2d mesoscopic modelling of bar–concrete bond

Engineering Structures 49 (2013) 696–706

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Engineering Structures

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2D mesoscopic modelling of bar–concrete bond

0141-0296/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.11.018

⇑ Corresponding author.E-mail address: [email protected] (O. Maurel).

Atef Daoud a,b, Olivier Maurel b,⇑, Christian Laborderie b

a LGC, Laboratoire de Génie Civil, ENIT, Université de Tunis El Manar, Tunisiab SIAME, Laboratoire de Sciences Appliquées à la Mécanique et au Génie Electrique, Université de Pau et des Pays de l’Adour, France

a r t i c l e i n f o

Article history:Received 6 March 2012Revised 27 September 2012Accepted 19 November 2012Available online 24 January 2013

Keywords:BondMesoscopic approachConcrete damageBar roughnessConfinement stress

a b s t r a c t

The degradation of the steel–concrete connection is a complex phenomenon depending on both the inter-face behaviour and the surrounding concrete damage. The purpose of this paper is to study bar/concreteinterface behaviour by performing a numerical analysis at the mesoscopic scale. The interest of thisapproach is to provide local information unavailable from experimental investigations. The presentedresults concern the 2D modelling of LMT pull-out test (to identify bond and friction) with the assumptionof perfect bond between the mesoscopic mortar and the steel bar. The influences of bar roughness and thelateral stresses were analysed.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The bond between reinforcing bars and concrete has beenacknowledged as a key factor to the proper performance of rein-forced concrete structures for well over 100 years. Bond responsemay be modelled at three different scales: the dimensions of thestructural element, the reinforcing bar and the lugs on the bar.Modelling bond behaviour at the structural scale implies thedevelopment of a model that characterises the effect of bond-zoneresponse on beam, column or connection response.

At the bar scale, the bond zone is represented as a homogenouscontinuum. The state of the bond zone may be characterised byconcrete and steel material properties [1,2] that are defined bystandardised tests (pull-out test, beam test, etc.).

Bond response can also be considered at the scale of the lugs onthe reinforcing bar. At this scale, the response is determined by thematerial properties of the concrete mortar and aggregate [3,4], thedeformation pattern of the steel reinforcing bar [1,5,6], load trans-fer between concrete mortar and aggregate [7] and the rate of en-ergy dissipation through fracture and crushing of the concretemortar and aggregate. The numerical concrete (initially proposedby Wittman [8]) can be a good solution to represent concretemeso-structure numerically. Mesoscopic models have proven tobe the most practicable and useful approach for studying the influ-ence of the concrete composition on the macroscopic propertiesand also to gain insight into the origin and nature of the nonlinearbehaviour of concrete [9].

To perform the mesoscopic representation of concrete material,both discrete element methods, such as truss model [10] and lat-tice model [11,12], and continuum finite element methods[13,14] have been adopted. In most of the mesoscopic models,the concrete is subdivided into three phases: the coarse aggregates,the mortar matrix with fine aggregate and the interfacial transitionzone (ITZ) which affects the initiation and propagation of cracks.The mechanical behaviour of the concrete belonging to the ITZ issignificantly different from that of the bulk concrete. However, itis very difficult to obtain the mechanical parameters of ITZ. There-fore including ITZ in the model introduces some uncertainties.Moreover, considering ITZ in the model increases the computa-tional time and computer memory requirement. For these reasons,in some models the ITZs are not considered in the numerical sim-ulations [9,13,16,15,14,30] and recently [34]. Different meshingtechniques have been applied for the discretisation of the complexmicrostructure. Aligned meshing approaches have the advantageof explicitly representing the boundaries between particles andmatrix. However, this method is rather tedious in 3D [13] devel-oped a mesh generation method based on the advancing front ap-proach in which the ITZ is modelled. Van Mier and Van Vliet [11]used a projection method of regular mesh onto the random aggre-gate structure. Zohdi and Wriggers [32] introduced an unalignedapproach in which the number of integration points is increasedin order to better capture the interfacial zone. A refined mesh closeto the geometrical boundaries has been proposed by Lohnert [33].Nguyen et al. [15] used a diffuse meshing technique which consistson the projection of the heterogeneous material properties on theshape functions of a finite element mesh. Each Gauss point takesthe corresponding material properties. Although this method does

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A. Daoud et al. / Engineering Structures 49 (2013) 696–706 697

not represent explicitly the ITZ, it allows a good reproduction ofdamage localisation around the aggregates.

In a mesoscale model, the most important parameters, such asthe shape, size and distribution of coarse aggregates within themortar matrix, significantly influence the mechanical behaviourof concrete. The simplest aggregate shape is circular [11,15] (2D)and spherical [9] (3D) [13] developed a procedure to generate ran-dom aggregate polygons for rounded and angular aggregates basedon the Monte Carlo random sampling principle. Garboczi et al. [17]developed an algorithm to generate realistic shapes of differentaggregate particles.

On the other hand, various material models for the aggregateand the mortar have been employed to study the concrete behav-iour, for example, 2D linear elastic analysis [18], nonlinear ortho-tropic fracture model [14] and isotropic damage model [9,19].

The mesoscale modelling of the connection between steel andconcrete have been very little studied. A lattice approach has beenused, by [20], to describe the mechanical interaction of a corrodingreinforcement bar, the surrounding concrete and the interface be-tween steel reinforcement and concrete. The effects of rib on sur-rounding concrete are taken into account with a cap-plasticityinterface model. The rust expansion is modelled as an Eigenstrain.This approach is very interesting and capable of representing manyof the important characteristics of corrosion-induced cracking andits influence on bond. Nevertheless, the post-peak response of thebond stress–slip curve is not in agreement with experimental re-sults: the parameter controlling the volume of rust expansion isdependant of local phenomenon: crushing of the rust and penetra-tion of rust in concrete.

1.1. Research significance

The present paper aims to analyse the bond force transfer, thedamage evolution, crack pattern and displacement fields on thesurrounding concrete by using a mesoscale modelling.

The bond test used in this study is the LMT pull-out test [21,22].This geometry does not induce lateral stresses due to confining ac-tions at the support of the testing machine. It enables local infor-mation (deformation and displacement fields) to be obtained onthe steel/concrete interface not available with the standardisedtests. Based on the geometry particularity of the LMT test, a planestress 2D model has been adopted to limit time computation andto simplify the modelling. A heterogeneous mesoscale model isconstructed for mortar material. The effect of bar roughness andlateral pressure are analysed to study the influence of aggregatedistribution on the crack pattern around the reinforcing bar andthe force transfer between steel and concrete.

2. Bond test

Various tests have been proposed to assess bond characteristicsin reinforced concrete structures. Most of the classical bond test(pull-out test) does not allow a local measure but only globaland make impossible to distinguish bond phenomena at the inter-face steel/concrete due to the confining actions at the support ofthe testing machine introducing a lateral stresses, which artificiallyincrease the bond strength.

In order to improve the classical bond tests, a modified pullouttest (called the LMT test: Laboratory of Mechanics and Technology,Cachan, France) has been proposed and designed by Ouglova et al.[21,22,35]. The aim of this test is to identify bond and friction atthe interface. It consists of a mortar plate with three squaredembedded bars (cross section 2 cm � 2 cm) (steel A56 usually usedin the civil engineering structure). The plate has a thickness of4 cm. The mortar has the following characteristics: maximum

aggregate size 4 mm, water to cement ratio equal to 0.46. The spec-imen have been cured during more than 1 week in a plastic vialsthen moved to the storage piece.

The geometry of the specimen and the experimental setup havebeen designed in order to eliminate the lateral stresses manifestingin the concentric pullout test (Fig. 1b). This test assesses the eval-uation of displacement and deformation fields during the test byusing a digital image correlation technique (Fig. 1c).

As a Teflon has been put around the central bar outside the zonecorresponding to the window, the bond length is only 12 cm. Thepullout load was applied at the free end of the central bar of thespecimen (Fig. 1a). The two other bars were loaded in the oppositedirection with respect to two bearings able to roll on a fixed beam,perpendicular to the loading direction, in order to avoid any lateralstress and parasite flexural actions. The central bar is in contactwith mortar on only two sides (Fig. 1a coupe A–A).

This test has been created to characterise the behaviour ofsteel–concrete interface particularly when the steel is corroded.To model the behaviour of corroded reinforced concrete structures,it is necessary to identify the behaviour of corroded steel, thebehaviour of concrete, the properties of rust and finally the behav-iour of steel–concrete interface. In a classical pull out test, there is alongitudinal displacement between steel and concrete (slip) due toapplied force. A triaxial state of stress is observed around the bar.The pull-out capacity is mainly characterised by a mean value ofresistance. In the development of the new LMT test, the main ideais to eliminate the complex triaxial state of stress and to have amore simple state of stress due to lateral stresses imposed ontwo faces of concrete specimen. The stress state is almost homog-enous. It prevents from an increase of confining forces during bothclassical pull out and tied tests. This new test that eliminates thelateral confinement also allows to impose very simply a known lat-eral pressure on the interface. This simplicity allows to determinevery easily the friction angle and cohesion of interface from theshearing stress-confining stress curves. The interface behaviourcan be then easily modelled.

3. Mechanical behaviour model for concrete components

Macroscopic models generally have many parameters to de-scribe the complex mechanical behaviour of concrete. The mainpart of this complexity is due to geometry, so we choose for pasteor aggregate a mechanical behaviour model as simple as possible,based on Mazars’s model [23]. The contact between paste andaggregate is considered as perfect. The model used is Fichant’smodel [24] which controls fracture energy Gf. The plasticity couldbe activated if necessary. Damage effects in compression at themesoscopic level are decreased with respect to macroscopic behav-iour. Since, it has been demonstrated that the behaviour in com-pression of concrete is partially due to the aggregate spatialdistribution. This simple model represents unilateral effects andprovides objective results whatever the mesh size as shown by[19]. In its original version, the model couples damage and plastic-ity; here, we present only the isotropic damage part of the model.

The effective stress ~r is obtained from strain e and initial char-acteristics of materials E and t:

~rij ¼E

ð1þ tÞ eij þEt

ð1þ tÞð1� 2tÞ ekkdij ð1Þ

Then the stresses rij are calculated from the damage variable D:

rij ¼ ð1� DÞh~riþij þ ð1� DÞa1 h~ri�ij ð2Þ

where hXiþ and hXi� design the positive and negative parts of tensorX defined by [25]. Damage is calculated from equivalent strain ~e de-fined by [23].

Fig. 1. (a) LMT specimen geometry [18], (b) lateral stress along the bar for the LMT test and classical Pull-out test [18], (c) specimen, testing machine (with CCD camera) [18].

698 A. Daoud et al. / Engineering Structures 49 (2013) 696–706

D ¼ 1� ft

E~eexp

hft

Gf

ft

E� ~e

� �� ð3Þ

For ~e >ft

Eand _D > 0

where ft is the tension strength, Gf is the fracture energy and h is thesize of the considered finite element [19]

For a regular isotropic mesh: h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRR

Xedxdy

q(in 2D computa-

tion), where Xe is the finite element.

Fig. 2. Mortar aggregate grading curve.

4. Mesostructure generation

The evaluation of the composite behaviour of mortar at meso-scopic level requires the generation of a numerical mortar withan aggregate structure, which consists in randomly distributedaggregates and cement matrix filling the space between the parti-cles. The numerical mortar model is developed in two dimensions:aggregates are represented by discs. Discs are randomly distrib-uted according to a given particle size distribution curve (mortarwith a 0/4 mm sand) (Fig. 3). We use a logarithmic distributionof class size and choose seven classes to describe the aggregates(Fig. 2).

In order to have a correct description of the geometry, thesmallest size of aggregate taken into account is 0.5 mm and thematrix represents the cement paste and small aggregates. Withthe volume percentage of the particle size distribution of aggre-gates (assuming same density of all aggregates, issued from same

Fig. 3. Numerical mortar.


rocks) and with the total volume of aggregates in the concretesample, the inclusion number for each granular class can be easilycalculated. A FORTRAN procedure, developed by [15], has beenadapted to randomly distribute aggregates in the specimen. Toavoid boundary effects which automatically increase the volumeof matrix between aggregates, the inclusions are placed in a largersample and the parts of aggregates located outside of the concretesample are cut. The principles of random aggregate distribution aresummarised as follows:

� The draw for the position of aggregates is made from the largestto the smallest aggregates in the mortar specimen with itsframe (all inclusions must have a non-null intersection withthe sample).� The coordinates (Xi,Yi) of the centre of a particle with Di diam-

eter are determined by a random function with no overlap withpreviously placed particles or steel–concrete boundary.� For each granular class, all aggregates must have a correct posi-

tion, and the process must continue to the following class untilthe last particle of the smallest class. If a particle cannot find aplace, the procedure stops and another starts, until it performsto place all aggregates.� The next step is to cut off all the aggregate parts outside of the

concrete specimen, and to verify the aggregates/paste ratio.

5. Simulations

The pull-out specimen is not entirely modelled with mesoscopicapproach: the mortar zone around the bars uses the mesoscopicmodel, the remains of concrete specimen has an elastic behaviourwith the characteristics of the homogenised mesoscopic model andis modelled with a coarse mesh. Such a procedure is undertaken totest ability of the model to be integrated in a global structural cal-culation. It enables the computation time to be decreased. Duringthe loading simulation, the damage zone must be included intothe mesoscopic zone. To determine the size of the mesoscopiczone, a macroscopic simulation has first been undertaken.

Despite the fact that the experiments are designed to be bi-dimensional, it must be noticed that the use of a mesoscopic modelwith a plane approach is idealised since the heterogeneities, in thiscase take the form of cylinders. The choice of a plane approach isjustified by our desire to best represent small heterogeneities thatare essential to the compactness of the material and by limiting thenumber of degrees of freedom given by the computer means (weused an 48 cores computer with 256Go of memory). The developedmodel is available for tridimensional approach but is now limitedto small structures.

If during the calculation the damage is near the borders of themacroscopic model, then the mesoscopic zone is increased so thatthe total energy dissipation due to cracking occurs in it. In the con-nection area between the macro-mesh and the mesoscopic mesh,the nodes of the two meshes are located on a common line butdo not necessarily correspond (Fig. 4a). The coupling is achievedby imposing the nodes of the mesoscopic mesh displacements con-sistent with those given by the shape functions of the macroscopicmodel. At the mesoscopic mortar–steel bar interface, a perfectbond is assumed.

5.1. Meshing method

The numerical concrete specimen is implemented in the FiniteElement code CAST3M: the mesoscopic mesh is obtained by usingthe diffuse meshing method [15]: projection of the heterogeneousmaterial properties on the shape functions of a finite elementmesh. Each Gauss point takes the corresponding material proper-ties (Fig. 4b). Calculation of the elementary stiffness matrix Ke

takes into account the distribution of different materials:

Keij ¼ZZ 1

�1Bt

ikðn;gÞCklðn;gÞBijdetðJÞdndg ð4Þ

Keij ¼XN

g¼1

Bkiðng ;ggÞCklðng ;ggÞBijðng ;ggÞxg ð5Þ

where xg is the weight of Gauss point (ng, gg) coordinates of Gausspoints and: Bij ¼ 1

2 ðVNij þ VNjiÞ, Nij shape function matrix.The degree of the polynomial function exactly integrated in Eq.

(5) depends on the Gauss point number. In our case, the approxi-mated function is piecewise constant and Eq. (5) gives an approx-imation of the elementary stiffness matrix. When considering aconstant element size, this method enables the representation ofa smaller aggregate size compared to the method that supposesconstant material properties into each element. As a consequence,this method has the advantage to allow a better description of thegrading curve of concrete and to reach a good granular compact-ness. However, it does not enable to model the ITZ. Nevertheless,the stiffness gradient between paste and aggregates (Fig. 4b) gen-erates a damage localisation at the interface between paste andaggregates which well describes the interfacial tension cracksaround the aggregates. A four-node quadrangular iso-parametricplane stress elements (0.33 mm), size is used.

5.2. Identification of the mechanical parameters for the mortar

The aggregates are divided into two different materials accord-ing to their diameter: the aggregates whose diameter is greater orequal than 0.5 mm are modelled as aggregates (they correspond toa volume fraction of 0.35) with a maximum aggregate diameter of4 mm; the ones whose diameter is lower or equal than 0.25 mm(they correspond to a volume fraction of 0.35 with a minimumaggregate diameter of 0.063 mm) are taken into account into thematrix. The matrix properties are obtained from another computa-tion of a ‘‘micro-mortar’’ made of pure cement paste and smallestaggregates (Fig. 5a). It is, of course, necessary to characterise themechanical behaviour of this ‘‘micro-mortar’’.

5.2.1. Micro-mortar identificationNumerical tension tests were carried out on a 4 � 4 � 4 mm3 spec-

imen. A small finite element size has been chosen (0.05 mm < small-est aggregate diameter 0.063 mm). The paste Young’s modulus hasbeen evaluated using a computation prediction proposed by [26].The value of Epaste is taken equal to 12 GPa corresponding to a w/c ra-tio equal to 0.5. The ratio between Young’s modulus of paste and

Fig. 4. (a) Connection between meso and macro-meshing and (b) diffuse meshing method.

(a)

(b) (c)

ft =2,25MPaE=21GPaGf=44J/m2

ft =3,43MPaE=31,8GPaGf=82J/m2

40mm

40mm4mm 4mm

MortarCement paste

+ fine aggregates(d 0,25mm)

Micro-mortar

Aggregates (d 0,5mm)

Homogenization of the micro-mortar properties

+ Aggregates(d 0,5mm)

Fig. 5. Numerical tension tests (a) modes of consideration of fine aggregates in numerical analyses, (b) micro-mortar and (c) mortar.


aggregate is assumed to be 3 (i.e. EAggregate = 36 GPa). The calculationshows that the ‘‘micro-mortar’’ has tension strength of 2.25 MPa, aYoung modulus of 22 GPa and fracture energy of 44 J/m2 (Fig. 5b).

5.2.2. Mortar identificationNumerical tension tests were carried out on a 4 � 4 � 4 cm3

specimen. The choice of the parameters defined in Section 5.2.1for the matrix and the same properties for the coarse aggregatethan the small aggregate allows a good approximation of theexperimental macroscopic mechanical properties of the mortar:

tensile strength ft of 3.0 MPa, Young modulus E of 30 GPa and frac-ture energy Gf of 80 J/m2 (Fig. 5c). For the elastic zone, mortar hasthe homogenised properties of the mesoscopic model.

The fracture energy of cement paste (matrix) including smallestparticles (diameter < 0.5 mm) has a relatively low value comparedto common values used in macroscopic simulations. Indeed, thecrack growth is governed by the aggregate distribution: for a lowstrength mortar, when a crack reaches the aggregate/cement pasteinterface, in most cases it has to get around the aggregate. Thematerial ductility increases and, of course, the fracture energy of

Table 1Mechanical properties of materials.

E (GPa) m ft (MPa) Gf (J/m2)

Mesoscopic zoneCement paste 12 0.2 2.00 15Cement paste + fine aggregates

(d 6 2.5 mm)21 0.2 2.25 44

Aggregate (d > 2.5 mm) 36 0.2 4.5 60Elastic zone 30 0.2 – –Steel 210 0.3 – –

Elastic zone

Mesoscopic zone

Pull-out load

Steel

Fig. 7. Definition of the mesoscopic zone.

Fig. 8. Meshing of the mesoscopic zone.


the homogenous material (cement paste and aggregates) increases.Table 1 summarizes the mechanical properties of materials used inthe calculation.

5.3. Macroscopic computation to determine the dimension ofmesoscopic zone

Macroscopic simulations are firstly carried out in order to deter-mine the size and the position of the mesoscopic zone given by thesize of the damaged zone. The same damage model used at themesoscopic level is used for concrete with the macroscopic proper-ties above established (ft = 3 MPa, E = 30 GPa and Gf = 80 J/m2). Thesteel is considered as an elastic material. For this computation, asmall finite element size is chosen (0.33 mm) for the entire speci-men. Fig. 6 shows the damage of the specimen from the macro-scopic computation. Damage is localised in the window aroundthe bar.

Fig. 7 shows the choice of the size and the position of the mes-oscopic zone. Fig. 8 shows the meshing of the mesoscopic zone inthe LMT test specimen using the diffuse method presented inSection 5.1.

5.4. Mesoscopic computation

5.4.1. Case of smooth bar without confinement5.4.1.1. Global behaviour. Fig. 9 presents the variations of the loadversus slip for macroscopic simulation, mesoscopic simulationand experiment. The mechanical properties for both the mesoscop-ic and macroscopic simulations are shown in Table 1. Finite ele-

Fig. 6. Concrete damage in macroscopic computation.

Fig. 9. Pull-out load versus slip curve.

ment size is set equal to the one used in the mesoscopic zone.Computed load versus slip curve using mesoscopic approach de-scribes accurately the global behaviour of the connection betweensmooth bar and mortar (stiffness and strength). However, com-puted load versus slip curve using macroscopic approach exhibitsa higher strength than the mesoscopic one. The mesoscopic simu-lation gives a better description of both pre-peak and peak behav-iour. Although fracture energy of macro-mortar (80 J/m2) is greaterthan the one of matrix (44 J/m2), the ductility obtained from themesoscopic computation reproduces better the experiment. Onthe other side, the rupture of the connection is more brittle inthe macroscopic computation. This assessment is due to the pre-cise geometry description of the meso-structure.

Fig. 11. Crack opening of smooth bar for slip = 1.5 � 10�2 mm (a) andslip = 3.3 � 10�2 mm (b).


5.4.1.2. Local behaviour. Local results are presented as damage pat-tern in the specimen and crack openings in the mesoscopic zone. Ata local scale, crack openings are studied with the two followinghypotheses: a damaged element is crossed only by a single crackand non-cracked material is elastic. Strain eij for a size element h(corresponding to a stress given by Eq. (2)) is composed of an elas-tic strain e0ij in the element and a displacement jump d representedby unitary crack opening tensor eoufij [31].

Fig. 10 shows that the damaged zone is localised around the bar.Damage is significant in the window which has a small thickness(2 cm) and tends to decrease in the 4 cm thickness zone. The cracksinitiate at the cement paste/aggregates interface and then propa-gate through the cement paste.

Then cracks continue to progress (Fig. 11a and b) when the pull-out load increases up to the failure of the specimen. This failure ischaracterised by a macroscopic longitudinal crack near the inter-face zone according to what was observed during the experiments(Fig. 12). Most of the cracks are parallel to the bar axis; along thebond length, no inclined cracks appear in the specimen. These find-ings matches the experimental results reported by [28] who hasshown that, for the smooth bar, no splitting of the specimen occurs.Experimental displacement fields obtained by digital imagecorrelation (DIC) are compared with numeric fields. The dimen-sions of observation zone are 3.5 cm2 � 3.0 cm2 and the image res-olution is 1200 � 1000 pixels providing a pixel size of 0.03 mm.The location of observation zone is shown in Fig. 13. Experimentaldisplacement fields are expressed in pixels. Displacements parallelto bar axis are named Ux and displacements normal to bar axis areUy. Figs. 14 and 15 give the iso-values of axial displacement Ux

between two stages (2 kN and 7 kN) from the mesoscopic compu-tation and the experimental results (from CORRELILMT [22]). Theexperimental and numerical displacement fields are comparableapart from the rigid-body motion displacement. It is possible to ob-serve the evolution of interface properties.

For the second stage (F = 7 kN), both numerical and experimen-tal iso-values (Fig. 14a and b) around the bar show a displacementdiscontinuity along the interface in the area close to the appliedload. There is a relative displacement between the reinforcementand the mortar. For the first stage (F = 2 kN), Fig. 15a and b show

Fig. 10. Damage of the mesoscopic zone.

Fig. 12. Experimental failure: interfacial longitudinal crack.

Fig. 13. Analysed zone for digital image correlation.

that these displacements are continuous and that there is a com-patibility of displacement between steel and mortar. The steel–mortar connection is not damaged yet.

Fig. 16a and b illustrates respectively numerical and experimen-tal deformation fields ey. Both fields indicate an opening of thesteel–mortar interface: the mortar located close to the interfacehas a great deformation value ey. In this discontinuity zone, ey

ranges respectively from 2 � 10�3 to 6 � 10�3 for experimentsand from 1.66 � 10�3 to 7.34 � 10�3 for numerical simulation.The proposed mesoscopic computation describes quantitatively

Fig. 14. Longitudinal displacement fields for smooth bar (F = 7 kN): (a) numerical and (b) experimental.

Fig. 15. Longitudinal displacement fields for smooth bar (F = 2 kN): (a) numerical and (b) experimental.

Fig. 16. Radial deformation fields for smooth bar (F = 7 kN): (a) numerical and (b) experimental.


the local deformation fields at the steel–mortar interface takinginto account the displacement discontinuity.

5.4.2. Effect of lateral pressureFig. 17 presents the variations of the pull-out load versus bar

slip for various lateral pressures. The increase in pull-out load re-sults from the increased frictional resistance caused by the pres-ence of the lateral stresses. This phenomenon has beendemonstrated by [28,29]. Local results show that there is no mac-roscopic crack around the bar/mortar interface as observed for theunconfined specimen for the same slip level (Fig. 18).

Fig. 17. Lateral stress effect on the pull-out load versus slip curve.

5.4.3. Effect of bar roughnessExperimentally, the roughness of the bar is obtained artificially

by welding (Fig. 19a). The ribs are made by successive coats ofwelding metal. However, the bar roughness effect has been evalu-ated numerically with a bar having the following idealised geome-try (Fig. 19b): rib face angle = 60�, rib length = 6 mm. innerdiameter = 10 mm, rib height = 2 mm and space between two con-secutive ribs = 11 mm.

Fig. 20 presents the pull-out load versus slip curve for ribbedbar. Experimental and numerical curves show similar ultimatepullout load. However, the numerical stiffness is higher then the

experimental one. This difference can be due to the irregularityof artificial rib production.

Global behaviour is presented as a load versus slip curve forribbed bar specimen at different confinement level (Fig. 21a). Asfor the smooth bar, lateral pressure increases significantly the

Fig. 18. Damage and crack opening patterns for a slip = 3.3 � 10�2 mm with a confining pressure of 2.5 MPa.

(a) (b)6mm

2mm

10mm

135°

11mm

Fig. 19. Ribbed bar: (a) real geometry and (b) idealised geometry.

Fig. 20. (a) Pull_out load versus slip curve for ribbed bar and (b) experimental crack.


pull-out strength. For smooth bars, friction ensures the main partof the strength but for the ribbed bar both friction and the bearingof the ribs against the concrete surface constitute the mechanicaltransfer of forces from reinforcement to the surrounding concrete.For the smooth bar, the bond failure is due to the development oflongitudinal macroscopic cracks at the steel–mortar interface.These cracks result from the shearing of the surrounding mortar

around the bar. The maximal pull-out force for the ribbed bar tomaximal pull-out force for smooth bar ratio is equal to 1.05(Fig. 21b). This low value compared to what is usually recognised(2.25) is due to the absence of concrete surrounding the bar whichnaturally induces a confinement. In case of low confinement, itseems logical that ribbed bars and smooth bars achieve an equiva-lent load capacity. The confining stresses in classical 3D pull out

Fig. 21. Load–slip curves for deformed bar at different confinement level (a) and comparison with smooth bars (b).

Fig. 23. ey deformation field for ribbed bar (F = 7 kN).


test with both smooth or ribbed bars are distributed all around thebar whereas, in the case of LMT test, given the reinforcing bar sec-tion is squared with ribs normal to bar axis and that the confine-ment is only applied on two opposite faces of concrete specimen,the confining stresses are much more greater in the case of classi-cal pull-out test.

The Fig. 22a and b illustrate the evolutions of longitudinal dis-placement Ux at two loading stages respectively set equal to 2 kNand 7 kN. Under a low force (F = 2 kN), the continuity of displace-ment through the interface is observed: there is no relative slip be-tween steel and concrete (Fig. 22a). Under a higher pull-out force(F = 7 kN), a discontinuity of longitudinal displacement Ux is high-lighted; the local slip increases and the interface is more and moredamaged as the pull-out force develops.

Fig. 23 shows the iso-values of deformation normal to bar axisey. It appears that, at this stage, the mortar close to interface is seri-ously damage. The deformation ey represents the opening of steel–concrete connection. As concerns the deformed bars, the mortarsituated between two lugs is compressed (ey is negative and rangesfrom �1.45 � 10�4 to 0). Damaged zones appear at the top of thelugs and propagate through surrounding mortar with a 45� inclina-tion. The tensile deformations in these zones reach 3.55 � 10�3.Between these zones, compressed struts can be observed. More-over, damaged zones parallel to bar axis can be detected initiatingthe shearing of mortar ‘‘teeth’’ between two lugs.

For the deformed bar, two kind of cracks can be distinguishedexperimentally (Fig. 20b) and observed numerically (Fig. 24):

� Cracks parallel to the bar axis: caused by shearing of the ‘‘ mortartooth’’ between two ribs, destroying the interface locally.

Fig. 22. Numerical longitudinal displacement fields

� The inclined cracks: mechanical interlock leads to inclined bear-ing forces which in turn lead to transverse tensile stresses andinternal splitting cracks along reinforcing bar; this cracks are ini-tiated in the second rib face (in the opposite of the loading side).

In Goto’s [7] experiment, crack patterns are recorded in photo-graphs which indicate the type of internal mechanism createdwhere bond stresses are large. The crack pattern, obtained in themesoscopic computation according to [27], reproduces accuratelythe initiation and the propagation of Goto’s cracks. These cracksdisappear when confinement stress is applied (Fig. 24).

The mesoscopic model is a convenient and useful tool to obtainthe characteristics of local cracking pattern around the bar (crackwidth, crack length and tortuosity). It is much more difficult andexpensive to obtain these characteristics from experimentalinvestigation.

Ux for ribbed bar: (a) F = 2 kN and (b) F = 7 kN.

Fig. 24. Evolution of crack opening of ribbed bar with confinement stressrespectively equal to 0 MPa (a) and 2.5 MPa (b).


6. Conclusions

Investigations on bond phenomena between steel and concretehave been carried out using the mesoscopic model. The bond testused in this study is the LMT test which has the particularity ofproviding local information on the interface between steel andconcrete that is not accessible from standardised tests and allowsthe confinement stress level to be varied. A numerical mortar hasbeen generated using a procedure which randomly distributesthe aggregates in the matrix from a real aggregate grading curve.

The global behaviour (load versus slip curve), the damage distri-bution and the crack pattern show the accuracy of the mesoscopicapproach to reproduce the force transfer between steel and con-crete. Moreover bond failure mechanisms have been accurately de-scribed for both smooth and ribbed bars. Finally, the lateral stresseffect on bond behaviour has been correctly reproduced. This firstwork on the modelling of steel–concrete interface validates our ap-proach on the basis of a simple geometry and accurate experimen-tal results. Future numerical work will focus on the 3Dcomputation and on modelling of the real deformation pattern ofthe reinforcement bar.

Acknowledgement

The authors would like to thank the French ANR-MEFISTO pro-ject for their financial support to carry out this study.

References

[1] Rehm G. The fundamentals of bond between steel reinforcement and concrete.Deutsche association for steel reinforcement-concrete, No = 138; 1961. p. 59.

[2] CEB-FIP model code. Bulletin d’information du CEB no. 203-204-205, finaldraft; 1991.

[3] Martin H. Bond performance of ribbed bars – influence of concretecomposition and consistency. In: Bartos P, editor. Bond in concrete. AppliedScience Publishers; 1982.

[4] Comité Européen du Béton (CEB). Adhérence – Ancrage- déformations. Bulletind’information N� = 48, avril (compte rendu de la 9ème session plénièreANKARA SEP 1964); 1965.

[5] Darwin D, Zuo J, Tholen ML, Idun EK. Development length criteria forconventional and high relative rib area reinforcing bars. ACI Struct J1996;93(3):347–59.

[6] Hamad BS. Bond strength improvement of reinforcing bars with speciallydesigned rib geometries. ACI Struct J 1995;92(1):3–13.

[7] Goto Y. Cracks formed in concrete around deformed tension bars. ACI J1971;68(4):244–51 [proceedings].

[8] Wittmann FH, Roelfstra PE, Sadouki H. Simulation and analysis of compositestructures. Mater Sci Eng 1984;68(2):2239–48.

[9] Wriggers P, Moftah SO. Mesoscale models for concrete: homogenization anddamage behaviour. Finite Elem Anal Des 2006;42:623–36.

[10] Bazant ZP, Tabbara MR, Kazemi MT, Pijaudier-Cabot G. Random particle modelfor fracture of aggregate composite. J Eng Mech – ASCE 1999;34:705–24.

[11] Van Mier JGM, Van Vliet MRA. Influence of microstructure of concrete on size/scale effects in the tensile fracture. Eng Fract Mech 2003;70:2281–306.

[12] Grassl P. A lattice approach to model flow in cracked concrete. Cem ConcrCompos 2009;31(7):454–60.

[13] Wang ZM, Kwan AKH, Chan HC. Mesoscopic study of concrete I: Generation ofaggregate structure and finite element mesh. Comput Struct1999;70(5):533–44.

[14] Kwan AKH, Wang ZM, Cham HC. Mesoscopic study of concrete II: Nonlinearfinite element analysis. Comput Struct 1999;70:545–56.

[15] Nguyen TD, Lawrence C, La Borderie C, Matallah M, Nahas G. A mesoscopicmodel for a better understanding of the transition from diffuse damage tolocalized damage. Eur J Environ Civil Eng 2010;14:751–76.

[16] Hafner S, Eckardt S, Luther T, Konke C. Mesoscale modelling of concrete:Geometry and numerics. Comput Struct 2006;84:450–61.

[17] Garboczi EJ, Cheok GS, Stone WC. Using LADAR to characterize the 3-D shapeof aggregates: preliminary results. Cem Concr Res 2006:1072–5.

[18] Agioutantis Z, Chatzopoulou E. A numerical investigation of the effect of theinterfacial zone in concrete mixtures: the case of the dilute limit. Cem ConcrRes 2000;30:715–23.

[19] La Borderie C, Lawrence C, Menou A. Approche mésoscopique ducomportement: Apport de la représentation géométrique. Eur J Environ CivEng 2007;11:407–21.

[20] Grassl P, Davies T. Lattice modelling of corrosion induced cracking and bond inreinforced concrete. Cem Concr Compos 2011;33:918–24.

[21] Ouglova A et al. The influence of corrosion on bond properties betweenconcrete and reinforcement in concrete structures. Mater Struct2008;41:969–80.

[22] Tran BH, Berthaud Y, Ragueneau F. Essais PIAF: Pour Identifier l’Adhérence et leFrottement. CFM 2007-18e Congrès Français de Mécanique; 2007.

[23] Mazars J. A description of micro- and macroscale damage of concretestructures. Eng Fract Mech 1986;25:729–37.

[24] Fichant S, La Borderie C, Pijaudier-Cabot G. Isotropic and anisotropicdescriptions of damage in concrete structures. Mech Cohes-Frict Mater1999;4:339.

[25] Ladevèze P. Sur une théorie de l’endommagement anisotrope. Technical report34. Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure deCachan; 1983.

[26] Stefan L, Benboudjema F, Torrenti JM, Bissonnette B. Prediction of elasticproperties of cement pastes at early ages. Comput Mater Sci 2010;47:775–84.

[27] Matallah M, La Borderie C, Maurel O. A method to estimate crack openings inconcrete structures. Int J Numer Anal Methods Geomech 2010;34:1615–33.

[28] Robins PJ, Standish IG. The effect of lateral pressure on bond of reinforcing barsin concrete. Int J Adhes Adhes 1982;2(2):129–33.

[29] Gambarova PG, Rosati GP. Bond and splitting in bar pull-out: behaviour lawsand concrete cover role. Concr Res Mag 1997;49(179).

[30] Zhou XQ, Hao H. Mesoscale modelling and analysis of damage of concrete slabunder contact detonation. Int J Impact Eng 2009;36:1315–26.

[31] Matallah M, La Borderie C. Inelasticity-damage-based model for numericalmodelling of concrete cracking. Eng Fract Mech 2009;76(8):1087–108.

[32] Zohdi TI, Wriggers P. Aspects of the computational testing of the mechanicalproperties of micro-heterogeneous material samples. Int J Numer Methods Eng2001:2573–99.

[33] Lohnert S. Computational homogenization of micro-heterogeneous materialsat finite strains including damage. PhD thesis, Hanover, IBNM; 2004.

[34] Snozzi L, Cabellero A, Molinari JF. Influence of the meso-structure in dynamicfracture of concrete under tensile loading. Cem Concr Res 2011;41:1130–42.

[35] Cremona C, Adélaide L, Berthaud Y, Bouteiller V, L’Hostis V, Poyet S, et al.‘‘Probabilistic and predictive performance-based approach for assessingreinforced concrete structures lifetime: the applet project. National Frenchproject (2007–2010).

2d mesoscopic modelling of bar–concrete bond

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