Engineering Structures 39 (2012) 66–78

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Bond slip model for the simulation of reinforced concrete structures

A. Casanova a,b, L. Jason a,d,⇑, L. Davenne c

a CEA, DEN, DANS, DM2S, SEMT, LM2S, F-91191 Gif sur Yvette, Franceb Laboratoire de Mécanique et Technologie (LMT), ENS Cachan, F-94235 Cachan, Francec IUT de Ville d’Avray, F-92410 Ville d’Avray, Franced LaMSID, UMR CNRS-EDF-CEA 2832, F-92141 Clamart, France

a r t i c l e i n f o

Article history:Received 14 March 2011Revised 23 December 2011Accepted 5 February 2012

Keywords:Reinforced concreteBond slipModel

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.engstruct.2012.02.007

⇑ Corresponding author at: CEA, DEN, DANS, DM2S,Yvette, France.

E-mail address: ludovic.jason@cea.fr (L. Jason).

a b s t r a c t

This paper presents a new finite element approach to model the steel–concrete bond effects. This modelproposes to relate steel, represented by truss elements, with the surrounding concrete in the case wherethe two meshes are not necessary coincident. The theoretical formulation is described and the model isapplied on a reinforced concrete tie. A characteristic stress distribution is observed, related to the transferof bond forces from steel to concrete. The results of this simulation are compared with a computation inwhich a perfect relation between steel and concrete is supposed. It clearly shows how the introduction ofthe bond model can improve the description of the cracking process (finite number of cracks).

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction elements were developed. These zero thickness elements,

Reinforced concrete structures may have to fulfill functions thatgo beyond their simple mechanical resistance. In some cases, infor-mation about the cracking behavior related to the quasi-brittleevolution of concrete can also become essential. For example, inthe case of containment buildings for nuclear power plants, crack-ing has a direct impact on the transfer properties that govern thepotential leakage rate (damage – permeability law [21] among oth-ers). Predicting the mechanical behavior but also characterizingthe crack evolution (opening and spacing) are thus key points inthe evaluation of this type of reinforced concrete structures.

Cracking in reinforced concrete structures is generally influ-enced by the stress distribution along the interface between steeland concrete. For example, in the case of a reinforced tie, oncethe first crack appears in the weakest point of the structure, theconcrete stress in the cracked zone drops to zero while the loadis totally supported by the steel reinforcement. The stresses arethen progressively transferred from steel to concrete (Fig. 1). Thistransition zone has an impact on the crack properties and isdirectly influenced by the steel–concrete interface [10]. Taking intoaccount these effects seems thus essential to predict correctly thecracking of reinforced concrete structures.

Different models exist to represent the steel–concrete bondbehavior. For example, Ngo and Scordelis [19] proposed a spring ele-ment, associated with a linear law, to relate concrete and steelnodes. To improve the description of the bond behavior, joint

ll rights reserved.

SEMT, LM2S, F-91191 Gif sur

introduced at the interface between steel and concrete, allow theuse of a nonlinear law [4,5,15,7,1,23]. Finally, special finite elementsare proposed to enclose, in a same element, the material behavior(steel or/and concrete) and the bond effects [18]. Dominguez [6]and Ibrahimbegovic et al. [12,13], among others, developed embed-ded elements whose principle is to describe the steel–concrete bondbehavior through an enrichment of the degrees of freedom (Fig. 2).

Even if these solutions give appropriate results, one of theirmain drawbacks, in the context of industrial applications, is theneed to explicitly consider the interface between steel and con-crete. It may impose meshing difficulties and heavy computationalcost which are not compatible with large scale simulations.

To overcome these difficulties, a perfect relation between steeland concrete is generally considered for structural applications andthe reinforcement is modeled using truss elements. This hypothe-sis imposes the same strain in both materials. Although this ap-proach simplifies the simulation, it is not fully representative ofthe experimental behavior (Fig. 1).

In this contribution, a model is proposed to combine the advan-tages of the two approaches. Contrary to refined models, theformulation does not require an explicit representation of thesteel–concrete interface (reduction of the computational effort)and supposes the use of steel truss elements in the case where steeland concrete nodes are not necessary coincident (Fig. 3). Moreover,contrary to classical industrial hypothesis (perfect relation), thepresented model is also able to represent the bond effects (theevolution of the slip between steel and concrete especially) in thesame way as ‘‘fine’’ approaches. In this sense, it represents a goodalternative to classical refined models and industrial classicalcomputations.

Fig. 1. Distribution of stresses in steel and concrete in a reinforced concrete tie afterthe first crack (rc and rs are the stresses in concrete and steel respectively, and rst

c isthe concrete strength).

Fig. 4. Shape of the bond stress–slip law [9].

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 67

The theoretical formulation of the model is presented in Section2. It is then validated in Section 3 on the case of a reinforced con-crete tie. Finally, the influence of the bond effect is investigated bya comparison with a simulation using the hypothesis of a perfectrelation.

2. Description of the bond model

In reinforced concrete structures, the relation between steel andconcrete is governed by a bond stress distributed along the inter-face. The evolution of this bond stress is generally studied in theform of a bond stress-relative displacement (slip between steeland concrete) curve (Fig. 4). Therefore, the bond behavior can berepresented by a bond stress–slip (s–s) law fb defined by:

s ¼ fbðsÞ ð1Þ

kn

kl

Concrete

Steel

Zerothickness

kn

kl th

Embedded elementEmbedded element

D

D

Fig. 2. Examples of bond elements (at the top left: spring element, at

Fig. 3. Description of the problem (steel an

To take this effect into account, the principle of our approach isto introduce internal forces on both steel and concrete in the direc-tion of the steel truss element. As concrete and steel nodes are notnecessary coincident, each steel node is thus associated with theconcrete element in which it is included and with a tangentialdirection~t.

In this contribution, equations will be developed in the particu-lar case of a unique steel node inside a cubic concrete element asillustrated in Fig. 5 (initial configuration). In order to simplify theequations,~t coincides with the principal direction~x

The formulation of the model is divided into four steps:

– The evaluation of the steel–concrete slip with non-coincidentmeshes.

– The calculation of the nodal bond forces in the steel elementdirection.

Concrete

Steel

Concrete

Steel

Joint element

Zeroickness

egrees of freedom of concrete

egrees of freedom of concrete+ steel-concrete slip

the top right: joint element, at the bottom: embedded element).

d concrete nodes are not coincident).

Fig. 5. Evaluation of the relative displacement between steel and concrete.

68 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

– The introduction of kinematic relations in the normaldirections.

– The implementation of the model within the small displace-ments theory.

2.1. Evaluation of the relative displacement between steel and concrete

The steel–concrete slip is represented by the relative displace-ment in the x direction between the steel (~us1 at node s1) and theconcrete at the steel node (~ucs1 at virtual point cs1) (Fig. 5)

s ¼ jus1;x � ucs1;xj ð2Þ

with us1;x ¼~us1 �~x and ucs1;x ¼~ucs1 �~x.As the meshes are not necessary coincident, the displacement in

concrete at the steel node is obtained using the shape functions,following the equation:

ucs1;x ¼X8

i¼1

Niðxcs1; ycs1; zcs1Þuci;x ð3Þ

where (xcs1, ycs1, zcs1) are the coordinates of cs1 point.Ni(xcs1, ycs1, zcs1) represents the shape function of the ith concretenode evaluated at cs1 point and uci,x its displacement in the ~xdirection.

2.2. Nodal forces in the tangential direction

From Eqs. (1)–(3), the bond stress is computed:

Fig. 6. Definition of l.

s ¼ fb us1;x �X8

i¼1

Niðxcs1; ycs1; zcs1Þuci;x

����������

!ð4Þ

The internal nodal force induced by concrete on steel, ~Fcs1=s1, iscalculated using the equation:

~Fcs1=s1¼dp dsls �~x

¼dp dsl� fb us1;x�X8

i¼1

Niðxcs1;ycs1;zcs1Þuci;x

����������

!�~x

withd¼1 if us1;x�

P8i¼1

Niðxcs1;ycs1;zcs1Þuci;x P 0

d¼�1 if us1;x�P8i¼1

Niðxcs1;ycs1;zcs1Þuci;x<0

8>>><>>>:

ð5Þ

where ds is the diameter of the steel bar and l is derived from thelength of the steel elements connected to the node s1. For example,in Fig. 6:

l ¼ l1 þ l2

2ð6Þ

The balance of the internal bond forces involves ~Fs1=cs1

� �:

~Fs1=cs1 ¼ �~Fcs1=s1 ð7Þ

~Fs1=cs1 is converted in 8 nodal equivalent forces~Fs1=cj applied on eachconcrete node (Fig. 7).

~Fs1=cj ¼ Njðxcs1; ycs1; zcs1Þ~Fs1=cs1 ð8Þ

In this first approach, internal bond forces are redistributed onthe concrete thanks to the shape functions Nj of the concrete finiteelement. This choice can have some limits, especially when the sizeof the concrete becomes significantly higher than the steel diame-ter. In this situation, other functions could be used, in order to takeinto account the decrease of the bond effect with the size of theconcrete element.

Fig. 7. Balance of the internal bond forces.

Fig. 8. Iterative diagram for the numerical resolution.

La

0 x

u

Cylinder of concrete

Steel reinforcement

Fig. 9. Presentation of the problem.

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 69

2.3. Kinematics relations in the normal directions

Eqs. (5) and (8) relate the degrees of freedom of concrete andsteel in the tangential direction of the reinforcement. To closethe system, a perfect relation is supposed for the displacement inthe other directions. Even if this hypothesis is probably too strong,it enables to represent, in a simple way, the main component of thebond effect which is the tangential one. It could be improved in fu-ture work by additional relations to represent the potential bondeffects in the normal directions (Dowel effect for example).

Two equations are thus added:

us1;y ¼ ucs1;y ¼P8i¼1

Niðxcs1; ycs1; zcs1Þuci;y

us1;z ¼ ucs1;z ¼P8i¼1

Niðxcs1; ycs1; zcs1Þuci;z

8>>><>>>:

ð9Þ

2.4. Implementation of the bond element model

This bond model has been implemented in the finite elementcode Cast3M [3]. For nonlinear problems (material and/or bond

law), a Newton Raphson iterative method is used. In its classicalform, the nodal displacement {un+1} at the (n + 1)th iteration iscalculated following equation:

Knmat

� �dunþ1� �

¼ Fextf g � Fnint;mat

n oð10Þ

where {un+1} = {un} + {dun+1}.

Fig. 10. Mesh of the tie.

Table 1Steel parameters (Ss is the cross section of steel).

Es (GPa) Ss (m2) rsts (MPa)

210 7.854 � 10�5 400

Strain

Stress

stcσ

cE

uε Strain

Stress

stcσ

σ

εcE

uεFig. 11. Form of the softening concrete law.

Table 2Concrete parameters (Sc is the cross section of concrete).

Ec (GPa) Sc (m2) eu

30 10�2 0.045%

1,8

1,9

2,0

2,1

2,2

0 0,5 1 1,5 2 2,5 3

x (m)

Con

cret

e St

reng

th (

MPa

)

Fig. 12. Distribution of the tensile strength values along the tie.

Table 3Parameters of the scattered distribution.

rstc;med (MPa) ry (MPa)

2 0:03 � rstc;med

70 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

In this expression {Fext} represents the external forces, fFnint;matg

the usual internal forces induced by concrete and steel and ½Knmat�

the stiffness matrix at iteration n.As describe in the previous sections, to take into account the

bond effects, kinematics relations and bond forces are added. Thekinematics Eq. (9) are introduced using the Lagrange’s multipliermethod. Global internal bond forces fFn

int;bondg are calculated fromEqs. (5) and (8) using the distribution of the nodal displacementsat iteration n. Internal forces thus become:

Fnint

� �¼ Fn

int;mat

n oþ Fn

int;bond

n oð11Þ

The global stiffness matrix [Kn] is also updated:

½Kn� ¼ ½Knmat� þ ½K

nbond� ð12Þ

where

½Knbond� ¼

@Fnint;bond

@u

�����un

ð13Þ

The system finally becomes:

½Kn�fdunþ1g ¼ fFextg � fFnintg ð14Þ

The steps for the resolution of the system are summarized inFig. 8.

3. Validation and application of the bond model

In this part, the model is applied on a uniaxial problem. A rein-forced concrete tie is studied in order to evaluate the influence ofthe bond effects on the mechanical behavior.

3.1. Presentation of the problem

A concrete cylinder reinforced with a longitudinal steel bar isconsidered. A displacement is imposed at the first end of the steelreinforcement while the other end is blocked (Fig. 9). There are noboundary conditions on the concrete.

For the simulation, concrete and steel are meshed by truss ele-ments. The total length L (equal to 3 m) is divided into 100 identi-cal trusses as illustrated in Fig. 10. The steel behavior is modeled bya perfect plastic law (Young modulus Es and yield strength rst

s gi-ven in Table 1). Concrete is represented by a softening law, usingan isotropic plastic model illustrated in Fig. 11 (Table 2). To repre-sent the heterogeneity of concrete properties and in order to local-ize the mechanical degradation, a scattered distribution of the

concrete strength rstc is chosen. Fig. 12 represents the evolution

rstc along concrete (from x = 0 to x = 3 m). The average strength

rstc;med and the standard deviation ry are given in Table 3.

The bond stress–slip law is represented by a linear function(s = kb � s where kb is equal to 1010 Pa m�1).

0

5000

10000

15000

20000

25000

30000

35000

0 0,001 0,002 0,003 0,004 0,005

Imposed displacement (m)

For

ce (

N)

Elastic phase

Development of cracking in concrete

Steel behavior

Fig. 13. Force–displacement curve.

0,00E+00

2,00E+05

4,00E+05

6,00E+05

8,00E+05

1,00E+06

1,20E+06

1,40E+06

1,60E+06

1,80E+06

2,00E+06

0 0,5 1 1,5 2 2,5 3

x (m)

Con

cret

e st

ress

(P

a)

0,00E+00

5,00E+07

1,00E+08

1,50E+08

2,00E+08

2,50E+08

Stee

l str

ess

(Pa)

concrete steel

Fig. 14. Distribution of the stress along the tie (elastic phase).

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 71

3.2. Global mechanical behavior

Fig. 13 represents the global behavior of the tie. This curve canbe divided in three parts:

– An elastic phase where the behavior of both materials remainslinear.

– A phase of concrete cracking which is characterized by severalpeaks and drops of force.

– A perfect plastic phase where the behavior is only governed bythe steel plastic law (constant force).

It is to be noted that these three phases are representative of theexperimental behavior [17].

In the following sections, each phase is going to be discussed.

3.3. Elastic phase

Fig. 14 represents an example of stress distribution in bothmaterials along the tie during the elastic phase. As the loading is

applied directly on the reinforcement, at each end the force isessentially supported by the steel (rs � F

SSwith rs the steel stress

and F the applied load). Then, in a transition zone, forces are grad-ually transferred from steel to concrete through the interface:stresses in concrete increase while stresses in steel decrease fromthe end toward the middle of the tie. Finally, in the central zone,stresses are homogeneous.

To validate the results of the simulation, an analytical resolu-tion is proposed. This solution is obtained from the balance offorces between steel and concrete from a = x to a = L (Fig. 15).

The external force F applied on steel is balanced by an internalforce Fs(x) and a bond force Fbond/s(x).

FsðxÞ þ Fbond=sðxÞ ¼ F ð15Þ

with, considering the small displacement theory,

FsðxÞ ¼ SsEsdusðxÞ

dxð16Þ

where us(x) is the steel displacement at a = x.

Fig. 15. Force balance between a = x and a = L.

72 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

The bond force Fbond/s(x) is evaluated integrating the bond stressbetween a = x and a = L. Therefore:

Fbond=sðxÞ ¼ pdskb

Z L

x½usðaÞ � ucðaÞ�da ð17Þ

where ds is the diameter of the steel bar and uc(a) represents thedisplacement in concrete at a.

In the same way, the balance on concrete is given by

FcðxÞ þ Fbond=cðxÞ ¼ 0 ð18Þ

0,00E+00

2,00E+05

4,00E+05

6,00E+05

8,00E+05

1,00E+06

1,20E+06

1,40E+06

1,60E+06

1,80E+06

2,00E+06

0 0,5 1

Stre

ss (

Pa)

nume

Fig. 16. Stress distribution in

0,00E+00

5,00E+05

1,00E+06

1,50E+06

2,00E+06

2,50E+06

0 0,5 1

Stre

ss (

Pa)

elastic phase 1

Fig. 17. Distribution of the stress

with

FcðxÞ ¼ ScEcducðxÞ

dxð19Þ

and

Fbond=cðxÞ ¼ �Fbond=sðxÞ ð20Þ

After development, Eqs. (15) and (18) give the following system:

d3usðxÞdx3 � K1

dusðxÞdx

¼ K2

ducðxÞdx ¼ � EsSs

EcSc

dusðxÞdx þ F

EcSc

8>><>>: ð21Þ

with

K1 ¼ kbpds1

EsSsþ 1

EcSc

K2 ¼ � kbpdsFEsSsEcSc

8>><>>: ð22Þ

After resolution, it comes:

1,5 2 2,5 3

x (m)

rical analytical

concrete along the tie.

1,5 2 2,5 3

x (m)

st crack distribution of strength

in the concrete along the tie.

x (m)

0,00E+00

5,00E+07

1,00E+08

1,50E+08

2,00E+08

2,50E+08

0,00E+00 5,00E-01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00

Stre

ss (

Pa)

elastic phase 1st crack

Fig. 18. Distribution of the stress in the steel along the tie.

0,00E+00

5,00E+07

1,00E+08

1,50E+08

2,00E+08

2,50E+08

3,00E+08

3,50E+08

4,00E+08

4,50E+08

0,00E+00 5,00E-01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00

x (m)

Stre

ss (

Pa)

Yield strength of steel Yield strength of steel

Fig. 19. Stress distribution in the steel at the beginning of the plastic phase.

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 73

dusðxÞdx

¼ Ae�ffiffiffiffiK1

px þ Be

ffiffiffiKp

1x � K2

K1

ducðxÞdx ¼ � EsSs

EcScAe�

ffiffiffiffiK1

px þ Be

ffiffiffiKp

1x � K2K1

� �þ F

EcSc

8>><>>: ð23Þ

where

A ¼ effiffiffiffiK1

pL � 1

effiffiffiffiK1

pL � e�

ffiffiffiffiK1

pL

FEsSsþ K2

K1

B ¼ FEsSsþ K2

K1� A

8>>>><>>>>:

ð24Þ

Steel and concrete stresses along the tie are finally calculatedusing equation

rsðxÞ ¼ EsdusðxÞ

dx

rcðxÞ ¼ EcducðxÞ

dx

8>>><>>>:

ð25Þ

The numerical results are compared with the analytical solu-tion. Fig. 16 represents the stress evolution in concrete along thetie. The results are similar and validate the numerical implementa-tion in this particular case.

3.4. Cracking process

During the elastic phase, the stresses increase in concrete withthe imposed displacement. When the strength is reached in a con-crete element, an unloading is observed in this element and thestress drops to zero (Fig. 17). The load is then fully supported bythe corresponding steel element. Its stress becomes equal to F

SS

(Fig. 18). It is to be noted that this phenomenon is amplified bythe use of 1D truss element for concrete. Contrary to what happensin 3D simulation, no evolution of the strain distribution is possiblein the concrete cross section. It has nevertheless the advantage todirectly highlight the bond effect model on the structural behavior.

0,00E+00

5,00E+05

1,00E+06

1,50E+06

2,00E+06

2,50E+06

0 0,5 1 1,5 2 2,5 3

x (m)

Stre

ss (

Pa)

final distribution of stress distribution of strength

Fig. 20. Stress distribution in the concrete at the beginning of the plastic phase.

1

2

3

4

5

6

7

8

9

10

0 0,5 1 1,5 2 2,5 3

x(m)

tie

num

ber

1

1

1

2

2

2

3

3

3

34

4 45

5

3

1 2 43 5

1 425 3

5 1 3 2 4

3 3 1 4 2 5

5 2 4 1 3

5 2 3 1 4

2 5 1 4 3

Fig. 21. Position of cracks for the 10 ties. The numbers on the figure correspond to the order of apparition (1 for the first crack until 5 for the fifth crack). The dotted linescorrespond to the average positions of each crack.

74 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

When a concrete element is damaged, the stresses are redistrib-uted on each side of the crack. On both sides the stress distributionis similar what was observed during the elastic phase with a grad-ual transfer from steel to concrete. On a global point of view, theapparition of the first crack coincides with the first partial unload-ing on the force displacement curve (Fig. 13).

After the apparition of the first crack, stresses in concrete con-tinue to increase until a tensile strength is reached in a second ele-ment. Another crack appears, and a total unloading of theconsidered element is obtained. The same effect is observed untilthe steel stress in the cracked zones reaches it yield limit(Fig. 19). At this step, the global force cannot increase anymoreand the distribution of Fig. 20 is obtained (five cracks).

The crack length corresponds to the length of one finite element.This effect can be explained by the well-known stress localizationdue to the use of a softening law. It could be solved by including aregularization technique on the concrete constitutive law

[22,20,11]. This point will not be discussed in this paper and do notimpact the interest of the proposed approach on the bond slip model.

It is to be noted that the crack position is governed, as expected,by the strength distribution but also by the stress evolution im-posed by the bond effects. To underline this effect, 10 computa-tions have been carried out considering 10 different distributionsof the initial concrete tensile strength.

Fig. 21 shows the crack position along the 10 ties. In most cases(except for one tie), 5 cracks are observed. These cracks are gener-ally localized on one concrete element except when the distribu-tion of strength imposes a simultaneous cracking of twoelements (the 3rd crack of tie 1 for example). It is to be noted that,due to the transfer zone between steel and concrete, no crack candevelop at both ends. Finally, the crack spacing is constant with anaverage value of about 50 cm.

As a conclusion, the order of crack apparition is rather governedby the distribution of the mechanical strength whereas the final

Steel behavior

Development of cracking in concrete

Elastic phase

0

5000

10000

15000

20000

25000

30000

35000

0 0,001 0,002 0,003 0,004 0,005 0,006

Imposed displacement (m)

For

ce (

N)

Steel behavior

Development of cracking in concrete

Elastic phase

Fig. 22. Force–displacement curve at the loaded end of the steel.

0,00E+00

2,00E+06

4,00E+06

6,00E+06

8,00E+06

1,00E+07

1,20E+07

1,40E+07

0,00E+00 5,00E-01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00

x (m)

Stre

ss (

Pa)

concrete steel

Fig. 23. Stress distribution in both materials along the tie (elastic phase).

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 75

state (constant average spacing) is imposed by the stress distribu-tion related to the bond effects.

4. Influence of the bond model

In order to evaluate the influence of the bond effects on thebehavior of the tie, the results of the previous simulation are com-pared with the case where steel and concrete are perfectly bound.In this situation, steel and concrete nodes are identical and thesimulation is carried out in the same conditions as described inSection 3. Fig. 22 illustrates the global behavior. It is once again di-vided in three main phases (elastic phase, development of crackingand plastic zone). During the development of cracks, a higher num-ber of partial unloading is observed.

Fig. 23 describes the distribution of stress in both materials dur-ing the elastic phase. Contrary to what was observed with the bondmodel, stresses in steel and concrete are homogeneous along thetie. This can be explained by the perfect relation that imposes anidentical strain in both materials. It comes:

rs ¼Es

EsSs þ EcScF

rc ¼Ec

EsSs þ EcScF

ð26Þ

rc increases with the load until the minimal value of the concretestrength is reached. A crack appears in the element. Fig. 24 repre-sents the stress distribution in concrete after the first crack. It isto be noted that, with the perfect relation, the stress remains homo-geneous in the uncracked concrete. In this zone, the stress can in-crease until the second concrete strength limit.

Contrary to what was observed with the bond model, the appa-rition of the cracks is only governed by the strength distribution(Fig. 25). At the final state (Fig. 26), concrete is totally cracked be-cause the concrete strength has been reached in every element.It explains the high number of partial unloading on the force–displacement curve (Fig. 22). The loading is then only supportedby steel.

As expected, the perfect relation is not able to represent theexperimental situation where a finite number of cracks is observed.

0,00E+00

5,00E+05

1,00E+06

1,50E+06

2,00E+06

2,50E+06

0,00E+00 5,00E-01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00

x (m)

Stre

ss (

Pa)

perfect relation with bond model strength distribution

Fig. 24. Comparison of stress distributions in concrete after the first crack apparition.

0,00E+00

5,00E+05

1,00E+06

1,50E+06

2,00E+06

2,50E+06

0 0,5 1 1,5 2 2,5 3

x (m)

Stre

ss (

Pa)

with bond model perfect relation strength distribution

Fig. 25. Comparison of stress distributions in concrete after the third crack apparition.

0,00E+00

5,00E+07

1,00E+08

1,50E+08

2,00E+08

2,50E+08

3,00E+08

3,50E+08

0,00E+00 5,00E-01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00x (m)

Stre

ss (

Pa)

concrete steel

Fig. 26. Stress distribution in both materials when concrete is totally cracked (perfect relation).

76 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

Fig. 27. Axial displacement in concrete along the tie (for an imposed displacement of 1.6 mm on steel).

Table 4Comparison of the crack properties for an imposed displacement of 1.6 mm.

Number of cracks Average crack opening (mm) Minimal crack opening (mm) Maximal crack opening (mm)

Bond model 2 0.475 0.47 0.48Perfect relation 23 0.061 0.037 0.187

A. Casanova et al. / Engineering Structures 39 (2012) 66–78 77

The introduction of the bond effects, as presented in Section 2,solves this problem.

To highlight the differences between the two approaches (per-fect relation vs. bond model), the crack properties are compared.From continuum damage mechanics, one way to evaluate the crackopening is to represent the evolution of the axial displacement inthe concrete (Fig. 27). The position of the displacement jumps cor-responds to the position of the cracks while their values enable tocompute each crack opening. It is to be noted that other techniquesexist to obtain information about the crack properties and could beused in more complex cases (post processing methods using crack-ing strain [16], comparison with discrete computation [8], use ofcontinuum–discrete model including displacement discontinuity[2], etc.).

The crack properties are compared for an imposed displacementof 1.6 mm (Table 4). As expected, a higher number of cracks is ob-tained with the perfect relation (23 cracks vs. two with the bondmodel). As a consequence, the average crack opening is higher inthe case of the bond model. The distribution of the crack openingis also more regular.

5. Conclusions

In this contribution, a new finite element model is proposed torepresent bond effects between steel, modeled with truss ele-ments, and the surrounding concrete. These bond effects are takeninto account through additional internal forces calculated from thesteel–concrete slip. The proposed model has been applied on areinforced concrete tie. A characteristic stress distribution has beenobserved, related to the transfer of bond forces from steel to con-crete. In this case, crack apparition is both induced by the hetero-geneous characteristics of concrete (distribution of the tensilestrength along the tie) and by the transition zones where stressesare redistributed between the materials. At the end of the simula-

tion (mechanical behavior only governed by the steel plastic law), afinite number of crack is observed which corresponds qualitativelyto experimental observations [17].

Another simulation, using the hypothesis of a perfect relationbetween steel and concrete (classical hypothesis used in structuralapplications) has been also proposed. In this case, the evolution ofcracking is only governed by the distribution of the concretestrength since the stresses distribution is homogeneous in un-cracked zones. At the final state, the concrete is totally cracked. Itshows how the introduction of the bond model can improve thedescription of the cracking process.

Forthcoming 3D simulations will allow to also evaluate the ef-fects of the stress distribution in the concrete cross section.

Even if these results are promising, crack evolution is stronglyrelated to the choice of the bond stress–slip law which has a directinfluence on the final number of cracks.

To take this influence into account, an investigation on the bondstress–slip law will thus be launched, based either on literature[9,14] or on an experimental campaign to be more representativeof the experimental behavior.

Finally, the limitations of the proposed finite element model arerelated to the main hypothesis. As the steel is modeled using trusselements, one has to face the associated difficulties (stress concen-tration for example). Nevertheless, this could probably be avoidedby using regularization techniques or by proposing meshing rules(for example, not to consider concrete elements around steel smal-ler than the steel diameter). These points will also be investigatedin future work when considering 3D simulations.

Acknowledgement

The authors would like to thank Dr. S. Ghavamian for his helpfuldiscussions in the elaboration of the model.

78 A. Casanova et al. / Engineering Structures 39 (2012) 66–78

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