likewise, the new types ofhigh-performance concrete usually display brittle fracture behav-ior and, thus, they require theories that are able to model suchbehavior.

The formulation that we develop in this paper coincides withthat of Jenq and Shah[18]and Carpinteri et al. [19]in assumingthat diagonal tension failure is caused by a propagation of flexuralcracks. Beam failure occurs when a flexural crack reaches a certaindepth, which we callcritical depth[28]. It depends on the crack po-sition and on the boundary and loading conditions. This failure cri-terion is based on experimental observations[12]and on resultsobtained with analytical models[19]. In this regard, it has been ob-

served that cracks initially progress in a stable manner [19]. Whenone of them reaches its critical depth it becomes unstable, which

Nomenclature

a non-dimensional horizontal distance from the supportto the crack tip

a0 non-dimensional initial crack mouth positionbH Hillerborg brittleness numberg non-dimensional bond

c non-dimensional compression force depthk shear span slenderness ratiol trajectory exponentxc crack opening at the reinforcement levelxc non-dimensional crack opening at reinforcement levelrs reinforcement stressrs non-dimensional reinforcement stressry minimum between the yielding and sliding stress for

the barssc bond strengthn non-dimensional crack depthncrit non-dimensional critical crack depthf non-dimensional reinforcement coverAs bars cross section areab beam width

c reinforcement coverEc concrete Youngs modulusEs steel Youngs modulusfc concrete compression strengthfct concrete tensile strength

GF concrete fracture energyh beam heightl shear spanlch concrete characteristic lengthMc bending moment during crack progress

Mc non-dimensional bending moment during crack pro-gress

P reinforcement reactionpe bar perimeterTc traction at crack frontTc non-dimensional traction at crack frontTs bars reactionTs non-dimensional bars reactionV applied shear forceVc shear for concrete crack propagationVc non-dimensional shear for concrete crack propagationVF shear for element failureVt shear for crack propagationVt non-dimensional shear for crack propagationx crack tip horizontal position

x0 crack mouth horizontal positiony compression force depthq reinforcement ratioc non-dimensional depth of the compression forcey depth of the compression force

46 J.R. Carmona, G. Ruiz / Engineering Structures 66 (2014) 4556
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means that the crack length increases thereby consuming energystored solely in the specimen with no need for additional externalenergy to propagate the crack. As the model is based on FractureMechanics concepts, it reproduces the size-effect observed in theexperiments and the influence of the variables that govern the fail-ure. It also introduces the effect of the bond between the reinforc-ing bars and the concrete matrix on shear behavior, which provesto be influential in allowing or restraining crack propagation.

The papers is structured as follows. The subsequent section de-scribes the assumptions made as to model shear crack propagation

both in plain and reinforced concrete. Section3deals with the fail-ure criterion that is adopted in the model. A discussion on theshape of the crack is included in Section4, followed by a detaileddescription of the response of the model plus its validation againstexperimental results, in Section5. Finally, Section6summarizesthe results of the paper and draws several conclusions.

2. Modeling shear crack propagation

2.1. Plain concrete

A three point bending beam (TPB) is deemed to exist where avertical crack develops at a point in the shear span. The different

geometric variables relevant to the problem are displayed inFigs. 1a and 2a. The beam has a depthh, a widthband a shear span

equal tol, (which is the horizontal distance between the load pointand the closest support). The depth of the tensile stresses resultantis represented aszand the depth of the compressive stresses resul-tant as y . All these dimensions can be expressed in a non-dimen-sional way by dividing by the depth h . In this manner, we definen z

has the depth of the tensile force expressed in a non-dimen-

sional form and c yh

as the depth of the compressive force ex-pressed in a non-dimensional form; these parameters have avalue between 0 and 1. We will also consider two additionalparameters, the shear span slenderness, which is defined as k l

h,

and another that indicates the distance of the crack to the beamsupport, a x

l, where x is the horizontal distance from the crack

to the support.The depth of the compressive force, y, is evaluated by the equi-

librium of horizontal forces, seeFig. 1a:

Tc Cc; 1

We assume a rectangular distribution for the compressive stresses.The compressive force is also considered to be situated at the cen-troid of the distribution of compressive stresses so (1) can berewritten as:

Tc 2bhyfc: 2

From Eq.(2)the depth of the compressive force is expressed as:

y h Tc2bfc

: 3

Tensile and compressive forces can be expressed in a non-dimen-sional form by dividing them by the cross section area, bh, multi-plied by the concrete tensile strength, fct.

Tc Tcbhfct

; 4

Cc Ccbhfct

: 5

Thus, the result from expressing Eq. (3)in a non-dimensional formis:

c 1 1

2nfTc; 6

wherenf is the ratio between the concrete compression strength,fcand the concrete tensile strength, fct.

To represent the different non-linear processes located at thefront of the crack, the model assumes a cohesive behavior of thefractured concrete. During the crack development a damage zoneis generated at the front of the crack, along which tensile stressesare transferred between the crack faces during the crack opening.The main consequence of the existence of a non-linear area atthe crack front is the Fracture Mechanics Size Effect, which isdue to the release of the stored energy of the concrete element into

the fracture zone. Thus, in this work it is considered that the Frac-ture Mechanics size effect is the most important source of strengthscaling, as it is proposed in reference [29]. The force needed forcrack growth at a given crack depth and accounting for non-lineareffects is represented through Bazants law[30], which is well-known and widely-disseminated among researchers in concretefracture mechanics. Thus, it is considered that Tcfollows Bazantslaw, since it is a non-dimensional nominal strength that incorpo-rates the cohesive stresses in the crack. Therefore, it can be writtenas:

Tc Bffiffiffiffiffiffiffiffiffiffiffiffiffi1 bH

b0

q ; 7

where bH is the Hillerborgs brittleness number[29], which is de-fined as the ratio between the depth of the beam,h, and the mate-

(a)

(b)

Fig. 1. Bending analysis: (a) plain concrete section and (b) reinforced concretesection.

(a)

(b)

Fig. 2. Beam geometry: (a) plain concrete and (b) reinforced concrete.

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Re-writing Eq.(16)in a non-dimensional form:

Mt Mc M

s

c n2ffiffiffiffiffiffiffiffiffiffiffiffiffi1 bH

b0

q qrs c f: 17Since the reinforcement introduces a new force in the cross section,the equation for the horizontal equilibrium of forces is:

Cc TcTs: 18

The non-dimensional depth for the compressive force is now:

c 1 1

2nfTc qr

s

: 19

Finally, using Eq.(13), the shear that leads to equilibrium for a givencrack depth can be expressed as:

Vt Vtbhfct

1

akc n2ffiffiffiffiffiffiffiffiffiffiffiffiffi1 bH

b0

q qrsc f264

375: 20

Note that Eq.(20)indicates that the shear borne by the beam is thesum of two terms. The first one depends on the concrete properties,

whereas the other one depends on the reinforcement ratio, the ten-sion in the steel bars and the concrete cover. The sum of these termsis multiplied by another term which takes into account the slender-ness of the beam and the position of the crack front. All of the valuesin(20)are known except for the steel tension.

In order to evaluate the steel tension during crack growth, weneed to consider an additional equation by enforcing the compat-ibility of displacements between concrete and steel, which impliesthat the crack opening is equal to the stretching of the reinforce-ment bars. It is written as:

xc2

Dl; 21

wherexcis the crack opening and Dlis the stretching of the steelbar. It is assumed that the traction force of the reinforcement is

equal on both sides of the crack, as shown inFig. 4and Eq.(21).Other compatibility equations could be considered, for example,based on the Navier hypothesis or depending on the crack opening,as done in[25].

Eq.(21)can be expressed in a non-dimensional way by dividingboth terms by the depth of the beam:

xc2h

Dl

h )

xc2

Dl: 22

The non-dimensional crack opening, xc, can be evaluated by theexpression given by Tada et al. [35]. In order to take into accountthe concrete cover, an additional term, 1 f

n, has been introduced:

xc2

xc2h

12 akVtfctEc

nfn 1 f

n ; 23

wherefnis:

0:76 2:28n 3:87n2 2:04n3 0:66

1 n2: 24

The stretching of the bar (Fig. 4) can be expressed as:

Dl

Dl

h

r2sAs2scEspeh

rs 2 f2ct2scEs

Aspeh

; 25

wherescis the bond strength between steel and concrete, which isconsidered constant along the adherence length, and pe is the barperimeter. By substituting Eqs.(23) and (25)in Eq.(22), the non-dimensional tension on reinforcement can be expressed as:

rs 2

24Vtakg2bHnfn 1

f

n

; 26

wheregis the non-dimensional bond strength defined by Ruiz[25],which can be written as:

g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinEscfct

pechAs

s ; 27

wherenEis the ratio between the elastic modulus of steel and thatof concrete. Note thatgdepends not only onscbut also on the ratiopech=As, i.e.g is inversely proportional to the square root of the bardiameter,

ffiffiffiffi

/p

. Thus, strong bonds can be achieved by using thin bars(or fibers) with a regular bond strength.

For the sake of simplicity, elasticperfectly-plastic behavior forthe steel of the reinforcement bars is chosen in this work. Thus,once the tension in the reinforcement reaches the yield strength,fy, it remains constant during crack propagation. Tension in thereinforcement bars will then be given by:

rs 24Vtakg

2bHnfn 1 fn

h i1=2if . . .

1=2 < fyfct

fyfct

otherwise:

8>>>:28

In order to evaluate shear force and tension in steel bars, we finallyhave a system of three equations, Eqs.(19), (20) and (28), which canbe solved analytically. When the reinforcement traction reaches theyield strength, the shear strength is obtained just from Eqs.(19) and(20). It should be noted that the proposed model is valid for low andmedium reinforcement ratios. For high ratios, the failure is causedby excessive compressions below the load bearing point, and study-ing this type of failure is beyond the scope of this paper. Finally, weassume that a crack may form at any point along the shear span,which is especially true in the case of ribbed bars.

InFig. 5bd, the model response is shown for the midspan sec-tion of the beam in Fig. 5a. A beam under three point bending ismodelled.Fig. 5a also shows material properties. It is assumed thatcracks grow in a vertical manner. InFig. 5b the x-axis represents

the non-dimensional depth,n, and they-axis, the non-dimensionalshear strength during crack growth, Vt. A curve is determined foreach reinforcement ratio. As the reinforcement ratio increases,the shear for steel yielding increases.

It is well-known that crack growth may present stable or unsta-ble behavior. When crack growth is stable, an increase in crackdepth requires a load increase. Conversely, unstable crack growthis accompanied by a load decrease. If we observe the curve forq 0:78% inFig. 5b, an unstable branch is observed after crack ini-tiation. When the crack reaches the reinforcement an increase inthe load is detected and, afterwards, another unstable branch oc-curs. There exists a minimum for Vtatn 0:6 beyond which crackgrowth becomes stable. From this point on, shear increases untilthe reinforcement yields and the flexural capacity of this beam sec-

tion is reached. The behavior obtained with our model is analogousto the one described by Carpinteri using the shear version of hisFig. 4. Compatibility of displacements.

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Bridged Crack Model [36]. Nevertheless, the model proposed intro-duces concepts of cohesive theories in order to analyze size effect

and it allows studying the influence of bond between concrete andsteel during the crack growth.InFig. 5c and d, thex-axis represents the non-dimensional crack

opening at the reinforcement level,x , and the y-axis correspondsto the non-dimensional shear during crack growth, Vt. As the rein-forcement ratio increases, the shear also increases for the samecrack width.Fig. 5d zooms in onFig. 5c for low values of the crackwidth. In the case of plain concrete (q 0), it can be observed thatshear rapidly decreases to zero and that cracks never advance in astable manner.

The methodology presented here analyzes shear load during thepropagation of cracks in the case of three point bending beams.Beams subject to uniformly distributed loads and other load andboundary conditions, may be studied with this methodology there-

by changing the equilibrium equations in order to achieve a newrelationship between shear and applied load. Moreover, the model,conceptually simple as it is, shows that the equilibrium shear for agiven crack depth and in a given position along the shear span isgoverned by nine non-dimensional parameters. These parameterscan divided into three categories.

Geometrical parameters:

nz

h; k

l

h; a0

x

l; f

c

h: 29

Parameters referring to size and concrete properties:

bH h

ch; nf

fcfct

: 30

Parameters referring to steel properties and bond between steel andconcrete:

qAsAc

; g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinEscfct

pechAs

s ; fy

fyfct

: 31

Eq. (20) demonstrates that these three parameters are com-bined through the product qrs . Before yielding, the response ofthe model depends on the product qg, whereas it is qfy that gov-erns the second term in Eq.(20)after the reinforcement yields.

3. Failure criterion

Eq.(20)calculates the shear force for crack propagation, Vt, butshear failure, VF, will occur only for a definite crack depth. Thisdepth will be referred to henceforth as critical depth, ncrit. A crite-rion to determine ncritcan be derived from experimental observa-tions, namely those made by Carmona et al. [12]. Thisexperimental program was designed so that only one single

mixed-mode crack generated and propagated through the speci-men, as opposed to the usual dense crack pattern found in mostof the tests reported in the literature. In Fig. 6, we illustrate twoof the results that will help to explain the failure criterion.

Fig. 6a shows the crack pattern in two of the tests. The marksand figures on the sketch refer to the corresponding points in thePdcurves, as shown inFig. 6b, and to the load in kN that the beamwas resisting when the crack tip reached that position. During thecrack progress, a change in the nature of the crack propagation wasobserved and a subsequent unstable crack branch began leading tothe beam failure. This phenomenon was associated with the so-called diagonal tension failure. This change in the nature of crackpropagation can be observed in point C of beam L40 and point Dof beam L80. These points shown inFig. 6a are approximately lo-

cated on the line that joins the loading point with the point wherethe reinforcement reaches the support. This line coincides with an

5000mm

(a)

Materials properties

Concrete

f (MPa)c f (MPa)t G (N/m)F E (GPa)c

30.0 3.2 120.0 33.0

Ribbed steel

f (MPa)y 0,2% (MPa)cE (GPa)c520 200 2.0

200

Dimensions in mm

225220212)%22.1()%0.0( (0.78%)(0.28%)

400 350

Reinforcementratio

Notation

cross section

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1.0

=0.00%

=0.28%

=0.78%

=1.22%

V*t

0

0.1

0.2

0.3

0 0.02 0.04 0.06 0.08 0.1

V

*

t

0

0.025

0.05

0 0.0005 0.001

V

*

t

(b)

(c) (d)

=0.00%

=0.28%

=0.78%

=1.22%

=0.00%

=0.28%

=0.78%

=1.22%

c* c*

Fig. 5. Model response: (a) geometry and properties of the materials; (b) Vtncurves; (c)Vtx

ccurves; and (d) detail Vtx

ccurves.


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ideal strut that connects the loading point to the support. Theseexperimental observations were also assumed in an analyticalmodel proposed by Carpinteri et al.[19]. Therefore, for three pointbending flexure, the critical depth is defined by the line connectingthe loading point to the point where the reinforcement reaches thesupport. This definition was previously proposed by Zararis andPapadakis[28]. Mathematically, it can be expressed as follows:

ncrit f a1 f: 32

This failure criterion is associated with three point bending flexure,but can be easily generalized for any boundary and load conditions.

For example, for a simply-supported beam subjected to a uni-formly-distributed load, the bending moment diagram variation is

parabolic with a maximum at midspan. The critical crack variationis also parabolic. At the maximum bending moment point, the crit-ical crack is equal to the beam depth, and in the support it is equalto the cover; see Fig. 7b. Therefore, we propose that the critical

depth be related to the bending moment diagram: For the maxi-mum bending moment position, the critical depth is equal to thebeam depth (ncrit 1), and when the bending moment is equal tozero, the critical depth is equal to the reinforcement concrete cover.In between, the critical depth is proportional to the value of thebending moment at that position.

4. Crack shape

So far we have assumed that cracks grow in a vertical mannerfrom the initiation point. In reality, cracks have curved trajectoriesapproximately following the lines indicating the direction of max-imum compression. This effect means that the initial abscissa ofthe crack,a

0, is not the same as the abscissa of the crack tip when

it reaches critical depth, a. In order to consider this mismatch,

L80-9-2

14

15.25.7

615.5

15

12

12

15

15.6

15.6

1514.514

13

16.4

D

B

E

B

AC

D

0

4

8

12

16

0 1.5 3

(mm)

P(

kN)

L80-9-2

F

E

D

BA C

(b)

(a)

L40-7-29.4

9.58.6.1

9.85.

9.79.5A

0

2

46

8

10

12

0 1 2

(mm)

P(

kN)

L40-7-2

D

CB

A

9

failure

failure

failure

failure

Fig. 6. Failure criterion, experimental results: (a) crack patterns and (b) Pdcurves.

crit

crit

(a)

(b)

P

q

Fig. 7. Failure criterion: (a) three point bending and (b) uniformly distributed load.

crit

0

Fig. 8. Crack shape.

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equations defining trajectory patterns can be used, as the onesshown below, which were proposed in reference[19]:

af; n a0 0 6 n 6 f

a0 nf1f

l1 a0 f 6 n 6 1:

( 33

The above formula assumes a straight trajectory from the initiationpoint to the reinforcement and a parabolic trajectory which reachesthe load-bearing point. Through Eq.(33), a relationship between thecrack depth n and the initiation point, a0, is determined. For three

point bending, exponentl is equal to:

l 1

1 a0: 34

This equation was obtained from the experimental tests performedby Carpinteri et al.[37]. The failure criteria for curved cracks remainthe same as for straight cracks: Once the critical depth is reached,the element fails, (seeFig. 8).

5. Results and discussion

5.1. Model response and experimental validation

In this section it will be shown how the value of the initial crack

position,a0, affects the shear strength. A beam under three point

bending is modelled. Fig. 9a shows the geometry and materialproperties. In Fig. 9b the x-axis represents the non-dimensionaldepth, n, and they-axis the nondimensional shear strength duringcrack growth, Vt. A curve is determined for each crack initiationposition. The black part of the curve describes the shear resistancevariation during the crack process until the critical depth isreached (seeFig. 9b). The point where the crack front reaches thecritical depth indicates shear strength.

As it can be observed inFig. 9b, failure occurs prior to reinforce-ment yielding (brittle failure, diagonal tension) for cracks whichform close to the support, whereas failure occurs after the rein-forcement yields (flexural failure) for the ones farther off the sup-port. It must be noted that once reinforcement has yielded therecan also be brittle failure, as shown by Rodrigues et al. [38]. Never-theless, when the crack is located under the loading point, as thecritical depth is equal to the depth, a brittle failure caused by diag-onal tension cannot occur.

To validate the response of the model, we have compared theresults obtained with it to those obtained by a recent experimentalprogram performed by Carpinteri et al.[37]. A total of sixteen geo-metrically similar beams reinforced with 4 different reinforcementratios were tested (4 beams for each ratio), see Fig. 10a. In thisexperimental program, the initial position and the shape of criticalcracks were studied. The residual bond strength between concreteand steel was not measured in the experimental program and,therefore, it has been estimated using the formulation establishedin the Model Code (CEB-FIB), seeFig. 10a.Fig. 10b shows the crit-ical crack depth considered and the notation in order to facilitatethe understanding of the plots inFig. 10c.

Fig. 10c shows the comparison between the theoretical andexperimental results. Thex-axis corresponds to the initial positionof the cracks, whereas the y-axis represents the shear when thecrack reaches the critical depth (shear strength), VF. In order tofacilitate the comparison, selected crack patterns correspondingto one of each kind of the tested specimens are drawn at the bot-tom of the figure. The continuous lines represent the models re-

sponse while the symbols represent the experimental results.In order to describe the models response, we are going to focus

on the curve corresponding to a beam reinforced with 2/12, start-ing from the section corresponding to the load application point(a0 1). For the crack under the loading point, the rebar yields be-fore the crack reaches the critical depth and the failure is due toflexure. Distancing the initiation of the crack from the loadingpoint, we observe that the shear strength increases, since the rein-forcement gets more load and may even yield before the criticaldepth is attained. For a certain point, aY, a maximum value onthe curve is detected. This maximum indicates the point at whichthe rebar reaches its yield strength exactly at the same time thatthe crack reaches its critical depth. For values ofa0 lower thanaY, the rebar has not yielded when the crack tip reaches the critical

depth and the beam failure occurs by diagonal tension. The shearstrength decreases until reaching a minimum for a certain initia-tion point, situated in our case at a0 0:4. This minimum has alsobeen found in experimental results performed by Kim and White[39,40]. For cracks with an initiation point closer to the support,the shear strength starts to increase, although the critical depthis low. It should be noted that the actual shear strength of the beamis the smallest shear found by varying a0.

The shape of the curve obtained with the model, demonstratinga minimum in the central part of the shear span, coincides with thedescription proposed by Kani and Wittkopp[20]. In the zone nearthe support, we find an area where the failure is produced due tothe yielding of the rebars (flexure); but as we move away fromthe support along the shear span, the failure occurs during the

development of a crack (diagonal tension).

P

0

crit

1400mm

1200mm

100

Dimensions in mm.

28(0.50%)

200 180

Reinforcementratio

Notation

(a)

Materials propertiesConcrete


49.4 3.2 111.5 33.1Ribbed steel

f (MPa)y 0,2% (MPa)cE (GPa)c519 200 1.1

0

1

0

0.5

1.0

1.5

0 0.2 0.4 0.6 0.8 1.0

0=1.00=0.80=0.60=0.4

0=0.2

V = V ( )critF*

V*

t

(b)

1.00

t

*

Fig. 9. Model response: (a) geometry and properties of the materials and (b) Vtncurves.


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Fig. 10c shows that for low reinforcement ratios, e.g. 1/8, thefailure takes place in sections near the loading point, where thesteel reaches the elastic limit before the crack grows to the criticaldepth. Upon increasing the reinforcement ratio, the critical sectionmoves away from the point of application of the load to positionswhere the steel does not yield. The experimental tendency is cap-tured by the model and even the shear loads obtained coincidequite reasonably with the experimental ones. Moreover, the modelexplains the valley in the shear resistance along the beams lon-gitudinal axis. In the results obtained for the largest reinforcement

ratio (2/20), the differences between the model and the experi-ments are conditioned by the type of failure, since in highly-rein-forced beams the failure is produced by excessive compressionand not by diagonal tension.

5.2. Size effect

The sensitivity of the models response to size is validatedagainst experimental results on notched reinforced concrete beamsof several sizes performed by Carmona et al. [12],Fig. 11. The x-axis represents the size in terms of the Hillerborgs brittlenessnumber and the y-axis, the non-dimensional concrete shear forcrack propagation, Vc. The model follows a trend similar to thatof the tests analyzed.

Fig. 12shows the models response when varying the elementsize.Fig. 12a displays the size effect when the reinforcement ratio

is increased. The x-axis represents the size in terms of the Hiller-borgs brittleness number and the y-axis, the non-dimensionalshear strength, Vt. A curve is determined for each reinforcementratio. When there is no reinforcement, the curve follows Bazantslaw. As the reinforcement ratio is increased, the size effect changes.For a given reinforcement, the curve is similar to that of plain con-crete only for small values ofbH. As size increases, the contributionof the reinforcement grows. A minimum in the non-dimensionalshear strength is reached and, henceforth VFgrows with size until

*VF

18

28

212

220

1400mm

1200mm

100

Dimensions in mm

18 22021228)%41.3()%52.0( (1.13%)(0.50%)

200 180

Reinforcementratio

Notation

(a)

0

0.5

1

1.5

0

tests

model results

18

28

212

220

Diagonal tension

failure Flexural failure

18-3

212-2

28-3

220-2

llll

(c)

P

crit

0 0.25 0.5 0.75 1

0

Materials properties

Concrete


49.4 3.2 111.5 33.1

Ribbed steel

f (MPa)y 0,2% (MPa)cE (GPa)c519 200 1.1

(b)

0 0.25 0.5 10.75

Fig. 10. Model response: (a) beam geometry and (b) experimental results vs. model, VFa0 curves.

=0.026% test =0.026%test =0.013%=0.013%

=20=0.5 n =9.5ff =148y* =15

V *F

0.05

0.1

0.15

0.2

0.5 1 2

H

5

Fig. 11. Size effect: experimental results vs. model.

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the reinforcement bars yield. Of course, afterwards VF becomesinsensitive to size. A similar trend in size effect for reinforced con-crete beams was obtained by Ruiz[25]

Fig. 12b compares the response for two different values of thecrack initiation position: The first crack starts at the middle of

shear span,a0 0:5 (diagonal tension failure), and the other initi-ates under the load point, a0 1:0 (flexural failure). Thex-axis rep-resents the element size in terms of Hillerborgs brittlenessnumber. The usual range of this parameter in structures is indi-cated with two vertical lines. They-axis represents the non-dimen-sional shear strength. For the crack located at a0 0:5, there existsa strong size effect, i.e. shear strength depends on element size,which is caused by the existence of a fracture process zone onthe crack front. Nevertheless, size effect disappears both for verysmall sizes as well as for large ones. This can be explained withthe ratio between the length of the damage zone generated atthe front of the crack and the beam height. Recall that the lengthof the damage zone is a material characteristic and thus, its valuetends to be constant for a given concrete as size increases, i.e. it

tends to be very short compared with the beam depth for largebeams. In other words, the first term in Eq.(20)tends to be 0 assize increases, and then the equilibrium shear is given only bythe second term. This second term gives a constant value for largeelements because the reinforcement bars yield and thus the nom-

inal stress is constant and size independent. On the other hand, forsmall beams the damage zone tends to occupy the entire beamheight and size effect is greatly reduced. Therefore, the first termin Eq.(20)tends to a constant value as size decreases.

For the crack located at midspan, a0 1:0, shear strength doesnot present any size effect, since the term (c ncrit) in Eq.(20)isequal to zero. This means that non-linear stresses at the crack frontdo not contribute to the equilibrium shear and thus, size effect dis-appears. Moreover, for a completely-developed crack the rebarsyield and so the stress is constant and independent of the size.

The results obtained with the model coincide with the experi-mental observations made by Collins and Kuchma [6]. Theyshowed that the size effect is greatly reduced in beams containingwell-distributed longitudinal reinforcement along the beam

(a) (b)0=0.50=1.0

usual structural

dimensions range

=3

=5n =10f

f =125y*

=0.5%

0.1 1 10 100H

=0.0%=0.1%

=0.25%=0.5%=1.0%

H

0.001 0.01 0.1 1 10 100 1000

V*F

=3 0=0,5

0.001

0.01

0.1

1

0.01

0.1

1

V *F

Fig. 12. Size effect: (a) influence of reinforcement ratio and (b) influence of crack initiation position.

(a) (b)

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

V*t

(c)

0

0.5

1.0

1.5

0 0.25 0.5 0.75 1.0

0

V*F

(d)

100

112

200

0

0.2

0.4

0.6

0 0.0025 0.005 0.0075 0.01

V*t

0

0.0025

0.005

0.0075

0,01

0 0.2 0.4 0.6 0.8 1

=5=15=50=150

=5=15=50=150

=5=15=50=150

=33=50

=10

=150

c*

c*

Fig. 13. Bond influence: (a) Vtncurves; (b) Vtx

ccurves; (c)x

cncurves; and (d) V

Fa0 curves.


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height. The presence of reinforcement avoids the unstable crackpropagation and thus changes the type of failure from diagonaltension to flexure, where size effect is less noticeable.

5.3. Bond effect

The influence of bond between concrete and reinforcing bars isintroduced in the model (Eq. 28) using the non-dimensional bondstrength, g (Eq.27), defined by Ruiz[25]. As already mentioned,this parameter depends not only on sc, but on the ratio pech=As,i.e.g is inversely proportional to the square root of the bar diame-ter,

ffiffiffiffi

/p

. This section addresses how bond characteristics affectshear strength and size effect.

Fig. 13 shows the model response for variations in the bondcharacteristics between concrete and steel. The assumed rangeforg is set to visualize the bond effect, since the study of asymp-totic or extreme cases helps to understand the influence of bondin the behavior of reinforced concrete elements.

Fig. 13a represents crack depth versus shear. We modelled thebeam inFig. 12a reinforced with 1/12. All the parameters werekept constant except for the bond strength. When bond is in-

creased, the crack depth at which the steel yields decreases. Forlow bond strength conditions (smooth bars), cracks have to devel-op fully before the steel yields. In the case of high bond strength(ribbed bars), steel yielding occurs shortly after the crack crossesthe rebar. InFig. 13b, the x-axis displays the crack width and they-axis, the shear strength. As the bond strength increases thebehavior of the reinforced concrete section is stiffer, since steelyielding is reached with shorter crack widths.Fig. 13c representscrack depth versus crack width. It reveals that bond strength con-trols crack width. Stronger interfaces lead to wider openings ifcompared to the same crack length. Of course, in this case the sec-tion with stronger bond strength resists considerably more shear.

InFig. 13d, thex-axis displays the initial crack position and they-axis represents the shear strength. As bond strength increases,

the value ofaYdecreases and the shear strength increases. In theextreme case that the bond strength has an infinite value, the ele-ment fails by flexure because, for all possible cracks, the steelyields just after crossing the rebar position and therefore, beforereaching the critical depth. In this case, the failure takes place atthe section under load bearing point (a0 1:0)

Fig. 13b also shows that the minimum of the curve moves to-wards the support as the bond strength increases. Low values forthe bond strength lead to low values for the failure loads, evenlower than the load that is required for fracture initiation at thatparticular point, Vfis. This means that the diagonal tension failurewould be very unlikely for beams in which the steel-to-concreteinterface is weak. Indeed, in this case, there would be cracks onlyunder the load-bearing point and there would be neither crack

generation nor propagation along the shear span. Therefore, insuch conditions, shear failure would not occur.

Fig. 14shows the influence of non-dimensional bond on size ef-fect. The mechanical properties in these examples are shown in thegraphic. Thex-axis represents the size in terms of the Hillerborgsbrittleness number and the y-axis, the nondimensional shearstrength, Vt. As bond increases, size effects tend to disappear be-cause reinforcement bars yield at a lower crack depth. The effectof an increase in bond conditions is similar to the increase of rein-forcement ratio.

6. Conclusions

In this study, a new model based on nonlinear Fracture Mechan-ics concepts is introduced to analyze shear strength in reinforcedconcrete beams without stirrups. It identifies the variables thatgovern the failure, including the bond between concrete and steel.The following main points can be drawn from the study:

1. A failure criterion based on crack growth is introduced. When acrack reaches a given depth, which we refer to as critical depth,

beam failure takes place. This critical depth depends on the typeof loading, the boundary conditions and the beam geometry.

2. Shear strength depends on the section position along the shearspan, thereby showing a minimum around the mid-shear span(critical section). This variation is explained based on the rela-tionship which exists between crack propagation and failure.

3. Size effect in shear strength is also reproduced by this approach,since it stems from non-linear Fracture Mechanics concepts. Italso explains how reinforcement ratio, initial crack locationand bond between concrete and steel affect size effect and theirrespective asymptotic behavior.

4. Transition between flexural and shear failure is dependent onreinforcement ratio and steel strength. As reinforcement ratioincreases the critical section moves away from the point of

application of the load to positions where the steel does notyield.5. The bond strength of steel-to-concrete interface is also

accounted for in the model through the parameter g whichdepends not only on sc, but on the ratio pech=As. It has beenobserved that stronger interfaces lead to narrower openingsand shorter cracks for the same applied load. Therefore, shearfailure tends to disappear as bond strength increases becausea stronger bond leads the steel bars to yield before the crackreaches the critical depth thereby causing shear failure.

The presented model can contribute to a better understandingof the nature of shear strength in reinforced concrete elementswithout stirrups. Furthermore, it can be extended to other loadingand support conditions. Finally, expressions derived from thisstudy may be used to improve shear analysis of RC beams in designcodes.

Acknowledgement

Financial support from the Subdireccin General de Proyectos deInvestigacin, Spain, through Grant MAT2012-35416 is greatlyappreciated.

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=0.1=1=10=25

=3

0=0,5

n =10ff =125y*

H

=0.5%

0.001 0.01 0 .1 1 10 100 1000

=500.01

0.1

1

V*F

Fig. 14. Influence of non-dimensional bond on size effect.

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