propagation of a cohesive crack crossing a reinforcement layer

International Journal of Fracture 111: 265–282, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Propagation of a cohesive crack crossing a reinforcement layer

GONZALO RUIZE.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de Castilla-La Mancha, Paseo de laUniversidad 4, 13071 Ciudad Real, Spain (e-mail: [email protected])

Received 1 August 2000; accepted in revised form 10 May 2001

Abstract. This paper analyzes the propagation of a cohesive crack through a reinforcement layer and gives asolution that can be used for any specimen and loading condition. Here it faces the case of a reinforced prismaticbeam loaded at three points. Reinforcement is represented by means of a free-slip bar bridging the cracked section,anchored at both sides of the crack at a certain distance that is called the effective slip length. This length is obtainedby making the free-slip bar mechanically equivalent to the actual adherent reinforcement. With this model, thecrack development depends on three parameters (apart from those that represent the specimen geometry): thesize of the specimen, the cover thickness of the layer and the reinforcement strength. The latter depends on thereinforcement ratio and its adherence to the matrix while the reinforcement is in the elastic regime; otherwise,on the reinforcement ratio and its yielding strength. The thickness of the reinforcement cover influences the firststages of the development of the cohesive crack, and thus it also affects the value of the load peak. The computedload-displacement curves display a noticeable size effect, as real cohesive materials do. Finally, the model is ableto fit the available experimental results, and accurately reproduces the influence of size, amount of reinforcementand adherence variations in the tests.

Key words: Adherence, fracture of cohesive materials, reinforcement, size effect.

1. Introduction

Cementitious materials have little tensile strength, and thus are rarely used without somereinforcement distributed within the matrix, at least in the areas in which tensile stresses areexpected to take place. The presence of the reinforcement modifies the mechanical response ofthe matrix, specially when the level of stress increases up to generating cracks that develop andcoalesce across the fibres or the bars reinforcing the material. The analysis of such crackingprocesses and the subsequent load transfer between the cohesive matrix and the reinforcementis relevant to model the mechanical response of reinforced brittle materials, as can be seen asfollows.

One approach to study the overall response of reinforced brittle materials consists of bury-ing that complex cracking process into the constitutive equations of a homogenized continuum(e.g., see McDowell, 1997). This tends to be convenient if the scale of the reinforcement issmall compared to the size of the constituents of the plain matrix, and thus the reinforcementcan be considered as another phase of the composite material. Nevertheless, the mechanismsof deformation, crack propagation and stress transfer between the matrix and the reinforce-ment have to be accounted for at a mesoscopic or microscopic level, so that later on theireffects may be homogenized in the mean effective behaviour of the reinforced matrix.

On the other hand, the action of the reinforcement can be separated from the mechanicalresponse of the matrix, and doing so is appropriate if the reinforcement does not constitutea homogeneously distributed phase with a scale comparable to the other phases, but rather

266 G. Ruiz

it is a localized reinforcement to avoid tensile damage. This is the role given to steel inreinforced concrete: the steel rebars are located so as to sew the concrete cracks up, andonly if the concrete breaks can the full mechanical response of the specimen be achieved. Infact, the problem of fracture processes in reinforced areas is particularly important in concretetechnology, and some models to study it have been developed within this field. They focus onthe problem of a single crack crossing a reinforcement layer, which is important for preventingsudden catastrophic failure without warning and can be handled with relative ease by usingfracture mechanics. These models have been classified into three main groups by Bažant andPlanas (1998), on the basis of the constitutive equations that govern the matrix mechanicalbehaviour.

The more elementary models are based on linear elastic fracture mechanics (LEFM) (Bos-co and Carpinteri, 1990, 1992; Baluch et al., 1992; Massabò, 1994), in spite of the fact thatconcrete is remarkably non-linear, and of the inability of LEFM to deal with fracture nucle-ation. This implies that it is necessary to analyze an initially notched specimen so that a crackcan propagate from it. On the other hand, these models substitute a reinforcing layer by a pairof closing forces applied to the edges of the crack, which introduces some other difficultiesinto the analysis, namely that according to LEFM a crack with punctual forces applied atits edges cannot propagate, since the stress intensity factor becomes zero. Actually all thesemodels extend the notch ahead of the reinforcement layer to avoid this problem. In addition,a punctual force would produce an infinite displacement of the point where it is applied, andthus the closing forces have to be distributed as tractions over a certain area so as to controlthe flexibility of the specimen. All things considered, the LEFM based models have to beunderstood as roughly describing the evolution of the fracture after the crack has formed.

A second group of models accounts for non-linearity by including strain-softening in asmeared crack band, which together with the assumption of plane deformed sections can giveanalytically the specimen response (Hededal and Kroon, 1991; Gerstle et al., 1992; Ulfkjaeret al., 1994). Nevertheless, they make use of geometrical and mechanical parameters – likethe width of the band – that are not clearly related to the actual properties of the materials. Asregards the modelization of the reinforcement-matrix interaction, they associate the deforma-bility of the band at the level of the layer with the mechanical action of the reinforcement.Fantilli et al. (1999) use a more sophisticated approach that takes bond into account and evenconsiders the possibility of some kind of smeared cracking around the main crack by lettingthe concrete plastify in tension, although they keep the assumption of plane deformed sections.The simulations they present accurately fit some experimental results and give some hints as tothe extension of the cracking process. Ožbolt and Bruckner (1999) handle the fracture processby allowing strain-softening within a finite element framework and so their approach cannotactually be counted among the aforementioned simple analytical models.

The models in the third group (Hededal and Kroon, 1991; Ruiz et al., 1993; Hawkins andHjorsetet, 1992; Brincker et al., 1999) are based on the cohesive crack, a discrete non-linearapproach to fracture of concrete developed by Hillerborg et al. (1976). The cohesive modelprovides a closed theory for fracture: it allows a crack to open when the tensile strength ofthe material is reached, and from then on relates crack openings to cohesive stresses througha softening function which is considered a material property. Although these models rely onnearly the same basic assumptions, they use quite different formulations, spanning from thesuperposition of elastic influence matrices (Hededal and Kroon, 1991; Ruiz et al., 1993) tothe use of finite element codes (Hawkins and Hjorsetet, 1992; Brincker et al., 1999). Never-theless, the main difference between these models is the way load is transferred between the

Propagation of a cohesive crack crossing a reinforcement layer 267

reinforcement and the cohesive matrix: some of the models only account for the limit casesof perfect or of null bonding (Ruiz et al., 1993; Hawkins and Hjorsetet, 1992), while the restof them (Hededal and Kroon, 1991; Brincker et al., 1999) consider the interface properties byregulating the modulus of closing forces applied to the edges of the crack (and thus the resultsare dependent on the applying area or on the mesh used to solve the problem).

This paper presents a model that can be classified within this last group, but it applies theclosing forces bridging the crack inside the material. The intensity and point of application ofthe closing forces are related to the specimen deformation and to the adherence law betweenmatrix and reinforcement. The model is based on one of the existing methods to study acohesive crack growing through a plain cohesive material: the smeared crack tip method ofPlanas and Elices (1991). A direct application of this work is determining the three pointbending behaviour of a prismatic beam with a reinforcement layer when a cohesive crackcrosses its middle section (Figure 1). Nevertheless, the procedure this paper establishes isvalid for any specimen and any cementitious material reinforced with a single layer of fibres,and it could also be extended to multiple layers. The model was briefly outlined by Ruizand Planas (1994), and Ruiz et al. (1999), and has already been used to study the fracturebehaviour of lightly reinforced concrete beams (Ruiz, 1998; Ruiz et al., 1998).

The paper is structured as follows. Section 2 explains how the model approaches theproblem, and gives some general aspects of the analysis method. Section 3 gives a shortdescription of the calculation method (smeared crack tip), and Section 4 describes the modelfor the reinforcement, which is called effective slip length model: Section 4.1 the hypothesisestablished; Section 4.2 the analytic method used to incorporate it into the smeared crack tipmethod; and Section 4.3 the equations used to determine the basic parameters influencing thefailure. In Section 5 we study the sensitivity of the model to these parameters by performingsome numerical examples, and in Section 6 we compare numerical tests with experimentalresults. Finally, Section 7 contains some conclusions based on the results given by the model.Let us anticipate here that the load peak is strongly dependent on the adherence and on thethickness of the reinforcement cover, and so their influence should be accounted for in themodels for this kind of materials.

2. Layout of the problem

We want to develop a general model for a beam made out of a cohesive matrix and lightlyreinforced with a single layer of reinforcement, when subjected to flexural strength at threepoints (Figure 1a), so that it accounts for the matrix cohesive properties as well as for theinfluence of adherence and relative slip between the reinforcement and the cohesive matrix.

The following basic hypotheses are assumed:− Only one crack progresses at the central cross section of the beam, provided the rein-

forcement is light enough (Figure 1a).− The crack is cohesive: stresses are transferred between the crack faces.

A cohesive crack (Figure 1b) initiates where the maximum principal stress reaches thetensile strength of the material, ft . The crack is perpendicular to the direction of the majorprincipal stress and opens while transferring stress from one face to another. For monotonicmode I opening, the stress transferred is normal to the crack faces and is a single function ofthe crack opening:

268 G. Ruiz

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

σ = f (w), and

{f (0) = ft ,

f (w) ≥ 0.(1)

The function f (w) is known as softening function. It completely characterizes the mechan-ical behaviour of the crack. The values of f (w) become zero when the crack opening exceedsa critical value wc (Figure 1c).

The specific fracture energy, GF , is the work needed to separate the two faces of a unitsurface of crack, and is represented by the area under the softening curve (Figure 1c). An-other important parameter is the characteristic length, lch, which characterizes the intrinsicbrittleness of the cohesive material (Hillerborg et al., 1976) and is defined as:

lch = EcGF

f 2t

, (2)

where Ec is the elastic modulus of the cohesive material.− The cohesive behaviour of the matrix outside the fracture zone is linear elastic. This

assumption implies only a second order error, because in a lightly reinforced beam theinelastic deformations before the crack is formed are small.

− The shear stress τ transferred between reinforcement and matrix is a function of therelative slip, s, between both materials.

The resultant of the shear stresses acting on the matrix per unit length of reinforcement isdenoted as q, and is given by:

q = p τ(s), (3)

where p is the perimeter of the reinforcement cross section (Figure 2a), and τ(s) is the shearstress. The action of the reinforcement on the matrix is completely determined by the distrib-ution of q, q(x), along the reinforcement (Figure 2b). Here, x is the distance measured fromthe center of the beam. The distribution of q must be calculated according to the particularstress-slip relationship and stress-strain curve of reinforcement.

In this paper the shear stresses acting along the contact with the reinforcement are replacedby their resultant force applied at a point inside the cohesive matrix. This point is the center


Figure 2. Bond shear stresses in the contact surface between reinforcement and matrix (a), and force per unitlength they generate (b).

of gravity of function q in Equation (3). We further assume that the shear stress-slip law is ofthe rigid-plastic type. This way of accounting for the reinforcement is directly related to thebond-slip law and gives no singularities, thus solving some of the difficulties arising in othermodels (Ruiz, 1998).

The aforementioned assumptions are implemented in a finite element based programmeusing the smeared crack tip method with internal stresses (Bažant and Planas, 1998; Planasand Elices, 1992, 1993), whose fundamentals and the details of the implementation of theaction of the reinforcement (via internal stresses) are explained in the next section.

3. Calculation method: smeared crack tip with internal stresses

This method is meant to study the cracking process of plain concrete specimens with internalstresses generated by shrinkage or by temperature gradient across the specimen. The virtue ofthe method is that it leads to a triangular system of equations that can be directly solved byforward substitution. In the present case the internal stresses are caused by the reinforcementin the middle cross section of an uncracked beam and have to be calculated for each itera-tion, unlike the internal stresses due to shrinkage, which would remain constant throughoutthe cracking process. Here we describe the essentials of the method, while the next sectionexplains how those internal stresses can be obtained analytically.

The smeared crack tip method sees the solution as the superposition of the elastic casesdepicted in Figure 3. The stresses and displacements for the actual nonlinear state (Figure 3,left) are written as the sum of N elastic cases corresponding toN different crack lengths, eachwith a stress-free crack with its tip at node j (j = 1, 2, . . . , N) and an external load equalto �Pj (Figure 3, middle), plus a further elastic case corresponding to the uncracked bodysubjected to the internal stresses due to the reinforcement (Figure 3, right). According to thisscheme, the load, the displacement under the load-point, the nodal stresses and the nodal crackopenings are:

P =N∑j=1

�Pj , (4)

δ =N∑j=1

Cj�Pj + δ, (5)

σi =N∑j=1

Rij�Pj + σ i i = 1, 2, . . . , N, (6)

270 G. Ruiz

Figure 3. Elastic decomposition to formulate the equations of the smeared crack tip method with internal stresses.

wi =N∑j=1

Dij�Pj i = 1, 2, . . . , N, (7)

where Cj is the displacement under the load point generated by an external unit force for acrack reaching node j , δ is the load-point displacement generated by the internal stress fieldσ ( σ i at node i ; Figure 3, right), and Rij (matrix R) and Dij (matrix D) are, respectively,the nodal stress and the crack opening at node i when an external unit force is applied with acrack reaching node j . Notice that Rij = 0 for i < j (R is a lower triangular matrix), and thatDij = 0 for i ≥ j (D is an upper triangular matrix with zero diagonal elements). The internalstress field σ contributes neither to the external load in Equation (4), nor to the crack openingsin (7), since there is no crack in this elastic case, but they are accounted for to attain the finalstate of equilibrium through Equations (5) and (6).

To write the equations governing the nonlinear problem, it is considered that the cohesivezone has extended up to node m (Figure 3, left), and Equations (6) and (7) are used to writethat the stresses and crack openings over the cohesive zone satisfy the softening equationσi = f (wi), and that on the uncracked ligament wi = 0, i.e.:

N∑j=1

Rij�Pj + σ i = f

N∑j=1

Dij�Pj

i = 1, 2, . . . , m, (8)

wi =N∑j=1

Dij�Pj = 0 i = m,m+ 1, . . . , N. (9)

To solve the system we start with Equation (9), whose solution is trivial because the systemis triangular:

�Pj = 0 for j = m+ 1,m+ 2, . . . , N. (10)

Inserting this result in (8) and rearranging, we get:

m∑j=1

Rij�Pj = f

m∑j=1

Dij�Pj

− σ i i = 1, 2, . . . , m. (11)


This is a system of equations for the m unknowns �Pj (j = 1, . . . , m), which can besolved iteratively as follows: starting from an estimate for �Pj , the right-hand member isevaluated, and the system is solved for a better estimate. Since R is triangular, the iteration isvery fast. At each iteration the internal stresses σ i introduced by the reinforcement must beevaluated, which is done analytically as described in the following section. The total exteriorload, the displacement under the load point, and the stresses and openings along the crack areobtained entering with the solution of Equation (11) and with (10) in Equations (4)–(7).

4. The effective slip-length model

As indicated in the introduction, the reinforcement-to-matrix interaction is approximated bymeans of closing forces spanning the crack. The value and location of these forces are re-lated to the adherence law between the reinforcement and the cohesive matrix (Section 4.1)following Bažant and Cedolin (1980). The procedure to obtain and implement the internalstresses that these forces generate is described in Section 4.2. Finally, a parametric analysisof the model is performed, to point out which are the parameters that influence its behaviour(Section 4.3).

4.1. MODULUS AND POSITION OF THE PAIR OF FORCES REPRESENTING THE

REINFORCEMENT

The model assumes, based on well-known results of steel-concrete pull-out tests, that theadherence follows a bond shear stress-slip law (Bažant and Sener, 1988; Malvar, 1992). Fur-ther experimental evidence (Mazars et al., 1992; Morita et al., 1994) suggests that the axialstress law in steel bars is approximately triangular (Figure 4a, up), which implies a constantbond shear stress acting on the contact surface (Figure 4a, center). Therefore, a rigid-perfectlyplastic shear stress-slip law is a simple and reasonable approximation:{

τ ≤ τc if s = 0,τ = τc if s �= 0,

(12)

where τ is the shear stress at the reinforcement-matrix interface, τc is the bond shear strengthand s is the relative slip between both materials (Figure 4a, down). This τ − s law leads tosimple analytical solutions, though is not essential to the model, and more complex stress-sliplaws could be implemented.

To obtain an explicit formulation for the model, we assume that the reinforcement crosssection is a small fraction of the beam cross-section, and so the cohesive matrix can be con-sidered as rigid in this context. Therefore, the reinforcement is essentially unstrained exceptover the slip zones, of length Ls (see Figure 4a).

According to the hypotheses previously stated, the shear stress is constant over the sliplength, and the equilibrium of the horizontal forces acting on an arbitrary portion of thereinforcement requires that σ (x) = τcp(Ls−x) where p is the perimeter of the reinforcement.Global equilibrium further requires that Fr = σ (0)Ar = σrAr = τcpLs (whereAr is the crosssection of the reinforcement) from which it follows that the slip length is given by:

Ls = Arσr

τc p. (13)

272 G. Ruiz

Figure 4. Stresses in the reinforcing bars (a), and mechanical actions due to them on the beam bulk, that arereplaced by a pair of closing forces (b).

The relationship between the force and wr (the elongation of the reinforcement, which welater identify with the crack opening at the crossing point) is obtained by integrating the strainalong the slip length. For an elastic-perfectly plastic behaviour of the reinforcement the resultis:

Fr = Arσr =

√ArEr τcp wr if wr < wy = Arf

2y

Er τcp,

Arfy if wr ≥ wy,

(14)

where fy is the yield stress of the reinforcement and Er its elastic modulus, and the rest of theparameters are already known.

The foregoing model is implemented in the lightly reinforced beam as depicted in Fig-ure 4b. The actual situation is sketched in Figure 4b left, in which the shear stresses extend overthe slip length Ls at both sides of the crack. To simplify the formulation, this is approximatedby the arrangement in Figure 4b right, in which the shear transmission is concentrated atpoints located at a certain effective slip length Le at both sides of the crack. This is equivalent


Figure 5. Method to obtain the stresses generated by the reinforcement over the central cross section of the beam.

to substituting the adherent reinforcement by an unbonded bar of length 2Le anchored in theconcrete at its ends. The effective slip length is calculated to keep the relation between Fr andwr identical to that for the original model, which turns out to be equivalent to replacing the qdistribution by its resultant applied at its center of gravity. So, in our case:

Le = Ls

2. (15)

From (13) and (14) it follows that the effective slip length is given, in terms of the crackopening wr by:

Le =

√ArEr

4 τcpwr for wr < wy,

Arfy

2 τcpfor wr ≥ wy.

(16)

The effective slip length model allows analytical calculation of the internal stresses neededin Equation (11), as described next.

4.2. INTERNAL STRESSES CAUSED BY THE REINFORCEMENT AND NUMERICAL

IMPLEMENTATION

The stresses generated by the closing forces Fr at the central cross section of an uncrackedelastic beam (Figure 5a), can be calculated by superposing the stresses they would generatein an elastic halfspace (Figure 5b), with the stresses corresponding to the case that restoresthe equilibrium over the beam (Figure 5c). The first case was solved by Melan (1932). Thesecond is approximately handled by assuming that it produces a linear stress distribution onthe central cross-section, which preserves overall equilibrium of each half of the beam, (i.e.,the surface tractions are reduced to a force and a bending moment on the central cross section)as indicated in Figure 5e.

The stress distribution corresponding to the case in Figure 5b (the halfspace) can be writtenas:

σ 1 = Arσr

BF(y,Le, c), (17)

whereB is the beam width; y,Le and c are the dimensions defined in Figure 5a, and F(y,Le, c)is the stress distribution for a pair of unit closing forces (per unit thickness), derived by Melan(1932) as:

274 G. Ruiz

Le

2π (1 − ν)

[L2e

r41

+ L2e + 4cy − 2c2

r42

+

+8cyL2e

r62

+ 1 − 2ν

2

(1

r21

+ 3

r22

− 4y (c + y)

r42

)],

(18)

wherer1 =

√(y − c)2 + L2

e,

r2 =√(y + c)2 + L2

e .

(19)

Here it may be further noticed that the resultant per unit thickness of the stress distributionin Equation (17) must be Arσr/B, and thus the function F(y,Le, c) satisfies:∫ ∞

0F(y,Le, c) dy = 1. (20)

Likewise, the moment per unit thickness of the stress distribution relative to a point on thelower free surface must be cArσr/B, and therefore, F(y,Le, c) also satisfies the condition:∫ ∞

0y F(y,Le, c) dy = c. (21)

To get the overall stress distribution it is necessary to calculate the stresses in Figure 5e,whose resultant R and bending moment M are:

R = Arσr R∗, M = Arσr DM

∗, (22)

where:R∗ =

∫ ∞

D

F(y,Le, c) dy = 1 −∫ D

0F(y,Le, c) dy,

M∗ = 1

D

∫ ∞

D

y F(y,Le, c) dy = 1

D

[c −

∫ D

0y F(y,Le, c) dy

],

(23)

and then the stress distribution in Figure 5e is readily found to be:

σ 2 = σ1 + σ2y

D, where

σ1 = Arσr

BD(4R∗ − 6M∗),

σ2 = Arσr

BD(12M∗ − 6R∗),

(24)

from which it follows that, putting together Equations (17) and (24), the net stress distributionover the central cross section is given by:

σ = σ 1 + σ 2 = Arσr

BD

[DF + (4R∗ − 6M∗)+ (12M∗ − 6R∗)

y

D

]. (25)

The expression in brackets in (25) is a dimensionless function dependent on D, y, c and Le.So, Equation (25) can be written as:

σ = ρσr F′(D, y,Le, c), (26)


where ρ is the reinforcement ratio (ρ = Ar/BD) and F ′(D, y,Le, c) is the expression inbrackets in (25). To stress that F ′(D, y,Le, c) gives a dimensionless coefficient, it is conve-nient to use dimensionless arguments as well, that can easily be done by referring them tothe beam depth (since all of them are geometrical dimensions). This leads to the followingexpression of Equation (26):

σ = ρσr +(ψ, λe, γ ), (27)

where ψ = y/D, λe = Le/D, and γ = c/D. This formula allows the calculation of theinternal stresses at node i, required in each iteration performed to solve Equation (26), byusing the value of ψ corresponding to it, ψi :

σ i = ρσr +(ψi, λe, γ ). (28)

4.3. PARAMETRIC ANALYSIS

To reveal the effect of the various material and geometrical parameters involved in the prob-lem, a dimensional analysis has been carried out to identify the basic dimensionless groupsinfluencing the result, and various computational runs have been performed to see their relativeinfluence.

4.3.1. Basic dimensionless groupsTo write the equations in dimensionless form, the following dimensionless load, stresses, crackopenings and displacements have been defined (identified by a star):

σ ∗N = 3

2

P/

BD2ft, σ ∗ = σ

ft, w∗ = w

ft

GF

, δ∗ = δft

GF

(29)

(in σ ∗N , / and B are respectively the beam span and width). All geometrical lengths (coordi-

nates, cover thickness, etc) are referred to the beam depth, e.g.:

ψ = y

D, γ = c

D. (30)

When all the variables appearing in the governing Equations (11), (14), (16) and (27) are re-duced to dimensionless form according to the foregoing rules, and the analysis is restricted togeometrically similar structures, it turns out that the size effect is controlled by the Hillerborg’sbrittleness number βH , defined as:

βH = D

lch, (31)

which is the essential characteristic of fracture processes in plain cohesive specimens (Peters-son, 1981; Bache, 1994).

The presence of the reinforcement introduces five additional dimensionless groups appear-ing naturally in the equations, namely:

ρ = Ar

Ac, n = Er

Ec, f ∗

y = fy

ft, η =

√nτc

ft

plch

Ar, and γ = c

D. (32)

The dimensionless internal stresses can be easily written using these dimensionless groups:

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Figure 6. Beam dimensions (a), and softening function (b) used throughout the calculations.

σ ∗ = ρσ ∗r +(ψ, λe, γ ), (33)

where σ ∗r and λe have different expressions depending on the regime of the reinforcement

(elastic or plastic), which are:σ ∗r = η

√w∗r and λe = n

√w∗r

2βHη, if w∗

r ≤ w∗y =

(f ∗y

η

)2

,

σ ∗r = f ∗

y and λe = nf ∗y

2βHη2, if w∗

r > w∗y.

(34)

The behaviour of the model, according to (33) and (34), is selectively dependent on thedimensionless groups in (32), and only three combinations of them are active (besides βH ,that is always active). In the regime in which the reinforcement remains elastic, the behaviouris controlled by the four groups:

βH, ρη,µ1 = n

βHηand γ. (35)

When the reinforcement enters the plastic regime, the behaviour is controlled by the fourgroups:

βH, ρf∗y , µ2 = nf ∗

y

βHη2, and γ. (36)

5. Numerical results

We will now present some numerical results to show how the foregoing dimensionless groups,and their associate variables, influence the behaviour. All of the calculations have been per-formed using the beam geometry shown in Figure 6a, and the quasi-exponential softening dueto Planas and Elices (1991) with a critical crack opening wc = 5GF/ft :

f (w) =ft

{(1 + A) exp

(−Bw ft

GF

)}0 ≤ w ≤ wc,

0 wc < w,

(37)

where A and B are dimensionless constants whose values are 0.0082896 and 0.96020 respec-tively (Figure 6b). For n in (32) a typical value of 7 has been used throughout the runs of themodel except in Section 5.3, where precisely its influence on the model results is studied.


Figure 7. Size and reinforcement ratio effects on dimensionless load-displacement curves.

5.1. INFLUENCE OF THE SIZE AND THE REINFORCEMENT RATIO

Figures 7a–b show dimensionless load-displacement σ ∗N -δ∗ curves obtained for two sizes and

several reinforcement ratios. The reinforcement ratio increases the ultimate strength of thebeams proportionally to the dimensionless group ρf ∗

y , while the zone of the curves beforethe reinforcement yielding depends on the size, as seen more clearly in Figure 7c where theresults for four different sizes are superimposed (notice that δ∗ is divided by βH to facilitatethe comparison among the curves): as size increases, the dimensionless load peak decreases.Before reaching the reinforcement yielding the beam response is also dependent on the in-tensity of the internal stresses generated by the reinforcement, which depends directly on theproduct of the reinforcement ratio by the dimensionless adherence, ρη (see Equation (33) andthe upper part of (34)).

5.2. INFLUENCE OF THE BOND AND YIELDING STRENGTHS

To study the influence of the bond strength several calculations varying only the η parameterhave been carried out. Figure 8a shows the results obtained on a beam reinforced with linearelastic reinforcement without plastic branch (fy = ∞): for stronger adherences the beamwithstands a higher peak load and shows a more resistant and stiffer behaviour after the peak.Figures 8b–c represents σ ∗

N -δ∗ curves for beams with the same depth and reinforcement ratio,but where the reinforcement yields at two different yield strengths, fy = 150ft and 100ft ,respectively. In these curves, two well differentiated branches are observed. In the first branchthe reinforcement is elastic, and the curves are identical to those of Figure 8a for the sameadherences. The second branch, after the reinforcement yielding, is almost independent onthe adherence, thus mainly controlled by only the reinforcement ratio and its yield strength.

5.3. INFLUENCE OF µ1, µ2 AND n

Figure 9 shows σ ∗N − δ∗ curves in which only the dimensionless groups µ1 and µ2 in (35) and

(36) are varied, which is achieved by varying n while keeping βH , ρ, η and f ∗y constant. The

results show that the initial elastic response and the plastic branch are essentially unaffectedby varying n in the range 4 to 13. The effect is noticeable in the intermediate range, in whichthe matrix cracks, but the reinforcement remains elastic. It is, however, very small for practical

278 G. Ruiz

Figure 8. Adherence and yield strength effects on dimensionless load-displacement curves.

Figure 9. Dimensionless load-displacement curves for different values of n (and of µ1 and µ2).

purposes, since the maximum relative effect is of the order of 2% (at the relative minimum)while the effect on the peak load is only 0.4%.

It can, thus, be concluded that the influence of the dimensionless groups µ1 and µ2 (andn) is very small, and, therefore, the reinforcement controls the behaviour mainly through thegroups ρη (for reinforcement in elastic regime) and ρf ∗

y (for the reinforcement in plasticregime).

5.4. INFLUENCE OF THE REINFORCEMENT COVER

Figure 10 draws dimensionless load-displacement curves for beams with different reinforce-ment cover thicknesses but with the same effective depth and span (D − c = 0.5 lch and/ = 3 lch), which keep the beam proportions used throughout this numerical study (Figure 6a).Notice that the ordinates axis represents the dimensionless load divided by the factor (1 − γ ),to obtain values strictly proportional to the actual loads.

The peak diminishes as the cover length increases until reaching a minimum for c =0.20 lch, and then increases if the cover length goes on increasing. This behaviour can be


Figure 10. Dimensionless load-displacement curves for different thickness of the reinforcement cover but thesame effective depth (0.5lch) and span (3lch), and the dimensionless (σ/ft ) cohesive and internal stresses due tothe reinforcement for the peak load in each case.

explained by the stress profiles corresponding to the peaks (Figures 10a–f). If the coverthickness is short, the crack has crossed the reinforcement when the beam resistance reachesits peak, and so the reinforcement is already heavily strained at this point, which leads to agreat reinforcement contribution to the peak load – hyper-strength – (Figure 10a). The peakload diminishes when the cover is longer because the cohesive stresses developed below thereinforcement are stronger, and so the peak is reached with less reinforcement contribution(Figures 10b–d). The hyper-strength due to the layer of reinforcement is null for a criticalcover thickness cc, i.e., in this case the beam behaves as if it were unreinforced until thecracking load is reached. If the cover is thicker than cc the load peak increases in spite of thenull hyper-strength, because the total depth is higher which leads to higher bending moments(Figures 10e–f).

280 G. Ruiz

Various cases were run to determine the value of the critical cover thickness cc. This valuedepends solely on the beam depth and on the parameters that characterize the first stages ofthe cracking process, and can be written as:

cc = 0.28D1/4(αlch)3/4, (38)

where α is the ratio between the fracture energies corresponding to two softening functionsgiving the same load peak for the beam: the linear softening and the actual softening function.In the examples run in this paper α = 0.58.

6. Experimental verification

To compare the predictions of the model to experimental results it is necessary to know thevalues of the material parameters in (32), which implies a complete characterization of thecohesive matrix, the reinforcement, and the adherence between both materials.

We use the experimental programme described by Ruiz (1998) and Ruiz et al. (1998),which was meant to find out how the response of lightly reinforced beams is modified whenvarying βH , ρ and η, and that, unlike other experimental data available in the Bibliography,provides material properties obtained by independent tests.

Beams of 75, 150 and 300 mm depth, with proportional dimensions were made out of amicro-concrete (lch = 130 mm) whose microstructure scaled that of bigger size specimensof conventional concrete (lch = 200 to 300 mm). The beam proportions are sketched inFigure 11b.

The reinforcement was also made to scale out of steel wire of 2.5 mm diameter. The beamswere reinforced with three steel ratio levels, obtained by using 1, 2 or 4 wires, always ofthe same diameter to maintain constant the ratio p/Ar (perimeter/cross section). Half of thebeams were reinforced with smooth wire and the other half with wire superficially treated toincrease the adherence between the reinforcement and the matrix.

The curves given by this theoretical model adequately fit the test results, as shown inFigure 11. The results confirm that the model reproduces the characteristic size effect of thelightly reinforced beams as well as the behaviour of the beams when varying the reinforcementratio (Figure 11a) and the adherence (Figure 11b). Thus it is a good tool with which to studythis type of element by the use of ‘theoretical tests’, cheaper and more flexible than the realtests.

7. Conclusions

The paper describes the so-called effective slip length model that solves the propagation ofa cohesive crack through a single reinforcement layer. As an example, it solves the case of aprismatic beam with a single layer of reinforcement and loaded at three points, although themodel can be applied to any specimen geometry with multiple reinforcement layers.

The response of the effective slip length model depends on five parameters: beam size,reinforcement cover, reinforcement ratio, bond shear strength between reinforcement andcohesive matrix, and reinforcement yield strength.

The modelling of the adherence is decisive in the model’s response before reaching thereinforcement yielding, and it acts coupled with the reinforcement ratio. Once the reinforce-ment yields, the influence of the adherence vanishes, and it is the yield strength together with


Figure 11. Experimental load-displacement curves (P−δ) – dashed lines – compared to the results of the effectiveslip length model, for 150 mm depth beams and various steel ratios (a), and 300 mm depth beams with the sameratio, but different steel-concrete bond properties.

the reinforcement ratio which governs the model. Although we have considered a very simpleadherence law, the model could be applied using more complicated bond-slip laws.

Another important remark is that, for a given beam depth, there is a critical cover thicknessthat gives the minimum peak load; this is the cover thickness at which the fracture zone reachesthe reinforcement just at the peak load. Below this critical value, the reinforcement contributesto the peak load, thus generating some hyper-strength.

Finally, the effects of size, reinforcement ratio and adherence shown by real tests areaccurately caught by the model.

Acknowledgements

The author is indebted to Professors Manuel Elices and Jaime Planas for their advice on thisparticular application of their smeared crack model. He also wants to thank Professor JaimeGálvez for his remarks on the manuscript. Financial help from the Ministerio de Ciencia y Tec-nología, Spain, through Grant MAT2000-0705, and from the Vicerrectorado de Investigación,Universidad de Castilla-La Mancha, through Grant 011,1075, is gratefully acknowledged.

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propagation of a cohesive crack crossing a reinforcement layer

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