establishment of coupled analysis of interaction between

Establishment of Coupled Analysis of Interaction betweenStructural Deterioration and Reinforcement Corrosion by SaltDamageMika Suzuki, Naoyuki Fukuura, Hitoshi Takeda, MaruyaTsuyoshi

Journal of Advanced Concrete Technology, volume ( ), pp.14 2016 559-572

Multi-Mechanical Approach to Structural Performance Assessment of Corroded RC Members in ShearKukrit Toongoenthong , Koichi MaekawaJournal of Advanced Concrete Technology, volume ( ), pp.3 2005 107-122

Simulation of Coupled Corrosive Product Formation, Migration into Crack and Propagation in ReinforcedConcrete SectionsKoichi Maekawa, Kukrit ToongoenthongJournal of Advanced Concrete Technology, volume ( ), pp.3 2005 253-265

Time-Dependent Structural Analysis Considering Mass Transfer to Evaluate Deterioration Process of RCStructuresHikaru Nakamura , Worapong Srisoros Ryosuke Yashiro, , Minoru KuniedaJournal of Advanced Concrete Technology, volume ( ), pp.4 2006 147-158

Nonlinear Gel Migration in Cracked Concrete and Broken Symmetry of Corrosion ProfilesEsayas Gebreyouhannes , Koichi MaekawaJournal of Advanced Concrete Technology, volume ( ), pp.14 2016 271-286

Simulation of Steel Corrosion in Concrete Based on the Model of Macro-cell Corrosion CircuitTsuyoshi Maruya, Hitoshi Takeda Kenichi Horiguchi, , Satoru Koyama, Kai-Lin HsuJournal of Advanced Concrete Technology, volume ( ), pp.5 2007 343-362

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Journal of Advanced Concrete Technology Vol. 14, 559-572 September 2016 / Copyright © 2016 Japan Concrete Institute 559

Scientific paper

Establishment of Coupled Analysis of Interaction between Structural Deterioration and Reinforcement Corrosion by Salt Damage Mika Suzuki1, Naoyuki Fukuura2, Hitoshi Takeda3 and Tsuyoshi Maruya4

Received 5 July 2016, accepted 6 September 2016 doi:10.3151/jact.14.559 Abstract This study develops an analysis method for estimating the process of corrosion in concrete, including initial corrosion and the onset of corrosion-induced cracking. The method is suitable for application in rationalizing the verification of the durability of salt-damaged RC structures. Corrosion deterioration is computed by coupling the analysis of structure with the analysis of reinforcement corrosion. A method of calculating macro-cell corrosion in consideration of macro-cell corrosion current density is also proposed, focusing on cathodic elements of a reinforcing bar. The proposed analytical method is validated against dry-wet cyclic tests with salt solution to simulate macro-cell corrosion. We verify the accuracy of the method by confirming the non-uniformity of the concrete before cracking and, by coupling the analysis with structural analysis, investigating how the expansion ratio of corrosion products and diffusion coefficient of chloride ions affect the onset of corrosion, the time of initial corrosion cracking, the chloride ion density and the corrosion amount. This paper is based on an original paper (Suzuki et al. 2014) written in Japanese.

1. Introduction

Recently, it is becoming more and more important to predict the structural performance of reinforced concrete (RC) structures after the onset of corrosion-induced cracking on a long-term basis. That is, an analysis method that can predict structural performance is re-quired. It is suggested that limit state for the durability verification of RC structures should be the occurrence of corrosion cracking, not the occurrence of steel corrosion in the concrete (JSCE 2012).

The structural performance of RC structures in a salt environment deteriorates over time through the interac-tion of ongoing concrete cracking and steel corrosion (Fig. 1). Therefore, it is vitally important to take these interactions into consideration. However, there are cur-rently few analysis methods with such capabilities (Maekawa et al. 2003; Fukuura et al. 2007).

In this paper, a corrosion expansion model of steel in concrete is presented for finite elements, which are used in the model of the interaction between structural analy-sis and reinforcement corrosion analysis. This is ex-tremely important to the proper prediction of corrosion deterioration in RC structures from the onset of corro-sion.

The authors have established a coupled analysis be-tween structural deterioration and reinforcement corro-sion by salt damage. Nonlinear structural analysis of RC members is used to determine structural deterioration. The determination of reinforcement corrosion takes into account the diffusion of chloride ions and models of both micro-cell and macro-cell corrosion. Here, we introduce a distribution diagram of macro-cell corrosion current density. The method enables more accurate modelling of macro-cell corrosion in RC structures.

2. Design of coupled analysis of interaction between structural deterioration and rein-forcement corrosion

2.1 Outline of coupled analysis of interaction between structural deterioration and reinforce-ment corrosion The coupled analysis method being developed by the authors, which involves analysis of structure, analysis of reinforcement corrosion and analysis of how the two interact, aims to predict the deterioration of a reinforced concrete structure under salt attack. Structural analysis is carried out with a nonlinear finite element method using a three-dimensional model. First, chloride ion diffusion is determined. Then, based on the concentration of dif-fused chloride ions at the position of the reinforcement, three-dimensional reinforcement corrosion modeling is carried out. Interaction models are proposed to coodinate these two analyses.

Figure 2 is a flow diagram for this coupled analysis process. First, stresses and strains of concrete and rein-forcement bars are calculated through structural analysis. Next, the coefficient of diffusion of chloride ions is de-fined as dependent on the cracked concrete strain at a particular time. Third, corrosion of steel is computed in a

1Research Engineer, Civil Engineering Research InstituteTaisei Corporation, Yokohama, Japan. *Corresponding author, E-mail: [email protected], 2nd Division Manager, Coms EngineeringCorporation, Tokyo, Japan. 3Chief Research Engineer, Civil Engineering Research Institute, Taisei Corporation, Yokohama, Japan. 4General Manager, Civil Engineering Research Institute, Taisei Corporation, Yokohama, Japan.

M. Suzuki, N. Fukuura, H. Takeda and T. Maruya / Journal of Advanced Concrete Technology Vol. 14, 559-572, 2016 560

time-step corrosion analysis. Finally, structural analysis is carried out for the next time step, taking into account expansion strain of the concrete due to steel corrosion in the current time step. Loading can be added in any time step.

2.2 Structural analysis method Concrete is modelled as a solid element, while a rein-forcing bar is considered to be a truss element with an additional interface element modelling the mutual in-teraction between the bar and the surrounding concrete. Figure 3 illustrates the finite element model of the re-inforcing bar and the concrete and the rigid matrix rep-resenting the interface element. The truss element is rigid in the axial direction. The interface element has not rigid in the axial direction and has the same rigidity as the reinforcing bar in the plane normal to the reinforcing bar. By varying the rigidity in the shear direction along the reinforcing bar, deterioration in bonding could be taken into account. However, in this paper, interface rigidity is

constant and equal to that of the reinforcing bar. The fixed crack model (JSCE 2003) is used to model

the behaviour of cracked concrete. The relationship be-tween stress and strain is determined from the standard specification of concrete structures. Tensile softening of the concrete with an exponential function is included in the concrete tensile model. Coefficient C, the tensile softening characteristic, is set to 2.0 to correspond to plain concrete.

2.3 Corrosion analysis method The analysis of corrosion consists of 1) a diffusion model for chloride ions and 2) a corrosion model of steel in concrete. The diffusion model is described here, while Chapter 3 gives details of the corrosion model.

The diffusion analysis of chloride ions is carried out by a finite element method based on Fick’s law (Eq. (1)) in three dimensions. All elements in this analysis are solid elements. Equation (2), which is empirically formulated to give a margin of safety for durability evaluations, is

[Cl-]At the position of steel

Micro-cell corrosion density imicroand

Macro-cell corrosion densitybetween 2 element imacro,M-N

Accumulated corrosion density icor

Corrosion amount bymacro-cell in ΔT

Time=T

Cumulative corrosionamount at T+ΔT

Relationbetween

stressand

strain of

steeland

concrete

Reinforcement corrosion analysis

Structural analysis

Interaction model

Corrosion deterioration

Steelbond strength

Model of corrosionof RC member

Faraday’s Law

Loading

Time=T+ΔT

Mass transport modelin concrete

Change of diffusion

Coefficient of cracked element

Corrosion expansion

Time=T

Time=T

Fig. 2 Flow diagram of coupled analysis.

Interface(solid) element

x

yz

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

xz

yz

xy

z

y

x

xz

yz

xy

z

y

x

D

DD

D

γγγεεε

τττσσσ

2

2

1

1

00

00

b)Relation between σ-ε of interface element

x : Axial reinforcementy : Radial direction of the reinforcementz : Normal direction of x-y planeD1: Rigidity equivalent to a reinforcementD2: Rigidity of bond slip

(Initial value is equivalent to a reinforcement)

Concrete(solid) element

Reinforcing bar(truss) element

a)Finite element method

Fig. 3 Finite element model between reinforcing bar and concrete and rigid matrix.

load

b) Cracking in concrete due to corrosiona) Cracking in concrete due to load c) Reduction of load bearing capacity

Expanding corrosion crack due to corrosion expansion

chloride ion

Increasing chloridediffusion in crackedconcrete

timeD

ispl

acem

ent

Initiationstage

a) b)c)

Propagationstage

Accelerationstage

Deteriorationstage

Fig. 1 Process of RC structure deterioration due to salt damage.


used for the initial value of the diffusivity of chloride ions in the concrete.

According to the indoor test results on which Eq. (2) is based, the diffusion coefficient for concrete with a wa-ter-cement ratio of 0.6 is between 0.4cm2/year and 3.0cm2/year, which is quite a wide range. In modelling an RC member, the initial diffusion coefficient should not be set as constant throughout the corrosion process.

A further consideration is that the chloride ion con-centration in the pore solution influences corrosion. This is expressed using Eq. (3). On this occasion, we consider the influence of chloride ion immobilization with Eqs. (4) and (5) (Maruya et al. 2007).

2 2 2

2 2 2total total total total

x y zC C C C

D D Dt x y z

⎛ ⎞∂ ∂ ∂ ∂= ⋅ + ⋅ + ⋅⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(1)

10 , ,log 3.0( / ) 1.8x y zD W C= − (2)

1000[ ]

35.5free

e

CCl

W− ⋅

=⋅

(3)

(1 )free fixed totalC Cα= − ⋅ (4)

0.251 0.11 0.35 ( 0.1) 0.1 3.00.543 3.0

tot

fixed tot tot

tot

CC C

Cα

≤⎧⎪= − ⋅ −⎨≤⎪⎩

　　　　　　　　　＜＜

　　　　　　　　 (5)

where, Ctotal is all chloride ion concentration (kg/m3) , Dx,

y, z is diffisivity for the chloride ion of the concrete (cm2/year), C is unit quantity of cement (kg/m3), W: unit quantity of water (kg/m3), [Cl‐] is chloride ion concen-tration in pore solution (mol/l), Cfree is quantity of free chloride ion (kg/m3), We is unit quantity of non-hydration of watar (kg/m3), α fixed is immobilized coefficient, Ctot is quantity of all chloride ions (cement mass %) 2.4 Model of Interaction between structural de-terioration and reinforcement corrosion 2.4.1 Chloride diffusion model for cracked con-crete Cracks in concrete have a significant impact on the dif-fusion of chloride ions (Takewaka et al. 2003; Djerbi et al. 2008). The relationship between crack width and coefficient of diffusion has been elucidated in recent research (Takewaka et al. 2003). However, there is con-siderable variation among the results by various re-searchers (Fig. 4 (JSCE 2009)). For the finite element method presented in this paper, a diffusion coefficient is required for finite elements representing cracked con-crete. An apparent coefficient of diffusion is set up using coefficients for cracked and non-cracked concrete in reference to Djerbi’s method (Fig. 5). If a single crack occurs in one element and crack width is obtained by multiplying tensile strain by average crack length, the diffusion coefficient for the finite element is equationted as in Eq. (6). Because Dcr is the coefficient of diffusion through the finite element with the crack, dcr is set 1.0 until the crack width is 0.03mm.

0( ) ( )

1 ( )1 ( ) 1 ( )

cr d

crd cr

cr cr

D Dd

dd d

αω ω ω ωα ω

ε ω ε ε

= ×− + ⋅

= ≅ + ⋅

≅ + ⋅ ≅ + ⋅ ⋅

(6)

where D0 is the diffusion coefficient of the non-cracked concrete, Dcr is the diffusion coefficient through the finite element with the crack, and αd is the proportional increase in diffusion coefficient due to the crack. And dcr is the proportional increase in diffusion coefficient of the cracked part compared with D0 (a function of ω), ω is crack width (=ε · ), ε is the orthogonal direction to crack tensile strain (>0), and λ is the average length of a finite element ( 3 V= , where V is the volume of a finite ele-ment). 2.4.2 Model of the corrosion expansion of steel For the modelling of corrosion expansion, various tech-niques have been proposed in the past, including FEM analysis, analysis using a cylindrical model and analysis using a three-dimensional rigid spring model. Suda et al. (1993) arrenged the value of expansion ratio of corrosion products in concrete given by formula weight and density of corrosion product. Lundgren (2002) assumed a free expansion ratio of 2.0 and attempted to obtain the rela-tion between stress and strain of corrosion products in the radial direction. Toongoenthong and Maekawa (2005) also assumed a free expansion ratio of 2.0 and defined compositional rigidity values between the reinforcing bar and the corrosion products.

The effective expansion ratio of corrosion products is set by assuming the radial expansion of corrosion prod-

＝＋

D0Dcr0)( Ddcr ⋅⋅ ωω

ωω−

Fig. 5 Partition hypothesis of chloride diffusion through finite element with a single crack.

1.0E-081.0E-071.0E-061.0E-051.0E-041.0E-031.0E-021.0E-01

0.0 0.1 0.2 0.3 0.4

diffu

sion

coe

ffici

ent（

cm2 /s）

crack width(mm)

0.39-40℃(Tsukahara) 0.55-40℃(Tsukahara)Steel-40℃(Tsukahara) Steel-20℃(Tsukahara)Takewaka(ACT) 0.7（Maeda_JSCE）0.5（Maeda_JSCE） A.Djerbi(CCR)dilute solution(handbook)

Fig. 4 Crack width versus coefficient of diffusion through crack.


ucts. The effective expansion ratio of corrosion products in this paper is not the value given by formula weight and density of corrosion product but radial increase in vol-ume act effectively. And corrosion expansion is modelled as a single element (interface element), representing the steel including the corrosion products.

In treating corrosion product expansion as a radial in-crease in steel volume (Fig. 6), the steel bar effective expansion strain ε caused by corrosion expansion is the increase in volume V of a unit length of steel caused by corrosion divided into the volume of a unit length of steel before corrosion (Suzuki et al. 2015). This effective expansion strain ε is passed as an initial strain to the interface element. The interface element has the same rigidity as the steel. The rigidity of the concrete is re-duced after cracking.

3. Model of steel corrosion in concrete

3.1 Steel corrosion In this chapter, we first propose basic models for steel corrosion in concrete, one for micro-cell corrosion and one for macro-cell corrosion. Thereafter, a model is pro-posed for the quantity of rational corrosion calcutation for steel located as a bar in RC member.

The cathodal and anodal reactions of steel undergoing corrosion in concrete are expressed by Eqs. (7) and (8), respectively. When the cathodal and anodal reactions both occur in the same element, the reaction is defined as micro-cell corrosion. Similarly, macro-cell corrosion is when the two reactions occur in two different elements.

2 21 O H O 2 2OH2

e− −+ + → (7)

2F F 2e e e+ −→ + (8)

Both micro-cell corrosion and macro-cell corrosion in an RC member are modelled with an internal polarization curve and an external polarization curve. The internal polarization curve is not measured directly. Then, The internal polarization curve is represented by virtual ca-thodal and anodal reactions, as given by Eqs. (9) and (10) (Hsu et al. 2000), respectively.

10log cc c co

co

iE E

iβ

⎛ ⎞= × +⎜ ⎟

⎝ ⎠ (9)

10log aa a pit

pass

iE E

iβ

⎛ ⎞= × +⎜ ⎟⎜ ⎟

⎝ ⎠ (10)

where Ec is the half-cell potential in the internal cathodal polarization curve (V), Eco is the equilibrium potential of the internal cathodal reaction (V), β c is the Tafel constant of the internal cathodal polarization curve (V/decade), ic is the current density in the internal cathodal polarization curve (A/cm2), ico is the exchange current density (6.25x10-10 (A/cm2)), Ea is the half-cell potential in the internal anodal polarization curve (V), Epit is the pitting potential (V), β a is the Tafel constant of the internal anodal polarization curve (V/decade), ia is the current density in the internal anodal polarization curve (A/cm2), and ipass is the passivity current density, 5.8x10-8 (A/cm2) (Kranc et al. 1997).

External polarization curves are generally obtained directly by measurement using electrochemical instru-ments such as a potentiostat/ galvanostat. In this work, the external polarization curve is theoretically formu-lated from the internal polarization curve in reference to Jones (1996). By this means, consistent modeling be-tween internal and external polarization curves is en-sured.

Figure 7 illustrates how the external polarization curve is set. The half-cell potential of the element is forced to fall from Emicro to Ec (see Fig. 7a). The anodal reaction is restrained from imicro to ia on the internal an-odal polarization curve, as given by Eq. (10). Here, imicro is the point of intersection between the internal polari-zation curves, given by Eqs. (9) and (10). The cathodal reaction is promoted from imicro to ic on the internal ca-thodal polarization curve given by Eq. (9). The external cathodal current density iapp,c is then expressed by Eq. (11) according to the principle of charge conservation. In other words, the external cathodal current density iapp,c is the difference between the increment in the cathodal reaction and the decrement in the anodal reaction caused by overvoltage εc. Fig. 7b) shows plots (iapp,c, Ec) of external cathodal polarization curve for various εc. Similarly, the external anodal current density iapp,a is equal to the difference between the increment in the anodal reaction and the decrement in the cathodal reac-

Reinforcing bar

Corrosion product

Δt

(α-1)× Δt

Steel surface before corrosion

Diameter D

ΔW=πmD, ΔV=ΔW(α-1)/ γFe, V=πD2/4 , ε=ΔV/V

ΔW: Increase in unit length by corrosion (mg/cm)m : Quantity of corrosion product (mg/cm2)D : Diameter of reinforcing bar (mm)Δt : Decrease in steel diameter (mm)ΔV : Increase volume in unit length by corrosion (cm3/cm)α : Effective expansion ratio of corrosion productγFe : Unit weight of steel (7870mg/cm3)V : Volume in unit length of steel before corrosion (cm3/cm)ε : Effective expansion strain by corrosion expansion

Loss of steel

Fig. 6 Model of corrosion expansion.


tion (ia - ic) caused by overvoltage εa, as expressed by Eq. (12) (Jones 1996).

,app c c ai i i= − (11)

,app a a ci i i= − (12)

where, iapp,c is the external cathodal current den-sity(A/cm2), iapp,a is the external anthodal current den-sity(A/cm2) 3.2 Micro-cell corrosion model The model of micro-cell corrosion is illustrated in Fig. 8. The basic arrangement refers to (Hsu et al. 2000). The model is faithfully fitted to actual behaviour using coef-ficient values based on past experimental data (Bird et al. 1988; Funahashi et al. 1991; Kranc et al. 1997). The micro-cell corrosion current density is at the intersection between the internal cathodal polarization curve and the internal anodal polarization curve. In the case of imicro < ipass or where [Cl‐]/[OH‐] is below the threshold, the steel retains its passivity film and micro-cell corrosion does not occur. However, when the macro-cell corrosion calculation is carried out, as mentioned later, we assume imicro = ipass and Emicro = Ec (ipass). The equilibrium poten-tial, which constitutes an internal cathodal polarization curve, is shown in Eq. (13) and the pitting potential, which constitutes an internal anodal polarization curve, is shown in Eq. (14) (Bird et al. 1988). The half-cell potential is assumed to be the electric potential for a saturated copper sulphate electrode (V vs. CSE). Tafel constants for the internal cathodal polarization curves are set constant at -0.20 (V/decade). Tafel constants for the internal anodal polarization curve are set using Eq. (15) using an approximate experimantal formulation (Funa-hashi et al. 1991). Considering the domain of density polarization, ilim, the limit corrosion current density is set to 1.0x10-4A/cm2.

0.912 0.059coE pH= − × (13)

100.092 0.31 log ([Cl ] /[OH ])pitE − −= − − × (14)

2.0)(log04.0 10 +⋅−= totala Cβ (15)

where pH is the pH of the liquid phase (12.0) and [OH‐] is the hydroxyl ion density (mol/l). 3.3 Macro-cell corrosion model The macro-cell corrosion is calculated using the internal and external polarization curves. The macro-cell corro-sion between two elements (cathodal element M and anodal element N) of steel in the concrete is shown in Fig. 9 under the assumption that the area of the anodal ele-ment equals the area of the cathodal element. The inter-nal cathodal polarization curve of element M is notated as CM and the internal anodal polarization curve is AM. The equivalent internal cathodal and anodal polarization curves of element N are notated as CN and AN. When the current flow between element M and element N stops, micro-cell corrosion takes place and the respective cor-rosion current densities are imicro,M and imicro,N. Elements M and N are represented by Mmicro and Nmicro (the open circles in Fig. 9). When current flows between element

自然

電位

E

Internal cathodalpolarization curve

Internal anodal polarization curve

Epit decreases with increment of [Cl-] /[OH-]

Electric potential decreases with a drop of pitting potential

Pitting occurrence

Equilibrium state

Intersection point between internal anodal polarization curve and internal cathodal polarization curve

Epit

Eco

imicroipassico

Pitting limit

βa

βc

To take a passivity film into consideration,In case of imicro< ipass or [Cl-] /[OH-] < limit [Cl-] /[OH-],imicro＝ipass，Emicro=Ec( ipass ) (at macro-cell calculation),imicro＝0 (at micro-cell calculation)

Emicro

Hal

f-ce

ll po

tent

ialE

Current density log10 i Fig. 8 Micro-cell corrosion model.

Current density log10 i

Hal

f-ce

ll po

tent

ialE

Internal cathodalpolarization curve

Internal anodal polarization curve

imicroipassico

βa

βc

iapp,c=ic-ia

iapp,a=ia-ic

icia

iaic

εc

εa

Low

High

Emicro

Ec

Ea

Current density log10 i

Hal

f-ce

ll po

tent

ialE

External cathodalpolarization curve

External anodal polarization curve

imicroipassico

βa

βc

(iapp,c ,Ec)

(iapp,a ,Ea)

Emicro

a) i

app by overvoltage ε b) The external polarization curve

Fig. 7 The setting method of the external polarization curve.


M and element N, macro-cell corrosion begins. Current density ig,M―N at that time is the current density when the potential difference between the external cathodal po-larization curve (iapp,c,M, Ec) and the external anodal po-larization curve (iapp,a,N, Ea), as given by Eqs. (11) and (12), equals the potential difference ΔEM－N according to the concrete resistance between element M and element N (Eq. (16)). In this situation, element M and element N are as indicated by Mmacro and Nmacro (filled circles in Fig. 9). The corrosion current densities in element M and element N are then Emacro,M and Emacro,N. Thus, the macro-cell corrosion current density in element N in-creases by imacro, M-N (=imacro,N – imicro,N) and the macro-cell corrosion current density in element N decreases by imicro,M – imacro,M.

, , ,( )M N macro M macro N M N con g M NE E E L iΔ ρ− − −= − = ⋅ ⋅ (16)

where LM－N is the distance between element M and ele-ment N (cm), ρcon is the apparent resistivity of the con-crete (Ω·cm), ig,M－N is the current density between ele-ment M and element N when the current is flowing (A/cm2) 3.4 Modelling of steel material in RC member There are multiple elements when a reinforcing bar in concrete is modelled with multiple axial elements. It is clear from both theory and previous experiments (Na-kagawa et al, 2008) that the macro-cell corrosion current

density depends on the ratio of cathodal area to anodal area (see Fig. 10). In modelling a reinforcing bar as multiple axial elements, the influence of the ratio of cathodal area to anodal area is considered in this work. A distribution diagram of macro-cell corrosion current density β is determined and the macro-cell corrosion current between a certain cathodal element is distributed among each opposing anode.

The macro-cell corrosion current density between a certain anodal element (Ni, i=1~n), when there are n anodal elements in total, and a certain cathodal element (M) is given by Eq. (17). The distribution diagram βM－Ni is set according to Eq. (18). The distribution diagram βM

－Ni set it for a function of ratio of the cathodal element against anodal element group and macro-cell corrosion current density imacro,M-Ni ,which is calculated cathodal element area equals anodal element area in chapter 3.3. The influence of the distance of mutual interval by the ratio resistance is considered in Eq. (16), which calcu-lates the corrosion current density between independent elements M and Ni. The relations between cathodal element M and the opposing anodal element group are considered only in terms of area ratio. We assume here that water and dissolved oxygen are in adequate supply, and corrosion of the reinforcing bar mainly depends on the chloride ion concentration distribution. Where suffi-cient water and dissolved oxygen are not available, it would be necessary to consider these influences sepa-rately.

The macro-cell corrosion current density of an element K is not only the current density of element K acting as an anode, but is reduced by the macro-cell corrosion current density of element K acting as the cathode. Be-cause this is not fully clear theoretically, we model the total macro-cell corrosion current density as the contri-bution of multiple opposing anodal elements less the element’s own cathodal current density.

The total corrosion current density in an element K of the steel bar in an RC member is illustrated in Fig. 11. It can be calculated using Eq. (19). This is the micro-cell corrosion current density imicro,K plus the macro-cell cor-rosion current density between element K and the other elements when element K is acting as the anodal element, minus the macro-cell corrosion current density between element K and the other elements when element K be-comes the cathodal element. This cannot be negative.

, ,i icor M N i M N macro M Ni iβ− − −= ⋅ (17)

, ,

, ,1 1

i i i

i

ij j j j

N macro M N M macro M NMM N n n

NN macro M N N macro M N

j j

S i S iSSS i S i

β − −−

− −= =

⋅ ⋅= ⋅ =

⋅ ⋅∑ ∑(18)

, , , , ,1 1

( 0)j j

n n

cor K micro K cor M K cor K N cor Kj j

i i i i i− −= =

= + − ≥∑ ∑ (19)

where icor,M－Ni is the macro-cell corrosion current density between element M and element Ni of the steel bar in the RC member (A/cm2), βM－Ni is the distribution diagram of

Current density log i

Hal

f-ce

ll po

tent

ialE

External cathodalpolarization curve

External anodal polarization curve

ΔEM‐N

imicro,Nimicro,M

Emacro,N

Mmicro

AM

ANEmacro,M

CM，CN

ig,M - N

（iapp,c,M ,Ec,M）

（iapp,a,N ,Ea,N）

imacro,Nimacro,M

imacro,M-N

Mmacro

Nmacro

Nmicro

Fig. 9 The macro-cell corrosion between two elements

(element M, N) of steel in concrete.

Cathodal Area Anodal Area Current density

Current density : 1/2unit

Current density : 2unit

Area : 1unit Area : 2unit

：

：

a) Cathodal Area : Anodal Area =1:2

b) Cathodal Area : Anodal Area =2:1

Area : 1unitArea : 2unit

Fig. 10 Dependence of macro-cell corrosion current den-sity on ratio of cathodal area to anodal area.


macro-cell corrosion current density between element M and Ni, SM is the surface area of element M (cathodal part) (cm2), SN is the surface area of element N (anodal part) (cm2), and icor,K is the total corrosion current density of element K (A/cm2). 3.5 Method of calculating quantity of corrosion from current density The increase in amount of corrosion of element K with time, which is given by Eq. (20), is calculated using Faraday’s law. The reaction represented by Eq. (8) is assumed.

,cor KK

i t am

n FΔ

Δ⋅ ⋅

=⋅

(20)

where Δ mK is the increment in the amount of corrosion of element K(g/cm2), Δ t is the time increment, a is the atomic weight of iron (55.845), n is the ion equivalent of dissolved iron (2mol eq), and F is the Faraday constant (96,500C/mol eq). 4. Verification of coupled analysis method for interaction between structural deterio-ration and reinforcement corrosion

In this chapter, we verify the method of coupled analysis for the interaction between structural deterioration and reinforcement corrosion as detailed in Chapters two and three. Verification includes 1) comparison of limit cor-rosion quantity at corrosion crack occurrence with ex-perimental results, 2) comparison of time of corrosion occurrence and corrosion cracking with experimental results and 3) comparison of corrosion amount after corrosion cracking with experimental results.

4.1 Verification of limit corrosion quantity at corrosion crack occurrence 4.1.1 Overview of analysis There have been experiments aimed at measuring the limit corrosion quantity at initial corrosion crack occur-rence. Generally, corrosion has been promoted in these experiments by electrolytic (Suzuki et al. 2009; Mori-kawa et al. 1987; Tamori et al. 1988) and cyclic dry-ing-wetting methods (Maruya et al. 2007; Nakagawa et al. 2008). With the former, corrosion cracking is judged from the value of strain at the concrete surface. With the latter, visual observation is the method used. Generally, the limit corrosion quantity at initiation of corrosion cracking is higher when obtained by the wet-dry cyclic method, with the difference presumed to result from the different corrosion products each method produces. Al-ternatively, it is thought that the method of corrosion quantity measurement and the method to judge of cor-rosion crack have an influence upon limit corrosion quantity at corrosion crack occurrence. Also, the limit corrosion quantity may be on the high side with this method, because cracks will already have reached a width of around 0.1mm by the time they can be observed.

For these reasons, the method of corrosion adopted in this study was electrolytic. It is thought that initial cracking in the concrete is judged accurately because it is obtained from the value of strain at the concrete surface.

In this analysis, the effective expansion ratio of cor-rosion products is used in a model of the corrosion ex-pansion of steel. To obtain the limit corrosion quantity at corrosion crack occurrence, this expansion ratio must be set appropriately. To this end, the limit corrosion quantity at corrosion crack occurrence was investigated with effective expansion ratio of corrosion product α as one parameter (=1.10, 1.25, 1.50, 2.00) and cover C as a second parameter (=20, 40, 60, 100mm). The rebar di-ameter was 16mm and concrete compressive strength was 30N/mm2. Concrete tensile strength was set ac-cording to the standard concrete specification. The 3D analysis model is illustrated in Fig. 12.

The direction of the arrowCathode←Anode

Kmicroi ,

N1 N2 ・・・・・・ Nn

M1 M2 ・・・・・・ Mn

∑=

−

n

jNKcor ji

1,

∑=

−

n

jKMcor ji

1,

a) Micro-cell corrosion current density of element K

b)Total macro-cell corrosion current density between anodal element K and all the other cathodal elements

c) Total macro-cell corrosion current density between cathodal element K and all the other anodal elements

: Anodal element

: Cathodal element

anode

Element K

cathod

Micro-cell

Element K

Element K

Fig. 11 Total corrosion current density in element K of steel bar in RC member.

X

Y

Concrete (solid) element

Steel (truss) element

Interface (solid) element

ZBoundary conditionsX direction : center of reinforcing bar

andbottom center of the concrete surface

Y direction : center of reinforcing barZ direction : All nodes of one side

2C+D=96

2C+D

=96

Cove

r CSt

eel D

iam

eter

D

Thickness : 20mm (Z direction)

Fig. 12 An example of FEM model (D=16mm,C=40mm).


4.1.2 Comparison of results between analysis and experiment As examples of the analysis results, the relations between corrosion quantity and tensile strain at the concrete sur-face are illustrated in Fig. 13 and the maximum tensile main strain contour of the concrete is shown in Fig. 14. In the analysis, the corrosion crack criteria is the time at which the strain at the concrete surface reaches 100×10-6, representing the point when the concrete surface strain increases rapidly.

The relation between the ratio of cover thickness to bar diameter (C/D) and limit corrosion quantity at crack occurrence is shown in Fig. 15. There is an association between C/D and limit corrosion quantity at crack oc-currence. A similar tendency is seen in this analysis. Further, the limit corrosion quantity at crack occurrence corresponds closely to the values in the experiment when the effective expansion ratio of corrosion products is set between 1.10 and 1.50 in this analysis. And the effective expansion ratio in 3.2 is in the range 1.0 to 2.5 for cor-rosion quantities from 0 to 30 mg/cm2. Limit quantity of corrosion product at crack is in the range 1 to 40mg/cm2

in recent researches (Suzuki et al. 2009; Morikawa et al. 1987; Tamori et al. 1988). However, the limit corrosion quantity at crack occurrence is relatively scattered and it is difficult to set an expansion ratio. It appears that a value in the range 1.10 to 1.50 is suitable in the corrosion expansion model presented here.

4.2 Verification of when corrosion and corro-sion cracking occurs, and the amount of corro-sion after cracking In this section, the proposed analytical method is validated through dry-wet cyclic tests with salt solu-tion to simulate macro-cell corrosion (Maruya et al. 2007). 4.2.1 Overview of macro-cell corrosion test Test specimens (1200mm long, 100mm deep and 100mm high with concrete cover thickness of 20mm and rebar diameter of 13mm (SD295A)) were exposed at an indoor site. The experimental setup of the specimens is illus-trated in Fig. 16. To limit the movement of chloride ions to the vertical direction only, all parts of each specimen except the two exposure surfaces were coated with acrylic resin. On the exposure faces, areas within 200mm of the end margin were also coated with acrylic resin to avoid corrosion of the rebar ends. The concrete was air-entrained (AE) concrete with a water-cement ratio of 0.6 using ordinary Portland cement. The specimens were exposed under cyclic wet-dry exposure (3-day immer-sion in 20 degrees C water with 3% NaCl and 4-day drying at 20 degrees C and 65%RH).

Figure 17 shows half-cell electric potential history and Table 1 outlines the deterioration history for each

0

100

200

300

400

500

0 10 20 30 40 50

Conc

rete

surf

ace

stra

in(×

10-6

)

Quantity of corrosion product（mg/cm2）

C/D=1.25 C/D=2.5C/D=3.75 C/D=6.25

Corrosin crack criteria in the analysis

Fig. 13 Quantity of corrosive product versus concrete surface strain.

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7

Lim

it qu

antit

y of

cor

rosi

on p

rodu

ct a

t cra

ckm

lim(m

g/cm

2 )

C/DCover/Diameter of steel

α=1.10 α=1.25α=1.50 α=2.00Ref. Suzuki Ref. MorikawaRef. Tamori

Fig. 15 Relationship between C/D and limit quantity of corrosive product at crack.

Table 1 Corrosion deterioration history. Specimen Onset of initial corrosion (year) Onset of corrosion-induced crack (year) Experiment period (year)

Specimen-1 1.2 1.7 4.3 Specimen-2 0.9 3.9 7.9

Target element of max tensile main strain

D=16mm，C=60mm，α=1.250

1000

500

(1×10-6)

At quantity of corrosive product is 20mg/cm2 Fig. 14 Max tensile main strain contour.


specimen. Corrosion, assumed to start when there was a fall in the initial half-cell potential, began in Specimen-1 after 1.2 years of exposure and in Specimen-2 after 0.9 years of exposure.

Corrosion cracks were confirmed in Specimen-1 after 1.7 years of exposure. The crack width at that time was 0.04mm. Similarly corrosion cracks were confirmed in

Specimen-2 after 3.9 years of exposure. The crack width at that time was 0.06mm. Specimen-1 was exposed for 4.3 years and found to have a maximum corrosion crack width of 0.2mm. Specimen-2 was exposed for 7.9 years and found to have a maximum corrosion crack width of 0.25mm. Variation such as this in the onset of corrosion and initiation of corrosion cracks are known to arise in reinforced concrete even under identical conditions.

Figure 18 shows the crack patterns on the exposure surface with 20mm cover. Figure 19 shows choloride ion concentration distribution of specimen-1 and specimen-2. The red lines are the first corrosion cracks and the blue lines are the additional cracks until end of experiment. It is clear that corrosion cracks progressed from the initial cracks. Reinforced concrete has variations in crack pat-tern at occurring corrosion cracks.

4.2.2 FEM analysis model Table 2 shows the FEM analysis cases. Three analytical

Water-cement ratio : 60%Exposure condition : 20℃ in salt water with 3 % NaCl：3-day, 20℃65％RH：4-day

100

200 800 200

1200

100

アクリル樹脂被覆暴露面（上下面800mm区間開放）

異形鉄筋D13（かぶり20mm）

単位：mm

20100

Exposure surface(uncoated length =800mm)

Surface coating with acrylic resin Deformed reinforcing steel bar(diameter=13mm, cover thickness=20mm)

unit : mmFig. 16 Overview of experiment.

Specimen-1

Red line : First corrosion cracks(Specimen-1 : 1.7year, Specimen-2 : 3.9year)

Blue line : Additional cracks until end of experiment(Specimen-1 : 4.3year, Specimen-2 : 7.9year)

:Measured position of chloride ion concentration

800(Exposed surface) unit : mm

CR-1 N-1 N-2 N-3

CR-2 N-4

Specimen-2

Fig. 18 Cracks on exposed surface with 20mm cover.

Table 2 Analysis cases.

Analysis case initial value of diffusion coefficient Do (cm2/year)

Model of diffusion coeficient of cracked concrete

Linkage to structural analysis corrosion expansion ratio

CASE-1 uniform (1.0) - - CASE-2 -

Corrosion analysis only -

CASE-3 1.1 CASE-4 1.25 CASE-5

Model-1 (Takewaka) 2

CASE-6

variation (Area-B:2.6, Area-B:0.4)

Model-2 (Maeda)

Coupled analysis with stractural

analysis 1.25

-800

-700

-600

-500

-400

-300

-200

-100

0

0 2 4 6 8 10

Hal

f-cel

l pot

entia

l (m

V v

s CSE

)

Exposuring time (year)

Specimen-1 Specimen-2

:Corrosion occurence:Corrosion crack occurence:End of test

Fig. 17 Exposure time versus half-cell electric potential.

0

5

10

15

20

25

0 20 40 60 80 100

Chl

orid

e io

n co

ncen

tratio

n(kg

/m3 )

Distance from concrete surface of 20mm cover side(mm)

CR-1N-1N-2N-3

4.3 year after exposuringCover position

0

5

10

15

20

25

0 20 40 60 80 100

Chl

orid

e io

n co

ncen

tratio

n(kg

/m3 )

Distance from concrete surface of 20mm cover side(mm)

CR-2

N-47.9 year after exposuring

Cover position

a) Specimen-1 b) Specimen-2

Fig. 19 Choloride ion concentration distribution.


cases were used: with and without structural analysis coupling (i.e. corrosion analysis alone or coupled with structural analysis), initial value of diffusion coefficient D0, coefficient of diffusion through the crack Dcr, and expansion ratio of corrosion product α. Figure 20 shows the 1/2 model used for analysis, which was adopted in consideration of the sectional symmetry and exposure conditions. The conditions used in the structural analysis were common in all cases except for expansion ratio of corrosion product. That is, concrete compressive strength was 32.6N/mm2, concrete tensile strength was decided based on the basic concrete specification, steel yield strength was 345 N/mm2, and Young’s modulus of the steel was 210,000N/mm2. The conditions used in the reinforcement corrosion analysis were common except for the diffusion coefficient through the crack Dcr. The chloride ion concentration at the surface of the 800mm exposed area was assumed to be 25kg/m3. The threshold of [Cl-]/[OH-] was set at 16.3 under pH value of 12.0 condition. Therefore, the threshold chloride concentra-tion for corrosion of steel in concrete was 2.0kg/m3.

The diffusion coefficient of chloride ions was set separately for Area-A (the supposed anode area of the exposure surface) and Area-B (a typical part of the ex-posure surface) considering the non-uniformity of diffu-sion. The diffusion coefficient of chloride ions in CASE-1 was 1.0cm2/year, which equates to a wa-ter-cement ratio of 0.6. The diffusion coefficient of chloride ions in Area-A was assumed to be 2.6cm2/year and that in Area-B was 0.4cm2/year in CASE-2 to CASE-6. The degree of heterogeneity of the diffusion coefficient, which is based on Eq. (2), is within the variance of the experimental results.

Corrosion analysis only is carried out for CASE-1 to CASE2, while coupled analysis is employed for CASE-3 to CASE-6.

The proportional increase in diffusion coefficient against D0 in the cracked part, dcr, was set according to Model-1 and Model-2 (in Fig. 21). Experimental results obtained by Takewaka et al. (2003) are used for Model-1, and the experimental results for a water-cement ratio of 0.5 by Maeda et al. (2002) are used for Model-2. The

value of dcr is set at 1.0 until the crack width reaches 0.03mm in accordance with recent research. The lower condition of the cracked diffusion coefficient was adopted in Model-2. Corrosion expansion ratio is a pa-rameter in CASE-3 to CASE-5, while the corrosion ex-pansion ratio in CASE-6 is 1.25.

4.2.3 Corrosion occurrence and corrosion cracking time Corrosion occurrence times are shown in Table 3. For the analysis, corrosion occurrence time is taken to be the time at which a certain quantity of corrosion had oc-curred. The initial value of diffusivity influences these

Reinforcing steel bar

Chloride ion concentration on concrete surface Co

Area-A

Reinforcingsteel bar

Area-B

CL

X

Y

Z

Y

1200

100

20

unit : mm

50

100

800 200200

*1 : Area-A is the position of initial corrosion cracks in Specimen-1.*2 : Analysis model for Specimen-2 is same as that for Specimen-1.

a)Side view b) Cross-sectional view Fig. 20 Analysis model.

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 0.1 0.2 0.3The

prop

ortio

nal i

ncre

ase

in d

iffus

ion

coef

ficie

nt o

f the

cra

cked

par

t co

mpa

red

with

D0

d cr

Crack width(mm)

Model-1Model-2Experimental value by Takewaka

0.03

Fig. 21 Proportional increase in diffusion coefficient of cracked part.

Table 3 Analysis results.

Analysis case

Corrosion occurrence time (year)

corrosion cracking

time (year)

limit corrosion quantity at initial corrosion crack

occurrence (mg/cm2)CASE-1 1.35 - - CASE-2 0.8 - - CASE-3 0.8 1.17 20 CASE-4 0.8 0.98 9 CASE-5 0.8 0.85 4 CASE-6 0.8 0.98 9


corrosion occurrence times. In CASE-1, corrosion oc-currence is at 1.35 year and in CASE-2 to CASE-6 it is 0.80 year. There is a tendency for half-cell potential to fall prior to corrosion occurrence, so the half-cell poten-tial data is used to indicate the onset of corrosion (Fig. 17). Considering the non-homogeneity of concrete ma-terials, the corrosion occurrence times given by the analysis can be considered correct.

Corrosion cracking time and the limit corrosion quan-tity at the occurrence of the corrosion crack are shown in Table 3. Further, the time history of strain perpendicular to the axial direction (X-axis direction) is shown in Fig. 22. Tensile strain is perpendicular to the axial (X-axis direction) strain at the concrete surface at the assumed anode of the reinforcing bar. In the analysis, the tensile strain increases rapidly after cracking, but the sudden change such as that mentioned in 4.1 does not occur. The un-cracked concrete surrounding the cracked part re-stricts sudden crack progress, because corrosion expan-sion occurs locally from around Area-A in the analysis model.

The quantity of corrosion at the time of corrosion cracking cannot be compared directly between analysis and experiment, because corrosion quantity was not measured at this moment in the experiment. The ratio of cover thickness to bar diameter (C/D) in the experiment was around 1.5. According to the experimental results shown in Fig. 18, the quantity of corrosion at the time of

corrosion cracking in the experiment can be assumed to be less than approximately 30mg/cm2. This is close to the result judged from the strain.

4.2.4 Chloride ion concentration distribution Figure 23 compares the chloride ion concentration in the experiment and analysis. For Specimen-2, the position of analysis was adjusted by 100mm to match the position of the corrosion cracks in the experiment with Area-A in the analysis in Fig. 23. The chloride ion concentration around the corrosion cracks was locally high while in other areas it was relatively low as a result of the non-uniformity of initial diffusion coefficient D0. CASE-1 does not match the experimental results due to the uniform distribution of chloride ions in the exposure section. CASE-2 to CASE-6, which consider the non-homogeneity of the initial diffusion coefficient D0, follow the same tendency as the experimental results, with a locally high chloride ion concentration around the crack position and a relatively low concentration else-where.

Comparing the coupled analysis of CASE-3 to CASE-5, where the parameter is corrosion expansion ratio, a higher corrosion expansion ratio correlates with a higher chloride ion concentration. The increase area of diffusion coefficient and the area of corrosion cracks are large due to the corrosion expansion ratio.

The chloride ion concentration around Area-A in Case-3 is lower than that in Case-2, because the value of diffusion coefficient through the crack, Dcr, in Model-2 used for Case-3 is lower than that of Model-1 used for Case-2.

4.2.5 Evolution of corrosion amount after cor-rosion cracking Figure 24 compares the amount of corrosion in the ex-periment and analysis. In CASE-1, there is hardly any macro-cell corrosion. The increase in corrosion in Area-A observed in the experiments is not reflected in the results. In Case-2, where reinforcement corrosion analysis is not coupled with structural analysis, macro-cell corrosion is limited to Area-A. That is, the spread of the corrosion area is not replicated. However, with regard to maximum corrosion amount, there is not

0

1000

2000

3000

4000

5000

6000

0 1 2 3

Axi

al d

erec

tion

stra

in (*

10-6

)


Case-3 Case-4

Case-5 Case-6

crackcriteria

Fig. 22 Time history of strain perpendicular to the axial direction (X-axis direction).

0

5

10

15

20

25

0 200 400 600 800 1000 1200Chl

orid

e io

n co

ncen

tratio

n at

stee

l ba

r (kg

/m3 )

Position of analysis and measurement(mm)

Experiment Case-1Case-2 Case-3Case-4 Case-5Case-6

Exposed for 4.3 year

0

5

10

15

20

25

0 200 400 600 800 1000 1200Chl

orid

e io

n co

ncen

tratio

n at

stee

l ba

r (kg

/m3 )




a) Specimen-1 (Exposed for 4.3 years) b) Specimen-2 (Exposed for 7.9 years)

Fig. 23 Comparison of chloride concentration between experiment and analysis.


much difference between Case-2 and the other cases (from Case-3 to Case-6). Further, there is not much dif-ference between Case-1 and the experimental results. The area of increased chloride ion concentration ex-tended due to the onset of corrosion cracks in Case-3 to Case-5, which is the coupled analysis case using Model-1 for the diffusion coefficient of the cracked concrete. The anode area of macro-cell corrosion and the corrosion area also expanded. The analysis results accord with the experimental trend.

The explanation for the similar maximum corrosion amount in the non-coupled analysis in Case-2 and the coupled analysis is that the anode area in Case-3 to Case-5 is spread and there is no concentration of corro-sion in Area-A. In Case-6, using Model-2, the corrosion area is very short and almost as limited as in Case-1 (non-coupled analysis). The reason for this thought to be that the value of diffusion coefficient of the cracked concrete in Model-2 was lower than that in Model-1.

The increase in amount of corrosion in CASE-3 and CASE-5 is shown in Fig. 25 for Area-A and a point 60mm away from Area-A. In CASE-5, where the corro-sion expansion ratio is large, the amount of corrosion increased both in Area-A and, three years after initial exposure, 60mm away. In CASE-3, where the corrosion expansion ratio is small, the corrosion rate in Area-A is constant but corrosion does not take place 60mm away from Area-A. There is not enough data to compare ac-celeration of corrosion in this experiment. The rate of macro-cell corrosion around the cracked concrete fell

with time in experiments using induced 0.25mm to 0.60mm cracks. The pertinent factor here is thought to be that the ratio of anode area to cathode area increases over time. The analysis allows the corrosion rate trend to be expressed in terms of corrosion expansion ratio. Using coupled analysis, the expansion of the corroded area can be expressed.

These results indicate that it is possible to express the spreading of the corrosion area seen in the experimental results if the coefficient of diffusion through cracked concrete is taken into account. Although this analytical method is as yet not fully developed, the matching of Case-4 and Case-5 using Model-1 with the experimental results demonstrates that it accords with actual corrosion progress mechanism better than previous models.

4.2.6 Conclusion of comparison between ex-periment and analysis We have compared analytical and experimental results for the time of corrosion occurrence and corrosion crack occurrence as well as the amount of corrosion after cor-rosion cracking. Analytical CASE-4 using Model-1 gives the best consistency with the experimental results over most of the range of comparison. Although this result is reasonably good, there remain inadequacies in the ana-lytical method and it is not able to fully replicate the experimental results. On the other hand, using this ra-tional macro-cell corrosion model and coupled analysis between structural deterioration and reinforcement cor-rosion, improved simulation of the actual corrosion progress mechanism has come to be possible. 5. Conclusions

The followings are the summary of conclusions. 1. The coupled analysis method being developed by the

authors, which involves analysis of structure, analysis of reinforcement corrosion and analysis of how the two interact, aims to predict the deterioration of a re-inforced concrete structure under salt attack.

2. The external polarization curve is theoretically for-mulated from the internal polarization curve. By this means, consistent modeling between internal and ex-ternal polarization curves is ensured.

0

200

400

600

800

200 400 600 800

Cor

rosi

on a

mou

nt (m

g/cm

2 )




0

200

400

600

800

200 400 600 800

Cor

rosi

on a

mou

nt (m

g/cm

2 )




a) Specimen-1 (Exposed for 4.3 years) b) Specimen-2 (Exposed for 7.9 years)

Fig. 24 Amount of corrosion in experiment and analysis.

0

200

400

600

800

1000

0 2 4 6 8Cor

rosi

on a

mou

nt (m

g/cm

2 )


Case-3 Area-ACase-5 Area-ACase-3 around Area-ACase-5 around Area-A

Max corrosion amount ofspecimen-1 and specimen-2

Fig. 25 Corrosion amount with exposure time.


3. The distribution diagram βM－Ni set it for a function of ratio of the cathodal element against anodal element group and macro-cell corrosion current density imacro,M-Ni ,which is calculated cathodal element area equals anodal element area. The method enables more accurate modelling of corrosion in RC structures.

4. The diffusion coefficient for finite elements repre-senting cracked concrete was modeled using the rela-tionship between diffusion coefficient and strain, as-suming that a single crack occurs in a finite element. Thus, it was possible to set a diffusion coefficient for a cracked finite element by applying the relationship between the coefficient of diffusion of cracked con-crete and crack width. This relationship has been ob-tained in a number of experimental studies reported in the literature.

5. Corrosion expansion in RC member is modelled sim-ple. Expansion ratio of corrosive product is set in-crease rate of radial direction of corrosive product under confining pressure, not under free expansion. Corrosion expansion is modelled single element (in-terface element), which steel including the corrosive product. It is thought that a range of from 1.10 to 1.50 is proper as the expansion ratio of corrosive product in the corrosion expansion model of this article.

6. The proposed analytical method was validated by carrying out cyclic wet-dry tests with salt solution to simulate macro-cell corrosion. By taking the diffusion coefficient of the cracked concrete into consideration, the model simulates the experimentally observed spread of corrosion area. Although this analytical method is not yet fully developed, this study shows that using the relationship obtained by Model-1 for the diffusion coefficient through cracks yields a good match with the experimental results.

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“The break-down of passive films on iron.” Corrosion Science, 28(1), 81-86.

Djerbi, A., Bonnet, S., Khelidj, A. and Baroghel-bouny, V., (2008). “Influence of traversing crack on chloride diffusion into concrete.” Cement and Concrete Research, 38, 877-883.

Fukuura, N., Maruya, T., Takeda, H. and Koyama, S., (2007). “Examination about the establishment of coupled analysis system between material deterioration caused by reinforcement corrosion and load action.” JSCE 62th annual meeting summaries, 5, 145-146. (in Japanese)

Funahashi, M. and Bushman, J. B., (1991). “Technical review of 100mV polarization shift criterion for reinforcing steel in concrete.” Corrosion, 376-386.

Hsu, K., Takeda, H. and Maruya, T., (2000). “Numerical simulation on corrosion of steel in concrete structures under chloride attack.” J. Materials, Conc. Struct., Pavements, JSCE, 655, 5-48, 143-157.

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Committee, (2003). “Concrete Engineering Series 50.” (in Japanese)

Japan society of civil engineers (JSCE) Concrete Committee, (2009). “Concrete Engineering Series 86.” (in Japanese)

Japan society of civil engineers (JSCE) Concrete Committee, (2012). “Concrete Engineering Series 99.” (in Japanese)

Japan society of civil engineers (JSCE) Concrete Committee, (2013). “Standard specifications of concrete structures –Design.” (in Japanese)

Jones, D. A., (1996). “Principles and prevention of corrosion.” Second edition, 86-98, 168-198.

Kranc, S. C. and Sagues, A. A., (1997). “Modeling the time-dependent response to external polarization of a corrosion macrocell on steel in concrete.” J. Electrochem. Soc., 144(8), 2643-2652.

Lundgren, K., (2002). “Modelling the effect of corrosion on bond in reinforced concrete.” Magazine of Concrete Research, 54(3), 165-173.

Maekawa, K., Ishida, T. and Kishi, T., (2003). “Multi-scale modeling of concrete performance- integrated material and structural mechanics.” Journal of Advanced Concrete Technology, 1(2), 91-126.

Maruya, T., Takeda, H., Horiguchi, K., Koyama, S. and Hsu, K. L., (2007). “Simulation of steel corrosion in concrete based on the model of macro-cell corrosion circuit.” Journal of Advanced Concrete Technology, 5(3), 343-362.

Morikawa, M., Seki, H., and Okumura, T., (1987). “Basic study on cracking of concrete due to expansion by rebar corrosion.” Journal of JSCE, 378, 97-105. (in Japanese)

Nakagawa, H., Tanaka, T., Yokota, M. and Matsushima, M., (2008). “Experimental study on modes of concrete cracking and threshold amounts of corrosion for crack onset.” Journal of JSCE, 64(1), 110-121. (in Japanese)

Suda, K., Misra, S. and Motohashi, K., (1993). “Corrosion products of reinforcing bars embedded in concrete.” Corrosion Science, 35(5-8), 1543-1549.

Suzuki, M., Horiguchi, K., Fukuura, N. and Maruya, T., (2009). “Influence on crack due to corrosion expansion by corrosion promotion condition.” Proceedings of the Japan Concrete Institute, 31(1), 1081-1086. (in Japanese)

Suzuki, M., Fukuura, N. and Maruya, T., (2014). “Establishment of a coupled analysis between structure and reinforcement corrosion for predicting deterioration of reinforced concrete structure by salt damage.” Journal of JSCE, 70(3), 301-319. (in Japanese)

Suzuki, M., Fukuura, N., Takeda, H. and Maruya, T., (2015). “Establishment of a corrosion expansion model of steel in concrete.” Fifth international conference on construction materials, 94-104.

Takewaka, K., Yamaguchi, T. and Maeda, S., (2003). “Simulation model for deterioration of concrete structures due to chloride attack.” Journal of


Advanced Concrete Technology, 1(2), 139-146. Tamori, K., Maruyama, K., Odagawa, M. and Hashimoto,

T., (1988). “Cracking behavior of reinforced concrete members due to corrosion of reinforcing bars.” Proceedings of the Japan Concrete Institute, 10(2), 505-510. (in Japanese)

Toongoenthong, K. and Maekawa, K., (2005). “Simulation of coupled corrosive product formation, Migration into crack and propagation in reinforced concrete sections.” Journal of Advanced Concrete Technology, 3(2), 253-265.

establishment of coupled analysis of interaction between

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