propagation of a cohesive crack through adherent reinforcement layers

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Comput. Methods Appl. Mech. Engrg. 195 (2006) 7237–7248

Propagation of a cohesive crack through adherent reinforcement layers

Gonzalo Ruiz a,*, Jacinto R. Carmona a, David A. Cendon b

a E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spainb E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad Politecnica de Madrid, 28040 Madrid, Spain

Received 30 July 2004; received in revised form 22 December 2004; accepted 17 January 2005

Abstract

This paper analyzes the propagation of a cohesive crack crossing one or several reinforcement layers by a simple model valid for anyspecimen or loading condition. In this instance the case of reinforced concrete beams loaded at three points is considered. In thisapproach steel bars do not constitute a physical barrier to the propagation of the crack, as concrete continuity is preserved by a com-putational strategy consisting of overlapping both materials in the same spatial position. The model uses cohesive elements to representthe crack and interface elements to simulate the decohesion and shear generated in the steel–concrete debonding process. The resultsgiven by this model are compared to experimental results on rectangular and T-shaped beams reinforced by bars arranged in one orseveral layers. The model closely follows the experimental trends when changing the parameters that control the type of fracture; suchas the steel ratio, the bond strength, the position and arrangement of the bars, the size of the specimen and the shape of the beam cross-section. In spite of its simplicity, this model can be useful when studying local fracture and decohesion phenomena wherever they maytake place within a reinforced or prestressed concrete structure.� 2005 Elsevier B.V. All rights reserved.

Keywords: Fracture of structural concrete; Cohesive models; Interface deterioration

1. Introduction

For most ordinary beam design, the bending failure of reinforced concrete beams is ductile, but in practice it may be thatbeam dimensions are dictated by criteria other than strictly mechanical ones and then beams may require very little rein-forcement for a strict application of ultimate strength design (yield of steel and no tension for concrete). When the tensilestrength of concrete is taken into account, the cracking strength of the beam may be greater than the ultimate strength,which can lead to brittle behavior, i.e., once the load reaches the cracking strength collapse is instantaneous because thisload is greater than the ultimate (plastic) strength. Indeed, the risk of brittle behavior has fostered a detailed study on theprocesses of concrete fracture and subsequent load transfer between the matrix and the reinforcing steel bars (e.g. see [1]).

The models developed to study this problem [2–14] focus on the example of a crack that propagates through a reinforce-ment layer and that, at opening, causes the steel bars to be pulled-out. The main difficulty consists in modeling the prop-agation of the crack through the reinforcement layer, as such a layer—in 2D models—constitutes a discontinuity that stopsthe crack advance. Modeling the steel–concrete interaction is not simple either, since the pull-out process produces damageat the interface and it may also generate secondary fracture processes within the concrete bulk [15]. Some of the aforemen-tioned models solve both problems by substituting the rebars for closing forces applied at the crack lips whose intensityrelates to the strength of the reinforcement and of the interface [5,12]. Some authors model the steel bars explicitly, howeverthey only account for the extreme cases of perfect or null adherence [4].

0045-7825/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2005.01.029

* Corresponding author.E-mail address: [email protected] (G. Ruiz).

mailto:[email protected]

7238 G. Ruiz et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 7237–7248

This paper presents a model that enables propagation of the crack through the reinforcement as well as interface dete-rioration. We adopt a computational strategy that consists of overlapping both materials in the same spatial position. Con-crete is modeled as a continuum, whereas steel is overlapped and connected by interface elements to nodes occupying thesame initial position [16–18]. Concrete continuity allows crack propagation through the reinforcement and interface ele-ments transmit shear stresses depending on the relative displacement between the two materials. As a first approach tothe problem, we use a rigid-perfectly plastic adherence law, without accounting for more refined effects such as, for exam-ple; longitudinal cracking along the rebar due to the dilation caused by the bar protrusion. Nevertheless, the model canimplement any adherence law. Indeed, we try with the one given by the Model Code [19] so as to check how the adherencelaw influences the global beam behavior.

The model is validated against fracture tests on reinforced beams in flexure [20–22]. The numerical results closely mirrorthe experimental data on rectangular and T-beams of several sizes, reinforced with ribbed or smooth bars arranged in oneor two layers. It is noteworthy that the numerical tests reproduce the experimental size effect, which confirms the feasibilityof cohesive models to deal with concrete fracture. The strength of the reinforcement; which combines the mechanical prop-erties of the steel and of the interface, and the number of layers, also influences the response of the model.

This paper is structured as follows. Section 2 outlines the problem and explains the hypothesis relative to the materialsbehavior and to the mechanical model that we propose. Section 3 focuses on the details of the numerical implementation andvalidates the numerical outcome with experimental results from reinforced beam tests. Finally, Section 4 summarizes thework and gives some insights into the development of fracture processes in structural concrete based on the model results.

2. Layout of the problem

We want to model the collapse of a reinforced concrete beam when subjected to three point bending (Fig. 1a) accountingfor the fracture properties of concrete as well as the ongoing deterioration of the interface between the concrete and the steel.

2.1. Hypothesis relative to the materials behavior

Fracture of concrete is represented by means of a cohesive model. A cohesive crack initiates when and where the max-imum principal stress reaches the tensile strength of the material, ft (Fig. 1b). The crack is perpendicular to the direction ofthe major principal stress and opens while transferring stress from one face to another. For monotonic mode I opening, thestress transferred is perpendicular to the crack faces and is a single function of the crack opening:

r ¼ f ðwÞ andf ð0Þ ¼ ft;

f ðwÞP 0.

�ð1Þ

The function f(w) (sketched in Fig. 1c) is known as softening function or cohesive law. It completely characterizes themechanical behavior of the crack. The values of f(w) become zero when the crack opening exceeds a critical value wc. Thearea under the softening curve is the work needed to separate the two faces of a unit crack surface, i.e., the specific fractureenergy GF. Another important parameter is the characteristic length, ‘ch, which is used to define the intrinsic brittleness ofthe cohesive material [23], and is defined as

‘ch ¼EcGF

f 2t

; ð2Þ

where Ec is the elastic modulus of the cohesive material.

crack width, wO

stress, σ

σ = f (w)

ft

σ = f(w)

σ = 0

linear elastic

ft

GF

wc

w

(a)

(b) (c)

σ

Fig. 1. (a) Reinforced beam subjected to three point bending; (b) cohesive crack; and (c) softening function.

G. Ruiz et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 7237–7248 7239

In this work we assume that the reinforcement ratio is low or, at most, moderate. Under these conditions the fractureprocess starts by developing microcracks in the tensioned fiber near the central cross-section, but very soon a single cracknucleates from this microcracked zone [20]. Therefore, we reach the hypothesis that a single cohesive crack in the centralcross-section represents the fracture behavior of concrete adequately. The behavior of concrete outside the fracture zone istaken as linear elastic, since second order effects due to material non-linearity are small when the reinforcement ratio is nottoo high.

The mechanics of the steel–concrete interface are embedded in a law relating the shear stresses that are transmittedbetween reinforcement and concrete, s, to the relative slip, s, between both materials. The resultant force per unit lengthalong the reinforcement layer number i of the shear stresses acting on the matrix is given by

qi ¼ pisðsÞ; ð3Þwhere pi is the perimeter of the cross-section of all the bars in layer i (Fig. 2a). We assume that all the bars are alike and thusthe same s(s) applies to all of them. The action of reinforcement layer i on the concrete matrix is determined by the dis-tribution of qi along the layer (Fig. 2b). In this work we let the reinforcement follow any stress–strain law, although when itcomes to disclosing the effect of the steel on s and therefore on qi it is convenient to use a simple elastic–perfectly plasticlaw. Likewise, we admit any interface law, but a very simple s(s) will help us to understand the role of adherence in thecollapse process. For that purpose we use a bond-slip law that is rigid up unto the attainment of a critical value of theshear stress sc and which from then on gives a constant response.

The shear stresses acting along the contact with the reinforcement could be replaced by their resultant force acting at apoint inside the concrete, namely the center of gravity of function qi in Eq. (3). This is what Ruiz [14] did to avoid thepropagation of the cohesive crack being stopped due to the presence of reinforcement. Here we solve this problem by mod-eling concrete as a continuum, while the steel reinforcement layers are overlapped and connected to it through interfaceelements.

2.2. Mechanical model

Fig. 3 sketches the mechanical model that we use to study this problem. Specifically it depicts the case of a T-beam rein-forced by two layers of bars. The problem is three dimensional, mainly because of the flow of stresses around the connec-tions with the reinforcement layers and between the body and the head of the beam. We choose a 2D modelization in planestress though, as the 3D effects are very localized under the hypothesis stated in Section 2.1. As Fig. 3 suggests, we takeadvantage of the symmetry of the beam and of the loading conditions (Fig. 1) to model only half of the beam.

The cohesive crack is modeled by 1D spring elements that follow the softening function at opening. The forces theytransmit also depend on the crack surface assigned to each spring. For instance, the force transmitted by a spring attachedto a node in the head of a T-beam is computed as

F coh ¼ cbfr; ð4Þwhere c is the distance between nodes, bf is the width of the head of the beam and r is given by the cohesive law in (1) (c andbf are sketched in Fig. 3). The number of elements has to be related to the characteristic length to resolve the fracture pro-cess zone properly.

The concrete bulk is divided into two subbodies—the body and the head of the beam—and each of them are discretizedusing six-node triangular elements; the width of the triangles is bw for the body and bf for the head. We take the precautionof generating lines of nodes along the position of the reinforcement layers. The nodes along each layer are duplicated andconnected to the neighboring counterparts by truss elements modeling the steel rebars. The first node is blocked to simulatethe restriction of movement due to the other half of the reinforcement which is not modeled. The duplicated nodes repre-senting the steel are connected to the original ones within the concrete bulk by interface elements that transmit a forcebetween them. These interface elements are 1D springs provided with the bond-slip law at sliding. As in the case of the

p

sτ (s)

τ (s)

s

(a) (b)

x

qq = p τ(s)

q(x)

Arσr

steel rebar

Fig. 2. Mechanics of the steel–concrete interface of each reinforcement bar: (a) load transfer through the bar interface; and (b) shear force transmitted perunit length.

dc

D

tf

L/2

bw bf

Fig. 3. Mechanical model.


cohesive springs, the modulus of the force transmitted by the interface springs depends on the extent of interface assignedto them. For the elements sketched in Fig. 3 it would be

F int ¼ dpisðsÞ; ð5Þwhere d is the distance between adjacent nodes along the reinforcement layer, pi is the perimeter of the rebars pertaining tothe reinforcement layer being studied—number i—and s(s) is the bond-slip law. As in the case of the concrete fracture pro-cess, it is also necessary that the zone of load transfer between the steel rebar and the concrete bulk is resolved by a suf-ficient number of nodes.

This way of modeling preserves concrete continuity and, therefore, permits the crack propagation through the reinforce-ment layers. In addition, it would be of use for some other reinforcement arrangements, e.g., layers of reinforcement thatare not parallel to each other, prestressed reinforcement etc. It is also compatible with a more refined modelization of thecohesive matrix, for instance one accounting for microcracking [24–26]. Likewise, the model would work perfectly using3D elements to model the concrete bulk. For instance, Lettow et al. have recently used a similar approach in 3D to studythe debonding process between steel and concrete [18].

The numerical implementation of this model has been carried out using standard elements of commercial codes; specif-ically we have used ABAQUS and ANSYS. An algorithm based on the modified arc length method searches for the solu-tion at each step, to obtain the softening after the peak load. We also use the numerical stabilizer proposed by Cendon [15]when it comes to beam sizes for which snap-back is expected. This simply consists of loading the numerical specimenthrough a linear spring of negative stiffness. It is adjusted to change the global behavior of the system from unstable tostable crack propagation under displacement increments. Specific details of the meshes used in this paper are presentedas follows.

3. Validation of the model

3.1. Description of the tests

A complete mechanical characterization of concrete, steel and the interface in between is necessary to validate our modelagainst experimental results. We use two series of tests. The first one was performed by Ruiz et al. [20] and was meant tostudy the sensitivity of the response of lightly reinforced beams to size, steel ratio and steel to concrete adherence. The sec-ond one completed the former study by investigating the influence of the shape of the beam cross-section and of the rebararrangement on lightly reinforced beams [21,22]. Unlike other experimental data available in the Bibliography, thesepapers provide material properties obtained by independent tests (see Tables 1 and 2).

The specimens in series 1 were beams of 75, 150 and 300 mm in depth; the rest of the dimensions—with the exception ofthe width, which was 50 mm for all of the sizes—were proportional to the depth. The beam proportions are sketched inFig. 4a. They were made out of a microconcrete (‘ch = 130 mm) whose microstructure scaled that of bigger size specimensof conventional concrete (‘ch = 200–300 mm). The reinforcement was also made to scale out of steel wire of 2.5 mm indiameter. The beams were reinforced with three steel ratio levels, obtained by using 1, 2 or 4 wires, always of the samediameter, to maintain a constant ratio of p

As(perimeter/cross-section). Half of the beams were reinforced with smooth wires

Table 1Concrete mechanical properties

fca, MPa fts

b, MPa Ec, GPa GF, N/m ‘ch, mm

Series 1 39.5 3.8 30.5 62.5 130Series 2 40.4 4.3 26.3 66.5 95

a Cylindrical specimens, compression.b Cylinder splitting (Brazilian).

Table 2Mechanical characteristics of the reinforcement bars and of the steel–concrete interface

Es, GPa ru, MPa r0.2, MPa �r, % sc, MPa

Series 1–smooth 200 608 568 3.5 2.8Series 1–ribbed 162 587 538 2.3 5.2Series 2–ribbed 190 750 698 2.4 10.4a

a Estimated with the model.


and the other half with wires superficially treated to increase the strength at the interface. In this study we only model thebeams with medium reinforcement, i.e., one wire for the small beams, two wires for the medium beams and four wires forthe big beams (Fig. 4b); as there are three sizes and two levels of adherence, we end up with six different cases (Fig. 4c).

The beams in series 2 were all of 150 mm in depth. Half of them had a rectangular cross-section, whereas the other halfwere T-beams obtained by thickening by 50% the compressive head over the upper 2

3of the depth. The beams were rein-

forced as in series 1, other than for the arrangement of the wires: for each type of reinforcement there were some specimenswith the wires aligned horizontally and some others with the wires aligned vertically, the centroid being in the same posi-tion (22.5 mm to the bottom surface of the specimen). Again, beams were reinforced with smooth and ribbed wires,although bond-slip properties were not directly measured. Fig. 5 sketches the specimens resulting from the variationsand gives them a name for future reference. In this study we only model the beams reinforced with two adherent (ribbed)wires, which comprises four cases: rectangular and T-beams with two adherent wires aligned horizontally or vertically, i.e.,specimens R2H, R2V, T2H and T2V according to Fig. 5c.

0.15

D

25 25 15 1520 10 10= = =(b)

(c)

D

4 D4.5 D

B==50 mm

(a)

(bar spacing in mm)

ρ = 0.13%

D = 75 mm 150 mm 300 mm

(smooth and ribbed)

Fig. 4. Series 1: (a) beam dimensions; (b) rebar distribution; and (c) specimens with intermediate reinforcement (the ones modeled in this work).

(b)

(c)

(bar spacing in mm)

22.5

16.25 16.2517.5

17.5

13.7

5

R2H

T2V

T2H

R2V

600675

(a)

150

50 50

15068.

5

75

50

(dimensions in mm)

Fig. 5. Series 2: (a) rectangular and T-cross-section dimensions; (b) rebar arrangements; and (c) specimen nomenclature.


3.2. Numerical implementation

As concerns the numerical implementation, we use ABAQUS in the cases pertaining to series 1. We discretize the crackinto 40 cohesive elements. For the sake of simplicity, a linear softening function is used (Fig. 7a). The nominally infiniteinitial stiffness of the cohesive law is actually implemented as a linear ramp-up whose slope is chosen to avoid blockage ofthe numerical solver. In this case a stiffness of 198 MPa/lm enables the solver to work correctly while keeps the beam glo-bal flexibility almost unchanged. (Following the procedures developed by Planas et al. [27] we estimate that the 150 mmbeams in series 1 deflect only 0.06 lm at maximum load due to this fictitious initial stiffness.) The reinforcement is dividedinto 45 three-node truss elements attached to the concrete bulk by 90 interface elements. The truss elements implement theexperimental stress–strain law of the reinforcing wires, whereas the interface elements use a rigid-perfectly plastic lawwhose plateau reaches the experimental bond strength (Fig. 7b and c). Again, the infinite stiffness of the rigid initial behav-ior is substituted by a linear ramp-up—with a stiffness of 115 MPa/lm—for the sake of numerical stability. Concrete bulkis represented by 1800 six-node triangular elements. As mentioned above, linear elasticity with no yield limit or damagemodelization whatsoever is chosen for the bulk. Both fracture process and interface deterioration zones are properlyresolved by this discretization. For instance, in the case of the 150 mm deep beams there are 35 cohesive elements spanninga distance equal to the characteristic length and thus the results are not mesh dependent.

The cases pertaining to series 2 are studied using ANSYS with the mesh shown in Fig. 6. This mesh is valid for all thespecimens in series 2, which includes (a) rectangular and T-beams, and (b) reinforcement arranged in one or two layers. TheT-beams need a division in the concrete bulk that allows assignation of a different value for thickness to the elements in thecompression head. The discretization of both parts is done with six-node triangular elements, 774 for the body—whosethickness is 50 mm—and 324 for the head—50 mm thick for rectangular beams and 75 mm thick for T-beams. As in series1, the crack is discretized into 40 cohesive elements provided with a linear softening law, although now their mechanicalproperties are adjusted to the new material and to the change in thickness between the two parts of the T-beams (asexplained in Section 2.2, Eq. (4); the linear cohesive law is depicted in Fig. 7a; the rigid initial segment of the cohesivelaw is substituted by a linear ramp-up whose slope is 42 MPa/lm). Unlike in series 1, specimens in series 2 have layersof reinforcement at three levels, as indicated in Fig. 5. Our discretization defines three lines of 80 nodes along the locationof the layers; the nodes in each line are duplicated, united by truss elements and attached to the bulk by interface elementsas was carried out for a single layer of reinforcement in series 1. The layers are activated or not depending on the specimenthat is being modeled: R2H and T2H have only one layer of reinforcement and thus only the central layer is activated forthese cases, the springs being stronger as they represent the interface of two bars (Section 2.2, Eq. (5)); R2V and T2V havetwo layers of reinforcement corresponding to the lowermost and uppermost layers in the model and, consequently, these

X

Y

Z

Fig. 6. Mesh used for specimens of series 2.

0

1

2

3

4

0 10 20 30 40

σ (M

Pa)

w (μm)

(a)

seri

es 1

ribbedsmooth

0

200

400

600

800

0 0.5 1 1.5 2

σ (M

Pa)

ε (%)

(b)

0

2

4

6

8

10

12

0 0.3 0.6 0.9 1.2

τ (M

Pa)

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σ (M

Pa)

w (μm)

seri

es 2

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0

2

4

6

8

10

12

0 0.3 0.6 0.9 1.2

τ (M

Pa)

s (mm)

ModelCode

0 10 20 30 40

Fig. 7. Laws defining the behavior of materials in series 1 (upper row) and series 2 (lower row): (a) cohesive law for the concrete; (b) stress–strain law forthe steel; and (c) bond-slip law for the interface between steel and concrete.


cases require that these two layers are activated whereas the central one is not. In these latter cases the interface springs areweaker because they are only modeling the interface of one single bar. As concerns the constitutive laws, for the truss ele-ments we adopt a simple linear-perfectly plastic law (Fig. 7b). Finally, interface elements are provided with two bond-sliplaws. On the one hand, the same one used for series 1 but for sc, which in this case was estimated so that the results of themodel best fit the experimental results; in the computations the theoretically infinite stiffness is actually changed to660 MPa/mm. On the other hand, the bond-slip law recommended by the Model Code [19] (Fig. 7c).

3.3. Numerical results

The curves given by this model fit the experimental results well, as Figs. 8 and 9 show.

3.3.1. Size effect

The model reproduces the experimental size effect of these structural elements, since the maximum load withstood by thebeams increases at a slower rate than the beam dimensions. Please note that in Fig. 8 the axes are scaled as are the spec-imens in order to allow a direct visualization of the size effect. Numerical peak loads are slightly bigger than their exper-imental counterparts in Fig. 8, particularly for the smaller beams, which could be corrected by using a more complex

0

1

2

3

4

0 0.05 0.1 0.15 0.2

P (k

N)

δ (mm)

D = 75 mmribbed

0

2

4

6

8

0 0.1 0.2 0.3 0.4

testsmodel

P (k

N)

δ (mm)

D = 150 mmribbed

0

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0 0.2 0.4 0.6 0.8

P (k

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δ (mm)

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0

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P (k

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δ (mm)

D = 75 mmsmooth

0

2

4

6

8

0 0.1 0.2 0.3 0.4

P (k

N)

δ (mm)

D = 150 mmsmooth

0

4

8

12

16

0 0.2 0.4 0.6 0.8

P (k

N)

δ (mm)

D = 300 mmsmooth

Fig. 8. Results corresponding to series 1: P–d curves (load–displacement under the load point) given by the model compared with the curves obtained inthe tests (series 1).

R2H

0

2

4

6

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

testsmodel

P (k

N)

δ (mm)

MC

T2H

0

2

4

6

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P (k

N)

δ (mm)

R2V

0

2

4

6

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P (k

N)

δ (mm)

T2V

0

2

4

6

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P (k

N)

δ (mm)

Fig. 9. Results corresponding to series 2: P–d curves (load–displacement under the load point) given by the model compared with the curves obtained inthe tests. The dashed-grey curve in the top-left graphic (marked as MC) corresponds to using the bond-slip law in the Model Code.


cohesive law; to be specific, by a law having a bigger absolute value of the initial slope and a bigger critical opening wc [28].Indeed, the numerical simulations of the series 1 beams performed by Ruiz et al. [13] using a quasi-exponential curve gavemaximum loads that were closer to the experimental ones. The numerical prediction for the peak load in the simulationspertaining to series 2 is good too, other than for the T2H beam, although in this case it is the real beam that resists morethan the average of its class.


The accuracy in the prediction of the peak load is specially noteworthy when having in mind the very simple assump-tions we made regarding crack formation (Section 2.1) and the fact that we use linear softening. In general the absolutevalue of the initial slope for concrete and mortar is about twice as much as the initial slope of the linear softening withidentical fracture energy. This implies that the predicted peak, when linear softening is used, is larger than the peak loadobtained with initially steeper laws. Nevertheless, if we had used a law of this kind the model could have underestimatedthe peak. This can be explained reminding that experimental determination of cohesive laws require that the damaged zonebefore crack formation is as small as possible [29], but such condition is not met in unnotched specimens. In addition, thepresence of reinforcing bars may contribute to spreading the damage and, as a consequence, to increasing the consumptionof energy before a main crack nucleates.

3.3.2. Effect of the strength of the interface

The sensitivity of the model to the strength of the interface is also noteworthy. Please observe that the upper row inFig. 8 corresponds to beams reinforced with ribbed wires, while the lower row depicts the results for the beams with smoothwires (the steel ratio being 0.13% for all of them). In both cases the numerical results are good in the U-shaped portion ofthe P–d curve, which is the stretch that depends the most on the fracture and interface properties. This part of the curveshows initially some softening until the crack opens wide enough to allow load transfer to the reinforcement. Then thebeam recovers strength and the load withstood by the beam grows up until the yielding of the steel (ribbed wires) orthe complete slipping of the bar (smooth wires).

3.3.3. Analysis of the decohesion process

In order to see graphically the interaction between steel and concrete, we repeat the process for the 150 mm deep beamwith ribbed wires of series 1, but change the experimental stress–strain law of the steel for a simple bi-linear law (the result-ing P–d curve is depicted using a thick dashed line in Fig. 8). Fig. 10 represents the tensile stresses in the rebars and theshear stresses on the interface at four shots during the load process. The first shot is taken while both materials are still inthe elastic regime; the stresses along the rebar and the interface are very weak and originated by the difference in stiffnessbetween the two materials. The second shot is taken at the load peak; the crack has already propagated through the rein-forcement and, therefore, the load transfer process has already started; the interface has reached its critical value aroundthe crack, which causes an abrupt stress jump in the bar. As the crack continues to open we see that both the interfaceand the steel get loaded up to attain the yielding of the bars. In fact, note that, in the last shot depicted in Fig. 10, thetension in the bar has reached the yield stress. The shape of the shear stress profile along the interface is basically dueto the deterioration law we chose. Similarly it determines the shape of the tension profile along the reinforcement. In thiscase the profile is of a bi-linear type, which is in clear agreement with experimental observations in pull-out tests [30,31].

Some additional insights on the role played by the interface deterioration law can be drawn from Fig. 9. The resultsgiven for beam R2H when using the Model Code law fall under the curve obtained with the rigid-perfectly plastic adher-ence law (thick dashed-grey line labeled MC in Fig. 9). The initial slope of the former is gentler than that of the latter andthen a wider crack opening is needed to get the same level of resistance along the interface. In other words, the energyconsumed along the interface during the initial steps of the loading process is lower when using the Model Code law.

3.3.4. Effect of the shape of the cross-section

Fig. 9 shows that the model accurately reflects the response of the real beams when the shape of the cross-sectionchanges. T-beams have a more rigid initial ramp-up and the maximum load is slightly higher than it is for the rectangularbeams. Nevertheless, we found that the increase does not follow the predictions made using simple linear theories. Indeed,the additional resistance provided by the compression head should be of the order of 7% according to plain strength ofmaterials, while the experiments gave only a 3% increase in the maximum load and the percentage of increase given bythe calculations with the model falls in between these two figures. Additionally, the Ts show a protuberance at the bottomof the U-shaped stretch of their P–d curves that corresponds to the initial resistance of the reinforcement layers. This fea-ture is characteristic of large sizes, as pointed out by Ruiz et al. [20,13]. This observation can be simply explained: the pres-ence of the head makes the neutral axis go up and then the crack zone develops as if the T-beam were a larger rectangularbeam. Therefore, there is a kind of shape effect that is properly reproduced by this model and that resembles the well-known size effect.

3.3.5. Effect of the rebar arrangement

Fig. 9 compares P–d curves corresponding to beams that are identical but for the arrangement of the steel bars aroundthe centroid of the reinforcement. For beams R2H and T2H (first two curves in Fig. 9) the two reinforcing bars are placedhorizontally—thus in one single layer, while their counterparts R2V and T2V (second line of curves in Fig. 9) have the twobars arranged vertically—forming two layers of reinforcement. The differences in the curves given by the two arrangementsare noticeable in the transition indicating the plastification of the reinforcement. The yielding of the single layer of

P

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Fig. 10. Tensile stresses along the bars and shear stress on the interface (150 mm deep beam reinforced with ribbed wires, series 1).


reinforcement in the Hs is more abrupt than in the Vs. Indeed, the yielding in the results given by the model for the Hstakes place at a specific point of the P–d curve according to the bi-linear constitutive law given to the steel. In the Vsthe yielding of the two layers does not happen at the same time and then a transition is generated between the U-shapedstretch and the final plateau.

Ruiz and Carmona [21,22] report another interesting and somewhat subtle feature that differentiates the results obtainedwith both kinds of reinforcement arrangement. It consists of the experimental evidence that the Vs consume more energyright after the peak load than the Hs. Regrettably, the numerical model is not able to reproduce such behavior, mainlybecause of the very simple fracture process modelization that we adopt in this work. Undoubtedly, our results wouldbe even better if we modeled microcracking and/or damage around the main crack [24–26] and along the steel–concreteinterface [32].

4. Summary and conclusions

We have analyzed the propagation of a cohesive crack through one or several adherent reinforcement layers in a rein-forced concrete beam. To do so, we account for the fracture of the concrete matrix as well as for the ongoing deteriorationon the interface between the steel and the concrete at each layer.


In order to model the fracture of concrete, we have used a discrete crack model that follows a cohesive law. The rein-forcement layers have been modeled as being overlapped to the concrete bulk and connected to the matrix by interfaceelements that simulate the deterioration due to the decohesion process. As the concrete matrix is modeled keeping its con-tinuity, i.e., avoiding the discontinuity that the presence of the reinforcement layers could mean in a standard 2D model-ization, the crack can propagate across the layers only encountering opposition from the closing forces transmitted by theinterface elements. The implementation of the model has been carried out using commercial finite element codes, which canbe listed as one of the practical advantages of the method.

The model has been validated against three point bending tests on lightly reinforced beams. The numerical results clo-sely predict experimental data from rectangular and T-beams of different sizes and with several reinforcement layersformed by smooth and ribbed bars. The experimental size effect has been numerically reproduced, which confirms the apti-tude of cohesive models to deal with concrete fracture. The strength of the reinforcement; in which the properties of thebars and of the interface with concrete are combined, also influences the results of the model. We have also obtained thetensile stresses along the bars and the shear stresses at the interface during a numerical test and they are in accordance withpull-out data available in the Bibliography. Our modelization has allowed us to get the effect attributable to the shape ofthe cross-section of the beam, a shape effect that has been described in terms of an equivalent size effect. Finally, the modelhas shown sensitivity to the distribution of the reinforcing bars in one or two layers.

Consideration of the above allows us to conclude that this model can be of use to study other processes of local fracturethrough reinforcement areas combined with steel decohesion phenomena within structural concrete elements.

Acknowledgments

Gonzalo Ruiz and Jacinto R. Carmona acknowledge financial support from the Direccion General de Investigacion,Spain, through Grant MAT2003-00843, and from the Secretarıa de Estado de Infraestructuras, Spain, through GrantBOE305/2003. Likewise, the fellowship given by the Junta de Comunidades de Castilla-La Mancha to support the researchactivity of Jacinto R. Carmona is gratefully acknowledged.

References

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propagation of a cohesive crack through adherent reinforcement layers

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