STEEL-CONCRETE BOND ANALYSIS WITH

NONLOCAL CONTINUOUS DAMAGE

By Gilles Prjaudier-Cabot, ' Jacky Mazars , 2 Associate Members , ASCE, and Jacek Pulikowski3

ABSTRACT: This paper deals with the application of continuum damage mechan­ics to the description of the response of reinforced concrete members. Two sim­plifying assumptions are made: (1) The constitutive law of concrete is the isotropic damage law; and (2) there are no displacement discontinuities at the interface be­tween steel and concrete (this last statement is probably true only if ribbed rein­forcing bars are considered). Numerical implementations face the same patholog­ical difficulties related to strain softening and damage localization in shear bands near the steel bars. This problem and its consequence, mesh-dependency, were solved by using the "nonlocal continuum with local strain" approach. Predictions of pull-out tests are in very good agreement with test data, including size effect data. The study shows that it is possible to extend the nonlocal damage approach easily to the prediction of the response of reinforced concrete structures.

INTRODUCTION

Predicting the response of prestressed and reinforced concrete structures requires three major ingredients: (1) Constitutive equations for steel and con­crete that fit closely with experimental reality; (2) an objective computational method that allows predictions up to the failure of structures; and (3) a con­sistent model for the steel concrete interface.

The first item has been extensively studied over the past years and many models for concrete are available, e.g., fracturing strain models (Dougill 1979), continuous damage theories (Mazars and Pijaudier-Cabot 1989), or microplane models (Bazant and Prat 1988a, 1988b).

Finite element calculations in the presence of strain softening due to dis­tributed cracking in concrete is still a debated issue. From recent studies [see, e.g., Mazars and Bazant (1989)] it appears that the implementation of localization limiters provides physically realistic results with a finite energy dissipated at failure and objectivity with regard to the analyst's choice of finite element discretization. Results based on nonlocal continuum with local strain formulations agree with several size effect test data (Bazant and Lin 1988; Saouridis 1988).

The experimental aspects of the bond strength has been enlightened on numerous occasions [see, e.g., Somayaji and Shah (1981), Orangun et al. (1977)] but numerical aspects, and in particular the treatment of reinforcing bars in finite element calculations, may appear still unsatisfactory. Rein­forcements can be modeled in a finite element discretization by unidimen-

'Charge de Recherche au CNRS, Laboratoire de Mecanique et Technologie, ENS Cachan/CNRS/Universite Paris 6, 61, avenue du President Wilson, 94235 CA-CHAN Cedex, France.

2Prof., Laboratoire de Mecanique et Technologie, Cachan, France. 3Visiting scholar, Laboratoire de Mecanique et Technologie, Cachan, France; for­

merly, Sci., Technical University, Poznan, Poland. Note. Discussion open until August 1, 1991. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 26, 1990. This paper is part of the Journal of Structural Engineering, Vol. 117, No. 3, March, 1991. ©ASCE, ISSN 0733-9445/91/0OO3-0862/$1.00 + $.15 per page. Paper No. 25628.

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sional rods attached to the nodes of the mesh in which 2-D or 3-D elements represent the concrete. An adequate combination of shape functions provides the continuity of the displacement at the interface between steel and concrete [see, e.g., Hibbitt et al. (1987)]. In this approach the steel concrete inter­action is neglected, especially the differential deformations resulting from Poisson's effects. Recently a trend to combine interface elements with reg­ular bidimensional elements representing steel or concrete has emerged (Rots 1988; Rots 1985; Clement 1987). The nonlinear response of concrete near the steel bar is lumped into a fictitious interface that has special constitutive equations. This method has certainly become a powerful tool for predicting the response of reinforced concrete. In addition, some mechanisms such as friction that depends on the shape of the steel bars (deformed or not), have been included into the behavior of interface elements. Implementing special bond-slip relations raise the problem of the identification of additional ma­terial properties. These characteristics are usually obtained from the analysis of pull-out tests or reinforced concrete tension pull specimens [see, e.g., Dragosavic and Groeneveld (1984), Goto (1971)]. It is not surprising that the identification is based upon the same lumping method of the damage zone to an interface of zero thickness. In Dragosavic and Groeneveld (1984), the thickness of the damaged layer around the reinforcement is equal to the radius of the bar. Clement justified this technique from homogenization in 1985. Thus, omitting interface elements and considering that concrete is pro­gressively damaged around the steel bar is strictly equivalent to using in­terface elements if the bar presents surface deformations, i.e., if displace­ments are continuous at the interface. In this paper, we will restrict the analysis to ribbed bars. In particular, the large relative slip at the interface observed when nondeformed bars are used will be left out of consideration. In this latter case interface elements are needed to represent friction between steel and concrete.

In finite element applications, omitting interface elements should certainly reduce the computational cost. However, it requires the implementation of accurate constitutive equations for concrete and this is sometimes too deli­cate, although consistent results can be obtained with smeared or discrete crak concepts (Rots 1988; Ingraffea and Saouma 1985).

A scalar damage model for concrete has been developed over the past years at the Laboratoire de M6canique et Technologie. It has been applied to the analysis of plain concrete, reinforced concrete and prestressed concrete structures (Mazars and Pijaudier-Cabot 1989; Clement 1987; Breysse 1989; Bazant et al. 1987). Using three-dimensional analysis, Sun (1989) proved recently that reinforcing bars could be modeled in a 2-D calculation by an equivalent homogenized layer made of steel and concrete in which continuity of the displacements at the interface is assumed. The implementation of strain-softening constitutive equations for concrete, however, raise a major dif­ficulty in numerical analyses. From calculations on plain and reinforced concrete elements (Saouridis 1988), it appears that spurious localization instability and a strong mesh sensitivity renders the use of softening consti­tutive equations extremely delicate and sometimes physically unrealistic. A nonlocal damage model was developed to circumvent this problem (Pijau­dier-Cabot and Bazant 1987). Its application to reinforced concrete, how­ever, has never been addressed because analyses must combine local and nonlocal constitutive equations. We will show that this question can be treated

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in a simple and efficient manner. Our discussion will be mainly focused on the analysis of the pull-out test. In our opinion, this study is a prerequisite before starting on analyses on full-size structures. Comparisons with ex­perimental data will be presented and the size effect inherent to the fracture aspect of the pull-out failure will be studied from the numerical standpoint.

Davenne et al. (1989) recently applied this methodology to the analysis of reinforced concrete elements. It turned out in fact that the nonlocal model was not computer-time consuming (compared to local calculations) as the algorithm used in solving the nonlinear equations of equilibrium converges faster with the nonlocal constitutive equations. Davenne et al. (1989) also demonstrated that the model predicts the failure of reinforced concrete beams satisfactorily (maximum load, position of the cracks).

CONSTITUTIVE EQUATIONS

The geometry for the pull-out test recommended by ACI (American Con­crete Institute) or European standards is such that the steel bar embedded in concrete never yields. Therefore, linear elasticity is sufficient to describe the reinforcing bar. In usual computations on reinforced concrete members, elas-toplasticity would be required. This refinement, which is rather classical in nonlinear finite element computations, is not of interest in the scope of the present discussion.

Among the possibilities offered by the continuous damage theory (Mazars and Pijaudier-Cabot 1989), we choose this isotropic (scalar) damage model. This choice is a simplifying assumption rather than an accurate description of the behavior of concrete. Indeed, damage which represents the density of microcracks in the medium develops in specific directions: microcracks propagate perpendicular to the load in tension, and parallel to it in compres­sion. In past studies, it was demonstrated, however, that the scalar model is sufficient to describe accurately the response of plain concrete notched plates and laboratory-type fracture specimens (Saouridis 1988). The stress-strain relationship is:

<r„ = (1 - CiyCvutH (1) where ov, and eu = the components of the stress and strain tensors, respec­tively (i,j,k,l G [1,3]), Cjju = the initial stiffness moduli, and O, = the damage coefficient. The material is initially isotropic with E and v the Young's modulus and Poisson's ratio, respectively. According to this scalar descrip­tion of damage, concrete is assumed to remain isotropic up to failure. The damage coefficient il ranges from 0 for the virgin material to 1 at asymptotic failure (e,7 —» °°,o-,j —» 0).

In the presence of strain-softening spurious localization modes and mesh inobjectivity may occur. To eliminate these unrealistic features, we use the nonlocal damage formulation (Pijaudier-Cabot and Bazant 1987). The vari­able that controls the evolution of O is nonlocal. In the model, the positive strains control the growth of damage which is mainly due to microcracks opening in mode 1. The following norm called equivalent strain is defined (Mazars 1984)

*=yi>.-»2 (2) 864

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where (e,) = 0 if e, < 0 and (e,) = e, if e, > 0; and e, = the principal strains, e represents physically a measure of the amount of tensile strain for any arbitrary state of deformation in concrete. Next, we introduce the nonlocal variable e, which represents the average of € over the representative volume surrounding each point x in the material, i will be the variable that controls the growth of damage

i(x) = e(s)a(s — x)dv (3) Vr(x) Jv

V = the volume of the structure, Vr(x) = the representative volume at point x, and a(s — x) = a weight function

a(s - x) = exp —ids — x\

(4)

i 1

a(s — x)dv = Vr(x) (5)

In two dimensions k = 2. I is the so-called characteristic length of the nonlocal continuum. It is proportional to the smallest size of the damage localization zone. This length was measured experimentally (Bazant and Pi-jaudier-Cabot 1989): / = 2.8d„ in which da = the maximum size of the aggregate in concrete. The evolution of il is specified according to a for­malism that is similar to plasticity

F(l) = l-K (6)

If F(e) =. 0 and F(e) = 0 then il = / (e) (loading) and if F(e) < 0 or F(e) = 0 and F(e) < 0 then tl = 0 (unloading). Eq. 6 defines the damage surface, and in / (e ) = the evolution law of damage. K = the softening parameter and takes the largest value of e ever reached at the considered point in the medium. Initially K - K0, in which K0 = the threshold of damage. K0 may be regarded as the tensile strain at which damage is initiated. We consider that the threshold K0 is reached when the stress is maximum in uniaxial tension

K°Ji (7)

in which / , = the tensile strength of concrete. From experimental data, the growth of damage is different in tension (microcracks are directly opened by the load) from compression (microcracks open due to Poisson's effects). Thus, damage is decomposed into two parts, il, for tension and ilc for compression, il is computed from the relation in Eq. 8 which combines these two types of damage

il = a,il, + acilc (8)

The factors a, and ac are such that in uniaxial tension a, = 1, ac = 0, and il = il„ and in compression ac = 1, a, = 0, and il = ilc. The damage variables il, and ilc are functions of e

K0(l - A,) A, il, = 1 - — ; (9a)

e exp [B,(e - K0)]

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a = i -K0(l - Ac) Ac

exp [flc(€ - K0)] (9b)

and constants Ac, Bc, A„ and B, = material parameters. Eqs. 9a and 9b were introduced by Mazars (1984). He studied the sensitivity of the results to each material parameter and proposed average values of A„ B„ Ac, and Bc for concretes made with usual mix proportions. These coefficients characterize the softening response of concrete. Their range of variation is

1 < A C < 1.5 . . . .

103 < Bc < 2 • 103

0.7 < A, < 1 .2 . . .

104 < 5, < 5 • 10"

(10a)

(10b)

(10c)

(10d)

The material parameters E, v, K0, A„ Ac, B„ and Bc can be identified from standardized experiments: The response of a standard cylinder (diameter 16 cm, height 32 cm) in compression yields E, v, Ac, and Bc and data from three-point bending beams (40 x 40 X 10 cm3) provide the remaining un­knowns. The identification procedure was detailed by Saouridis (1988). Fig. 1(a) and (b) show the stress-strain law in tension and compression, respec­tively. The material parameters are E = 40,000 MPa, v = 0.2, K0 = 1.1 • 10~4, A, = \,B, = 2- 104, Ac = 1.2, and Bc = 1,515.

The factors a, and ac are expressed as nondimensional functions of the principal strains. We introduce the notation

<e,-> = ec / + e,; (11)

in which e„ = the positive strain due to positive stresses and ecl = the positive strain due to negative stresses (Poisson's effect), a, and ac are defined as (Mazars 1984)

3

€«(«„• + ec i) -iP

;=1 L

3 Crffe + Ccl)

(12a)

(12ft)

If all the stress components have the same sign, we obtain a, = 1, ac = 0 or ac = 1, a, = 0. These coefficients are less than 1 if the state of stress is tension and compression combined, that is a shear-type loading. For an arbitrary tridimensional state of stress a, + ac ¥= 1.

Let us consider pure shear: Eq. 1 reads T = [£(1 — il)/l + v]e12, where e12 = the shear strain, T = the shear stress and e = e = e12. Eq. 12a becomes

(13a) 1

1 + v,

and Eq. 12(b) becomes

A + v am 866

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' (T(MPa)

6.10-3

FIG. 1. Responses of Model to: (a) Tension; (b) Compression; (c) Shear

For large values of e12 we have fl, = D,c = 1, and we obtain

n~ I

1 + V 1 + V (14)

When P = 1, the material has failed i.e. fl = 1. The shear strength TC, which can be computed from Eqs. 1 and 8 is lower than the tensile strength: Tc = /(/(I + v), and the energy dissipated at failure compared to the energy dissipated in direct tension goes with the same ratio. This is not realistic:

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Because of aggregate interlock and internal friction the material should be more ductile in shear than in tension and the shear strength should be higher than the tensile strength. In the isotropic damage model, the directional as­pect of the microcrack propagation is lost and the effect of damage does not depend on the type of sollicitation. The purpose of exponent (3 (0 > 1) is to reduce the effect of damage on the response of the material under shear compared to tension where a, = 1. For similar reasons, the shear retention factor was introduced in smeared crack models (Rots 1988; Schnobrich 1985). In these constitutive equations the effect of damage is unidirectional, i.e., damage affects the longitudinal stiffness of a specimen loaded in compres­sion or tension. The shear modulus is kept constant in the original smeared crack concept and the shear retention factor was introduced to account for the effect of damage on the shear modulus of concrete.

It is appropriate at this point to address the question of the identification of p. From the model, experiments on concrete subjected to shear stresses would be required. Such tests are not usual and are very difficult to set up properly. An indirect procedure of identification from pull-out tests may be preferred. In the third part of this paper, we will show that numerical sim­ulations on pull-out tests are extremely sensitive to values of (J, which can then be easily identified. Fig. 1(c) shows the response of the model to pure shear for various values of p. Once the peak stress has been reached, damage starts to grow, especially O,. Accordingly, the response exhibits first a strain-softening portion, then hardening (regarded as the effect of internal friction and aggregate interlock) may occur.

FINITE ELEMENT IMPLEMENTATION: MESH OBJECTIVITY

These constitutive equations have been implemented in the finite element code CESAR developed at Labaratoire Central des Pouts et Chaussees (LCPC) ("Cesar" 1986). A secant stiffness matrix algorithm is used to solve the nonlinear equations of equilibrium. This algorithm ensures convergence of the iterative process on points located beyond the maximum load of a struc­ture, on the softening portion of the response. The nonlocal model does not require any special finite element formulation (Bazant and Pijaudier-Cabot 1988). For each integration point, the other gauss points that belong to its representative volume are identified before the calculation starts. This in­formation, which is needed in the calculation of the average effective strain, is stored once and for all in a connectivity table.

In reinforced concrete applications we need to combine local (steel) and partially nonlocal (concrete) stress-strain relations. In the boundary layer of concrete touching steel, regions of the representative volume of concrete attached to an integration point lie in steel. A similar problem is faced for the boundary layer located near a free surface: Regions of the representative volume lie outside the structure. In these cases the averaging of the equiv­alent strain in Eq. 3 must be reconsidered: The shape of the ribs and the material properties of steel should not influence the calculation of e in the interface region. Therefore, the presence of steel can be regarded as a special boundary condition applied to concrete, which does not modify the consti­tutive equations of the material. The computation of i near the interface is identical to the computation of e near a free boundary, and regions of the representative volume protruding outside concrete are deleted from the in-

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2.5 cm

L

'H

d 9 i •^v;'

* . ' • • ' . . - •

'il-"t. #1' •

#3;

1

t ; " : • • ; • ;

•. » ..

• ' ' •'* V

?&

J

t steel

(a)

7 1 L 1 I l\

-. i 2 -

1 1 ' 1

35 elements

i

1 1

1

1

1

7 •(b)

/

1

1

1

1

/ 1

/' / .

'/

•

7 1

i — /

',

120 elements | 460 elements

. / •

?.. - 1

,

. i " - "

-'-- --- -. ?' ~

-1

_,.,.-- _ .t.-~

. r ~ " ----'-- --- - - - --

. ' - ? —

-/-- — — . "> -'--- -

- r " --- -w ;::::::::::::::::::::::

FIG. 2. Pull-out Test: Specimen Geometry; and Finite Element Meshes for Con­vergence Study

tegrals in Eqs. 3 and 5. Physical or theoretical reasons for this special av­eraging have not been established yet. In a recent work, Pijaudier-Cabot and Berthaud (1989) showed from the study of an interacting crack system that damage should be nonlocal. Justifications for the calculation of the nonlocal damage in the boundary layer near a free surface require the study of the interaction of a system of cracks with a free surface. This work is currently in progress.

Computations on a pull-out test with various meshes were carried out to investigate mesh sensitivity. The specimen is a cylinder of concrete cast in a rigid metal hollow cylinder of internal diameter 0 = 20 cm and height h = 15 cm (Fig. 2). A steel bar of diameter db = 40 mm is embedded at the center of the specimen. The pull-out force is applied at one end of the bar and displacements on the circumferential surface of the cylinder of concrete are prevented. The material parameters for concrete are E = 29,000 MPa, v = 0.20, Ac = 1.07, Bc = 1,512, A, = 1, B, = 2 • 104, (3 = 1, and / = 60 mm. For steel we took E = 210,000 MPa and v = 0.3. The finite element discretizations are two-dimensional axisymetric. Five meshes made of four-noded elements were considered (15, 35, 120, 460, 810 elements [see Fig. 2(b), 2(c), and 2(d)]). Computations with the local (/ = 0) and nonlocal damage model were compared.

Fig. 3 shows the load versus displacement of the steel bar at the point of application of the load. In Fig. 3(a) we have reported the results from the local calculation. We see that convergence with mesh refinement is very slow in this case. The maximum load decreases from 19.2 kN (mesh with 15 elements) to 7 kN for the mesh with the largest number of elements. The nonlocal calculation exhibits almost identical results for the meshes with 460 and 810 elements [Fig. 3(b)]: The maximum load changes just slightly, and overall the curves are similar with an identical softening response after the peak load. The difference between local and nonlocal calculations is much

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20.00 -i

0.00 0.00 5.00 10.00 15.00 20.00

displacement (mm x 1000)

FIG. 3. Convergence of F.E. Results with Mesh Refinement, Comparison with: (a) Local Model; (h) Nonlocal Model

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II pj;>: | l | j ' ' :

F IS; PA'-mi-

m BBSS'

', x\* $ if m •

Damage from to 0.143 0.286

0.286 0.429 0.429 0.S71 0.571 0.714 0.714 0.857 0.857 1.0

local damage

(a)

nonlocal damage

(b)

fi O N

T3

o

T3 • I — I

4 _ nonlocal damage model

local damage model

- - - ^

200 400 600

number of elements

FIG. 4. Damage Zone at Peak Load: (a) Local and Calculation for Mesh with 120 Elements; (b) Nonlocal Calculation for Mesh with 120 Elements; (c) Width of Lo­calization Zone at Failure versus Number of Elements

more significant when we look at the predicted distribution of damage at failure. Fig. 4 shows the local and nonlocal results for the mesh with 120 elements. The dark region represents concrete that has failed (0.85 < ft < 1). In the local calculation damage remains confined to the row of element located next to the steel bar. This is not the case for the nonlocal compu­tation. We see that damage has propagated over a large number of elements and the shape of the damage zone is consistent with previous computations and experimental observations (Rots 1988). We have reported on Fig. 4(c)

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a 6

•O 0.2

</> Pi o o

M> 0 .1

S

0.0

nonlocal damage model

local damage model

I I I 200 400 600

number of elements

FIG. 5. Convergence of Energy Dissipated at Failure with Mesh Refinement: Comparison for Local and Nonlocal Calculations

the evolution of the thickness of the damage zone at failure with mesh re­finement. H is defined as the minimum thickness of the damage layer around the bar in which O. > 0.85. As expected H tends to zero when the element size goes to zero in the local analyses, and the width of the band is clearly controlled by the size of the finite elements. H is almost constant in the nonlocal calculation for meshes made with 120 elements and more. A three-dimensional solution to this damage localization problem would be required to establish accurately the relation between the width of the localization zone and the characteristic length of the material.

Finally we have plotted on Fig. 5 the evolution of the energy dissipated due to damage W with the number of finite elements in the mesh. This energy corresponds to the area under the load-displacement curves on Fig. 3 and it represents the amount of energy consumed when the steel bar is completely pulled out. From this plot, we can see that in local computations W goes to zero when the number of element increases. It means that bond breaking occurs without energy consumption, which is physically unrealistic. The nonlocal formulation remedies to this problem and a proper convergence toward a nonzero energy dissipated at failure is observed.

COMPARISON WITH TEST DATA

Adrouche and Lorrain (1987) performed a series of pull-out tests on spec­imens similar to Fig. 2(a). The diameter of the concrete cylinder was re­duced to 0 = 85 mm, with h = 25 cm and db = 12 mm. Tests were con­ducted at an early age (three days). Strain gauges were glued on the steel bars at regular intervals (see Fig. 6), and the strain profiles along the bar were recorded at different load levels. From compression and bending tests,

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F*

Damage from to A B C D E

FIG. 6. Comparison of Predicted Strain Distribution in Reinforcing Bar with Ad-rouche's (1987) Results

the following material properties were measured: fc = 16 MPa (compressive strength),/, = MPa, and E = 20,909 MPa. In the present model we assumed that/, was reached at the onset of damage in tension (for e = K0), thus we obtained K0 = 1.1 • 10~4. The parameters Ac and Bc were fitted from the reported data in compression: Ac = 1.07 and Bc = 2,000, with v = 0.2. The experimental stress-strain curve of concrete in tension was not reported and usual values of A, and B, were selected: A, = 0.8, B, = 2 • 104. Finally, the maximum size of the aggregates was da = 3.15 mm, which yields I = 8.8 mm.

We used a mesh with 160 degrees of freedom for the axisymmetric com­putation and fitted the coefficient p so that numerical results would give the same strain at gauge A when the maximum load applied in the experiments is reached. We obtained p = 1.1.

Fig. 6 presents the comparison between the calculated and measured strain profiles for the loads P = 2, 14, 18, and 24 kN. Considering the lack of information provided, especially the response of concrete in tension at early ages, the agreement is good. Fig. 6 shows also the progression of damage with increasing loads. In these experiments [see also Adrouche (1987)] dam-

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P = 1.0 P= 1.1 P=l-4

FIG. 7. Effect of p on Damage Zone at Constant Load: (a) (3 = 1.0; (b) (3 = 1.1; (c) p = 1.4

age was first noticed on the global response at P = 14 kN (onset of nonlinear behavior). Our calculation agrees with this observation. The experiments were not conducted up to the maximum load; we calculated it at approximately 28 kN. Our comparison further suggests that it is possible to obtain p with reasonable accuracy from load versus slip (or strain in the steel bar) rela­tions.

It is also of interest to investigate the sensitivity of our predictions to variations of p. Fig. 7 shows the result of calculations for (3 = 1.0, 1.1, and 1.4 under the load P = 14 kN. For (3 = 1, the distribution of damage shows that P is almost maximum. The shear band has propagated through the entire length of the specimen. If (3 = 1.1 or p = 1.4, the distribution of damage is quite different. It has affected less than one-third of the inter­face length. The maximum loads are P = 14.5 kN, P = 28 kN, and P > 40 kN for p = 1, 1.1, and 1.4 respectively. A 10% increase of p from 1 to 1.1 results in a 100% increase of the pull-out strength. Since p reduces the effect of damage when concrete is subjected to shear, such a conclusion was expected. These results point out that the original damage model, in which ( 3 = 1 , underestimated the maximum carrying capacity of structural

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members loaded in shear. We may further remark that the assumption of isotropic damage also precludes any variations of Poisson's ration, and di-latancy of concrete is neglected. In the present comparisons, this assumption does not seem to be too restrictive.

SIZE EFFECT AND PULL-OUT TESTS

In most building codes ("Building" 1983; "Code" 1978), the steel-con­crete bond is characterized by a strength criterion. On the other side, various experiments [see, e.g., Goto (1971)], point out the fracture aspect of the pull-out failure, which can be decomposed into two mechanisms:

• The splitting failure is caused by the radial pressure applied by the lugs of the bar to the concrete. This pressure is balanced by circumferential tensile forces causing cracking in concrete.

• The shear failure occurs if the concrete cover is much larger. Concrete is crushed in front of the lugs and cracking occurs in a band (the "shear band") located near the interface. Debonding does not appear, however; the lugs prevent any steel concrete relative slip. Large slip is the conse­quence of microcracking in the damage zone around steel. This crack opening is represented by shear strains (fracturing strains) in the homogenized ma­terial equivalent to damaged concrete (Ladeveze 1983).

Since cracking does exist, a size effect should certainly be expected in both types of failure. Bazant and Sener (1988) studied the size effect on the pull-out strength from experiments on geometrically similar specimens. They proposed an approximate prediction formula

vu ( . d„\~l/2

-? = *, l + \o' 7 (15) Ci \ dj where

C, = ( l . 23 + 3 . 2 3 - + 53 — jVfc (16)

Vu = the maximum shear stress, db = the diameter of the steel bar, da = the maximum aggregate size, Cx = the minimum cover, and ku and X0 are material properties. The constant d in Eq. 15 is computed in psi (1 psi = 6,895 N/mm2) and corresponds to the empirical formula of Orangun et al. (1977) Eq. 16. V„ is computed from test data

VU = P-^ (17)

where S = the nominal surface area of the embedded bar and Pmax = the maximum pull-out load. Eq. 15 may be rewritten in a form suitable for linear regression from which kx and X0 are identified

C\ 1 1 db

~i = -2 + : - (18) Vl kj \0kj da

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FIG. 8. BaSant and Sener (1988) Pull-out Tests: (a) Specimen Geometry, (b, c, d) Finite Element Meshes

The specimen tested were concrete cubes of side length 38.1 mm, 76.2 mm, and 152.4 mm in which a steel bar of diameter 1.6 mm, 3.2 mm, and 6.4 mm, respectively, was embedded. The embeddement length ld was 12.7 mm, 25.4 mm, and 50.8 mm, respectively. The load was applied on the steel bar, and the cube was held down by a square sleeve [see Fig. 8(a)]. The side of the square respected also the similitude (6.35, 12.7, and 25.4 mm). The compressive strength of concrete reported was/c = 45.8 MPa (measured on companion cylinders), and the maximum aggregate size was da = 6.4 mm. The material parameters that were not reported were fitted so that the closest possible prediction of the response of the only medium size specimen was achieved. We obtained:/, « / c / 10 = 4.5 MPa, E = 40,000 MPa, v = 0.2, /(To = 1.1 • 10~4, A, = 1, B, = 2-104, Ac = 1.2, Bc = 1,515, and 0 = 1.065 (with 1 ~ 2.8, da = 17.7 mm). It must be stressed that this set of material properties was obtained by fitting the model with results on spec­imens of one size. Results on the two other sizes are true predictions. A same procedure was followed by Saouridis (1988) and yielded a size effect

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FIG. 9. Pull-out Tests; Predicted Force versus Slip Curves for Three Specimen Sizes

on the tensile strength of concrete consistent with experimental data. For the three specimen sizes, axisymmetric finite element computations

were carried out. The concrete block was represented as a cylinder of radius R (R = 21.5 mm, 43 mm, and 83 mm, respectively) such that the cross sections were kept identical to those in the experiments. Variations of R certainly affect the cover of the bar. However, we did not observe substantial differences between computations in which R is equal to half the side of the square and our computation. The finite element meshes are presented in Fig. 8(b), 8(c), and 8(d) and the numbers of degree of freedom were 160, 880, and 3,200 for the small, medium, and large size specimens, respectively. In the calculation displacements of the steel bar (at points A on Fig. 8) were controlled.

Fig. 9 shows the force versus displacement curves for the three sizes and the damage zones around the reinforcing bars are reported on Fig. 10. We remark that the extent of the damage zone in the radial direction decreases as the size of the specimen increases. The damage zone (0.85 s D ^ 1) has the shape of a hook for the large specimen, which may be explained by the presence of compressive stresses under the sleeve which delay the pro­gression of damage. From one size to another, the shape of this zone, which is also the localization zone, changes depending on whether it includes the retaining sleeve or not. It shows the well-known influence of the boundary conditions on the localization zone. For tests in which these conditions are remotely applied, e.g., three points bending on notched specimens, the lo­calization zone kept the same size and shape (Bazant and Lin 1988; Saouridis 1988). This observation further suggests that the size effect in this type of pull-out test results from a much more complicated localization of damage.

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Damage from 0.143 0.286 0.429 0.S71 0.714 0.857

to 0.286 0.429 0.571 0.714 0.857 1.0

(b)

FIG. 10. Pull-out Tests; Damage Zones at Peak Load for Specimens: (a) Small; (to) Medium; (c) and Large Specimens

It is not possible, unfortunately, to infer from our calculation which type of failure mechanism occurs. Qualitatively Fig. 10 suggests that for the large specimen, damage propagates close the interface. This is typically the case of shear failure. For the two smaller specimens the damage zone extends well over half the concrete block. Because our computation is axisymmetric and because we use an isotropic damage model, conclusions cannot be drawn. For this purpose a three-dimensional computation in which anisotropy due to damage is accounted for, would be more appropriate. Fig. 11 shows the identification of the size effect relation in Eqs. 15-18 and its comparison with experimental data. We found fc, = 4.47 and X0 = 0.303. On the usual l°g(V„/C,) versus log(db/da) plot [see Fig. ll(fc)], the numerical predictions are consistent with the experiments. Due to the size effect, the maximum shear strength V„ varies from 17 MPa for the small specimen to 14 MPa for the larger specimen. This 20% variation supports the applicability of the size

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.

0.5

0.4

03

02

0.1

0

ICf/Vut2

A Bazant ,Sener 1988 * F.E. Calculation

^*>

a A

A

*

A

. . i

0.5 1 * I* 1S

Log!Vu/Cj )

0.6

0.5

0.i

0.3

0.2

0.1

A Bazant, Sener 1988

""•"" F.E. Calculation

-0.4 -0.2 0 0.2

Log (dDId9 )

0.4

FIG. 11. Comparison with Experimental Data: (a) Identification of Size Effect Law; (b) Size Effect in Pull-out Test

effect formula and brings up questions as to whether it should be accounted for in current engineering calculations and design.

CONCLUSIONS

1. The proposed method of analysis of the response of reinforced concrete elements is based on two assumptions: (1) There is no discontinuity of displace-

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ments at the steel-concrete interface; and (2) the isotropic (scalar) damage model is accurate enough to model the response of concrete under shear stresses. A simple modification dealing with the combination of damage due to tensile stresses, and damage due to compressive stresses was proposed. The original formulation of the model clearly overestimated the growth of damage due to shear stresses and consequently it underestimated the shear strength of concrete. Effects such as internal friction or aggregate interlock were not represented. An additional material parameter B was introduced for this purpose.

2. The implementation of the nonlocal damage model in the analysis of rein­forced concrete elements must combine local and nonlocal constitutive relations. This is achieved by modifying the computation of the nonlocal damage variable: when the domain of integration intersects the steel bars, the zone lying in steel is chopped off so that the average is computed from the local effective strain in concrete only. Finite element computations on an axisymmetric pull-out test show that the distribution of damage at failure is not sensitive to mesh refinement. The width of the damaged layer of concrete around the bar does not depend on the size of the finite elements and should be controlled by the characteristic length of concrete. Finally, the energy dissipated at failure (i.e., when the bar is completely pulled out) converges toward a nonzero value with mesh refine­ment. These properties are not observed in calculations when local strain-soft­ening constitutive equations are used and they are essential in the scope of future applications on reinforced concrete beams or frames because computations should not be affected by a spurious damage localization due to the finite element dis­cretization.

3. The size and shape of the damage zone are consistent with experiments and with previous calculations with smeared or discrete crack models. Compar­isons with existing test results show that the present technique allows accurate predictions of the bond strength and of the evolution of damage near the inter­face. The formulation is also able to predict a size effect consistent with ex­perimental observations.

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