a finite macro element for corroded reinforced concrete

8/11/2019 A Finite Macro Element for Corroded Reinforced Concrete

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Materials and Structures (2006) 39:571584

DOI 10.1617/s11527-006-9096-x

O R I G I N A L A R T I C L E

A finite macro-element for corroded reinforced concrete

Raoul Francois Arnaud Castel Thierry Vidal

Received: 14 February 2005 / Accepted: 22 July 2005C RILEM 2006

Abstract This paper proposes a model of the mechani-

cal behaviour of corroded reinforced concrete members

subjected to bending under service load. The model is

based on the formulation of a macro-element to be used

in FEM analysis, having a length equal to the distance

between two consecutive flexural cracks and a cross-

section equal to the member cross-section. The me-

chanical formulation is directly written in generalized

variables (bending moment and curvature) and is based

on the concept of the transfer length necessary for the

transmission of tensile load from re-bar to tensile con-crete thanks to the bond. It is thus possible to take into

account the effect of reinforcement corrosion on the

bond between re-bar and concrete, by increasing the

transfer length versus intensity of corrosion. The varia-

tion of the transfer length versus corrosion is expressed

using a scalar damage parameter. A first experimental

validation is performed on a 17-year-old beam kept in

a chloride environment under its service load.

Resume Cet article propose un modele de fonction-

nement mecanique en service delement de beton armed egrad e par corrosion des armatures. Le modele est

base sur la d efinition dun macro-element de largeur

egalea la distance entrefissures de flexion et de hauteur

R. Francois

LMDC, INSA Toulouse, France

A. Castel T. Vidal

LMDC, UPS Toulouse, France

egale a celle de la poutre. La formulation m ecanique

appliquee a ce macro-element est calculee en vari-

ables generalisees (moment-courbure) en prenant en

comptela longueur de transfert necessairea larmature

pour transmettre une partie des efforts de traction au

beton. La prise en compte de leffet de la corrosion

sur ladherence acier-beton est alors realisee en aug-

mentant cette longueur de transfert en fonction dun

parametre scalaire dendommagement de corrosion.

Un premier exemple de validation du modeleestr ealise

sur une poutre en beton arme vieillie pendant 17 ansdans une ambiance saline.

1. Introduction

The prediction of changes in the mechanical behaviour

of reinforced concrete structures during their ageing is

an objective of major importance for buildingowners. It

will help in the decision to plan repair or reinforcement

of the structure, set up a maintenance program or, on

the contrary, envisage the demolition and re-buildingof the structure.

The main cause of ageing damage in reinforced con-

crete structures is reinforcement corrosion. Damage

can be detected visually as coincident cracks along the

reinforcement, which are significant of both reduction

of the re-bar cross-section and loss of bond between

reinforcement and concrete.

Building design standards neglect the tensile con-

crete located between flexural cracks because it does


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572 Materials and Structures (2006) 39:571584

not have any effect on the load-bearing capacity (Ul-

timate Limit State). So, by using these standards, it is

not possible to evaluate the effect of bond degradation

due to reinforcement corrosion on the serviceability

of a reinforced structure [1, 2]. In this paper, a model

of reinforced concrete behaviour in the post-cracking

state is proposed based on a reduced-inertia approachwhichtakes tension stiffeningand corrosion effects into

account.

In normal conditions, reinforced concrete elements

subjected to bending are always cracked in their tensile

zones because the tensile strength of concrete is low.

Consequently, usual mechanical models for reinforced

concrete design do not take into account the tensile

concrete, since it does not significantly influence the

load-bearing capacity of a structure (ULS) [3]. How-

ever the tensile concrete located between two flexural

cracks contributes to the flexural stiffness of the struc-tural element. Indeed, the bond between the re-bars

and the concrete is still active in these areas and leads

to a mechanical interaction between the reinforcements

and the concrete. This is called the tension-stiffening

effect, and is well known to structural engineers. To

take this phenomenon into account in order to correct

the calculations carried out neglecting concrete tensile

strength, two main approaches have been adopted.

In the first approach, the tension stiffening is mod-

elled using a decrease of the concrete elastic modulus

based on the constitutive laws of concrete under ten-sion, which include a strain softening curve in the post-

cracking phase [46]. Bond-slip based models are also

included in this approach. These models are based on

the assumed bond stress distribution along the tension

zone and are constructed from the force equilibrium

and strain compatibility condition at the cracked con-

crete matrix [716].

The second approach is based on the moment-

curvature relationship [1720], which takes into ac-

count the development of flexural cracks during load-

ing. The tension-stiffening effect is related to variousexperimental parameters: tensile strength of concrete,

creep, ratio between rebars and concrete, rebar diame-

ter, type of re-bars [21, 22].

The global behaviour is then calculated by integra-

tion of the moment-curvature law.

The use of the moment-curvature law is also the

CEB-FIP approach [23] with two alternatives. The first

is empirical (Chapter 3.6 CEB-FIP Model Code [23])

and the second is based on the concept of an effective

tensile member which replaces the tensile concrete

between two cracks. The cross-section of the effec-

tive tensile member determined by FEM analysis is

2.5(h d)b where h is the cross-section height, dthe

effective height (distance between the centre of gravity

of the rebars and the compressive fibre of the beam)and

b the thickness. The moment-curvature law is correctedto consider the tension-stiffening effect by using the av-

erage strain in the effective tension member resulting

from the force equilibrium.

Recently, Kwak et al. [24] proposed an analytical

approach to model the tension stiffening effect in the

case of tension members and beams. Unlike previous

approaches based on the assumed bond stress distribu-

tion function, this model represents the normal strain

distribution in the concrete by using a polynomial func-

tion on thetransfer length, which is thelength necessary

for the full transmission of the tensile load from rebarto concrete.

In the present paper, an approach based on a linear

variation of concrete and re-bar strain over the trans-

fer length is adopted. This approach is the same as

the CEB-FIP model which assumes a constant bond

strength between reinforcement and concrete. It is then

possible to take into account the development of cor-

rosion and the resulting de-bonding by increasing the

transfer length between concrete and re-bars. The im-

plementation of the model is based on the definition

of macro-elements having a length equal to the dis-tance between two flexural cracks and the same height

as the beam height. Such global approach in general-

ized variables wasalready performed in previous works

[2527] dealing with reinforced concrete element or

soil-structure interaction. The main interest of the pro-

posed model is that perfect bonding between concrete

and steel is not assumed which is the case for previous

studies. As a result, the bond damage due to corro-

sion is taking into account thanks to the increase of the

transfer length between rebar and concrete. To estab-

lish the diagnosis of a corroded element, it is neces-sary to perform numerous non-destructive tests on the

structures, so as to know the exact spacing between

flexural cracks. The reduced inertia of these macro-

elements then depends on the transfer length and its

increase due to corrosion. The inertia of the macro-

element is then implemented in elastic finite element

analysis which calculates the global stiffness of cor-

roded reinforced concrete members. The validityof this

approach is established in the absence of corrosion by


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Materials and Structures (2006) 39:571584 573

Fig. 1 Lay-out of A beams,

all dimensions in mm.

comparison with previous models and experimental re-

sults, and in presence of corrosion with experimental

results.

2. Experimental program

This paper is based on a large experimental program

dedicated to the study of reinforcement corrosion ini-

tiated in 1984 at the LMDC laboratory in Toulouse(France) [28]. This research project, which is still in

progress, consists in periodic inspection and testing of

reinforced concrete members stored in a loaded state in

a chloride environment. The main interests of the ex-

perimental program are the dimensions of the members

tested, which are representative of real structures (3 m

long, 150 280 mm cross-section), the storage in the

loaded state (members are cracked due to bending mo-

ment), the natural corrosion (no accelerated corrosion

using electrical field) and the existence of numerous

control members also aged in a loaded state but in anon-aggressive environment.

This paper focuses on two beams aged for 17

years and denoted A1CL and A1T for the corroded

and the control beam respectively. Both beams were

loaded in 3-point bending at the design service load

(Mser= 13.5 kNm, normal stress in reinforcement

s 160 MPa) corresponding to aggressive environ-

mental conditions according to French standards. In

this standard, the crack width is controlled by the nor-

mal stress limit in the reinforcement. The flexural load

was maintained using an adequate device [29] through-out the experiment.

The reinforcement lay-out for A beams is shown in

Fig. 1.

The reinforcement was provided by ribbed bars with

a 500 MPa yield strength. The mechanical characteris-

tics of the aged concrete were: compressive strength

= 63 MPa, tensile strength (through splitting tests) =

6.8 MPa, and elastic modulus=35 GPa. Water porosity

was 15.2%.

3. Mechanical formulation for one

macro-element without corrosion

In the post-cracking state of reinforced concrete and in

a cracked cross-section, all tensile stresses are concen-

trated on the reinforcement. A part of this tensile load is

transferred to the concrete located between two cracks

because of the bond between the steel and the concrete.

However, it takes a minimal bar length, called the trans-

fer length and noted L t, to achieve the full transfer of

stresses between steel and concrete. Figure 2 shows thetypical strain profile in reinforcement located between

two flexural cracks.

In this figure,L tis assumed to be lower than the half

spacing between two cracks, even though the CEB-FIP

code considers this to be impossible in real life because

a new crack will appear when the full strength transfer

is obtained. Of course, the model take into account both

possibilities: L tlarger or lower than the half-spacing

between two cracks.

The transfer length is difficult to evaluate by direct

experiment on beams but it is possible to measure itindirectly using the strain variation at the concrete sur-

face on a tension member.

3.1. Evaluation of transfer length using tension

member

Experimental tests were performed on a 500 mm long

tension member with 100 100 mm cross-section,

Lelem

sncbefore cracking

Crack face

xLt

(x)s

Fig. 2 Typical reinforcement strain profile between two cracks

[30].


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Lt Lelem

x

N

Tension Member

N

s

Strain distributionin re-bar

(x)

Fig. 3 Strain profile in reinforcement in tension member.

Fig. 4 Prismatic sample used as a tension member and locations

of the eight strain gauges on the half-length. (four gauges werelocated on the opposite face length in mm).

reinforced with a 12-mm-diameter ribbed bar (Fig. 3).

The concrete compressive characteristics strength was

40 MPa.

In these tests, both end surfaces of the tension mem-

ber played the role of a crack surface for the tensile

concrete located between two cracks. Then, the trans-

fer length could be lower than the half-length of the

tension member, which allowed a strain distribution in

the reinforcement to be obtained as in Fig. 2.The experimental program was built to study the

case of a perfect bond corresponding to physicochem-

ical adhesion between steel and concrete without any

relative slip by using smooth bars, and the case of ad-

hesion failure where the bond is due to the mechanical

interaction between ribs and concrete. Since the de-

vice was symmetrical, only a half tension member was

instrumented, using eight 30-mm long strain gauges.

To eliminate flexural effects during the test, the strain

gauges were placed on opposite sides of the tension

member. The strain gauge spacing was 50 mm (Fig. 4).Figures 5 and Fig. 6 show two different behaviours

for both tension members below and above a load

threshold of around 67 kN. Firstly, below 67 kN, all

strains measured on both tension members are identi-

cal and correspond to the theoretical behaviour. It is

then possible to conclude that the transfer length is

less than 100 mm, which corresponds to the distance

of the first pair of gauges from the edge of the ten-

sion member. The re-bar geometry (ribbed or smooth)

Fig. 5 Strain measurement on the tension member reinforced

by smooth bar12.

Fig. 6 Strain measurement on the tension member reinforced

by ribbed bar12.

has no influence because the bond is perfect (adhe-

sion). Secondly, above 10 kN load, the behaviours ofthe two members are different. For the one reinforced

with smooth bar, the gauges located near the edge (G7

and G8) do not continue to measure the same value as

the theoretical curve (Fig. 5).

This behaviour is due to debondingbetween the steel

and the concrete which began at the edge of the member

and has now reached the location of the first gauges.

This debonding corresponds to an adhesion failure.

When the load increases, the debonding progresses

along the bar to reach the second pair of gauges (G5 and

G6 at 150 mm from the edge), then the third (200 mm)and finally the middle of the member, which leads to

total debonding (Fig. 5). An interaction between the

steel bar and the concrete still exists due to friction.

Figure 7 shows the relative strain (gauge measurement

over perfect bond value) variation along the member

half-length and then the progressive debonding of the

reinforcement.Ncris the load which corresponds to the

appearance of a crack in the tension member reinforced

with ribbed bar (Ncr= 36 kN). Figure 7 shows that the


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0

20

40

60

80

100

120

0 50 100 150 200 250 300

0.25Ncr

0.5Ncr

0.75Ncr

Ncr

abcissa on the tie (mm)

Relative strain (%)

Smooth re-bar

Fig. 7 Variation of relative strain profile along the tension mem-

ber reinforced with smooth bar: the transfer length increases

strongly versus the load.

0

20

40

60

80

100

120

0 50 100 150 200 250 300

0.25Ncr

0.5Ncr

0.75Ncr

Ncr


Relative strain (%)

Ribbed re-bar

Fig. 8 Variation of relative strain profile along the tension mem-ber reinforced with ribbed bar: the transfer length is almost con-

stant versus the load.

transfer length is significantly increased for a load cor-

responding to 50% of the cracking load (0.5 Ncr) of a

ribbed reinforced member.

As mentioned previously, there is also a behaviour

change for the tension member reinforced with ribbed

bar about 67 kN load (Fig. 6). The gauges located near

the edge (G7 and G8) do not continue to measure the

value given by the theoretical curve. The change in theslope is clear and is coincident (total load and strain

gauge) with the debonding at the same location for the

member reinforced by smooth bar. However, there is

still a transmission of stress between re-bar and con-

crete since the strain increases with the load. This is due

to the interaction of the ribs with the concrete. When the

load is increased, the behaviour remains constant until

the appearance of a crack at 36 kN, which means that

the transfer length is constant whatever the load when

rib interaction with concrete occurs. Figure 8 shows

that the strain profile along the tension member is al-most constant versus the load level; there is only a slight

difference between the behaviour corresponding to the

bond due to adhesion and the bond due to rib interac-

tion, so the transfer length remains almost constant.

Thus the load threshold corresponding to the adhe-

sion failure has little influence on the transmission of

stresses between ribbed bars and concrete. Contrary to

the case of smooth bars, even when the adhesion fail-

ure is generalized along the whole tension member, the

-20

0

20

40

60

80

100

120

0 50 100 150 200 250 300


relative strain

finite element

analysis concrete

experimental measurement on c oncrete surface :

case of perfect bond (plain re-bar)

(strain gauge 30 mm long)

finite element

analysis

reinforcement

Lt

95%

experimental measurement on concrete surface

: over the perfect bond threshold (ribbed re-bar)

(strain gauge 30 mm long)

Fig. 9 Transfer length: comparison between FEM analysis and

experimental results obtained on tension members.

transfer of tensile strength between re-bar and concrete

is still possible and is due to the mechanical interactionbetween ribs and concrete. This transfer exists until a

new crack appears.

Figure 9 shows the results of a finite element anal-

ysis performed on the same tension member as those

previously tested. For this analysis, a perfect bond is

assumed between the smooth rebar and concrete, the

materials are assumed elastics and the computation is

2D. The unbroken line represents the relative strain cal-

culated for the reinforcement (100% is the full transfer

of stresses from re-bar to concrete) and the dotted line

represents the relative strain measured on the concretesurface, which corresponds to the strain measured by

the gauge in the experimental tests also plotted on the

same figure.

According to Fig. 9, the transfer length correspond-

ing to full transfer in the reinforcement is the same

as the one corresponding to the concrete surface. Fur-

thermore, the experimental measurements confirm the

shape of the strain profile calculated by FEM. Above

the load threshold leading to adhesion failure, there is

a slight increase of transfer length, and experimental

measurements are still close to the profile calculatedby FEM.

These profiles allow the transfer length to be cal-

culated. It will be evaluated as the distance from the

edge of the tension member necessary to reach 95%

of the value corresponding to full transfer. For the ten-

sion members tested, this distance was L t= 105 mm

(Fig. 9). This transfer length was associated with a

0.01 m2 tensile cross-section of concrete and a ratio

of concrete cross-section over re-bar cross-section of


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0

10

20

3040

50

60

70

80

90

100

110

120

130

0 50 100 150

Lt (mm)

Ratio Tensile concrete section

over re-bar section

Beam ATie 100x100 mm

FEM Analysis

Fig. 10 Variation of transfer length versus ratio of concrete

cross-section to re-bar cross-section.

about 88. Indeed, the transfer length would be expected

to depend on these parameters.

When the FEM analysis is performed for the case

of perfect bond, results show that the transfer length isnot related to the elastic modulus of the concrete (even

for a large variation of E: from 20 GPa to 100 GPa).

In the case of bonding due to interaction with the ribs,

the transfer length could be influenced by the concrete

characteristics, such as the compressive strength. This

phenomenon will be studied in further experiments.

However, for the C40 concrete experimentally tested,

the transfer length was similar for the adhesion state

and for the rib interaction state.

The transfer length depends on the ratio of concrete

cross-section to re-bar cross-section. Numerical simu-lations performed by FEM show that the transfer length

is sensitive to low ratios but is fairly constant for high

ratios (Fig. 10). For usual reinforced concrete elements,

the concrete cross-section is large with respect to the

reinforcement cross-section, so the transfer length cor-

responds to the stabilized length for high ratios. Nev-

ertheless, we have to bear in mind that it could be have

a boundary effect for the low value of cover. This point

will be discuss in a next paper.

According to the variation of the transfer length with

respect to the ratio of concrete cross-section over re-bar cross-section (Fig. 10), it is possible to evaluate

the transfer length for a given flexural member. To do

this, the equivalent tension member cross-section used

by CEB [23] will be used. In the case of a rectangular

cross-section, this equivalent cross-section according

to CEB is 2.5(h d)b where h is the depth of the

beam,dthe effective depth and b the thickness. In the

case of the A beam, the equivalent tension member

cross-section is 2.5(h d)b(0.021 m2, which leads

h-y

Lt

h-y

x

Lelem

snc

Lt

Lelem

x

sh

o

onc

Crack faceCrack face Crack face

Fig. 11 Macro-element hypotheses: linear variationof both neu-

tral axis location andre-bar tensilestressover the transfer length.

to a ratio of concrete cross-section over re-bar cross-

section equal to 51. The transfer length is then L t=

105 mm.

3.2. Calculation of average inertia of the

macro-element

The calculation of the average inertia of the macro-

element is performed by assuming that strain variationis linear along the transfer length. This hypothesis is the

same as the one used in the CEB-FIP code model. This

hypothesis is clearly false since the profile is more ex-

ponentially shaped. Nevertheless, an polynomial func-

tion (third order) was tested in a first approach and the

result in term of generalized variable, moment and cur-

vature was not significantly different. Then, the linear

profile was chosen and it correspond to the CEB ap-

proach.s is the re-bar strain in cracked cross-section

and sc is the re-bar strain before cracking. Another

hypothesis is that the variation of the height of the neu-tral axis is also linear over the transfer length between

the value corresponding to the classical reinforced con-

crete calculation in cracked cross-section and the one

calculated before cracking (Fig. 11).

The re-bar strains and the location of the neutral axis

are used to calculate the flexural curvature (x) for a

given abscissa of the macro-element. According to the

L tvalue, the equation of the flexural curvature is modi-

fied. In the following, all calculations will be performed

on a half macro-element because of the symmetry.

ifL t

Lelem

2

then (x) =

s (x)

d y0(x)

when x L t and when L t xLelem

2

then (x) = nc =sn c

d y0nc

ifL t

Lelem

2

then (x) =

s (x)

d y0(x)


7/14


8/14


Table 1 Mesh and characteristics of the macro-

elements for the control beam A1T

A1T

Nodes Location (m) Lelem(m) Im (m4)

1 0.000 0.575 2.899e-4

2 0.575 0.105 1.02e-4

3 0.680 0.210 1.02e-44 0.890 0.240 1.114e-4

5 1.130 0.230 1.085e-4

6 1.360 0.200 0.994e-4

7 1.560 0.250 1.142e-4

8 1.810 0.230 1.085e-4

9 2.040 0.220 1.055e-4

10 2.260 0.105 1.02e-4

11 2.365 0.435 2.899e-4

12 2.800

cracking map of A1T Beam

3000LOAD

Lt Lt1

2 3 4 5 6 7 8 9 10 11

12

Fig. 14 Cracking map due to bending of the control beam A1T.

experimentalbehaviorof the A beam corresponds to the

post-cracked behavior. This is not a problem because

the aim of the proposed model using macro-elements

is then to re-evaluate the behaviour of existing struc-

tures, which are already cracked by their service load.

As a result, the cracking map of the A1T beam, which

is shown Fig. 14, allows the mesh of the finite element

model to be defined. In the cracked zone, each macro-

element depends on the distance between two consec-

utive cracks and Table 1 shows the average bendinginertia of the macro-element calculated with the 105-

mm transfer length.

The global stiffness of the cracked beam (A1T) ex-

hibits a strong correlation with the proposed model

based on macro-element and transfer length (Fig. 15).

The CEB-FIPmodel slightlyoverestimates the stiffness

of the beam but is close to the experimental behaviour.

The next step of this paper is to take into account the

corrosion in the macro-element inertia.

0

2

4

6

8

10

12

14

16

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

deflection (CEB)

deflection (Lt)

deflection(experimental)

Moment

(kN.m)

Deflection

(mm)

Beam

A1T

Fig. 15 Comparison between experimental behavior of A1T

beam (already cracked), the model using CEB-FIP code, and

the proposed model.

4. Extention of the model to the mechanicalbehavior of corroded Rc members

The approach developed in this paper consists of in-

creasing the transfer length as a function of the inten-

sity of the re-bar corrosion (Fig. 16). The new transfer

length taking into account corrosion is called L tcor.

According to the intensityof the corrosion, thetrans-

fer lengthvaries from theinitial valueLt (without corro-

sion) and tends to infinity for serious corrosion leading

to total debonding between re-bar and concrete.

In order to be more in touch with other studies tak-ing into consideration environmental effects on the me-

chanical behavior of reinforced concrete [3234], a

scalar bond damage parameter Dc is introduced which

varies from 0 (no corrosion) to 1 (total debonding due

to corrosion). The relation between L tcorand Dc is the

following:

L tcor=

L t

1 Dc

When Dc = 0, there is no damage due to corrosion andL tcor= L t. When Dc = 1, there is total damage of the

bond between re-bar and concrete, and L tcor tends to

infinity (Fig. 17).

The deterioration of the bond due to corrosion of

the reinforcement is linked to the section loss of the

re-bars. The appearance of cracks and their width are

directly correlated to the amount of corrosion products

present due to the oxidation process [35]. The loss of

bond is correlated with the cracking process because of


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before cracking

Ltcor

Lt

Lelem

snc

x

(x)

Crack face

s

Fig. 16 Typical view of variation of transfer length to take into

account the loss of bond due to corrosion.

Dc = 0 : Ltcor = Lt

Dc = 1 : Ltcor =

Strain distributionin re-bar

(x)s

x

Fig. 17 Variation of transfer length and bond damage parameter

versus intensity of re-bar corrosion.

the loss of confinement of the re-bars in the surround-

ing concrete. Then, the loss of bond varies because of

the development of corrosion cracks (length and width)

but also because of the increase in the amount of cor-

rosion oxides which are less resistant than steel [36].

The bond damage can then be expressed as a functionof the section loss as follows.

Dc = 1

As As

As As0

n

forAs As0, elsewhere Dc = 0.

where As0 is the section loss threshold which initi-

ates the first crack; As is the section loss of re-bar;

n is a parameter describing the quantitative variation

of progressive debonding versus the section loss of the

reinforcement.

4.1. Variation of the bond damage parameter with

the section loss of re-bar due to corrosion

It is difficult to evaluate the variation of the Dc param-

eter versus the section loss of re-bars experimentally.

Mechanical tests could be done on corroded tension

members to measure the transfer length. But, unfor-

tunately, the transfer length is affected by corrosion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

n=5Mangat-ElgarfC.Fang et alCabreraH-S. Lee et alA. Almusallam et al

corrosion (%)

Dp0

Fig. 18 Fitting of the damage parameter model with the exper-

imental results from [3640].

once corrosion cracks have appeared. The use of straingauges on such cracked tension members would not

provide significant measurements.

Much research hasbeen conductedon corroded sam-

ples using the pull-out test to estimate the bond strength

versus corrosion. In most cases, the corrosion was not

natural corrosion but was accelerated by an electric

field. Of course, doubts could be emitted as to whether

such accelerated corrosion is representative but these

results allow the effect of corrosion cracks on the bond

to be estimated. Figure 18 shows a synthesis of experi-

mentalresults obtained by some researchers [3640]. Inthese research works, the progressive debonding with

increasing intensity of corrosion is expressed as a re-

duction of the failure bond stress in a pull-out test. On

Fig. 18, the experimental results are expressed using

a scalar damage parameter Dpo which represents the

relative loss of failure bond stress in pull-out tests:

Dpo =uo uc

uo

whereu0 is the failure bond stress in the pull-out teston a non-corroded sample and uc is the failure bond

stress in the pull-out test on a c% corroded sample (c%

is the relative section loss of the reinforcement).

Of course, we have to bear in mind that the use of

a damage parameter calculated from pull-out tests to

evaluate the variation in transfer length for tension tests

is open to question. Nevertheless, the results of pull-out

tests will be influenced by the rebar confinement loss

due to corrosion cracks in the same way as the variation


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0.00E+000 1.00E-004 2.00E-004 3.00E-004 4.00E-004

0.00E+000

1.00E-005

2.00E-005

3.00E-005

4.00E-005

5.00E-005

6.00E-005

7.00E-005

8.00E-005

9.00E-005

1.00E-004

Macro-elementLelem = 200 mm - Lt = 105 mm

Inertia in crackedcross-section

Inertia of macro-element

loss of rebar cross-section (m2)

Bendin

ginertia(m4)

Fig. 19 Variation of the

average bending inertia of a

200 mm length

macro-element of A beam

versus the section loss of

re-bar due to corrosion.

of transfer length will be influenced by the same loss

of confinement. This is a strong assumption since Dc

is linked to the increase in transfer length and Dp0 is

linked to the decrease in bond strength.Figure 18 shows a strongdecrease of the failure bond

strength versus the intensity of corrosion. For relatively

low percentages of section loss, there is a large amount

of bonding damage. This phenomenon is best fitted ac-

cording to the recursive least squares method, by using

n = 5 in the expression for the scalar damage Dc. The

error according to the least mean squares is minored for

n = 5 and increase of 8%forn = 6 and 22% forn = 4.

Other values of n lead to more than 35% in increase.

The parameter Dc was calculated using the section loss

corresponding to the percentage of corrosion c%.

4.2. Variation of the average bending inertia of a

macro-element with the intensity of rebar

corrosion

The average bending inertia of the macro-element de-

creases when corrosion of the reinforcement increases.

The coupled effect of section loss and loss of bond

due to corrosion are involved in this variation of bend-

ing inertia. Figure 19 shows the variation of the average

bending inertia of a 200 mm-long macro-element of theA beam versus the intensity of corrosion. On the same

figure, the inertia in a cracked cross-section (tensile

concrete neglected) and its variation versus corrosion

is also plotted. For large section loss, both inertias are

identical but for small loss of section (i.e. the first stage

of the corrosion process), there is a large difference

between the variation of average inertia taking into ac-

count the tension-stiffening effect and the cracked in-

ertia. Thus the loss of bond due to corrosion is the most

important factor for low corrosion intensities which are

the more usual in real degraded structures.

The model of corroded reinforced concrete be-

haviour based on the transfer length and its increaseversus corrosion intensity, was validated on the A1CL

beam, a 17-year-old beam that had been stored in a

chloride environment. After recording the bending me-

chanical behaviour of the A1CL beam, the reinforce-

ments were observed after removal of the concrete to

precisely establish the map of section loss of tensile

re-bar due to corrosion (Fig. 20). Details on the record-

ing of the section loss can be found in a previous work

[35]. All the re-bars were cut into small pieces (10 mm)

and weighed to evaluate the loss of mass relative to a

non-corroded re-bar. The loss of mass is then trans-lated in terms of section loss. If corrosion is recorded

and does not lead to the appearance of cracks, then

the intensity of corrosion which is quantified by the

section loss is less than As0 (the threshold for crack

appearance). So, the comparison between the cracking

map (Fig. 21) and the map of section loss allows this

threshold to be determined [35]. Because the 3-point

loading was simultaneous to the corrosion process, the

A1CL beam exhibits more flexural cracks than the con-

trol beam A1T. This phenomenon was also noticed by

Y. Ballim and J.C. Reid [41].Table 2 shows the discretization of the A1CL beam

which takes into account the location of the flexural

cracks. For each macro-element, the damage parame-

ter Dc is calculated as a function of the section loss

recorded at the location of the macro-element. The

bending average inertia is then deduced from the length

of the macro-element, the damage parameter and the

loss of section of the reinforcement. When the macro-

element is not located between two consecutive cracks,


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0

5

10

15

20

25

30

35

40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Location along the beam (m)

Steelcross-section

loss(mm

2)

Front reinforcem ent Back reinforc ementFig. 20 Variation of section

loss in the tensile

reinforcement due to

corrosion along the A1CL1

beam.

Table 2 Mesh and

characteristics of the

macro-elements for the

corroded beam A1Cl1

A1Cl1

Nodes Location (m) Lelem (m) As (m2) Dc Im (m4)

1 0.000 0.205 3.00E-05 0.296 2.899e-4

2 0.205 0.105 3.90E-05 0.378 0.791e-4

3 0.310 0.210 4.00E-06 0.014 1.007e-4

4 0.520 0.130 4.00E-06 0.014 0.842e-4

5 0.650 0.250 4.20E-05 0.403 0.812e-4

6 0.900 0.250 3.50E-05 0.342 0.851e-4

7 1.150 0.250 4.50E-05 0.427 0.798e-4

8 1.400 0.220 4.20E-05 0.403 0.785e-4

9 1.620 0.200 3.60E-05 0.351 0.793e-4

10 1.820 0.230 0.00E+00 0 0.108e-4

11 2.050 0.200 6.00E-06 0.038 0.963e-4

12 2.250 0.270 1.20E-05 0.109 1.093e-4

13 2.520 0.105 2.50E-05 0.248 0.805e-4

14 2.625 0.175 2.50E-05 0.248 2.899e-4

15 2.800

for example at either end of the beam, only the loss of

section of the reinforcement reduces the average bend-

ing inertia. As a result, the inertia is very close to that

corresponding to the non-corroded state.

4.3. Comparison between experimental resultsand the model extended to include corrosion

effects

Figure 22 shows the comparison between the experi-

mental behaviour of the corroded beam (A1CL) and

the proposed model. As mentioned previously, it is

the behaviour of the pre-cracked beam which is com-

pared. There is a strong correlation between experi-

mental stiffness of the corroded beam and the model.

Figure 22 shows that the decrease of the stiffness

between the corroded beam and the control beam is

about 37%. This variation is important and corresponds

to only 11% of section loss of the re-bars in the middle

part of the corroded beam according to the map of sec-

tion loss (Fig. 22 and Table 2). Such an 11% sectionloss of tensile reinforcement will lead to a 1% decrease

in stiffness if the transfer length is not changed and only

the section loss of reinforcement is used to decrease the

average inertia of the macro-element. This calculation

then corresponds to a bond not affected by corrosion.

If the tension-stiffening effect is neglected, the 11% of

section loss of re-bars will lead to a 10% decrease in the

stiffness of the beam. Then in these two opposite cases

(bond not affected by corrosion or no tension-stiffening


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0

2

4

6

8

10

12

14

16

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

Ltcor model

Lt model

experimental

corroded experimental

Moment

(kN.m)

Deflection (mm)

Beams A

Fig. 21 Comparison between experimental behaviour of the

A1T control beam and the A1CL1 corroded beam and the pro-

posed model.

cracking map of A1Cl1 Beam

3000 LOAD

Lt Lt

1

2 3 4 5 6 7 8 9 10 11

15

12 13 14

0.8 0.7 0.5 0.6 0.5

1.8 0.9 1.5 1.0 0.4

Fig. 22 Cracking map of the A1CL1 beam, the flexural cracks

are transverse and the corrosion cracks are longitudinal (width

in mm).

effect), the influence of the section loss due to corrosion

of the reinforcement is largely insufficient to explain

the decrease in stiffness shown by experimental data.

As a result, the loss of bond due to reinforcement cor-

rosion appears to be the main parameter allowing the

mechanical behaviour of reinforced concrete membersunder the service load to be described. There is a lack

of knowledge on the effect of corrosion on the deflec-

tion behaviour of RC members. Most test methods used

in the literature are performed by separating corrosion

and mechanical tests as sequential process [3, 3640,

42, 43]. In this study, as in real structures, the corro-

sion takes place while the beam carries load and the

two effects act synergistically. Furthermore corrosion

is not accelerated by impressing an electrical current on

the steel. In an study assessing the structural effects of

reinforcement corrosion under simultaneous load and

corrosion conditions, Y. Ballim and J.C. Read[42] have

founded that when 6% of the mass of steel is corroded,

then beam stiffness is decreased by 4070% relative

to the stiffness of the control samples. This difference

may be explained by the fact that Y. Ballim and J.C.Read used an electrical current to accelerated corro-

sion which leads to a generalized corrosion which is

not the case in real structures and in our study. Another

explanation could be the fact that the intensity of the

sustained loading has a effect on the coupled process

of progressive corrosion and the stiffness reduction of

the beam.

5. Conclusion

This paper deals with the mechanical behaviour of rein-

forced concrete members under their service load when

damaged by reinforcement corrosion. The model is

based on the discretization of reinforced concrete mem-

bers using macro-elements whose length is the distance

between two consecutive flexural cracks. Because the

model will be used to re-evaluate the behaviour of cor-

roded members, the location of existing flexural cracks

will be known during the diagnostic phase. Neverthe-

less, the model could be used for predictive analysis

by using a cracking pattern from models found in de-sign codes. A such predictive calculation was recently

done for an international benchmark and predictive re-

sults were very closed to experimental results [44]. The

tension-stiffening effect is taken into account by using

a transfer length. The local intensity of the corrosion

of re-bars is plugged into the model as both reduc-

tion of the cross-section of the re-bars and increase of

the transfer length to take into account the progres-

sive debonding due to corrosion. Thus, this approach

allows the combined effect of the reduction of cross-

section and the debonding due to corrosion to be takeninto account in the mechanical behaviour of reinforced

concrete members.

The use of the model requires a knowledge of the

transfer length, which could be evaluated for a given

concrete by experimental tests on tension members.

The model is a good fit for the experimental results

obtained on a non-corroded reinforced concrete beam.

The transfer length needs to be increased to take the

corrosion effect on the bond into consideration. This


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modification of the transfer length is modelled through

a damage parameter, the variation of which versus the

intensity of corrosion is based on data from pull-out

tests on accelerated corroded samples found in the lit-

erature. The proposed model was validated experimen-

tally using a naturally corroded 17-year-old beam.

An important point to notice is that the experimentalprogram allows for assessing the structural effects of

reinforcement corrosion under simultaneous load and

corrosion conditions. A strong correlationwas obtained

between experiments and the model. This first result

is interesting even though it is necessary to test the

model on other reinforced concrete members with dif-

ferent geometries to draw conclusions about its relia-

bility. The model was recently tested in an international

Benchmark. Reinforced concrete members tested were

reinforced by smooth bars and the experimental results

appears to be well fitted by using the same transferlength as the one use for ribbed bars.

Results show that the variation of the mechanical

behaviour of reinforced concrete members under ser-

vice load is mainly caused by loss of bond due to re-bar

corrosion.

Another interest of the proposed model based on a

local approach (at the scale of a macro-element) for the

corrosion is that it allows the variation of the intensity

of corrosion along the re-bars to be taken into account.

Indeed, on a reinforced concrete structure, the develop-

ment of corrosion is random in terms of both locationand intensity.

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